# Properties

 Label 26.2.a.b.1.1 Level $26$ Weight $2$ Character 26.1 Self dual yes Analytic conductor $0.208$ 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] = [26,2,Mod(1,26)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(26, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("26.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$26 = 2 \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 26.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: $$0.207611045255$$ 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 $$\chi$$ $$=$$ 26.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+1.00000 q^{2} -3.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} -3.00000 q^{6} +1.00000 q^{7} +1.00000 q^{8} +6.00000 q^{9} +O(q^{10})$$ $$q+1.00000 q^{2} -3.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} -3.00000 q^{6} +1.00000 q^{7} +1.00000 q^{8} +6.00000 q^{9} -1.00000 q^{10} -2.00000 q^{11} -3.00000 q^{12} -1.00000 q^{13} +1.00000 q^{14} +3.00000 q^{15} +1.00000 q^{16} -3.00000 q^{17} +6.00000 q^{18} +6.00000 q^{19} -1.00000 q^{20} -3.00000 q^{21} -2.00000 q^{22} -4.00000 q^{23} -3.00000 q^{24} -4.00000 q^{25} -1.00000 q^{26} -9.00000 q^{27} +1.00000 q^{28} +2.00000 q^{29} +3.00000 q^{30} +4.00000 q^{31} +1.00000 q^{32} +6.00000 q^{33} -3.00000 q^{34} -1.00000 q^{35} +6.00000 q^{36} +3.00000 q^{37} +6.00000 q^{38} +3.00000 q^{39} -1.00000 q^{40} -3.00000 q^{42} -5.00000 q^{43} -2.00000 q^{44} -6.00000 q^{45} -4.00000 q^{46} +13.0000 q^{47} -3.00000 q^{48} -6.00000 q^{49} -4.00000 q^{50} +9.00000 q^{51} -1.00000 q^{52} +12.0000 q^{53} -9.00000 q^{54} +2.00000 q^{55} +1.00000 q^{56} -18.0000 q^{57} +2.00000 q^{58} -10.0000 q^{59} +3.00000 q^{60} -8.00000 q^{61} +4.00000 q^{62} +6.00000 q^{63} +1.00000 q^{64} +1.00000 q^{65} +6.00000 q^{66} -2.00000 q^{67} -3.00000 q^{68} +12.0000 q^{69} -1.00000 q^{70} -5.00000 q^{71} +6.00000 q^{72} -10.0000 q^{73} +3.00000 q^{74} +12.0000 q^{75} +6.00000 q^{76} -2.00000 q^{77} +3.00000 q^{78} -4.00000 q^{79} -1.00000 q^{80} +9.00000 q^{81} -3.00000 q^{84} +3.00000 q^{85} -5.00000 q^{86} -6.00000 q^{87} -2.00000 q^{88} +6.00000 q^{89} -6.00000 q^{90} -1.00000 q^{91} -4.00000 q^{92} -12.0000 q^{93} +13.0000 q^{94} -6.00000 q^{95} -3.00000 q^{96} +14.0000 q^{97} -6.00000 q^{98} -12.0000 q^{99} +O(q^{100})$$

## Coefficient data

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

Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 1.00000 0.707107
$$3$$ −3.00000 −1.73205 −0.866025 0.500000i $$-0.833333\pi$$
−0.866025 + 0.500000i $$0.833333\pi$$
$$4$$ 1.00000 0.500000
$$5$$ −1.00000 −0.447214 −0.223607 0.974679i $$-0.571783\pi$$
−0.223607 + 0.974679i $$0.571783\pi$$
$$6$$ −3.00000 −1.22474
$$7$$ 1.00000 0.377964 0.188982 0.981981i $$-0.439481\pi$$
0.188982 + 0.981981i $$0.439481\pi$$
$$8$$ 1.00000 0.353553
$$9$$ 6.00000 2.00000
$$10$$ −1.00000 −0.316228
$$11$$ −2.00000 −0.603023 −0.301511 0.953463i $$-0.597491\pi$$
−0.301511 + 0.953463i $$0.597491\pi$$
$$12$$ −3.00000 −0.866025
$$13$$ −1.00000 −0.277350
$$14$$ 1.00000 0.267261
$$15$$ 3.00000 0.774597
$$16$$ 1.00000 0.250000
$$17$$ −3.00000 −0.727607 −0.363803 0.931476i $$-0.618522\pi$$
−0.363803 + 0.931476i $$0.618522\pi$$
$$18$$ 6.00000 1.41421
$$19$$ 6.00000 1.37649 0.688247 0.725476i $$-0.258380\pi$$
0.688247 + 0.725476i $$0.258380\pi$$
$$20$$ −1.00000 −0.223607
$$21$$ −3.00000 −0.654654
$$22$$ −2.00000 −0.426401
$$23$$ −4.00000 −0.834058 −0.417029 0.908893i $$-0.636929\pi$$
−0.417029 + 0.908893i $$0.636929\pi$$
$$24$$ −3.00000 −0.612372
$$25$$ −4.00000 −0.800000
$$26$$ −1.00000 −0.196116
$$27$$ −9.00000 −1.73205
$$28$$ 1.00000 0.188982
$$29$$ 2.00000 0.371391 0.185695 0.982607i $$-0.440546\pi$$
0.185695 + 0.982607i $$0.440546\pi$$
$$30$$ 3.00000 0.547723
$$31$$ 4.00000 0.718421 0.359211 0.933257i $$-0.383046\pi$$
0.359211 + 0.933257i $$0.383046\pi$$
$$32$$ 1.00000 0.176777
$$33$$ 6.00000 1.04447
$$34$$ −3.00000 −0.514496
$$35$$ −1.00000 −0.169031
$$36$$ 6.00000 1.00000
$$37$$ 3.00000 0.493197 0.246598 0.969118i $$-0.420687\pi$$
0.246598 + 0.969118i $$0.420687\pi$$
$$38$$ 6.00000 0.973329
$$39$$ 3.00000 0.480384
$$40$$ −1.00000 −0.158114
$$41$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$42$$ −3.00000 −0.462910
$$43$$ −5.00000 −0.762493 −0.381246 0.924473i $$-0.624505\pi$$
−0.381246 + 0.924473i $$0.624505\pi$$
$$44$$ −2.00000 −0.301511
$$45$$ −6.00000 −0.894427
$$46$$ −4.00000 −0.589768
$$47$$ 13.0000 1.89624 0.948122 0.317905i $$-0.102979\pi$$
0.948122 + 0.317905i $$0.102979\pi$$
$$48$$ −3.00000 −0.433013
$$49$$ −6.00000 −0.857143
$$50$$ −4.00000 −0.565685
$$51$$ 9.00000 1.26025
$$52$$ −1.00000 −0.138675
$$53$$ 12.0000 1.64833 0.824163 0.566352i $$-0.191646\pi$$
0.824163 + 0.566352i $$0.191646\pi$$
$$54$$ −9.00000 −1.22474
$$55$$ 2.00000 0.269680
$$56$$ 1.00000 0.133631
$$57$$ −18.0000 −2.38416
$$58$$ 2.00000 0.262613
$$59$$ −10.0000 −1.30189 −0.650945 0.759125i $$-0.725627\pi$$
−0.650945 + 0.759125i $$0.725627\pi$$
$$60$$ 3.00000 0.387298
$$61$$ −8.00000 −1.02430 −0.512148 0.858898i $$-0.671150\pi$$
−0.512148 + 0.858898i $$0.671150\pi$$
$$62$$ 4.00000 0.508001
$$63$$ 6.00000 0.755929
$$64$$ 1.00000 0.125000
$$65$$ 1.00000 0.124035
$$66$$ 6.00000 0.738549
$$67$$ −2.00000 −0.244339 −0.122169 0.992509i $$-0.538985\pi$$
−0.122169 + 0.992509i $$0.538985\pi$$
$$68$$ −3.00000 −0.363803
$$69$$ 12.0000 1.44463
$$70$$ −1.00000 −0.119523
$$71$$ −5.00000 −0.593391 −0.296695 0.954972i $$-0.595885\pi$$
−0.296695 + 0.954972i $$0.595885\pi$$
$$72$$ 6.00000 0.707107
$$73$$ −10.0000 −1.17041 −0.585206 0.810885i $$-0.698986\pi$$
−0.585206 + 0.810885i $$0.698986\pi$$
$$74$$ 3.00000 0.348743
$$75$$ 12.0000 1.38564
$$76$$ 6.00000 0.688247
$$77$$ −2.00000 −0.227921
$$78$$ 3.00000 0.339683
$$79$$ −4.00000 −0.450035 −0.225018 0.974355i $$-0.572244\pi$$
−0.225018 + 0.974355i $$0.572244\pi$$
$$80$$ −1.00000 −0.111803
$$81$$ 9.00000 1.00000
$$82$$ 0 0
$$83$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$84$$ −3.00000 −0.327327
$$85$$ 3.00000 0.325396
$$86$$ −5.00000 −0.539164
$$87$$ −6.00000 −0.643268
$$88$$ −2.00000 −0.213201
$$89$$ 6.00000 0.635999 0.317999 0.948091i $$-0.396989\pi$$
0.317999 + 0.948091i $$0.396989\pi$$
$$90$$ −6.00000 −0.632456
$$91$$ −1.00000 −0.104828
$$92$$ −4.00000 −0.417029
$$93$$ −12.0000 −1.24434
$$94$$ 13.0000 1.34085
$$95$$ −6.00000 −0.615587
$$96$$ −3.00000 −0.306186
$$97$$ 14.0000 1.42148 0.710742 0.703452i $$-0.248359\pi$$
0.710742 + 0.703452i $$0.248359\pi$$
$$98$$ −6.00000 −0.606092
$$99$$ −12.0000 −1.20605
$$100$$ −4.00000 −0.400000
$$101$$ 4.00000 0.398015 0.199007 0.979998i $$-0.436228\pi$$
0.199007 + 0.979998i $$0.436228\pi$$
$$102$$ 9.00000 0.891133
$$103$$ −8.00000 −0.788263 −0.394132 0.919054i $$-0.628955\pi$$
−0.394132 + 0.919054i $$0.628955\pi$$
$$104$$ −1.00000 −0.0980581
$$105$$ 3.00000 0.292770
$$106$$ 12.0000 1.16554
$$107$$ −4.00000 −0.386695 −0.193347 0.981130i $$-0.561934\pi$$
−0.193347 + 0.981130i $$0.561934\pi$$
$$108$$ −9.00000 −0.866025
$$109$$ 19.0000 1.81987 0.909935 0.414751i $$-0.136131\pi$$
0.909935 + 0.414751i $$0.136131\pi$$
$$110$$ 2.00000 0.190693
$$111$$ −9.00000 −0.854242
$$112$$ 1.00000 0.0944911
$$113$$ 2.00000 0.188144 0.0940721 0.995565i $$-0.470012\pi$$
0.0940721 + 0.995565i $$0.470012\pi$$
$$114$$ −18.0000 −1.68585
$$115$$ 4.00000 0.373002
$$116$$ 2.00000 0.185695
$$117$$ −6.00000 −0.554700
$$118$$ −10.0000 −0.920575
$$119$$ −3.00000 −0.275010
$$120$$ 3.00000 0.273861
$$121$$ −7.00000 −0.636364
$$122$$ −8.00000 −0.724286
$$123$$ 0 0
$$124$$ 4.00000 0.359211
$$125$$ 9.00000 0.804984
$$126$$ 6.00000 0.534522
$$127$$ 16.0000 1.41977 0.709885 0.704317i $$-0.248747\pi$$
0.709885 + 0.704317i $$0.248747\pi$$
$$128$$ 1.00000 0.0883883
$$129$$ 15.0000 1.32068
$$130$$ 1.00000 0.0877058
$$131$$ −1.00000 −0.0873704 −0.0436852 0.999045i $$-0.513910\pi$$
−0.0436852 + 0.999045i $$0.513910\pi$$
$$132$$ 6.00000 0.522233
$$133$$ 6.00000 0.520266
$$134$$ −2.00000 −0.172774
$$135$$ 9.00000 0.774597
$$136$$ −3.00000 −0.257248
$$137$$ 12.0000 1.02523 0.512615 0.858619i $$-0.328677\pi$$
0.512615 + 0.858619i $$0.328677\pi$$
$$138$$ 12.0000 1.02151
$$139$$ 7.00000 0.593732 0.296866 0.954919i $$-0.404058\pi$$
0.296866 + 0.954919i $$0.404058\pi$$
$$140$$ −1.00000 −0.0845154
$$141$$ −39.0000 −3.28439
$$142$$ −5.00000 −0.419591
$$143$$ 2.00000 0.167248
$$144$$ 6.00000 0.500000
$$145$$ −2.00000 −0.166091
$$146$$ −10.0000 −0.827606
$$147$$ 18.0000 1.48461
$$148$$ 3.00000 0.246598
$$149$$ −18.0000 −1.47462 −0.737309 0.675556i $$-0.763904\pi$$
−0.737309 + 0.675556i $$0.763904\pi$$
$$150$$ 12.0000 0.979796
$$151$$ −9.00000 −0.732410 −0.366205 0.930534i $$-0.619343\pi$$
−0.366205 + 0.930534i $$0.619343\pi$$
$$152$$ 6.00000 0.486664
$$153$$ −18.0000 −1.45521
$$154$$ −2.00000 −0.161165
$$155$$ −4.00000 −0.321288
$$156$$ 3.00000 0.240192
$$157$$ −10.0000 −0.798087 −0.399043 0.916932i $$-0.630658\pi$$
−0.399043 + 0.916932i $$0.630658\pi$$
$$158$$ −4.00000 −0.318223
$$159$$ −36.0000 −2.85499
$$160$$ −1.00000 −0.0790569
$$161$$ −4.00000 −0.315244
$$162$$ 9.00000 0.707107
$$163$$ −4.00000 −0.313304 −0.156652 0.987654i $$-0.550070\pi$$
−0.156652 + 0.987654i $$0.550070\pi$$
$$164$$ 0 0
$$165$$ −6.00000 −0.467099
$$166$$ 0 0
$$167$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$168$$ −3.00000 −0.231455
$$169$$ 1.00000 0.0769231
$$170$$ 3.00000 0.230089
$$171$$ 36.0000 2.75299
$$172$$ −5.00000 −0.381246
$$173$$ 20.0000 1.52057 0.760286 0.649589i $$-0.225059\pi$$
0.760286 + 0.649589i $$0.225059\pi$$
$$174$$ −6.00000 −0.454859
$$175$$ −4.00000 −0.302372
$$176$$ −2.00000 −0.150756
$$177$$ 30.0000 2.25494
$$178$$ 6.00000 0.449719
$$179$$ −9.00000 −0.672692 −0.336346 0.941739i $$-0.609191\pi$$
−0.336346 + 0.941739i $$0.609191\pi$$
$$180$$ −6.00000 −0.447214
$$181$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$182$$ −1.00000 −0.0741249
$$183$$ 24.0000 1.77413
$$184$$ −4.00000 −0.294884
$$185$$ −3.00000 −0.220564
$$186$$ −12.0000 −0.879883
$$187$$ 6.00000 0.438763
$$188$$ 13.0000 0.948122
$$189$$ −9.00000 −0.654654
$$190$$ −6.00000 −0.435286
$$191$$ 10.0000 0.723575 0.361787 0.932261i $$-0.382167\pi$$
0.361787 + 0.932261i $$0.382167\pi$$
$$192$$ −3.00000 −0.216506
$$193$$ −16.0000 −1.15171 −0.575853 0.817554i $$-0.695330\pi$$
−0.575853 + 0.817554i $$0.695330\pi$$
$$194$$ 14.0000 1.00514
$$195$$ −3.00000 −0.214834
$$196$$ −6.00000 −0.428571
$$197$$ 9.00000 0.641223 0.320612 0.947211i $$-0.396112\pi$$
0.320612 + 0.947211i $$0.396112\pi$$
$$198$$ −12.0000 −0.852803
$$199$$ −10.0000 −0.708881 −0.354441 0.935079i $$-0.615329\pi$$
−0.354441 + 0.935079i $$0.615329\pi$$
$$200$$ −4.00000 −0.282843
$$201$$ 6.00000 0.423207
$$202$$ 4.00000 0.281439
$$203$$ 2.00000 0.140372
$$204$$ 9.00000 0.630126
$$205$$ 0 0
$$206$$ −8.00000 −0.557386
$$207$$ −24.0000 −1.66812
$$208$$ −1.00000 −0.0693375
$$209$$ −12.0000 −0.830057
$$210$$ 3.00000 0.207020
$$211$$ 23.0000 1.58339 0.791693 0.610920i $$-0.209200\pi$$
0.791693 + 0.610920i $$0.209200\pi$$
$$212$$ 12.0000 0.824163
$$213$$ 15.0000 1.02778
$$214$$ −4.00000 −0.273434
$$215$$ 5.00000 0.340997
$$216$$ −9.00000 −0.612372
$$217$$ 4.00000 0.271538
$$218$$ 19.0000 1.28684
$$219$$ 30.0000 2.02721
$$220$$ 2.00000 0.134840
$$221$$ 3.00000 0.201802
$$222$$ −9.00000 −0.604040
$$223$$ −21.0000 −1.40626 −0.703132 0.711059i $$-0.748216\pi$$
−0.703132 + 0.711059i $$0.748216\pi$$
$$224$$ 1.00000 0.0668153
$$225$$ −24.0000 −1.60000
$$226$$ 2.00000 0.133038
$$227$$ −24.0000 −1.59294 −0.796468 0.604681i $$-0.793301\pi$$
−0.796468 + 0.604681i $$0.793301\pi$$
$$228$$ −18.0000 −1.19208
$$229$$ −15.0000 −0.991228 −0.495614 0.868543i $$-0.665057\pi$$
−0.495614 + 0.868543i $$0.665057\pi$$
$$230$$ 4.00000 0.263752
$$231$$ 6.00000 0.394771
$$232$$ 2.00000 0.131306
$$233$$ −11.0000 −0.720634 −0.360317 0.932830i $$-0.617331\pi$$
−0.360317 + 0.932830i $$0.617331\pi$$
$$234$$ −6.00000 −0.392232
$$235$$ −13.0000 −0.848026
$$236$$ −10.0000 −0.650945
$$237$$ 12.0000 0.779484
$$238$$ −3.00000 −0.194461
$$239$$ 9.00000 0.582162 0.291081 0.956698i $$-0.405985\pi$$
0.291081 + 0.956698i $$0.405985\pi$$
$$240$$ 3.00000 0.193649
$$241$$ 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 $$0.5\pi$$
$$252$$ 6.00000 0.377964
$$253$$ 8.00000 0.502956
$$254$$ 16.0000 1.00393
$$255$$ −9.00000 −0.563602
$$256$$ 1.00000 0.0625000
$$257$$ −15.0000 −0.935674 −0.467837 0.883815i $$-0.654967\pi$$
−0.467837 + 0.883815i $$0.654967\pi$$
$$258$$ 15.0000 0.933859
$$259$$ 3.00000 0.186411
$$260$$ 1.00000 0.0620174
$$261$$ 12.0000 0.742781
$$262$$ −1.00000 −0.0617802
$$263$$ 12.0000 0.739952 0.369976 0.929041i $$-0.379366\pi$$
0.369976 + 0.929041i $$0.379366\pi$$
$$264$$ 6.00000 0.369274
$$265$$ −12.0000 −0.737154
$$266$$ 6.00000 0.367884
$$267$$ −18.0000 −1.10158
$$268$$ −2.00000 −0.122169
$$269$$ −24.0000 −1.46331 −0.731653 0.681677i $$-0.761251\pi$$
−0.731653 + 0.681677i $$0.761251\pi$$
$$270$$ 9.00000 0.547723
$$271$$ 13.0000 0.789694 0.394847 0.918747i $$-0.370798\pi$$
0.394847 + 0.918747i $$0.370798\pi$$
$$272$$ −3.00000 −0.181902
$$273$$ 3.00000 0.181568
$$274$$ 12.0000 0.724947
$$275$$ 8.00000 0.482418
$$276$$ 12.0000 0.722315
$$277$$ 12.0000 0.721010 0.360505 0.932757i $$-0.382604\pi$$
0.360505 + 0.932757i $$0.382604\pi$$
$$278$$ 7.00000 0.419832
$$279$$ 24.0000 1.43684
$$280$$ −1.00000 −0.0597614
$$281$$ −26.0000 −1.55103 −0.775515 0.631329i $$-0.782510\pi$$
−0.775515 + 0.631329i $$0.782510\pi$$
$$282$$ −39.0000 −2.32242
$$283$$ 4.00000 0.237775 0.118888 0.992908i $$-0.462067\pi$$
0.118888 + 0.992908i $$0.462067\pi$$
$$284$$ −5.00000 −0.296695
$$285$$ 18.0000 1.06623
$$286$$ 2.00000 0.118262
$$287$$ 0 0
$$288$$ 6.00000 0.353553
$$289$$ −8.00000 −0.470588
$$290$$ −2.00000 −0.117444
$$291$$ −42.0000 −2.46208
$$292$$ −10.0000 −0.585206
$$293$$ 7.00000 0.408944 0.204472 0.978872i $$-0.434452\pi$$
0.204472 + 0.978872i $$0.434452\pi$$
$$294$$ 18.0000 1.04978
$$295$$ 10.0000 0.582223
$$296$$ 3.00000 0.174371
$$297$$ 18.0000 1.04447
$$298$$ −18.0000 −1.04271
$$299$$ 4.00000 0.231326
$$300$$ 12.0000 0.692820
$$301$$ −5.00000 −0.288195
$$302$$ −9.00000 −0.517892
$$303$$ −12.0000 −0.689382
$$304$$ 6.00000 0.344124
$$305$$ 8.00000 0.458079
$$306$$ −18.0000 −1.02899
$$307$$ 14.0000 0.799022 0.399511 0.916728i $$-0.369180\pi$$
0.399511 + 0.916728i $$0.369180\pi$$
$$308$$ −2.00000 −0.113961
$$309$$ 24.0000 1.36531
$$310$$ −4.00000 −0.227185
$$311$$ 18.0000 1.02069 0.510343 0.859971i $$-0.329518\pi$$
0.510343 + 0.859971i $$0.329518\pi$$
$$312$$ 3.00000 0.169842
$$313$$ −1.00000 −0.0565233 −0.0282617 0.999601i $$-0.508997\pi$$
−0.0282617 + 0.999601i $$0.508997\pi$$
$$314$$ −10.0000 −0.564333
$$315$$ −6.00000 −0.338062
$$316$$ −4.00000 −0.225018
$$317$$ −18.0000 −1.01098 −0.505490 0.862832i $$-0.668688\pi$$
−0.505490 + 0.862832i $$0.668688\pi$$
$$318$$ −36.0000 −2.01878
$$319$$ −4.00000 −0.223957
$$320$$ −1.00000 −0.0559017
$$321$$ 12.0000 0.669775
$$322$$ −4.00000 −0.222911
$$323$$ −18.0000 −1.00155
$$324$$ 9.00000 0.500000
$$325$$ 4.00000 0.221880
$$326$$ −4.00000 −0.221540
$$327$$ −57.0000 −3.15211
$$328$$ 0 0
$$329$$ 13.0000 0.716713
$$330$$ −6.00000 −0.330289
$$331$$ −4.00000 −0.219860 −0.109930 0.993939i $$-0.535063\pi$$
−0.109930 + 0.993939i $$0.535063\pi$$
$$332$$ 0 0
$$333$$ 18.0000 0.986394
$$334$$ 0 0
$$335$$ 2.00000 0.109272
$$336$$ −3.00000 −0.163663
$$337$$ 23.0000 1.25289 0.626445 0.779466i $$-0.284509\pi$$
0.626445 + 0.779466i $$0.284509\pi$$
$$338$$ 1.00000 0.0543928
$$339$$ −6.00000 −0.325875
$$340$$ 3.00000 0.162698
$$341$$ −8.00000 −0.433224
$$342$$ 36.0000 1.94666
$$343$$ −13.0000 −0.701934
$$344$$ −5.00000 −0.269582
$$345$$ −12.0000 −0.646058
$$346$$ 20.0000 1.07521
$$347$$ −9.00000 −0.483145 −0.241573 0.970383i $$-0.577663\pi$$
−0.241573 + 0.970383i $$0.577663\pi$$
$$348$$ −6.00000 −0.321634
$$349$$ 7.00000 0.374701 0.187351 0.982293i $$-0.440010\pi$$
0.187351 + 0.982293i $$0.440010\pi$$
$$350$$ −4.00000 −0.213809
$$351$$ 9.00000 0.480384
$$352$$ −2.00000 −0.106600
$$353$$ 4.00000 0.212899 0.106449 0.994318i $$-0.466052\pi$$
0.106449 + 0.994318i $$0.466052\pi$$
$$354$$ 30.0000 1.59448
$$355$$ 5.00000 0.265372
$$356$$ 6.00000 0.317999
$$357$$ 9.00000 0.476331
$$358$$ −9.00000 −0.475665
$$359$$ 24.0000 1.26667 0.633336 0.773877i $$-0.281685\pi$$
0.633336 + 0.773877i $$0.281685\pi$$
$$360$$ −6.00000 −0.316228
$$361$$ 17.0000 0.894737
$$362$$ 0 0
$$363$$ 21.0000 1.10221
$$364$$ −1.00000 −0.0524142
$$365$$ 10.0000 0.523424
$$366$$ 24.0000 1.25450
$$367$$ −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
$$372$$ −12.0000 −0.622171
$$373$$ −4.00000 −0.207112 −0.103556 0.994624i $$-0.533022\pi$$
−0.103556 + 0.994624i $$0.533022\pi$$
$$374$$ 6.00000 0.310253
$$375$$ −27.0000 −1.39427
$$376$$ 13.0000 0.670424
$$377$$ −2.00000 −0.103005
$$378$$ −9.00000 −0.462910
$$379$$ 16.0000 0.821865 0.410932 0.911666i $$-0.365203\pi$$
0.410932 + 0.911666i $$0.365203\pi$$
$$380$$ −6.00000 −0.307794
$$381$$ −48.0000 −2.45911
$$382$$ 10.0000 0.511645
$$383$$ 27.0000 1.37964 0.689818 0.723983i $$-0.257691\pi$$
0.689818 + 0.723983i $$0.257691\pi$$
$$384$$ −3.00000 −0.153093
$$385$$ 2.00000 0.101929
$$386$$ −16.0000 −0.814379
$$387$$ −30.0000 −1.52499
$$388$$ 14.0000 0.710742
$$389$$ −30.0000 −1.52106 −0.760530 0.649303i $$-0.775061\pi$$
−0.760530 + 0.649303i $$0.775061\pi$$
$$390$$ −3.00000 −0.151911
$$391$$ 12.0000 0.606866
$$392$$ −6.00000 −0.303046
$$393$$ 3.00000 0.151330
$$394$$ 9.00000 0.453413
$$395$$ 4.00000 0.201262
$$396$$ −12.0000 −0.603023
$$397$$ −22.0000 −1.10415 −0.552074 0.833795i $$-0.686163\pi$$
−0.552074 + 0.833795i $$0.686163\pi$$
$$398$$ −10.0000 −0.501255
$$399$$ −18.0000 −0.901127
$$400$$ −4.00000 −0.200000
$$401$$ 24.0000 1.19850 0.599251 0.800561i $$-0.295465\pi$$
0.599251 + 0.800561i $$0.295465\pi$$
$$402$$ 6.00000 0.299253
$$403$$ −4.00000 −0.199254
$$404$$ 4.00000 0.199007
$$405$$ −9.00000 −0.447214
$$406$$ 2.00000 0.0992583
$$407$$ −6.00000 −0.297409
$$408$$ 9.00000 0.445566
$$409$$ 4.00000 0.197787 0.0988936 0.995098i $$-0.468470\pi$$
0.0988936 + 0.995098i $$0.468470\pi$$
$$410$$ 0 0
$$411$$ −36.0000 −1.77575
$$412$$ −8.00000 −0.394132
$$413$$ −10.0000 −0.492068
$$414$$ −24.0000 −1.17954
$$415$$ 0 0
$$416$$ −1.00000 −0.0490290
$$417$$ −21.0000 −1.02837
$$418$$ −12.0000 −0.586939
$$419$$ 21.0000 1.02592 0.512959 0.858413i $$-0.328549\pi$$
0.512959 + 0.858413i $$0.328549\pi$$
$$420$$ 3.00000 0.146385
$$421$$ −5.00000 −0.243685 −0.121843 0.992549i $$-0.538880\pi$$
−0.121843 + 0.992549i $$0.538880\pi$$
$$422$$ 23.0000 1.11962
$$423$$ 78.0000 3.79249
$$424$$ 12.0000 0.582772
$$425$$ 12.0000 0.582086
$$426$$ 15.0000 0.726752
$$427$$ −8.00000 −0.387147
$$428$$ −4.00000 −0.193347
$$429$$ −6.00000 −0.289683
$$430$$ 5.00000 0.241121
$$431$$ 33.0000 1.58955 0.794777 0.606902i $$-0.207588\pi$$
0.794777 + 0.606902i $$0.207588\pi$$
$$432$$ −9.00000 −0.433013
$$433$$ 7.00000 0.336399 0.168199 0.985753i $$-0.446205\pi$$
0.168199 + 0.985753i $$0.446205\pi$$
$$434$$ 4.00000 0.192006
$$435$$ 6.00000 0.287678
$$436$$ 19.0000 0.909935
$$437$$ −24.0000 −1.14808
$$438$$ 30.0000 1.43346
$$439$$ −22.0000 −1.05000 −0.525001 0.851101i $$-0.675935\pi$$
−0.525001 + 0.851101i $$0.675935\pi$$
$$440$$ 2.00000 0.0953463
$$441$$ −36.0000 −1.71429
$$442$$ 3.00000 0.142695
$$443$$ −39.0000 −1.85295 −0.926473 0.376361i $$-0.877175\pi$$
−0.926473 + 0.376361i $$0.877175\pi$$
$$444$$ −9.00000 −0.427121
$$445$$ −6.00000 −0.284427
$$446$$ −21.0000 −0.994379
$$447$$ 54.0000 2.55411
$$448$$ 1.00000 0.0472456
$$449$$ −26.0000 −1.22702 −0.613508 0.789689i $$-0.710242\pi$$
−0.613508 + 0.789689i $$0.710242\pi$$
$$450$$ −24.0000 −1.13137
$$451$$ 0 0
$$452$$ 2.00000 0.0940721
$$453$$ 27.0000 1.26857
$$454$$ −24.0000 −1.12638
$$455$$ 1.00000 0.0468807
$$456$$ −18.0000 −0.842927
$$457$$ 10.0000 0.467780 0.233890 0.972263i $$-0.424854\pi$$
0.233890 + 0.972263i $$0.424854\pi$$
$$458$$ −15.0000 −0.700904
$$459$$ 27.0000 1.26025
$$460$$ 4.00000 0.186501
$$461$$ −21.0000 −0.978068 −0.489034 0.872265i $$-0.662651\pi$$
−0.489034 + 0.872265i $$0.662651\pi$$
$$462$$ 6.00000 0.279145
$$463$$ 16.0000 0.743583 0.371792 0.928316i $$-0.378744\pi$$
0.371792 + 0.928316i $$0.378744\pi$$
$$464$$ 2.00000 0.0928477
$$465$$ 12.0000 0.556487
$$466$$ −11.0000 −0.509565
$$467$$ 20.0000 0.925490 0.462745 0.886492i $$-0.346865\pi$$
0.462745 + 0.886492i $$0.346865\pi$$
$$468$$ −6.00000 −0.277350
$$469$$ −2.00000 −0.0923514
$$470$$ −13.0000 −0.599645
$$471$$ 30.0000 1.38233
$$472$$ −10.0000 −0.460287
$$473$$ 10.0000 0.459800
$$474$$ 12.0000 0.551178
$$475$$ −24.0000 −1.10120
$$476$$ −3.00000 −0.137505
$$477$$ 72.0000 3.29665
$$478$$ 9.00000 0.411650
$$479$$ −3.00000 −0.137073 −0.0685367 0.997649i $$-0.521833\pi$$
−0.0685367 + 0.997649i $$0.521833\pi$$
$$480$$ 3.00000 0.136931
$$481$$ −3.00000 −0.136788
$$482$$ 18.0000 0.819878
$$483$$ 12.0000 0.546019
$$484$$ −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
$$489$$ 12.0000 0.542659
$$490$$ 6.00000 0.271052
$$491$$ −5.00000 −0.225647 −0.112823 0.993615i $$-0.535989\pi$$
−0.112823 + 0.993615i $$0.535989\pi$$
$$492$$ 0 0
$$493$$ −6.00000 −0.270226
$$494$$ −6.00000 −0.269953
$$495$$ 12.0000 0.539360
$$496$$ 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 $$-0.601037\pi$$
−0.312115 + 0.950044i $$0.601037\pi$$
$$504$$ 6.00000 0.267261
$$505$$ −4.00000 −0.177998
$$506$$ 8.00000 0.355643
$$507$$ −3.00000 −0.133235
$$508$$ 16.0000 0.709885
$$509$$ 34.0000 1.50702 0.753512 0.657434i $$-0.228358\pi$$
0.753512 + 0.657434i $$0.228358\pi$$
$$510$$ −9.00000 −0.398527
$$511$$ −10.0000 −0.442374
$$512$$ 1.00000 0.0441942
$$513$$ −54.0000 −2.38416
$$514$$ −15.0000 −0.661622
$$515$$ 8.00000 0.352522
$$516$$ 15.0000 0.660338
$$517$$ −26.0000 −1.14348
$$518$$ 3.00000 0.131812
$$519$$ −60.0000 −2.63371
$$520$$ 1.00000 0.0438529
$$521$$ 39.0000 1.70862 0.854311 0.519763i $$-0.173980\pi$$
0.854311 + 0.519763i $$0.173980\pi$$
$$522$$ 12.0000 0.525226
$$523$$ −36.0000 −1.57417 −0.787085 0.616844i $$-0.788411\pi$$
−0.787085 + 0.616844i $$0.788411\pi$$
$$524$$ −1.00000 −0.0436852
$$525$$ 12.0000 0.523723
$$526$$ 12.0000 0.523225
$$527$$ −12.0000 −0.522728
$$528$$ 6.00000 0.261116
$$529$$ −7.00000 −0.304348
$$530$$ −12.0000 −0.521247
$$531$$ −60.0000 −2.60378
$$532$$ 6.00000 0.260133
$$533$$ 0 0
$$534$$ −18.0000 −0.778936
$$535$$ 4.00000 0.172935
$$536$$ −2.00000 −0.0863868
$$537$$ 27.0000 1.16514
$$538$$ −24.0000 −1.03471
$$539$$ 12.0000 0.516877
$$540$$ 9.00000 0.387298
$$541$$ 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
$$546$$ 3.00000 0.128388
$$547$$ 37.0000 1.58201 0.791003 0.611812i $$-0.209559\pi$$
0.791003 + 0.611812i $$0.209559\pi$$
$$548$$ 12.0000 0.512615
$$549$$ −48.0000 −2.04859
$$550$$ 8.00000 0.341121
$$551$$ 12.0000 0.511217
$$552$$ 12.0000 0.510754
$$553$$ −4.00000 −0.170097
$$554$$ 12.0000 0.509831
$$555$$ 9.00000 0.382029
$$556$$ 7.00000 0.296866
$$557$$ 33.0000 1.39825 0.699127 0.714997i $$-0.253572\pi$$
0.699127 + 0.714997i $$0.253572\pi$$
$$558$$ 24.0000 1.01600
$$559$$ 5.00000 0.211477
$$560$$ −1.00000 −0.0422577
$$561$$ −18.0000 −0.759961
$$562$$ −26.0000 −1.09674
$$563$$ 11.0000 0.463595 0.231797 0.972764i $$-0.425539\pi$$
0.231797 + 0.972764i $$0.425539\pi$$
$$564$$ −39.0000 −1.64220
$$565$$ −2.00000 −0.0841406
$$566$$ 4.00000 0.168133
$$567$$ 9.00000 0.377964
$$568$$ −5.00000 −0.209795
$$569$$ 31.0000 1.29959 0.649794 0.760111i $$-0.274855\pi$$
0.649794 + 0.760111i $$0.274855\pi$$
$$570$$ 18.0000 0.753937
$$571$$ 33.0000 1.38101 0.690504 0.723329i $$-0.257389\pi$$
0.690504 + 0.723329i $$0.257389\pi$$
$$572$$ 2.00000 0.0836242
$$573$$ −30.0000 −1.25327
$$574$$ 0 0
$$575$$ 16.0000 0.667246
$$576$$ 6.00000 0.250000
$$577$$ 18.0000 0.749350 0.374675 0.927156i $$-0.377754\pi$$
0.374675 + 0.927156i $$0.377754\pi$$
$$578$$ −8.00000 −0.332756
$$579$$ 48.0000 1.99481
$$580$$ −2.00000 −0.0830455
$$581$$ 0 0
$$582$$ −42.0000 −1.74096
$$583$$ −24.0000 −0.993978
$$584$$ −10.0000 −0.413803
$$585$$ 6.00000 0.248069
$$586$$ 7.00000 0.289167
$$587$$ −28.0000 −1.15568 −0.577842 0.816149i $$-0.696105\pi$$
−0.577842 + 0.816149i $$0.696105\pi$$
$$588$$ 18.0000 0.742307
$$589$$ 24.0000 0.988903
$$590$$ 10.0000 0.411693
$$591$$ −27.0000 −1.11063
$$592$$ 3.00000 0.123299
$$593$$ −22.0000 −0.903432 −0.451716 0.892162i $$-0.649188\pi$$
−0.451716 + 0.892162i $$0.649188\pi$$
$$594$$ 18.0000 0.738549
$$595$$ 3.00000 0.122988
$$596$$ −18.0000 −0.737309
$$597$$ 30.0000 1.22782
$$598$$ 4.00000 0.163572
$$599$$ −2.00000 −0.0817178 −0.0408589 0.999165i $$-0.513009\pi$$
−0.0408589 + 0.999165i $$0.513009\pi$$
$$600$$ 12.0000 0.489898
$$601$$ −35.0000 −1.42768 −0.713840 0.700309i $$-0.753046\pi$$
−0.713840 + 0.700309i $$0.753046\pi$$
$$602$$ −5.00000 −0.203785
$$603$$ −12.0000 −0.488678
$$604$$ −9.00000 −0.366205
$$605$$ 7.00000 0.284590
$$606$$ −12.0000 −0.487467
$$607$$ 6.00000 0.243532 0.121766 0.992559i $$-0.461144\pi$$
0.121766 + 0.992559i $$0.461144\pi$$
$$608$$ 6.00000 0.243332
$$609$$ −6.00000 −0.243132
$$610$$ 8.00000 0.323911
$$611$$ −13.0000 −0.525924
$$612$$ −18.0000 −0.727607
$$613$$ 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 $$-0.395622\pi$$
0.322068 + 0.946717i $$0.395622\pi$$
$$618$$ 24.0000 0.965422
$$619$$ 4.00000 0.160774 0.0803868 0.996764i $$-0.474384\pi$$
0.0803868 + 0.996764i $$0.474384\pi$$
$$620$$ −4.00000 −0.160644
$$621$$ 36.0000 1.44463
$$622$$ 18.0000 0.721734
$$623$$ 6.00000 0.240385
$$624$$ 3.00000 0.120096
$$625$$ 11.0000 0.440000
$$626$$ −1.00000 −0.0399680
$$627$$ 36.0000 1.43770
$$628$$ −10.0000 −0.399043
$$629$$ −9.00000 −0.358854
$$630$$ −6.00000 −0.239046
$$631$$ −5.00000 −0.199047 −0.0995234 0.995035i $$-0.531732\pi$$
−0.0995234 + 0.995035i $$0.531732\pi$$
$$632$$ −4.00000 −0.159111
$$633$$ −69.0000 −2.74250
$$634$$ −18.0000 −0.714871
$$635$$ −16.0000 −0.634941
$$636$$ −36.0000 −1.42749
$$637$$ 6.00000 0.237729
$$638$$ −4.00000 −0.158362
$$639$$ −30.0000 −1.18678
$$640$$ −1.00000 −0.0395285
$$641$$ −2.00000 −0.0789953 −0.0394976 0.999220i $$-0.512576\pi$$
−0.0394976 + 0.999220i $$0.512576\pi$$
$$642$$ 12.0000 0.473602
$$643$$ 14.0000 0.552106 0.276053 0.961142i $$-0.410973\pi$$
0.276053 + 0.961142i $$0.410973\pi$$
$$644$$ −4.00000 −0.157622
$$645$$ −15.0000 −0.590624
$$646$$ −18.0000 −0.708201
$$647$$ −38.0000 −1.49393 −0.746967 0.664861i $$-0.768491\pi$$
−0.746967 + 0.664861i $$0.768491\pi$$
$$648$$ 9.00000 0.353553
$$649$$ 20.0000 0.785069
$$650$$ 4.00000 0.156893
$$651$$ −12.0000 −0.470317
$$652$$ −4.00000 −0.156652
$$653$$ 24.0000 0.939193 0.469596 0.882881i $$-0.344399\pi$$
0.469596 + 0.882881i $$0.344399\pi$$
$$654$$ −57.0000 −2.22888
$$655$$ 1.00000 0.0390732
$$656$$ 0 0
$$657$$ −60.0000 −2.34082
$$658$$ 13.0000 0.506793
$$659$$ −12.0000 −0.467454 −0.233727 0.972302i $$-0.575092\pi$$
−0.233727 + 0.972302i $$0.575092\pi$$
$$660$$ −6.00000 −0.233550
$$661$$ −10.0000 −0.388955 −0.194477 0.980907i $$-0.562301\pi$$
−0.194477 + 0.980907i $$0.562301\pi$$
$$662$$ −4.00000 −0.155464
$$663$$ −9.00000 −0.349531
$$664$$ 0 0
$$665$$ −6.00000 −0.232670
$$666$$ 18.0000 0.697486
$$667$$ −8.00000 −0.309761
$$668$$ 0 0
$$669$$ 63.0000 2.43572
$$670$$ 2.00000 0.0772667
$$671$$ 16.0000 0.617673
$$672$$ −3.00000 −0.115728
$$673$$ 37.0000 1.42625 0.713123 0.701039i $$-0.247280\pi$$
0.713123 + 0.701039i $$0.247280\pi$$
$$674$$ 23.0000 0.885927
$$675$$ 36.0000 1.38564
$$676$$ 1.00000 0.0384615
$$677$$ −36.0000 −1.38359 −0.691796 0.722093i $$-0.743180\pi$$
−0.691796 + 0.722093i $$0.743180\pi$$
$$678$$ −6.00000 −0.230429
$$679$$ 14.0000 0.537271
$$680$$ 3.00000 0.115045
$$681$$ 72.0000 2.75905
$$682$$ −8.00000 −0.306336
$$683$$ −44.0000 −1.68361 −0.841807 0.539779i $$-0.818508\pi$$
−0.841807 + 0.539779i $$0.818508\pi$$
$$684$$ 36.0000 1.37649
$$685$$ −12.0000 −0.458496
$$686$$ −13.0000 −0.496342
$$687$$ 45.0000 1.71686
$$688$$ −5.00000 −0.190623
$$689$$ −12.0000 −0.457164
$$690$$ −12.0000 −0.456832
$$691$$ −8.00000 −0.304334 −0.152167 0.988355i $$-0.548625\pi$$
−0.152167 + 0.988355i $$0.548625\pi$$
$$692$$ 20.0000 0.760286
$$693$$ −12.0000 −0.455842
$$694$$ −9.00000 −0.341635
$$695$$ −7.00000 −0.265525
$$696$$ −6.00000 −0.227429
$$697$$ 0 0
$$698$$ 7.00000 0.264954
$$699$$ 33.0000 1.24817
$$700$$ −4.00000 −0.151186
$$701$$ −12.0000 −0.453234 −0.226617 0.973984i $$-0.572767\pi$$
−0.226617 + 0.973984i $$0.572767\pi$$
$$702$$ 9.00000 0.339683
$$703$$ 18.0000 0.678883
$$704$$ −2.00000 −0.0753778
$$705$$ 39.0000 1.46882
$$706$$ 4.00000 0.150542
$$707$$ 4.00000 0.150435
$$708$$ 30.0000 1.12747
$$709$$ 38.0000 1.42712 0.713560 0.700594i $$-0.247082\pi$$
0.713560 + 0.700594i $$0.247082\pi$$
$$710$$ 5.00000 0.187647
$$711$$ −24.0000 −0.900070
$$712$$ 6.00000 0.224860
$$713$$ −16.0000 −0.599205
$$714$$ 9.00000 0.336817
$$715$$ −2.00000 −0.0747958
$$716$$ −9.00000 −0.336346
$$717$$ −27.0000 −1.00833
$$718$$ 24.0000 0.895672
$$719$$ −22.0000 −0.820462 −0.410231 0.911982i $$-0.634552\pi$$
−0.410231 + 0.911982i $$0.634552\pi$$
$$720$$ −6.00000 −0.223607
$$721$$ −8.00000 −0.297936
$$722$$ 17.0000 0.632674
$$723$$ −54.0000 −2.00828
$$724$$ 0 0
$$725$$ −8.00000 −0.297113
$$726$$ 21.0000 0.779383
$$727$$ −14.0000 −0.519231 −0.259616 0.965712i $$-0.583596\pi$$
−0.259616 + 0.965712i $$0.583596\pi$$
$$728$$ −1.00000 −0.0370625
$$729$$ −27.0000 −1.00000
$$730$$ 10.0000 0.370117
$$731$$ 15.0000 0.554795
$$732$$ 24.0000 0.887066
$$733$$ −43.0000 −1.58824 −0.794121 0.607760i $$-0.792068\pi$$
−0.794121 + 0.607760i $$0.792068\pi$$
$$734$$ −10.0000 −0.369107
$$735$$ −18.0000 −0.663940
$$736$$ −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
$$741$$ 18.0000 0.661247
$$742$$ 12.0000 0.440534
$$743$$ −47.0000 −1.72426 −0.862131 0.506685i $$-0.830871\pi$$
−0.862131 + 0.506685i $$0.830871\pi$$
$$744$$ −12.0000 −0.439941
$$745$$ 18.0000 0.659469
$$746$$ −4.00000 −0.146450
$$747$$ 0 0
$$748$$ 6.00000 0.219382
$$749$$ −4.00000 −0.146157
$$750$$ −27.0000 −0.985901
$$751$$ 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
$$756$$ −9.00000 −0.327327
$$757$$ −12.0000 −0.436147 −0.218074 0.975932i $$-0.569977\pi$$
−0.218074 + 0.975932i $$0.569977\pi$$
$$758$$ 16.0000 0.581146
$$759$$ −24.0000 −0.871145
$$760$$ −6.00000 −0.217643
$$761$$ 6.00000 0.217500 0.108750 0.994069i $$-0.465315\pi$$
0.108750 + 0.994069i $$0.465315\pi$$
$$762$$ −48.0000 −1.73886
$$763$$ 19.0000 0.687846
$$764$$ 10.0000 0.361787
$$765$$ 18.0000 0.650791
$$766$$ 27.0000 0.975550
$$767$$ 10.0000 0.361079
$$768$$ −3.00000 −0.108253
$$769$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$770$$ 2.00000 0.0720750
$$771$$ 45.0000 1.62064
$$772$$ −16.0000 −0.575853
$$773$$ 11.0000 0.395643 0.197821 0.980238i $$-0.436613\pi$$
0.197821 + 0.980238i $$0.436613\pi$$
$$774$$ −30.0000 −1.07833
$$775$$ −16.0000 −0.574737
$$776$$ 14.0000 0.502571
$$777$$ −9.00000 −0.322873
$$778$$ −30.0000 −1.07555
$$779$$ 0 0
$$780$$ −3.00000 −0.107417
$$781$$ 10.0000 0.357828
$$782$$ 12.0000 0.429119
$$783$$ −18.0000 −0.643268
$$784$$ −6.00000 −0.214286
$$785$$ 10.0000 0.356915
$$786$$ 3.00000 0.107006
$$787$$ 32.0000 1.14068 0.570338 0.821410i $$-0.306812\pi$$
0.570338 + 0.821410i $$0.306812\pi$$
$$788$$ 9.00000 0.320612
$$789$$ −36.0000 −1.28163
$$790$$ 4.00000 0.142314
$$791$$ 2.00000 0.0711118
$$792$$ −12.0000 −0.426401
$$793$$ 8.00000 0.284088
$$794$$ −22.0000 −0.780751
$$795$$ 36.0000 1.27679
$$796$$ −10.0000 −0.354441
$$797$$ −42.0000 −1.48772 −0.743858 0.668338i $$-0.767006\pi$$
−0.743858 + 0.668338i $$0.767006\pi$$
$$798$$ −18.0000 −0.637193
$$799$$ −39.0000 −1.37972
$$800$$ −4.00000 −0.141421
$$801$$ 36.0000 1.27200
$$802$$ 24.0000 0.847469
$$803$$ 20.0000 0.705785
$$804$$ 6.00000 0.211604
$$805$$ 4.00000 0.140981
$$806$$ −4.00000 −0.140894
$$807$$ 72.0000 2.53452
$$808$$ 4.00000 0.140720
$$809$$ −9.00000 −0.316423 −0.158212 0.987405i $$-0.550573\pi$$
−0.158212 + 0.987405i $$0.550573\pi$$
$$810$$ −9.00000 −0.316228
$$811$$ −28.0000 −0.983213 −0.491606 0.870817i $$-0.663590\pi$$
−0.491606 + 0.870817i $$0.663590\pi$$
$$812$$ 2.00000 0.0701862
$$813$$ −39.0000 −1.36779
$$814$$ −6.00000 −0.210300
$$815$$ 4.00000 0.140114
$$816$$ 9.00000 0.315063
$$817$$ −30.0000 −1.04957
$$818$$ 4.00000 0.139857
$$819$$ −6.00000 −0.209657
$$820$$ 0 0
$$821$$ −25.0000 −0.872506 −0.436253 0.899824i $$-0.643695\pi$$
−0.436253 + 0.899824i $$0.643695\pi$$
$$822$$ −36.0000 −1.25564
$$823$$ 54.0000 1.88232 0.941161 0.337959i $$-0.109737\pi$$
0.941161 + 0.337959i $$0.109737\pi$$
$$824$$ −8.00000 −0.278693
$$825$$ −24.0000 −0.835573
$$826$$ −10.0000 −0.347945
$$827$$ 30.0000 1.04320 0.521601 0.853189i $$-0.325335\pi$$
0.521601 + 0.853189i $$0.325335\pi$$
$$828$$ −24.0000 −0.834058
$$829$$ −38.0000 −1.31979 −0.659897 0.751356i $$-0.729400\pi$$
−0.659897 + 0.751356i $$0.729400\pi$$
$$830$$ 0 0
$$831$$ −36.0000 −1.24883
$$832$$ −1.00000 −0.0346688
$$833$$ 18.0000 0.623663
$$834$$ −21.0000 −0.727171
$$835$$ 0 0
$$836$$ −12.0000 −0.415029
$$837$$ −36.0000 −1.24434
$$838$$ 21.0000 0.725433
$$839$$ 56.0000 1.93333 0.966667 0.256036i $$-0.0824164\pi$$
0.966667 + 0.256036i $$0.0824164\pi$$
$$840$$ 3.00000 0.103510
$$841$$ −25.0000 −0.862069
$$842$$ −5.00000 −0.172311
$$843$$ 78.0000 2.68646
$$844$$ 23.0000 0.791693
$$845$$ −1.00000 −0.0344010
$$846$$ 78.0000 2.68170
$$847$$ −7.00000 −0.240523
$$848$$ 12.0000 0.412082
$$849$$ −12.0000 −0.411839
$$850$$ 12.0000 0.411597
$$851$$ −12.0000 −0.411355
$$852$$ 15.0000 0.513892
$$853$$ 49.0000 1.67773 0.838864 0.544341i $$-0.183220\pi$$
0.838864 + 0.544341i $$0.183220\pi$$
$$854$$ −8.00000 −0.273754
$$855$$ −36.0000 −1.23117
$$856$$ −4.00000 −0.136717
$$857$$ 46.0000 1.57133 0.785665 0.618652i $$-0.212321\pi$$
0.785665 + 0.618652i $$0.212321\pi$$
$$858$$ −6.00000 −0.204837
$$859$$ 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 $$-0.559948\pi$$
−0.187222 + 0.982318i $$0.559948\pi$$
$$864$$ −9.00000 −0.306186
$$865$$ −20.0000 −0.680020
$$866$$ 7.00000 0.237870
$$867$$ 24.0000 0.815083
$$868$$ 4.00000 0.135769
$$869$$ 8.00000 0.271381
$$870$$ 6.00000 0.203419
$$871$$ 2.00000 0.0677674
$$872$$ 19.0000 0.643421
$$873$$ 84.0000 2.84297
$$874$$ −24.0000 −0.811812
$$875$$ 9.00000 0.304256
$$876$$ 30.0000 1.01361
$$877$$ −39.0000 −1.31694 −0.658468 0.752609i $$-0.728795\pi$$
−0.658468 + 0.752609i $$0.728795\pi$$
$$878$$ −22.0000 −0.742464
$$879$$ −21.0000 −0.708312
$$880$$ 2.00000 0.0674200
$$881$$ 21.0000 0.707508 0.353754 0.935339i $$-0.384905\pi$$
0.353754 + 0.935339i $$0.384905\pi$$
$$882$$ −36.0000 −1.21218
$$883$$ −47.0000 −1.58168 −0.790838 0.612026i $$-0.790355\pi$$
−0.790838 + 0.612026i $$0.790355\pi$$
$$884$$ 3.00000 0.100901
$$885$$ −30.0000 −1.00844
$$886$$ −39.0000 −1.31023
$$887$$ −8.00000 −0.268614 −0.134307 0.990940i $$-0.542881\pi$$
−0.134307 + 0.990940i $$0.542881\pi$$
$$888$$ −9.00000 −0.302020
$$889$$ 16.0000 0.536623
$$890$$ −6.00000 −0.201120
$$891$$ −18.0000 −0.603023
$$892$$ −21.0000 −0.703132
$$893$$ 78.0000 2.61017
$$894$$ 54.0000 1.80603
$$895$$ 9.00000 0.300837
$$896$$ 1.00000 0.0334077
$$897$$ −12.0000 −0.400668
$$898$$ −26.0000 −0.867631
$$899$$ 8.00000 0.266815
$$900$$ −24.0000 −0.800000
$$901$$ −36.0000 −1.19933
$$902$$ 0 0
$$903$$ 15.0000 0.499169
$$904$$ 2.00000 0.0665190
$$905$$ 0 0
$$906$$ 27.0000 0.897015
$$907$$ −9.00000 −0.298840 −0.149420 0.988774i $$-0.547741\pi$$
−0.149420 + 0.988774i $$0.547741\pi$$
$$908$$ −24.0000 −0.796468
$$909$$ 24.0000 0.796030
$$910$$ 1.00000 0.0331497
$$911$$ −54.0000 −1.78910 −0.894550 0.446968i $$-0.852504\pi$$
−0.894550 + 0.446968i $$0.852504\pi$$
$$912$$ −18.0000 −0.596040
$$913$$ 0 0
$$914$$ 10.0000 0.330771
$$915$$ −24.0000 −0.793416
$$916$$ −15.0000 −0.495614
$$917$$ −1.00000 −0.0330229
$$918$$ 27.0000 0.891133
$$919$$ 24.0000 0.791687 0.395843 0.918318i $$-0.370452\pi$$
0.395843 + 0.918318i $$0.370452\pi$$
$$920$$ 4.00000 0.131876
$$921$$ −42.0000 −1.38395
$$922$$ −21.0000 −0.691598
$$923$$ 5.00000 0.164577
$$924$$ 6.00000 0.197386
$$925$$ −12.0000 −0.394558
$$926$$ 16.0000 0.525793
$$927$$ −48.0000 −1.57653
$$928$$ 2.00000 0.0656532
$$929$$ −36.0000 −1.18112 −0.590561 0.806993i $$-0.701093\pi$$
−0.590561 + 0.806993i $$0.701093\pi$$
$$930$$ 12.0000 0.393496
$$931$$ −36.0000 −1.17985
$$932$$ −11.0000 −0.360317
$$933$$ −54.0000 −1.76788
$$934$$ 20.0000 0.654420
$$935$$ −6.00000 −0.196221
$$936$$ −6.00000 −0.196116
$$937$$ −42.0000 −1.37208 −0.686040 0.727564i $$-0.740653\pi$$
−0.686040 + 0.727564i $$0.740653\pi$$
$$938$$ −2.00000 −0.0653023
$$939$$ 3.00000 0.0979013
$$940$$ −13.0000 −0.424013
$$941$$ 25.0000 0.814977 0.407488 0.913210i $$-0.366405\pi$$
0.407488 + 0.913210i $$0.366405\pi$$
$$942$$ 30.0000 0.977453
$$943$$ 0 0
$$944$$ −10.0000 −0.325472
$$945$$ 9.00000 0.292770
$$946$$ 10.0000 0.325128
$$947$$ −18.0000 −0.584921 −0.292461 0.956278i $$-0.594474\pi$$
−0.292461 + 0.956278i $$0.594474\pi$$
$$948$$ 12.0000 0.389742
$$949$$ 10.0000 0.324614
$$950$$ −24.0000 −0.778663
$$951$$ 54.0000 1.75107
$$952$$ −3.00000 −0.0972306
$$953$$ 23.0000 0.745043 0.372522 0.928024i $$-0.378493\pi$$
0.372522 + 0.928024i $$0.378493\pi$$
$$954$$ 72.0000 2.33109
$$955$$ −10.0000 −0.323592
$$956$$ 9.00000 0.291081
$$957$$ 12.0000 0.387905
$$958$$ −3.00000 −0.0969256
$$959$$ 12.0000 0.387500
$$960$$ 3.00000 0.0968246
$$961$$ −15.0000 −0.483871
$$962$$ −3.00000 −0.0967239
$$963$$ −24.0000 −0.773389
$$964$$ 18.0000 0.579741
$$965$$ 16.0000 0.515058
$$966$$ 12.0000 0.386094
$$967$$ 23.0000 0.739630 0.369815 0.929105i $$-0.379421\pi$$
0.369815 + 0.929105i $$0.379421\pi$$
$$968$$ −7.00000 −0.224989
$$969$$ 54.0000 1.73473
$$970$$ −14.0000 −0.449513
$$971$$ −15.0000 −0.481373 −0.240686 0.970603i $$-0.577373\pi$$
−0.240686 + 0.970603i $$0.577373\pi$$
$$972$$ 0 0
$$973$$ 7.00000 0.224410
$$974$$ −16.0000 −0.512673
$$975$$ −12.0000 −0.384308
$$976$$ −8.00000 −0.256074
$$977$$ −30.0000 −0.959785 −0.479893 0.877327i $$-0.659324\pi$$
−0.479893 + 0.877327i $$0.659324\pi$$
$$978$$ 12.0000 0.383718
$$979$$ −12.0000 −0.383522
$$980$$ 6.00000 0.191663
$$981$$ 114.000 3.63974
$$982$$ −5.00000 −0.159556
$$983$$ −31.0000 −0.988746 −0.494373 0.869250i $$-0.664602\pi$$
−0.494373 + 0.869250i $$0.664602\pi$$
$$984$$ 0 0
$$985$$ −9.00000 −0.286764
$$986$$ −6.00000 −0.191079
$$987$$ −39.0000 −1.24138
$$988$$ −6.00000 −0.190885
$$989$$ 20.0000 0.635963
$$990$$ 12.0000 0.381385
$$991$$ −30.0000 −0.952981 −0.476491 0.879180i $$-0.658091\pi$$
−0.476491 + 0.879180i $$0.658091\pi$$
$$992$$ 4.00000 0.127000
$$993$$ 12.0000 0.380808
$$994$$ −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
$$999$$ −27.0000 −0.854242
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 26.2.a.b.1.1 1
3.2 odd 2 234.2.a.b.1.1 1
4.3 odd 2 208.2.a.d.1.1 1
5.2 odd 4 650.2.b.a.599.2 2
5.3 odd 4 650.2.b.a.599.1 2
5.4 even 2 650.2.a.g.1.1 1
7.2 even 3 1274.2.f.l.1145.1 2
7.3 odd 6 1274.2.f.a.79.1 2
7.4 even 3 1274.2.f.l.79.1 2
7.5 odd 6 1274.2.f.a.1145.1 2
7.6 odd 2 1274.2.a.o.1.1 1
8.3 odd 2 832.2.a.a.1.1 1
8.5 even 2 832.2.a.j.1.1 1
9.2 odd 6 2106.2.e.t.1405.1 2
9.4 even 3 2106.2.e.h.703.1 2
9.5 odd 6 2106.2.e.t.703.1 2
9.7 even 3 2106.2.e.h.1405.1 2
11.10 odd 2 3146.2.a.a.1.1 1
12.11 even 2 1872.2.a.m.1.1 1
13.2 odd 12 338.2.e.d.147.2 4
13.3 even 3 338.2.c.c.191.1 2
13.4 even 6 338.2.c.g.315.1 2
13.5 odd 4 338.2.b.a.337.1 2
13.6 odd 12 338.2.e.d.23.1 4
13.7 odd 12 338.2.e.d.23.2 4
13.8 odd 4 338.2.b.a.337.2 2
13.9 even 3 338.2.c.c.315.1 2
13.10 even 6 338.2.c.g.191.1 2
13.11 odd 12 338.2.e.d.147.1 4
13.12 even 2 338.2.a.a.1.1 1
15.2 even 4 5850.2.e.v.5149.1 2
15.8 even 4 5850.2.e.v.5149.2 2
15.14 odd 2 5850.2.a.bn.1.1 1
16.3 odd 4 3328.2.b.k.1665.2 2
16.5 even 4 3328.2.b.g.1665.2 2
16.11 odd 4 3328.2.b.k.1665.1 2
16.13 even 4 3328.2.b.g.1665.1 2
17.16 even 2 7514.2.a.i.1.1 1
19.18 odd 2 9386.2.a.f.1.1 1
20.19 odd 2 5200.2.a.c.1.1 1
24.5 odd 2 7488.2.a.w.1.1 1
24.11 even 2 7488.2.a.v.1.1 1
39.5 even 4 3042.2.b.f.1351.2 2
39.8 even 4 3042.2.b.f.1351.1 2
39.38 odd 2 3042.2.a.l.1.1 1
52.31 even 4 2704.2.f.j.337.2 2
52.47 even 4 2704.2.f.j.337.1 2
52.51 odd 2 2704.2.a.n.1.1 1
65.64 even 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 1.1 even 1 trivial
208.2.a.d.1.1 1 4.3 odd 2
234.2.a.b.1.1 1 3.2 odd 2
338.2.a.a.1.1 1 13.12 even 2
338.2.b.a.337.1 2 13.5 odd 4
338.2.b.a.337.2 2 13.8 odd 4
338.2.c.c.191.1 2 13.3 even 3
338.2.c.c.315.1 2 13.9 even 3
338.2.c.g.191.1 2 13.10 even 6
338.2.c.g.315.1 2 13.4 even 6
338.2.e.d.23.1 4 13.6 odd 12
338.2.e.d.23.2 4 13.7 odd 12
338.2.e.d.147.1 4 13.11 odd 12
338.2.e.d.147.2 4 13.2 odd 12
650.2.a.g.1.1 1 5.4 even 2
650.2.b.a.599.1 2 5.3 odd 4
650.2.b.a.599.2 2 5.2 odd 4
832.2.a.a.1.1 1 8.3 odd 2
832.2.a.j.1.1 1 8.5 even 2
1274.2.a.o.1.1 1 7.6 odd 2
1274.2.f.a.79.1 2 7.3 odd 6
1274.2.f.a.1145.1 2 7.5 odd 6
1274.2.f.l.79.1 2 7.4 even 3
1274.2.f.l.1145.1 2 7.2 even 3
1872.2.a.m.1.1 1 12.11 even 2
2106.2.e.h.703.1 2 9.4 even 3
2106.2.e.h.1405.1 2 9.7 even 3
2106.2.e.t.703.1 2 9.5 odd 6
2106.2.e.t.1405.1 2 9.2 odd 6
2704.2.a.n.1.1 1 52.51 odd 2
2704.2.f.j.337.1 2 52.47 even 4
2704.2.f.j.337.2 2 52.31 even 4
3042.2.a.l.1.1 1 39.38 odd 2
3042.2.b.f.1351.1 2 39.8 even 4
3042.2.b.f.1351.2 2 39.5 even 4
3146.2.a.a.1.1 1 11.10 odd 2
3328.2.b.g.1665.1 2 16.13 even 4
3328.2.b.g.1665.2 2 16.5 even 4
3328.2.b.k.1665.1 2 16.11 odd 4
3328.2.b.k.1665.2 2 16.3 odd 4
5200.2.a.c.1.1 1 20.19 odd 2
5850.2.a.bn.1.1 1 15.14 odd 2
5850.2.e.v.5149.1 2 15.2 even 4
5850.2.e.v.5149.2 2 15.8 even 4
7488.2.a.v.1.1 1 24.11 even 2
7488.2.a.w.1.1 1 24.5 odd 2
7514.2.a.i.1.1 1 17.16 even 2
8450.2.a.y.1.1 1 65.64 even 2
9386.2.a.f.1.1 1 19.18 odd 2