# Properties

 Label 234.2.a.b.1.1 Level $234$ Weight $2$ Character 234.1 Self dual yes Analytic conductor $1.868$ 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] = [234,2,Mod(1,234)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(234, 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("234.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$234 = 2 \cdot 3^{2} \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 234.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: $$1.86849940730$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 26) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Character $$\chi$$ $$=$$ 234.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{5} +1.00000 q^{7} -1.00000 q^{8} +O(q^{10})$$ $$q-1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{5} +1.00000 q^{7} -1.00000 q^{8} -1.00000 q^{10} +2.00000 q^{11} -1.00000 q^{13} -1.00000 q^{14} +1.00000 q^{16} +3.00000 q^{17} +6.00000 q^{19} +1.00000 q^{20} -2.00000 q^{22} +4.00000 q^{23} -4.00000 q^{25} +1.00000 q^{26} +1.00000 q^{28} -2.00000 q^{29} +4.00000 q^{31} -1.00000 q^{32} -3.00000 q^{34} +1.00000 q^{35} +3.00000 q^{37} -6.00000 q^{38} -1.00000 q^{40} -5.00000 q^{43} +2.00000 q^{44} -4.00000 q^{46} -13.0000 q^{47} -6.00000 q^{49} +4.00000 q^{50} -1.00000 q^{52} -12.0000 q^{53} +2.00000 q^{55} -1.00000 q^{56} +2.00000 q^{58} +10.0000 q^{59} -8.00000 q^{61} -4.00000 q^{62} +1.00000 q^{64} -1.00000 q^{65} -2.00000 q^{67} +3.00000 q^{68} -1.00000 q^{70} +5.00000 q^{71} -10.0000 q^{73} -3.00000 q^{74} +6.00000 q^{76} +2.00000 q^{77} -4.00000 q^{79} +1.00000 q^{80} +3.00000 q^{85} +5.00000 q^{86} -2.00000 q^{88} -6.00000 q^{89} -1.00000 q^{91} +4.00000 q^{92} +13.0000 q^{94} +6.00000 q^{95} +14.0000 q^{97} +6.00000 q^{98} +O(q^{100})$$

## 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
$$3$$ 0 0
$$4$$ 1.00000 0.500000
$$5$$ 1.00000 0.447214 0.223607 0.974679i $$-0.428217\pi$$
0.223607 + 0.974679i $$0.428217\pi$$
$$6$$ 0 0
$$7$$ 1.00000 0.377964 0.188982 0.981981i $$-0.439481\pi$$
0.188982 + 0.981981i $$0.439481\pi$$
$$8$$ −1.00000 −0.353553
$$9$$ 0 0
$$10$$ −1.00000 −0.316228
$$11$$ 2.00000 0.603023 0.301511 0.953463i $$-0.402509\pi$$
0.301511 + 0.953463i $$0.402509\pi$$
$$12$$ 0 0
$$13$$ −1.00000 −0.277350
$$14$$ −1.00000 −0.267261
$$15$$ 0 0
$$16$$ 1.00000 0.250000
$$17$$ 3.00000 0.727607 0.363803 0.931476i $$-0.381478\pi$$
0.363803 + 0.931476i $$0.381478\pi$$
$$18$$ 0 0
$$19$$ 6.00000 1.37649 0.688247 0.725476i $$-0.258380\pi$$
0.688247 + 0.725476i $$0.258380\pi$$
$$20$$ 1.00000 0.223607
$$21$$ 0 0
$$22$$ −2.00000 −0.426401
$$23$$ 4.00000 0.834058 0.417029 0.908893i $$-0.363071\pi$$
0.417029 + 0.908893i $$0.363071\pi$$
$$24$$ 0 0
$$25$$ −4.00000 −0.800000
$$26$$ 1.00000 0.196116
$$27$$ 0 0
$$28$$ 1.00000 0.188982
$$29$$ −2.00000 −0.371391 −0.185695 0.982607i $$-0.559454\pi$$
−0.185695 + 0.982607i $$0.559454\pi$$
$$30$$ 0 0
$$31$$ 4.00000 0.718421 0.359211 0.933257i $$-0.383046\pi$$
0.359211 + 0.933257i $$0.383046\pi$$
$$32$$ −1.00000 −0.176777
$$33$$ 0 0
$$34$$ −3.00000 −0.514496
$$35$$ 1.00000 0.169031
$$36$$ 0 0
$$37$$ 3.00000 0.493197 0.246598 0.969118i $$-0.420687\pi$$
0.246598 + 0.969118i $$0.420687\pi$$
$$38$$ −6.00000 −0.973329
$$39$$ 0 0
$$40$$ −1.00000 −0.158114
$$41$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$42$$ 0 0
$$43$$ −5.00000 −0.762493 −0.381246 0.924473i $$-0.624505\pi$$
−0.381246 + 0.924473i $$0.624505\pi$$
$$44$$ 2.00000 0.301511
$$45$$ 0 0
$$46$$ −4.00000 −0.589768
$$47$$ −13.0000 −1.89624 −0.948122 0.317905i $$-0.897021\pi$$
−0.948122 + 0.317905i $$0.897021\pi$$
$$48$$ 0 0
$$49$$ −6.00000 −0.857143
$$50$$ 4.00000 0.565685
$$51$$ 0 0
$$52$$ −1.00000 −0.138675
$$53$$ −12.0000 −1.64833 −0.824163 0.566352i $$-0.808354\pi$$
−0.824163 + 0.566352i $$0.808354\pi$$
$$54$$ 0 0
$$55$$ 2.00000 0.269680
$$56$$ −1.00000 −0.133631
$$57$$ 0 0
$$58$$ 2.00000 0.262613
$$59$$ 10.0000 1.30189 0.650945 0.759125i $$-0.274373\pi$$
0.650945 + 0.759125i $$0.274373\pi$$
$$60$$ 0 0
$$61$$ −8.00000 −1.02430 −0.512148 0.858898i $$-0.671150\pi$$
−0.512148 + 0.858898i $$0.671150\pi$$
$$62$$ −4.00000 −0.508001
$$63$$ 0 0
$$64$$ 1.00000 0.125000
$$65$$ −1.00000 −0.124035
$$66$$ 0 0
$$67$$ −2.00000 −0.244339 −0.122169 0.992509i $$-0.538985\pi$$
−0.122169 + 0.992509i $$0.538985\pi$$
$$68$$ 3.00000 0.363803
$$69$$ 0 0
$$70$$ −1.00000 −0.119523
$$71$$ 5.00000 0.593391 0.296695 0.954972i $$-0.404115\pi$$
0.296695 + 0.954972i $$0.404115\pi$$
$$72$$ 0 0
$$73$$ −10.0000 −1.17041 −0.585206 0.810885i $$-0.698986\pi$$
−0.585206 + 0.810885i $$0.698986\pi$$
$$74$$ −3.00000 −0.348743
$$75$$ 0 0
$$76$$ 6.00000 0.688247
$$77$$ 2.00000 0.227921
$$78$$ 0 0
$$79$$ −4.00000 −0.450035 −0.225018 0.974355i $$-0.572244\pi$$
−0.225018 + 0.974355i $$0.572244\pi$$
$$80$$ 1.00000 0.111803
$$81$$ 0 0
$$82$$ 0 0
$$83$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$84$$ 0 0
$$85$$ 3.00000 0.325396
$$86$$ 5.00000 0.539164
$$87$$ 0 0
$$88$$ −2.00000 −0.213201
$$89$$ −6.00000 −0.635999 −0.317999 0.948091i $$-0.603011\pi$$
−0.317999 + 0.948091i $$0.603011\pi$$
$$90$$ 0 0
$$91$$ −1.00000 −0.104828
$$92$$ 4.00000 0.417029
$$93$$ 0 0
$$94$$ 13.0000 1.34085
$$95$$ 6.00000 0.615587
$$96$$ 0 0
$$97$$ 14.0000 1.42148 0.710742 0.703452i $$-0.248359\pi$$
0.710742 + 0.703452i $$0.248359\pi$$
$$98$$ 6.00000 0.606092
$$99$$ 0 0
$$100$$ −4.00000 −0.400000
$$101$$ −4.00000 −0.398015 −0.199007 0.979998i $$-0.563772\pi$$
−0.199007 + 0.979998i $$0.563772\pi$$
$$102$$ 0 0
$$103$$ −8.00000 −0.788263 −0.394132 0.919054i $$-0.628955\pi$$
−0.394132 + 0.919054i $$0.628955\pi$$
$$104$$ 1.00000 0.0980581
$$105$$ 0 0
$$106$$ 12.0000 1.16554
$$107$$ 4.00000 0.386695 0.193347 0.981130i $$-0.438066\pi$$
0.193347 + 0.981130i $$0.438066\pi$$
$$108$$ 0 0
$$109$$ 19.0000 1.81987 0.909935 0.414751i $$-0.136131\pi$$
0.909935 + 0.414751i $$0.136131\pi$$
$$110$$ −2.00000 −0.190693
$$111$$ 0 0
$$112$$ 1.00000 0.0944911
$$113$$ −2.00000 −0.188144 −0.0940721 0.995565i $$-0.529988\pi$$
−0.0940721 + 0.995565i $$0.529988\pi$$
$$114$$ 0 0
$$115$$ 4.00000 0.373002
$$116$$ −2.00000 −0.185695
$$117$$ 0 0
$$118$$ −10.0000 −0.920575
$$119$$ 3.00000 0.275010
$$120$$ 0 0
$$121$$ −7.00000 −0.636364
$$122$$ 8.00000 0.724286
$$123$$ 0 0
$$124$$ 4.00000 0.359211
$$125$$ −9.00000 −0.804984
$$126$$ 0 0
$$127$$ 16.0000 1.41977 0.709885 0.704317i $$-0.248747\pi$$
0.709885 + 0.704317i $$0.248747\pi$$
$$128$$ −1.00000 −0.0883883
$$129$$ 0 0
$$130$$ 1.00000 0.0877058
$$131$$ 1.00000 0.0873704 0.0436852 0.999045i $$-0.486090\pi$$
0.0436852 + 0.999045i $$0.486090\pi$$
$$132$$ 0 0
$$133$$ 6.00000 0.520266
$$134$$ 2.00000 0.172774
$$135$$ 0 0
$$136$$ −3.00000 −0.257248
$$137$$ −12.0000 −1.02523 −0.512615 0.858619i $$-0.671323\pi$$
−0.512615 + 0.858619i $$0.671323\pi$$
$$138$$ 0 0
$$139$$ 7.00000 0.593732 0.296866 0.954919i $$-0.404058\pi$$
0.296866 + 0.954919i $$0.404058\pi$$
$$140$$ 1.00000 0.0845154
$$141$$ 0 0
$$142$$ −5.00000 −0.419591
$$143$$ −2.00000 −0.167248
$$144$$ 0 0
$$145$$ −2.00000 −0.166091
$$146$$ 10.0000 0.827606
$$147$$ 0 0
$$148$$ 3.00000 0.246598
$$149$$ 18.0000 1.47462 0.737309 0.675556i $$-0.236096\pi$$
0.737309 + 0.675556i $$0.236096\pi$$
$$150$$ 0 0
$$151$$ −9.00000 −0.732410 −0.366205 0.930534i $$-0.619343\pi$$
−0.366205 + 0.930534i $$0.619343\pi$$
$$152$$ −6.00000 −0.486664
$$153$$ 0 0
$$154$$ −2.00000 −0.161165
$$155$$ 4.00000 0.321288
$$156$$ 0 0
$$157$$ −10.0000 −0.798087 −0.399043 0.916932i $$-0.630658\pi$$
−0.399043 + 0.916932i $$0.630658\pi$$
$$158$$ 4.00000 0.318223
$$159$$ 0 0
$$160$$ −1.00000 −0.0790569
$$161$$ 4.00000 0.315244
$$162$$ 0 0
$$163$$ −4.00000 −0.313304 −0.156652 0.987654i $$-0.550070\pi$$
−0.156652 + 0.987654i $$0.550070\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ 0 0
$$167$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$168$$ 0 0
$$169$$ 1.00000 0.0769231
$$170$$ −3.00000 −0.230089
$$171$$ 0 0
$$172$$ −5.00000 −0.381246
$$173$$ −20.0000 −1.52057 −0.760286 0.649589i $$-0.774941\pi$$
−0.760286 + 0.649589i $$0.774941\pi$$
$$174$$ 0 0
$$175$$ −4.00000 −0.302372
$$176$$ 2.00000 0.150756
$$177$$ 0 0
$$178$$ 6.00000 0.449719
$$179$$ 9.00000 0.672692 0.336346 0.941739i $$-0.390809\pi$$
0.336346 + 0.941739i $$0.390809\pi$$
$$180$$ 0 0
$$181$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$182$$ 1.00000 0.0741249
$$183$$ 0 0
$$184$$ −4.00000 −0.294884
$$185$$ 3.00000 0.220564
$$186$$ 0 0
$$187$$ 6.00000 0.438763
$$188$$ −13.0000 −0.948122
$$189$$ 0 0
$$190$$ −6.00000 −0.435286
$$191$$ −10.0000 −0.723575 −0.361787 0.932261i $$-0.617833\pi$$
−0.361787 + 0.932261i $$0.617833\pi$$
$$192$$ 0 0
$$193$$ −16.0000 −1.15171 −0.575853 0.817554i $$-0.695330\pi$$
−0.575853 + 0.817554i $$0.695330\pi$$
$$194$$ −14.0000 −1.00514
$$195$$ 0 0
$$196$$ −6.00000 −0.428571
$$197$$ −9.00000 −0.641223 −0.320612 0.947211i $$-0.603888\pi$$
−0.320612 + 0.947211i $$0.603888\pi$$
$$198$$ 0 0
$$199$$ −10.0000 −0.708881 −0.354441 0.935079i $$-0.615329\pi$$
−0.354441 + 0.935079i $$0.615329\pi$$
$$200$$ 4.00000 0.282843
$$201$$ 0 0
$$202$$ 4.00000 0.281439
$$203$$ −2.00000 −0.140372
$$204$$ 0 0
$$205$$ 0 0
$$206$$ 8.00000 0.557386
$$207$$ 0 0
$$208$$ −1.00000 −0.0693375
$$209$$ 12.0000 0.830057
$$210$$ 0 0
$$211$$ 23.0000 1.58339 0.791693 0.610920i $$-0.209200\pi$$
0.791693 + 0.610920i $$0.209200\pi$$
$$212$$ −12.0000 −0.824163
$$213$$ 0 0
$$214$$ −4.00000 −0.273434
$$215$$ −5.00000 −0.340997
$$216$$ 0 0
$$217$$ 4.00000 0.271538
$$218$$ −19.0000 −1.28684
$$219$$ 0 0
$$220$$ 2.00000 0.134840
$$221$$ −3.00000 −0.201802
$$222$$ 0 0
$$223$$ −21.0000 −1.40626 −0.703132 0.711059i $$-0.748216\pi$$
−0.703132 + 0.711059i $$0.748216\pi$$
$$224$$ −1.00000 −0.0668153
$$225$$ 0 0
$$226$$ 2.00000 0.133038
$$227$$ 24.0000 1.59294 0.796468 0.604681i $$-0.206699\pi$$
0.796468 + 0.604681i $$0.206699\pi$$
$$228$$ 0 0
$$229$$ −15.0000 −0.991228 −0.495614 0.868543i $$-0.665057\pi$$
−0.495614 + 0.868543i $$0.665057\pi$$
$$230$$ −4.00000 −0.263752
$$231$$ 0 0
$$232$$ 2.00000 0.131306
$$233$$ 11.0000 0.720634 0.360317 0.932830i $$-0.382669\pi$$
0.360317 + 0.932830i $$0.382669\pi$$
$$234$$ 0 0
$$235$$ −13.0000 −0.848026
$$236$$ 10.0000 0.650945
$$237$$ 0 0
$$238$$ −3.00000 −0.194461
$$239$$ −9.00000 −0.582162 −0.291081 0.956698i $$-0.594015\pi$$
−0.291081 + 0.956698i $$0.594015\pi$$
$$240$$ 0 0
$$241$$ 18.0000 1.15948 0.579741 0.814801i $$-0.303154\pi$$
0.579741 + 0.814801i $$0.303154\pi$$
$$242$$ 7.00000 0.449977
$$243$$ 0 0
$$244$$ −8.00000 −0.512148
$$245$$ −6.00000 −0.383326
$$246$$ 0 0
$$247$$ −6.00000 −0.381771
$$248$$ −4.00000 −0.254000
$$249$$ 0 0
$$250$$ 9.00000 0.569210
$$251$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$252$$ 0 0
$$253$$ 8.00000 0.502956
$$254$$ −16.0000 −1.00393
$$255$$ 0 0
$$256$$ 1.00000 0.0625000
$$257$$ 15.0000 0.935674 0.467837 0.883815i $$-0.345033\pi$$
0.467837 + 0.883815i $$0.345033\pi$$
$$258$$ 0 0
$$259$$ 3.00000 0.186411
$$260$$ −1.00000 −0.0620174
$$261$$ 0 0
$$262$$ −1.00000 −0.0617802
$$263$$ −12.0000 −0.739952 −0.369976 0.929041i $$-0.620634\pi$$
−0.369976 + 0.929041i $$0.620634\pi$$
$$264$$ 0 0
$$265$$ −12.0000 −0.737154
$$266$$ −6.00000 −0.367884
$$267$$ 0 0
$$268$$ −2.00000 −0.122169
$$269$$ 24.0000 1.46331 0.731653 0.681677i $$-0.238749\pi$$
0.731653 + 0.681677i $$0.238749\pi$$
$$270$$ 0 0
$$271$$ 13.0000 0.789694 0.394847 0.918747i $$-0.370798\pi$$
0.394847 + 0.918747i $$0.370798\pi$$
$$272$$ 3.00000 0.181902
$$273$$ 0 0
$$274$$ 12.0000 0.724947
$$275$$ −8.00000 −0.482418
$$276$$ 0 0
$$277$$ 12.0000 0.721010 0.360505 0.932757i $$-0.382604\pi$$
0.360505 + 0.932757i $$0.382604\pi$$
$$278$$ −7.00000 −0.419832
$$279$$ 0 0
$$280$$ −1.00000 −0.0597614
$$281$$ 26.0000 1.55103 0.775515 0.631329i $$-0.217490\pi$$
0.775515 + 0.631329i $$0.217490\pi$$
$$282$$ 0 0
$$283$$ 4.00000 0.237775 0.118888 0.992908i $$-0.462067\pi$$
0.118888 + 0.992908i $$0.462067\pi$$
$$284$$ 5.00000 0.296695
$$285$$ 0 0
$$286$$ 2.00000 0.118262
$$287$$ 0 0
$$288$$ 0 0
$$289$$ −8.00000 −0.470588
$$290$$ 2.00000 0.117444
$$291$$ 0 0
$$292$$ −10.0000 −0.585206
$$293$$ −7.00000 −0.408944 −0.204472 0.978872i $$-0.565548\pi$$
−0.204472 + 0.978872i $$0.565548\pi$$
$$294$$ 0 0
$$295$$ 10.0000 0.582223
$$296$$ −3.00000 −0.174371
$$297$$ 0 0
$$298$$ −18.0000 −1.04271
$$299$$ −4.00000 −0.231326
$$300$$ 0 0
$$301$$ −5.00000 −0.288195
$$302$$ 9.00000 0.517892
$$303$$ 0 0
$$304$$ 6.00000 0.344124
$$305$$ −8.00000 −0.458079
$$306$$ 0 0
$$307$$ 14.0000 0.799022 0.399511 0.916728i $$-0.369180\pi$$
0.399511 + 0.916728i $$0.369180\pi$$
$$308$$ 2.00000 0.113961
$$309$$ 0 0
$$310$$ −4.00000 −0.227185
$$311$$ −18.0000 −1.02069 −0.510343 0.859971i $$-0.670482\pi$$
−0.510343 + 0.859971i $$0.670482\pi$$
$$312$$ 0 0
$$313$$ −1.00000 −0.0565233 −0.0282617 0.999601i $$-0.508997\pi$$
−0.0282617 + 0.999601i $$0.508997\pi$$
$$314$$ 10.0000 0.564333
$$315$$ 0 0
$$316$$ −4.00000 −0.225018
$$317$$ 18.0000 1.01098 0.505490 0.862832i $$-0.331312\pi$$
0.505490 + 0.862832i $$0.331312\pi$$
$$318$$ 0 0
$$319$$ −4.00000 −0.223957
$$320$$ 1.00000 0.0559017
$$321$$ 0 0
$$322$$ −4.00000 −0.222911
$$323$$ 18.0000 1.00155
$$324$$ 0 0
$$325$$ 4.00000 0.221880
$$326$$ 4.00000 0.221540
$$327$$ 0 0
$$328$$ 0 0
$$329$$ −13.0000 −0.716713
$$330$$ 0 0
$$331$$ −4.00000 −0.219860 −0.109930 0.993939i $$-0.535063\pi$$
−0.109930 + 0.993939i $$0.535063\pi$$
$$332$$ 0 0
$$333$$ 0 0
$$334$$ 0 0
$$335$$ −2.00000 −0.109272
$$336$$ 0 0
$$337$$ 23.0000 1.25289 0.626445 0.779466i $$-0.284509\pi$$
0.626445 + 0.779466i $$0.284509\pi$$
$$338$$ −1.00000 −0.0543928
$$339$$ 0 0
$$340$$ 3.00000 0.162698
$$341$$ 8.00000 0.433224
$$342$$ 0 0
$$343$$ −13.0000 −0.701934
$$344$$ 5.00000 0.269582
$$345$$ 0 0
$$346$$ 20.0000 1.07521
$$347$$ 9.00000 0.483145 0.241573 0.970383i $$-0.422337\pi$$
0.241573 + 0.970383i $$0.422337\pi$$
$$348$$ 0 0
$$349$$ 7.00000 0.374701 0.187351 0.982293i $$-0.440010\pi$$
0.187351 + 0.982293i $$0.440010\pi$$
$$350$$ 4.00000 0.213809
$$351$$ 0 0
$$352$$ −2.00000 −0.106600
$$353$$ −4.00000 −0.212899 −0.106449 0.994318i $$-0.533948\pi$$
−0.106449 + 0.994318i $$0.533948\pi$$
$$354$$ 0 0
$$355$$ 5.00000 0.265372
$$356$$ −6.00000 −0.317999
$$357$$ 0 0
$$358$$ −9.00000 −0.475665
$$359$$ −24.0000 −1.26667 −0.633336 0.773877i $$-0.718315\pi$$
−0.633336 + 0.773877i $$0.718315\pi$$
$$360$$ 0 0
$$361$$ 17.0000 0.894737
$$362$$ 0 0
$$363$$ 0 0
$$364$$ −1.00000 −0.0524142
$$365$$ −10.0000 −0.523424
$$366$$ 0 0
$$367$$ −10.0000 −0.521996 −0.260998 0.965339i $$-0.584052\pi$$
−0.260998 + 0.965339i $$0.584052\pi$$
$$368$$ 4.00000 0.208514
$$369$$ 0 0
$$370$$ −3.00000 −0.155963
$$371$$ −12.0000 −0.623009
$$372$$ 0 0
$$373$$ −4.00000 −0.207112 −0.103556 0.994624i $$-0.533022\pi$$
−0.103556 + 0.994624i $$0.533022\pi$$
$$374$$ −6.00000 −0.310253
$$375$$ 0 0
$$376$$ 13.0000 0.670424
$$377$$ 2.00000 0.103005
$$378$$ 0 0
$$379$$ 16.0000 0.821865 0.410932 0.911666i $$-0.365203\pi$$
0.410932 + 0.911666i $$0.365203\pi$$
$$380$$ 6.00000 0.307794
$$381$$ 0 0
$$382$$ 10.0000 0.511645
$$383$$ −27.0000 −1.37964 −0.689818 0.723983i $$-0.742309\pi$$
−0.689818 + 0.723983i $$0.742309\pi$$
$$384$$ 0 0
$$385$$ 2.00000 0.101929
$$386$$ 16.0000 0.814379
$$387$$ 0 0
$$388$$ 14.0000 0.710742
$$389$$ 30.0000 1.52106 0.760530 0.649303i $$-0.224939\pi$$
0.760530 + 0.649303i $$0.224939\pi$$
$$390$$ 0 0
$$391$$ 12.0000 0.606866
$$392$$ 6.00000 0.303046
$$393$$ 0 0
$$394$$ 9.00000 0.453413
$$395$$ −4.00000 −0.201262
$$396$$ 0 0
$$397$$ −22.0000 −1.10415 −0.552074 0.833795i $$-0.686163\pi$$
−0.552074 + 0.833795i $$0.686163\pi$$
$$398$$ 10.0000 0.501255
$$399$$ 0 0
$$400$$ −4.00000 −0.200000
$$401$$ −24.0000 −1.19850 −0.599251 0.800561i $$-0.704535\pi$$
−0.599251 + 0.800561i $$0.704535\pi$$
$$402$$ 0 0
$$403$$ −4.00000 −0.199254
$$404$$ −4.00000 −0.199007
$$405$$ 0 0
$$406$$ 2.00000 0.0992583
$$407$$ 6.00000 0.297409
$$408$$ 0 0
$$409$$ 4.00000 0.197787 0.0988936 0.995098i $$-0.468470\pi$$
0.0988936 + 0.995098i $$0.468470\pi$$
$$410$$ 0 0
$$411$$ 0 0
$$412$$ −8.00000 −0.394132
$$413$$ 10.0000 0.492068
$$414$$ 0 0
$$415$$ 0 0
$$416$$ 1.00000 0.0490290
$$417$$ 0 0
$$418$$ −12.0000 −0.586939
$$419$$ −21.0000 −1.02592 −0.512959 0.858413i $$-0.671451\pi$$
−0.512959 + 0.858413i $$0.671451\pi$$
$$420$$ 0 0
$$421$$ −5.00000 −0.243685 −0.121843 0.992549i $$-0.538880\pi$$
−0.121843 + 0.992549i $$0.538880\pi$$
$$422$$ −23.0000 −1.11962
$$423$$ 0 0
$$424$$ 12.0000 0.582772
$$425$$ −12.0000 −0.582086
$$426$$ 0 0
$$427$$ −8.00000 −0.387147
$$428$$ 4.00000 0.193347
$$429$$ 0 0
$$430$$ 5.00000 0.241121
$$431$$ −33.0000 −1.58955 −0.794777 0.606902i $$-0.792412\pi$$
−0.794777 + 0.606902i $$0.792412\pi$$
$$432$$ 0 0
$$433$$ 7.00000 0.336399 0.168199 0.985753i $$-0.446205\pi$$
0.168199 + 0.985753i $$0.446205\pi$$
$$434$$ −4.00000 −0.192006
$$435$$ 0 0
$$436$$ 19.0000 0.909935
$$437$$ 24.0000 1.14808
$$438$$ 0 0
$$439$$ −22.0000 −1.05000 −0.525001 0.851101i $$-0.675935\pi$$
−0.525001 + 0.851101i $$0.675935\pi$$
$$440$$ −2.00000 −0.0953463
$$441$$ 0 0
$$442$$ 3.00000 0.142695
$$443$$ 39.0000 1.85295 0.926473 0.376361i $$-0.122825\pi$$
0.926473 + 0.376361i $$0.122825\pi$$
$$444$$ 0 0
$$445$$ −6.00000 −0.284427
$$446$$ 21.0000 0.994379
$$447$$ 0 0
$$448$$ 1.00000 0.0472456
$$449$$ 26.0000 1.22702 0.613508 0.789689i $$-0.289758\pi$$
0.613508 + 0.789689i $$0.289758\pi$$
$$450$$ 0 0
$$451$$ 0 0
$$452$$ −2.00000 −0.0940721
$$453$$ 0 0
$$454$$ −24.0000 −1.12638
$$455$$ −1.00000 −0.0468807
$$456$$ 0 0
$$457$$ 10.0000 0.467780 0.233890 0.972263i $$-0.424854\pi$$
0.233890 + 0.972263i $$0.424854\pi$$
$$458$$ 15.0000 0.700904
$$459$$ 0 0
$$460$$ 4.00000 0.186501
$$461$$ 21.0000 0.978068 0.489034 0.872265i $$-0.337349\pi$$
0.489034 + 0.872265i $$0.337349\pi$$
$$462$$ 0 0
$$463$$ 16.0000 0.743583 0.371792 0.928316i $$-0.378744\pi$$
0.371792 + 0.928316i $$0.378744\pi$$
$$464$$ −2.00000 −0.0928477
$$465$$ 0 0
$$466$$ −11.0000 −0.509565
$$467$$ −20.0000 −0.925490 −0.462745 0.886492i $$-0.653135\pi$$
−0.462745 + 0.886492i $$0.653135\pi$$
$$468$$ 0 0
$$469$$ −2.00000 −0.0923514
$$470$$ 13.0000 0.599645
$$471$$ 0 0
$$472$$ −10.0000 −0.460287
$$473$$ −10.0000 −0.459800
$$474$$ 0 0
$$475$$ −24.0000 −1.10120
$$476$$ 3.00000 0.137505
$$477$$ 0 0
$$478$$ 9.00000 0.411650
$$479$$ 3.00000 0.137073 0.0685367 0.997649i $$-0.478167\pi$$
0.0685367 + 0.997649i $$0.478167\pi$$
$$480$$ 0 0
$$481$$ −3.00000 −0.136788
$$482$$ −18.0000 −0.819878
$$483$$ 0 0
$$484$$ −7.00000 −0.318182
$$485$$ 14.0000 0.635707
$$486$$ 0 0
$$487$$ −16.0000 −0.725029 −0.362515 0.931978i $$-0.618082\pi$$
−0.362515 + 0.931978i $$0.618082\pi$$
$$488$$ 8.00000 0.362143
$$489$$ 0 0
$$490$$ 6.00000 0.271052
$$491$$ 5.00000 0.225647 0.112823 0.993615i $$-0.464011\pi$$
0.112823 + 0.993615i $$0.464011\pi$$
$$492$$ 0 0
$$493$$ −6.00000 −0.270226
$$494$$ 6.00000 0.269953
$$495$$ 0 0
$$496$$ 4.00000 0.179605
$$497$$ 5.00000 0.224281
$$498$$ 0 0
$$499$$ −32.0000 −1.43252 −0.716258 0.697835i $$-0.754147\pi$$
−0.716258 + 0.697835i $$0.754147\pi$$
$$500$$ −9.00000 −0.402492
$$501$$ 0 0
$$502$$ 0 0
$$503$$ 14.0000 0.624229 0.312115 0.950044i $$-0.398963\pi$$
0.312115 + 0.950044i $$0.398963\pi$$
$$504$$ 0 0
$$505$$ −4.00000 −0.177998
$$506$$ −8.00000 −0.355643
$$507$$ 0 0
$$508$$ 16.0000 0.709885
$$509$$ −34.0000 −1.50702 −0.753512 0.657434i $$-0.771642\pi$$
−0.753512 + 0.657434i $$0.771642\pi$$
$$510$$ 0 0
$$511$$ −10.0000 −0.442374
$$512$$ −1.00000 −0.0441942
$$513$$ 0 0
$$514$$ −15.0000 −0.661622
$$515$$ −8.00000 −0.352522
$$516$$ 0 0
$$517$$ −26.0000 −1.14348
$$518$$ −3.00000 −0.131812
$$519$$ 0 0
$$520$$ 1.00000 0.0438529
$$521$$ −39.0000 −1.70862 −0.854311 0.519763i $$-0.826020\pi$$
−0.854311 + 0.519763i $$0.826020\pi$$
$$522$$ 0 0
$$523$$ −36.0000 −1.57417 −0.787085 0.616844i $$-0.788411\pi$$
−0.787085 + 0.616844i $$0.788411\pi$$
$$524$$ 1.00000 0.0436852
$$525$$ 0 0
$$526$$ 12.0000 0.523225
$$527$$ 12.0000 0.522728
$$528$$ 0 0
$$529$$ −7.00000 −0.304348
$$530$$ 12.0000 0.521247
$$531$$ 0 0
$$532$$ 6.00000 0.260133
$$533$$ 0 0
$$534$$ 0 0
$$535$$ 4.00000 0.172935
$$536$$ 2.00000 0.0863868
$$537$$ 0 0
$$538$$ −24.0000 −1.03471
$$539$$ −12.0000 −0.516877
$$540$$ 0 0
$$541$$ 17.0000 0.730887 0.365444 0.930834i $$-0.380917\pi$$
0.365444 + 0.930834i $$0.380917\pi$$
$$542$$ −13.0000 −0.558398
$$543$$ 0 0
$$544$$ −3.00000 −0.128624
$$545$$ 19.0000 0.813871
$$546$$ 0 0
$$547$$ 37.0000 1.58201 0.791003 0.611812i $$-0.209559\pi$$
0.791003 + 0.611812i $$0.209559\pi$$
$$548$$ −12.0000 −0.512615
$$549$$ 0 0
$$550$$ 8.00000 0.341121
$$551$$ −12.0000 −0.511217
$$552$$ 0 0
$$553$$ −4.00000 −0.170097
$$554$$ −12.0000 −0.509831
$$555$$ 0 0
$$556$$ 7.00000 0.296866
$$557$$ −33.0000 −1.39825 −0.699127 0.714997i $$-0.746428\pi$$
−0.699127 + 0.714997i $$0.746428\pi$$
$$558$$ 0 0
$$559$$ 5.00000 0.211477
$$560$$ 1.00000 0.0422577
$$561$$ 0 0
$$562$$ −26.0000 −1.09674
$$563$$ −11.0000 −0.463595 −0.231797 0.972764i $$-0.574461\pi$$
−0.231797 + 0.972764i $$0.574461\pi$$
$$564$$ 0 0
$$565$$ −2.00000 −0.0841406
$$566$$ −4.00000 −0.168133
$$567$$ 0 0
$$568$$ −5.00000 −0.209795
$$569$$ −31.0000 −1.29959 −0.649794 0.760111i $$-0.725145\pi$$
−0.649794 + 0.760111i $$0.725145\pi$$
$$570$$ 0 0
$$571$$ 33.0000 1.38101 0.690504 0.723329i $$-0.257389\pi$$
0.690504 + 0.723329i $$0.257389\pi$$
$$572$$ −2.00000 −0.0836242
$$573$$ 0 0
$$574$$ 0 0
$$575$$ −16.0000 −0.667246
$$576$$ 0 0
$$577$$ 18.0000 0.749350 0.374675 0.927156i $$-0.377754\pi$$
0.374675 + 0.927156i $$0.377754\pi$$
$$578$$ 8.00000 0.332756
$$579$$ 0 0
$$580$$ −2.00000 −0.0830455
$$581$$ 0 0
$$582$$ 0 0
$$583$$ −24.0000 −0.993978
$$584$$ 10.0000 0.413803
$$585$$ 0 0
$$586$$ 7.00000 0.289167
$$587$$ 28.0000 1.15568 0.577842 0.816149i $$-0.303895\pi$$
0.577842 + 0.816149i $$0.303895\pi$$
$$588$$ 0 0
$$589$$ 24.0000 0.988903
$$590$$ −10.0000 −0.411693
$$591$$ 0 0
$$592$$ 3.00000 0.123299
$$593$$ 22.0000 0.903432 0.451716 0.892162i $$-0.350812\pi$$
0.451716 + 0.892162i $$0.350812\pi$$
$$594$$ 0 0
$$595$$ 3.00000 0.122988
$$596$$ 18.0000 0.737309
$$597$$ 0 0
$$598$$ 4.00000 0.163572
$$599$$ 2.00000 0.0817178 0.0408589 0.999165i $$-0.486991\pi$$
0.0408589 + 0.999165i $$0.486991\pi$$
$$600$$ 0 0
$$601$$ −35.0000 −1.42768 −0.713840 0.700309i $$-0.753046\pi$$
−0.713840 + 0.700309i $$0.753046\pi$$
$$602$$ 5.00000 0.203785
$$603$$ 0 0
$$604$$ −9.00000 −0.366205
$$605$$ −7.00000 −0.284590
$$606$$ 0 0
$$607$$ 6.00000 0.243532 0.121766 0.992559i $$-0.461144\pi$$
0.121766 + 0.992559i $$0.461144\pi$$
$$608$$ −6.00000 −0.243332
$$609$$ 0 0
$$610$$ 8.00000 0.323911
$$611$$ 13.0000 0.525924
$$612$$ 0 0
$$613$$ 26.0000 1.05013 0.525065 0.851062i $$-0.324041\pi$$
0.525065 + 0.851062i $$0.324041\pi$$
$$614$$ −14.0000 −0.564994
$$615$$ 0 0
$$616$$ −2.00000 −0.0805823
$$617$$ −16.0000 −0.644136 −0.322068 0.946717i $$-0.604378\pi$$
−0.322068 + 0.946717i $$0.604378\pi$$
$$618$$ 0 0
$$619$$ 4.00000 0.160774 0.0803868 0.996764i $$-0.474384\pi$$
0.0803868 + 0.996764i $$0.474384\pi$$
$$620$$ 4.00000 0.160644
$$621$$ 0 0
$$622$$ 18.0000 0.721734
$$623$$ −6.00000 −0.240385
$$624$$ 0 0
$$625$$ 11.0000 0.440000
$$626$$ 1.00000 0.0399680
$$627$$ 0 0
$$628$$ −10.0000 −0.399043
$$629$$ 9.00000 0.358854
$$630$$ 0 0
$$631$$ −5.00000 −0.199047 −0.0995234 0.995035i $$-0.531732\pi$$
−0.0995234 + 0.995035i $$0.531732\pi$$
$$632$$ 4.00000 0.159111
$$633$$ 0 0
$$634$$ −18.0000 −0.714871
$$635$$ 16.0000 0.634941
$$636$$ 0 0
$$637$$ 6.00000 0.237729
$$638$$ 4.00000 0.158362
$$639$$ 0 0
$$640$$ −1.00000 −0.0395285
$$641$$ 2.00000 0.0789953 0.0394976 0.999220i $$-0.487424\pi$$
0.0394976 + 0.999220i $$0.487424\pi$$
$$642$$ 0 0
$$643$$ 14.0000 0.552106 0.276053 0.961142i $$-0.410973\pi$$
0.276053 + 0.961142i $$0.410973\pi$$
$$644$$ 4.00000 0.157622
$$645$$ 0 0
$$646$$ −18.0000 −0.708201
$$647$$ 38.0000 1.49393 0.746967 0.664861i $$-0.231509\pi$$
0.746967 + 0.664861i $$0.231509\pi$$
$$648$$ 0 0
$$649$$ 20.0000 0.785069
$$650$$ −4.00000 −0.156893
$$651$$ 0 0
$$652$$ −4.00000 −0.156652
$$653$$ −24.0000 −0.939193 −0.469596 0.882881i $$-0.655601\pi$$
−0.469596 + 0.882881i $$0.655601\pi$$
$$654$$ 0 0
$$655$$ 1.00000 0.0390732
$$656$$ 0 0
$$657$$ 0 0
$$658$$ 13.0000 0.506793
$$659$$ 12.0000 0.467454 0.233727 0.972302i $$-0.424908\pi$$
0.233727 + 0.972302i $$0.424908\pi$$
$$660$$ 0 0
$$661$$ −10.0000 −0.388955 −0.194477 0.980907i $$-0.562301\pi$$
−0.194477 + 0.980907i $$0.562301\pi$$
$$662$$ 4.00000 0.155464
$$663$$ 0 0
$$664$$ 0 0
$$665$$ 6.00000 0.232670
$$666$$ 0 0
$$667$$ −8.00000 −0.309761
$$668$$ 0 0
$$669$$ 0 0
$$670$$ 2.00000 0.0772667
$$671$$ −16.0000 −0.617673
$$672$$ 0 0
$$673$$ 37.0000 1.42625 0.713123 0.701039i $$-0.247280\pi$$
0.713123 + 0.701039i $$0.247280\pi$$
$$674$$ −23.0000 −0.885927
$$675$$ 0 0
$$676$$ 1.00000 0.0384615
$$677$$ 36.0000 1.38359 0.691796 0.722093i $$-0.256820\pi$$
0.691796 + 0.722093i $$0.256820\pi$$
$$678$$ 0 0
$$679$$ 14.0000 0.537271
$$680$$ −3.00000 −0.115045
$$681$$ 0 0
$$682$$ −8.00000 −0.306336
$$683$$ 44.0000 1.68361 0.841807 0.539779i $$-0.181492\pi$$
0.841807 + 0.539779i $$0.181492\pi$$
$$684$$ 0 0
$$685$$ −12.0000 −0.458496
$$686$$ 13.0000 0.496342
$$687$$ 0 0
$$688$$ −5.00000 −0.190623
$$689$$ 12.0000 0.457164
$$690$$ 0 0
$$691$$ −8.00000 −0.304334 −0.152167 0.988355i $$-0.548625\pi$$
−0.152167 + 0.988355i $$0.548625\pi$$
$$692$$ −20.0000 −0.760286
$$693$$ 0 0
$$694$$ −9.00000 −0.341635
$$695$$ 7.00000 0.265525
$$696$$ 0 0
$$697$$ 0 0
$$698$$ −7.00000 −0.264954
$$699$$ 0 0
$$700$$ −4.00000 −0.151186
$$701$$ 12.0000 0.453234 0.226617 0.973984i $$-0.427233\pi$$
0.226617 + 0.973984i $$0.427233\pi$$
$$702$$ 0 0
$$703$$ 18.0000 0.678883
$$704$$ 2.00000 0.0753778
$$705$$ 0 0
$$706$$ 4.00000 0.150542
$$707$$ −4.00000 −0.150435
$$708$$ 0 0
$$709$$ 38.0000 1.42712 0.713560 0.700594i $$-0.247082\pi$$
0.713560 + 0.700594i $$0.247082\pi$$
$$710$$ −5.00000 −0.187647
$$711$$ 0 0
$$712$$ 6.00000 0.224860
$$713$$ 16.0000 0.599205
$$714$$ 0 0
$$715$$ −2.00000 −0.0747958
$$716$$ 9.00000 0.336346
$$717$$ 0 0
$$718$$ 24.0000 0.895672
$$719$$ 22.0000 0.820462 0.410231 0.911982i $$-0.365448\pi$$
0.410231 + 0.911982i $$0.365448\pi$$
$$720$$ 0 0
$$721$$ −8.00000 −0.297936
$$722$$ −17.0000 −0.632674
$$723$$ 0 0
$$724$$ 0 0
$$725$$ 8.00000 0.297113
$$726$$ 0 0
$$727$$ −14.0000 −0.519231 −0.259616 0.965712i $$-0.583596\pi$$
−0.259616 + 0.965712i $$0.583596\pi$$
$$728$$ 1.00000 0.0370625
$$729$$ 0 0
$$730$$ 10.0000 0.370117
$$731$$ −15.0000 −0.554795
$$732$$ 0 0
$$733$$ −43.0000 −1.58824 −0.794121 0.607760i $$-0.792068\pi$$
−0.794121 + 0.607760i $$0.792068\pi$$
$$734$$ 10.0000 0.369107
$$735$$ 0 0
$$736$$ −4.00000 −0.147442
$$737$$ −4.00000 −0.147342
$$738$$ 0 0
$$739$$ 12.0000 0.441427 0.220714 0.975339i $$-0.429161\pi$$
0.220714 + 0.975339i $$0.429161\pi$$
$$740$$ 3.00000 0.110282
$$741$$ 0 0
$$742$$ 12.0000 0.440534
$$743$$ 47.0000 1.72426 0.862131 0.506685i $$-0.169129\pi$$
0.862131 + 0.506685i $$0.169129\pi$$
$$744$$ 0 0
$$745$$ 18.0000 0.659469
$$746$$ 4.00000 0.146450
$$747$$ 0 0
$$748$$ 6.00000 0.219382
$$749$$ 4.00000 0.146157
$$750$$ 0 0
$$751$$ 24.0000 0.875772 0.437886 0.899030i $$-0.355727\pi$$
0.437886 + 0.899030i $$0.355727\pi$$
$$752$$ −13.0000 −0.474061
$$753$$ 0 0
$$754$$ −2.00000 −0.0728357
$$755$$ −9.00000 −0.327544
$$756$$ 0 0
$$757$$ −12.0000 −0.436147 −0.218074 0.975932i $$-0.569977\pi$$
−0.218074 + 0.975932i $$0.569977\pi$$
$$758$$ −16.0000 −0.581146
$$759$$ 0 0
$$760$$ −6.00000 −0.217643
$$761$$ −6.00000 −0.217500 −0.108750 0.994069i $$-0.534685\pi$$
−0.108750 + 0.994069i $$0.534685\pi$$
$$762$$ 0 0
$$763$$ 19.0000 0.687846
$$764$$ −10.0000 −0.361787
$$765$$ 0 0
$$766$$ 27.0000 0.975550
$$767$$ −10.0000 −0.361079
$$768$$ 0 0
$$769$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$770$$ −2.00000 −0.0720750
$$771$$ 0 0
$$772$$ −16.0000 −0.575853
$$773$$ −11.0000 −0.395643 −0.197821 0.980238i $$-0.563387\pi$$
−0.197821 + 0.980238i $$0.563387\pi$$
$$774$$ 0 0
$$775$$ −16.0000 −0.574737
$$776$$ −14.0000 −0.502571
$$777$$ 0 0
$$778$$ −30.0000 −1.07555
$$779$$ 0 0
$$780$$ 0 0
$$781$$ 10.0000 0.357828
$$782$$ −12.0000 −0.429119
$$783$$ 0 0
$$784$$ −6.00000 −0.214286
$$785$$ −10.0000 −0.356915
$$786$$ 0 0
$$787$$ 32.0000 1.14068 0.570338 0.821410i $$-0.306812\pi$$
0.570338 + 0.821410i $$0.306812\pi$$
$$788$$ −9.00000 −0.320612
$$789$$ 0 0
$$790$$ 4.00000 0.142314
$$791$$ −2.00000 −0.0711118
$$792$$ 0 0
$$793$$ 8.00000 0.284088
$$794$$ 22.0000 0.780751
$$795$$ 0 0
$$796$$ −10.0000 −0.354441
$$797$$ 42.0000 1.48772 0.743858 0.668338i $$-0.232994\pi$$
0.743858 + 0.668338i $$0.232994\pi$$
$$798$$ 0 0
$$799$$ −39.0000 −1.37972
$$800$$ 4.00000 0.141421
$$801$$ 0 0
$$802$$ 24.0000 0.847469
$$803$$ −20.0000 −0.705785
$$804$$ 0 0
$$805$$ 4.00000 0.140981
$$806$$ 4.00000 0.140894
$$807$$ 0 0
$$808$$ 4.00000 0.140720
$$809$$ 9.00000 0.316423 0.158212 0.987405i $$-0.449427\pi$$
0.158212 + 0.987405i $$0.449427\pi$$
$$810$$ 0 0
$$811$$ −28.0000 −0.983213 −0.491606 0.870817i $$-0.663590\pi$$
−0.491606 + 0.870817i $$0.663590\pi$$
$$812$$ −2.00000 −0.0701862
$$813$$ 0 0
$$814$$ −6.00000 −0.210300
$$815$$ −4.00000 −0.140114
$$816$$ 0 0
$$817$$ −30.0000 −1.04957
$$818$$ −4.00000 −0.139857
$$819$$ 0 0
$$820$$ 0 0
$$821$$ 25.0000 0.872506 0.436253 0.899824i $$-0.356305\pi$$
0.436253 + 0.899824i $$0.356305\pi$$
$$822$$ 0 0
$$823$$ 54.0000 1.88232 0.941161 0.337959i $$-0.109737\pi$$
0.941161 + 0.337959i $$0.109737\pi$$
$$824$$ 8.00000 0.278693
$$825$$ 0 0
$$826$$ −10.0000 −0.347945
$$827$$ −30.0000 −1.04320 −0.521601 0.853189i $$-0.674665\pi$$
−0.521601 + 0.853189i $$0.674665\pi$$
$$828$$ 0 0
$$829$$ −38.0000 −1.31979 −0.659897 0.751356i $$-0.729400\pi$$
−0.659897 + 0.751356i $$0.729400\pi$$
$$830$$ 0 0
$$831$$ 0 0
$$832$$ −1.00000 −0.0346688
$$833$$ −18.0000 −0.623663
$$834$$ 0 0
$$835$$ 0 0
$$836$$ 12.0000 0.415029
$$837$$ 0 0
$$838$$ 21.0000 0.725433
$$839$$ −56.0000 −1.93333 −0.966667 0.256036i $$-0.917584\pi$$
−0.966667 + 0.256036i $$0.917584\pi$$
$$840$$ 0 0
$$841$$ −25.0000 −0.862069
$$842$$ 5.00000 0.172311
$$843$$ 0 0
$$844$$ 23.0000 0.791693
$$845$$ 1.00000 0.0344010
$$846$$ 0 0
$$847$$ −7.00000 −0.240523
$$848$$ −12.0000 −0.412082
$$849$$ 0 0
$$850$$ 12.0000 0.411597
$$851$$ 12.0000 0.411355
$$852$$ 0 0
$$853$$ 49.0000 1.67773 0.838864 0.544341i $$-0.183220\pi$$
0.838864 + 0.544341i $$0.183220\pi$$
$$854$$ 8.00000 0.273754
$$855$$ 0 0
$$856$$ −4.00000 −0.136717
$$857$$ −46.0000 −1.57133 −0.785665 0.618652i $$-0.787679\pi$$
−0.785665 + 0.618652i $$0.787679\pi$$
$$858$$ 0 0
$$859$$ 20.0000 0.682391 0.341196 0.939992i $$-0.389168\pi$$
0.341196 + 0.939992i $$0.389168\pi$$
$$860$$ −5.00000 −0.170499
$$861$$ 0 0
$$862$$ 33.0000 1.12398
$$863$$ 11.0000 0.374444 0.187222 0.982318i $$-0.440052\pi$$
0.187222 + 0.982318i $$0.440052\pi$$
$$864$$ 0 0
$$865$$ −20.0000 −0.680020
$$866$$ −7.00000 −0.237870
$$867$$ 0 0
$$868$$ 4.00000 0.135769
$$869$$ −8.00000 −0.271381
$$870$$ 0 0
$$871$$ 2.00000 0.0677674
$$872$$ −19.0000 −0.643421
$$873$$ 0 0
$$874$$ −24.0000 −0.811812
$$875$$ −9.00000 −0.304256
$$876$$ 0 0
$$877$$ −39.0000 −1.31694 −0.658468 0.752609i $$-0.728795\pi$$
−0.658468 + 0.752609i $$0.728795\pi$$
$$878$$ 22.0000 0.742464
$$879$$ 0 0
$$880$$ 2.00000 0.0674200
$$881$$ −21.0000 −0.707508 −0.353754 0.935339i $$-0.615095\pi$$
−0.353754 + 0.935339i $$0.615095\pi$$
$$882$$ 0 0
$$883$$ −47.0000 −1.58168 −0.790838 0.612026i $$-0.790355\pi$$
−0.790838 + 0.612026i $$0.790355\pi$$
$$884$$ −3.00000 −0.100901
$$885$$ 0 0
$$886$$ −39.0000 −1.31023
$$887$$ 8.00000 0.268614 0.134307 0.990940i $$-0.457119\pi$$
0.134307 + 0.990940i $$0.457119\pi$$
$$888$$ 0 0
$$889$$ 16.0000 0.536623
$$890$$ 6.00000 0.201120
$$891$$ 0 0
$$892$$ −21.0000 −0.703132
$$893$$ −78.0000 −2.61017
$$894$$ 0 0
$$895$$ 9.00000 0.300837
$$896$$ −1.00000 −0.0334077
$$897$$ 0 0
$$898$$ −26.0000 −0.867631
$$899$$ −8.00000 −0.266815
$$900$$ 0 0
$$901$$ −36.0000 −1.19933
$$902$$ 0 0
$$903$$ 0 0
$$904$$ 2.00000 0.0665190
$$905$$ 0 0
$$906$$ 0 0
$$907$$ −9.00000 −0.298840 −0.149420 0.988774i $$-0.547741\pi$$
−0.149420 + 0.988774i $$0.547741\pi$$
$$908$$ 24.0000 0.796468
$$909$$ 0 0
$$910$$ 1.00000 0.0331497
$$911$$ 54.0000 1.78910 0.894550 0.446968i $$-0.147496\pi$$
0.894550 + 0.446968i $$0.147496\pi$$
$$912$$ 0 0
$$913$$ 0 0
$$914$$ −10.0000 −0.330771
$$915$$ 0 0
$$916$$ −15.0000 −0.495614
$$917$$ 1.00000 0.0330229
$$918$$ 0 0
$$919$$ 24.0000 0.791687 0.395843 0.918318i $$-0.370452\pi$$
0.395843 + 0.918318i $$0.370452\pi$$
$$920$$ −4.00000 −0.131876
$$921$$ 0 0
$$922$$ −21.0000 −0.691598
$$923$$ −5.00000 −0.164577
$$924$$ 0 0
$$925$$ −12.0000 −0.394558
$$926$$ −16.0000 −0.525793
$$927$$ 0 0
$$928$$ 2.00000 0.0656532
$$929$$ 36.0000 1.18112 0.590561 0.806993i $$-0.298907\pi$$
0.590561 + 0.806993i $$0.298907\pi$$
$$930$$ 0 0
$$931$$ −36.0000 −1.17985
$$932$$ 11.0000 0.360317
$$933$$ 0 0
$$934$$ 20.0000 0.654420
$$935$$ 6.00000 0.196221
$$936$$ 0 0
$$937$$ −42.0000 −1.37208 −0.686040 0.727564i $$-0.740653\pi$$
−0.686040 + 0.727564i $$0.740653\pi$$
$$938$$ 2.00000 0.0653023
$$939$$ 0 0
$$940$$ −13.0000 −0.424013
$$941$$ −25.0000 −0.814977 −0.407488 0.913210i $$-0.633595\pi$$
−0.407488 + 0.913210i $$0.633595\pi$$
$$942$$ 0 0
$$943$$ 0 0
$$944$$ 10.0000 0.325472
$$945$$ 0 0
$$946$$ 10.0000 0.325128
$$947$$ 18.0000 0.584921 0.292461 0.956278i $$-0.405526\pi$$
0.292461 + 0.956278i $$0.405526\pi$$
$$948$$ 0 0
$$949$$ 10.0000 0.324614
$$950$$ 24.0000 0.778663
$$951$$ 0 0
$$952$$ −3.00000 −0.0972306
$$953$$ −23.0000 −0.745043 −0.372522 0.928024i $$-0.621507\pi$$
−0.372522 + 0.928024i $$0.621507\pi$$
$$954$$ 0 0
$$955$$ −10.0000 −0.323592
$$956$$ −9.00000 −0.291081
$$957$$ 0 0
$$958$$ −3.00000 −0.0969256
$$959$$ −12.0000 −0.387500
$$960$$ 0 0
$$961$$ −15.0000 −0.483871
$$962$$ 3.00000 0.0967239
$$963$$ 0 0
$$964$$ 18.0000 0.579741
$$965$$ −16.0000 −0.515058
$$966$$ 0 0
$$967$$ 23.0000 0.739630 0.369815 0.929105i $$-0.379421\pi$$
0.369815 + 0.929105i $$0.379421\pi$$
$$968$$ 7.00000 0.224989
$$969$$ 0 0
$$970$$ −14.0000 −0.449513
$$971$$ 15.0000 0.481373 0.240686 0.970603i $$-0.422627\pi$$
0.240686 + 0.970603i $$0.422627\pi$$
$$972$$ 0 0
$$973$$ 7.00000 0.224410
$$974$$ 16.0000 0.512673
$$975$$ 0 0
$$976$$ −8.00000 −0.256074
$$977$$ 30.0000 0.959785 0.479893 0.877327i $$-0.340676\pi$$
0.479893 + 0.877327i $$0.340676\pi$$
$$978$$ 0 0
$$979$$ −12.0000 −0.383522
$$980$$ −6.00000 −0.191663
$$981$$ 0 0
$$982$$ −5.00000 −0.159556
$$983$$ 31.0000 0.988746 0.494373 0.869250i $$-0.335398\pi$$
0.494373 + 0.869250i $$0.335398\pi$$
$$984$$ 0 0
$$985$$ −9.00000 −0.286764
$$986$$ 6.00000 0.191079
$$987$$ 0 0
$$988$$ −6.00000 −0.190885
$$989$$ −20.0000 −0.635963
$$990$$ 0 0
$$991$$ −30.0000 −0.952981 −0.476491 0.879180i $$-0.658091\pi$$
−0.476491 + 0.879180i $$0.658091\pi$$
$$992$$ −4.00000 −0.127000
$$993$$ 0 0
$$994$$ −5.00000 −0.158590
$$995$$ −10.0000 −0.317021
$$996$$ 0 0
$$997$$ −10.0000 −0.316703 −0.158352 0.987383i $$-0.550618\pi$$
−0.158352 + 0.987383i $$0.550618\pi$$
$$998$$ 32.0000 1.01294
$$999$$ 0 0
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 234.2.a.b.1.1 1
3.2 odd 2 26.2.a.b.1.1 1
4.3 odd 2 1872.2.a.m.1.1 1
5.2 odd 4 5850.2.e.v.5149.1 2
5.3 odd 4 5850.2.e.v.5149.2 2
5.4 even 2 5850.2.a.bn.1.1 1
8.3 odd 2 7488.2.a.v.1.1 1
8.5 even 2 7488.2.a.w.1.1 1
9.2 odd 6 2106.2.e.h.1405.1 2
9.4 even 3 2106.2.e.t.703.1 2
9.5 odd 6 2106.2.e.h.703.1 2
9.7 even 3 2106.2.e.t.1405.1 2
12.11 even 2 208.2.a.d.1.1 1
13.5 odd 4 3042.2.b.f.1351.2 2
13.8 odd 4 3042.2.b.f.1351.1 2
13.12 even 2 3042.2.a.l.1.1 1
15.2 even 4 650.2.b.a.599.2 2
15.8 even 4 650.2.b.a.599.1 2
15.14 odd 2 650.2.a.g.1.1 1
21.2 odd 6 1274.2.f.l.1145.1 2
21.5 even 6 1274.2.f.a.1145.1 2
21.11 odd 6 1274.2.f.l.79.1 2
21.17 even 6 1274.2.f.a.79.1 2
21.20 even 2 1274.2.a.o.1.1 1
24.5 odd 2 832.2.a.j.1.1 1
24.11 even 2 832.2.a.a.1.1 1
33.32 even 2 3146.2.a.a.1.1 1
39.2 even 12 338.2.e.d.147.2 4
39.5 even 4 338.2.b.a.337.1 2
39.8 even 4 338.2.b.a.337.2 2
39.11 even 12 338.2.e.d.147.1 4
39.17 odd 6 338.2.c.g.315.1 2
39.20 even 12 338.2.e.d.23.2 4
39.23 odd 6 338.2.c.g.191.1 2
39.29 odd 6 338.2.c.c.191.1 2
39.32 even 12 338.2.e.d.23.1 4
39.35 odd 6 338.2.c.c.315.1 2
39.38 odd 2 338.2.a.a.1.1 1
48.5 odd 4 3328.2.b.g.1665.2 2
48.11 even 4 3328.2.b.k.1665.1 2
48.29 odd 4 3328.2.b.g.1665.1 2
48.35 even 4 3328.2.b.k.1665.2 2
51.50 odd 2 7514.2.a.i.1.1 1
57.56 even 2 9386.2.a.f.1.1 1
60.59 even 2 5200.2.a.c.1.1 1
156.47 odd 4 2704.2.f.j.337.1 2
156.83 odd 4 2704.2.f.j.337.2 2
156.155 even 2 2704.2.a.n.1.1 1
195.194 odd 2 8450.2.a.y.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
26.2.a.b.1.1 1 3.2 odd 2
208.2.a.d.1.1 1 12.11 even 2
234.2.a.b.1.1 1 1.1 even 1 trivial
338.2.a.a.1.1 1 39.38 odd 2
338.2.b.a.337.1 2 39.5 even 4
338.2.b.a.337.2 2 39.8 even 4
338.2.c.c.191.1 2 39.29 odd 6
338.2.c.c.315.1 2 39.35 odd 6
338.2.c.g.191.1 2 39.23 odd 6
338.2.c.g.315.1 2 39.17 odd 6
338.2.e.d.23.1 4 39.32 even 12
338.2.e.d.23.2 4 39.20 even 12
338.2.e.d.147.1 4 39.11 even 12
338.2.e.d.147.2 4 39.2 even 12
650.2.a.g.1.1 1 15.14 odd 2
650.2.b.a.599.1 2 15.8 even 4
650.2.b.a.599.2 2 15.2 even 4
832.2.a.a.1.1 1 24.11 even 2
832.2.a.j.1.1 1 24.5 odd 2
1274.2.a.o.1.1 1 21.20 even 2
1274.2.f.a.79.1 2 21.17 even 6
1274.2.f.a.1145.1 2 21.5 even 6
1274.2.f.l.79.1 2 21.11 odd 6
1274.2.f.l.1145.1 2 21.2 odd 6
1872.2.a.m.1.1 1 4.3 odd 2
2106.2.e.h.703.1 2 9.5 odd 6
2106.2.e.h.1405.1 2 9.2 odd 6
2106.2.e.t.703.1 2 9.4 even 3
2106.2.e.t.1405.1 2 9.7 even 3
2704.2.a.n.1.1 1 156.155 even 2
2704.2.f.j.337.1 2 156.47 odd 4
2704.2.f.j.337.2 2 156.83 odd 4
3042.2.a.l.1.1 1 13.12 even 2
3042.2.b.f.1351.1 2 13.8 odd 4
3042.2.b.f.1351.2 2 13.5 odd 4
3146.2.a.a.1.1 1 33.32 even 2
3328.2.b.g.1665.1 2 48.29 odd 4
3328.2.b.g.1665.2 2 48.5 odd 4
3328.2.b.k.1665.1 2 48.11 even 4
3328.2.b.k.1665.2 2 48.35 even 4
5200.2.a.c.1.1 1 60.59 even 2
5850.2.a.bn.1.1 1 5.4 even 2
5850.2.e.v.5149.1 2 5.2 odd 4
5850.2.e.v.5149.2 2 5.3 odd 4
7488.2.a.v.1.1 1 8.3 odd 2
7488.2.a.w.1.1 1 8.5 even 2
7514.2.a.i.1.1 1 51.50 odd 2
8450.2.a.y.1.1 1 195.194 odd 2
9386.2.a.f.1.1 1 57.56 even 2