# Properties

 Label 195.2.a.b.1.1 Level $195$ Weight $2$ Character 195.1 Self dual yes Analytic conductor $1.557$ Analytic rank $0$ Dimension $1$ CM no Inner twists $1$

# Learn more

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [195,2,Mod(1,195)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(195, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("195.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$195 = 3 \cdot 5 \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 195.a (trivial)

## Newform invariants

comment: select newform

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

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

 Self dual: yes Analytic conductor: $$1.55708283941$$ 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 $$\chi$$ $$=$$ 195.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+2.00000 q^{2} -1.00000 q^{3} +2.00000 q^{4} +1.00000 q^{5} -2.00000 q^{6} +3.00000 q^{7} +1.00000 q^{9} +O(q^{10})$$ $$q+2.00000 q^{2} -1.00000 q^{3} +2.00000 q^{4} +1.00000 q^{5} -2.00000 q^{6} +3.00000 q^{7} +1.00000 q^{9} +2.00000 q^{10} -1.00000 q^{11} -2.00000 q^{12} -1.00000 q^{13} +6.00000 q^{14} -1.00000 q^{15} -4.00000 q^{16} -1.00000 q^{17} +2.00000 q^{18} -2.00000 q^{19} +2.00000 q^{20} -3.00000 q^{21} -2.00000 q^{22} -3.00000 q^{23} +1.00000 q^{25} -2.00000 q^{26} -1.00000 q^{27} +6.00000 q^{28} -2.00000 q^{29} -2.00000 q^{30} -6.00000 q^{31} -8.00000 q^{32} +1.00000 q^{33} -2.00000 q^{34} +3.00000 q^{35} +2.00000 q^{36} +11.0000 q^{37} -4.00000 q^{38} +1.00000 q^{39} -5.00000 q^{41} -6.00000 q^{42} +4.00000 q^{43} -2.00000 q^{44} +1.00000 q^{45} -6.00000 q^{46} -10.0000 q^{47} +4.00000 q^{48} +2.00000 q^{49} +2.00000 q^{50} +1.00000 q^{51} -2.00000 q^{52} +11.0000 q^{53} -2.00000 q^{54} -1.00000 q^{55} +2.00000 q^{57} -4.00000 q^{58} +8.00000 q^{59} -2.00000 q^{60} +13.0000 q^{61} -12.0000 q^{62} +3.00000 q^{63} -8.00000 q^{64} -1.00000 q^{65} +2.00000 q^{66} +12.0000 q^{67} -2.00000 q^{68} +3.00000 q^{69} +6.00000 q^{70} -5.00000 q^{71} +10.0000 q^{73} +22.0000 q^{74} -1.00000 q^{75} -4.00000 q^{76} -3.00000 q^{77} +2.00000 q^{78} -3.00000 q^{79} -4.00000 q^{80} +1.00000 q^{81} -10.0000 q^{82} -12.0000 q^{83} -6.00000 q^{84} -1.00000 q^{85} +8.00000 q^{86} +2.00000 q^{87} -15.0000 q^{89} +2.00000 q^{90} -3.00000 q^{91} -6.00000 q^{92} +6.00000 q^{93} -20.0000 q^{94} -2.00000 q^{95} +8.00000 q^{96} +17.0000 q^{97} +4.00000 q^{98} -1.00000 q^{99} +O(q^{100})$$

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 2.00000 1.41421 0.707107 0.707107i $$-0.250000\pi$$
0.707107 + 0.707107i $$0.250000\pi$$
$$3$$ −1.00000 −0.577350
$$4$$ 2.00000 1.00000
$$5$$ 1.00000 0.447214
$$6$$ −2.00000 −0.816497
$$7$$ 3.00000 1.13389 0.566947 0.823754i $$-0.308125\pi$$
0.566947 + 0.823754i $$0.308125\pi$$
$$8$$ 0 0
$$9$$ 1.00000 0.333333
$$10$$ 2.00000 0.632456
$$11$$ −1.00000 −0.301511 −0.150756 0.988571i $$-0.548171\pi$$
−0.150756 + 0.988571i $$0.548171\pi$$
$$12$$ −2.00000 −0.577350
$$13$$ −1.00000 −0.277350
$$14$$ 6.00000 1.60357
$$15$$ −1.00000 −0.258199
$$16$$ −4.00000 −1.00000
$$17$$ −1.00000 −0.242536 −0.121268 0.992620i $$-0.538696\pi$$
−0.121268 + 0.992620i $$0.538696\pi$$
$$18$$ 2.00000 0.471405
$$19$$ −2.00000 −0.458831 −0.229416 0.973329i $$-0.573682\pi$$
−0.229416 + 0.973329i $$0.573682\pi$$
$$20$$ 2.00000 0.447214
$$21$$ −3.00000 −0.654654
$$22$$ −2.00000 −0.426401
$$23$$ −3.00000 −0.625543 −0.312772 0.949828i $$-0.601257\pi$$
−0.312772 + 0.949828i $$0.601257\pi$$
$$24$$ 0 0
$$25$$ 1.00000 0.200000
$$26$$ −2.00000 −0.392232
$$27$$ −1.00000 −0.192450
$$28$$ 6.00000 1.13389
$$29$$ −2.00000 −0.371391 −0.185695 0.982607i $$-0.559454\pi$$
−0.185695 + 0.982607i $$0.559454\pi$$
$$30$$ −2.00000 −0.365148
$$31$$ −6.00000 −1.07763 −0.538816 0.842424i $$-0.681128\pi$$
−0.538816 + 0.842424i $$0.681128\pi$$
$$32$$ −8.00000 −1.41421
$$33$$ 1.00000 0.174078
$$34$$ −2.00000 −0.342997
$$35$$ 3.00000 0.507093
$$36$$ 2.00000 0.333333
$$37$$ 11.0000 1.80839 0.904194 0.427121i $$-0.140472\pi$$
0.904194 + 0.427121i $$0.140472\pi$$
$$38$$ −4.00000 −0.648886
$$39$$ 1.00000 0.160128
$$40$$ 0 0
$$41$$ −5.00000 −0.780869 −0.390434 0.920631i $$-0.627675\pi$$
−0.390434 + 0.920631i $$0.627675\pi$$
$$42$$ −6.00000 −0.925820
$$43$$ 4.00000 0.609994 0.304997 0.952353i $$-0.401344\pi$$
0.304997 + 0.952353i $$0.401344\pi$$
$$44$$ −2.00000 −0.301511
$$45$$ 1.00000 0.149071
$$46$$ −6.00000 −0.884652
$$47$$ −10.0000 −1.45865 −0.729325 0.684167i $$-0.760166\pi$$
−0.729325 + 0.684167i $$0.760166\pi$$
$$48$$ 4.00000 0.577350
$$49$$ 2.00000 0.285714
$$50$$ 2.00000 0.282843
$$51$$ 1.00000 0.140028
$$52$$ −2.00000 −0.277350
$$53$$ 11.0000 1.51097 0.755483 0.655168i $$-0.227402\pi$$
0.755483 + 0.655168i $$0.227402\pi$$
$$54$$ −2.00000 −0.272166
$$55$$ −1.00000 −0.134840
$$56$$ 0 0
$$57$$ 2.00000 0.264906
$$58$$ −4.00000 −0.525226
$$59$$ 8.00000 1.04151 0.520756 0.853706i $$-0.325650\pi$$
0.520756 + 0.853706i $$0.325650\pi$$
$$60$$ −2.00000 −0.258199
$$61$$ 13.0000 1.66448 0.832240 0.554416i $$-0.187058\pi$$
0.832240 + 0.554416i $$0.187058\pi$$
$$62$$ −12.0000 −1.52400
$$63$$ 3.00000 0.377964
$$64$$ −8.00000 −1.00000
$$65$$ −1.00000 −0.124035
$$66$$ 2.00000 0.246183
$$67$$ 12.0000 1.46603 0.733017 0.680211i $$-0.238112\pi$$
0.733017 + 0.680211i $$0.238112\pi$$
$$68$$ −2.00000 −0.242536
$$69$$ 3.00000 0.361158
$$70$$ 6.00000 0.717137
$$71$$ −5.00000 −0.593391 −0.296695 0.954972i $$-0.595885\pi$$
−0.296695 + 0.954972i $$0.595885\pi$$
$$72$$ 0 0
$$73$$ 10.0000 1.17041 0.585206 0.810885i $$-0.301014\pi$$
0.585206 + 0.810885i $$0.301014\pi$$
$$74$$ 22.0000 2.55745
$$75$$ −1.00000 −0.115470
$$76$$ −4.00000 −0.458831
$$77$$ −3.00000 −0.341882
$$78$$ 2.00000 0.226455
$$79$$ −3.00000 −0.337526 −0.168763 0.985657i $$-0.553977\pi$$
−0.168763 + 0.985657i $$0.553977\pi$$
$$80$$ −4.00000 −0.447214
$$81$$ 1.00000 0.111111
$$82$$ −10.0000 −1.10432
$$83$$ −12.0000 −1.31717 −0.658586 0.752506i $$-0.728845\pi$$
−0.658586 + 0.752506i $$0.728845\pi$$
$$84$$ −6.00000 −0.654654
$$85$$ −1.00000 −0.108465
$$86$$ 8.00000 0.862662
$$87$$ 2.00000 0.214423
$$88$$ 0 0
$$89$$ −15.0000 −1.59000 −0.794998 0.606612i $$-0.792528\pi$$
−0.794998 + 0.606612i $$0.792528\pi$$
$$90$$ 2.00000 0.210819
$$91$$ −3.00000 −0.314485
$$92$$ −6.00000 −0.625543
$$93$$ 6.00000 0.622171
$$94$$ −20.0000 −2.06284
$$95$$ −2.00000 −0.205196
$$96$$ 8.00000 0.816497
$$97$$ 17.0000 1.72609 0.863044 0.505128i $$-0.168555\pi$$
0.863044 + 0.505128i $$0.168555\pi$$
$$98$$ 4.00000 0.404061
$$99$$ −1.00000 −0.100504
$$100$$ 2.00000 0.200000
$$101$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$102$$ 2.00000 0.198030
$$103$$ −16.0000 −1.57653 −0.788263 0.615338i $$-0.789020\pi$$
−0.788263 + 0.615338i $$0.789020\pi$$
$$104$$ 0 0
$$105$$ −3.00000 −0.292770
$$106$$ 22.0000 2.13683
$$107$$ 9.00000 0.870063 0.435031 0.900415i $$-0.356737\pi$$
0.435031 + 0.900415i $$0.356737\pi$$
$$108$$ −2.00000 −0.192450
$$109$$ −16.0000 −1.53252 −0.766261 0.642529i $$-0.777885\pi$$
−0.766261 + 0.642529i $$0.777885\pi$$
$$110$$ −2.00000 −0.190693
$$111$$ −11.0000 −1.04407
$$112$$ −12.0000 −1.13389
$$113$$ 14.0000 1.31701 0.658505 0.752577i $$-0.271189\pi$$
0.658505 + 0.752577i $$0.271189\pi$$
$$114$$ 4.00000 0.374634
$$115$$ −3.00000 −0.279751
$$116$$ −4.00000 −0.371391
$$117$$ −1.00000 −0.0924500
$$118$$ 16.0000 1.47292
$$119$$ −3.00000 −0.275010
$$120$$ 0 0
$$121$$ −10.0000 −0.909091
$$122$$ 26.0000 2.35393
$$123$$ 5.00000 0.450835
$$124$$ −12.0000 −1.07763
$$125$$ 1.00000 0.0894427
$$126$$ 6.00000 0.534522
$$127$$ 10.0000 0.887357 0.443678 0.896186i $$-0.353673\pi$$
0.443678 + 0.896186i $$0.353673\pi$$
$$128$$ 0 0
$$129$$ −4.00000 −0.352180
$$130$$ −2.00000 −0.175412
$$131$$ −6.00000 −0.524222 −0.262111 0.965038i $$-0.584419\pi$$
−0.262111 + 0.965038i $$0.584419\pi$$
$$132$$ 2.00000 0.174078
$$133$$ −6.00000 −0.520266
$$134$$ 24.0000 2.07328
$$135$$ −1.00000 −0.0860663
$$136$$ 0 0
$$137$$ 18.0000 1.53784 0.768922 0.639343i $$-0.220793\pi$$
0.768922 + 0.639343i $$0.220793\pi$$
$$138$$ 6.00000 0.510754
$$139$$ −1.00000 −0.0848189 −0.0424094 0.999100i $$-0.513503\pi$$
−0.0424094 + 0.999100i $$0.513503\pi$$
$$140$$ 6.00000 0.507093
$$141$$ 10.0000 0.842152
$$142$$ −10.0000 −0.839181
$$143$$ 1.00000 0.0836242
$$144$$ −4.00000 −0.333333
$$145$$ −2.00000 −0.166091
$$146$$ 20.0000 1.65521
$$147$$ −2.00000 −0.164957
$$148$$ 22.0000 1.80839
$$149$$ 13.0000 1.06500 0.532501 0.846430i $$-0.321252\pi$$
0.532501 + 0.846430i $$0.321252\pi$$
$$150$$ −2.00000 −0.163299
$$151$$ 16.0000 1.30206 0.651031 0.759051i $$-0.274337\pi$$
0.651031 + 0.759051i $$0.274337\pi$$
$$152$$ 0 0
$$153$$ −1.00000 −0.0808452
$$154$$ −6.00000 −0.483494
$$155$$ −6.00000 −0.481932
$$156$$ 2.00000 0.160128
$$157$$ −10.0000 −0.798087 −0.399043 0.916932i $$-0.630658\pi$$
−0.399043 + 0.916932i $$0.630658\pi$$
$$158$$ −6.00000 −0.477334
$$159$$ −11.0000 −0.872357
$$160$$ −8.00000 −0.632456
$$161$$ −9.00000 −0.709299
$$162$$ 2.00000 0.157135
$$163$$ −13.0000 −1.01824 −0.509119 0.860696i $$-0.670029\pi$$
−0.509119 + 0.860696i $$0.670029\pi$$
$$164$$ −10.0000 −0.780869
$$165$$ 1.00000 0.0778499
$$166$$ −24.0000 −1.86276
$$167$$ −12.0000 −0.928588 −0.464294 0.885681i $$-0.653692\pi$$
−0.464294 + 0.885681i $$0.653692\pi$$
$$168$$ 0 0
$$169$$ 1.00000 0.0769231
$$170$$ −2.00000 −0.153393
$$171$$ −2.00000 −0.152944
$$172$$ 8.00000 0.609994
$$173$$ −6.00000 −0.456172 −0.228086 0.973641i $$-0.573247\pi$$
−0.228086 + 0.973641i $$0.573247\pi$$
$$174$$ 4.00000 0.303239
$$175$$ 3.00000 0.226779
$$176$$ 4.00000 0.301511
$$177$$ −8.00000 −0.601317
$$178$$ −30.0000 −2.24860
$$179$$ 2.00000 0.149487 0.0747435 0.997203i $$-0.476186\pi$$
0.0747435 + 0.997203i $$0.476186\pi$$
$$180$$ 2.00000 0.149071
$$181$$ −7.00000 −0.520306 −0.260153 0.965567i $$-0.583773\pi$$
−0.260153 + 0.965567i $$0.583773\pi$$
$$182$$ −6.00000 −0.444750
$$183$$ −13.0000 −0.960988
$$184$$ 0 0
$$185$$ 11.0000 0.808736
$$186$$ 12.0000 0.879883
$$187$$ 1.00000 0.0731272
$$188$$ −20.0000 −1.45865
$$189$$ −3.00000 −0.218218
$$190$$ −4.00000 −0.290191
$$191$$ −8.00000 −0.578860 −0.289430 0.957199i $$-0.593466\pi$$
−0.289430 + 0.957199i $$0.593466\pi$$
$$192$$ 8.00000 0.577350
$$193$$ −13.0000 −0.935760 −0.467880 0.883792i $$-0.654982\pi$$
−0.467880 + 0.883792i $$0.654982\pi$$
$$194$$ 34.0000 2.44106
$$195$$ 1.00000 0.0716115
$$196$$ 4.00000 0.285714
$$197$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$198$$ −2.00000 −0.142134
$$199$$ 4.00000 0.283552 0.141776 0.989899i $$-0.454719\pi$$
0.141776 + 0.989899i $$0.454719\pi$$
$$200$$ 0 0
$$201$$ −12.0000 −0.846415
$$202$$ 0 0
$$203$$ −6.00000 −0.421117
$$204$$ 2.00000 0.140028
$$205$$ −5.00000 −0.349215
$$206$$ −32.0000 −2.22955
$$207$$ −3.00000 −0.208514
$$208$$ 4.00000 0.277350
$$209$$ 2.00000 0.138343
$$210$$ −6.00000 −0.414039
$$211$$ −4.00000 −0.275371 −0.137686 0.990476i $$-0.543966\pi$$
−0.137686 + 0.990476i $$0.543966\pi$$
$$212$$ 22.0000 1.51097
$$213$$ 5.00000 0.342594
$$214$$ 18.0000 1.23045
$$215$$ 4.00000 0.272798
$$216$$ 0 0
$$217$$ −18.0000 −1.22192
$$218$$ −32.0000 −2.16731
$$219$$ −10.0000 −0.675737
$$220$$ −2.00000 −0.134840
$$221$$ 1.00000 0.0672673
$$222$$ −22.0000 −1.47654
$$223$$ 8.00000 0.535720 0.267860 0.963458i $$-0.413684\pi$$
0.267860 + 0.963458i $$0.413684\pi$$
$$224$$ −24.0000 −1.60357
$$225$$ 1.00000 0.0666667
$$226$$ 28.0000 1.86253
$$227$$ 22.0000 1.46019 0.730096 0.683345i $$-0.239475\pi$$
0.730096 + 0.683345i $$0.239475\pi$$
$$228$$ 4.00000 0.264906
$$229$$ 18.0000 1.18947 0.594737 0.803921i $$-0.297256\pi$$
0.594737 + 0.803921i $$0.297256\pi$$
$$230$$ −6.00000 −0.395628
$$231$$ 3.00000 0.197386
$$232$$ 0 0
$$233$$ −27.0000 −1.76883 −0.884414 0.466702i $$-0.845442\pi$$
−0.884414 + 0.466702i $$0.845442\pi$$
$$234$$ −2.00000 −0.130744
$$235$$ −10.0000 −0.652328
$$236$$ 16.0000 1.04151
$$237$$ 3.00000 0.194871
$$238$$ −6.00000 −0.388922
$$239$$ −13.0000 −0.840900 −0.420450 0.907316i $$-0.638128\pi$$
−0.420450 + 0.907316i $$0.638128\pi$$
$$240$$ 4.00000 0.258199
$$241$$ −2.00000 −0.128831 −0.0644157 0.997923i $$-0.520518\pi$$
−0.0644157 + 0.997923i $$0.520518\pi$$
$$242$$ −20.0000 −1.28565
$$243$$ −1.00000 −0.0641500
$$244$$ 26.0000 1.66448
$$245$$ 2.00000 0.127775
$$246$$ 10.0000 0.637577
$$247$$ 2.00000 0.127257
$$248$$ 0 0
$$249$$ 12.0000 0.760469
$$250$$ 2.00000 0.126491
$$251$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$252$$ 6.00000 0.377964
$$253$$ 3.00000 0.188608
$$254$$ 20.0000 1.25491
$$255$$ 1.00000 0.0626224
$$256$$ 16.0000 1.00000
$$257$$ −18.0000 −1.12281 −0.561405 0.827541i $$-0.689739\pi$$
−0.561405 + 0.827541i $$0.689739\pi$$
$$258$$ −8.00000 −0.498058
$$259$$ 33.0000 2.05052
$$260$$ −2.00000 −0.124035
$$261$$ −2.00000 −0.123797
$$262$$ −12.0000 −0.741362
$$263$$ −8.00000 −0.493301 −0.246651 0.969104i $$-0.579330\pi$$
−0.246651 + 0.969104i $$0.579330\pi$$
$$264$$ 0 0
$$265$$ 11.0000 0.675725
$$266$$ −12.0000 −0.735767
$$267$$ 15.0000 0.917985
$$268$$ 24.0000 1.46603
$$269$$ −4.00000 −0.243884 −0.121942 0.992537i $$-0.538912\pi$$
−0.121942 + 0.992537i $$0.538912\pi$$
$$270$$ −2.00000 −0.121716
$$271$$ 22.0000 1.33640 0.668202 0.743980i $$-0.267064\pi$$
0.668202 + 0.743980i $$0.267064\pi$$
$$272$$ 4.00000 0.242536
$$273$$ 3.00000 0.181568
$$274$$ 36.0000 2.17484
$$275$$ −1.00000 −0.0603023
$$276$$ 6.00000 0.361158
$$277$$ −18.0000 −1.08152 −0.540758 0.841178i $$-0.681862\pi$$
−0.540758 + 0.841178i $$0.681862\pi$$
$$278$$ −2.00000 −0.119952
$$279$$ −6.00000 −0.359211
$$280$$ 0 0
$$281$$ 30.0000 1.78965 0.894825 0.446417i $$-0.147300\pi$$
0.894825 + 0.446417i $$0.147300\pi$$
$$282$$ 20.0000 1.19098
$$283$$ −12.0000 −0.713326 −0.356663 0.934233i $$-0.616086\pi$$
−0.356663 + 0.934233i $$0.616086\pi$$
$$284$$ −10.0000 −0.593391
$$285$$ 2.00000 0.118470
$$286$$ 2.00000 0.118262
$$287$$ −15.0000 −0.885422
$$288$$ −8.00000 −0.471405
$$289$$ −16.0000 −0.941176
$$290$$ −4.00000 −0.234888
$$291$$ −17.0000 −0.996558
$$292$$ 20.0000 1.17041
$$293$$ 24.0000 1.40209 0.701047 0.713115i $$-0.252716\pi$$
0.701047 + 0.713115i $$0.252716\pi$$
$$294$$ −4.00000 −0.233285
$$295$$ 8.00000 0.465778
$$296$$ 0 0
$$297$$ 1.00000 0.0580259
$$298$$ 26.0000 1.50614
$$299$$ 3.00000 0.173494
$$300$$ −2.00000 −0.115470
$$301$$ 12.0000 0.691669
$$302$$ 32.0000 1.84139
$$303$$ 0 0
$$304$$ 8.00000 0.458831
$$305$$ 13.0000 0.744378
$$306$$ −2.00000 −0.114332
$$307$$ −5.00000 −0.285365 −0.142683 0.989769i $$-0.545573\pi$$
−0.142683 + 0.989769i $$0.545573\pi$$
$$308$$ −6.00000 −0.341882
$$309$$ 16.0000 0.910208
$$310$$ −12.0000 −0.681554
$$311$$ 24.0000 1.36092 0.680458 0.732787i $$-0.261781\pi$$
0.680458 + 0.732787i $$0.261781\pi$$
$$312$$ 0 0
$$313$$ −10.0000 −0.565233 −0.282617 0.959233i $$-0.591202\pi$$
−0.282617 + 0.959233i $$0.591202\pi$$
$$314$$ −20.0000 −1.12867
$$315$$ 3.00000 0.169031
$$316$$ −6.00000 −0.337526
$$317$$ 28.0000 1.57264 0.786318 0.617822i $$-0.211985\pi$$
0.786318 + 0.617822i $$0.211985\pi$$
$$318$$ −22.0000 −1.23370
$$319$$ 2.00000 0.111979
$$320$$ −8.00000 −0.447214
$$321$$ −9.00000 −0.502331
$$322$$ −18.0000 −1.00310
$$323$$ 2.00000 0.111283
$$324$$ 2.00000 0.111111
$$325$$ −1.00000 −0.0554700
$$326$$ −26.0000 −1.44001
$$327$$ 16.0000 0.884802
$$328$$ 0 0
$$329$$ −30.0000 −1.65395
$$330$$ 2.00000 0.110096
$$331$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$332$$ −24.0000 −1.31717
$$333$$ 11.0000 0.602796
$$334$$ −24.0000 −1.31322
$$335$$ 12.0000 0.655630
$$336$$ 12.0000 0.654654
$$337$$ 4.00000 0.217894 0.108947 0.994048i $$-0.465252\pi$$
0.108947 + 0.994048i $$0.465252\pi$$
$$338$$ 2.00000 0.108786
$$339$$ −14.0000 −0.760376
$$340$$ −2.00000 −0.108465
$$341$$ 6.00000 0.324918
$$342$$ −4.00000 −0.216295
$$343$$ −15.0000 −0.809924
$$344$$ 0 0
$$345$$ 3.00000 0.161515
$$346$$ −12.0000 −0.645124
$$347$$ −19.0000 −1.01997 −0.509987 0.860182i $$-0.670350\pi$$
−0.509987 + 0.860182i $$0.670350\pi$$
$$348$$ 4.00000 0.214423
$$349$$ −8.00000 −0.428230 −0.214115 0.976808i $$-0.568687\pi$$
−0.214115 + 0.976808i $$0.568687\pi$$
$$350$$ 6.00000 0.320713
$$351$$ 1.00000 0.0533761
$$352$$ 8.00000 0.426401
$$353$$ 6.00000 0.319348 0.159674 0.987170i $$-0.448956\pi$$
0.159674 + 0.987170i $$0.448956\pi$$
$$354$$ −16.0000 −0.850390
$$355$$ −5.00000 −0.265372
$$356$$ −30.0000 −1.59000
$$357$$ 3.00000 0.158777
$$358$$ 4.00000 0.211407
$$359$$ 24.0000 1.26667 0.633336 0.773877i $$-0.281685\pi$$
0.633336 + 0.773877i $$0.281685\pi$$
$$360$$ 0 0
$$361$$ −15.0000 −0.789474
$$362$$ −14.0000 −0.735824
$$363$$ 10.0000 0.524864
$$364$$ −6.00000 −0.314485
$$365$$ 10.0000 0.523424
$$366$$ −26.0000 −1.35904
$$367$$ 36.0000 1.87918 0.939592 0.342296i $$-0.111204\pi$$
0.939592 + 0.342296i $$0.111204\pi$$
$$368$$ 12.0000 0.625543
$$369$$ −5.00000 −0.260290
$$370$$ 22.0000 1.14373
$$371$$ 33.0000 1.71327
$$372$$ 12.0000 0.622171
$$373$$ 4.00000 0.207112 0.103556 0.994624i $$-0.466978\pi$$
0.103556 + 0.994624i $$0.466978\pi$$
$$374$$ 2.00000 0.103418
$$375$$ −1.00000 −0.0516398
$$376$$ 0 0
$$377$$ 2.00000 0.103005
$$378$$ −6.00000 −0.308607
$$379$$ −14.0000 −0.719132 −0.359566 0.933120i $$-0.617075\pi$$
−0.359566 + 0.933120i $$0.617075\pi$$
$$380$$ −4.00000 −0.205196
$$381$$ −10.0000 −0.512316
$$382$$ −16.0000 −0.818631
$$383$$ −30.0000 −1.53293 −0.766464 0.642287i $$-0.777986\pi$$
−0.766464 + 0.642287i $$0.777986\pi$$
$$384$$ 0 0
$$385$$ −3.00000 −0.152894
$$386$$ −26.0000 −1.32337
$$387$$ 4.00000 0.203331
$$388$$ 34.0000 1.72609
$$389$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$390$$ 2.00000 0.101274
$$391$$ 3.00000 0.151717
$$392$$ 0 0
$$393$$ 6.00000 0.302660
$$394$$ 0 0
$$395$$ −3.00000 −0.150946
$$396$$ −2.00000 −0.100504
$$397$$ −29.0000 −1.45547 −0.727734 0.685859i $$-0.759427\pi$$
−0.727734 + 0.685859i $$0.759427\pi$$
$$398$$ 8.00000 0.401004
$$399$$ 6.00000 0.300376
$$400$$ −4.00000 −0.200000
$$401$$ 30.0000 1.49813 0.749064 0.662497i $$-0.230503\pi$$
0.749064 + 0.662497i $$0.230503\pi$$
$$402$$ −24.0000 −1.19701
$$403$$ 6.00000 0.298881
$$404$$ 0 0
$$405$$ 1.00000 0.0496904
$$406$$ −12.0000 −0.595550
$$407$$ −11.0000 −0.545250
$$408$$ 0 0
$$409$$ 2.00000 0.0988936 0.0494468 0.998777i $$-0.484254\pi$$
0.0494468 + 0.998777i $$0.484254\pi$$
$$410$$ −10.0000 −0.493865
$$411$$ −18.0000 −0.887875
$$412$$ −32.0000 −1.57653
$$413$$ 24.0000 1.18096
$$414$$ −6.00000 −0.294884
$$415$$ −12.0000 −0.589057
$$416$$ 8.00000 0.392232
$$417$$ 1.00000 0.0489702
$$418$$ 4.00000 0.195646
$$419$$ −26.0000 −1.27018 −0.635092 0.772437i $$-0.719038\pi$$
−0.635092 + 0.772437i $$0.719038\pi$$
$$420$$ −6.00000 −0.292770
$$421$$ 4.00000 0.194948 0.0974740 0.995238i $$-0.468924\pi$$
0.0974740 + 0.995238i $$0.468924\pi$$
$$422$$ −8.00000 −0.389434
$$423$$ −10.0000 −0.486217
$$424$$ 0 0
$$425$$ −1.00000 −0.0485071
$$426$$ 10.0000 0.484502
$$427$$ 39.0000 1.88734
$$428$$ 18.0000 0.870063
$$429$$ −1.00000 −0.0482805
$$430$$ 8.00000 0.385794
$$431$$ −24.0000 −1.15604 −0.578020 0.816023i $$-0.696174\pi$$
−0.578020 + 0.816023i $$0.696174\pi$$
$$432$$ 4.00000 0.192450
$$433$$ 4.00000 0.192228 0.0961139 0.995370i $$-0.469359\pi$$
0.0961139 + 0.995370i $$0.469359\pi$$
$$434$$ −36.0000 −1.72806
$$435$$ 2.00000 0.0958927
$$436$$ −32.0000 −1.53252
$$437$$ 6.00000 0.287019
$$438$$ −20.0000 −0.955637
$$439$$ −17.0000 −0.811366 −0.405683 0.914014i $$-0.632966\pi$$
−0.405683 + 0.914014i $$0.632966\pi$$
$$440$$ 0 0
$$441$$ 2.00000 0.0952381
$$442$$ 2.00000 0.0951303
$$443$$ 9.00000 0.427603 0.213801 0.976877i $$-0.431415\pi$$
0.213801 + 0.976877i $$0.431415\pi$$
$$444$$ −22.0000 −1.04407
$$445$$ −15.0000 −0.711068
$$446$$ 16.0000 0.757622
$$447$$ −13.0000 −0.614879
$$448$$ −24.0000 −1.13389
$$449$$ −13.0000 −0.613508 −0.306754 0.951789i $$-0.599243\pi$$
−0.306754 + 0.951789i $$0.599243\pi$$
$$450$$ 2.00000 0.0942809
$$451$$ 5.00000 0.235441
$$452$$ 28.0000 1.31701
$$453$$ −16.0000 −0.751746
$$454$$ 44.0000 2.06502
$$455$$ −3.00000 −0.140642
$$456$$ 0 0
$$457$$ −11.0000 −0.514558 −0.257279 0.966337i $$-0.582826\pi$$
−0.257279 + 0.966337i $$0.582826\pi$$
$$458$$ 36.0000 1.68217
$$459$$ 1.00000 0.0466760
$$460$$ −6.00000 −0.279751
$$461$$ 15.0000 0.698620 0.349310 0.937007i $$-0.386416\pi$$
0.349310 + 0.937007i $$0.386416\pi$$
$$462$$ 6.00000 0.279145
$$463$$ 27.0000 1.25480 0.627398 0.778699i $$-0.284120\pi$$
0.627398 + 0.778699i $$0.284120\pi$$
$$464$$ 8.00000 0.371391
$$465$$ 6.00000 0.278243
$$466$$ −54.0000 −2.50150
$$467$$ −23.0000 −1.06431 −0.532157 0.846646i $$-0.678618\pi$$
−0.532157 + 0.846646i $$0.678618\pi$$
$$468$$ −2.00000 −0.0924500
$$469$$ 36.0000 1.66233
$$470$$ −20.0000 −0.922531
$$471$$ 10.0000 0.460776
$$472$$ 0 0
$$473$$ −4.00000 −0.183920
$$474$$ 6.00000 0.275589
$$475$$ −2.00000 −0.0917663
$$476$$ −6.00000 −0.275010
$$477$$ 11.0000 0.503655
$$478$$ −26.0000 −1.18921
$$479$$ 9.00000 0.411220 0.205610 0.978634i $$-0.434082\pi$$
0.205610 + 0.978634i $$0.434082\pi$$
$$480$$ 8.00000 0.365148
$$481$$ −11.0000 −0.501557
$$482$$ −4.00000 −0.182195
$$483$$ 9.00000 0.409514
$$484$$ −20.0000 −0.909091
$$485$$ 17.0000 0.771930
$$486$$ −2.00000 −0.0907218
$$487$$ −7.00000 −0.317200 −0.158600 0.987343i $$-0.550698\pi$$
−0.158600 + 0.987343i $$0.550698\pi$$
$$488$$ 0 0
$$489$$ 13.0000 0.587880
$$490$$ 4.00000 0.180702
$$491$$ 28.0000 1.26362 0.631811 0.775122i $$-0.282312\pi$$
0.631811 + 0.775122i $$0.282312\pi$$
$$492$$ 10.0000 0.450835
$$493$$ 2.00000 0.0900755
$$494$$ 4.00000 0.179969
$$495$$ −1.00000 −0.0449467
$$496$$ 24.0000 1.07763
$$497$$ −15.0000 −0.672842
$$498$$ 24.0000 1.07547
$$499$$ −14.0000 −0.626726 −0.313363 0.949633i $$-0.601456\pi$$
−0.313363 + 0.949633i $$0.601456\pi$$
$$500$$ 2.00000 0.0894427
$$501$$ 12.0000 0.536120
$$502$$ 0 0
$$503$$ 28.0000 1.24846 0.624229 0.781241i $$-0.285413\pi$$
0.624229 + 0.781241i $$0.285413\pi$$
$$504$$ 0 0
$$505$$ 0 0
$$506$$ 6.00000 0.266733
$$507$$ −1.00000 −0.0444116
$$508$$ 20.0000 0.887357
$$509$$ −7.00000 −0.310270 −0.155135 0.987893i $$-0.549581\pi$$
−0.155135 + 0.987893i $$0.549581\pi$$
$$510$$ 2.00000 0.0885615
$$511$$ 30.0000 1.32712
$$512$$ 32.0000 1.41421
$$513$$ 2.00000 0.0883022
$$514$$ −36.0000 −1.58789
$$515$$ −16.0000 −0.705044
$$516$$ −8.00000 −0.352180
$$517$$ 10.0000 0.439799
$$518$$ 66.0000 2.89987
$$519$$ 6.00000 0.263371
$$520$$ 0 0
$$521$$ −14.0000 −0.613351 −0.306676 0.951814i $$-0.599217\pi$$
−0.306676 + 0.951814i $$0.599217\pi$$
$$522$$ −4.00000 −0.175075
$$523$$ 16.0000 0.699631 0.349816 0.936819i $$-0.386244\pi$$
0.349816 + 0.936819i $$0.386244\pi$$
$$524$$ −12.0000 −0.524222
$$525$$ −3.00000 −0.130931
$$526$$ −16.0000 −0.697633
$$527$$ 6.00000 0.261364
$$528$$ −4.00000 −0.174078
$$529$$ −14.0000 −0.608696
$$530$$ 22.0000 0.955619
$$531$$ 8.00000 0.347170
$$532$$ −12.0000 −0.520266
$$533$$ 5.00000 0.216574
$$534$$ 30.0000 1.29823
$$535$$ 9.00000 0.389104
$$536$$ 0 0
$$537$$ −2.00000 −0.0863064
$$538$$ −8.00000 −0.344904
$$539$$ −2.00000 −0.0861461
$$540$$ −2.00000 −0.0860663
$$541$$ −30.0000 −1.28980 −0.644900 0.764267i $$-0.723101\pi$$
−0.644900 + 0.764267i $$0.723101\pi$$
$$542$$ 44.0000 1.88996
$$543$$ 7.00000 0.300399
$$544$$ 8.00000 0.342997
$$545$$ −16.0000 −0.685365
$$546$$ 6.00000 0.256776
$$547$$ −32.0000 −1.36822 −0.684111 0.729378i $$-0.739809\pi$$
−0.684111 + 0.729378i $$0.739809\pi$$
$$548$$ 36.0000 1.53784
$$549$$ 13.0000 0.554826
$$550$$ −2.00000 −0.0852803
$$551$$ 4.00000 0.170406
$$552$$ 0 0
$$553$$ −9.00000 −0.382719
$$554$$ −36.0000 −1.52949
$$555$$ −11.0000 −0.466924
$$556$$ −2.00000 −0.0848189
$$557$$ −30.0000 −1.27114 −0.635570 0.772043i $$-0.719235\pi$$
−0.635570 + 0.772043i $$0.719235\pi$$
$$558$$ −12.0000 −0.508001
$$559$$ −4.00000 −0.169182
$$560$$ −12.0000 −0.507093
$$561$$ −1.00000 −0.0422200
$$562$$ 60.0000 2.53095
$$563$$ 21.0000 0.885044 0.442522 0.896758i $$-0.354084\pi$$
0.442522 + 0.896758i $$0.354084\pi$$
$$564$$ 20.0000 0.842152
$$565$$ 14.0000 0.588984
$$566$$ −24.0000 −1.00880
$$567$$ 3.00000 0.125988
$$568$$ 0 0
$$569$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$570$$ 4.00000 0.167542
$$571$$ −31.0000 −1.29731 −0.648655 0.761083i $$-0.724668\pi$$
−0.648655 + 0.761083i $$0.724668\pi$$
$$572$$ 2.00000 0.0836242
$$573$$ 8.00000 0.334205
$$574$$ −30.0000 −1.25218
$$575$$ −3.00000 −0.125109
$$576$$ −8.00000 −0.333333
$$577$$ −19.0000 −0.790980 −0.395490 0.918470i $$-0.629425\pi$$
−0.395490 + 0.918470i $$0.629425\pi$$
$$578$$ −32.0000 −1.33102
$$579$$ 13.0000 0.540262
$$580$$ −4.00000 −0.166091
$$581$$ −36.0000 −1.49353
$$582$$ −34.0000 −1.40935
$$583$$ −11.0000 −0.455573
$$584$$ 0 0
$$585$$ −1.00000 −0.0413449
$$586$$ 48.0000 1.98286
$$587$$ −18.0000 −0.742940 −0.371470 0.928445i $$-0.621146\pi$$
−0.371470 + 0.928445i $$0.621146\pi$$
$$588$$ −4.00000 −0.164957
$$589$$ 12.0000 0.494451
$$590$$ 16.0000 0.658710
$$591$$ 0 0
$$592$$ −44.0000 −1.80839
$$593$$ 4.00000 0.164260 0.0821302 0.996622i $$-0.473828\pi$$
0.0821302 + 0.996622i $$0.473828\pi$$
$$594$$ 2.00000 0.0820610
$$595$$ −3.00000 −0.122988
$$596$$ 26.0000 1.06500
$$597$$ −4.00000 −0.163709
$$598$$ 6.00000 0.245358
$$599$$ 16.0000 0.653742 0.326871 0.945069i $$-0.394006\pi$$
0.326871 + 0.945069i $$0.394006\pi$$
$$600$$ 0 0
$$601$$ −37.0000 −1.50926 −0.754631 0.656150i $$-0.772184\pi$$
−0.754631 + 0.656150i $$0.772184\pi$$
$$602$$ 24.0000 0.978167
$$603$$ 12.0000 0.488678
$$604$$ 32.0000 1.30206
$$605$$ −10.0000 −0.406558
$$606$$ 0 0
$$607$$ −32.0000 −1.29884 −0.649420 0.760430i $$-0.724988\pi$$
−0.649420 + 0.760430i $$0.724988\pi$$
$$608$$ 16.0000 0.648886
$$609$$ 6.00000 0.243132
$$610$$ 26.0000 1.05271
$$611$$ 10.0000 0.404557
$$612$$ −2.00000 −0.0808452
$$613$$ 13.0000 0.525065 0.262533 0.964923i $$-0.415442\pi$$
0.262533 + 0.964923i $$0.415442\pi$$
$$614$$ −10.0000 −0.403567
$$615$$ 5.00000 0.201619
$$616$$ 0 0
$$617$$ −42.0000 −1.69086 −0.845428 0.534089i $$-0.820655\pi$$
−0.845428 + 0.534089i $$0.820655\pi$$
$$618$$ 32.0000 1.28723
$$619$$ −34.0000 −1.36658 −0.683288 0.730149i $$-0.739451\pi$$
−0.683288 + 0.730149i $$0.739451\pi$$
$$620$$ −12.0000 −0.481932
$$621$$ 3.00000 0.120386
$$622$$ 48.0000 1.92462
$$623$$ −45.0000 −1.80289
$$624$$ −4.00000 −0.160128
$$625$$ 1.00000 0.0400000
$$626$$ −20.0000 −0.799361
$$627$$ −2.00000 −0.0798723
$$628$$ −20.0000 −0.798087
$$629$$ −11.0000 −0.438599
$$630$$ 6.00000 0.239046
$$631$$ 40.0000 1.59237 0.796187 0.605050i $$-0.206847\pi$$
0.796187 + 0.605050i $$0.206847\pi$$
$$632$$ 0 0
$$633$$ 4.00000 0.158986
$$634$$ 56.0000 2.22404
$$635$$ 10.0000 0.396838
$$636$$ −22.0000 −0.872357
$$637$$ −2.00000 −0.0792429
$$638$$ 4.00000 0.158362
$$639$$ −5.00000 −0.197797
$$640$$ 0 0
$$641$$ −12.0000 −0.473972 −0.236986 0.971513i $$-0.576159\pi$$
−0.236986 + 0.971513i $$0.576159\pi$$
$$642$$ −18.0000 −0.710403
$$643$$ 15.0000 0.591542 0.295771 0.955259i $$-0.404423\pi$$
0.295771 + 0.955259i $$0.404423\pi$$
$$644$$ −18.0000 −0.709299
$$645$$ −4.00000 −0.157500
$$646$$ 4.00000 0.157378
$$647$$ 47.0000 1.84776 0.923880 0.382682i $$-0.124999\pi$$
0.923880 + 0.382682i $$0.124999\pi$$
$$648$$ 0 0
$$649$$ −8.00000 −0.314027
$$650$$ −2.00000 −0.0784465
$$651$$ 18.0000 0.705476
$$652$$ −26.0000 −1.01824
$$653$$ 22.0000 0.860927 0.430463 0.902608i $$-0.358350\pi$$
0.430463 + 0.902608i $$0.358350\pi$$
$$654$$ 32.0000 1.25130
$$655$$ −6.00000 −0.234439
$$656$$ 20.0000 0.780869
$$657$$ 10.0000 0.390137
$$658$$ −60.0000 −2.33904
$$659$$ 20.0000 0.779089 0.389545 0.921008i $$-0.372632\pi$$
0.389545 + 0.921008i $$0.372632\pi$$
$$660$$ 2.00000 0.0778499
$$661$$ 4.00000 0.155582 0.0777910 0.996970i $$-0.475213\pi$$
0.0777910 + 0.996970i $$0.475213\pi$$
$$662$$ 0 0
$$663$$ −1.00000 −0.0388368
$$664$$ 0 0
$$665$$ −6.00000 −0.232670
$$666$$ 22.0000 0.852483
$$667$$ 6.00000 0.232321
$$668$$ −24.0000 −0.928588
$$669$$ −8.00000 −0.309298
$$670$$ 24.0000 0.927201
$$671$$ −13.0000 −0.501859
$$672$$ 24.0000 0.925820
$$673$$ −6.00000 −0.231283 −0.115642 0.993291i $$-0.536892\pi$$
−0.115642 + 0.993291i $$0.536892\pi$$
$$674$$ 8.00000 0.308148
$$675$$ −1.00000 −0.0384900
$$676$$ 2.00000 0.0769231
$$677$$ 3.00000 0.115299 0.0576497 0.998337i $$-0.481639\pi$$
0.0576497 + 0.998337i $$0.481639\pi$$
$$678$$ −28.0000 −1.07533
$$679$$ 51.0000 1.95720
$$680$$ 0 0
$$681$$ −22.0000 −0.843042
$$682$$ 12.0000 0.459504
$$683$$ −24.0000 −0.918334 −0.459167 0.888350i $$-0.651852\pi$$
−0.459167 + 0.888350i $$0.651852\pi$$
$$684$$ −4.00000 −0.152944
$$685$$ 18.0000 0.687745
$$686$$ −30.0000 −1.14541
$$687$$ −18.0000 −0.686743
$$688$$ −16.0000 −0.609994
$$689$$ −11.0000 −0.419067
$$690$$ 6.00000 0.228416
$$691$$ 22.0000 0.836919 0.418460 0.908235i $$-0.362570\pi$$
0.418460 + 0.908235i $$0.362570\pi$$
$$692$$ −12.0000 −0.456172
$$693$$ −3.00000 −0.113961
$$694$$ −38.0000 −1.44246
$$695$$ −1.00000 −0.0379322
$$696$$ 0 0
$$697$$ 5.00000 0.189389
$$698$$ −16.0000 −0.605609
$$699$$ 27.0000 1.02123
$$700$$ 6.00000 0.226779
$$701$$ −20.0000 −0.755390 −0.377695 0.925930i $$-0.623283\pi$$
−0.377695 + 0.925930i $$0.623283\pi$$
$$702$$ 2.00000 0.0754851
$$703$$ −22.0000 −0.829746
$$704$$ 8.00000 0.301511
$$705$$ 10.0000 0.376622
$$706$$ 12.0000 0.451626
$$707$$ 0 0
$$708$$ −16.0000 −0.601317
$$709$$ 4.00000 0.150223 0.0751116 0.997175i $$-0.476069\pi$$
0.0751116 + 0.997175i $$0.476069\pi$$
$$710$$ −10.0000 −0.375293
$$711$$ −3.00000 −0.112509
$$712$$ 0 0
$$713$$ 18.0000 0.674105
$$714$$ 6.00000 0.224544
$$715$$ 1.00000 0.0373979
$$716$$ 4.00000 0.149487
$$717$$ 13.0000 0.485494
$$718$$ 48.0000 1.79134
$$719$$ −24.0000 −0.895049 −0.447524 0.894272i $$-0.647694\pi$$
−0.447524 + 0.894272i $$0.647694\pi$$
$$720$$ −4.00000 −0.149071
$$721$$ −48.0000 −1.78761
$$722$$ −30.0000 −1.11648
$$723$$ 2.00000 0.0743808
$$724$$ −14.0000 −0.520306
$$725$$ −2.00000 −0.0742781
$$726$$ 20.0000 0.742270
$$727$$ 38.0000 1.40934 0.704671 0.709534i $$-0.251095\pi$$
0.704671 + 0.709534i $$0.251095\pi$$
$$728$$ 0 0
$$729$$ 1.00000 0.0370370
$$730$$ 20.0000 0.740233
$$731$$ −4.00000 −0.147945
$$732$$ −26.0000 −0.960988
$$733$$ −49.0000 −1.80986 −0.904928 0.425564i $$-0.860076\pi$$
−0.904928 + 0.425564i $$0.860076\pi$$
$$734$$ 72.0000 2.65757
$$735$$ −2.00000 −0.0737711
$$736$$ 24.0000 0.884652
$$737$$ −12.0000 −0.442026
$$738$$ −10.0000 −0.368105
$$739$$ 10.0000 0.367856 0.183928 0.982940i $$-0.441119\pi$$
0.183928 + 0.982940i $$0.441119\pi$$
$$740$$ 22.0000 0.808736
$$741$$ −2.00000 −0.0734718
$$742$$ 66.0000 2.42294
$$743$$ −34.0000 −1.24734 −0.623670 0.781688i $$-0.714359\pi$$
−0.623670 + 0.781688i $$0.714359\pi$$
$$744$$ 0 0
$$745$$ 13.0000 0.476283
$$746$$ 8.00000 0.292901
$$747$$ −12.0000 −0.439057
$$748$$ 2.00000 0.0731272
$$749$$ 27.0000 0.986559
$$750$$ −2.00000 −0.0730297
$$751$$ −5.00000 −0.182453 −0.0912263 0.995830i $$-0.529079\pi$$
−0.0912263 + 0.995830i $$0.529079\pi$$
$$752$$ 40.0000 1.45865
$$753$$ 0 0
$$754$$ 4.00000 0.145671
$$755$$ 16.0000 0.582300
$$756$$ −6.00000 −0.218218
$$757$$ −8.00000 −0.290765 −0.145382 0.989376i $$-0.546441\pi$$
−0.145382 + 0.989376i $$0.546441\pi$$
$$758$$ −28.0000 −1.01701
$$759$$ −3.00000 −0.108893
$$760$$ 0 0
$$761$$ −6.00000 −0.217500 −0.108750 0.994069i $$-0.534685\pi$$
−0.108750 + 0.994069i $$0.534685\pi$$
$$762$$ −20.0000 −0.724524
$$763$$ −48.0000 −1.73772
$$764$$ −16.0000 −0.578860
$$765$$ −1.00000 −0.0361551
$$766$$ −60.0000 −2.16789
$$767$$ −8.00000 −0.288863
$$768$$ −16.0000 −0.577350
$$769$$ −40.0000 −1.44244 −0.721218 0.692708i $$-0.756418\pi$$
−0.721218 + 0.692708i $$0.756418\pi$$
$$770$$ −6.00000 −0.216225
$$771$$ 18.0000 0.648254
$$772$$ −26.0000 −0.935760
$$773$$ 36.0000 1.29483 0.647415 0.762138i $$-0.275850\pi$$
0.647415 + 0.762138i $$0.275850\pi$$
$$774$$ 8.00000 0.287554
$$775$$ −6.00000 −0.215526
$$776$$ 0 0
$$777$$ −33.0000 −1.18387
$$778$$ 0 0
$$779$$ 10.0000 0.358287
$$780$$ 2.00000 0.0716115
$$781$$ 5.00000 0.178914
$$782$$ 6.00000 0.214560
$$783$$ 2.00000 0.0714742
$$784$$ −8.00000 −0.285714
$$785$$ −10.0000 −0.356915
$$786$$ 12.0000 0.428026
$$787$$ 52.0000 1.85360 0.926800 0.375555i $$-0.122548\pi$$
0.926800 + 0.375555i $$0.122548\pi$$
$$788$$ 0 0
$$789$$ 8.00000 0.284808
$$790$$ −6.00000 −0.213470
$$791$$ 42.0000 1.49335
$$792$$ 0 0
$$793$$ −13.0000 −0.461644
$$794$$ −58.0000 −2.05834
$$795$$ −11.0000 −0.390130
$$796$$ 8.00000 0.283552
$$797$$ −47.0000 −1.66483 −0.832413 0.554156i $$-0.813041\pi$$
−0.832413 + 0.554156i $$0.813041\pi$$
$$798$$ 12.0000 0.424795
$$799$$ 10.0000 0.353775
$$800$$ −8.00000 −0.282843
$$801$$ −15.0000 −0.529999
$$802$$ 60.0000 2.11867
$$803$$ −10.0000 −0.352892
$$804$$ −24.0000 −0.846415
$$805$$ −9.00000 −0.317208
$$806$$ 12.0000 0.422682
$$807$$ 4.00000 0.140807
$$808$$ 0 0
$$809$$ −26.0000 −0.914111 −0.457056 0.889438i $$-0.651096\pi$$
−0.457056 + 0.889438i $$0.651096\pi$$
$$810$$ 2.00000 0.0702728
$$811$$ 36.0000 1.26413 0.632065 0.774915i $$-0.282207\pi$$
0.632065 + 0.774915i $$0.282207\pi$$
$$812$$ −12.0000 −0.421117
$$813$$ −22.0000 −0.771574
$$814$$ −22.0000 −0.771100
$$815$$ −13.0000 −0.455370
$$816$$ −4.00000 −0.140028
$$817$$ −8.00000 −0.279885
$$818$$ 4.00000 0.139857
$$819$$ −3.00000 −0.104828
$$820$$ −10.0000 −0.349215
$$821$$ 27.0000 0.942306 0.471153 0.882051i $$-0.343838\pi$$
0.471153 + 0.882051i $$0.343838\pi$$
$$822$$ −36.0000 −1.25564
$$823$$ 20.0000 0.697156 0.348578 0.937280i $$-0.386665\pi$$
0.348578 + 0.937280i $$0.386665\pi$$
$$824$$ 0 0
$$825$$ 1.00000 0.0348155
$$826$$ 48.0000 1.67013
$$827$$ 26.0000 0.904109 0.452054 0.891990i $$-0.350691\pi$$
0.452054 + 0.891990i $$0.350691\pi$$
$$828$$ −6.00000 −0.208514
$$829$$ 30.0000 1.04194 0.520972 0.853574i $$-0.325570\pi$$
0.520972 + 0.853574i $$0.325570\pi$$
$$830$$ −24.0000 −0.833052
$$831$$ 18.0000 0.624413
$$832$$ 8.00000 0.277350
$$833$$ −2.00000 −0.0692959
$$834$$ 2.00000 0.0692543
$$835$$ −12.0000 −0.415277
$$836$$ 4.00000 0.138343
$$837$$ 6.00000 0.207390
$$838$$ −52.0000 −1.79631
$$839$$ −5.00000 −0.172619 −0.0863096 0.996268i $$-0.527507\pi$$
−0.0863096 + 0.996268i $$0.527507\pi$$
$$840$$ 0 0
$$841$$ −25.0000 −0.862069
$$842$$ 8.00000 0.275698
$$843$$ −30.0000 −1.03325
$$844$$ −8.00000 −0.275371
$$845$$ 1.00000 0.0344010
$$846$$ −20.0000 −0.687614
$$847$$ −30.0000 −1.03081
$$848$$ −44.0000 −1.51097
$$849$$ 12.0000 0.411839
$$850$$ −2.00000 −0.0685994
$$851$$ −33.0000 −1.13123
$$852$$ 10.0000 0.342594
$$853$$ 45.0000 1.54077 0.770385 0.637579i $$-0.220064\pi$$
0.770385 + 0.637579i $$0.220064\pi$$
$$854$$ 78.0000 2.66911
$$855$$ −2.00000 −0.0683986
$$856$$ 0 0
$$857$$ 29.0000 0.990621 0.495311 0.868716i $$-0.335054\pi$$
0.495311 + 0.868716i $$0.335054\pi$$
$$858$$ −2.00000 −0.0682789
$$859$$ −29.0000 −0.989467 −0.494734 0.869045i $$-0.664734\pi$$
−0.494734 + 0.869045i $$0.664734\pi$$
$$860$$ 8.00000 0.272798
$$861$$ 15.0000 0.511199
$$862$$ −48.0000 −1.63489
$$863$$ 34.0000 1.15737 0.578687 0.815550i $$-0.303565\pi$$
0.578687 + 0.815550i $$0.303565\pi$$
$$864$$ 8.00000 0.272166
$$865$$ −6.00000 −0.204006
$$866$$ 8.00000 0.271851
$$867$$ 16.0000 0.543388
$$868$$ −36.0000 −1.22192
$$869$$ 3.00000 0.101768
$$870$$ 4.00000 0.135613
$$871$$ −12.0000 −0.406604
$$872$$ 0 0
$$873$$ 17.0000 0.575363
$$874$$ 12.0000 0.405906
$$875$$ 3.00000 0.101419
$$876$$ −20.0000 −0.675737
$$877$$ 18.0000 0.607817 0.303908 0.952701i $$-0.401708\pi$$
0.303908 + 0.952701i $$0.401708\pi$$
$$878$$ −34.0000 −1.14744
$$879$$ −24.0000 −0.809500
$$880$$ 4.00000 0.134840
$$881$$ 2.00000 0.0673817 0.0336909 0.999432i $$-0.489274\pi$$
0.0336909 + 0.999432i $$0.489274\pi$$
$$882$$ 4.00000 0.134687
$$883$$ −16.0000 −0.538443 −0.269221 0.963078i $$-0.586766\pi$$
−0.269221 + 0.963078i $$0.586766\pi$$
$$884$$ 2.00000 0.0672673
$$885$$ −8.00000 −0.268917
$$886$$ 18.0000 0.604722
$$887$$ −21.0000 −0.705111 −0.352555 0.935791i $$-0.614687\pi$$
−0.352555 + 0.935791i $$0.614687\pi$$
$$888$$ 0 0
$$889$$ 30.0000 1.00617
$$890$$ −30.0000 −1.00560
$$891$$ −1.00000 −0.0335013
$$892$$ 16.0000 0.535720
$$893$$ 20.0000 0.669274
$$894$$ −26.0000 −0.869570
$$895$$ 2.00000 0.0668526
$$896$$ 0 0
$$897$$ −3.00000 −0.100167
$$898$$ −26.0000 −0.867631
$$899$$ 12.0000 0.400222
$$900$$ 2.00000 0.0666667
$$901$$ −11.0000 −0.366463
$$902$$ 10.0000 0.332964
$$903$$ −12.0000 −0.399335
$$904$$ 0 0
$$905$$ −7.00000 −0.232688
$$906$$ −32.0000 −1.06313
$$907$$ 6.00000 0.199227 0.0996134 0.995026i $$-0.468239\pi$$
0.0996134 + 0.995026i $$0.468239\pi$$
$$908$$ 44.0000 1.46019
$$909$$ 0 0
$$910$$ −6.00000 −0.198898
$$911$$ 44.0000 1.45779 0.728893 0.684628i $$-0.240035\pi$$
0.728893 + 0.684628i $$0.240035\pi$$
$$912$$ −8.00000 −0.264906
$$913$$ 12.0000 0.397142
$$914$$ −22.0000 −0.727695
$$915$$ −13.0000 −0.429767
$$916$$ 36.0000 1.18947
$$917$$ −18.0000 −0.594412
$$918$$ 2.00000 0.0660098
$$919$$ 37.0000 1.22052 0.610259 0.792202i $$-0.291065\pi$$
0.610259 + 0.792202i $$0.291065\pi$$
$$920$$ 0 0
$$921$$ 5.00000 0.164756
$$922$$ 30.0000 0.987997
$$923$$ 5.00000 0.164577
$$924$$ 6.00000 0.197386
$$925$$ 11.0000 0.361678
$$926$$ 54.0000 1.77455
$$927$$ −16.0000 −0.525509
$$928$$ 16.0000 0.525226
$$929$$ 1.00000 0.0328089 0.0164045 0.999865i $$-0.494778\pi$$
0.0164045 + 0.999865i $$0.494778\pi$$
$$930$$ 12.0000 0.393496
$$931$$ −4.00000 −0.131095
$$932$$ −54.0000 −1.76883
$$933$$ −24.0000 −0.785725
$$934$$ −46.0000 −1.50517
$$935$$ 1.00000 0.0327035
$$936$$ 0 0
$$937$$ 30.0000 0.980057 0.490029 0.871706i $$-0.336986\pi$$
0.490029 + 0.871706i $$0.336986\pi$$
$$938$$ 72.0000 2.35088
$$939$$ 10.0000 0.326338
$$940$$ −20.0000 −0.652328
$$941$$ 37.0000 1.20617 0.603083 0.797679i $$-0.293939\pi$$
0.603083 + 0.797679i $$0.293939\pi$$
$$942$$ 20.0000 0.651635
$$943$$ 15.0000 0.488467
$$944$$ −32.0000 −1.04151
$$945$$ −3.00000 −0.0975900
$$946$$ −8.00000 −0.260102
$$947$$ −24.0000 −0.779895 −0.389948 0.920837i $$-0.627507\pi$$
−0.389948 + 0.920837i $$0.627507\pi$$
$$948$$ 6.00000 0.194871
$$949$$ −10.0000 −0.324614
$$950$$ −4.00000 −0.129777
$$951$$ −28.0000 −0.907962
$$952$$ 0 0
$$953$$ −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
$$956$$ −26.0000 −0.840900
$$957$$ −2.00000 −0.0646508
$$958$$ 18.0000 0.581554
$$959$$ 54.0000 1.74375
$$960$$ 8.00000 0.258199
$$961$$ 5.00000 0.161290
$$962$$ −22.0000 −0.709308
$$963$$ 9.00000 0.290021
$$964$$ −4.00000 −0.128831
$$965$$ −13.0000 −0.418485
$$966$$ 18.0000 0.579141
$$967$$ −16.0000 −0.514525 −0.257263 0.966342i $$-0.582821\pi$$
−0.257263 + 0.966342i $$0.582821\pi$$
$$968$$ 0 0
$$969$$ −2.00000 −0.0642493
$$970$$ 34.0000 1.09167
$$971$$ 12.0000 0.385098 0.192549 0.981287i $$-0.438325\pi$$
0.192549 + 0.981287i $$0.438325\pi$$
$$972$$ −2.00000 −0.0641500
$$973$$ −3.00000 −0.0961756
$$974$$ −14.0000 −0.448589
$$975$$ 1.00000 0.0320256
$$976$$ −52.0000 −1.66448
$$977$$ 32.0000 1.02377 0.511885 0.859054i $$-0.328947\pi$$
0.511885 + 0.859054i $$0.328947\pi$$
$$978$$ 26.0000 0.831388
$$979$$ 15.0000 0.479402
$$980$$ 4.00000 0.127775
$$981$$ −16.0000 −0.510841
$$982$$ 56.0000 1.78703
$$983$$ −12.0000 −0.382741 −0.191370 0.981518i $$-0.561293\pi$$
−0.191370 + 0.981518i $$0.561293\pi$$
$$984$$ 0 0
$$985$$ 0 0
$$986$$ 4.00000 0.127386
$$987$$ 30.0000 0.954911
$$988$$ 4.00000 0.127257
$$989$$ −12.0000 −0.381578
$$990$$ −2.00000 −0.0635642
$$991$$ −25.0000 −0.794151 −0.397076 0.917786i $$-0.629975\pi$$
−0.397076 + 0.917786i $$0.629975\pi$$
$$992$$ 48.0000 1.52400
$$993$$ 0 0
$$994$$ −30.0000 −0.951542
$$995$$ 4.00000 0.126809
$$996$$ 24.0000 0.760469
$$997$$ −36.0000 −1.14013 −0.570066 0.821599i $$-0.693082\pi$$
−0.570066 + 0.821599i $$0.693082\pi$$
$$998$$ −28.0000 −0.886325
$$999$$ −11.0000 −0.348025
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 195.2.a.b.1.1 1
3.2 odd 2 585.2.a.b.1.1 1
4.3 odd 2 3120.2.a.u.1.1 1
5.2 odd 4 975.2.c.a.274.2 2
5.3 odd 4 975.2.c.a.274.1 2
5.4 even 2 975.2.a.c.1.1 1
7.6 odd 2 9555.2.a.v.1.1 1
12.11 even 2 9360.2.a.d.1.1 1
13.12 even 2 2535.2.a.a.1.1 1
15.2 even 4 2925.2.c.c.2224.1 2
15.8 even 4 2925.2.c.c.2224.2 2
15.14 odd 2 2925.2.a.q.1.1 1
39.38 odd 2 7605.2.a.u.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
195.2.a.b.1.1 1 1.1 even 1 trivial
585.2.a.b.1.1 1 3.2 odd 2
975.2.a.c.1.1 1 5.4 even 2
975.2.c.a.274.1 2 5.3 odd 4
975.2.c.a.274.2 2 5.2 odd 4
2535.2.a.a.1.1 1 13.12 even 2
2925.2.a.q.1.1 1 15.14 odd 2
2925.2.c.c.2224.1 2 15.2 even 4
2925.2.c.c.2224.2 2 15.8 even 4
3120.2.a.u.1.1 1 4.3 odd 2
7605.2.a.u.1.1 1 39.38 odd 2
9360.2.a.d.1.1 1 12.11 even 2
9555.2.a.v.1.1 1 7.6 odd 2