# Properties

 Label 418.2.a.a.1.1 Level $418$ Weight $2$ Character 418.1 Self dual yes Analytic conductor $3.338$ Analytic rank $1$ 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] = [418,2,Mod(1,418)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(418, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("418.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$418 = 2 \cdot 11 \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 418.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: $$3.33774680449$$ Analytic rank: $$1$$ 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 $$\chi$$ $$=$$ 418.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -2.00000 q^{5} -1.00000 q^{6} -3.00000 q^{7} +1.00000 q^{8} -2.00000 q^{9} +O(q^{10})$$ $$q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -2.00000 q^{5} -1.00000 q^{6} -3.00000 q^{7} +1.00000 q^{8} -2.00000 q^{9} -2.00000 q^{10} -1.00000 q^{11} -1.00000 q^{12} +1.00000 q^{13} -3.00000 q^{14} +2.00000 q^{15} +1.00000 q^{16} -7.00000 q^{17} -2.00000 q^{18} +1.00000 q^{19} -2.00000 q^{20} +3.00000 q^{21} -1.00000 q^{22} -5.00000 q^{23} -1.00000 q^{24} -1.00000 q^{25} +1.00000 q^{26} +5.00000 q^{27} -3.00000 q^{28} +1.00000 q^{29} +2.00000 q^{30} +10.0000 q^{31} +1.00000 q^{32} +1.00000 q^{33} -7.00000 q^{34} +6.00000 q^{35} -2.00000 q^{36} -6.00000 q^{37} +1.00000 q^{38} -1.00000 q^{39} -2.00000 q^{40} +6.00000 q^{41} +3.00000 q^{42} -4.00000 q^{43} -1.00000 q^{44} +4.00000 q^{45} -5.00000 q^{46} -1.00000 q^{48} +2.00000 q^{49} -1.00000 q^{50} +7.00000 q^{51} +1.00000 q^{52} -1.00000 q^{53} +5.00000 q^{54} +2.00000 q^{55} -3.00000 q^{56} -1.00000 q^{57} +1.00000 q^{58} +3.00000 q^{59} +2.00000 q^{60} -12.0000 q^{61} +10.0000 q^{62} +6.00000 q^{63} +1.00000 q^{64} -2.00000 q^{65} +1.00000 q^{66} +3.00000 q^{67} -7.00000 q^{68} +5.00000 q^{69} +6.00000 q^{70} -10.0000 q^{71} -2.00000 q^{72} +3.00000 q^{73} -6.00000 q^{74} +1.00000 q^{75} +1.00000 q^{76} +3.00000 q^{77} -1.00000 q^{78} +8.00000 q^{79} -2.00000 q^{80} +1.00000 q^{81} +6.00000 q^{82} +8.00000 q^{83} +3.00000 q^{84} +14.0000 q^{85} -4.00000 q^{86} -1.00000 q^{87} -1.00000 q^{88} -8.00000 q^{89} +4.00000 q^{90} -3.00000 q^{91} -5.00000 q^{92} -10.0000 q^{93} -2.00000 q^{95} -1.00000 q^{96} +8.00000 q^{97} +2.00000 q^{98} +2.00000 q^{99} +O(q^{100})$$

## 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 $$-0.593215\pi$$
−0.288675 + 0.957427i $$0.593215\pi$$
$$4$$ 1.00000 0.500000
$$5$$ −2.00000 −0.894427 −0.447214 0.894427i $$-0.647584\pi$$
−0.447214 + 0.894427i $$0.647584\pi$$
$$6$$ −1.00000 −0.408248
$$7$$ −3.00000 −1.13389 −0.566947 0.823754i $$-0.691875\pi$$
−0.566947 + 0.823754i $$0.691875\pi$$
$$8$$ 1.00000 0.353553
$$9$$ −2.00000 −0.666667
$$10$$ −2.00000 −0.632456
$$11$$ −1.00000 −0.301511
$$12$$ −1.00000 −0.288675
$$13$$ 1.00000 0.277350 0.138675 0.990338i $$-0.455716\pi$$
0.138675 + 0.990338i $$0.455716\pi$$
$$14$$ −3.00000 −0.801784
$$15$$ 2.00000 0.516398
$$16$$ 1.00000 0.250000
$$17$$ −7.00000 −1.69775 −0.848875 0.528594i $$-0.822719\pi$$
−0.848875 + 0.528594i $$0.822719\pi$$
$$18$$ −2.00000 −0.471405
$$19$$ 1.00000 0.229416
$$20$$ −2.00000 −0.447214
$$21$$ 3.00000 0.654654
$$22$$ −1.00000 −0.213201
$$23$$ −5.00000 −1.04257 −0.521286 0.853382i $$-0.674548\pi$$
−0.521286 + 0.853382i $$0.674548\pi$$
$$24$$ −1.00000 −0.204124
$$25$$ −1.00000 −0.200000
$$26$$ 1.00000 0.196116
$$27$$ 5.00000 0.962250
$$28$$ −3.00000 −0.566947
$$29$$ 1.00000 0.185695 0.0928477 0.995680i $$-0.470403\pi$$
0.0928477 + 0.995680i $$0.470403\pi$$
$$30$$ 2.00000 0.365148
$$31$$ 10.0000 1.79605 0.898027 0.439941i $$-0.145001\pi$$
0.898027 + 0.439941i $$0.145001\pi$$
$$32$$ 1.00000 0.176777
$$33$$ 1.00000 0.174078
$$34$$ −7.00000 −1.20049
$$35$$ 6.00000 1.01419
$$36$$ −2.00000 −0.333333
$$37$$ −6.00000 −0.986394 −0.493197 0.869918i $$-0.664172\pi$$
−0.493197 + 0.869918i $$0.664172\pi$$
$$38$$ 1.00000 0.162221
$$39$$ −1.00000 −0.160128
$$40$$ −2.00000 −0.316228
$$41$$ 6.00000 0.937043 0.468521 0.883452i $$-0.344787\pi$$
0.468521 + 0.883452i $$0.344787\pi$$
$$42$$ 3.00000 0.462910
$$43$$ −4.00000 −0.609994 −0.304997 0.952353i $$-0.598656\pi$$
−0.304997 + 0.952353i $$0.598656\pi$$
$$44$$ −1.00000 −0.150756
$$45$$ 4.00000 0.596285
$$46$$ −5.00000 −0.737210
$$47$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$48$$ −1.00000 −0.144338
$$49$$ 2.00000 0.285714
$$50$$ −1.00000 −0.141421
$$51$$ 7.00000 0.980196
$$52$$ 1.00000 0.138675
$$53$$ −1.00000 −0.137361 −0.0686803 0.997639i $$-0.521879\pi$$
−0.0686803 + 0.997639i $$0.521879\pi$$
$$54$$ 5.00000 0.680414
$$55$$ 2.00000 0.269680
$$56$$ −3.00000 −0.400892
$$57$$ −1.00000 −0.132453
$$58$$ 1.00000 0.131306
$$59$$ 3.00000 0.390567 0.195283 0.980747i $$-0.437437\pi$$
0.195283 + 0.980747i $$0.437437\pi$$
$$60$$ 2.00000 0.258199
$$61$$ −12.0000 −1.53644 −0.768221 0.640184i $$-0.778858\pi$$
−0.768221 + 0.640184i $$0.778858\pi$$
$$62$$ 10.0000 1.27000
$$63$$ 6.00000 0.755929
$$64$$ 1.00000 0.125000
$$65$$ −2.00000 −0.248069
$$66$$ 1.00000 0.123091
$$67$$ 3.00000 0.366508 0.183254 0.983066i $$-0.441337\pi$$
0.183254 + 0.983066i $$0.441337\pi$$
$$68$$ −7.00000 −0.848875
$$69$$ 5.00000 0.601929
$$70$$ 6.00000 0.717137
$$71$$ −10.0000 −1.18678 −0.593391 0.804914i $$-0.702211\pi$$
−0.593391 + 0.804914i $$0.702211\pi$$
$$72$$ −2.00000 −0.235702
$$73$$ 3.00000 0.351123 0.175562 0.984468i $$-0.443826\pi$$
0.175562 + 0.984468i $$0.443826\pi$$
$$74$$ −6.00000 −0.697486
$$75$$ 1.00000 0.115470
$$76$$ 1.00000 0.114708
$$77$$ 3.00000 0.341882
$$78$$ −1.00000 −0.113228
$$79$$ 8.00000 0.900070 0.450035 0.893011i $$-0.351411\pi$$
0.450035 + 0.893011i $$0.351411\pi$$
$$80$$ −2.00000 −0.223607
$$81$$ 1.00000 0.111111
$$82$$ 6.00000 0.662589
$$83$$ 8.00000 0.878114 0.439057 0.898459i $$-0.355313\pi$$
0.439057 + 0.898459i $$0.355313\pi$$
$$84$$ 3.00000 0.327327
$$85$$ 14.0000 1.51851
$$86$$ −4.00000 −0.431331
$$87$$ −1.00000 −0.107211
$$88$$ −1.00000 −0.106600
$$89$$ −8.00000 −0.847998 −0.423999 0.905663i $$-0.639374\pi$$
−0.423999 + 0.905663i $$0.639374\pi$$
$$90$$ 4.00000 0.421637
$$91$$ −3.00000 −0.314485
$$92$$ −5.00000 −0.521286
$$93$$ −10.0000 −1.03695
$$94$$ 0 0
$$95$$ −2.00000 −0.205196
$$96$$ −1.00000 −0.102062
$$97$$ 8.00000 0.812277 0.406138 0.913812i $$-0.366875\pi$$
0.406138 + 0.913812i $$0.366875\pi$$
$$98$$ 2.00000 0.202031
$$99$$ 2.00000 0.201008
$$100$$ −1.00000 −0.100000
$$101$$ 6.00000 0.597022 0.298511 0.954406i $$-0.403510\pi$$
0.298511 + 0.954406i $$0.403510\pi$$
$$102$$ 7.00000 0.693103
$$103$$ 12.0000 1.18240 0.591198 0.806527i $$-0.298655\pi$$
0.591198 + 0.806527i $$0.298655\pi$$
$$104$$ 1.00000 0.0980581
$$105$$ −6.00000 −0.585540
$$106$$ −1.00000 −0.0971286
$$107$$ −13.0000 −1.25676 −0.628379 0.777908i $$-0.716281\pi$$
−0.628379 + 0.777908i $$0.716281\pi$$
$$108$$ 5.00000 0.481125
$$109$$ −13.0000 −1.24517 −0.622587 0.782551i $$-0.713918\pi$$
−0.622587 + 0.782551i $$0.713918\pi$$
$$110$$ 2.00000 0.190693
$$111$$ 6.00000 0.569495
$$112$$ −3.00000 −0.283473
$$113$$ −12.0000 −1.12887 −0.564433 0.825479i $$-0.690905\pi$$
−0.564433 + 0.825479i $$0.690905\pi$$
$$114$$ −1.00000 −0.0936586
$$115$$ 10.0000 0.932505
$$116$$ 1.00000 0.0928477
$$117$$ −2.00000 −0.184900
$$118$$ 3.00000 0.276172
$$119$$ 21.0000 1.92507
$$120$$ 2.00000 0.182574
$$121$$ 1.00000 0.0909091
$$122$$ −12.0000 −1.08643
$$123$$ −6.00000 −0.541002
$$124$$ 10.0000 0.898027
$$125$$ 12.0000 1.07331
$$126$$ 6.00000 0.534522
$$127$$ 16.0000 1.41977 0.709885 0.704317i $$-0.248747\pi$$
0.709885 + 0.704317i $$0.248747\pi$$
$$128$$ 1.00000 0.0883883
$$129$$ 4.00000 0.352180
$$130$$ −2.00000 −0.175412
$$131$$ −22.0000 −1.92215 −0.961074 0.276289i $$-0.910895\pi$$
−0.961074 + 0.276289i $$0.910895\pi$$
$$132$$ 1.00000 0.0870388
$$133$$ −3.00000 −0.260133
$$134$$ 3.00000 0.259161
$$135$$ −10.0000 −0.860663
$$136$$ −7.00000 −0.600245
$$137$$ −17.0000 −1.45241 −0.726204 0.687479i $$-0.758717\pi$$
−0.726204 + 0.687479i $$0.758717\pi$$
$$138$$ 5.00000 0.425628
$$139$$ −14.0000 −1.18746 −0.593732 0.804663i $$-0.702346\pi$$
−0.593732 + 0.804663i $$0.702346\pi$$
$$140$$ 6.00000 0.507093
$$141$$ 0 0
$$142$$ −10.0000 −0.839181
$$143$$ −1.00000 −0.0836242
$$144$$ −2.00000 −0.166667
$$145$$ −2.00000 −0.166091
$$146$$ 3.00000 0.248282
$$147$$ −2.00000 −0.164957
$$148$$ −6.00000 −0.493197
$$149$$ −18.0000 −1.47462 −0.737309 0.675556i $$-0.763904\pi$$
−0.737309 + 0.675556i $$0.763904\pi$$
$$150$$ 1.00000 0.0816497
$$151$$ 12.0000 0.976546 0.488273 0.872691i $$-0.337627\pi$$
0.488273 + 0.872691i $$0.337627\pi$$
$$152$$ 1.00000 0.0811107
$$153$$ 14.0000 1.13183
$$154$$ 3.00000 0.241747
$$155$$ −20.0000 −1.60644
$$156$$ −1.00000 −0.0800641
$$157$$ −22.0000 −1.75579 −0.877896 0.478852i $$-0.841053\pi$$
−0.877896 + 0.478852i $$0.841053\pi$$
$$158$$ 8.00000 0.636446
$$159$$ 1.00000 0.0793052
$$160$$ −2.00000 −0.158114
$$161$$ 15.0000 1.18217
$$162$$ 1.00000 0.0785674
$$163$$ −2.00000 −0.156652 −0.0783260 0.996928i $$-0.524958\pi$$
−0.0783260 + 0.996928i $$0.524958\pi$$
$$164$$ 6.00000 0.468521
$$165$$ −2.00000 −0.155700
$$166$$ 8.00000 0.620920
$$167$$ −22.0000 −1.70241 −0.851206 0.524832i $$-0.824128\pi$$
−0.851206 + 0.524832i $$0.824128\pi$$
$$168$$ 3.00000 0.231455
$$169$$ −12.0000 −0.923077
$$170$$ 14.0000 1.07375
$$171$$ −2.00000 −0.152944
$$172$$ −4.00000 −0.304997
$$173$$ 14.0000 1.06440 0.532200 0.846619i $$-0.321365\pi$$
0.532200 + 0.846619i $$0.321365\pi$$
$$174$$ −1.00000 −0.0758098
$$175$$ 3.00000 0.226779
$$176$$ −1.00000 −0.0753778
$$177$$ −3.00000 −0.225494
$$178$$ −8.00000 −0.599625
$$179$$ 4.00000 0.298974 0.149487 0.988764i $$-0.452238\pi$$
0.149487 + 0.988764i $$0.452238\pi$$
$$180$$ 4.00000 0.298142
$$181$$ −2.00000 −0.148659 −0.0743294 0.997234i $$-0.523682\pi$$
−0.0743294 + 0.997234i $$0.523682\pi$$
$$182$$ −3.00000 −0.222375
$$183$$ 12.0000 0.887066
$$184$$ −5.00000 −0.368605
$$185$$ 12.0000 0.882258
$$186$$ −10.0000 −0.733236
$$187$$ 7.00000 0.511891
$$188$$ 0 0
$$189$$ −15.0000 −1.09109
$$190$$ −2.00000 −0.145095
$$191$$ 3.00000 0.217072 0.108536 0.994092i $$-0.465384\pi$$
0.108536 + 0.994092i $$0.465384\pi$$
$$192$$ −1.00000 −0.0721688
$$193$$ −4.00000 −0.287926 −0.143963 0.989583i $$-0.545985\pi$$
−0.143963 + 0.989583i $$0.545985\pi$$
$$194$$ 8.00000 0.574367
$$195$$ 2.00000 0.143223
$$196$$ 2.00000 0.142857
$$197$$ −12.0000 −0.854965 −0.427482 0.904024i $$-0.640599\pi$$
−0.427482 + 0.904024i $$0.640599\pi$$
$$198$$ 2.00000 0.142134
$$199$$ 23.0000 1.63043 0.815213 0.579161i $$-0.196620\pi$$
0.815213 + 0.579161i $$0.196620\pi$$
$$200$$ −1.00000 −0.0707107
$$201$$ −3.00000 −0.211604
$$202$$ 6.00000 0.422159
$$203$$ −3.00000 −0.210559
$$204$$ 7.00000 0.490098
$$205$$ −12.0000 −0.838116
$$206$$ 12.0000 0.836080
$$207$$ 10.0000 0.695048
$$208$$ 1.00000 0.0693375
$$209$$ −1.00000 −0.0691714
$$210$$ −6.00000 −0.414039
$$211$$ 1.00000 0.0688428 0.0344214 0.999407i $$-0.489041\pi$$
0.0344214 + 0.999407i $$0.489041\pi$$
$$212$$ −1.00000 −0.0686803
$$213$$ 10.0000 0.685189
$$214$$ −13.0000 −0.888662
$$215$$ 8.00000 0.545595
$$216$$ 5.00000 0.340207
$$217$$ −30.0000 −2.03653
$$218$$ −13.0000 −0.880471
$$219$$ −3.00000 −0.202721
$$220$$ 2.00000 0.134840
$$221$$ −7.00000 −0.470871
$$222$$ 6.00000 0.402694
$$223$$ 10.0000 0.669650 0.334825 0.942280i $$-0.391323\pi$$
0.334825 + 0.942280i $$0.391323\pi$$
$$224$$ −3.00000 −0.200446
$$225$$ 2.00000 0.133333
$$226$$ −12.0000 −0.798228
$$227$$ −15.0000 −0.995585 −0.497792 0.867296i $$-0.665856\pi$$
−0.497792 + 0.867296i $$0.665856\pi$$
$$228$$ −1.00000 −0.0662266
$$229$$ 20.0000 1.32164 0.660819 0.750546i $$-0.270209\pi$$
0.660819 + 0.750546i $$0.270209\pi$$
$$230$$ 10.0000 0.659380
$$231$$ −3.00000 −0.197386
$$232$$ 1.00000 0.0656532
$$233$$ −10.0000 −0.655122 −0.327561 0.944830i $$-0.606227\pi$$
−0.327561 + 0.944830i $$0.606227\pi$$
$$234$$ −2.00000 −0.130744
$$235$$ 0 0
$$236$$ 3.00000 0.195283
$$237$$ −8.00000 −0.519656
$$238$$ 21.0000 1.36123
$$239$$ 25.0000 1.61712 0.808558 0.588417i $$-0.200249\pi$$
0.808558 + 0.588417i $$0.200249\pi$$
$$240$$ 2.00000 0.129099
$$241$$ −4.00000 −0.257663 −0.128831 0.991667i $$-0.541123\pi$$
−0.128831 + 0.991667i $$0.541123\pi$$
$$242$$ 1.00000 0.0642824
$$243$$ −16.0000 −1.02640
$$244$$ −12.0000 −0.768221
$$245$$ −4.00000 −0.255551
$$246$$ −6.00000 −0.382546
$$247$$ 1.00000 0.0636285
$$248$$ 10.0000 0.635001
$$249$$ −8.00000 −0.506979
$$250$$ 12.0000 0.758947
$$251$$ −6.00000 −0.378717 −0.189358 0.981908i $$-0.560641\pi$$
−0.189358 + 0.981908i $$0.560641\pi$$
$$252$$ 6.00000 0.377964
$$253$$ 5.00000 0.314347
$$254$$ 16.0000 1.00393
$$255$$ −14.0000 −0.876714
$$256$$ 1.00000 0.0625000
$$257$$ −22.0000 −1.37232 −0.686161 0.727450i $$-0.740706\pi$$
−0.686161 + 0.727450i $$0.740706\pi$$
$$258$$ 4.00000 0.249029
$$259$$ 18.0000 1.11847
$$260$$ −2.00000 −0.124035
$$261$$ −2.00000 −0.123797
$$262$$ −22.0000 −1.35916
$$263$$ 4.00000 0.246651 0.123325 0.992366i $$-0.460644\pi$$
0.123325 + 0.992366i $$0.460644\pi$$
$$264$$ 1.00000 0.0615457
$$265$$ 2.00000 0.122859
$$266$$ −3.00000 −0.183942
$$267$$ 8.00000 0.489592
$$268$$ 3.00000 0.183254
$$269$$ 18.0000 1.09748 0.548740 0.835993i $$-0.315108\pi$$
0.548740 + 0.835993i $$0.315108\pi$$
$$270$$ −10.0000 −0.608581
$$271$$ −15.0000 −0.911185 −0.455593 0.890188i $$-0.650573\pi$$
−0.455593 + 0.890188i $$0.650573\pi$$
$$272$$ −7.00000 −0.424437
$$273$$ 3.00000 0.181568
$$274$$ −17.0000 −1.02701
$$275$$ 1.00000 0.0603023
$$276$$ 5.00000 0.300965
$$277$$ −16.0000 −0.961347 −0.480673 0.876900i $$-0.659608\pi$$
−0.480673 + 0.876900i $$0.659608\pi$$
$$278$$ −14.0000 −0.839664
$$279$$ −20.0000 −1.19737
$$280$$ 6.00000 0.358569
$$281$$ 16.0000 0.954480 0.477240 0.878773i $$-0.341637\pi$$
0.477240 + 0.878773i $$0.341637\pi$$
$$282$$ 0 0
$$283$$ −14.0000 −0.832214 −0.416107 0.909316i $$-0.636606\pi$$
−0.416107 + 0.909316i $$0.636606\pi$$
$$284$$ −10.0000 −0.593391
$$285$$ 2.00000 0.118470
$$286$$ −1.00000 −0.0591312
$$287$$ −18.0000 −1.06251
$$288$$ −2.00000 −0.117851
$$289$$ 32.0000 1.88235
$$290$$ −2.00000 −0.117444
$$291$$ −8.00000 −0.468968
$$292$$ 3.00000 0.175562
$$293$$ −9.00000 −0.525786 −0.262893 0.964825i $$-0.584677\pi$$
−0.262893 + 0.964825i $$0.584677\pi$$
$$294$$ −2.00000 −0.116642
$$295$$ −6.00000 −0.349334
$$296$$ −6.00000 −0.348743
$$297$$ −5.00000 −0.290129
$$298$$ −18.0000 −1.04271
$$299$$ −5.00000 −0.289157
$$300$$ 1.00000 0.0577350
$$301$$ 12.0000 0.691669
$$302$$ 12.0000 0.690522
$$303$$ −6.00000 −0.344691
$$304$$ 1.00000 0.0573539
$$305$$ 24.0000 1.37424
$$306$$ 14.0000 0.800327
$$307$$ −16.0000 −0.913168 −0.456584 0.889680i $$-0.650927\pi$$
−0.456584 + 0.889680i $$0.650927\pi$$
$$308$$ 3.00000 0.170941
$$309$$ −12.0000 −0.682656
$$310$$ −20.0000 −1.13592
$$311$$ 3.00000 0.170114 0.0850572 0.996376i $$-0.472893\pi$$
0.0850572 + 0.996376i $$0.472893\pi$$
$$312$$ −1.00000 −0.0566139
$$313$$ 21.0000 1.18699 0.593495 0.804838i $$-0.297748\pi$$
0.593495 + 0.804838i $$0.297748\pi$$
$$314$$ −22.0000 −1.24153
$$315$$ −12.0000 −0.676123
$$316$$ 8.00000 0.450035
$$317$$ 21.0000 1.17948 0.589739 0.807594i $$-0.299231\pi$$
0.589739 + 0.807594i $$0.299231\pi$$
$$318$$ 1.00000 0.0560772
$$319$$ −1.00000 −0.0559893
$$320$$ −2.00000 −0.111803
$$321$$ 13.0000 0.725589
$$322$$ 15.0000 0.835917
$$323$$ −7.00000 −0.389490
$$324$$ 1.00000 0.0555556
$$325$$ −1.00000 −0.0554700
$$326$$ −2.00000 −0.110770
$$327$$ 13.0000 0.718902
$$328$$ 6.00000 0.331295
$$329$$ 0 0
$$330$$ −2.00000 −0.110096
$$331$$ 29.0000 1.59398 0.796992 0.603990i $$-0.206423\pi$$
0.796992 + 0.603990i $$0.206423\pi$$
$$332$$ 8.00000 0.439057
$$333$$ 12.0000 0.657596
$$334$$ −22.0000 −1.20379
$$335$$ −6.00000 −0.327815
$$336$$ 3.00000 0.163663
$$337$$ −20.0000 −1.08947 −0.544735 0.838608i $$-0.683370\pi$$
−0.544735 + 0.838608i $$0.683370\pi$$
$$338$$ −12.0000 −0.652714
$$339$$ 12.0000 0.651751
$$340$$ 14.0000 0.759257
$$341$$ −10.0000 −0.541530
$$342$$ −2.00000 −0.108148
$$343$$ 15.0000 0.809924
$$344$$ −4.00000 −0.215666
$$345$$ −10.0000 −0.538382
$$346$$ 14.0000 0.752645
$$347$$ −26.0000 −1.39575 −0.697877 0.716218i $$-0.745872\pi$$
−0.697877 + 0.716218i $$0.745872\pi$$
$$348$$ −1.00000 −0.0536056
$$349$$ 10.0000 0.535288 0.267644 0.963518i $$-0.413755\pi$$
0.267644 + 0.963518i $$0.413755\pi$$
$$350$$ 3.00000 0.160357
$$351$$ 5.00000 0.266880
$$352$$ −1.00000 −0.0533002
$$353$$ 1.00000 0.0532246 0.0266123 0.999646i $$-0.491528\pi$$
0.0266123 + 0.999646i $$0.491528\pi$$
$$354$$ −3.00000 −0.159448
$$355$$ 20.0000 1.06149
$$356$$ −8.00000 −0.423999
$$357$$ −21.0000 −1.11144
$$358$$ 4.00000 0.211407
$$359$$ −17.0000 −0.897226 −0.448613 0.893726i $$-0.648082\pi$$
−0.448613 + 0.893726i $$0.648082\pi$$
$$360$$ 4.00000 0.210819
$$361$$ 1.00000 0.0526316
$$362$$ −2.00000 −0.105118
$$363$$ −1.00000 −0.0524864
$$364$$ −3.00000 −0.157243
$$365$$ −6.00000 −0.314054
$$366$$ 12.0000 0.627250
$$367$$ 16.0000 0.835193 0.417597 0.908633i $$-0.362873\pi$$
0.417597 + 0.908633i $$0.362873\pi$$
$$368$$ −5.00000 −0.260643
$$369$$ −12.0000 −0.624695
$$370$$ 12.0000 0.623850
$$371$$ 3.00000 0.155752
$$372$$ −10.0000 −0.518476
$$373$$ 11.0000 0.569558 0.284779 0.958593i $$-0.408080\pi$$
0.284779 + 0.958593i $$0.408080\pi$$
$$374$$ 7.00000 0.361961
$$375$$ −12.0000 −0.619677
$$376$$ 0 0
$$377$$ 1.00000 0.0515026
$$378$$ −15.0000 −0.771517
$$379$$ −5.00000 −0.256833 −0.128416 0.991720i $$-0.540989\pi$$
−0.128416 + 0.991720i $$0.540989\pi$$
$$380$$ −2.00000 −0.102598
$$381$$ −16.0000 −0.819705
$$382$$ 3.00000 0.153493
$$383$$ 2.00000 0.102195 0.0510976 0.998694i $$-0.483728\pi$$
0.0510976 + 0.998694i $$0.483728\pi$$
$$384$$ −1.00000 −0.0510310
$$385$$ −6.00000 −0.305788
$$386$$ −4.00000 −0.203595
$$387$$ 8.00000 0.406663
$$388$$ 8.00000 0.406138
$$389$$ 18.0000 0.912636 0.456318 0.889817i $$-0.349168\pi$$
0.456318 + 0.889817i $$0.349168\pi$$
$$390$$ 2.00000 0.101274
$$391$$ 35.0000 1.77003
$$392$$ 2.00000 0.101015
$$393$$ 22.0000 1.10975
$$394$$ −12.0000 −0.604551
$$395$$ −16.0000 −0.805047
$$396$$ 2.00000 0.100504
$$397$$ 34.0000 1.70641 0.853206 0.521575i $$-0.174655\pi$$
0.853206 + 0.521575i $$0.174655\pi$$
$$398$$ 23.0000 1.15289
$$399$$ 3.00000 0.150188
$$400$$ −1.00000 −0.0500000
$$401$$ −2.00000 −0.0998752 −0.0499376 0.998752i $$-0.515902\pi$$
−0.0499376 + 0.998752i $$0.515902\pi$$
$$402$$ −3.00000 −0.149626
$$403$$ 10.0000 0.498135
$$404$$ 6.00000 0.298511
$$405$$ −2.00000 −0.0993808
$$406$$ −3.00000 −0.148888
$$407$$ 6.00000 0.297409
$$408$$ 7.00000 0.346552
$$409$$ 34.0000 1.68119 0.840596 0.541663i $$-0.182205\pi$$
0.840596 + 0.541663i $$0.182205\pi$$
$$410$$ −12.0000 −0.592638
$$411$$ 17.0000 0.838548
$$412$$ 12.0000 0.591198
$$413$$ −9.00000 −0.442861
$$414$$ 10.0000 0.491473
$$415$$ −16.0000 −0.785409
$$416$$ 1.00000 0.0490290
$$417$$ 14.0000 0.685583
$$418$$ −1.00000 −0.0489116
$$419$$ 10.0000 0.488532 0.244266 0.969708i $$-0.421453\pi$$
0.244266 + 0.969708i $$0.421453\pi$$
$$420$$ −6.00000 −0.292770
$$421$$ 15.0000 0.731055 0.365528 0.930800i $$-0.380889\pi$$
0.365528 + 0.930800i $$0.380889\pi$$
$$422$$ 1.00000 0.0486792
$$423$$ 0 0
$$424$$ −1.00000 −0.0485643
$$425$$ 7.00000 0.339550
$$426$$ 10.0000 0.484502
$$427$$ 36.0000 1.74216
$$428$$ −13.0000 −0.628379
$$429$$ 1.00000 0.0482805
$$430$$ 8.00000 0.385794
$$431$$ 34.0000 1.63772 0.818861 0.573992i $$-0.194606\pi$$
0.818861 + 0.573992i $$0.194606\pi$$
$$432$$ 5.00000 0.240563
$$433$$ 24.0000 1.15337 0.576683 0.816968i $$-0.304347\pi$$
0.576683 + 0.816968i $$0.304347\pi$$
$$434$$ −30.0000 −1.44005
$$435$$ 2.00000 0.0958927
$$436$$ −13.0000 −0.622587
$$437$$ −5.00000 −0.239182
$$438$$ −3.00000 −0.143346
$$439$$ −14.0000 −0.668184 −0.334092 0.942541i $$-0.608430\pi$$
−0.334092 + 0.942541i $$0.608430\pi$$
$$440$$ 2.00000 0.0953463
$$441$$ −4.00000 −0.190476
$$442$$ −7.00000 −0.332956
$$443$$ 18.0000 0.855206 0.427603 0.903967i $$-0.359358\pi$$
0.427603 + 0.903967i $$0.359358\pi$$
$$444$$ 6.00000 0.284747
$$445$$ 16.0000 0.758473
$$446$$ 10.0000 0.473514
$$447$$ 18.0000 0.851371
$$448$$ −3.00000 −0.141737
$$449$$ 24.0000 1.13263 0.566315 0.824189i $$-0.308369\pi$$
0.566315 + 0.824189i $$0.308369\pi$$
$$450$$ 2.00000 0.0942809
$$451$$ −6.00000 −0.282529
$$452$$ −12.0000 −0.564433
$$453$$ −12.0000 −0.563809
$$454$$ −15.0000 −0.703985
$$455$$ 6.00000 0.281284
$$456$$ −1.00000 −0.0468293
$$457$$ −37.0000 −1.73079 −0.865393 0.501093i $$-0.832931\pi$$
−0.865393 + 0.501093i $$0.832931\pi$$
$$458$$ 20.0000 0.934539
$$459$$ −35.0000 −1.63366
$$460$$ 10.0000 0.466252
$$461$$ −8.00000 −0.372597 −0.186299 0.982493i $$-0.559649\pi$$
−0.186299 + 0.982493i $$0.559649\pi$$
$$462$$ −3.00000 −0.139573
$$463$$ 16.0000 0.743583 0.371792 0.928316i $$-0.378744\pi$$
0.371792 + 0.928316i $$0.378744\pi$$
$$464$$ 1.00000 0.0464238
$$465$$ 20.0000 0.927478
$$466$$ −10.0000 −0.463241
$$467$$ 20.0000 0.925490 0.462745 0.886492i $$-0.346865\pi$$
0.462745 + 0.886492i $$0.346865\pi$$
$$468$$ −2.00000 −0.0924500
$$469$$ −9.00000 −0.415581
$$470$$ 0 0
$$471$$ 22.0000 1.01371
$$472$$ 3.00000 0.138086
$$473$$ 4.00000 0.183920
$$474$$ −8.00000 −0.367452
$$475$$ −1.00000 −0.0458831
$$476$$ 21.0000 0.962533
$$477$$ 2.00000 0.0915737
$$478$$ 25.0000 1.14347
$$479$$ −12.0000 −0.548294 −0.274147 0.961688i $$-0.588395\pi$$
−0.274147 + 0.961688i $$0.588395\pi$$
$$480$$ 2.00000 0.0912871
$$481$$ −6.00000 −0.273576
$$482$$ −4.00000 −0.182195
$$483$$ −15.0000 −0.682524
$$484$$ 1.00000 0.0454545
$$485$$ −16.0000 −0.726523
$$486$$ −16.0000 −0.725775
$$487$$ −12.0000 −0.543772 −0.271886 0.962329i $$-0.587647\pi$$
−0.271886 + 0.962329i $$0.587647\pi$$
$$488$$ −12.0000 −0.543214
$$489$$ 2.00000 0.0904431
$$490$$ −4.00000 −0.180702
$$491$$ 10.0000 0.451294 0.225647 0.974209i $$-0.427550\pi$$
0.225647 + 0.974209i $$0.427550\pi$$
$$492$$ −6.00000 −0.270501
$$493$$ −7.00000 −0.315264
$$494$$ 1.00000 0.0449921
$$495$$ −4.00000 −0.179787
$$496$$ 10.0000 0.449013
$$497$$ 30.0000 1.34568
$$498$$ −8.00000 −0.358489
$$499$$ −28.0000 −1.25345 −0.626726 0.779240i $$-0.715605\pi$$
−0.626726 + 0.779240i $$0.715605\pi$$
$$500$$ 12.0000 0.536656
$$501$$ 22.0000 0.982888
$$502$$ −6.00000 −0.267793
$$503$$ −19.0000 −0.847168 −0.423584 0.905857i $$-0.639228\pi$$
−0.423584 + 0.905857i $$0.639228\pi$$
$$504$$ 6.00000 0.267261
$$505$$ −12.0000 −0.533993
$$506$$ 5.00000 0.222277
$$507$$ 12.0000 0.532939
$$508$$ 16.0000 0.709885
$$509$$ −34.0000 −1.50702 −0.753512 0.657434i $$-0.771642\pi$$
−0.753512 + 0.657434i $$0.771642\pi$$
$$510$$ −14.0000 −0.619930
$$511$$ −9.00000 −0.398137
$$512$$ 1.00000 0.0441942
$$513$$ 5.00000 0.220755
$$514$$ −22.0000 −0.970378
$$515$$ −24.0000 −1.05757
$$516$$ 4.00000 0.176090
$$517$$ 0 0
$$518$$ 18.0000 0.790875
$$519$$ −14.0000 −0.614532
$$520$$ −2.00000 −0.0877058
$$521$$ −20.0000 −0.876216 −0.438108 0.898922i $$-0.644351\pi$$
−0.438108 + 0.898922i $$0.644351\pi$$
$$522$$ −2.00000 −0.0875376
$$523$$ −21.0000 −0.918266 −0.459133 0.888368i $$-0.651840\pi$$
−0.459133 + 0.888368i $$0.651840\pi$$
$$524$$ −22.0000 −0.961074
$$525$$ −3.00000 −0.130931
$$526$$ 4.00000 0.174408
$$527$$ −70.0000 −3.04925
$$528$$ 1.00000 0.0435194
$$529$$ 2.00000 0.0869565
$$530$$ 2.00000 0.0868744
$$531$$ −6.00000 −0.260378
$$532$$ −3.00000 −0.130066
$$533$$ 6.00000 0.259889
$$534$$ 8.00000 0.346194
$$535$$ 26.0000 1.12408
$$536$$ 3.00000 0.129580
$$537$$ −4.00000 −0.172613
$$538$$ 18.0000 0.776035
$$539$$ −2.00000 −0.0861461
$$540$$ −10.0000 −0.430331
$$541$$ −16.0000 −0.687894 −0.343947 0.938989i $$-0.611764\pi$$
−0.343947 + 0.938989i $$0.611764\pi$$
$$542$$ −15.0000 −0.644305
$$543$$ 2.00000 0.0858282
$$544$$ −7.00000 −0.300123
$$545$$ 26.0000 1.11372
$$546$$ 3.00000 0.128388
$$547$$ 8.00000 0.342055 0.171028 0.985266i $$-0.445291\pi$$
0.171028 + 0.985266i $$0.445291\pi$$
$$548$$ −17.0000 −0.726204
$$549$$ 24.0000 1.02430
$$550$$ 1.00000 0.0426401
$$551$$ 1.00000 0.0426014
$$552$$ 5.00000 0.212814
$$553$$ −24.0000 −1.02058
$$554$$ −16.0000 −0.679775
$$555$$ −12.0000 −0.509372
$$556$$ −14.0000 −0.593732
$$557$$ −6.00000 −0.254228 −0.127114 0.991888i $$-0.540571\pi$$
−0.127114 + 0.991888i $$0.540571\pi$$
$$558$$ −20.0000 −0.846668
$$559$$ −4.00000 −0.169182
$$560$$ 6.00000 0.253546
$$561$$ −7.00000 −0.295540
$$562$$ 16.0000 0.674919
$$563$$ 24.0000 1.01148 0.505740 0.862686i $$-0.331220\pi$$
0.505740 + 0.862686i $$0.331220\pi$$
$$564$$ 0 0
$$565$$ 24.0000 1.00969
$$566$$ −14.0000 −0.588464
$$567$$ −3.00000 −0.125988
$$568$$ −10.0000 −0.419591
$$569$$ −12.0000 −0.503066 −0.251533 0.967849i $$-0.580935\pi$$
−0.251533 + 0.967849i $$0.580935\pi$$
$$570$$ 2.00000 0.0837708
$$571$$ −14.0000 −0.585882 −0.292941 0.956131i $$-0.594634\pi$$
−0.292941 + 0.956131i $$0.594634\pi$$
$$572$$ −1.00000 −0.0418121
$$573$$ −3.00000 −0.125327
$$574$$ −18.0000 −0.751305
$$575$$ 5.00000 0.208514
$$576$$ −2.00000 −0.0833333
$$577$$ 23.0000 0.957503 0.478751 0.877951i $$-0.341090\pi$$
0.478751 + 0.877951i $$0.341090\pi$$
$$578$$ 32.0000 1.33102
$$579$$ 4.00000 0.166234
$$580$$ −2.00000 −0.0830455
$$581$$ −24.0000 −0.995688
$$582$$ −8.00000 −0.331611
$$583$$ 1.00000 0.0414158
$$584$$ 3.00000 0.124141
$$585$$ 4.00000 0.165380
$$586$$ −9.00000 −0.371787
$$587$$ −12.0000 −0.495293 −0.247647 0.968850i $$-0.579657\pi$$
−0.247647 + 0.968850i $$0.579657\pi$$
$$588$$ −2.00000 −0.0824786
$$589$$ 10.0000 0.412043
$$590$$ −6.00000 −0.247016
$$591$$ 12.0000 0.493614
$$592$$ −6.00000 −0.246598
$$593$$ 42.0000 1.72473 0.862367 0.506284i $$-0.168981\pi$$
0.862367 + 0.506284i $$0.168981\pi$$
$$594$$ −5.00000 −0.205152
$$595$$ −42.0000 −1.72183
$$596$$ −18.0000 −0.737309
$$597$$ −23.0000 −0.941327
$$598$$ −5.00000 −0.204465
$$599$$ −4.00000 −0.163436 −0.0817178 0.996656i $$-0.526041\pi$$
−0.0817178 + 0.996656i $$0.526041\pi$$
$$600$$ 1.00000 0.0408248
$$601$$ 8.00000 0.326327 0.163163 0.986599i $$-0.447830\pi$$
0.163163 + 0.986599i $$0.447830\pi$$
$$602$$ 12.0000 0.489083
$$603$$ −6.00000 −0.244339
$$604$$ 12.0000 0.488273
$$605$$ −2.00000 −0.0813116
$$606$$ −6.00000 −0.243733
$$607$$ −22.0000 −0.892952 −0.446476 0.894795i $$-0.647321\pi$$
−0.446476 + 0.894795i $$0.647321\pi$$
$$608$$ 1.00000 0.0405554
$$609$$ 3.00000 0.121566
$$610$$ 24.0000 0.971732
$$611$$ 0 0
$$612$$ 14.0000 0.565916
$$613$$ −28.0000 −1.13091 −0.565455 0.824779i $$-0.691299\pi$$
−0.565455 + 0.824779i $$0.691299\pi$$
$$614$$ −16.0000 −0.645707
$$615$$ 12.0000 0.483887
$$616$$ 3.00000 0.120873
$$617$$ −30.0000 −1.20775 −0.603877 0.797077i $$-0.706378\pi$$
−0.603877 + 0.797077i $$0.706378\pi$$
$$618$$ −12.0000 −0.482711
$$619$$ −20.0000 −0.803868 −0.401934 0.915669i $$-0.631662\pi$$
−0.401934 + 0.915669i $$0.631662\pi$$
$$620$$ −20.0000 −0.803219
$$621$$ −25.0000 −1.00322
$$622$$ 3.00000 0.120289
$$623$$ 24.0000 0.961540
$$624$$ −1.00000 −0.0400320
$$625$$ −19.0000 −0.760000
$$626$$ 21.0000 0.839329
$$627$$ 1.00000 0.0399362
$$628$$ −22.0000 −0.877896
$$629$$ 42.0000 1.67465
$$630$$ −12.0000 −0.478091
$$631$$ −8.00000 −0.318475 −0.159237 0.987240i $$-0.550904\pi$$
−0.159237 + 0.987240i $$0.550904\pi$$
$$632$$ 8.00000 0.318223
$$633$$ −1.00000 −0.0397464
$$634$$ 21.0000 0.834017
$$635$$ −32.0000 −1.26988
$$636$$ 1.00000 0.0396526
$$637$$ 2.00000 0.0792429
$$638$$ −1.00000 −0.0395904
$$639$$ 20.0000 0.791188
$$640$$ −2.00000 −0.0790569
$$641$$ 40.0000 1.57991 0.789953 0.613168i $$-0.210105\pi$$
0.789953 + 0.613168i $$0.210105\pi$$
$$642$$ 13.0000 0.513069
$$643$$ −26.0000 −1.02534 −0.512670 0.858586i $$-0.671344\pi$$
−0.512670 + 0.858586i $$0.671344\pi$$
$$644$$ 15.0000 0.591083
$$645$$ −8.00000 −0.315000
$$646$$ −7.00000 −0.275411
$$647$$ 3.00000 0.117942 0.0589711 0.998260i $$-0.481218\pi$$
0.0589711 + 0.998260i $$0.481218\pi$$
$$648$$ 1.00000 0.0392837
$$649$$ −3.00000 −0.117760
$$650$$ −1.00000 −0.0392232
$$651$$ 30.0000 1.17579
$$652$$ −2.00000 −0.0783260
$$653$$ 4.00000 0.156532 0.0782660 0.996933i $$-0.475062\pi$$
0.0782660 + 0.996933i $$0.475062\pi$$
$$654$$ 13.0000 0.508340
$$655$$ 44.0000 1.71922
$$656$$ 6.00000 0.234261
$$657$$ −6.00000 −0.234082
$$658$$ 0 0
$$659$$ 11.0000 0.428499 0.214250 0.976779i $$-0.431269\pi$$
0.214250 + 0.976779i $$0.431269\pi$$
$$660$$ −2.00000 −0.0778499
$$661$$ 29.0000 1.12797 0.563985 0.825785i $$-0.309268\pi$$
0.563985 + 0.825785i $$0.309268\pi$$
$$662$$ 29.0000 1.12712
$$663$$ 7.00000 0.271857
$$664$$ 8.00000 0.310460
$$665$$ 6.00000 0.232670
$$666$$ 12.0000 0.464991
$$667$$ −5.00000 −0.193601
$$668$$ −22.0000 −0.851206
$$669$$ −10.0000 −0.386622
$$670$$ −6.00000 −0.231800
$$671$$ 12.0000 0.463255
$$672$$ 3.00000 0.115728
$$673$$ −24.0000 −0.925132 −0.462566 0.886585i $$-0.653071\pi$$
−0.462566 + 0.886585i $$0.653071\pi$$
$$674$$ −20.0000 −0.770371
$$675$$ −5.00000 −0.192450
$$676$$ −12.0000 −0.461538
$$677$$ −29.0000 −1.11456 −0.557280 0.830324i $$-0.688155\pi$$
−0.557280 + 0.830324i $$0.688155\pi$$
$$678$$ 12.0000 0.460857
$$679$$ −24.0000 −0.921035
$$680$$ 14.0000 0.536875
$$681$$ 15.0000 0.574801
$$682$$ −10.0000 −0.382920
$$683$$ −8.00000 −0.306111 −0.153056 0.988218i $$-0.548911\pi$$
−0.153056 + 0.988218i $$0.548911\pi$$
$$684$$ −2.00000 −0.0764719
$$685$$ 34.0000 1.29907
$$686$$ 15.0000 0.572703
$$687$$ −20.0000 −0.763048
$$688$$ −4.00000 −0.152499
$$689$$ −1.00000 −0.0380970
$$690$$ −10.0000 −0.380693
$$691$$ −14.0000 −0.532585 −0.266293 0.963892i $$-0.585799\pi$$
−0.266293 + 0.963892i $$0.585799\pi$$
$$692$$ 14.0000 0.532200
$$693$$ −6.00000 −0.227921
$$694$$ −26.0000 −0.986947
$$695$$ 28.0000 1.06210
$$696$$ −1.00000 −0.0379049
$$697$$ −42.0000 −1.59086
$$698$$ 10.0000 0.378506
$$699$$ 10.0000 0.378235
$$700$$ 3.00000 0.113389
$$701$$ 24.0000 0.906467 0.453234 0.891392i $$-0.350270\pi$$
0.453234 + 0.891392i $$0.350270\pi$$
$$702$$ 5.00000 0.188713
$$703$$ −6.00000 −0.226294
$$704$$ −1.00000 −0.0376889
$$705$$ 0 0
$$706$$ 1.00000 0.0376355
$$707$$ −18.0000 −0.676960
$$708$$ −3.00000 −0.112747
$$709$$ −26.0000 −0.976450 −0.488225 0.872718i $$-0.662356\pi$$
−0.488225 + 0.872718i $$0.662356\pi$$
$$710$$ 20.0000 0.750587
$$711$$ −16.0000 −0.600047
$$712$$ −8.00000 −0.299813
$$713$$ −50.0000 −1.87251
$$714$$ −21.0000 −0.785905
$$715$$ 2.00000 0.0747958
$$716$$ 4.00000 0.149487
$$717$$ −25.0000 −0.933642
$$718$$ −17.0000 −0.634434
$$719$$ −33.0000 −1.23069 −0.615346 0.788257i $$-0.710984\pi$$
−0.615346 + 0.788257i $$0.710984\pi$$
$$720$$ 4.00000 0.149071
$$721$$ −36.0000 −1.34071
$$722$$ 1.00000 0.0372161
$$723$$ 4.00000 0.148762
$$724$$ −2.00000 −0.0743294
$$725$$ −1.00000 −0.0371391
$$726$$ −1.00000 −0.0371135
$$727$$ 47.0000 1.74313 0.871567 0.490277i $$-0.163104\pi$$
0.871567 + 0.490277i $$0.163104\pi$$
$$728$$ −3.00000 −0.111187
$$729$$ 13.0000 0.481481
$$730$$ −6.00000 −0.222070
$$731$$ 28.0000 1.03562
$$732$$ 12.0000 0.443533
$$733$$ −14.0000 −0.517102 −0.258551 0.965998i $$-0.583245\pi$$
−0.258551 + 0.965998i $$0.583245\pi$$
$$734$$ 16.0000 0.590571
$$735$$ 4.00000 0.147542
$$736$$ −5.00000 −0.184302
$$737$$ −3.00000 −0.110506
$$738$$ −12.0000 −0.441726
$$739$$ −28.0000 −1.03000 −0.514998 0.857191i $$-0.672207\pi$$
−0.514998 + 0.857191i $$0.672207\pi$$
$$740$$ 12.0000 0.441129
$$741$$ −1.00000 −0.0367359
$$742$$ 3.00000 0.110133
$$743$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$744$$ −10.0000 −0.366618
$$745$$ 36.0000 1.31894
$$746$$ 11.0000 0.402739
$$747$$ −16.0000 −0.585409
$$748$$ 7.00000 0.255945
$$749$$ 39.0000 1.42503
$$750$$ −12.0000 −0.438178
$$751$$ −50.0000 −1.82453 −0.912263 0.409605i $$-0.865667\pi$$
−0.912263 + 0.409605i $$0.865667\pi$$
$$752$$ 0 0
$$753$$ 6.00000 0.218652
$$754$$ 1.00000 0.0364179
$$755$$ −24.0000 −0.873449
$$756$$ −15.0000 −0.545545
$$757$$ 38.0000 1.38113 0.690567 0.723269i $$-0.257361\pi$$
0.690567 + 0.723269i $$0.257361\pi$$
$$758$$ −5.00000 −0.181608
$$759$$ −5.00000 −0.181489
$$760$$ −2.00000 −0.0725476
$$761$$ −11.0000 −0.398750 −0.199375 0.979923i $$-0.563891\pi$$
−0.199375 + 0.979923i $$0.563891\pi$$
$$762$$ −16.0000 −0.579619
$$763$$ 39.0000 1.41189
$$764$$ 3.00000 0.108536
$$765$$ −28.0000 −1.01234
$$766$$ 2.00000 0.0722629
$$767$$ 3.00000 0.108324
$$768$$ −1.00000 −0.0360844
$$769$$ −49.0000 −1.76699 −0.883493 0.468445i $$-0.844814\pi$$
−0.883493 + 0.468445i $$0.844814\pi$$
$$770$$ −6.00000 −0.216225
$$771$$ 22.0000 0.792311
$$772$$ −4.00000 −0.143963
$$773$$ −23.0000 −0.827253 −0.413626 0.910447i $$-0.635738\pi$$
−0.413626 + 0.910447i $$0.635738\pi$$
$$774$$ 8.00000 0.287554
$$775$$ −10.0000 −0.359211
$$776$$ 8.00000 0.287183
$$777$$ −18.0000 −0.645746
$$778$$ 18.0000 0.645331
$$779$$ 6.00000 0.214972
$$780$$ 2.00000 0.0716115
$$781$$ 10.0000 0.357828
$$782$$ 35.0000 1.25160
$$783$$ 5.00000 0.178685
$$784$$ 2.00000 0.0714286
$$785$$ 44.0000 1.57043
$$786$$ 22.0000 0.784714
$$787$$ 49.0000 1.74666 0.873331 0.487128i $$-0.161955\pi$$
0.873331 + 0.487128i $$0.161955\pi$$
$$788$$ −12.0000 −0.427482
$$789$$ −4.00000 −0.142404
$$790$$ −16.0000 −0.569254
$$791$$ 36.0000 1.28001
$$792$$ 2.00000 0.0710669
$$793$$ −12.0000 −0.426132
$$794$$ 34.0000 1.20661
$$795$$ −2.00000 −0.0709327
$$796$$ 23.0000 0.815213
$$797$$ 19.0000 0.673015 0.336507 0.941681i $$-0.390754\pi$$
0.336507 + 0.941681i $$0.390754\pi$$
$$798$$ 3.00000 0.106199
$$799$$ 0 0
$$800$$ −1.00000 −0.0353553
$$801$$ 16.0000 0.565332
$$802$$ −2.00000 −0.0706225
$$803$$ −3.00000 −0.105868
$$804$$ −3.00000 −0.105802
$$805$$ −30.0000 −1.05736
$$806$$ 10.0000 0.352235
$$807$$ −18.0000 −0.633630
$$808$$ 6.00000 0.211079
$$809$$ −13.0000 −0.457056 −0.228528 0.973537i $$-0.573391\pi$$
−0.228528 + 0.973537i $$0.573391\pi$$
$$810$$ −2.00000 −0.0702728
$$811$$ −25.0000 −0.877869 −0.438934 0.898519i $$-0.644644\pi$$
−0.438934 + 0.898519i $$0.644644\pi$$
$$812$$ −3.00000 −0.105279
$$813$$ 15.0000 0.526073
$$814$$ 6.00000 0.210300
$$815$$ 4.00000 0.140114
$$816$$ 7.00000 0.245049
$$817$$ −4.00000 −0.139942
$$818$$ 34.0000 1.18878
$$819$$ 6.00000 0.209657
$$820$$ −12.0000 −0.419058
$$821$$ −16.0000 −0.558404 −0.279202 0.960232i $$-0.590070\pi$$
−0.279202 + 0.960232i $$0.590070\pi$$
$$822$$ 17.0000 0.592943
$$823$$ −43.0000 −1.49889 −0.749443 0.662069i $$-0.769679\pi$$
−0.749443 + 0.662069i $$0.769679\pi$$
$$824$$ 12.0000 0.418040
$$825$$ −1.00000 −0.0348155
$$826$$ −9.00000 −0.313150
$$827$$ 37.0000 1.28662 0.643308 0.765607i $$-0.277561\pi$$
0.643308 + 0.765607i $$0.277561\pi$$
$$828$$ 10.0000 0.347524
$$829$$ 25.0000 0.868286 0.434143 0.900844i $$-0.357051\pi$$
0.434143 + 0.900844i $$0.357051\pi$$
$$830$$ −16.0000 −0.555368
$$831$$ 16.0000 0.555034
$$832$$ 1.00000 0.0346688
$$833$$ −14.0000 −0.485071
$$834$$ 14.0000 0.484780
$$835$$ 44.0000 1.52268
$$836$$ −1.00000 −0.0345857
$$837$$ 50.0000 1.72825
$$838$$ 10.0000 0.345444
$$839$$ 40.0000 1.38095 0.690477 0.723355i $$-0.257401\pi$$
0.690477 + 0.723355i $$0.257401\pi$$
$$840$$ −6.00000 −0.207020
$$841$$ −28.0000 −0.965517
$$842$$ 15.0000 0.516934
$$843$$ −16.0000 −0.551069
$$844$$ 1.00000 0.0344214
$$845$$ 24.0000 0.825625
$$846$$ 0 0
$$847$$ −3.00000 −0.103081
$$848$$ −1.00000 −0.0343401
$$849$$ 14.0000 0.480479
$$850$$ 7.00000 0.240098
$$851$$ 30.0000 1.02839
$$852$$ 10.0000 0.342594
$$853$$ 6.00000 0.205436 0.102718 0.994711i $$-0.467246\pi$$
0.102718 + 0.994711i $$0.467246\pi$$
$$854$$ 36.0000 1.23189
$$855$$ 4.00000 0.136797
$$856$$ −13.0000 −0.444331
$$857$$ −30.0000 −1.02478 −0.512390 0.858753i $$-0.671240\pi$$
−0.512390 + 0.858753i $$0.671240\pi$$
$$858$$ 1.00000 0.0341394
$$859$$ −30.0000 −1.02359 −0.511793 0.859109i $$-0.671019\pi$$
−0.511793 + 0.859109i $$0.671019\pi$$
$$860$$ 8.00000 0.272798
$$861$$ 18.0000 0.613438
$$862$$ 34.0000 1.15804
$$863$$ −6.00000 −0.204242 −0.102121 0.994772i $$-0.532563\pi$$
−0.102121 + 0.994772i $$0.532563\pi$$
$$864$$ 5.00000 0.170103
$$865$$ −28.0000 −0.952029
$$866$$ 24.0000 0.815553
$$867$$ −32.0000 −1.08678
$$868$$ −30.0000 −1.01827
$$869$$ −8.00000 −0.271381
$$870$$ 2.00000 0.0678064
$$871$$ 3.00000 0.101651
$$872$$ −13.0000 −0.440236
$$873$$ −16.0000 −0.541518
$$874$$ −5.00000 −0.169128
$$875$$ −36.0000 −1.21702
$$876$$ −3.00000 −0.101361
$$877$$ 27.0000 0.911725 0.455863 0.890050i $$-0.349331\pi$$
0.455863 + 0.890050i $$0.349331\pi$$
$$878$$ −14.0000 −0.472477
$$879$$ 9.00000 0.303562
$$880$$ 2.00000 0.0674200
$$881$$ 42.0000 1.41502 0.707508 0.706705i $$-0.249819\pi$$
0.707508 + 0.706705i $$0.249819\pi$$
$$882$$ −4.00000 −0.134687
$$883$$ −44.0000 −1.48072 −0.740359 0.672212i $$-0.765344\pi$$
−0.740359 + 0.672212i $$0.765344\pi$$
$$884$$ −7.00000 −0.235435
$$885$$ 6.00000 0.201688
$$886$$ 18.0000 0.604722
$$887$$ −4.00000 −0.134307 −0.0671534 0.997743i $$-0.521392\pi$$
−0.0671534 + 0.997743i $$0.521392\pi$$
$$888$$ 6.00000 0.201347
$$889$$ −48.0000 −1.60987
$$890$$ 16.0000 0.536321
$$891$$ −1.00000 −0.0335013
$$892$$ 10.0000 0.334825
$$893$$ 0 0
$$894$$ 18.0000 0.602010
$$895$$ −8.00000 −0.267411
$$896$$ −3.00000 −0.100223
$$897$$ 5.00000 0.166945
$$898$$ 24.0000 0.800890
$$899$$ 10.0000 0.333519
$$900$$ 2.00000 0.0666667
$$901$$ 7.00000 0.233204
$$902$$ −6.00000 −0.199778
$$903$$ −12.0000 −0.399335
$$904$$ −12.0000 −0.399114
$$905$$ 4.00000 0.132964
$$906$$ −12.0000 −0.398673
$$907$$ 13.0000 0.431658 0.215829 0.976431i $$-0.430755\pi$$
0.215829 + 0.976431i $$0.430755\pi$$
$$908$$ −15.0000 −0.497792
$$909$$ −12.0000 −0.398015
$$910$$ 6.00000 0.198898
$$911$$ −6.00000 −0.198789 −0.0993944 0.995048i $$-0.531691\pi$$
−0.0993944 + 0.995048i $$0.531691\pi$$
$$912$$ −1.00000 −0.0331133
$$913$$ −8.00000 −0.264761
$$914$$ −37.0000 −1.22385
$$915$$ −24.0000 −0.793416
$$916$$ 20.0000 0.660819
$$917$$ 66.0000 2.17951
$$918$$ −35.0000 −1.15517
$$919$$ 3.00000 0.0989609 0.0494804 0.998775i $$-0.484243\pi$$
0.0494804 + 0.998775i $$0.484243\pi$$
$$920$$ 10.0000 0.329690
$$921$$ 16.0000 0.527218
$$922$$ −8.00000 −0.263466
$$923$$ −10.0000 −0.329154
$$924$$ −3.00000 −0.0986928
$$925$$ 6.00000 0.197279
$$926$$ 16.0000 0.525793
$$927$$ −24.0000 −0.788263
$$928$$ 1.00000 0.0328266
$$929$$ 29.0000 0.951459 0.475730 0.879592i $$-0.342184\pi$$
0.475730 + 0.879592i $$0.342184\pi$$
$$930$$ 20.0000 0.655826
$$931$$ 2.00000 0.0655474
$$932$$ −10.0000 −0.327561
$$933$$ −3.00000 −0.0982156
$$934$$ 20.0000 0.654420
$$935$$ −14.0000 −0.457849
$$936$$ −2.00000 −0.0653720
$$937$$ −5.00000 −0.163343 −0.0816714 0.996659i $$-0.526026\pi$$
−0.0816714 + 0.996659i $$0.526026\pi$$
$$938$$ −9.00000 −0.293860
$$939$$ −21.0000 −0.685309
$$940$$ 0 0
$$941$$ 33.0000 1.07577 0.537885 0.843018i $$-0.319224\pi$$
0.537885 + 0.843018i $$0.319224\pi$$
$$942$$ 22.0000 0.716799
$$943$$ −30.0000 −0.976934
$$944$$ 3.00000 0.0976417
$$945$$ 30.0000 0.975900
$$946$$ 4.00000 0.130051
$$947$$ −48.0000 −1.55979 −0.779895 0.625910i $$-0.784728\pi$$
−0.779895 + 0.625910i $$0.784728\pi$$
$$948$$ −8.00000 −0.259828
$$949$$ 3.00000 0.0973841
$$950$$ −1.00000 −0.0324443
$$951$$ −21.0000 −0.680972
$$952$$ 21.0000 0.680614
$$953$$ 60.0000 1.94359 0.971795 0.235826i $$-0.0757795\pi$$
0.971795 + 0.235826i $$0.0757795\pi$$
$$954$$ 2.00000 0.0647524
$$955$$ −6.00000 −0.194155
$$956$$ 25.0000 0.808558
$$957$$ 1.00000 0.0323254
$$958$$ −12.0000 −0.387702
$$959$$ 51.0000 1.64688
$$960$$ 2.00000 0.0645497
$$961$$ 69.0000 2.22581
$$962$$ −6.00000 −0.193448
$$963$$ 26.0000 0.837838
$$964$$ −4.00000 −0.128831
$$965$$ 8.00000 0.257529
$$966$$ −15.0000 −0.482617
$$967$$ 32.0000 1.02905 0.514525 0.857475i $$-0.327968\pi$$
0.514525 + 0.857475i $$0.327968\pi$$
$$968$$ 1.00000 0.0321412
$$969$$ 7.00000 0.224872
$$970$$ −16.0000 −0.513729
$$971$$ −44.0000 −1.41203 −0.706014 0.708198i $$-0.749508\pi$$
−0.706014 + 0.708198i $$0.749508\pi$$
$$972$$ −16.0000 −0.513200
$$973$$ 42.0000 1.34646
$$974$$ −12.0000 −0.384505
$$975$$ 1.00000 0.0320256
$$976$$ −12.0000 −0.384111
$$977$$ −60.0000 −1.91957 −0.959785 0.280736i $$-0.909421\pi$$
−0.959785 + 0.280736i $$0.909421\pi$$
$$978$$ 2.00000 0.0639529
$$979$$ 8.00000 0.255681
$$980$$ −4.00000 −0.127775
$$981$$ 26.0000 0.830116
$$982$$ 10.0000 0.319113
$$983$$ −46.0000 −1.46717 −0.733586 0.679597i $$-0.762155\pi$$
−0.733586 + 0.679597i $$0.762155\pi$$
$$984$$ −6.00000 −0.191273
$$985$$ 24.0000 0.764704
$$986$$ −7.00000 −0.222925
$$987$$ 0 0
$$988$$ 1.00000 0.0318142
$$989$$ 20.0000 0.635963
$$990$$ −4.00000 −0.127128
$$991$$ 44.0000 1.39771 0.698853 0.715265i $$-0.253694\pi$$
0.698853 + 0.715265i $$0.253694\pi$$
$$992$$ 10.0000 0.317500
$$993$$ −29.0000 −0.920287
$$994$$ 30.0000 0.951542
$$995$$ −46.0000 −1.45830
$$996$$ −8.00000 −0.253490
$$997$$ −10.0000 −0.316703 −0.158352 0.987383i $$-0.550618\pi$$
−0.158352 + 0.987383i $$0.550618\pi$$
$$998$$ −28.0000 −0.886325
$$999$$ −30.0000 −0.949158
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 418.2.a.a.1.1 1
3.2 odd 2 3762.2.a.g.1.1 1
4.3 odd 2 3344.2.a.h.1.1 1
11.10 odd 2 4598.2.a.b.1.1 1
19.18 odd 2 7942.2.a.i.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
418.2.a.a.1.1 1 1.1 even 1 trivial
3344.2.a.h.1.1 1 4.3 odd 2
3762.2.a.g.1.1 1 3.2 odd 2
4598.2.a.b.1.1 1 11.10 odd 2
7942.2.a.i.1.1 1 19.18 odd 2