# Properties

 Label 195.2.a.d.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 $1$

# Related objects

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)

 $$f(q)$$ $$=$$ $$q+2.00000 q^{2} +1.00000 q^{3} +2.00000 q^{4} +1.00000 q^{5} +2.00000 q^{6} -3.00000 q^{7} +1.00000 q^{9} +O(q^{10})$$ $$q+2.00000 q^{2} +1.00000 q^{3} +2.00000 q^{4} +1.00000 q^{5} +2.00000 q^{6} -3.00000 q^{7} +1.00000 q^{9} +2.00000 q^{10} -5.00000 q^{11} +2.00000 q^{12} +1.00000 q^{13} -6.00000 q^{14} +1.00000 q^{15} -4.00000 q^{16} +5.00000 q^{17} +2.00000 q^{18} +2.00000 q^{19} +2.00000 q^{20} -3.00000 q^{21} -10.0000 q^{22} -1.00000 q^{23} +1.00000 q^{25} +2.00000 q^{26} +1.00000 q^{27} -6.00000 q^{28} +10.0000 q^{29} +2.00000 q^{30} -2.00000 q^{31} -8.00000 q^{32} -5.00000 q^{33} +10.0000 q^{34} -3.00000 q^{35} +2.00000 q^{36} -3.00000 q^{37} +4.00000 q^{38} +1.00000 q^{39} -9.00000 q^{41} -6.00000 q^{42} -4.00000 q^{43} -10.0000 q^{44} +1.00000 q^{45} -2.00000 q^{46} +10.0000 q^{47} -4.00000 q^{48} +2.00000 q^{49} +2.00000 q^{50} +5.00000 q^{51} +2.00000 q^{52} +9.00000 q^{53} +2.00000 q^{54} -5.00000 q^{55} +2.00000 q^{57} +20.0000 q^{58} +2.00000 q^{60} -11.0000 q^{61} -4.00000 q^{62} -3.00000 q^{63} -8.00000 q^{64} +1.00000 q^{65} -10.0000 q^{66} -4.00000 q^{67} +10.0000 q^{68} -1.00000 q^{69} -6.00000 q^{70} +15.0000 q^{71} +6.00000 q^{73} -6.00000 q^{74} +1.00000 q^{75} +4.00000 q^{76} +15.0000 q^{77} +2.00000 q^{78} -11.0000 q^{79} -4.00000 q^{80} +1.00000 q^{81} -18.0000 q^{82} +8.00000 q^{83} -6.00000 q^{84} +5.00000 q^{85} -8.00000 q^{86} +10.0000 q^{87} -11.0000 q^{89} +2.00000 q^{90} -3.00000 q^{91} -2.00000 q^{92} -2.00000 q^{93} +20.0000 q^{94} +2.00000 q^{95} -8.00000 q^{96} -9.00000 q^{97} +4.00000 q^{98} -5.00000 q^{99} +O(q^{100})$$

## 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 $$-0.691875\pi$$
−0.566947 + 0.823754i $$0.691875\pi$$
$$8$$ 0 0
$$9$$ 1.00000 0.333333
$$10$$ 2.00000 0.632456
$$11$$ −5.00000 −1.50756 −0.753778 0.657129i $$-0.771771\pi$$
−0.753778 + 0.657129i $$0.771771\pi$$
$$12$$ 2.00000 0.577350
$$13$$ 1.00000 0.277350
$$14$$ −6.00000 −1.60357
$$15$$ 1.00000 0.258199
$$16$$ −4.00000 −1.00000
$$17$$ 5.00000 1.21268 0.606339 0.795206i $$-0.292637\pi$$
0.606339 + 0.795206i $$0.292637\pi$$
$$18$$ 2.00000 0.471405
$$19$$ 2.00000 0.458831 0.229416 0.973329i $$-0.426318\pi$$
0.229416 + 0.973329i $$0.426318\pi$$
$$20$$ 2.00000 0.447214
$$21$$ −3.00000 −0.654654
$$22$$ −10.0000 −2.13201
$$23$$ −1.00000 −0.208514 −0.104257 0.994550i $$-0.533247\pi$$
−0.104257 + 0.994550i $$0.533247\pi$$
$$24$$ 0 0
$$25$$ 1.00000 0.200000
$$26$$ 2.00000 0.392232
$$27$$ 1.00000 0.192450
$$28$$ −6.00000 −1.13389
$$29$$ 10.0000 1.85695 0.928477 0.371391i $$-0.121119\pi$$
0.928477 + 0.371391i $$0.121119\pi$$
$$30$$ 2.00000 0.365148
$$31$$ −2.00000 −0.359211 −0.179605 0.983739i $$-0.557482\pi$$
−0.179605 + 0.983739i $$0.557482\pi$$
$$32$$ −8.00000 −1.41421
$$33$$ −5.00000 −0.870388
$$34$$ 10.0000 1.71499
$$35$$ −3.00000 −0.507093
$$36$$ 2.00000 0.333333
$$37$$ −3.00000 −0.493197 −0.246598 0.969118i $$-0.579313\pi$$
−0.246598 + 0.969118i $$0.579313\pi$$
$$38$$ 4.00000 0.648886
$$39$$ 1.00000 0.160128
$$40$$ 0 0
$$41$$ −9.00000 −1.40556 −0.702782 0.711405i $$-0.748059\pi$$
−0.702782 + 0.711405i $$0.748059\pi$$
$$42$$ −6.00000 −0.925820
$$43$$ −4.00000 −0.609994 −0.304997 0.952353i $$-0.598656\pi$$
−0.304997 + 0.952353i $$0.598656\pi$$
$$44$$ −10.0000 −1.50756
$$45$$ 1.00000 0.149071
$$46$$ −2.00000 −0.294884
$$47$$ 10.0000 1.45865 0.729325 0.684167i $$-0.239834\pi$$
0.729325 + 0.684167i $$0.239834\pi$$
$$48$$ −4.00000 −0.577350
$$49$$ 2.00000 0.285714
$$50$$ 2.00000 0.282843
$$51$$ 5.00000 0.700140
$$52$$ 2.00000 0.277350
$$53$$ 9.00000 1.23625 0.618123 0.786082i $$-0.287894\pi$$
0.618123 + 0.786082i $$0.287894\pi$$
$$54$$ 2.00000 0.272166
$$55$$ −5.00000 −0.674200
$$56$$ 0 0
$$57$$ 2.00000 0.264906
$$58$$ 20.0000 2.62613
$$59$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$60$$ 2.00000 0.258199
$$61$$ −11.0000 −1.40841 −0.704203 0.709999i $$-0.748695\pi$$
−0.704203 + 0.709999i $$0.748695\pi$$
$$62$$ −4.00000 −0.508001
$$63$$ −3.00000 −0.377964
$$64$$ −8.00000 −1.00000
$$65$$ 1.00000 0.124035
$$66$$ −10.0000 −1.23091
$$67$$ −4.00000 −0.488678 −0.244339 0.969690i $$-0.578571\pi$$
−0.244339 + 0.969690i $$0.578571\pi$$
$$68$$ 10.0000 1.21268
$$69$$ −1.00000 −0.120386
$$70$$ −6.00000 −0.717137
$$71$$ 15.0000 1.78017 0.890086 0.455792i $$-0.150644\pi$$
0.890086 + 0.455792i $$0.150644\pi$$
$$72$$ 0 0
$$73$$ 6.00000 0.702247 0.351123 0.936329i $$-0.385800\pi$$
0.351123 + 0.936329i $$0.385800\pi$$
$$74$$ −6.00000 −0.697486
$$75$$ 1.00000 0.115470
$$76$$ 4.00000 0.458831
$$77$$ 15.0000 1.70941
$$78$$ 2.00000 0.226455
$$79$$ −11.0000 −1.23760 −0.618798 0.785550i $$-0.712380\pi$$
−0.618798 + 0.785550i $$0.712380\pi$$
$$80$$ −4.00000 −0.447214
$$81$$ 1.00000 0.111111
$$82$$ −18.0000 −1.98777
$$83$$ 8.00000 0.878114 0.439057 0.898459i $$-0.355313\pi$$
0.439057 + 0.898459i $$0.355313\pi$$
$$84$$ −6.00000 −0.654654
$$85$$ 5.00000 0.542326
$$86$$ −8.00000 −0.862662
$$87$$ 10.0000 1.07211
$$88$$ 0 0
$$89$$ −11.0000 −1.16600 −0.582999 0.812473i $$-0.698121\pi$$
−0.582999 + 0.812473i $$0.698121\pi$$
$$90$$ 2.00000 0.210819
$$91$$ −3.00000 −0.314485
$$92$$ −2.00000 −0.208514
$$93$$ −2.00000 −0.207390
$$94$$ 20.0000 2.06284
$$95$$ 2.00000 0.205196
$$96$$ −8.00000 −0.816497
$$97$$ −9.00000 −0.913812 −0.456906 0.889515i $$-0.651042\pi$$
−0.456906 + 0.889515i $$0.651042\pi$$
$$98$$ 4.00000 0.404061
$$99$$ −5.00000 −0.502519
$$100$$ 2.00000 0.200000
$$101$$ −12.0000 −1.19404 −0.597022 0.802225i $$-0.703650\pi$$
−0.597022 + 0.802225i $$0.703650\pi$$
$$102$$ 10.0000 0.990148
$$103$$ 4.00000 0.394132 0.197066 0.980390i $$-0.436859\pi$$
0.197066 + 0.980390i $$0.436859\pi$$
$$104$$ 0 0
$$105$$ −3.00000 −0.292770
$$106$$ 18.0000 1.74831
$$107$$ 3.00000 0.290021 0.145010 0.989430i $$-0.453678\pi$$
0.145010 + 0.989430i $$0.453678\pi$$
$$108$$ 2.00000 0.192450
$$109$$ 16.0000 1.53252 0.766261 0.642529i $$-0.222115\pi$$
0.766261 + 0.642529i $$0.222115\pi$$
$$110$$ −10.0000 −0.953463
$$111$$ −3.00000 −0.284747
$$112$$ 12.0000 1.13389
$$113$$ 2.00000 0.188144 0.0940721 0.995565i $$-0.470012\pi$$
0.0940721 + 0.995565i $$0.470012\pi$$
$$114$$ 4.00000 0.374634
$$115$$ −1.00000 −0.0932505
$$116$$ 20.0000 1.85695
$$117$$ 1.00000 0.0924500
$$118$$ 0 0
$$119$$ −15.0000 −1.37505
$$120$$ 0 0
$$121$$ 14.0000 1.27273
$$122$$ −22.0000 −1.99179
$$123$$ −9.00000 −0.811503
$$124$$ −4.00000 −0.359211
$$125$$ 1.00000 0.0894427
$$126$$ −6.00000 −0.534522
$$127$$ 14.0000 1.24230 0.621150 0.783692i $$-0.286666\pi$$
0.621150 + 0.783692i $$0.286666\pi$$
$$128$$ 0 0
$$129$$ −4.00000 −0.352180
$$130$$ 2.00000 0.175412
$$131$$ 6.00000 0.524222 0.262111 0.965038i $$-0.415581\pi$$
0.262111 + 0.965038i $$0.415581\pi$$
$$132$$ −10.0000 −0.870388
$$133$$ −6.00000 −0.520266
$$134$$ −8.00000 −0.691095
$$135$$ 1.00000 0.0860663
$$136$$ 0 0
$$137$$ 6.00000 0.512615 0.256307 0.966595i $$-0.417494\pi$$
0.256307 + 0.966595i $$0.417494\pi$$
$$138$$ −2.00000 −0.170251
$$139$$ −17.0000 −1.44192 −0.720961 0.692976i $$-0.756299\pi$$
−0.720961 + 0.692976i $$0.756299\pi$$
$$140$$ −6.00000 −0.507093
$$141$$ 10.0000 0.842152
$$142$$ 30.0000 2.51754
$$143$$ −5.00000 −0.418121
$$144$$ −4.00000 −0.333333
$$145$$ 10.0000 0.830455
$$146$$ 12.0000 0.993127
$$147$$ 2.00000 0.164957
$$148$$ −6.00000 −0.493197
$$149$$ −7.00000 −0.573462 −0.286731 0.958011i $$-0.592569\pi$$
−0.286731 + 0.958011i $$0.592569\pi$$
$$150$$ 2.00000 0.163299
$$151$$ −12.0000 −0.976546 −0.488273 0.872691i $$-0.662373\pi$$
−0.488273 + 0.872691i $$0.662373\pi$$
$$152$$ 0 0
$$153$$ 5.00000 0.404226
$$154$$ 30.0000 2.41747
$$155$$ −2.00000 −0.160644
$$156$$ 2.00000 0.160128
$$157$$ 22.0000 1.75579 0.877896 0.478852i $$-0.158947\pi$$
0.877896 + 0.478852i $$0.158947\pi$$
$$158$$ −22.0000 −1.75023
$$159$$ 9.00000 0.713746
$$160$$ −8.00000 −0.632456
$$161$$ 3.00000 0.236433
$$162$$ 2.00000 0.157135
$$163$$ −11.0000 −0.861586 −0.430793 0.902451i $$-0.641766\pi$$
−0.430793 + 0.902451i $$0.641766\pi$$
$$164$$ −18.0000 −1.40556
$$165$$ −5.00000 −0.389249
$$166$$ 16.0000 1.24184
$$167$$ −8.00000 −0.619059 −0.309529 0.950890i $$-0.600171\pi$$
−0.309529 + 0.950890i $$0.600171\pi$$
$$168$$ 0 0
$$169$$ 1.00000 0.0769231
$$170$$ 10.0000 0.766965
$$171$$ 2.00000 0.152944
$$172$$ −8.00000 −0.609994
$$173$$ −2.00000 −0.152057 −0.0760286 0.997106i $$-0.524224\pi$$
−0.0760286 + 0.997106i $$0.524224\pi$$
$$174$$ 20.0000 1.51620
$$175$$ −3.00000 −0.226779
$$176$$ 20.0000 1.50756
$$177$$ 0 0
$$178$$ −22.0000 −1.64897
$$179$$ −6.00000 −0.448461 −0.224231 0.974536i $$-0.571987\pi$$
−0.224231 + 0.974536i $$0.571987\pi$$
$$180$$ 2.00000 0.149071
$$181$$ −23.0000 −1.70958 −0.854788 0.518977i $$-0.826313\pi$$
−0.854788 + 0.518977i $$0.826313\pi$$
$$182$$ −6.00000 −0.444750
$$183$$ −11.0000 −0.813143
$$184$$ 0 0
$$185$$ −3.00000 −0.220564
$$186$$ −4.00000 −0.293294
$$187$$ −25.0000 −1.82818
$$188$$ 20.0000 1.45865
$$189$$ −3.00000 −0.218218
$$190$$ 4.00000 0.290191
$$191$$ −20.0000 −1.44715 −0.723575 0.690246i $$-0.757502\pi$$
−0.723575 + 0.690246i $$0.757502\pi$$
$$192$$ −8.00000 −0.577350
$$193$$ 13.0000 0.935760 0.467880 0.883792i $$-0.345018\pi$$
0.467880 + 0.883792i $$0.345018\pi$$
$$194$$ −18.0000 −1.29232
$$195$$ 1.00000 0.0716115
$$196$$ 4.00000 0.285714
$$197$$ −12.0000 −0.854965 −0.427482 0.904024i $$-0.640599\pi$$
−0.427482 + 0.904024i $$0.640599\pi$$
$$198$$ −10.0000 −0.710669
$$199$$ 4.00000 0.283552 0.141776 0.989899i $$-0.454719\pi$$
0.141776 + 0.989899i $$0.454719\pi$$
$$200$$ 0 0
$$201$$ −4.00000 −0.282138
$$202$$ −24.0000 −1.68863
$$203$$ −30.0000 −2.10559
$$204$$ 10.0000 0.700140
$$205$$ −9.00000 −0.628587
$$206$$ 8.00000 0.557386
$$207$$ −1.00000 −0.0695048
$$208$$ −4.00000 −0.277350
$$209$$ −10.0000 −0.691714
$$210$$ −6.00000 −0.414039
$$211$$ −4.00000 −0.275371 −0.137686 0.990476i $$-0.543966\pi$$
−0.137686 + 0.990476i $$0.543966\pi$$
$$212$$ 18.0000 1.23625
$$213$$ 15.0000 1.02778
$$214$$ 6.00000 0.410152
$$215$$ −4.00000 −0.272798
$$216$$ 0 0
$$217$$ 6.00000 0.407307
$$218$$ 32.0000 2.16731
$$219$$ 6.00000 0.405442
$$220$$ −10.0000 −0.674200
$$221$$ 5.00000 0.336336
$$222$$ −6.00000 −0.402694
$$223$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$224$$ 24.0000 1.60357
$$225$$ 1.00000 0.0666667
$$226$$ 4.00000 0.266076
$$227$$ −18.0000 −1.19470 −0.597351 0.801980i $$-0.703780\pi$$
−0.597351 + 0.801980i $$0.703780\pi$$
$$228$$ 4.00000 0.264906
$$229$$ −14.0000 −0.925146 −0.462573 0.886581i $$-0.653074\pi$$
−0.462573 + 0.886581i $$0.653074\pi$$
$$230$$ −2.00000 −0.131876
$$231$$ 15.0000 0.986928
$$232$$ 0 0
$$233$$ −25.0000 −1.63780 −0.818902 0.573933i $$-0.805417\pi$$
−0.818902 + 0.573933i $$0.805417\pi$$
$$234$$ 2.00000 0.130744
$$235$$ 10.0000 0.652328
$$236$$ 0 0
$$237$$ −11.0000 −0.714527
$$238$$ −30.0000 −1.94461
$$239$$ 15.0000 0.970269 0.485135 0.874439i $$-0.338771\pi$$
0.485135 + 0.874439i $$0.338771\pi$$
$$240$$ −4.00000 −0.258199
$$241$$ −14.0000 −0.901819 −0.450910 0.892570i $$-0.648900\pi$$
−0.450910 + 0.892570i $$0.648900\pi$$
$$242$$ 28.0000 1.79991
$$243$$ 1.00000 0.0641500
$$244$$ −22.0000 −1.40841
$$245$$ 2.00000 0.127775
$$246$$ −18.0000 −1.14764
$$247$$ 2.00000 0.127257
$$248$$ 0 0
$$249$$ 8.00000 0.506979
$$250$$ 2.00000 0.126491
$$251$$ 20.0000 1.26239 0.631194 0.775625i $$-0.282565\pi$$
0.631194 + 0.775625i $$0.282565\pi$$
$$252$$ −6.00000 −0.377964
$$253$$ 5.00000 0.314347
$$254$$ 28.0000 1.75688
$$255$$ 5.00000 0.313112
$$256$$ 16.0000 1.00000
$$257$$ 18.0000 1.12281 0.561405 0.827541i $$-0.310261\pi$$
0.561405 + 0.827541i $$0.310261\pi$$
$$258$$ −8.00000 −0.498058
$$259$$ 9.00000 0.559233
$$260$$ 2.00000 0.124035
$$261$$ 10.0000 0.618984
$$262$$ 12.0000 0.741362
$$263$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$264$$ 0 0
$$265$$ 9.00000 0.552866
$$266$$ −12.0000 −0.735767
$$267$$ −11.0000 −0.673189
$$268$$ −8.00000 −0.488678
$$269$$ 32.0000 1.95107 0.975537 0.219834i $$-0.0705517\pi$$
0.975537 + 0.219834i $$0.0705517\pi$$
$$270$$ 2.00000 0.121716
$$271$$ −2.00000 −0.121491 −0.0607457 0.998153i $$-0.519348\pi$$
−0.0607457 + 0.998153i $$0.519348\pi$$
$$272$$ −20.0000 −1.21268
$$273$$ −3.00000 −0.181568
$$274$$ 12.0000 0.724947
$$275$$ −5.00000 −0.301511
$$276$$ −2.00000 −0.120386
$$277$$ 26.0000 1.56219 0.781094 0.624413i $$-0.214662\pi$$
0.781094 + 0.624413i $$0.214662\pi$$
$$278$$ −34.0000 −2.03918
$$279$$ −2.00000 −0.119737
$$280$$ 0 0
$$281$$ −10.0000 −0.596550 −0.298275 0.954480i $$-0.596411\pi$$
−0.298275 + 0.954480i $$0.596411\pi$$
$$282$$ 20.0000 1.19098
$$283$$ −8.00000 −0.475551 −0.237775 0.971320i $$-0.576418\pi$$
−0.237775 + 0.971320i $$0.576418\pi$$
$$284$$ 30.0000 1.78017
$$285$$ 2.00000 0.118470
$$286$$ −10.0000 −0.591312
$$287$$ 27.0000 1.59376
$$288$$ −8.00000 −0.471405
$$289$$ 8.00000 0.470588
$$290$$ 20.0000 1.17444
$$291$$ −9.00000 −0.527589
$$292$$ 12.0000 0.702247
$$293$$ 24.0000 1.40209 0.701047 0.713115i $$-0.252716\pi$$
0.701047 + 0.713115i $$0.252716\pi$$
$$294$$ 4.00000 0.233285
$$295$$ 0 0
$$296$$ 0 0
$$297$$ −5.00000 −0.290129
$$298$$ −14.0000 −0.810998
$$299$$ −1.00000 −0.0578315
$$300$$ 2.00000 0.115470
$$301$$ 12.0000 0.691669
$$302$$ −24.0000 −1.38104
$$303$$ −12.0000 −0.689382
$$304$$ −8.00000 −0.458831
$$305$$ −11.0000 −0.629858
$$306$$ 10.0000 0.571662
$$307$$ −19.0000 −1.08439 −0.542194 0.840254i $$-0.682406\pi$$
−0.542194 + 0.840254i $$0.682406\pi$$
$$308$$ 30.0000 1.70941
$$309$$ 4.00000 0.227552
$$310$$ −4.00000 −0.227185
$$311$$ 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 $$0.591202\pi$$
$$314$$ 44.0000 2.48306
$$315$$ −3.00000 −0.169031
$$316$$ −22.0000 −1.23760
$$317$$ 12.0000 0.673987 0.336994 0.941507i $$-0.390590\pi$$
0.336994 + 0.941507i $$0.390590\pi$$
$$318$$ 18.0000 1.00939
$$319$$ −50.0000 −2.79946
$$320$$ −8.00000 −0.447214
$$321$$ 3.00000 0.167444
$$322$$ 6.00000 0.334367
$$323$$ 10.0000 0.556415
$$324$$ 2.00000 0.111111
$$325$$ 1.00000 0.0554700
$$326$$ −22.0000 −1.21847
$$327$$ 16.0000 0.884802
$$328$$ 0 0
$$329$$ −30.0000 −1.65395
$$330$$ −10.0000 −0.550482
$$331$$ 32.0000 1.75888 0.879440 0.476011i $$-0.157918\pi$$
0.879440 + 0.476011i $$0.157918\pi$$
$$332$$ 16.0000 0.878114
$$333$$ −3.00000 −0.164399
$$334$$ −16.0000 −0.875481
$$335$$ −4.00000 −0.218543
$$336$$ 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
$$339$$ 2.00000 0.108625
$$340$$ 10.0000 0.542326
$$341$$ 10.0000 0.541530
$$342$$ 4.00000 0.216295
$$343$$ 15.0000 0.809924
$$344$$ 0 0
$$345$$ −1.00000 −0.0538382
$$346$$ −4.00000 −0.215041
$$347$$ −1.00000 −0.0536828 −0.0268414 0.999640i $$-0.508545\pi$$
−0.0268414 + 0.999640i $$0.508545\pi$$
$$348$$ 20.0000 1.07211
$$349$$ 20.0000 1.07058 0.535288 0.844670i $$-0.320203\pi$$
0.535288 + 0.844670i $$0.320203\pi$$
$$350$$ −6.00000 −0.320713
$$351$$ 1.00000 0.0533761
$$352$$ 40.0000 2.13201
$$353$$ −14.0000 −0.745145 −0.372572 0.928003i $$-0.621524\pi$$
−0.372572 + 0.928003i $$0.621524\pi$$
$$354$$ 0 0
$$355$$ 15.0000 0.796117
$$356$$ −22.0000 −1.16600
$$357$$ −15.0000 −0.793884
$$358$$ −12.0000 −0.634220
$$359$$ 16.0000 0.844448 0.422224 0.906492i $$-0.361250\pi$$
0.422224 + 0.906492i $$0.361250\pi$$
$$360$$ 0 0
$$361$$ −15.0000 −0.789474
$$362$$ −46.0000 −2.41771
$$363$$ 14.0000 0.734809
$$364$$ −6.00000 −0.314485
$$365$$ 6.00000 0.314054
$$366$$ −22.0000 −1.14996
$$367$$ 8.00000 0.417597 0.208798 0.977959i $$-0.433045\pi$$
0.208798 + 0.977959i $$0.433045\pi$$
$$368$$ 4.00000 0.208514
$$369$$ −9.00000 −0.468521
$$370$$ −6.00000 −0.311925
$$371$$ −27.0000 −1.40177
$$372$$ −4.00000 −0.207390
$$373$$ 16.0000 0.828449 0.414224 0.910175i $$-0.364053\pi$$
0.414224 + 0.910175i $$0.364053\pi$$
$$374$$ −50.0000 −2.58544
$$375$$ 1.00000 0.0516398
$$376$$ 0 0
$$377$$ 10.0000 0.515026
$$378$$ −6.00000 −0.308607
$$379$$ 6.00000 0.308199 0.154100 0.988055i $$-0.450752\pi$$
0.154100 + 0.988055i $$0.450752\pi$$
$$380$$ 4.00000 0.205196
$$381$$ 14.0000 0.717242
$$382$$ −40.0000 −2.04658
$$383$$ −18.0000 −0.919757 −0.459879 0.887982i $$-0.652107\pi$$
−0.459879 + 0.887982i $$0.652107\pi$$
$$384$$ 0 0
$$385$$ 15.0000 0.764471
$$386$$ 26.0000 1.32337
$$387$$ −4.00000 −0.203331
$$388$$ −18.0000 −0.913812
$$389$$ −24.0000 −1.21685 −0.608424 0.793612i $$-0.708198\pi$$
−0.608424 + 0.793612i $$0.708198\pi$$
$$390$$ 2.00000 0.101274
$$391$$ −5.00000 −0.252861
$$392$$ 0 0
$$393$$ 6.00000 0.302660
$$394$$ −24.0000 −1.20910
$$395$$ −11.0000 −0.553470
$$396$$ −10.0000 −0.502519
$$397$$ −19.0000 −0.953583 −0.476791 0.879017i $$-0.658200\pi$$
−0.476791 + 0.879017i $$0.658200\pi$$
$$398$$ 8.00000 0.401004
$$399$$ −6.00000 −0.300376
$$400$$ −4.00000 −0.200000
$$401$$ −18.0000 −0.898877 −0.449439 0.893311i $$-0.648376\pi$$
−0.449439 + 0.893311i $$0.648376\pi$$
$$402$$ −8.00000 −0.399004
$$403$$ −2.00000 −0.0996271
$$404$$ −24.0000 −1.19404
$$405$$ 1.00000 0.0496904
$$406$$ −60.0000 −2.97775
$$407$$ 15.0000 0.743522
$$408$$ 0 0
$$409$$ −26.0000 −1.28562 −0.642809 0.766027i $$-0.722231\pi$$
−0.642809 + 0.766027i $$0.722231\pi$$
$$410$$ −18.0000 −0.888957
$$411$$ 6.00000 0.295958
$$412$$ 8.00000 0.394132
$$413$$ 0 0
$$414$$ −2.00000 −0.0982946
$$415$$ 8.00000 0.392705
$$416$$ −8.00000 −0.392232
$$417$$ −17.0000 −0.832494
$$418$$ −20.0000 −0.978232
$$419$$ 26.0000 1.27018 0.635092 0.772437i $$-0.280962\pi$$
0.635092 + 0.772437i $$0.280962\pi$$
$$420$$ −6.00000 −0.292770
$$421$$ 32.0000 1.55958 0.779792 0.626038i $$-0.215325\pi$$
0.779792 + 0.626038i $$0.215325\pi$$
$$422$$ −8.00000 −0.389434
$$423$$ 10.0000 0.486217
$$424$$ 0 0
$$425$$ 5.00000 0.242536
$$426$$ 30.0000 1.45350
$$427$$ 33.0000 1.59698
$$428$$ 6.00000 0.290021
$$429$$ −5.00000 −0.241402
$$430$$ −8.00000 −0.385794
$$431$$ −16.0000 −0.770693 −0.385346 0.922772i $$-0.625918\pi$$
−0.385346 + 0.922772i $$0.625918\pi$$
$$432$$ −4.00000 −0.192450
$$433$$ 24.0000 1.15337 0.576683 0.816968i $$-0.304347\pi$$
0.576683 + 0.816968i $$0.304347\pi$$
$$434$$ 12.0000 0.576018
$$435$$ 10.0000 0.479463
$$436$$ 32.0000 1.53252
$$437$$ −2.00000 −0.0956730
$$438$$ 12.0000 0.573382
$$439$$ −33.0000 −1.57500 −0.787502 0.616312i $$-0.788626\pi$$
−0.787502 + 0.616312i $$0.788626\pi$$
$$440$$ 0 0
$$441$$ 2.00000 0.0952381
$$442$$ 10.0000 0.475651
$$443$$ 35.0000 1.66290 0.831450 0.555599i $$-0.187511\pi$$
0.831450 + 0.555599i $$0.187511\pi$$
$$444$$ −6.00000 −0.284747
$$445$$ −11.0000 −0.521450
$$446$$ 0 0
$$447$$ −7.00000 −0.331089
$$448$$ 24.0000 1.13389
$$449$$ 15.0000 0.707894 0.353947 0.935266i $$-0.384839\pi$$
0.353947 + 0.935266i $$0.384839\pi$$
$$450$$ 2.00000 0.0942809
$$451$$ 45.0000 2.11897
$$452$$ 4.00000 0.188144
$$453$$ −12.0000 −0.563809
$$454$$ −36.0000 −1.68956
$$455$$ −3.00000 −0.140642
$$456$$ 0 0
$$457$$ −13.0000 −0.608114 −0.304057 0.952654i $$-0.598341\pi$$
−0.304057 + 0.952654i $$0.598341\pi$$
$$458$$ −28.0000 −1.30835
$$459$$ 5.00000 0.233380
$$460$$ −2.00000 −0.0932505
$$461$$ 3.00000 0.139724 0.0698620 0.997557i $$-0.477744\pi$$
0.0698620 + 0.997557i $$0.477744\pi$$
$$462$$ 30.0000 1.39573
$$463$$ 5.00000 0.232370 0.116185 0.993228i $$-0.462933\pi$$
0.116185 + 0.993228i $$0.462933\pi$$
$$464$$ −40.0000 −1.85695
$$465$$ −2.00000 −0.0927478
$$466$$ −50.0000 −2.31621
$$467$$ −29.0000 −1.34196 −0.670980 0.741475i $$-0.734126\pi$$
−0.670980 + 0.741475i $$0.734126\pi$$
$$468$$ 2.00000 0.0924500
$$469$$ 12.0000 0.554109
$$470$$ 20.0000 0.922531
$$471$$ 22.0000 1.01371
$$472$$ 0 0
$$473$$ 20.0000 0.919601
$$474$$ −22.0000 −1.01049
$$475$$ 2.00000 0.0917663
$$476$$ −30.0000 −1.37505
$$477$$ 9.00000 0.412082
$$478$$ 30.0000 1.37217
$$479$$ 5.00000 0.228456 0.114228 0.993455i $$-0.463561\pi$$
0.114228 + 0.993455i $$0.463561\pi$$
$$480$$ −8.00000 −0.365148
$$481$$ −3.00000 −0.136788
$$482$$ −28.0000 −1.27537
$$483$$ 3.00000 0.136505
$$484$$ 28.0000 1.27273
$$485$$ −9.00000 −0.408669
$$486$$ 2.00000 0.0907218
$$487$$ 7.00000 0.317200 0.158600 0.987343i $$-0.449302\pi$$
0.158600 + 0.987343i $$0.449302\pi$$
$$488$$ 0 0
$$489$$ −11.0000 −0.497437
$$490$$ 4.00000 0.180702
$$491$$ 16.0000 0.722070 0.361035 0.932552i $$-0.382424\pi$$
0.361035 + 0.932552i $$0.382424\pi$$
$$492$$ −18.0000 −0.811503
$$493$$ 50.0000 2.25189
$$494$$ 4.00000 0.179969
$$495$$ −5.00000 −0.224733
$$496$$ 8.00000 0.359211
$$497$$ −45.0000 −2.01853
$$498$$ 16.0000 0.716977
$$499$$ 34.0000 1.52205 0.761025 0.648723i $$-0.224697\pi$$
0.761025 + 0.648723i $$0.224697\pi$$
$$500$$ 2.00000 0.0894427
$$501$$ −8.00000 −0.357414
$$502$$ 40.0000 1.78529
$$503$$ 12.0000 0.535054 0.267527 0.963550i $$-0.413794\pi$$
0.267527 + 0.963550i $$0.413794\pi$$
$$504$$ 0 0
$$505$$ −12.0000 −0.533993
$$506$$ 10.0000 0.444554
$$507$$ 1.00000 0.0444116
$$508$$ 28.0000 1.24230
$$509$$ 21.0000 0.930809 0.465404 0.885098i $$-0.345909\pi$$
0.465404 + 0.885098i $$0.345909\pi$$
$$510$$ 10.0000 0.442807
$$511$$ −18.0000 −0.796273
$$512$$ 32.0000 1.41421
$$513$$ 2.00000 0.0883022
$$514$$ 36.0000 1.58789
$$515$$ 4.00000 0.176261
$$516$$ −8.00000 −0.352180
$$517$$ −50.0000 −2.19900
$$518$$ 18.0000 0.790875
$$519$$ −2.00000 −0.0877903
$$520$$ 0 0
$$521$$ −22.0000 −0.963837 −0.481919 0.876216i $$-0.660060\pi$$
−0.481919 + 0.876216i $$0.660060\pi$$
$$522$$ 20.0000 0.875376
$$523$$ −20.0000 −0.874539 −0.437269 0.899331i $$-0.644054\pi$$
−0.437269 + 0.899331i $$0.644054\pi$$
$$524$$ 12.0000 0.524222
$$525$$ −3.00000 −0.130931
$$526$$ 0 0
$$527$$ −10.0000 −0.435607
$$528$$ 20.0000 0.870388
$$529$$ −22.0000 −0.956522
$$530$$ 18.0000 0.781870
$$531$$ 0 0
$$532$$ −12.0000 −0.520266
$$533$$ −9.00000 −0.389833
$$534$$ −22.0000 −0.952033
$$535$$ 3.00000 0.129701
$$536$$ 0 0
$$537$$ −6.00000 −0.258919
$$538$$ 64.0000 2.75924
$$539$$ −10.0000 −0.430730
$$540$$ 2.00000 0.0860663
$$541$$ 38.0000 1.63375 0.816874 0.576816i $$-0.195705\pi$$
0.816874 + 0.576816i $$0.195705\pi$$
$$542$$ −4.00000 −0.171815
$$543$$ −23.0000 −0.987024
$$544$$ −40.0000 −1.71499
$$545$$ 16.0000 0.685365
$$546$$ −6.00000 −0.256776
$$547$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$548$$ 12.0000 0.512615
$$549$$ −11.0000 −0.469469
$$550$$ −10.0000 −0.426401
$$551$$ 20.0000 0.852029
$$552$$ 0 0
$$553$$ 33.0000 1.40330
$$554$$ 52.0000 2.20927
$$555$$ −3.00000 −0.127343
$$556$$ −34.0000 −1.44192
$$557$$ −30.0000 −1.27114 −0.635570 0.772043i $$-0.719235\pi$$
−0.635570 + 0.772043i $$0.719235\pi$$
$$558$$ −4.00000 −0.169334
$$559$$ −4.00000 −0.169182
$$560$$ 12.0000 0.507093
$$561$$ −25.0000 −1.05550
$$562$$ −20.0000 −0.843649
$$563$$ −41.0000 −1.72794 −0.863972 0.503540i $$-0.832031\pi$$
−0.863972 + 0.503540i $$0.832031\pi$$
$$564$$ 20.0000 0.842152
$$565$$ 2.00000 0.0841406
$$566$$ −16.0000 −0.672530
$$567$$ −3.00000 −0.125988
$$568$$ 0 0
$$569$$ 16.0000 0.670755 0.335377 0.942084i $$-0.391136\pi$$
0.335377 + 0.942084i $$0.391136\pi$$
$$570$$ 4.00000 0.167542
$$571$$ 17.0000 0.711428 0.355714 0.934595i $$-0.384238\pi$$
0.355714 + 0.934595i $$0.384238\pi$$
$$572$$ −10.0000 −0.418121
$$573$$ −20.0000 −0.835512
$$574$$ 54.0000 2.25392
$$575$$ −1.00000 −0.0417029
$$576$$ −8.00000 −0.333333
$$577$$ −21.0000 −0.874241 −0.437121 0.899403i $$-0.644002\pi$$
−0.437121 + 0.899403i $$0.644002\pi$$
$$578$$ 16.0000 0.665512
$$579$$ 13.0000 0.540262
$$580$$ 20.0000 0.830455
$$581$$ −24.0000 −0.995688
$$582$$ −18.0000 −0.746124
$$583$$ −45.0000 −1.86371
$$584$$ 0 0
$$585$$ 1.00000 0.0413449
$$586$$ 48.0000 1.98286
$$587$$ 42.0000 1.73353 0.866763 0.498721i $$-0.166197\pi$$
0.866763 + 0.498721i $$0.166197\pi$$
$$588$$ 4.00000 0.164957
$$589$$ −4.00000 −0.164817
$$590$$ 0 0
$$591$$ −12.0000 −0.493614
$$592$$ 12.0000 0.493197
$$593$$ −36.0000 −1.47834 −0.739171 0.673517i $$-0.764783\pi$$
−0.739171 + 0.673517i $$0.764783\pi$$
$$594$$ −10.0000 −0.410305
$$595$$ −15.0000 −0.614940
$$596$$ −14.0000 −0.573462
$$597$$ 4.00000 0.163709
$$598$$ −2.00000 −0.0817861
$$599$$ 4.00000 0.163436 0.0817178 0.996656i $$-0.473959\pi$$
0.0817178 + 0.996656i $$0.473959\pi$$
$$600$$ 0 0
$$601$$ −5.00000 −0.203954 −0.101977 0.994787i $$-0.532517\pi$$
−0.101977 + 0.994787i $$0.532517\pi$$
$$602$$ 24.0000 0.978167
$$603$$ −4.00000 −0.162893
$$604$$ −24.0000 −0.976546
$$605$$ 14.0000 0.569181
$$606$$ −24.0000 −0.974933
$$607$$ −40.0000 −1.62355 −0.811775 0.583970i $$-0.801498\pi$$
−0.811775 + 0.583970i $$0.801498\pi$$
$$608$$ −16.0000 −0.648886
$$609$$ −30.0000 −1.21566
$$610$$ −22.0000 −0.890754
$$611$$ 10.0000 0.404557
$$612$$ 10.0000 0.404226
$$613$$ 3.00000 0.121169 0.0605844 0.998163i $$-0.480704\pi$$
0.0605844 + 0.998163i $$0.480704\pi$$
$$614$$ −38.0000 −1.53356
$$615$$ −9.00000 −0.362915
$$616$$ 0 0
$$617$$ 22.0000 0.885687 0.442843 0.896599i $$-0.353970\pi$$
0.442843 + 0.896599i $$0.353970\pi$$
$$618$$ 8.00000 0.321807
$$619$$ −2.00000 −0.0803868 −0.0401934 0.999192i $$-0.512797\pi$$
−0.0401934 + 0.999192i $$0.512797\pi$$
$$620$$ −4.00000 −0.160644
$$621$$ −1.00000 −0.0401286
$$622$$ 48.0000 1.92462
$$623$$ 33.0000 1.32212
$$624$$ −4.00000 −0.160128
$$625$$ 1.00000 0.0400000
$$626$$ −20.0000 −0.799361
$$627$$ −10.0000 −0.399362
$$628$$ 44.0000 1.75579
$$629$$ −15.0000 −0.598089
$$630$$ −6.00000 −0.239046
$$631$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$632$$ 0 0
$$633$$ −4.00000 −0.158986
$$634$$ 24.0000 0.953162
$$635$$ 14.0000 0.555573
$$636$$ 18.0000 0.713746
$$637$$ 2.00000 0.0792429
$$638$$ −100.000 −3.95904
$$639$$ 15.0000 0.593391
$$640$$ 0 0
$$641$$ −36.0000 −1.42191 −0.710957 0.703235i $$-0.751738\pi$$
−0.710957 + 0.703235i $$0.751738\pi$$
$$642$$ 6.00000 0.236801
$$643$$ 1.00000 0.0394362 0.0197181 0.999806i $$-0.493723\pi$$
0.0197181 + 0.999806i $$0.493723\pi$$
$$644$$ 6.00000 0.236433
$$645$$ −4.00000 −0.157500
$$646$$ 20.0000 0.786889
$$647$$ 21.0000 0.825595 0.412798 0.910823i $$-0.364552\pi$$
0.412798 + 0.910823i $$0.364552\pi$$
$$648$$ 0 0
$$649$$ 0 0
$$650$$ 2.00000 0.0784465
$$651$$ 6.00000 0.235159
$$652$$ −22.0000 −0.861586
$$653$$ −30.0000 −1.17399 −0.586995 0.809590i $$-0.699689\pi$$
−0.586995 + 0.809590i $$0.699689\pi$$
$$654$$ 32.0000 1.25130
$$655$$ 6.00000 0.234439
$$656$$ 36.0000 1.40556
$$657$$ 6.00000 0.234082
$$658$$ −60.0000 −2.33904
$$659$$ 12.0000 0.467454 0.233727 0.972302i $$-0.424908\pi$$
0.233727 + 0.972302i $$0.424908\pi$$
$$660$$ −10.0000 −0.389249
$$661$$ −16.0000 −0.622328 −0.311164 0.950356i $$-0.600719\pi$$
−0.311164 + 0.950356i $$0.600719\pi$$
$$662$$ 64.0000 2.48743
$$663$$ 5.00000 0.194184
$$664$$ 0 0
$$665$$ −6.00000 −0.232670
$$666$$ −6.00000 −0.232495
$$667$$ −10.0000 −0.387202
$$668$$ −16.0000 −0.619059
$$669$$ 0 0
$$670$$ −8.00000 −0.309067
$$671$$ 55.0000 2.12325
$$672$$ 24.0000 0.925820
$$673$$ 22.0000 0.848038 0.424019 0.905653i $$-0.360619\pi$$
0.424019 + 0.905653i $$0.360619\pi$$
$$674$$ 8.00000 0.308148
$$675$$ 1.00000 0.0384900
$$676$$ 2.00000 0.0769231
$$677$$ −7.00000 −0.269032 −0.134516 0.990911i $$-0.542948\pi$$
−0.134516 + 0.990911i $$0.542948\pi$$
$$678$$ 4.00000 0.153619
$$679$$ 27.0000 1.03616
$$680$$ 0 0
$$681$$ −18.0000 −0.689761
$$682$$ 20.0000 0.765840
$$683$$ 4.00000 0.153056 0.0765279 0.997067i $$-0.475617\pi$$
0.0765279 + 0.997067i $$0.475617\pi$$
$$684$$ 4.00000 0.152944
$$685$$ 6.00000 0.229248
$$686$$ 30.0000 1.14541
$$687$$ −14.0000 −0.534133
$$688$$ 16.0000 0.609994
$$689$$ 9.00000 0.342873
$$690$$ −2.00000 −0.0761387
$$691$$ 6.00000 0.228251 0.114125 0.993466i $$-0.463593\pi$$
0.114125 + 0.993466i $$0.463593\pi$$
$$692$$ −4.00000 −0.152057
$$693$$ 15.0000 0.569803
$$694$$ −2.00000 −0.0759190
$$695$$ −17.0000 −0.644847
$$696$$ 0 0
$$697$$ −45.0000 −1.70450
$$698$$ 40.0000 1.51402
$$699$$ −25.0000 −0.945587
$$700$$ −6.00000 −0.226779
$$701$$ −4.00000 −0.151078 −0.0755390 0.997143i $$-0.524068\pi$$
−0.0755390 + 0.997143i $$0.524068\pi$$
$$702$$ 2.00000 0.0754851
$$703$$ −6.00000 −0.226294
$$704$$ 40.0000 1.50756
$$705$$ 10.0000 0.376622
$$706$$ −28.0000 −1.05379
$$707$$ 36.0000 1.35392
$$708$$ 0 0
$$709$$ 20.0000 0.751116 0.375558 0.926799i $$-0.377451\pi$$
0.375558 + 0.926799i $$0.377451\pi$$
$$710$$ 30.0000 1.12588
$$711$$ −11.0000 −0.412532
$$712$$ 0 0
$$713$$ 2.00000 0.0749006
$$714$$ −30.0000 −1.12272
$$715$$ −5.00000 −0.186989
$$716$$ −12.0000 −0.448461
$$717$$ 15.0000 0.560185
$$718$$ 32.0000 1.19423
$$719$$ 12.0000 0.447524 0.223762 0.974644i $$-0.428166\pi$$
0.223762 + 0.974644i $$0.428166\pi$$
$$720$$ −4.00000 −0.149071
$$721$$ −12.0000 −0.446903
$$722$$ −30.0000 −1.11648
$$723$$ −14.0000 −0.520666
$$724$$ −46.0000 −1.70958
$$725$$ 10.0000 0.371391
$$726$$ 28.0000 1.03918
$$727$$ 6.00000 0.222528 0.111264 0.993791i $$-0.464510\pi$$
0.111264 + 0.993791i $$0.464510\pi$$
$$728$$ 0 0
$$729$$ 1.00000 0.0370370
$$730$$ 12.0000 0.444140
$$731$$ −20.0000 −0.739727
$$732$$ −22.0000 −0.813143
$$733$$ −15.0000 −0.554038 −0.277019 0.960864i $$-0.589346\pi$$
−0.277019 + 0.960864i $$0.589346\pi$$
$$734$$ 16.0000 0.590571
$$735$$ 2.00000 0.0737711
$$736$$ 8.00000 0.294884
$$737$$ 20.0000 0.736709
$$738$$ −18.0000 −0.662589
$$739$$ 38.0000 1.39785 0.698926 0.715194i $$-0.253662\pi$$
0.698926 + 0.715194i $$0.253662\pi$$
$$740$$ −6.00000 −0.220564
$$741$$ 2.00000 0.0734718
$$742$$ −54.0000 −1.98240
$$743$$ 6.00000 0.220119 0.110059 0.993925i $$-0.464896\pi$$
0.110059 + 0.993925i $$0.464896\pi$$
$$744$$ 0 0
$$745$$ −7.00000 −0.256460
$$746$$ 32.0000 1.17160
$$747$$ 8.00000 0.292705
$$748$$ −50.0000 −1.82818
$$749$$ −9.00000 −0.328853
$$750$$ 2.00000 0.0730297
$$751$$ −45.0000 −1.64207 −0.821037 0.570875i $$-0.806604\pi$$
−0.821037 + 0.570875i $$0.806604\pi$$
$$752$$ −40.0000 −1.45865
$$753$$ 20.0000 0.728841
$$754$$ 20.0000 0.728357
$$755$$ −12.0000 −0.436725
$$756$$ −6.00000 −0.218218
$$757$$ 36.0000 1.30844 0.654221 0.756303i $$-0.272997\pi$$
0.654221 + 0.756303i $$0.272997\pi$$
$$758$$ 12.0000 0.435860
$$759$$ 5.00000 0.181489
$$760$$ 0 0
$$761$$ −14.0000 −0.507500 −0.253750 0.967270i $$-0.581664\pi$$
−0.253750 + 0.967270i $$0.581664\pi$$
$$762$$ 28.0000 1.01433
$$763$$ −48.0000 −1.73772
$$764$$ −40.0000 −1.44715
$$765$$ 5.00000 0.180775
$$766$$ −36.0000 −1.30073
$$767$$ 0 0
$$768$$ 16.0000 0.577350
$$769$$ −12.0000 −0.432731 −0.216366 0.976312i $$-0.569420\pi$$
−0.216366 + 0.976312i $$0.569420\pi$$
$$770$$ 30.0000 1.08112
$$771$$ 18.0000 0.648254
$$772$$ 26.0000 0.935760
$$773$$ 24.0000 0.863220 0.431610 0.902060i $$-0.357946\pi$$
0.431610 + 0.902060i $$0.357946\pi$$
$$774$$ −8.00000 −0.287554
$$775$$ −2.00000 −0.0718421
$$776$$ 0 0
$$777$$ 9.00000 0.322873
$$778$$ −48.0000 −1.72088
$$779$$ −18.0000 −0.644917
$$780$$ 2.00000 0.0716115
$$781$$ −75.0000 −2.68371
$$782$$ −10.0000 −0.357599
$$783$$ 10.0000 0.357371
$$784$$ −8.00000 −0.285714
$$785$$ 22.0000 0.785214
$$786$$ 12.0000 0.428026
$$787$$ 44.0000 1.56843 0.784215 0.620489i $$-0.213066\pi$$
0.784215 + 0.620489i $$0.213066\pi$$
$$788$$ −24.0000 −0.854965
$$789$$ 0 0
$$790$$ −22.0000 −0.782725
$$791$$ −6.00000 −0.213335
$$792$$ 0 0
$$793$$ −11.0000 −0.390621
$$794$$ −38.0000 −1.34857
$$795$$ 9.00000 0.319197
$$796$$ 8.00000 0.283552
$$797$$ −5.00000 −0.177109 −0.0885545 0.996071i $$-0.528225\pi$$
−0.0885545 + 0.996071i $$0.528225\pi$$
$$798$$ −12.0000 −0.424795
$$799$$ 50.0000 1.76887
$$800$$ −8.00000 −0.282843
$$801$$ −11.0000 −0.388666
$$802$$ −36.0000 −1.27120
$$803$$ −30.0000 −1.05868
$$804$$ −8.00000 −0.282138
$$805$$ 3.00000 0.105736
$$806$$ −4.00000 −0.140894
$$807$$ 32.0000 1.12645
$$808$$ 0 0
$$809$$ −6.00000 −0.210949 −0.105474 0.994422i $$-0.533636\pi$$
−0.105474 + 0.994422i $$0.533636\pi$$
$$810$$ 2.00000 0.0702728
$$811$$ −4.00000 −0.140459 −0.0702295 0.997531i $$-0.522373\pi$$
−0.0702295 + 0.997531i $$0.522373\pi$$
$$812$$ −60.0000 −2.10559
$$813$$ −2.00000 −0.0701431
$$814$$ 30.0000 1.05150
$$815$$ −11.0000 −0.385313
$$816$$ −20.0000 −0.700140
$$817$$ −8.00000 −0.279885
$$818$$ −52.0000 −1.81814
$$819$$ −3.00000 −0.104828
$$820$$ −18.0000 −0.628587
$$821$$ −41.0000 −1.43091 −0.715455 0.698659i $$-0.753781\pi$$
−0.715455 + 0.698659i $$0.753781\pi$$
$$822$$ 12.0000 0.418548
$$823$$ −48.0000 −1.67317 −0.836587 0.547833i $$-0.815453\pi$$
−0.836587 + 0.547833i $$0.815453\pi$$
$$824$$ 0 0
$$825$$ −5.00000 −0.174078
$$826$$ 0 0
$$827$$ −42.0000 −1.46048 −0.730242 0.683189i $$-0.760592\pi$$
−0.730242 + 0.683189i $$0.760592\pi$$
$$828$$ −2.00000 −0.0695048
$$829$$ 14.0000 0.486240 0.243120 0.969996i $$-0.421829\pi$$
0.243120 + 0.969996i $$0.421829\pi$$
$$830$$ 16.0000 0.555368
$$831$$ 26.0000 0.901930
$$832$$ −8.00000 −0.277350
$$833$$ 10.0000 0.346479
$$834$$ −34.0000 −1.17732
$$835$$ −8.00000 −0.276851
$$836$$ −20.0000 −0.691714
$$837$$ −2.00000 −0.0691301
$$838$$ 52.0000 1.79631
$$839$$ 7.00000 0.241667 0.120833 0.992673i $$-0.461443\pi$$
0.120833 + 0.992673i $$0.461443\pi$$
$$840$$ 0 0
$$841$$ 71.0000 2.44828
$$842$$ 64.0000 2.20559
$$843$$ −10.0000 −0.344418
$$844$$ −8.00000 −0.275371
$$845$$ 1.00000 0.0344010
$$846$$ 20.0000 0.687614
$$847$$ −42.0000 −1.44314
$$848$$ −36.0000 −1.23625
$$849$$ −8.00000 −0.274559
$$850$$ 10.0000 0.342997
$$851$$ 3.00000 0.102839
$$852$$ 30.0000 1.02778
$$853$$ 51.0000 1.74621 0.873103 0.487535i $$-0.162104\pi$$
0.873103 + 0.487535i $$0.162104\pi$$
$$854$$ 66.0000 2.25847
$$855$$ 2.00000 0.0683986
$$856$$ 0 0
$$857$$ −17.0000 −0.580709 −0.290354 0.956919i $$-0.593773\pi$$
−0.290354 + 0.956919i $$0.593773\pi$$
$$858$$ −10.0000 −0.341394
$$859$$ 35.0000 1.19418 0.597092 0.802173i $$-0.296323\pi$$
0.597092 + 0.802173i $$0.296323\pi$$
$$860$$ −8.00000 −0.272798
$$861$$ 27.0000 0.920158
$$862$$ −32.0000 −1.08992
$$863$$ −22.0000 −0.748889 −0.374444 0.927249i $$-0.622167\pi$$
−0.374444 + 0.927249i $$0.622167\pi$$
$$864$$ −8.00000 −0.272166
$$865$$ −2.00000 −0.0680020
$$866$$ 48.0000 1.63111
$$867$$ 8.00000 0.271694
$$868$$ 12.0000 0.407307
$$869$$ 55.0000 1.86575
$$870$$ 20.0000 0.678064
$$871$$ −4.00000 −0.135535
$$872$$ 0 0
$$873$$ −9.00000 −0.304604
$$874$$ −4.00000 −0.135302
$$875$$ −3.00000 −0.101419
$$876$$ 12.0000 0.405442
$$877$$ 22.0000 0.742887 0.371444 0.928456i $$-0.378863\pi$$
0.371444 + 0.928456i $$0.378863\pi$$
$$878$$ −66.0000 −2.22739
$$879$$ 24.0000 0.809500
$$880$$ 20.0000 0.674200
$$881$$ −14.0000 −0.471672 −0.235836 0.971793i $$-0.575783\pi$$
−0.235836 + 0.971793i $$0.575783\pi$$
$$882$$ 4.00000 0.134687
$$883$$ 12.0000 0.403832 0.201916 0.979403i $$-0.435283\pi$$
0.201916 + 0.979403i $$0.435283\pi$$
$$884$$ 10.0000 0.336336
$$885$$ 0 0
$$886$$ 70.0000 2.35170
$$887$$ −15.0000 −0.503651 −0.251825 0.967773i $$-0.581031\pi$$
−0.251825 + 0.967773i $$0.581031\pi$$
$$888$$ 0 0
$$889$$ −42.0000 −1.40863
$$890$$ −22.0000 −0.737442
$$891$$ −5.00000 −0.167506
$$892$$ 0 0
$$893$$ 20.0000 0.669274
$$894$$ −14.0000 −0.468230
$$895$$ −6.00000 −0.200558
$$896$$ 0 0
$$897$$ −1.00000 −0.0333890
$$898$$ 30.0000 1.00111
$$899$$ −20.0000 −0.667037
$$900$$ 2.00000 0.0666667
$$901$$ 45.0000 1.49917
$$902$$ 90.0000 2.99667
$$903$$ 12.0000 0.399335
$$904$$ 0 0
$$905$$ −23.0000 −0.764546
$$906$$ −24.0000 −0.797347
$$907$$ 2.00000 0.0664089 0.0332045 0.999449i $$-0.489429\pi$$
0.0332045 + 0.999449i $$0.489429\pi$$
$$908$$ −36.0000 −1.19470
$$909$$ −12.0000 −0.398015
$$910$$ −6.00000 −0.198898
$$911$$ 8.00000 0.265052 0.132526 0.991180i $$-0.457691\pi$$
0.132526 + 0.991180i $$0.457691\pi$$
$$912$$ −8.00000 −0.264906
$$913$$ −40.0000 −1.32381
$$914$$ −26.0000 −0.860004
$$915$$ −11.0000 −0.363649
$$916$$ −28.0000 −0.925146
$$917$$ −18.0000 −0.594412
$$918$$ 10.0000 0.330049
$$919$$ 29.0000 0.956622 0.478311 0.878191i $$-0.341249\pi$$
0.478311 + 0.878191i $$0.341249\pi$$
$$920$$ 0 0
$$921$$ −19.0000 −0.626071
$$922$$ 6.00000 0.197599
$$923$$ 15.0000 0.493731
$$924$$ 30.0000 0.986928
$$925$$ −3.00000 −0.0986394
$$926$$ 10.0000 0.328620
$$927$$ 4.00000 0.131377
$$928$$ −80.0000 −2.62613
$$929$$ 21.0000 0.688988 0.344494 0.938789i $$-0.388051\pi$$
0.344494 + 0.938789i $$0.388051\pi$$
$$930$$ −4.00000 −0.131165
$$931$$ 4.00000 0.131095
$$932$$ −50.0000 −1.63780
$$933$$ 24.0000 0.785725
$$934$$ −58.0000 −1.89782
$$935$$ −25.0000 −0.817587
$$936$$ 0 0
$$937$$ 2.00000 0.0653372 0.0326686 0.999466i $$-0.489599\pi$$
0.0326686 + 0.999466i $$0.489599\pi$$
$$938$$ 24.0000 0.783628
$$939$$ −10.0000 −0.326338
$$940$$ 20.0000 0.652328
$$941$$ −23.0000 −0.749779 −0.374889 0.927070i $$-0.622319\pi$$
−0.374889 + 0.927070i $$0.622319\pi$$
$$942$$ 44.0000 1.43360
$$943$$ 9.00000 0.293080
$$944$$ 0 0
$$945$$ −3.00000 −0.0975900
$$946$$ 40.0000 1.30051
$$947$$ 36.0000 1.16984 0.584921 0.811090i $$-0.301125\pi$$
0.584921 + 0.811090i $$0.301125\pi$$
$$948$$ −22.0000 −0.714527
$$949$$ 6.00000 0.194768
$$950$$ 4.00000 0.129777
$$951$$ 12.0000 0.389127
$$952$$ 0 0
$$953$$ −11.0000 −0.356325 −0.178162 0.984001i $$-0.557015\pi$$
−0.178162 + 0.984001i $$0.557015\pi$$
$$954$$ 18.0000 0.582772
$$955$$ −20.0000 −0.647185
$$956$$ 30.0000 0.970269
$$957$$ −50.0000 −1.61627
$$958$$ 10.0000 0.323085
$$959$$ −18.0000 −0.581250
$$960$$ −8.00000 −0.258199
$$961$$ −27.0000 −0.870968
$$962$$ −6.00000 −0.193448
$$963$$ 3.00000 0.0966736
$$964$$ −28.0000 −0.901819
$$965$$ 13.0000 0.418485
$$966$$ 6.00000 0.193047
$$967$$ −40.0000 −1.28631 −0.643157 0.765735i $$-0.722376\pi$$
−0.643157 + 0.765735i $$0.722376\pi$$
$$968$$ 0 0
$$969$$ 10.0000 0.321246
$$970$$ −18.0000 −0.577945
$$971$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$972$$ 2.00000 0.0641500
$$973$$ 51.0000 1.63498
$$974$$ 14.0000 0.448589
$$975$$ 1.00000 0.0320256
$$976$$ 44.0000 1.40841
$$977$$ 12.0000 0.383914 0.191957 0.981403i $$-0.438517\pi$$
0.191957 + 0.981403i $$0.438517\pi$$
$$978$$ −22.0000 −0.703482
$$979$$ 55.0000 1.75781
$$980$$ 4.00000 0.127775
$$981$$ 16.0000 0.510841
$$982$$ 32.0000 1.02116
$$983$$ 36.0000 1.14822 0.574111 0.818778i $$-0.305348\pi$$
0.574111 + 0.818778i $$0.305348\pi$$
$$984$$ 0 0
$$985$$ −12.0000 −0.382352
$$986$$ 100.000 3.18465
$$987$$ −30.0000 −0.954911
$$988$$ 4.00000 0.127257
$$989$$ 4.00000 0.127193
$$990$$ −10.0000 −0.317821
$$991$$ 39.0000 1.23888 0.619438 0.785046i $$-0.287361\pi$$
0.619438 + 0.785046i $$0.287361\pi$$
$$992$$ 16.0000 0.508001
$$993$$ 32.0000 1.01549
$$994$$ −90.0000 −2.85463
$$995$$ 4.00000 0.126809
$$996$$ 16.0000 0.506979
$$997$$ −16.0000 −0.506725 −0.253363 0.967371i $$-0.581537\pi$$
−0.253363 + 0.967371i $$0.581537\pi$$
$$998$$ 68.0000 2.15250
$$999$$ −3.00000 −0.0949158
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.d.1.1 1
3.2 odd 2 585.2.a.a.1.1 1
4.3 odd 2 3120.2.a.n.1.1 1
5.2 odd 4 975.2.c.b.274.2 2
5.3 odd 4 975.2.c.b.274.1 2
5.4 even 2 975.2.a.b.1.1 1
7.6 odd 2 9555.2.a.t.1.1 1
12.11 even 2 9360.2.a.w.1.1 1
13.12 even 2 2535.2.a.b.1.1 1
15.2 even 4 2925.2.c.d.2224.1 2
15.8 even 4 2925.2.c.d.2224.2 2
15.14 odd 2 2925.2.a.t.1.1 1
39.38 odd 2 7605.2.a.v.1.1 1

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