# Properties

 Label 1682.2.a.d.1.1 Level $1682$ Weight $2$ Character 1682.1 Self dual yes Analytic conductor $13.431$ 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] = [1682,2,Mod(1,1682)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1682, 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("1682.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1682 = 2 \cdot 29^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1682.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: $$13.4308376200$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 58) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Character $$\chi$$ $$=$$ 1682.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} -1.00000 q^{6} -2.00000 q^{7} -1.00000 q^{8} -2.00000 q^{9} +O(q^{10})$$ $$q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} -1.00000 q^{6} -2.00000 q^{7} -1.00000 q^{8} -2.00000 q^{9} -1.00000 q^{10} +3.00000 q^{11} +1.00000 q^{12} -1.00000 q^{13} +2.00000 q^{14} +1.00000 q^{15} +1.00000 q^{16} -8.00000 q^{17} +2.00000 q^{18} +1.00000 q^{20} -2.00000 q^{21} -3.00000 q^{22} +4.00000 q^{23} -1.00000 q^{24} -4.00000 q^{25} +1.00000 q^{26} -5.00000 q^{27} -2.00000 q^{28} -1.00000 q^{30} +3.00000 q^{31} -1.00000 q^{32} +3.00000 q^{33} +8.00000 q^{34} -2.00000 q^{35} -2.00000 q^{36} -8.00000 q^{37} -1.00000 q^{39} -1.00000 q^{40} -2.00000 q^{41} +2.00000 q^{42} +11.0000 q^{43} +3.00000 q^{44} -2.00000 q^{45} -4.00000 q^{46} -13.0000 q^{47} +1.00000 q^{48} -3.00000 q^{49} +4.00000 q^{50} -8.00000 q^{51} -1.00000 q^{52} -11.0000 q^{53} +5.00000 q^{54} +3.00000 q^{55} +2.00000 q^{56} +1.00000 q^{60} +8.00000 q^{61} -3.00000 q^{62} +4.00000 q^{63} +1.00000 q^{64} -1.00000 q^{65} -3.00000 q^{66} -12.0000 q^{67} -8.00000 q^{68} +4.00000 q^{69} +2.00000 q^{70} +2.00000 q^{71} +2.00000 q^{72} -4.00000 q^{73} +8.00000 q^{74} -4.00000 q^{75} -6.00000 q^{77} +1.00000 q^{78} -15.0000 q^{79} +1.00000 q^{80} +1.00000 q^{81} +2.00000 q^{82} +4.00000 q^{83} -2.00000 q^{84} -8.00000 q^{85} -11.0000 q^{86} -3.00000 q^{88} +10.0000 q^{89} +2.00000 q^{90} +2.00000 q^{91} +4.00000 q^{92} +3.00000 q^{93} +13.0000 q^{94} -1.00000 q^{96} +2.00000 q^{97} +3.00000 q^{98} -6.00000 q^{99} +O(q^{100})$$

## Coefficient data

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

Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ −1.00000 −0.707107
$$3$$ 1.00000 0.577350 0.288675 0.957427i $$-0.406785\pi$$
0.288675 + 0.957427i $$0.406785\pi$$
$$4$$ 1.00000 0.500000
$$5$$ 1.00000 0.447214 0.223607 0.974679i $$-0.428217\pi$$
0.223607 + 0.974679i $$0.428217\pi$$
$$6$$ −1.00000 −0.408248
$$7$$ −2.00000 −0.755929 −0.377964 0.925820i $$-0.623376\pi$$
−0.377964 + 0.925820i $$0.623376\pi$$
$$8$$ −1.00000 −0.353553
$$9$$ −2.00000 −0.666667
$$10$$ −1.00000 −0.316228
$$11$$ 3.00000 0.904534 0.452267 0.891883i $$-0.350615\pi$$
0.452267 + 0.891883i $$0.350615\pi$$
$$12$$ 1.00000 0.288675
$$13$$ −1.00000 −0.277350 −0.138675 0.990338i $$-0.544284\pi$$
−0.138675 + 0.990338i $$0.544284\pi$$
$$14$$ 2.00000 0.534522
$$15$$ 1.00000 0.258199
$$16$$ 1.00000 0.250000
$$17$$ −8.00000 −1.94029 −0.970143 0.242536i $$-0.922021\pi$$
−0.970143 + 0.242536i $$0.922021\pi$$
$$18$$ 2.00000 0.471405
$$19$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$20$$ 1.00000 0.223607
$$21$$ −2.00000 −0.436436
$$22$$ −3.00000 −0.639602
$$23$$ 4.00000 0.834058 0.417029 0.908893i $$-0.363071\pi$$
0.417029 + 0.908893i $$0.363071\pi$$
$$24$$ −1.00000 −0.204124
$$25$$ −4.00000 −0.800000
$$26$$ 1.00000 0.196116
$$27$$ −5.00000 −0.962250
$$28$$ −2.00000 −0.377964
$$29$$ 0 0
$$30$$ −1.00000 −0.182574
$$31$$ 3.00000 0.538816 0.269408 0.963026i $$-0.413172\pi$$
0.269408 + 0.963026i $$0.413172\pi$$
$$32$$ −1.00000 −0.176777
$$33$$ 3.00000 0.522233
$$34$$ 8.00000 1.37199
$$35$$ −2.00000 −0.338062
$$36$$ −2.00000 −0.333333
$$37$$ −8.00000 −1.31519 −0.657596 0.753371i $$-0.728427\pi$$
−0.657596 + 0.753371i $$0.728427\pi$$
$$38$$ 0 0
$$39$$ −1.00000 −0.160128
$$40$$ −1.00000 −0.158114
$$41$$ −2.00000 −0.312348 −0.156174 0.987730i $$-0.549916\pi$$
−0.156174 + 0.987730i $$0.549916\pi$$
$$42$$ 2.00000 0.308607
$$43$$ 11.0000 1.67748 0.838742 0.544529i $$-0.183292\pi$$
0.838742 + 0.544529i $$0.183292\pi$$
$$44$$ 3.00000 0.452267
$$45$$ −2.00000 −0.298142
$$46$$ −4.00000 −0.589768
$$47$$ −13.0000 −1.89624 −0.948122 0.317905i $$-0.897021\pi$$
−0.948122 + 0.317905i $$0.897021\pi$$
$$48$$ 1.00000 0.144338
$$49$$ −3.00000 −0.428571
$$50$$ 4.00000 0.565685
$$51$$ −8.00000 −1.12022
$$52$$ −1.00000 −0.138675
$$53$$ −11.0000 −1.51097 −0.755483 0.655168i $$-0.772598\pi$$
−0.755483 + 0.655168i $$0.772598\pi$$
$$54$$ 5.00000 0.680414
$$55$$ 3.00000 0.404520
$$56$$ 2.00000 0.267261
$$57$$ 0 0
$$58$$ 0 0
$$59$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$60$$ 1.00000 0.129099
$$61$$ 8.00000 1.02430 0.512148 0.858898i $$-0.328850\pi$$
0.512148 + 0.858898i $$0.328850\pi$$
$$62$$ −3.00000 −0.381000
$$63$$ 4.00000 0.503953
$$64$$ 1.00000 0.125000
$$65$$ −1.00000 −0.124035
$$66$$ −3.00000 −0.369274
$$67$$ −12.0000 −1.46603 −0.733017 0.680211i $$-0.761888\pi$$
−0.733017 + 0.680211i $$0.761888\pi$$
$$68$$ −8.00000 −0.970143
$$69$$ 4.00000 0.481543
$$70$$ 2.00000 0.239046
$$71$$ 2.00000 0.237356 0.118678 0.992933i $$-0.462134\pi$$
0.118678 + 0.992933i $$0.462134\pi$$
$$72$$ 2.00000 0.235702
$$73$$ −4.00000 −0.468165 −0.234082 0.972217i $$-0.575209\pi$$
−0.234082 + 0.972217i $$0.575209\pi$$
$$74$$ 8.00000 0.929981
$$75$$ −4.00000 −0.461880
$$76$$ 0 0
$$77$$ −6.00000 −0.683763
$$78$$ 1.00000 0.113228
$$79$$ −15.0000 −1.68763 −0.843816 0.536633i $$-0.819696\pi$$
−0.843816 + 0.536633i $$0.819696\pi$$
$$80$$ 1.00000 0.111803
$$81$$ 1.00000 0.111111
$$82$$ 2.00000 0.220863
$$83$$ 4.00000 0.439057 0.219529 0.975606i $$-0.429548\pi$$
0.219529 + 0.975606i $$0.429548\pi$$
$$84$$ −2.00000 −0.218218
$$85$$ −8.00000 −0.867722
$$86$$ −11.0000 −1.18616
$$87$$ 0 0
$$88$$ −3.00000 −0.319801
$$89$$ 10.0000 1.06000 0.529999 0.847998i $$-0.322192\pi$$
0.529999 + 0.847998i $$0.322192\pi$$
$$90$$ 2.00000 0.210819
$$91$$ 2.00000 0.209657
$$92$$ 4.00000 0.417029
$$93$$ 3.00000 0.311086
$$94$$ 13.0000 1.34085
$$95$$ 0 0
$$96$$ −1.00000 −0.102062
$$97$$ 2.00000 0.203069 0.101535 0.994832i $$-0.467625\pi$$
0.101535 + 0.994832i $$0.467625\pi$$
$$98$$ 3.00000 0.303046
$$99$$ −6.00000 −0.603023
$$100$$ −4.00000 −0.400000
$$101$$ 8.00000 0.796030 0.398015 0.917379i $$-0.369699\pi$$
0.398015 + 0.917379i $$0.369699\pi$$
$$102$$ 8.00000 0.792118
$$103$$ 14.0000 1.37946 0.689730 0.724066i $$-0.257729\pi$$
0.689730 + 0.724066i $$0.257729\pi$$
$$104$$ 1.00000 0.0980581
$$105$$ −2.00000 −0.195180
$$106$$ 11.0000 1.06841
$$107$$ −2.00000 −0.193347 −0.0966736 0.995316i $$-0.530820\pi$$
−0.0966736 + 0.995316i $$0.530820\pi$$
$$108$$ −5.00000 −0.481125
$$109$$ 5.00000 0.478913 0.239457 0.970907i $$-0.423031\pi$$
0.239457 + 0.970907i $$0.423031\pi$$
$$110$$ −3.00000 −0.286039
$$111$$ −8.00000 −0.759326
$$112$$ −2.00000 −0.188982
$$113$$ 6.00000 0.564433 0.282216 0.959351i $$-0.408930\pi$$
0.282216 + 0.959351i $$0.408930\pi$$
$$114$$ 0 0
$$115$$ 4.00000 0.373002
$$116$$ 0 0
$$117$$ 2.00000 0.184900
$$118$$ 0 0
$$119$$ 16.0000 1.46672
$$120$$ −1.00000 −0.0912871
$$121$$ −2.00000 −0.181818
$$122$$ −8.00000 −0.724286
$$123$$ −2.00000 −0.180334
$$124$$ 3.00000 0.269408
$$125$$ −9.00000 −0.804984
$$126$$ −4.00000 −0.356348
$$127$$ −8.00000 −0.709885 −0.354943 0.934888i $$-0.615500\pi$$
−0.354943 + 0.934888i $$0.615500\pi$$
$$128$$ −1.00000 −0.0883883
$$129$$ 11.0000 0.968496
$$130$$ 1.00000 0.0877058
$$131$$ −12.0000 −1.04844 −0.524222 0.851581i $$-0.675644\pi$$
−0.524222 + 0.851581i $$0.675644\pi$$
$$132$$ 3.00000 0.261116
$$133$$ 0 0
$$134$$ 12.0000 1.03664
$$135$$ −5.00000 −0.430331
$$136$$ 8.00000 0.685994
$$137$$ 12.0000 1.02523 0.512615 0.858619i $$-0.328677\pi$$
0.512615 + 0.858619i $$0.328677\pi$$
$$138$$ −4.00000 −0.340503
$$139$$ −20.0000 −1.69638 −0.848189 0.529694i $$-0.822307\pi$$
−0.848189 + 0.529694i $$0.822307\pi$$
$$140$$ −2.00000 −0.169031
$$141$$ −13.0000 −1.09480
$$142$$ −2.00000 −0.167836
$$143$$ −3.00000 −0.250873
$$144$$ −2.00000 −0.166667
$$145$$ 0 0
$$146$$ 4.00000 0.331042
$$147$$ −3.00000 −0.247436
$$148$$ −8.00000 −0.657596
$$149$$ 15.0000 1.22885 0.614424 0.788976i $$-0.289388\pi$$
0.614424 + 0.788976i $$0.289388\pi$$
$$150$$ 4.00000 0.326599
$$151$$ 2.00000 0.162758 0.0813788 0.996683i $$-0.474068\pi$$
0.0813788 + 0.996683i $$0.474068\pi$$
$$152$$ 0 0
$$153$$ 16.0000 1.29352
$$154$$ 6.00000 0.483494
$$155$$ 3.00000 0.240966
$$156$$ −1.00000 −0.0800641
$$157$$ −18.0000 −1.43656 −0.718278 0.695756i $$-0.755069\pi$$
−0.718278 + 0.695756i $$0.755069\pi$$
$$158$$ 15.0000 1.19334
$$159$$ −11.0000 −0.872357
$$160$$ −1.00000 −0.0790569
$$161$$ −8.00000 −0.630488
$$162$$ −1.00000 −0.0785674
$$163$$ −9.00000 −0.704934 −0.352467 0.935824i $$-0.614657\pi$$
−0.352467 + 0.935824i $$0.614657\pi$$
$$164$$ −2.00000 −0.156174
$$165$$ 3.00000 0.233550
$$166$$ −4.00000 −0.310460
$$167$$ −2.00000 −0.154765 −0.0773823 0.997001i $$-0.524656\pi$$
−0.0773823 + 0.997001i $$0.524656\pi$$
$$168$$ 2.00000 0.154303
$$169$$ −12.0000 −0.923077
$$170$$ 8.00000 0.613572
$$171$$ 0 0
$$172$$ 11.0000 0.838742
$$173$$ −6.00000 −0.456172 −0.228086 0.973641i $$-0.573247\pi$$
−0.228086 + 0.973641i $$0.573247\pi$$
$$174$$ 0 0
$$175$$ 8.00000 0.604743
$$176$$ 3.00000 0.226134
$$177$$ 0 0
$$178$$ −10.0000 −0.749532
$$179$$ −10.0000 −0.747435 −0.373718 0.927543i $$-0.621917\pi$$
−0.373718 + 0.927543i $$0.621917\pi$$
$$180$$ −2.00000 −0.149071
$$181$$ 7.00000 0.520306 0.260153 0.965567i $$-0.416227\pi$$
0.260153 + 0.965567i $$0.416227\pi$$
$$182$$ −2.00000 −0.148250
$$183$$ 8.00000 0.591377
$$184$$ −4.00000 −0.294884
$$185$$ −8.00000 −0.588172
$$186$$ −3.00000 −0.219971
$$187$$ −24.0000 −1.75505
$$188$$ −13.0000 −0.948122
$$189$$ 10.0000 0.727393
$$190$$ 0 0
$$191$$ 8.00000 0.578860 0.289430 0.957199i $$-0.406534\pi$$
0.289430 + 0.957199i $$0.406534\pi$$
$$192$$ 1.00000 0.0721688
$$193$$ −14.0000 −1.00774 −0.503871 0.863779i $$-0.668091\pi$$
−0.503871 + 0.863779i $$0.668091\pi$$
$$194$$ −2.00000 −0.143592
$$195$$ −1.00000 −0.0716115
$$196$$ −3.00000 −0.214286
$$197$$ 18.0000 1.28245 0.641223 0.767354i $$-0.278427\pi$$
0.641223 + 0.767354i $$0.278427\pi$$
$$198$$ 6.00000 0.426401
$$199$$ −10.0000 −0.708881 −0.354441 0.935079i $$-0.615329\pi$$
−0.354441 + 0.935079i $$0.615329\pi$$
$$200$$ 4.00000 0.282843
$$201$$ −12.0000 −0.846415
$$202$$ −8.00000 −0.562878
$$203$$ 0 0
$$204$$ −8.00000 −0.560112
$$205$$ −2.00000 −0.139686
$$206$$ −14.0000 −0.975426
$$207$$ −8.00000 −0.556038
$$208$$ −1.00000 −0.0693375
$$209$$ 0 0
$$210$$ 2.00000 0.138013
$$211$$ 3.00000 0.206529 0.103264 0.994654i $$-0.467071\pi$$
0.103264 + 0.994654i $$0.467071\pi$$
$$212$$ −11.0000 −0.755483
$$213$$ 2.00000 0.137038
$$214$$ 2.00000 0.136717
$$215$$ 11.0000 0.750194
$$216$$ 5.00000 0.340207
$$217$$ −6.00000 −0.407307
$$218$$ −5.00000 −0.338643
$$219$$ −4.00000 −0.270295
$$220$$ 3.00000 0.202260
$$221$$ 8.00000 0.538138
$$222$$ 8.00000 0.536925
$$223$$ −26.0000 −1.74109 −0.870544 0.492090i $$-0.836233\pi$$
−0.870544 + 0.492090i $$0.836233\pi$$
$$224$$ 2.00000 0.133631
$$225$$ 8.00000 0.533333
$$226$$ −6.00000 −0.399114
$$227$$ 18.0000 1.19470 0.597351 0.801980i $$-0.296220\pi$$
0.597351 + 0.801980i $$0.296220\pi$$
$$228$$ 0 0
$$229$$ −10.0000 −0.660819 −0.330409 0.943838i $$-0.607187\pi$$
−0.330409 + 0.943838i $$0.607187\pi$$
$$230$$ −4.00000 −0.263752
$$231$$ −6.00000 −0.394771
$$232$$ 0 0
$$233$$ −1.00000 −0.0655122 −0.0327561 0.999463i $$-0.510428\pi$$
−0.0327561 + 0.999463i $$0.510428\pi$$
$$234$$ −2.00000 −0.130744
$$235$$ −13.0000 −0.848026
$$236$$ 0 0
$$237$$ −15.0000 −0.974355
$$238$$ −16.0000 −1.03713
$$239$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$240$$ 1.00000 0.0645497
$$241$$ 17.0000 1.09507 0.547533 0.836784i $$-0.315567\pi$$
0.547533 + 0.836784i $$0.315567\pi$$
$$242$$ 2.00000 0.128565
$$243$$ 16.0000 1.02640
$$244$$ 8.00000 0.512148
$$245$$ −3.00000 −0.191663
$$246$$ 2.00000 0.127515
$$247$$ 0 0
$$248$$ −3.00000 −0.190500
$$249$$ 4.00000 0.253490
$$250$$ 9.00000 0.569210
$$251$$ −27.0000 −1.70422 −0.852112 0.523359i $$-0.824679\pi$$
−0.852112 + 0.523359i $$0.824679\pi$$
$$252$$ 4.00000 0.251976
$$253$$ 12.0000 0.754434
$$254$$ 8.00000 0.501965
$$255$$ −8.00000 −0.500979
$$256$$ 1.00000 0.0625000
$$257$$ 13.0000 0.810918 0.405459 0.914113i $$-0.367112\pi$$
0.405459 + 0.914113i $$0.367112\pi$$
$$258$$ −11.0000 −0.684830
$$259$$ 16.0000 0.994192
$$260$$ −1.00000 −0.0620174
$$261$$ 0 0
$$262$$ 12.0000 0.741362
$$263$$ −9.00000 −0.554964 −0.277482 0.960731i $$-0.589500\pi$$
−0.277482 + 0.960731i $$0.589500\pi$$
$$264$$ −3.00000 −0.184637
$$265$$ −11.0000 −0.675725
$$266$$ 0 0
$$267$$ 10.0000 0.611990
$$268$$ −12.0000 −0.733017
$$269$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$270$$ 5.00000 0.304290
$$271$$ 13.0000 0.789694 0.394847 0.918747i $$-0.370798\pi$$
0.394847 + 0.918747i $$0.370798\pi$$
$$272$$ −8.00000 −0.485071
$$273$$ 2.00000 0.121046
$$274$$ −12.0000 −0.724947
$$275$$ −12.0000 −0.723627
$$276$$ 4.00000 0.240772
$$277$$ −2.00000 −0.120168 −0.0600842 0.998193i $$-0.519137\pi$$
−0.0600842 + 0.998193i $$0.519137\pi$$
$$278$$ 20.0000 1.19952
$$279$$ −6.00000 −0.359211
$$280$$ 2.00000 0.119523
$$281$$ 27.0000 1.61068 0.805342 0.592810i $$-0.201981\pi$$
0.805342 + 0.592810i $$0.201981\pi$$
$$282$$ 13.0000 0.774139
$$283$$ 4.00000 0.237775 0.118888 0.992908i $$-0.462067\pi$$
0.118888 + 0.992908i $$0.462067\pi$$
$$284$$ 2.00000 0.118678
$$285$$ 0 0
$$286$$ 3.00000 0.177394
$$287$$ 4.00000 0.236113
$$288$$ 2.00000 0.117851
$$289$$ 47.0000 2.76471
$$290$$ 0 0
$$291$$ 2.00000 0.117242
$$292$$ −4.00000 −0.234082
$$293$$ −14.0000 −0.817889 −0.408944 0.912559i $$-0.634103\pi$$
−0.408944 + 0.912559i $$0.634103\pi$$
$$294$$ 3.00000 0.174964
$$295$$ 0 0
$$296$$ 8.00000 0.464991
$$297$$ −15.0000 −0.870388
$$298$$ −15.0000 −0.868927
$$299$$ −4.00000 −0.231326
$$300$$ −4.00000 −0.230940
$$301$$ −22.0000 −1.26806
$$302$$ −2.00000 −0.115087
$$303$$ 8.00000 0.459588
$$304$$ 0 0
$$305$$ 8.00000 0.458079
$$306$$ −16.0000 −0.914659
$$307$$ 7.00000 0.399511 0.199756 0.979846i $$-0.435985\pi$$
0.199756 + 0.979846i $$0.435985\pi$$
$$308$$ −6.00000 −0.341882
$$309$$ 14.0000 0.796432
$$310$$ −3.00000 −0.170389
$$311$$ 8.00000 0.453638 0.226819 0.973937i $$-0.427167\pi$$
0.226819 + 0.973937i $$0.427167\pi$$
$$312$$ 1.00000 0.0566139
$$313$$ 9.00000 0.508710 0.254355 0.967111i $$-0.418137\pi$$
0.254355 + 0.967111i $$0.418137\pi$$
$$314$$ 18.0000 1.01580
$$315$$ 4.00000 0.225374
$$316$$ −15.0000 −0.843816
$$317$$ 12.0000 0.673987 0.336994 0.941507i $$-0.390590\pi$$
0.336994 + 0.941507i $$0.390590\pi$$
$$318$$ 11.0000 0.616849
$$319$$ 0 0
$$320$$ 1.00000 0.0559017
$$321$$ −2.00000 −0.111629
$$322$$ 8.00000 0.445823
$$323$$ 0 0
$$324$$ 1.00000 0.0555556
$$325$$ 4.00000 0.221880
$$326$$ 9.00000 0.498464
$$327$$ 5.00000 0.276501
$$328$$ 2.00000 0.110432
$$329$$ 26.0000 1.43343
$$330$$ −3.00000 −0.165145
$$331$$ 23.0000 1.26419 0.632097 0.774889i $$-0.282194\pi$$
0.632097 + 0.774889i $$0.282194\pi$$
$$332$$ 4.00000 0.219529
$$333$$ 16.0000 0.876795
$$334$$ 2.00000 0.109435
$$335$$ −12.0000 −0.655630
$$336$$ −2.00000 −0.109109
$$337$$ 32.0000 1.74315 0.871576 0.490261i $$-0.163099\pi$$
0.871576 + 0.490261i $$0.163099\pi$$
$$338$$ 12.0000 0.652714
$$339$$ 6.00000 0.325875
$$340$$ −8.00000 −0.433861
$$341$$ 9.00000 0.487377
$$342$$ 0 0
$$343$$ 20.0000 1.07990
$$344$$ −11.0000 −0.593080
$$345$$ 4.00000 0.215353
$$346$$ 6.00000 0.322562
$$347$$ −2.00000 −0.107366 −0.0536828 0.998558i $$-0.517096\pi$$
−0.0536828 + 0.998558i $$0.517096\pi$$
$$348$$ 0 0
$$349$$ −15.0000 −0.802932 −0.401466 0.915874i $$-0.631499\pi$$
−0.401466 + 0.915874i $$0.631499\pi$$
$$350$$ −8.00000 −0.427618
$$351$$ 5.00000 0.266880
$$352$$ −3.00000 −0.159901
$$353$$ −26.0000 −1.38384 −0.691920 0.721974i $$-0.743235\pi$$
−0.691920 + 0.721974i $$0.743235\pi$$
$$354$$ 0 0
$$355$$ 2.00000 0.106149
$$356$$ 10.0000 0.529999
$$357$$ 16.0000 0.846810
$$358$$ 10.0000 0.528516
$$359$$ 25.0000 1.31945 0.659725 0.751507i $$-0.270673\pi$$
0.659725 + 0.751507i $$0.270673\pi$$
$$360$$ 2.00000 0.105409
$$361$$ −19.0000 −1.00000
$$362$$ −7.00000 −0.367912
$$363$$ −2.00000 −0.104973
$$364$$ 2.00000 0.104828
$$365$$ −4.00000 −0.209370
$$366$$ −8.00000 −0.418167
$$367$$ 32.0000 1.67039 0.835193 0.549957i $$-0.185356\pi$$
0.835193 + 0.549957i $$0.185356\pi$$
$$368$$ 4.00000 0.208514
$$369$$ 4.00000 0.208232
$$370$$ 8.00000 0.415900
$$371$$ 22.0000 1.14218
$$372$$ 3.00000 0.155543
$$373$$ −21.0000 −1.08734 −0.543669 0.839299i $$-0.682965\pi$$
−0.543669 + 0.839299i $$0.682965\pi$$
$$374$$ 24.0000 1.24101
$$375$$ −9.00000 −0.464758
$$376$$ 13.0000 0.670424
$$377$$ 0 0
$$378$$ −10.0000 −0.514344
$$379$$ −20.0000 −1.02733 −0.513665 0.857991i $$-0.671713\pi$$
−0.513665 + 0.857991i $$0.671713\pi$$
$$380$$ 0 0
$$381$$ −8.00000 −0.409852
$$382$$ −8.00000 −0.409316
$$383$$ 14.0000 0.715367 0.357683 0.933843i $$-0.383567\pi$$
0.357683 + 0.933843i $$0.383567\pi$$
$$384$$ −1.00000 −0.0510310
$$385$$ −6.00000 −0.305788
$$386$$ 14.0000 0.712581
$$387$$ −22.0000 −1.11832
$$388$$ 2.00000 0.101535
$$389$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$390$$ 1.00000 0.0506370
$$391$$ −32.0000 −1.61831
$$392$$ 3.00000 0.151523
$$393$$ −12.0000 −0.605320
$$394$$ −18.0000 −0.906827
$$395$$ −15.0000 −0.754732
$$396$$ −6.00000 −0.301511
$$397$$ −17.0000 −0.853206 −0.426603 0.904439i $$-0.640290\pi$$
−0.426603 + 0.904439i $$0.640290\pi$$
$$398$$ 10.0000 0.501255
$$399$$ 0 0
$$400$$ −4.00000 −0.200000
$$401$$ 27.0000 1.34832 0.674158 0.738587i $$-0.264507\pi$$
0.674158 + 0.738587i $$0.264507\pi$$
$$402$$ 12.0000 0.598506
$$403$$ −3.00000 −0.149441
$$404$$ 8.00000 0.398015
$$405$$ 1.00000 0.0496904
$$406$$ 0 0
$$407$$ −24.0000 −1.18964
$$408$$ 8.00000 0.396059
$$409$$ −30.0000 −1.48340 −0.741702 0.670729i $$-0.765981\pi$$
−0.741702 + 0.670729i $$0.765981\pi$$
$$410$$ 2.00000 0.0987730
$$411$$ 12.0000 0.591916
$$412$$ 14.0000 0.689730
$$413$$ 0 0
$$414$$ 8.00000 0.393179
$$415$$ 4.00000 0.196352
$$416$$ 1.00000 0.0490290
$$417$$ −20.0000 −0.979404
$$418$$ 0 0
$$419$$ −10.0000 −0.488532 −0.244266 0.969708i $$-0.578547\pi$$
−0.244266 + 0.969708i $$0.578547\pi$$
$$420$$ −2.00000 −0.0975900
$$421$$ −32.0000 −1.55958 −0.779792 0.626038i $$-0.784675\pi$$
−0.779792 + 0.626038i $$0.784675\pi$$
$$422$$ −3.00000 −0.146038
$$423$$ 26.0000 1.26416
$$424$$ 11.0000 0.534207
$$425$$ 32.0000 1.55223
$$426$$ −2.00000 −0.0969003
$$427$$ −16.0000 −0.774294
$$428$$ −2.00000 −0.0966736
$$429$$ −3.00000 −0.144841
$$430$$ −11.0000 −0.530467
$$431$$ 32.0000 1.54139 0.770693 0.637207i $$-0.219910\pi$$
0.770693 + 0.637207i $$0.219910\pi$$
$$432$$ −5.00000 −0.240563
$$433$$ 16.0000 0.768911 0.384455 0.923144i $$-0.374389\pi$$
0.384455 + 0.923144i $$0.374389\pi$$
$$434$$ 6.00000 0.288009
$$435$$ 0 0
$$436$$ 5.00000 0.239457
$$437$$ 0 0
$$438$$ 4.00000 0.191127
$$439$$ 20.0000 0.954548 0.477274 0.878755i $$-0.341625\pi$$
0.477274 + 0.878755i $$0.341625\pi$$
$$440$$ −3.00000 −0.143019
$$441$$ 6.00000 0.285714
$$442$$ −8.00000 −0.380521
$$443$$ −4.00000 −0.190046 −0.0950229 0.995475i $$-0.530292\pi$$
−0.0950229 + 0.995475i $$0.530292\pi$$
$$444$$ −8.00000 −0.379663
$$445$$ 10.0000 0.474045
$$446$$ 26.0000 1.23114
$$447$$ 15.0000 0.709476
$$448$$ −2.00000 −0.0944911
$$449$$ 10.0000 0.471929 0.235965 0.971762i $$-0.424175\pi$$
0.235965 + 0.971762i $$0.424175\pi$$
$$450$$ −8.00000 −0.377124
$$451$$ −6.00000 −0.282529
$$452$$ 6.00000 0.282216
$$453$$ 2.00000 0.0939682
$$454$$ −18.0000 −0.844782
$$455$$ 2.00000 0.0937614
$$456$$ 0 0
$$457$$ −2.00000 −0.0935561 −0.0467780 0.998905i $$-0.514895\pi$$
−0.0467780 + 0.998905i $$0.514895\pi$$
$$458$$ 10.0000 0.467269
$$459$$ 40.0000 1.86704
$$460$$ 4.00000 0.186501
$$461$$ −2.00000 −0.0931493 −0.0465746 0.998915i $$-0.514831\pi$$
−0.0465746 + 0.998915i $$0.514831\pi$$
$$462$$ 6.00000 0.279145
$$463$$ 4.00000 0.185896 0.0929479 0.995671i $$-0.470371\pi$$
0.0929479 + 0.995671i $$0.470371\pi$$
$$464$$ 0 0
$$465$$ 3.00000 0.139122
$$466$$ 1.00000 0.0463241
$$467$$ 27.0000 1.24941 0.624705 0.780860i $$-0.285219\pi$$
0.624705 + 0.780860i $$0.285219\pi$$
$$468$$ 2.00000 0.0924500
$$469$$ 24.0000 1.10822
$$470$$ 13.0000 0.599645
$$471$$ −18.0000 −0.829396
$$472$$ 0 0
$$473$$ 33.0000 1.51734
$$474$$ 15.0000 0.688973
$$475$$ 0 0
$$476$$ 16.0000 0.733359
$$477$$ 22.0000 1.00731
$$478$$ 0 0
$$479$$ 5.00000 0.228456 0.114228 0.993455i $$-0.463561\pi$$
0.114228 + 0.993455i $$0.463561\pi$$
$$480$$ −1.00000 −0.0456435
$$481$$ 8.00000 0.364769
$$482$$ −17.0000 −0.774329
$$483$$ −8.00000 −0.364013
$$484$$ −2.00000 −0.0909091
$$485$$ 2.00000 0.0908153
$$486$$ −16.0000 −0.725775
$$487$$ −22.0000 −0.996915 −0.498458 0.866914i $$-0.666100\pi$$
−0.498458 + 0.866914i $$0.666100\pi$$
$$488$$ −8.00000 −0.362143
$$489$$ −9.00000 −0.406994
$$490$$ 3.00000 0.135526
$$491$$ 33.0000 1.48927 0.744635 0.667472i $$-0.232624\pi$$
0.744635 + 0.667472i $$0.232624\pi$$
$$492$$ −2.00000 −0.0901670
$$493$$ 0 0
$$494$$ 0 0
$$495$$ −6.00000 −0.269680
$$496$$ 3.00000 0.134704
$$497$$ −4.00000 −0.179425
$$498$$ −4.00000 −0.179244
$$499$$ −20.0000 −0.895323 −0.447661 0.894203i $$-0.647743\pi$$
−0.447661 + 0.894203i $$0.647743\pi$$
$$500$$ −9.00000 −0.402492
$$501$$ −2.00000 −0.0893534
$$502$$ 27.0000 1.20507
$$503$$ −19.0000 −0.847168 −0.423584 0.905857i $$-0.639228\pi$$
−0.423584 + 0.905857i $$0.639228\pi$$
$$504$$ −4.00000 −0.178174
$$505$$ 8.00000 0.355995
$$506$$ −12.0000 −0.533465
$$507$$ −12.0000 −0.532939
$$508$$ −8.00000 −0.354943
$$509$$ −15.0000 −0.664863 −0.332432 0.943127i $$-0.607869\pi$$
−0.332432 + 0.943127i $$0.607869\pi$$
$$510$$ 8.00000 0.354246
$$511$$ 8.00000 0.353899
$$512$$ −1.00000 −0.0441942
$$513$$ 0 0
$$514$$ −13.0000 −0.573405
$$515$$ 14.0000 0.616914
$$516$$ 11.0000 0.484248
$$517$$ −39.0000 −1.71522
$$518$$ −16.0000 −0.703000
$$519$$ −6.00000 −0.263371
$$520$$ 1.00000 0.0438529
$$521$$ −13.0000 −0.569540 −0.284770 0.958596i $$-0.591917\pi$$
−0.284770 + 0.958596i $$0.591917\pi$$
$$522$$ 0 0
$$523$$ 24.0000 1.04945 0.524723 0.851273i $$-0.324169\pi$$
0.524723 + 0.851273i $$0.324169\pi$$
$$524$$ −12.0000 −0.524222
$$525$$ 8.00000 0.349149
$$526$$ 9.00000 0.392419
$$527$$ −24.0000 −1.04546
$$528$$ 3.00000 0.130558
$$529$$ −7.00000 −0.304348
$$530$$ 11.0000 0.477809
$$531$$ 0 0
$$532$$ 0 0
$$533$$ 2.00000 0.0866296
$$534$$ −10.0000 −0.432742
$$535$$ −2.00000 −0.0864675
$$536$$ 12.0000 0.518321
$$537$$ −10.0000 −0.431532
$$538$$ 0 0
$$539$$ −9.00000 −0.387657
$$540$$ −5.00000 −0.215166
$$541$$ 8.00000 0.343947 0.171973 0.985102i $$-0.444986\pi$$
0.171973 + 0.985102i $$0.444986\pi$$
$$542$$ −13.0000 −0.558398
$$543$$ 7.00000 0.300399
$$544$$ 8.00000 0.342997
$$545$$ 5.00000 0.214176
$$546$$ −2.00000 −0.0855921
$$547$$ 38.0000 1.62476 0.812381 0.583127i $$-0.198171\pi$$
0.812381 + 0.583127i $$0.198171\pi$$
$$548$$ 12.0000 0.512615
$$549$$ −16.0000 −0.682863
$$550$$ 12.0000 0.511682
$$551$$ 0 0
$$552$$ −4.00000 −0.170251
$$553$$ 30.0000 1.27573
$$554$$ 2.00000 0.0849719
$$555$$ −8.00000 −0.339581
$$556$$ −20.0000 −0.848189
$$557$$ −2.00000 −0.0847427 −0.0423714 0.999102i $$-0.513491\pi$$
−0.0423714 + 0.999102i $$0.513491\pi$$
$$558$$ 6.00000 0.254000
$$559$$ −11.0000 −0.465250
$$560$$ −2.00000 −0.0845154
$$561$$ −24.0000 −1.01328
$$562$$ −27.0000 −1.13893
$$563$$ 11.0000 0.463595 0.231797 0.972764i $$-0.425539\pi$$
0.231797 + 0.972764i $$0.425539\pi$$
$$564$$ −13.0000 −0.547399
$$565$$ 6.00000 0.252422
$$566$$ −4.00000 −0.168133
$$567$$ −2.00000 −0.0839921
$$568$$ −2.00000 −0.0839181
$$569$$ 30.0000 1.25767 0.628833 0.777541i $$-0.283533\pi$$
0.628833 + 0.777541i $$0.283533\pi$$
$$570$$ 0 0
$$571$$ −28.0000 −1.17176 −0.585882 0.810397i $$-0.699252\pi$$
−0.585882 + 0.810397i $$0.699252\pi$$
$$572$$ −3.00000 −0.125436
$$573$$ 8.00000 0.334205
$$574$$ −4.00000 −0.166957
$$575$$ −16.0000 −0.667246
$$576$$ −2.00000 −0.0833333
$$577$$ −8.00000 −0.333044 −0.166522 0.986038i $$-0.553254\pi$$
−0.166522 + 0.986038i $$0.553254\pi$$
$$578$$ −47.0000 −1.95494
$$579$$ −14.0000 −0.581820
$$580$$ 0 0
$$581$$ −8.00000 −0.331896
$$582$$ −2.00000 −0.0829027
$$583$$ −33.0000 −1.36672
$$584$$ 4.00000 0.165521
$$585$$ 2.00000 0.0826898
$$586$$ 14.0000 0.578335
$$587$$ 28.0000 1.15568 0.577842 0.816149i $$-0.303895\pi$$
0.577842 + 0.816149i $$0.303895\pi$$
$$588$$ −3.00000 −0.123718
$$589$$ 0 0
$$590$$ 0 0
$$591$$ 18.0000 0.740421
$$592$$ −8.00000 −0.328798
$$593$$ 39.0000 1.60154 0.800769 0.598973i $$-0.204424\pi$$
0.800769 + 0.598973i $$0.204424\pi$$
$$594$$ 15.0000 0.615457
$$595$$ 16.0000 0.655936
$$596$$ 15.0000 0.614424
$$597$$ −10.0000 −0.409273
$$598$$ 4.00000 0.163572
$$599$$ 5.00000 0.204294 0.102147 0.994769i $$-0.467429\pi$$
0.102147 + 0.994769i $$0.467429\pi$$
$$600$$ 4.00000 0.163299
$$601$$ −2.00000 −0.0815817 −0.0407909 0.999168i $$-0.512988\pi$$
−0.0407909 + 0.999168i $$0.512988\pi$$
$$602$$ 22.0000 0.896653
$$603$$ 24.0000 0.977356
$$604$$ 2.00000 0.0813788
$$605$$ −2.00000 −0.0813116
$$606$$ −8.00000 −0.324978
$$607$$ −3.00000 −0.121766 −0.0608831 0.998145i $$-0.519392\pi$$
−0.0608831 + 0.998145i $$0.519392\pi$$
$$608$$ 0 0
$$609$$ 0 0
$$610$$ −8.00000 −0.323911
$$611$$ 13.0000 0.525924
$$612$$ 16.0000 0.646762
$$613$$ −31.0000 −1.25208 −0.626039 0.779792i $$-0.715325\pi$$
−0.626039 + 0.779792i $$0.715325\pi$$
$$614$$ −7.00000 −0.282497
$$615$$ −2.00000 −0.0806478
$$616$$ 6.00000 0.241747
$$617$$ 12.0000 0.483102 0.241551 0.970388i $$-0.422344\pi$$
0.241551 + 0.970388i $$0.422344\pi$$
$$618$$ −14.0000 −0.563163
$$619$$ 35.0000 1.40677 0.703384 0.710810i $$-0.251671\pi$$
0.703384 + 0.710810i $$0.251671\pi$$
$$620$$ 3.00000 0.120483
$$621$$ −20.0000 −0.802572
$$622$$ −8.00000 −0.320771
$$623$$ −20.0000 −0.801283
$$624$$ −1.00000 −0.0400320
$$625$$ 11.0000 0.440000
$$626$$ −9.00000 −0.359712
$$627$$ 0 0
$$628$$ −18.0000 −0.718278
$$629$$ 64.0000 2.55185
$$630$$ −4.00000 −0.159364
$$631$$ −38.0000 −1.51276 −0.756378 0.654135i $$-0.773033\pi$$
−0.756378 + 0.654135i $$0.773033\pi$$
$$632$$ 15.0000 0.596668
$$633$$ 3.00000 0.119239
$$634$$ −12.0000 −0.476581
$$635$$ −8.00000 −0.317470
$$636$$ −11.0000 −0.436178
$$637$$ 3.00000 0.118864
$$638$$ 0 0
$$639$$ −4.00000 −0.158238
$$640$$ −1.00000 −0.0395285
$$641$$ 8.00000 0.315981 0.157991 0.987441i $$-0.449498\pi$$
0.157991 + 0.987441i $$0.449498\pi$$
$$642$$ 2.00000 0.0789337
$$643$$ 34.0000 1.34083 0.670415 0.741987i $$-0.266116\pi$$
0.670415 + 0.741987i $$0.266116\pi$$
$$644$$ −8.00000 −0.315244
$$645$$ 11.0000 0.433125
$$646$$ 0 0
$$647$$ −32.0000 −1.25805 −0.629025 0.777385i $$-0.716546\pi$$
−0.629025 + 0.777385i $$0.716546\pi$$
$$648$$ −1.00000 −0.0392837
$$649$$ 0 0
$$650$$ −4.00000 −0.156893
$$651$$ −6.00000 −0.235159
$$652$$ −9.00000 −0.352467
$$653$$ 26.0000 1.01746 0.508729 0.860927i $$-0.330115\pi$$
0.508729 + 0.860927i $$0.330115\pi$$
$$654$$ −5.00000 −0.195515
$$655$$ −12.0000 −0.468879
$$656$$ −2.00000 −0.0780869
$$657$$ 8.00000 0.312110
$$658$$ −26.0000 −1.01359
$$659$$ −15.0000 −0.584317 −0.292159 0.956370i $$-0.594373\pi$$
−0.292159 + 0.956370i $$0.594373\pi$$
$$660$$ 3.00000 0.116775
$$661$$ −18.0000 −0.700119 −0.350059 0.936727i $$-0.613839\pi$$
−0.350059 + 0.936727i $$0.613839\pi$$
$$662$$ −23.0000 −0.893920
$$663$$ 8.00000 0.310694
$$664$$ −4.00000 −0.155230
$$665$$ 0 0
$$666$$ −16.0000 −0.619987
$$667$$ 0 0
$$668$$ −2.00000 −0.0773823
$$669$$ −26.0000 −1.00522
$$670$$ 12.0000 0.463600
$$671$$ 24.0000 0.926510
$$672$$ 2.00000 0.0771517
$$673$$ 9.00000 0.346925 0.173462 0.984841i $$-0.444505\pi$$
0.173462 + 0.984841i $$0.444505\pi$$
$$674$$ −32.0000 −1.23259
$$675$$ 20.0000 0.769800
$$676$$ −12.0000 −0.461538
$$677$$ −38.0000 −1.46046 −0.730229 0.683202i $$-0.760587\pi$$
−0.730229 + 0.683202i $$0.760587\pi$$
$$678$$ −6.00000 −0.230429
$$679$$ −4.00000 −0.153506
$$680$$ 8.00000 0.306786
$$681$$ 18.0000 0.689761
$$682$$ −9.00000 −0.344628
$$683$$ 24.0000 0.918334 0.459167 0.888350i $$-0.348148\pi$$
0.459167 + 0.888350i $$0.348148\pi$$
$$684$$ 0 0
$$685$$ 12.0000 0.458496
$$686$$ −20.0000 −0.763604
$$687$$ −10.0000 −0.381524
$$688$$ 11.0000 0.419371
$$689$$ 11.0000 0.419067
$$690$$ −4.00000 −0.152277
$$691$$ −28.0000 −1.06517 −0.532585 0.846376i $$-0.678779\pi$$
−0.532585 + 0.846376i $$0.678779\pi$$
$$692$$ −6.00000 −0.228086
$$693$$ 12.0000 0.455842
$$694$$ 2.00000 0.0759190
$$695$$ −20.0000 −0.758643
$$696$$ 0 0
$$697$$ 16.0000 0.606043
$$698$$ 15.0000 0.567758
$$699$$ −1.00000 −0.0378235
$$700$$ 8.00000 0.302372
$$701$$ 27.0000 1.01978 0.509888 0.860241i $$-0.329687\pi$$
0.509888 + 0.860241i $$0.329687\pi$$
$$702$$ −5.00000 −0.188713
$$703$$ 0 0
$$704$$ 3.00000 0.113067
$$705$$ −13.0000 −0.489608
$$706$$ 26.0000 0.978523
$$707$$ −16.0000 −0.601742
$$708$$ 0 0
$$709$$ 15.0000 0.563337 0.281668 0.959512i $$-0.409112\pi$$
0.281668 + 0.959512i $$0.409112\pi$$
$$710$$ −2.00000 −0.0750587
$$711$$ 30.0000 1.12509
$$712$$ −10.0000 −0.374766
$$713$$ 12.0000 0.449404
$$714$$ −16.0000 −0.598785
$$715$$ −3.00000 −0.112194
$$716$$ −10.0000 −0.373718
$$717$$ 0 0
$$718$$ −25.0000 −0.932992
$$719$$ −50.0000 −1.86469 −0.932343 0.361576i $$-0.882239\pi$$
−0.932343 + 0.361576i $$0.882239\pi$$
$$720$$ −2.00000 −0.0745356
$$721$$ −28.0000 −1.04277
$$722$$ 19.0000 0.707107
$$723$$ 17.0000 0.632237
$$724$$ 7.00000 0.260153
$$725$$ 0 0
$$726$$ 2.00000 0.0742270
$$727$$ −8.00000 −0.296704 −0.148352 0.988935i $$-0.547397\pi$$
−0.148352 + 0.988935i $$0.547397\pi$$
$$728$$ −2.00000 −0.0741249
$$729$$ 13.0000 0.481481
$$730$$ 4.00000 0.148047
$$731$$ −88.0000 −3.25480
$$732$$ 8.00000 0.295689
$$733$$ −24.0000 −0.886460 −0.443230 0.896408i $$-0.646168\pi$$
−0.443230 + 0.896408i $$0.646168\pi$$
$$734$$ −32.0000 −1.18114
$$735$$ −3.00000 −0.110657
$$736$$ −4.00000 −0.147442
$$737$$ −36.0000 −1.32608
$$738$$ −4.00000 −0.147242
$$739$$ 5.00000 0.183928 0.0919640 0.995762i $$-0.470686\pi$$
0.0919640 + 0.995762i $$0.470686\pi$$
$$740$$ −8.00000 −0.294086
$$741$$ 0 0
$$742$$ −22.0000 −0.807645
$$743$$ −44.0000 −1.61420 −0.807102 0.590412i $$-0.798965\pi$$
−0.807102 + 0.590412i $$0.798965\pi$$
$$744$$ −3.00000 −0.109985
$$745$$ 15.0000 0.549557
$$746$$ 21.0000 0.768865
$$747$$ −8.00000 −0.292705
$$748$$ −24.0000 −0.877527
$$749$$ 4.00000 0.146157
$$750$$ 9.00000 0.328634
$$751$$ −32.0000 −1.16770 −0.583848 0.811863i $$-0.698454\pi$$
−0.583848 + 0.811863i $$0.698454\pi$$
$$752$$ −13.0000 −0.474061
$$753$$ −27.0000 −0.983935
$$754$$ 0 0
$$755$$ 2.00000 0.0727875
$$756$$ 10.0000 0.363696
$$757$$ −8.00000 −0.290765 −0.145382 0.989376i $$-0.546441\pi$$
−0.145382 + 0.989376i $$0.546441\pi$$
$$758$$ 20.0000 0.726433
$$759$$ 12.0000 0.435572
$$760$$ 0 0
$$761$$ 42.0000 1.52250 0.761249 0.648459i $$-0.224586\pi$$
0.761249 + 0.648459i $$0.224586\pi$$
$$762$$ 8.00000 0.289809
$$763$$ −10.0000 −0.362024
$$764$$ 8.00000 0.289430
$$765$$ 16.0000 0.578481
$$766$$ −14.0000 −0.505841
$$767$$ 0 0
$$768$$ 1.00000 0.0360844
$$769$$ 20.0000 0.721218 0.360609 0.932717i $$-0.382569\pi$$
0.360609 + 0.932717i $$0.382569\pi$$
$$770$$ 6.00000 0.216225
$$771$$ 13.0000 0.468184
$$772$$ −14.0000 −0.503871
$$773$$ −14.0000 −0.503545 −0.251773 0.967786i $$-0.581013\pi$$
−0.251773 + 0.967786i $$0.581013\pi$$
$$774$$ 22.0000 0.790774
$$775$$ −12.0000 −0.431053
$$776$$ −2.00000 −0.0717958
$$777$$ 16.0000 0.573997
$$778$$ 0 0
$$779$$ 0 0
$$780$$ −1.00000 −0.0358057
$$781$$ 6.00000 0.214697
$$782$$ 32.0000 1.14432
$$783$$ 0 0
$$784$$ −3.00000 −0.107143
$$785$$ −18.0000 −0.642448
$$786$$ 12.0000 0.428026
$$787$$ −22.0000 −0.784215 −0.392108 0.919919i $$-0.628254\pi$$
−0.392108 + 0.919919i $$0.628254\pi$$
$$788$$ 18.0000 0.641223
$$789$$ −9.00000 −0.320408
$$790$$ 15.0000 0.533676
$$791$$ −12.0000 −0.426671
$$792$$ 6.00000 0.213201
$$793$$ −8.00000 −0.284088
$$794$$ 17.0000 0.603307
$$795$$ −11.0000 −0.390130
$$796$$ −10.0000 −0.354441
$$797$$ 32.0000 1.13350 0.566749 0.823890i $$-0.308201\pi$$
0.566749 + 0.823890i $$0.308201\pi$$
$$798$$ 0 0
$$799$$ 104.000 3.67926
$$800$$ 4.00000 0.141421
$$801$$ −20.0000 −0.706665
$$802$$ −27.0000 −0.953403
$$803$$ −12.0000 −0.423471
$$804$$ −12.0000 −0.423207
$$805$$ −8.00000 −0.281963
$$806$$ 3.00000 0.105670
$$807$$ 0 0
$$808$$ −8.00000 −0.281439
$$809$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$810$$ −1.00000 −0.0351364
$$811$$ −18.0000 −0.632065 −0.316033 0.948748i $$-0.602351\pi$$
−0.316033 + 0.948748i $$0.602351\pi$$
$$812$$ 0 0
$$813$$ 13.0000 0.455930
$$814$$ 24.0000 0.841200
$$815$$ −9.00000 −0.315256
$$816$$ −8.00000 −0.280056
$$817$$ 0 0
$$818$$ 30.0000 1.04893
$$819$$ −4.00000 −0.139771
$$820$$ −2.00000 −0.0698430
$$821$$ −33.0000 −1.15171 −0.575854 0.817553i $$-0.695330\pi$$
−0.575854 + 0.817553i $$0.695330\pi$$
$$822$$ −12.0000 −0.418548
$$823$$ 16.0000 0.557725 0.278862 0.960331i $$-0.410043\pi$$
0.278862 + 0.960331i $$0.410043\pi$$
$$824$$ −14.0000 −0.487713
$$825$$ −12.0000 −0.417786
$$826$$ 0 0
$$827$$ −13.0000 −0.452054 −0.226027 0.974121i $$-0.572574\pi$$
−0.226027 + 0.974121i $$0.572574\pi$$
$$828$$ −8.00000 −0.278019
$$829$$ −40.0000 −1.38926 −0.694629 0.719368i $$-0.744431\pi$$
−0.694629 + 0.719368i $$0.744431\pi$$
$$830$$ −4.00000 −0.138842
$$831$$ −2.00000 −0.0693792
$$832$$ −1.00000 −0.0346688
$$833$$ 24.0000 0.831551
$$834$$ 20.0000 0.692543
$$835$$ −2.00000 −0.0692129
$$836$$ 0 0
$$837$$ −15.0000 −0.518476
$$838$$ 10.0000 0.345444
$$839$$ −45.0000 −1.55357 −0.776786 0.629764i $$-0.783151\pi$$
−0.776786 + 0.629764i $$0.783151\pi$$
$$840$$ 2.00000 0.0690066
$$841$$ 0 0
$$842$$ 32.0000 1.10279
$$843$$ 27.0000 0.929929
$$844$$ 3.00000 0.103264
$$845$$ −12.0000 −0.412813
$$846$$ −26.0000 −0.893898
$$847$$ 4.00000 0.137442
$$848$$ −11.0000 −0.377742
$$849$$ 4.00000 0.137280
$$850$$ −32.0000 −1.09759
$$851$$ −32.0000 −1.09695
$$852$$ 2.00000 0.0685189
$$853$$ −14.0000 −0.479351 −0.239675 0.970853i $$-0.577041\pi$$
−0.239675 + 0.970853i $$0.577041\pi$$
$$854$$ 16.0000 0.547509
$$855$$ 0 0
$$856$$ 2.00000 0.0683586
$$857$$ −27.0000 −0.922302 −0.461151 0.887322i $$-0.652563\pi$$
−0.461151 + 0.887322i $$0.652563\pi$$
$$858$$ 3.00000 0.102418
$$859$$ 25.0000 0.852989 0.426494 0.904490i $$-0.359748\pi$$
0.426494 + 0.904490i $$0.359748\pi$$
$$860$$ 11.0000 0.375097
$$861$$ 4.00000 0.136320
$$862$$ −32.0000 −1.08992
$$863$$ −46.0000 −1.56586 −0.782929 0.622111i $$-0.786275\pi$$
−0.782929 + 0.622111i $$0.786275\pi$$
$$864$$ 5.00000 0.170103
$$865$$ −6.00000 −0.204006
$$866$$ −16.0000 −0.543702
$$867$$ 47.0000 1.59620
$$868$$ −6.00000 −0.203653
$$869$$ −45.0000 −1.52652
$$870$$ 0 0
$$871$$ 12.0000 0.406604
$$872$$ −5.00000 −0.169321
$$873$$ −4.00000 −0.135379
$$874$$ 0 0
$$875$$ 18.0000 0.608511
$$876$$ −4.00000 −0.135147
$$877$$ 13.0000 0.438979 0.219489 0.975615i $$-0.429561\pi$$
0.219489 + 0.975615i $$0.429561\pi$$
$$878$$ −20.0000 −0.674967
$$879$$ −14.0000 −0.472208
$$880$$ 3.00000 0.101130
$$881$$ −42.0000 −1.41502 −0.707508 0.706705i $$-0.750181\pi$$
−0.707508 + 0.706705i $$0.750181\pi$$
$$882$$ −6.00000 −0.202031
$$883$$ −26.0000 −0.874970 −0.437485 0.899226i $$-0.644131\pi$$
−0.437485 + 0.899226i $$0.644131\pi$$
$$884$$ 8.00000 0.269069
$$885$$ 0 0
$$886$$ 4.00000 0.134383
$$887$$ −33.0000 −1.10803 −0.554016 0.832506i $$-0.686905\pi$$
−0.554016 + 0.832506i $$0.686905\pi$$
$$888$$ 8.00000 0.268462
$$889$$ 16.0000 0.536623
$$890$$ −10.0000 −0.335201
$$891$$ 3.00000 0.100504
$$892$$ −26.0000 −0.870544
$$893$$ 0 0
$$894$$ −15.0000 −0.501675
$$895$$ −10.0000 −0.334263
$$896$$ 2.00000 0.0668153
$$897$$ −4.00000 −0.133556
$$898$$ −10.0000 −0.333704
$$899$$ 0 0
$$900$$ 8.00000 0.266667
$$901$$ 88.0000 2.93171
$$902$$ 6.00000 0.199778
$$903$$ −22.0000 −0.732114
$$904$$ −6.00000 −0.199557
$$905$$ 7.00000 0.232688
$$906$$ −2.00000 −0.0664455
$$907$$ −28.0000 −0.929725 −0.464862 0.885383i $$-0.653896\pi$$
−0.464862 + 0.885383i $$0.653896\pi$$
$$908$$ 18.0000 0.597351
$$909$$ −16.0000 −0.530687
$$910$$ −2.00000 −0.0662994
$$911$$ 13.0000 0.430709 0.215355 0.976536i $$-0.430909\pi$$
0.215355 + 0.976536i $$0.430909\pi$$
$$912$$ 0 0
$$913$$ 12.0000 0.397142
$$914$$ 2.00000 0.0661541
$$915$$ 8.00000 0.264472
$$916$$ −10.0000 −0.330409
$$917$$ 24.0000 0.792550
$$918$$ −40.0000 −1.32020
$$919$$ 30.0000 0.989609 0.494804 0.869004i $$-0.335240\pi$$
0.494804 + 0.869004i $$0.335240\pi$$
$$920$$ −4.00000 −0.131876
$$921$$ 7.00000 0.230658
$$922$$ 2.00000 0.0658665
$$923$$ −2.00000 −0.0658308
$$924$$ −6.00000 −0.197386
$$925$$ 32.0000 1.05215
$$926$$ −4.00000 −0.131448
$$927$$ −28.0000 −0.919641
$$928$$ 0 0
$$929$$ −30.0000 −0.984268 −0.492134 0.870519i $$-0.663783\pi$$
−0.492134 + 0.870519i $$0.663783\pi$$
$$930$$ −3.00000 −0.0983739
$$931$$ 0 0
$$932$$ −1.00000 −0.0327561
$$933$$ 8.00000 0.261908
$$934$$ −27.0000 −0.883467
$$935$$ −24.0000 −0.784884
$$936$$ −2.00000 −0.0653720
$$937$$ −2.00000 −0.0653372 −0.0326686 0.999466i $$-0.510401\pi$$
−0.0326686 + 0.999466i $$0.510401\pi$$
$$938$$ −24.0000 −0.783628
$$939$$ 9.00000 0.293704
$$940$$ −13.0000 −0.424013
$$941$$ 37.0000 1.20617 0.603083 0.797679i $$-0.293939\pi$$
0.603083 + 0.797679i $$0.293939\pi$$
$$942$$ 18.0000 0.586472
$$943$$ −8.00000 −0.260516
$$944$$ 0 0
$$945$$ 10.0000 0.325300
$$946$$ −33.0000 −1.07292
$$947$$ −33.0000 −1.07236 −0.536178 0.844105i $$-0.680132\pi$$
−0.536178 + 0.844105i $$0.680132\pi$$
$$948$$ −15.0000 −0.487177
$$949$$ 4.00000 0.129845
$$950$$ 0 0
$$951$$ 12.0000 0.389127
$$952$$ −16.0000 −0.518563
$$953$$ −1.00000 −0.0323932 −0.0161966 0.999869i $$-0.505156\pi$$
−0.0161966 + 0.999869i $$0.505156\pi$$
$$954$$ −22.0000 −0.712276
$$955$$ 8.00000 0.258874
$$956$$ 0 0
$$957$$ 0 0
$$958$$ −5.00000 −0.161543
$$959$$ −24.0000 −0.775000
$$960$$ 1.00000 0.0322749
$$961$$ −22.0000 −0.709677
$$962$$ −8.00000 −0.257930
$$963$$ 4.00000 0.128898
$$964$$ 17.0000 0.547533
$$965$$ −14.0000 −0.450676
$$966$$ 8.00000 0.257396
$$967$$ −13.0000 −0.418052 −0.209026 0.977910i $$-0.567029\pi$$
−0.209026 + 0.977910i $$0.567029\pi$$
$$968$$ 2.00000 0.0642824
$$969$$ 0 0
$$970$$ −2.00000 −0.0642161
$$971$$ 28.0000 0.898563 0.449281 0.893390i $$-0.351680\pi$$
0.449281 + 0.893390i $$0.351680\pi$$
$$972$$ 16.0000 0.513200
$$973$$ 40.0000 1.28234
$$974$$ 22.0000 0.704925
$$975$$ 4.00000 0.128103
$$976$$ 8.00000 0.256074
$$977$$ 13.0000 0.415907 0.207953 0.978139i $$-0.433320\pi$$
0.207953 + 0.978139i $$0.433320\pi$$
$$978$$ 9.00000 0.287788
$$979$$ 30.0000 0.958804
$$980$$ −3.00000 −0.0958315
$$981$$ −10.0000 −0.319275
$$982$$ −33.0000 −1.05307
$$983$$ −49.0000 −1.56286 −0.781429 0.623995i $$-0.785509\pi$$
−0.781429 + 0.623995i $$0.785509\pi$$
$$984$$ 2.00000 0.0637577
$$985$$ 18.0000 0.573528
$$986$$ 0 0
$$987$$ 26.0000 0.827589
$$988$$ 0 0
$$989$$ 44.0000 1.39912
$$990$$ 6.00000 0.190693
$$991$$ 22.0000 0.698853 0.349427 0.936964i $$-0.386376\pi$$
0.349427 + 0.936964i $$0.386376\pi$$
$$992$$ −3.00000 −0.0952501
$$993$$ 23.0000 0.729883
$$994$$ 4.00000 0.126872
$$995$$ −10.0000 −0.317021
$$996$$ 4.00000 0.126745
$$997$$ −8.00000 −0.253363 −0.126681 0.991943i $$-0.540433\pi$$
−0.126681 + 0.991943i $$0.540433\pi$$
$$998$$ 20.0000 0.633089
$$999$$ 40.0000 1.26554
Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000

## Twists

By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 1682.2.a.d.1.1 1
29.12 odd 4 1682.2.b.a.1681.1 2
29.17 odd 4 1682.2.b.a.1681.2 2
29.28 even 2 58.2.a.b.1.1 1
87.86 odd 2 522.2.a.b.1.1 1
116.115 odd 2 464.2.a.e.1.1 1
145.28 odd 4 1450.2.b.b.349.1 2
145.57 odd 4 1450.2.b.b.349.2 2
145.144 even 2 1450.2.a.c.1.1 1
203.202 odd 2 2842.2.a.e.1.1 1
232.115 odd 2 1856.2.a.f.1.1 1
232.173 even 2 1856.2.a.k.1.1 1
319.318 odd 2 7018.2.a.a.1.1 1
348.347 even 2 4176.2.a.n.1.1 1
377.376 even 2 9802.2.a.a.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
58.2.a.b.1.1 1 29.28 even 2
464.2.a.e.1.1 1 116.115 odd 2
522.2.a.b.1.1 1 87.86 odd 2
1450.2.a.c.1.1 1 145.144 even 2
1450.2.b.b.349.1 2 145.28 odd 4
1450.2.b.b.349.2 2 145.57 odd 4
1682.2.a.d.1.1 1 1.1 even 1 trivial
1682.2.b.a.1681.1 2 29.12 odd 4
1682.2.b.a.1681.2 2 29.17 odd 4
1856.2.a.f.1.1 1 232.115 odd 2
1856.2.a.k.1.1 1 232.173 even 2
2842.2.a.e.1.1 1 203.202 odd 2
4176.2.a.n.1.1 1 348.347 even 2
7018.2.a.a.1.1 1 319.318 odd 2
9802.2.a.a.1.1 1 377.376 even 2