Properties

 Label 338.2.a.a.1.1 Level $338$ Weight $2$ Character 338.1 Self dual yes Analytic conductor $2.699$ 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] = [338,2,Mod(1,338)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(338, 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("338.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$338 = 2 \cdot 13^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 338.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: $$2.69894358832$$ Analytic rank: $$1$$ 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 $$\chi$$ $$=$$ 338.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

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

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 $$-0.428217\pi$$
0.223607 + 0.974679i $$0.428217\pi$$
$$6$$ 3.00000 1.22474
$$7$$ −1.00000 −0.377964 −0.188982 0.981981i $$-0.560519\pi$$
−0.188982 + 0.981981i $$0.560519\pi$$
$$8$$ −1.00000 −0.353553
$$9$$ 6.00000 2.00000
$$10$$ −1.00000 −0.316228
$$11$$ 2.00000 0.603023 0.301511 0.953463i $$-0.402509\pi$$
0.301511 + 0.953463i $$0.402509\pi$$
$$12$$ −3.00000 −0.866025
$$13$$ 0 0
$$14$$ 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 $$-0.741620\pi$$
−0.688247 + 0.725476i $$0.741620\pi$$
$$20$$ 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
$$26$$ 0 0
$$27$$ −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 $$-0.616954\pi$$
−0.359211 + 0.933257i $$0.616954\pi$$
$$32$$ −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 $$-0.579313\pi$$
−0.246598 + 0.969118i $$0.579313\pi$$
$$38$$ 6.00000 0.973329
$$39$$ 0 0
$$40$$ −1.00000 −0.158114
$$41$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$42$$ −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 $$-0.897021\pi$$
−0.948122 + 0.317905i $$0.897021\pi$$
$$48$$ −3.00000 −0.433013
$$49$$ −6.00000 −0.857143
$$50$$ 4.00000 0.565685
$$51$$ 9.00000 1.26025
$$52$$ 0 0
$$53$$ 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 $$-0.274373\pi$$
0.650945 + 0.759125i $$0.274373\pi$$
$$60$$ −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
$$65$$ 0 0
$$66$$ 6.00000 0.738549
$$67$$ 2.00000 0.244339 0.122169 0.992509i $$-0.461015\pi$$
0.122169 + 0.992509i $$0.461015\pi$$
$$68$$ −3.00000 −0.363803
$$69$$ 12.0000 1.44463
$$70$$ 1.00000 0.119523
$$71$$ 5.00000 0.593391 0.296695 0.954972i $$-0.404115\pi$$
0.296695 + 0.954972i $$0.404115\pi$$
$$72$$ −6.00000 −0.707107
$$73$$ 10.0000 1.17041 0.585206 0.810885i $$-0.301014\pi$$
0.585206 + 0.810885i $$0.301014\pi$$
$$74$$ 3.00000 0.348743
$$75$$ 12.0000 1.38564
$$76$$ −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
$$81$$ 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 $$-0.603011\pi$$
−0.317999 + 0.948091i $$0.603011\pi$$
$$90$$ −6.00000 −0.632456
$$91$$ 0 0
$$92$$ −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 $$-0.751641\pi$$
−0.710742 + 0.703452i $$0.751641\pi$$
$$98$$ 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 $$0.628955\pi$$
$$104$$ 0 0
$$105$$ 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 $$-0.863869\pi$$
−0.909935 + 0.414751i $$0.863869\pi$$
$$110$$ −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
$$117$$ 0 0
$$118$$ −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
$$130$$ 0 0
$$131$$ −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 $$-0.671323\pi$$
−0.512615 + 0.858619i $$0.671323\pi$$
$$138$$ −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
$$143$$ 0 0
$$144$$ 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 $$-0.236096\pi$$
0.737309 + 0.675556i $$0.236096\pi$$
$$150$$ −12.0000 −0.979796
$$151$$ 9.00000 0.732410 0.366205 0.930534i $$-0.380657\pi$$
0.366205 + 0.930534i $$0.380657\pi$$
$$152$$ 6.00000 0.486664
$$153$$ −18.0000 −1.45521
$$154$$ 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
$$159$$ −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 $$-0.449930\pi$$
0.156652 + 0.987654i $$0.449930\pi$$
$$164$$ 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
$$169$$ 0 0
$$170$$ 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 $$0.5\pi$$
$$182$$ 0 0
$$183$$ 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 $$-0.304670\pi$$
0.575853 + 0.817554i $$0.304670\pi$$
$$194$$ 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 $$0.603888\pi$$
$$198$$ −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
$$208$$ 0 0
$$209$$ −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
$$221$$ 0 0
$$222$$ −9.00000 −0.604040
$$223$$ 21.0000 1.40626 0.703132 0.711059i $$-0.251784\pi$$
0.703132 + 0.711059i $$0.251784\pi$$
$$224$$ 1.00000 0.0668153
$$225$$ −24.0000 −1.60000
$$226$$ −2.00000 −0.133038
$$227$$ 24.0000 1.59294 0.796468 0.604681i $$-0.206699\pi$$
0.796468 + 0.604681i $$0.206699\pi$$
$$228$$ 18.0000 1.19208
$$229$$ 15.0000 0.991228 0.495614 0.868543i $$-0.334943\pi$$
0.495614 + 0.868543i $$0.334943\pi$$
$$230$$ 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 $$0.617331\pi$$
$$234$$ 0 0
$$235$$ −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 $$-0.594015\pi$$
−0.291081 + 0.956698i $$0.594015\pi$$
$$240$$ −3.00000 −0.193649
$$241$$ −18.0000 −1.15948 −0.579741 0.814801i $$-0.696846\pi$$
−0.579741 + 0.814801i $$0.696846\pi$$
$$242$$ 7.00000 0.449977
$$243$$ 0 0
$$244$$ −8.00000 −0.512148
$$245$$ −6.00000 −0.383326
$$246$$ 0 0
$$247$$ 0 0
$$248$$ 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
$$260$$ 0 0
$$261$$ 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 $$-0.629202\pi$$
−0.394847 + 0.918747i $$0.629202\pi$$
$$272$$ −3.00000 −0.181902
$$273$$ 0 0
$$274$$ 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 $$-0.217490\pi$$
0.775515 + 0.631329i $$0.217490\pi$$
$$282$$ −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
$$286$$ 0 0
$$287$$ 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 $$-0.565548\pi$$
−0.204472 + 0.978872i $$0.565548\pi$$
$$294$$ −18.0000 −1.04978
$$295$$ 10.0000 0.582223
$$296$$ 3.00000 0.174371
$$297$$ −18.0000 −1.04447
$$298$$ −18.0000 −1.04271
$$299$$ 0 0
$$300$$ 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 $$-0.630820\pi$$
−0.399511 + 0.916728i $$0.630820\pi$$
$$308$$ −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 $$0.329518\pi$$
$$312$$ 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
$$315$$ −6.00000 −0.338062
$$316$$ −4.00000 −0.225018
$$317$$ 18.0000 1.01098 0.505490 0.862832i $$-0.331312\pi$$
0.505490 + 0.862832i $$0.331312\pi$$
$$318$$ 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
$$325$$ 0 0
$$326$$ −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 $$-0.464937\pi$$
0.109930 + 0.993939i $$0.464937\pi$$
$$332$$ 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 $$0.284509\pi$$
$$338$$ 0 0
$$339$$ −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 $$-0.559990\pi$$
−0.187351 + 0.982293i $$0.559990\pi$$
$$350$$ −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 $$0.533948\pi$$
$$354$$ 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 $$-0.718315\pi$$
−0.633336 + 0.773877i $$0.718315\pi$$
$$360$$ −6.00000 −0.316228
$$361$$ 17.0000 0.894737
$$362$$ 0 0
$$363$$ 21.0000 1.10221
$$364$$ 0 0
$$365$$ 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
$$377$$ 0 0
$$378$$ −9.00000 −0.462910
$$379$$ −16.0000 −0.821865 −0.410932 0.911666i $$-0.634797\pi$$
−0.410932 + 0.911666i $$0.634797\pi$$
$$380$$ −6.00000 −0.307794
$$381$$ −48.0000 −2.45911
$$382$$ −10.0000 −0.511645
$$383$$ −27.0000 −1.37964 −0.689818 0.723983i $$-0.742309\pi$$
−0.689818 + 0.723983i $$0.742309\pi$$
$$384$$ 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 $$0.775061\pi$$
$$390$$ 0 0
$$391$$ 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 $$-0.313837\pi$$
0.552074 + 0.833795i $$0.313837\pi$$
$$398$$ 10.0000 0.501255
$$399$$ −18.0000 −0.901127
$$400$$ −4.00000 −0.200000
$$401$$ −24.0000 −1.19850 −0.599251 0.800561i $$-0.704535\pi$$
−0.599251 + 0.800561i $$0.704535\pi$$
$$402$$ 6.00000 0.299253
$$403$$ 0 0
$$404$$ 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 $$-0.531530\pi$$
−0.0988936 + 0.995098i $$0.531530\pi$$
$$410$$ 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
$$416$$ 0 0
$$417$$ −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 $$-0.461120\pi$$
0.121843 + 0.992549i $$0.461120\pi$$
$$422$$ −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
$$429$$ 0 0
$$430$$ 5.00000 0.241121
$$431$$ −33.0000 −1.58955 −0.794777 0.606902i $$-0.792412\pi$$
−0.794777 + 0.606902i $$0.792412\pi$$
$$432$$ −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
$$442$$ 0 0
$$443$$ −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 $$-0.289758\pi$$
0.613508 + 0.789689i $$0.289758\pi$$
$$450$$ 24.0000 1.13137
$$451$$ 0 0
$$452$$ 2.00000 0.0940721
$$453$$ −27.0000 −1.26857
$$454$$ −24.0000 −1.12638
$$455$$ 0 0
$$456$$ −18.0000 −0.842927
$$457$$ −10.0000 −0.467780 −0.233890 0.972263i $$-0.575146\pi$$
−0.233890 + 0.972263i $$0.575146\pi$$
$$458$$ −15.0000 −0.700904
$$459$$ 27.0000 1.26025
$$460$$ −4.00000 −0.186501
$$461$$ 21.0000 0.978068 0.489034 0.872265i $$-0.337349\pi$$
0.489034 + 0.872265i $$0.337349\pi$$
$$462$$ −6.00000 −0.279145
$$463$$ −16.0000 −0.743583 −0.371792 0.928316i $$-0.621256\pi$$
−0.371792 + 0.928316i $$0.621256\pi$$
$$464$$ 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 $$0.346865\pi$$
$$468$$ 0 0
$$469$$ −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 $$-0.478167\pi$$
0.0685367 + 0.997649i $$0.478167\pi$$
$$480$$ 3.00000 0.136931
$$481$$ 0 0
$$482$$ 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 $$-0.381918\pi$$
0.362515 + 0.931978i $$0.381918\pi$$
$$488$$ 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
$$494$$ 0 0
$$495$$ 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 $$-0.245853\pi$$
0.716258 + 0.697835i $$0.245853\pi$$
$$500$$ −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
$$507$$ 0 0
$$508$$ 16.0000 0.709885
$$509$$ −34.0000 −1.50702 −0.753512 0.657434i $$-0.771642\pi$$
−0.753512 + 0.657434i $$0.771642\pi$$
$$510$$ −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
$$520$$ 0 0
$$521$$ 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 $$-0.619083\pi$$
−0.365444 + 0.930834i $$0.619083\pi$$
$$542$$ 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
$$549$$ −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 $$-0.746428\pi$$
−0.699127 + 0.714997i $$0.746428\pi$$
$$558$$ 24.0000 1.01600
$$559$$ 0 0
$$560$$ −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 $$0.257389\pi$$
$$572$$ 0 0
$$573$$ −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 $$-0.622246\pi$$
−0.374675 + 0.927156i $$0.622246\pi$$
$$578$$ 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
$$585$$ 0 0
$$586$$ 7.00000 0.289167
$$587$$ 28.0000 1.15568 0.577842 0.816149i $$-0.303895\pi$$
0.577842 + 0.816149i $$0.303895\pi$$
$$588$$ 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 $$-0.350812\pi$$
0.451716 + 0.892162i $$0.350812\pi$$
$$594$$ 18.0000 0.738549
$$595$$ 3.00000 0.122988
$$596$$ 18.0000 0.737309
$$597$$ 30.0000 1.22782
$$598$$ 0 0
$$599$$ −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
$$611$$ 0 0
$$612$$ −18.0000 −0.727607
$$613$$ −26.0000 −1.05013 −0.525065 0.851062i $$-0.675959\pi$$
−0.525065 + 0.851062i $$0.675959\pi$$
$$614$$ 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 $$0.604378\pi$$
$$618$$ −24.0000 −0.965422
$$619$$ −4.00000 −0.160774 −0.0803868 0.996764i $$-0.525616\pi$$
−0.0803868 + 0.996764i $$0.525616\pi$$
$$620$$ −4.00000 −0.160644
$$621$$ 36.0000 1.44463
$$622$$ −18.0000 −0.721734
$$623$$ 6.00000 0.240385
$$624$$ 0 0
$$625$$ 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 $$-0.468268\pi$$
0.0995234 + 0.995035i $$0.468268\pi$$
$$632$$ 4.00000 0.159111
$$633$$ −69.0000 −2.74250
$$634$$ −18.0000 −0.714871
$$635$$ 16.0000 0.634941
$$636$$ −36.0000 −1.42749
$$637$$ 0 0
$$638$$ −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 $$-0.589027\pi$$
−0.276053 + 0.961142i $$0.589027\pi$$
$$644$$ 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
$$650$$ 0 0
$$651$$ −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 $$-0.437699\pi$$
0.194477 + 0.980907i $$0.437699\pi$$
$$662$$ −4.00000 −0.155464
$$663$$ 0 0
$$664$$ 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
$$676$$ 0 0
$$677$$ −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 $$-0.181492\pi$$
0.841807 + 0.539779i $$0.181492\pi$$
$$684$$ −36.0000 −1.37649
$$685$$ −12.0000 −0.458496
$$686$$ −13.0000 −0.496342
$$687$$ −45.0000 −1.71686
$$688$$ −5.00000 −0.190623
$$689$$ 0 0
$$690$$ −12.0000 −0.456832
$$691$$ 8.00000 0.304334 0.152167 0.988355i $$-0.451375\pi$$
0.152167 + 0.988355i $$0.451375\pi$$
$$692$$ 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 $$0.572767\pi$$
$$702$$ 0 0
$$703$$ 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 $$-0.752918\pi$$
−0.713560 + 0.700594i $$0.752918\pi$$
$$710$$ −5.00000 −0.187647
$$711$$ −24.0000 −0.900070
$$712$$ 6.00000 0.224860
$$713$$ 16.0000 0.599205
$$714$$ 9.00000 0.336817
$$715$$ 0 0
$$716$$ −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 $$0.583596\pi$$
$$728$$ 0 0
$$729$$ −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 $$-0.207932\pi$$
0.794121 + 0.607760i $$0.207932\pi$$
$$734$$ 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 $$-0.570839\pi$$
−0.220714 + 0.975339i $$0.570839\pi$$
$$740$$ −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 $$0.169129\pi$$
$$744$$ −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
$$754$$ 0 0
$$755$$ 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 $$-0.534685\pi$$
−0.108750 + 0.994069i $$0.534685\pi$$
$$762$$ 48.0000 1.73886
$$763$$ 19.0000 0.687846
$$764$$ 10.0000 0.361787
$$765$$ −18.0000 −0.650791
$$766$$ 27.0000 0.975550
$$767$$ 0 0
$$768$$ −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 $$-0.563387\pi$$
−0.197821 + 0.980238i $$0.563387\pi$$
$$774$$ 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
$$780$$ 0 0
$$781$$ 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 $$-0.693188\pi$$
−0.570338 + 0.821410i $$0.693188\pi$$
$$788$$ −9.00000 −0.320612
$$789$$ −36.0000 −1.28163
$$790$$ 4.00000 0.142314
$$791$$ −2.00000 −0.0711118
$$792$$ −12.0000 −0.426401
$$793$$ 0 0
$$794$$ −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
$$806$$ 0 0
$$807$$ 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 $$-0.336410\pi$$
0.491606 + 0.870817i $$0.336410\pi$$
$$812$$ −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
$$819$$ 0 0
$$820$$ 0 0
$$821$$ 25.0000 0.872506 0.436253 0.899824i $$-0.356305\pi$$
0.436253 + 0.899824i $$0.356305\pi$$
$$822$$ −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 $$-0.674665\pi$$
−0.521601 + 0.853189i $$0.674665\pi$$
$$828$$ −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
$$832$$ 0 0
$$833$$ 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 $$-0.917584\pi$$
−0.966667 + 0.256036i $$0.917584\pi$$
$$840$$ −3.00000 −0.103510
$$841$$ −25.0000 −0.862069
$$842$$ −5.00000 −0.172311
$$843$$ −78.0000 −2.68646
$$844$$ 23.0000 0.791693
$$845$$ 0 0
$$846$$ 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 $$-0.816780\pi$$
−0.838864 + 0.544341i $$0.816780\pi$$
$$854$$ −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 $$0.212321\pi$$
$$858$$ 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 $$0.440052\pi$$
$$864$$ 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
$$871$$ 0 0
$$872$$ 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 $$-0.271205\pi$$
0.658468 + 0.752609i $$0.271205\pi$$
$$878$$ 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 $$0.790355\pi$$
$$884$$ 0 0
$$885$$ −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
$$897$$ 0 0
$$898$$ −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
$$910$$ 0 0
$$911$$ −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
$$923$$ 0 0
$$924$$ 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 $$-0.298907\pi$$
0.590561 + 0.806993i $$0.298907\pi$$
$$930$$ −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
$$936$$ 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
$$939$$ 3.00000 0.0979013
$$940$$ −13.0000 −0.424013
$$941$$ −25.0000 −0.814977 −0.407488 0.913210i $$-0.633595\pi$$
−0.407488 + 0.913210i $$0.633595\pi$$
$$942$$ −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 $$-0.405526\pi$$
0.292461 + 0.956278i $$0.405526\pi$$
$$948$$ 12.0000 0.389742
$$949$$ 0 0
$$950$$ −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
$$962$$ 0 0
$$963$$ −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 $$-0.620579\pi$$
−0.369815 + 0.929105i $$0.620579\pi$$
$$968$$ 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
$$975$$ 0 0
$$976$$ −8.00000 −0.256074
$$977$$ 30.0000 0.959785 0.479893 0.877327i $$-0.340676\pi$$
0.479893 + 0.877327i $$0.340676\pi$$
$$978$$ 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 $$-0.335398\pi$$
0.494373 + 0.869250i $$0.335398\pi$$
$$984$$ 0 0
$$985$$ −9.00000 −0.286764
$$986$$ 6.00000 0.191079
$$987$$ −39.0000 −1.24138
$$988$$ 0 0
$$989$$ 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 338.2.a.a.1.1 1
3.2 odd 2 3042.2.a.l.1.1 1
4.3 odd 2 2704.2.a.n.1.1 1
5.4 even 2 8450.2.a.y.1.1 1
13.2 odd 12 338.2.e.d.147.1 4
13.3 even 3 338.2.c.g.191.1 2
13.4 even 6 338.2.c.c.315.1 2
13.5 odd 4 338.2.b.a.337.2 2
13.6 odd 12 338.2.e.d.23.2 4
13.7 odd 12 338.2.e.d.23.1 4
13.8 odd 4 338.2.b.a.337.1 2
13.9 even 3 338.2.c.g.315.1 2
13.10 even 6 338.2.c.c.191.1 2
13.11 odd 12 338.2.e.d.147.2 4
13.12 even 2 26.2.a.b.1.1 1
39.5 even 4 3042.2.b.f.1351.1 2
39.8 even 4 3042.2.b.f.1351.2 2
39.38 odd 2 234.2.a.b.1.1 1
52.31 even 4 2704.2.f.j.337.1 2
52.47 even 4 2704.2.f.j.337.2 2
52.51 odd 2 208.2.a.d.1.1 1
65.12 odd 4 650.2.b.a.599.2 2
65.38 odd 4 650.2.b.a.599.1 2
65.64 even 2 650.2.a.g.1.1 1
91.12 odd 6 1274.2.f.a.1145.1 2
91.25 even 6 1274.2.f.l.79.1 2
91.38 odd 6 1274.2.f.a.79.1 2
91.51 even 6 1274.2.f.l.1145.1 2
91.90 odd 2 1274.2.a.o.1.1 1
104.51 odd 2 832.2.a.a.1.1 1
104.77 even 2 832.2.a.j.1.1 1
117.25 even 6 2106.2.e.h.1405.1 2
117.38 odd 6 2106.2.e.t.1405.1 2
117.77 odd 6 2106.2.e.t.703.1 2
117.103 even 6 2106.2.e.h.703.1 2
143.142 odd 2 3146.2.a.a.1.1 1
156.155 even 2 1872.2.a.m.1.1 1
195.38 even 4 5850.2.e.v.5149.2 2
195.77 even 4 5850.2.e.v.5149.1 2
195.194 odd 2 5850.2.a.bn.1.1 1
208.51 odd 4 3328.2.b.k.1665.2 2
208.77 even 4 3328.2.b.g.1665.1 2
208.155 odd 4 3328.2.b.k.1665.1 2
208.181 even 4 3328.2.b.g.1665.2 2
221.220 even 2 7514.2.a.i.1.1 1
247.246 odd 2 9386.2.a.f.1.1 1
260.259 odd 2 5200.2.a.c.1.1 1
312.77 odd 2 7488.2.a.w.1.1 1
312.155 even 2 7488.2.a.v.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
26.2.a.b.1.1 1 13.12 even 2
208.2.a.d.1.1 1 52.51 odd 2
234.2.a.b.1.1 1 39.38 odd 2
338.2.a.a.1.1 1 1.1 even 1 trivial
338.2.b.a.337.1 2 13.8 odd 4
338.2.b.a.337.2 2 13.5 odd 4
338.2.c.c.191.1 2 13.10 even 6
338.2.c.c.315.1 2 13.4 even 6
338.2.c.g.191.1 2 13.3 even 3
338.2.c.g.315.1 2 13.9 even 3
338.2.e.d.23.1 4 13.7 odd 12
338.2.e.d.23.2 4 13.6 odd 12
338.2.e.d.147.1 4 13.2 odd 12
338.2.e.d.147.2 4 13.11 odd 12
650.2.a.g.1.1 1 65.64 even 2
650.2.b.a.599.1 2 65.38 odd 4
650.2.b.a.599.2 2 65.12 odd 4
832.2.a.a.1.1 1 104.51 odd 2
832.2.a.j.1.1 1 104.77 even 2
1274.2.a.o.1.1 1 91.90 odd 2
1274.2.f.a.79.1 2 91.38 odd 6
1274.2.f.a.1145.1 2 91.12 odd 6
1274.2.f.l.79.1 2 91.25 even 6
1274.2.f.l.1145.1 2 91.51 even 6
1872.2.a.m.1.1 1 156.155 even 2
2106.2.e.h.703.1 2 117.103 even 6
2106.2.e.h.1405.1 2 117.25 even 6
2106.2.e.t.703.1 2 117.77 odd 6
2106.2.e.t.1405.1 2 117.38 odd 6
2704.2.a.n.1.1 1 4.3 odd 2
2704.2.f.j.337.1 2 52.31 even 4
2704.2.f.j.337.2 2 52.47 even 4
3042.2.a.l.1.1 1 3.2 odd 2
3042.2.b.f.1351.1 2 39.5 even 4
3042.2.b.f.1351.2 2 39.8 even 4
3146.2.a.a.1.1 1 143.142 odd 2
3328.2.b.g.1665.1 2 208.77 even 4
3328.2.b.g.1665.2 2 208.181 even 4
3328.2.b.k.1665.1 2 208.155 odd 4
3328.2.b.k.1665.2 2 208.51 odd 4
5200.2.a.c.1.1 1 260.259 odd 2
5850.2.a.bn.1.1 1 195.194 odd 2
5850.2.e.v.5149.1 2 195.77 even 4
5850.2.e.v.5149.2 2 195.38 even 4
7488.2.a.v.1.1 1 312.155 even 2
7488.2.a.w.1.1 1 312.77 odd 2
7514.2.a.i.1.1 1 221.220 even 2
8450.2.a.y.1.1 1 5.4 even 2
9386.2.a.f.1.1 1 247.246 odd 2