# Properties

 Label 1425.2.a.j.1.1 Level $1425$ Weight $2$ Character 1425.1 Self dual yes Analytic conductor $11.379$ 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] = [1425,2,Mod(1,1425)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1425, 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("1425.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1425 = 3 \cdot 5^{2} \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1425.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: $$11.3786822880$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 57) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Character $$\chi$$ $$=$$ 1425.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+2.00000 q^{2} +1.00000 q^{3} +2.00000 q^{4} +2.00000 q^{6} +5.00000 q^{7} +1.00000 q^{9} +O(q^{10})$$ $$q+2.00000 q^{2} +1.00000 q^{3} +2.00000 q^{4} +2.00000 q^{6} +5.00000 q^{7} +1.00000 q^{9} +1.00000 q^{11} +2.00000 q^{12} -2.00000 q^{13} +10.0000 q^{14} -4.00000 q^{16} +1.00000 q^{17} +2.00000 q^{18} -1.00000 q^{19} +5.00000 q^{21} +2.00000 q^{22} +4.00000 q^{23} -4.00000 q^{26} +1.00000 q^{27} +10.0000 q^{28} -2.00000 q^{29} -6.00000 q^{31} -8.00000 q^{32} +1.00000 q^{33} +2.00000 q^{34} +2.00000 q^{36} -2.00000 q^{38} -2.00000 q^{39} +10.0000 q^{42} +1.00000 q^{43} +2.00000 q^{44} +8.00000 q^{46} +9.00000 q^{47} -4.00000 q^{48} +18.0000 q^{49} +1.00000 q^{51} -4.00000 q^{52} -10.0000 q^{53} +2.00000 q^{54} -1.00000 q^{57} -4.00000 q^{58} -8.00000 q^{59} -1.00000 q^{61} -12.0000 q^{62} +5.00000 q^{63} -8.00000 q^{64} +2.00000 q^{66} -8.00000 q^{67} +2.00000 q^{68} +4.00000 q^{69} -12.0000 q^{71} +11.0000 q^{73} -2.00000 q^{76} +5.00000 q^{77} -4.00000 q^{78} +16.0000 q^{79} +1.00000 q^{81} -12.0000 q^{83} +10.0000 q^{84} +2.00000 q^{86} -2.00000 q^{87} -6.00000 q^{89} -10.0000 q^{91} +8.00000 q^{92} -6.00000 q^{93} +18.0000 q^{94} -8.00000 q^{96} +10.0000 q^{97} +36.0000 q^{98} +1.00000 q^{99} +O(q^{100})$$

## 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
$$5$$ 0 0
$$6$$ 2.00000 0.816497
$$7$$ 5.00000 1.88982 0.944911 0.327327i $$-0.106148\pi$$
0.944911 + 0.327327i $$0.106148\pi$$
$$8$$ 0 0
$$9$$ 1.00000 0.333333
$$10$$ 0 0
$$11$$ 1.00000 0.301511 0.150756 0.988571i $$-0.451829\pi$$
0.150756 + 0.988571i $$0.451829\pi$$
$$12$$ 2.00000 0.577350
$$13$$ −2.00000 −0.554700 −0.277350 0.960769i $$-0.589456\pi$$
−0.277350 + 0.960769i $$0.589456\pi$$
$$14$$ 10.0000 2.67261
$$15$$ 0 0
$$16$$ −4.00000 −1.00000
$$17$$ 1.00000 0.242536 0.121268 0.992620i $$-0.461304\pi$$
0.121268 + 0.992620i $$0.461304\pi$$
$$18$$ 2.00000 0.471405
$$19$$ −1.00000 −0.229416
$$20$$ 0 0
$$21$$ 5.00000 1.09109
$$22$$ 2.00000 0.426401
$$23$$ 4.00000 0.834058 0.417029 0.908893i $$-0.363071\pi$$
0.417029 + 0.908893i $$0.363071\pi$$
$$24$$ 0 0
$$25$$ 0 0
$$26$$ −4.00000 −0.784465
$$27$$ 1.00000 0.192450
$$28$$ 10.0000 1.88982
$$29$$ −2.00000 −0.371391 −0.185695 0.982607i $$-0.559454\pi$$
−0.185695 + 0.982607i $$0.559454\pi$$
$$30$$ 0 0
$$31$$ −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
$$35$$ 0 0
$$36$$ 2.00000 0.333333
$$37$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$38$$ −2.00000 −0.324443
$$39$$ −2.00000 −0.320256
$$40$$ 0 0
$$41$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$42$$ 10.0000 1.54303
$$43$$ 1.00000 0.152499 0.0762493 0.997089i $$-0.475706\pi$$
0.0762493 + 0.997089i $$0.475706\pi$$
$$44$$ 2.00000 0.301511
$$45$$ 0 0
$$46$$ 8.00000 1.17954
$$47$$ 9.00000 1.31278 0.656392 0.754420i $$-0.272082\pi$$
0.656392 + 0.754420i $$0.272082\pi$$
$$48$$ −4.00000 −0.577350
$$49$$ 18.0000 2.57143
$$50$$ 0 0
$$51$$ 1.00000 0.140028
$$52$$ −4.00000 −0.554700
$$53$$ −10.0000 −1.37361 −0.686803 0.726844i $$-0.740986\pi$$
−0.686803 + 0.726844i $$0.740986\pi$$
$$54$$ 2.00000 0.272166
$$55$$ 0 0
$$56$$ 0 0
$$57$$ −1.00000 −0.132453
$$58$$ −4.00000 −0.525226
$$59$$ −8.00000 −1.04151 −0.520756 0.853706i $$-0.674350\pi$$
−0.520756 + 0.853706i $$0.674350\pi$$
$$60$$ 0 0
$$61$$ −1.00000 −0.128037 −0.0640184 0.997949i $$-0.520392\pi$$
−0.0640184 + 0.997949i $$0.520392\pi$$
$$62$$ −12.0000 −1.52400
$$63$$ 5.00000 0.629941
$$64$$ −8.00000 −1.00000
$$65$$ 0 0
$$66$$ 2.00000 0.246183
$$67$$ −8.00000 −0.977356 −0.488678 0.872464i $$-0.662521\pi$$
−0.488678 + 0.872464i $$0.662521\pi$$
$$68$$ 2.00000 0.242536
$$69$$ 4.00000 0.481543
$$70$$ 0 0
$$71$$ −12.0000 −1.42414 −0.712069 0.702109i $$-0.752242\pi$$
−0.712069 + 0.702109i $$0.752242\pi$$
$$72$$ 0 0
$$73$$ 11.0000 1.28745 0.643726 0.765256i $$-0.277388\pi$$
0.643726 + 0.765256i $$0.277388\pi$$
$$74$$ 0 0
$$75$$ 0 0
$$76$$ −2.00000 −0.229416
$$77$$ 5.00000 0.569803
$$78$$ −4.00000 −0.452911
$$79$$ 16.0000 1.80014 0.900070 0.435745i $$-0.143515\pi$$
0.900070 + 0.435745i $$0.143515\pi$$
$$80$$ 0 0
$$81$$ 1.00000 0.111111
$$82$$ 0 0
$$83$$ −12.0000 −1.31717 −0.658586 0.752506i $$-0.728845\pi$$
−0.658586 + 0.752506i $$0.728845\pi$$
$$84$$ 10.0000 1.09109
$$85$$ 0 0
$$86$$ 2.00000 0.215666
$$87$$ −2.00000 −0.214423
$$88$$ 0 0
$$89$$ −6.00000 −0.635999 −0.317999 0.948091i $$-0.603011\pi$$
−0.317999 + 0.948091i $$0.603011\pi$$
$$90$$ 0 0
$$91$$ −10.0000 −1.04828
$$92$$ 8.00000 0.834058
$$93$$ −6.00000 −0.622171
$$94$$ 18.0000 1.85656
$$95$$ 0 0
$$96$$ −8.00000 −0.816497
$$97$$ 10.0000 1.01535 0.507673 0.861550i $$-0.330506\pi$$
0.507673 + 0.861550i $$0.330506\pi$$
$$98$$ 36.0000 3.63655
$$99$$ 1.00000 0.100504
$$100$$ 0 0
$$101$$ 2.00000 0.199007 0.0995037 0.995037i $$-0.468274\pi$$
0.0995037 + 0.995037i $$0.468274\pi$$
$$102$$ 2.00000 0.198030
$$103$$ 2.00000 0.197066 0.0985329 0.995134i $$-0.468585\pi$$
0.0985329 + 0.995134i $$0.468585\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ −20.0000 −1.94257
$$107$$ −6.00000 −0.580042 −0.290021 0.957020i $$-0.593662\pi$$
−0.290021 + 0.957020i $$0.593662\pi$$
$$108$$ 2.00000 0.192450
$$109$$ 4.00000 0.383131 0.191565 0.981480i $$-0.438644\pi$$
0.191565 + 0.981480i $$0.438644\pi$$
$$110$$ 0 0
$$111$$ 0 0
$$112$$ −20.0000 −1.88982
$$113$$ −2.00000 −0.188144 −0.0940721 0.995565i $$-0.529988\pi$$
−0.0940721 + 0.995565i $$0.529988\pi$$
$$114$$ −2.00000 −0.187317
$$115$$ 0 0
$$116$$ −4.00000 −0.371391
$$117$$ −2.00000 −0.184900
$$118$$ −16.0000 −1.47292
$$119$$ 5.00000 0.458349
$$120$$ 0 0
$$121$$ −10.0000 −0.909091
$$122$$ −2.00000 −0.181071
$$123$$ 0 0
$$124$$ −12.0000 −1.07763
$$125$$ 0 0
$$126$$ 10.0000 0.890871
$$127$$ 2.00000 0.177471 0.0887357 0.996055i $$-0.471717\pi$$
0.0887357 + 0.996055i $$0.471717\pi$$
$$128$$ 0 0
$$129$$ 1.00000 0.0880451
$$130$$ 0 0
$$131$$ 7.00000 0.611593 0.305796 0.952097i $$-0.401077\pi$$
0.305796 + 0.952097i $$0.401077\pi$$
$$132$$ 2.00000 0.174078
$$133$$ −5.00000 −0.433555
$$134$$ −16.0000 −1.38219
$$135$$ 0 0
$$136$$ 0 0
$$137$$ 9.00000 0.768922 0.384461 0.923141i $$-0.374387\pi$$
0.384461 + 0.923141i $$0.374387\pi$$
$$138$$ 8.00000 0.681005
$$139$$ −13.0000 −1.10265 −0.551323 0.834292i $$-0.685877\pi$$
−0.551323 + 0.834292i $$0.685877\pi$$
$$140$$ 0 0
$$141$$ 9.00000 0.757937
$$142$$ −24.0000 −2.01404
$$143$$ −2.00000 −0.167248
$$144$$ −4.00000 −0.333333
$$145$$ 0 0
$$146$$ 22.0000 1.82073
$$147$$ 18.0000 1.48461
$$148$$ 0 0
$$149$$ −21.0000 −1.72039 −0.860194 0.509968i $$-0.829657\pi$$
−0.860194 + 0.509968i $$0.829657\pi$$
$$150$$ 0 0
$$151$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$152$$ 0 0
$$153$$ 1.00000 0.0808452
$$154$$ 10.0000 0.805823
$$155$$ 0 0
$$156$$ −4.00000 −0.320256
$$157$$ 18.0000 1.43656 0.718278 0.695756i $$-0.244931\pi$$
0.718278 + 0.695756i $$0.244931\pi$$
$$158$$ 32.0000 2.54578
$$159$$ −10.0000 −0.793052
$$160$$ 0 0
$$161$$ 20.0000 1.57622
$$162$$ 2.00000 0.157135
$$163$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ −24.0000 −1.86276
$$167$$ −10.0000 −0.773823 −0.386912 0.922117i $$-0.626458\pi$$
−0.386912 + 0.922117i $$0.626458\pi$$
$$168$$ 0 0
$$169$$ −9.00000 −0.692308
$$170$$ 0 0
$$171$$ −1.00000 −0.0764719
$$172$$ 2.00000 0.152499
$$173$$ −6.00000 −0.456172 −0.228086 0.973641i $$-0.573247\pi$$
−0.228086 + 0.973641i $$0.573247\pi$$
$$174$$ −4.00000 −0.303239
$$175$$ 0 0
$$176$$ −4.00000 −0.301511
$$177$$ −8.00000 −0.601317
$$178$$ −12.0000 −0.899438
$$179$$ −18.0000 −1.34538 −0.672692 0.739923i $$-0.734862\pi$$
−0.672692 + 0.739923i $$0.734862\pi$$
$$180$$ 0 0
$$181$$ −14.0000 −1.04061 −0.520306 0.853980i $$-0.674182\pi$$
−0.520306 + 0.853980i $$0.674182\pi$$
$$182$$ −20.0000 −1.48250
$$183$$ −1.00000 −0.0739221
$$184$$ 0 0
$$185$$ 0 0
$$186$$ −12.0000 −0.879883
$$187$$ 1.00000 0.0731272
$$188$$ 18.0000 1.31278
$$189$$ 5.00000 0.363696
$$190$$ 0 0
$$191$$ 9.00000 0.651217 0.325609 0.945505i $$-0.394431\pi$$
0.325609 + 0.945505i $$0.394431\pi$$
$$192$$ −8.00000 −0.577350
$$193$$ −4.00000 −0.287926 −0.143963 0.989583i $$-0.545985\pi$$
−0.143963 + 0.989583i $$0.545985\pi$$
$$194$$ 20.0000 1.43592
$$195$$ 0 0
$$196$$ 36.0000 2.57143
$$197$$ 2.00000 0.142494 0.0712470 0.997459i $$-0.477302\pi$$
0.0712470 + 0.997459i $$0.477302\pi$$
$$198$$ 2.00000 0.142134
$$199$$ −21.0000 −1.48865 −0.744325 0.667817i $$-0.767229\pi$$
−0.744325 + 0.667817i $$0.767229\pi$$
$$200$$ 0 0
$$201$$ −8.00000 −0.564276
$$202$$ 4.00000 0.281439
$$203$$ −10.0000 −0.701862
$$204$$ 2.00000 0.140028
$$205$$ 0 0
$$206$$ 4.00000 0.278693
$$207$$ 4.00000 0.278019
$$208$$ 8.00000 0.554700
$$209$$ −1.00000 −0.0691714
$$210$$ 0 0
$$211$$ 12.0000 0.826114 0.413057 0.910705i $$-0.364461\pi$$
0.413057 + 0.910705i $$0.364461\pi$$
$$212$$ −20.0000 −1.37361
$$213$$ −12.0000 −0.822226
$$214$$ −12.0000 −0.820303
$$215$$ 0 0
$$216$$ 0 0
$$217$$ −30.0000 −2.03653
$$218$$ 8.00000 0.541828
$$219$$ 11.0000 0.743311
$$220$$ 0 0
$$221$$ −2.00000 −0.134535
$$222$$ 0 0
$$223$$ −12.0000 −0.803579 −0.401790 0.915732i $$-0.631612\pi$$
−0.401790 + 0.915732i $$0.631612\pi$$
$$224$$ −40.0000 −2.67261
$$225$$ 0 0
$$226$$ −4.00000 −0.266076
$$227$$ −18.0000 −1.19470 −0.597351 0.801980i $$-0.703780\pi$$
−0.597351 + 0.801980i $$0.703780\pi$$
$$228$$ −2.00000 −0.132453
$$229$$ 25.0000 1.65205 0.826023 0.563636i $$-0.190598\pi$$
0.826023 + 0.563636i $$0.190598\pi$$
$$230$$ 0 0
$$231$$ 5.00000 0.328976
$$232$$ 0 0
$$233$$ −9.00000 −0.589610 −0.294805 0.955557i $$-0.595255\pi$$
−0.294805 + 0.955557i $$0.595255\pi$$
$$234$$ −4.00000 −0.261488
$$235$$ 0 0
$$236$$ −16.0000 −1.04151
$$237$$ 16.0000 1.03931
$$238$$ 10.0000 0.648204
$$239$$ −3.00000 −0.194054 −0.0970269 0.995282i $$-0.530933\pi$$
−0.0970269 + 0.995282i $$0.530933\pi$$
$$240$$ 0 0
$$241$$ 20.0000 1.28831 0.644157 0.764894i $$-0.277208\pi$$
0.644157 + 0.764894i $$0.277208\pi$$
$$242$$ −20.0000 −1.28565
$$243$$ 1.00000 0.0641500
$$244$$ −2.00000 −0.128037
$$245$$ 0 0
$$246$$ 0 0
$$247$$ 2.00000 0.127257
$$248$$ 0 0
$$249$$ −12.0000 −0.760469
$$250$$ 0 0
$$251$$ 7.00000 0.441836 0.220918 0.975292i $$-0.429095\pi$$
0.220918 + 0.975292i $$0.429095\pi$$
$$252$$ 10.0000 0.629941
$$253$$ 4.00000 0.251478
$$254$$ 4.00000 0.250982
$$255$$ 0 0
$$256$$ 16.0000 1.00000
$$257$$ 8.00000 0.499026 0.249513 0.968371i $$-0.419729\pi$$
0.249513 + 0.968371i $$0.419729\pi$$
$$258$$ 2.00000 0.124515
$$259$$ 0 0
$$260$$ 0 0
$$261$$ −2.00000 −0.123797
$$262$$ 14.0000 0.864923
$$263$$ −23.0000 −1.41824 −0.709120 0.705087i $$-0.750908\pi$$
−0.709120 + 0.705087i $$0.750908\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ −10.0000 −0.613139
$$267$$ −6.00000 −0.367194
$$268$$ −16.0000 −0.977356
$$269$$ −14.0000 −0.853595 −0.426798 0.904347i $$-0.640358\pi$$
−0.426798 + 0.904347i $$0.640358\pi$$
$$270$$ 0 0
$$271$$ 12.0000 0.728948 0.364474 0.931214i $$-0.381249\pi$$
0.364474 + 0.931214i $$0.381249\pi$$
$$272$$ −4.00000 −0.242536
$$273$$ −10.0000 −0.605228
$$274$$ 18.0000 1.08742
$$275$$ 0 0
$$276$$ 8.00000 0.481543
$$277$$ 11.0000 0.660926 0.330463 0.943819i $$-0.392795\pi$$
0.330463 + 0.943819i $$0.392795\pi$$
$$278$$ −26.0000 −1.55938
$$279$$ −6.00000 −0.359211
$$280$$ 0 0
$$281$$ 10.0000 0.596550 0.298275 0.954480i $$-0.403589\pi$$
0.298275 + 0.954480i $$0.403589\pi$$
$$282$$ 18.0000 1.07188
$$283$$ 13.0000 0.772770 0.386385 0.922338i $$-0.373724\pi$$
0.386385 + 0.922338i $$0.373724\pi$$
$$284$$ −24.0000 −1.42414
$$285$$ 0 0
$$286$$ −4.00000 −0.236525
$$287$$ 0 0
$$288$$ −8.00000 −0.471405
$$289$$ −16.0000 −0.941176
$$290$$ 0 0
$$291$$ 10.0000 0.586210
$$292$$ 22.0000 1.28745
$$293$$ 28.0000 1.63578 0.817889 0.575376i $$-0.195144\pi$$
0.817889 + 0.575376i $$0.195144\pi$$
$$294$$ 36.0000 2.09956
$$295$$ 0 0
$$296$$ 0 0
$$297$$ 1.00000 0.0580259
$$298$$ −42.0000 −2.43299
$$299$$ −8.00000 −0.462652
$$300$$ 0 0
$$301$$ 5.00000 0.288195
$$302$$ 0 0
$$303$$ 2.00000 0.114897
$$304$$ 4.00000 0.229416
$$305$$ 0 0
$$306$$ 2.00000 0.114332
$$307$$ 12.0000 0.684876 0.342438 0.939540i $$-0.388747\pi$$
0.342438 + 0.939540i $$0.388747\pi$$
$$308$$ 10.0000 0.569803
$$309$$ 2.00000 0.113776
$$310$$ 0 0
$$311$$ −21.0000 −1.19080 −0.595400 0.803429i $$-0.703007\pi$$
−0.595400 + 0.803429i $$0.703007\pi$$
$$312$$ 0 0
$$313$$ 2.00000 0.113047 0.0565233 0.998401i $$-0.481998\pi$$
0.0565233 + 0.998401i $$0.481998\pi$$
$$314$$ 36.0000 2.03160
$$315$$ 0 0
$$316$$ 32.0000 1.80014
$$317$$ 4.00000 0.224662 0.112331 0.993671i $$-0.464168\pi$$
0.112331 + 0.993671i $$0.464168\pi$$
$$318$$ −20.0000 −1.12154
$$319$$ −2.00000 −0.111979
$$320$$ 0 0
$$321$$ −6.00000 −0.334887
$$322$$ 40.0000 2.22911
$$323$$ −1.00000 −0.0556415
$$324$$ 2.00000 0.111111
$$325$$ 0 0
$$326$$ 0 0
$$327$$ 4.00000 0.221201
$$328$$ 0 0
$$329$$ 45.0000 2.48093
$$330$$ 0 0
$$331$$ −4.00000 −0.219860 −0.109930 0.993939i $$-0.535063\pi$$
−0.109930 + 0.993939i $$0.535063\pi$$
$$332$$ −24.0000 −1.31717
$$333$$ 0 0
$$334$$ −20.0000 −1.09435
$$335$$ 0 0
$$336$$ −20.0000 −1.09109
$$337$$ 14.0000 0.762629 0.381314 0.924445i $$-0.375472\pi$$
0.381314 + 0.924445i $$0.375472\pi$$
$$338$$ −18.0000 −0.979071
$$339$$ −2.00000 −0.108625
$$340$$ 0 0
$$341$$ −6.00000 −0.324918
$$342$$ −2.00000 −0.108148
$$343$$ 55.0000 2.96972
$$344$$ 0 0
$$345$$ 0 0
$$346$$ −12.0000 −0.645124
$$347$$ 25.0000 1.34207 0.671035 0.741426i $$-0.265850\pi$$
0.671035 + 0.741426i $$0.265850\pi$$
$$348$$ −4.00000 −0.214423
$$349$$ 9.00000 0.481759 0.240879 0.970555i $$-0.422564\pi$$
0.240879 + 0.970555i $$0.422564\pi$$
$$350$$ 0 0
$$351$$ −2.00000 −0.106752
$$352$$ −8.00000 −0.426401
$$353$$ 2.00000 0.106449 0.0532246 0.998583i $$-0.483050\pi$$
0.0532246 + 0.998583i $$0.483050\pi$$
$$354$$ −16.0000 −0.850390
$$355$$ 0 0
$$356$$ −12.0000 −0.635999
$$357$$ 5.00000 0.264628
$$358$$ −36.0000 −1.90266
$$359$$ 37.0000 1.95279 0.976393 0.216003i $$-0.0693022\pi$$
0.976393 + 0.216003i $$0.0693022\pi$$
$$360$$ 0 0
$$361$$ 1.00000 0.0526316
$$362$$ −28.0000 −1.47165
$$363$$ −10.0000 −0.524864
$$364$$ −20.0000 −1.04828
$$365$$ 0 0
$$366$$ −2.00000 −0.104542
$$367$$ 8.00000 0.417597 0.208798 0.977959i $$-0.433045\pi$$
0.208798 + 0.977959i $$0.433045\pi$$
$$368$$ −16.0000 −0.834058
$$369$$ 0 0
$$370$$ 0 0
$$371$$ −50.0000 −2.59587
$$372$$ −12.0000 −0.622171
$$373$$ −16.0000 −0.828449 −0.414224 0.910175i $$-0.635947\pi$$
−0.414224 + 0.910175i $$0.635947\pi$$
$$374$$ 2.00000 0.103418
$$375$$ 0 0
$$376$$ 0 0
$$377$$ 4.00000 0.206010
$$378$$ 10.0000 0.514344
$$379$$ 34.0000 1.74646 0.873231 0.487306i $$-0.162020\pi$$
0.873231 + 0.487306i $$0.162020\pi$$
$$380$$ 0 0
$$381$$ 2.00000 0.102463
$$382$$ 18.0000 0.920960
$$383$$ 34.0000 1.73732 0.868659 0.495410i $$-0.164982\pi$$
0.868659 + 0.495410i $$0.164982\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ −8.00000 −0.407189
$$387$$ 1.00000 0.0508329
$$388$$ 20.0000 1.01535
$$389$$ −27.0000 −1.36895 −0.684477 0.729034i $$-0.739969\pi$$
−0.684477 + 0.729034i $$0.739969\pi$$
$$390$$ 0 0
$$391$$ 4.00000 0.202289
$$392$$ 0 0
$$393$$ 7.00000 0.353103
$$394$$ 4.00000 0.201517
$$395$$ 0 0
$$396$$ 2.00000 0.100504
$$397$$ −25.0000 −1.25471 −0.627357 0.778732i $$-0.715863\pi$$
−0.627357 + 0.778732i $$0.715863\pi$$
$$398$$ −42.0000 −2.10527
$$399$$ −5.00000 −0.250313
$$400$$ 0 0
$$401$$ 36.0000 1.79775 0.898877 0.438201i $$-0.144384\pi$$
0.898877 + 0.438201i $$0.144384\pi$$
$$402$$ −16.0000 −0.798007
$$403$$ 12.0000 0.597763
$$404$$ 4.00000 0.199007
$$405$$ 0 0
$$406$$ −20.0000 −0.992583
$$407$$ 0 0
$$408$$ 0 0
$$409$$ −14.0000 −0.692255 −0.346128 0.938187i $$-0.612504\pi$$
−0.346128 + 0.938187i $$0.612504\pi$$
$$410$$ 0 0
$$411$$ 9.00000 0.443937
$$412$$ 4.00000 0.197066
$$413$$ −40.0000 −1.96827
$$414$$ 8.00000 0.393179
$$415$$ 0 0
$$416$$ 16.0000 0.784465
$$417$$ −13.0000 −0.636613
$$418$$ −2.00000 −0.0978232
$$419$$ 28.0000 1.36789 0.683945 0.729534i $$-0.260263\pi$$
0.683945 + 0.729534i $$0.260263\pi$$
$$420$$ 0 0
$$421$$ 26.0000 1.26716 0.633581 0.773676i $$-0.281584\pi$$
0.633581 + 0.773676i $$0.281584\pi$$
$$422$$ 24.0000 1.16830
$$423$$ 9.00000 0.437595
$$424$$ 0 0
$$425$$ 0 0
$$426$$ −24.0000 −1.16280
$$427$$ −5.00000 −0.241967
$$428$$ −12.0000 −0.580042
$$429$$ −2.00000 −0.0965609
$$430$$ 0 0
$$431$$ −34.0000 −1.63772 −0.818861 0.573992i $$-0.805394\pi$$
−0.818861 + 0.573992i $$0.805394\pi$$
$$432$$ −4.00000 −0.192450
$$433$$ −6.00000 −0.288342 −0.144171 0.989553i $$-0.546051\pi$$
−0.144171 + 0.989553i $$0.546051\pi$$
$$434$$ −60.0000 −2.88009
$$435$$ 0 0
$$436$$ 8.00000 0.383131
$$437$$ −4.00000 −0.191346
$$438$$ 22.0000 1.05120
$$439$$ 26.0000 1.24091 0.620456 0.784241i $$-0.286947\pi$$
0.620456 + 0.784241i $$0.286947\pi$$
$$440$$ 0 0
$$441$$ 18.0000 0.857143
$$442$$ −4.00000 −0.190261
$$443$$ 5.00000 0.237557 0.118779 0.992921i $$-0.462102\pi$$
0.118779 + 0.992921i $$0.462102\pi$$
$$444$$ 0 0
$$445$$ 0 0
$$446$$ −24.0000 −1.13643
$$447$$ −21.0000 −0.993266
$$448$$ −40.0000 −1.88982
$$449$$ −36.0000 −1.69895 −0.849473 0.527633i $$-0.823080\pi$$
−0.849473 + 0.527633i $$0.823080\pi$$
$$450$$ 0 0
$$451$$ 0 0
$$452$$ −4.00000 −0.188144
$$453$$ 0 0
$$454$$ −36.0000 −1.68956
$$455$$ 0 0
$$456$$ 0 0
$$457$$ 29.0000 1.35656 0.678281 0.734802i $$-0.262725\pi$$
0.678281 + 0.734802i $$0.262725\pi$$
$$458$$ 50.0000 2.33635
$$459$$ 1.00000 0.0466760
$$460$$ 0 0
$$461$$ 27.0000 1.25752 0.628758 0.777601i $$-0.283564\pi$$
0.628758 + 0.777601i $$0.283564\pi$$
$$462$$ 10.0000 0.465242
$$463$$ −17.0000 −0.790057 −0.395029 0.918669i $$-0.629265\pi$$
−0.395029 + 0.918669i $$0.629265\pi$$
$$464$$ 8.00000 0.371391
$$465$$ 0 0
$$466$$ −18.0000 −0.833834
$$467$$ 5.00000 0.231372 0.115686 0.993286i $$-0.463093\pi$$
0.115686 + 0.993286i $$0.463093\pi$$
$$468$$ −4.00000 −0.184900
$$469$$ −40.0000 −1.84703
$$470$$ 0 0
$$471$$ 18.0000 0.829396
$$472$$ 0 0
$$473$$ 1.00000 0.0459800
$$474$$ 32.0000 1.46981
$$475$$ 0 0
$$476$$ 10.0000 0.458349
$$477$$ −10.0000 −0.457869
$$478$$ −6.00000 −0.274434
$$479$$ −16.0000 −0.731059 −0.365529 0.930800i $$-0.619112\pi$$
−0.365529 + 0.930800i $$0.619112\pi$$
$$480$$ 0 0
$$481$$ 0 0
$$482$$ 40.0000 1.82195
$$483$$ 20.0000 0.910032
$$484$$ −20.0000 −0.909091
$$485$$ 0 0
$$486$$ 2.00000 0.0907218
$$487$$ 16.0000 0.725029 0.362515 0.931978i $$-0.381918\pi$$
0.362515 + 0.931978i $$0.381918\pi$$
$$488$$ 0 0
$$489$$ 0 0
$$490$$ 0 0
$$491$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$492$$ 0 0
$$493$$ −2.00000 −0.0900755
$$494$$ 4.00000 0.179969
$$495$$ 0 0
$$496$$ 24.0000 1.07763
$$497$$ −60.0000 −2.69137
$$498$$ −24.0000 −1.07547
$$499$$ 5.00000 0.223831 0.111915 0.993718i $$-0.464301\pi$$
0.111915 + 0.993718i $$0.464301\pi$$
$$500$$ 0 0
$$501$$ −10.0000 −0.446767
$$502$$ 14.0000 0.624851
$$503$$ −16.0000 −0.713405 −0.356702 0.934218i $$-0.616099\pi$$
−0.356702 + 0.934218i $$0.616099\pi$$
$$504$$ 0 0
$$505$$ 0 0
$$506$$ 8.00000 0.355643
$$507$$ −9.00000 −0.399704
$$508$$ 4.00000 0.177471
$$509$$ 6.00000 0.265945 0.132973 0.991120i $$-0.457548\pi$$
0.132973 + 0.991120i $$0.457548\pi$$
$$510$$ 0 0
$$511$$ 55.0000 2.43306
$$512$$ 32.0000 1.41421
$$513$$ −1.00000 −0.0441511
$$514$$ 16.0000 0.705730
$$515$$ 0 0
$$516$$ 2.00000 0.0880451
$$517$$ 9.00000 0.395820
$$518$$ 0 0
$$519$$ −6.00000 −0.263371
$$520$$ 0 0
$$521$$ 36.0000 1.57719 0.788594 0.614914i $$-0.210809\pi$$
0.788594 + 0.614914i $$0.210809\pi$$
$$522$$ −4.00000 −0.175075
$$523$$ 34.0000 1.48672 0.743358 0.668894i $$-0.233232\pi$$
0.743358 + 0.668894i $$0.233232\pi$$
$$524$$ 14.0000 0.611593
$$525$$ 0 0
$$526$$ −46.0000 −2.00570
$$527$$ −6.00000 −0.261364
$$528$$ −4.00000 −0.174078
$$529$$ −7.00000 −0.304348
$$530$$ 0 0
$$531$$ −8.00000 −0.347170
$$532$$ −10.0000 −0.433555
$$533$$ 0 0
$$534$$ −12.0000 −0.519291
$$535$$ 0 0
$$536$$ 0 0
$$537$$ −18.0000 −0.776757
$$538$$ −28.0000 −1.20717
$$539$$ 18.0000 0.775315
$$540$$ 0 0
$$541$$ 3.00000 0.128980 0.0644900 0.997918i $$-0.479458\pi$$
0.0644900 + 0.997918i $$0.479458\pi$$
$$542$$ 24.0000 1.03089
$$543$$ −14.0000 −0.600798
$$544$$ −8.00000 −0.342997
$$545$$ 0 0
$$546$$ −20.0000 −0.855921
$$547$$ 26.0000 1.11168 0.555840 0.831289i $$-0.312397\pi$$
0.555840 + 0.831289i $$0.312397\pi$$
$$548$$ 18.0000 0.768922
$$549$$ −1.00000 −0.0426790
$$550$$ 0 0
$$551$$ 2.00000 0.0852029
$$552$$ 0 0
$$553$$ 80.0000 3.40195
$$554$$ 22.0000 0.934690
$$555$$ 0 0
$$556$$ −26.0000 −1.10265
$$557$$ 41.0000 1.73723 0.868613 0.495491i $$-0.165012\pi$$
0.868613 + 0.495491i $$0.165012\pi$$
$$558$$ −12.0000 −0.508001
$$559$$ −2.00000 −0.0845910
$$560$$ 0 0
$$561$$ 1.00000 0.0422200
$$562$$ 20.0000 0.843649
$$563$$ 12.0000 0.505740 0.252870 0.967500i $$-0.418626\pi$$
0.252870 + 0.967500i $$0.418626\pi$$
$$564$$ 18.0000 0.757937
$$565$$ 0 0
$$566$$ 26.0000 1.09286
$$567$$ 5.00000 0.209980
$$568$$ 0 0
$$569$$ −18.0000 −0.754599 −0.377300 0.926091i $$-0.623147\pi$$
−0.377300 + 0.926091i $$0.623147\pi$$
$$570$$ 0 0
$$571$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$572$$ −4.00000 −0.167248
$$573$$ 9.00000 0.375980
$$574$$ 0 0
$$575$$ 0 0
$$576$$ −8.00000 −0.333333
$$577$$ −27.0000 −1.12402 −0.562012 0.827129i $$-0.689973\pi$$
−0.562012 + 0.827129i $$0.689973\pi$$
$$578$$ −32.0000 −1.33102
$$579$$ −4.00000 −0.166234
$$580$$ 0 0
$$581$$ −60.0000 −2.48922
$$582$$ 20.0000 0.829027
$$583$$ −10.0000 −0.414158
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 56.0000 2.31334
$$587$$ −7.00000 −0.288921 −0.144460 0.989511i $$-0.546145\pi$$
−0.144460 + 0.989511i $$0.546145\pi$$
$$588$$ 36.0000 1.48461
$$589$$ 6.00000 0.247226
$$590$$ 0 0
$$591$$ 2.00000 0.0822690
$$592$$ 0 0
$$593$$ 6.00000 0.246390 0.123195 0.992382i $$-0.460686\pi$$
0.123195 + 0.992382i $$0.460686\pi$$
$$594$$ 2.00000 0.0820610
$$595$$ 0 0
$$596$$ −42.0000 −1.72039
$$597$$ −21.0000 −0.859473
$$598$$ −16.0000 −0.654289
$$599$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$600$$ 0 0
$$601$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$602$$ 10.0000 0.407570
$$603$$ −8.00000 −0.325785
$$604$$ 0 0
$$605$$ 0 0
$$606$$ 4.00000 0.162489
$$607$$ −26.0000 −1.05531 −0.527654 0.849460i $$-0.676928\pi$$
−0.527654 + 0.849460i $$0.676928\pi$$
$$608$$ 8.00000 0.324443
$$609$$ −10.0000 −0.405220
$$610$$ 0 0
$$611$$ −18.0000 −0.728202
$$612$$ 2.00000 0.0808452
$$613$$ −33.0000 −1.33286 −0.666429 0.745569i $$-0.732178\pi$$
−0.666429 + 0.745569i $$0.732178\pi$$
$$614$$ 24.0000 0.968561
$$615$$ 0 0
$$616$$ 0 0
$$617$$ −27.0000 −1.08698 −0.543490 0.839416i $$-0.682897\pi$$
−0.543490 + 0.839416i $$0.682897\pi$$
$$618$$ 4.00000 0.160904
$$619$$ 4.00000 0.160774 0.0803868 0.996764i $$-0.474384\pi$$
0.0803868 + 0.996764i $$0.474384\pi$$
$$620$$ 0 0
$$621$$ 4.00000 0.160514
$$622$$ −42.0000 −1.68405
$$623$$ −30.0000 −1.20192
$$624$$ 8.00000 0.320256
$$625$$ 0 0
$$626$$ 4.00000 0.159872
$$627$$ −1.00000 −0.0399362
$$628$$ 36.0000 1.43656
$$629$$ 0 0
$$630$$ 0 0
$$631$$ 15.0000 0.597141 0.298570 0.954388i $$-0.403490\pi$$
0.298570 + 0.954388i $$0.403490\pi$$
$$632$$ 0 0
$$633$$ 12.0000 0.476957
$$634$$ 8.00000 0.317721
$$635$$ 0 0
$$636$$ −20.0000 −0.793052
$$637$$ −36.0000 −1.42637
$$638$$ −4.00000 −0.158362
$$639$$ −12.0000 −0.474713
$$640$$ 0 0
$$641$$ 18.0000 0.710957 0.355479 0.934684i $$-0.384318\pi$$
0.355479 + 0.934684i $$0.384318\pi$$
$$642$$ −12.0000 −0.473602
$$643$$ 1.00000 0.0394362 0.0197181 0.999806i $$-0.493723\pi$$
0.0197181 + 0.999806i $$0.493723\pi$$
$$644$$ 40.0000 1.57622
$$645$$ 0 0
$$646$$ −2.00000 −0.0786889
$$647$$ 39.0000 1.53325 0.766624 0.642096i $$-0.221935\pi$$
0.766624 + 0.642096i $$0.221935\pi$$
$$648$$ 0 0
$$649$$ −8.00000 −0.314027
$$650$$ 0 0
$$651$$ −30.0000 −1.17579
$$652$$ 0 0
$$653$$ −3.00000 −0.117399 −0.0586995 0.998276i $$-0.518695\pi$$
−0.0586995 + 0.998276i $$0.518695\pi$$
$$654$$ 8.00000 0.312825
$$655$$ 0 0
$$656$$ 0 0
$$657$$ 11.0000 0.429151
$$658$$ 90.0000 3.50857
$$659$$ −14.0000 −0.545363 −0.272681 0.962104i $$-0.587910\pi$$
−0.272681 + 0.962104i $$0.587910\pi$$
$$660$$ 0 0
$$661$$ 12.0000 0.466746 0.233373 0.972387i $$-0.425024\pi$$
0.233373 + 0.972387i $$0.425024\pi$$
$$662$$ −8.00000 −0.310929
$$663$$ −2.00000 −0.0776736
$$664$$ 0 0
$$665$$ 0 0
$$666$$ 0 0
$$667$$ −8.00000 −0.309761
$$668$$ −20.0000 −0.773823
$$669$$ −12.0000 −0.463947
$$670$$ 0 0
$$671$$ −1.00000 −0.0386046
$$672$$ −40.0000 −1.54303
$$673$$ 24.0000 0.925132 0.462566 0.886585i $$-0.346929\pi$$
0.462566 + 0.886585i $$0.346929\pi$$
$$674$$ 28.0000 1.07852
$$675$$ 0 0
$$676$$ −18.0000 −0.692308
$$677$$ −34.0000 −1.30673 −0.653363 0.757045i $$-0.726642\pi$$
−0.653363 + 0.757045i $$0.726642\pi$$
$$678$$ −4.00000 −0.153619
$$679$$ 50.0000 1.91882
$$680$$ 0 0
$$681$$ −18.0000 −0.689761
$$682$$ −12.0000 −0.459504
$$683$$ 6.00000 0.229584 0.114792 0.993390i $$-0.463380\pi$$
0.114792 + 0.993390i $$0.463380\pi$$
$$684$$ −2.00000 −0.0764719
$$685$$ 0 0
$$686$$ 110.000 4.19982
$$687$$ 25.0000 0.953809
$$688$$ −4.00000 −0.152499
$$689$$ 20.0000 0.761939
$$690$$ 0 0
$$691$$ −31.0000 −1.17930 −0.589648 0.807661i $$-0.700733\pi$$
−0.589648 + 0.807661i $$0.700733\pi$$
$$692$$ −12.0000 −0.456172
$$693$$ 5.00000 0.189934
$$694$$ 50.0000 1.89797
$$695$$ 0 0
$$696$$ 0 0
$$697$$ 0 0
$$698$$ 18.0000 0.681310
$$699$$ −9.00000 −0.340411
$$700$$ 0 0
$$701$$ −22.0000 −0.830929 −0.415464 0.909610i $$-0.636381\pi$$
−0.415464 + 0.909610i $$0.636381\pi$$
$$702$$ −4.00000 −0.150970
$$703$$ 0 0
$$704$$ −8.00000 −0.301511
$$705$$ 0 0
$$706$$ 4.00000 0.150542
$$707$$ 10.0000 0.376089
$$708$$ −16.0000 −0.601317
$$709$$ −42.0000 −1.57734 −0.788672 0.614815i $$-0.789231\pi$$
−0.788672 + 0.614815i $$0.789231\pi$$
$$710$$ 0 0
$$711$$ 16.0000 0.600047
$$712$$ 0 0
$$713$$ −24.0000 −0.898807
$$714$$ 10.0000 0.374241
$$715$$ 0 0
$$716$$ −36.0000 −1.34538
$$717$$ −3.00000 −0.112037
$$718$$ 74.0000 2.76166
$$719$$ 33.0000 1.23069 0.615346 0.788257i $$-0.289016\pi$$
0.615346 + 0.788257i $$0.289016\pi$$
$$720$$ 0 0
$$721$$ 10.0000 0.372419
$$722$$ 2.00000 0.0744323
$$723$$ 20.0000 0.743808
$$724$$ −28.0000 −1.04061
$$725$$ 0 0
$$726$$ −20.0000 −0.742270
$$727$$ 23.0000 0.853023 0.426511 0.904482i $$-0.359742\pi$$
0.426511 + 0.904482i $$0.359742\pi$$
$$728$$ 0 0
$$729$$ 1.00000 0.0370370
$$730$$ 0 0
$$731$$ 1.00000 0.0369863
$$732$$ −2.00000 −0.0739221
$$733$$ 14.0000 0.517102 0.258551 0.965998i $$-0.416755\pi$$
0.258551 + 0.965998i $$0.416755\pi$$
$$734$$ 16.0000 0.590571
$$735$$ 0 0
$$736$$ −32.0000 −1.17954
$$737$$ −8.00000 −0.294684
$$738$$ 0 0
$$739$$ −5.00000 −0.183928 −0.0919640 0.995762i $$-0.529314\pi$$
−0.0919640 + 0.995762i $$0.529314\pi$$
$$740$$ 0 0
$$741$$ 2.00000 0.0734718
$$742$$ −100.000 −3.67112
$$743$$ 8.00000 0.293492 0.146746 0.989174i $$-0.453120\pi$$
0.146746 + 0.989174i $$0.453120\pi$$
$$744$$ 0 0
$$745$$ 0 0
$$746$$ −32.0000 −1.17160
$$747$$ −12.0000 −0.439057
$$748$$ 2.00000 0.0731272
$$749$$ −30.0000 −1.09618
$$750$$ 0 0
$$751$$ 24.0000 0.875772 0.437886 0.899030i $$-0.355727\pi$$
0.437886 + 0.899030i $$0.355727\pi$$
$$752$$ −36.0000 −1.31278
$$753$$ 7.00000 0.255094
$$754$$ 8.00000 0.291343
$$755$$ 0 0
$$756$$ 10.0000 0.363696
$$757$$ 17.0000 0.617876 0.308938 0.951082i $$-0.400027\pi$$
0.308938 + 0.951082i $$0.400027\pi$$
$$758$$ 68.0000 2.46987
$$759$$ 4.00000 0.145191
$$760$$ 0 0
$$761$$ 15.0000 0.543750 0.271875 0.962333i $$-0.412356\pi$$
0.271875 + 0.962333i $$0.412356\pi$$
$$762$$ 4.00000 0.144905
$$763$$ 20.0000 0.724049
$$764$$ 18.0000 0.651217
$$765$$ 0 0
$$766$$ 68.0000 2.45694
$$767$$ 16.0000 0.577727
$$768$$ 16.0000 0.577350
$$769$$ 11.0000 0.396670 0.198335 0.980134i $$-0.436447\pi$$
0.198335 + 0.980134i $$0.436447\pi$$
$$770$$ 0 0
$$771$$ 8.00000 0.288113
$$772$$ −8.00000 −0.287926
$$773$$ −20.0000 −0.719350 −0.359675 0.933078i $$-0.617112\pi$$
−0.359675 + 0.933078i $$0.617112\pi$$
$$774$$ 2.00000 0.0718885
$$775$$ 0 0
$$776$$ 0 0
$$777$$ 0 0
$$778$$ −54.0000 −1.93599
$$779$$ 0 0
$$780$$ 0 0
$$781$$ −12.0000 −0.429394
$$782$$ 8.00000 0.286079
$$783$$ −2.00000 −0.0714742
$$784$$ −72.0000 −2.57143
$$785$$ 0 0
$$786$$ 14.0000 0.499363
$$787$$ −40.0000 −1.42585 −0.712923 0.701242i $$-0.752629\pi$$
−0.712923 + 0.701242i $$0.752629\pi$$
$$788$$ 4.00000 0.142494
$$789$$ −23.0000 −0.818822
$$790$$ 0 0
$$791$$ −10.0000 −0.355559
$$792$$ 0 0
$$793$$ 2.00000 0.0710221
$$794$$ −50.0000 −1.77443
$$795$$ 0 0
$$796$$ −42.0000 −1.48865
$$797$$ 44.0000 1.55856 0.779280 0.626676i $$-0.215585\pi$$
0.779280 + 0.626676i $$0.215585\pi$$
$$798$$ −10.0000 −0.353996
$$799$$ 9.00000 0.318397
$$800$$ 0 0
$$801$$ −6.00000 −0.212000
$$802$$ 72.0000 2.54241
$$803$$ 11.0000 0.388182
$$804$$ −16.0000 −0.564276
$$805$$ 0 0
$$806$$ 24.0000 0.845364
$$807$$ −14.0000 −0.492823
$$808$$ 0 0
$$809$$ −55.0000 −1.93370 −0.966849 0.255351i $$-0.917809\pi$$
−0.966849 + 0.255351i $$0.917809\pi$$
$$810$$ 0 0
$$811$$ −38.0000 −1.33436 −0.667180 0.744896i $$-0.732499\pi$$
−0.667180 + 0.744896i $$0.732499\pi$$
$$812$$ −20.0000 −0.701862
$$813$$ 12.0000 0.420858
$$814$$ 0 0
$$815$$ 0 0
$$816$$ −4.00000 −0.140028
$$817$$ −1.00000 −0.0349856
$$818$$ −28.0000 −0.978997
$$819$$ −10.0000 −0.349428
$$820$$ 0 0
$$821$$ −45.0000 −1.57051 −0.785255 0.619172i $$-0.787468\pi$$
−0.785255 + 0.619172i $$0.787468\pi$$
$$822$$ 18.0000 0.627822
$$823$$ −43.0000 −1.49889 −0.749443 0.662069i $$-0.769679\pi$$
−0.749443 + 0.662069i $$0.769679\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ −80.0000 −2.78356
$$827$$ −12.0000 −0.417281 −0.208640 0.977992i $$-0.566904\pi$$
−0.208640 + 0.977992i $$0.566904\pi$$
$$828$$ 8.00000 0.278019
$$829$$ 52.0000 1.80603 0.903017 0.429604i $$-0.141347\pi$$
0.903017 + 0.429604i $$0.141347\pi$$
$$830$$ 0 0
$$831$$ 11.0000 0.381586
$$832$$ 16.0000 0.554700
$$833$$ 18.0000 0.623663
$$834$$ −26.0000 −0.900306
$$835$$ 0 0
$$836$$ −2.00000 −0.0691714
$$837$$ −6.00000 −0.207390
$$838$$ 56.0000 1.93449
$$839$$ 54.0000 1.86429 0.932144 0.362089i $$-0.117936\pi$$
0.932144 + 0.362089i $$0.117936\pi$$
$$840$$ 0 0
$$841$$ −25.0000 −0.862069
$$842$$ 52.0000 1.79204
$$843$$ 10.0000 0.344418
$$844$$ 24.0000 0.826114
$$845$$ 0 0
$$846$$ 18.0000 0.618853
$$847$$ −50.0000 −1.71802
$$848$$ 40.0000 1.37361
$$849$$ 13.0000 0.446159
$$850$$ 0 0
$$851$$ 0 0
$$852$$ −24.0000 −0.822226
$$853$$ 14.0000 0.479351 0.239675 0.970853i $$-0.422959\pi$$
0.239675 + 0.970853i $$0.422959\pi$$
$$854$$ −10.0000 −0.342193
$$855$$ 0 0
$$856$$ 0 0
$$857$$ 8.00000 0.273275 0.136637 0.990621i $$-0.456370\pi$$
0.136637 + 0.990621i $$0.456370\pi$$
$$858$$ −4.00000 −0.136558
$$859$$ 27.0000 0.921228 0.460614 0.887601i $$-0.347629\pi$$
0.460614 + 0.887601i $$0.347629\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ −68.0000 −2.31609
$$863$$ 44.0000 1.49778 0.748889 0.662696i $$-0.230588\pi$$
0.748889 + 0.662696i $$0.230588\pi$$
$$864$$ −8.00000 −0.272166
$$865$$ 0 0
$$866$$ −12.0000 −0.407777
$$867$$ −16.0000 −0.543388
$$868$$ −60.0000 −2.03653
$$869$$ 16.0000 0.542763
$$870$$ 0 0
$$871$$ 16.0000 0.542139
$$872$$ 0 0
$$873$$ 10.0000 0.338449
$$874$$ −8.00000 −0.270604
$$875$$ 0 0
$$876$$ 22.0000 0.743311
$$877$$ 6.00000 0.202606 0.101303 0.994856i $$-0.467699\pi$$
0.101303 + 0.994856i $$0.467699\pi$$
$$878$$ 52.0000 1.75491
$$879$$ 28.0000 0.944417
$$880$$ 0 0
$$881$$ −37.0000 −1.24656 −0.623281 0.781998i $$-0.714201\pi$$
−0.623281 + 0.781998i $$0.714201\pi$$
$$882$$ 36.0000 1.21218
$$883$$ −35.0000 −1.17784 −0.588922 0.808190i $$-0.700447\pi$$
−0.588922 + 0.808190i $$0.700447\pi$$
$$884$$ −4.00000 −0.134535
$$885$$ 0 0
$$886$$ 10.0000 0.335957
$$887$$ 12.0000 0.402921 0.201460 0.979497i $$-0.435431\pi$$
0.201460 + 0.979497i $$0.435431\pi$$
$$888$$ 0 0
$$889$$ 10.0000 0.335389
$$890$$ 0 0
$$891$$ 1.00000 0.0335013
$$892$$ −24.0000 −0.803579
$$893$$ −9.00000 −0.301174
$$894$$ −42.0000 −1.40469
$$895$$ 0 0
$$896$$ 0 0
$$897$$ −8.00000 −0.267112
$$898$$ −72.0000 −2.40267
$$899$$ 12.0000 0.400222
$$900$$ 0 0
$$901$$ −10.0000 −0.333148
$$902$$ 0 0
$$903$$ 5.00000 0.166390
$$904$$ 0 0
$$905$$ 0 0
$$906$$ 0 0
$$907$$ 26.0000 0.863316 0.431658 0.902037i $$-0.357929\pi$$
0.431658 + 0.902037i $$0.357929\pi$$
$$908$$ −36.0000 −1.19470
$$909$$ 2.00000 0.0663358
$$910$$ 0 0
$$911$$ −6.00000 −0.198789 −0.0993944 0.995048i $$-0.531691\pi$$
−0.0993944 + 0.995048i $$0.531691\pi$$
$$912$$ 4.00000 0.132453
$$913$$ −12.0000 −0.397142
$$914$$ 58.0000 1.91847
$$915$$ 0 0
$$916$$ 50.0000 1.65205
$$917$$ 35.0000 1.15580
$$918$$ 2.00000 0.0660098
$$919$$ 8.00000 0.263896 0.131948 0.991257i $$-0.457877\pi$$
0.131948 + 0.991257i $$0.457877\pi$$
$$920$$ 0 0
$$921$$ 12.0000 0.395413
$$922$$ 54.0000 1.77840
$$923$$ 24.0000 0.789970
$$924$$ 10.0000 0.328976
$$925$$ 0 0
$$926$$ −34.0000 −1.11731
$$927$$ 2.00000 0.0656886
$$928$$ 16.0000 0.525226
$$929$$ −2.00000 −0.0656179 −0.0328089 0.999462i $$-0.510445\pi$$
−0.0328089 + 0.999462i $$0.510445\pi$$
$$930$$ 0 0
$$931$$ −18.0000 −0.589926
$$932$$ −18.0000 −0.589610
$$933$$ −21.0000 −0.687509
$$934$$ 10.0000 0.327210
$$935$$ 0 0
$$936$$ 0 0
$$937$$ −21.0000 −0.686040 −0.343020 0.939328i $$-0.611450\pi$$
−0.343020 + 0.939328i $$0.611450\pi$$
$$938$$ −80.0000 −2.61209
$$939$$ 2.00000 0.0652675
$$940$$ 0 0
$$941$$ 42.0000 1.36916 0.684580 0.728937i $$-0.259985\pi$$
0.684580 + 0.728937i $$0.259985\pi$$
$$942$$ 36.0000 1.17294
$$943$$ 0 0
$$944$$ 32.0000 1.04151
$$945$$ 0 0
$$946$$ 2.00000 0.0650256
$$947$$ 28.0000 0.909878 0.454939 0.890523i $$-0.349661\pi$$
0.454939 + 0.890523i $$0.349661\pi$$
$$948$$ 32.0000 1.03931
$$949$$ −22.0000 −0.714150
$$950$$ 0 0
$$951$$ 4.00000 0.129709
$$952$$ 0 0
$$953$$ 32.0000 1.03658 0.518291 0.855204i $$-0.326568\pi$$
0.518291 + 0.855204i $$0.326568\pi$$
$$954$$ −20.0000 −0.647524
$$955$$ 0 0
$$956$$ −6.00000 −0.194054
$$957$$ −2.00000 −0.0646508
$$958$$ −32.0000 −1.03387
$$959$$ 45.0000 1.45313
$$960$$ 0 0
$$961$$ 5.00000 0.161290
$$962$$ 0 0
$$963$$ −6.00000 −0.193347
$$964$$ 40.0000 1.28831
$$965$$ 0 0
$$966$$ 40.0000 1.28698
$$967$$ 32.0000 1.02905 0.514525 0.857475i $$-0.327968\pi$$
0.514525 + 0.857475i $$0.327968\pi$$
$$968$$ 0 0
$$969$$ −1.00000 −0.0321246
$$970$$ 0 0
$$971$$ 12.0000 0.385098 0.192549 0.981287i $$-0.438325\pi$$
0.192549 + 0.981287i $$0.438325\pi$$
$$972$$ 2.00000 0.0641500
$$973$$ −65.0000 −2.08380
$$974$$ 32.0000 1.02535
$$975$$ 0 0
$$976$$ 4.00000 0.128037
$$977$$ −54.0000 −1.72761 −0.863807 0.503824i $$-0.831926\pi$$
−0.863807 + 0.503824i $$0.831926\pi$$
$$978$$ 0 0
$$979$$ −6.00000 −0.191761
$$980$$ 0 0
$$981$$ 4.00000 0.127710
$$982$$ 0 0
$$983$$ 44.0000 1.40338 0.701691 0.712481i $$-0.252429\pi$$
0.701691 + 0.712481i $$0.252429\pi$$
$$984$$ 0 0
$$985$$ 0 0
$$986$$ −4.00000 −0.127386
$$987$$ 45.0000 1.43237
$$988$$ 4.00000 0.127257
$$989$$ 4.00000 0.127193
$$990$$ 0 0
$$991$$ −16.0000 −0.508257 −0.254128 0.967170i $$-0.581789\pi$$
−0.254128 + 0.967170i $$0.581789\pi$$
$$992$$ 48.0000 1.52400
$$993$$ −4.00000 −0.126936
$$994$$ −120.000 −3.80617
$$995$$ 0 0
$$996$$ −24.0000 −0.760469
$$997$$ 47.0000 1.48850 0.744252 0.667898i $$-0.232806\pi$$
0.744252 + 0.667898i $$0.232806\pi$$
$$998$$ 10.0000 0.316544
$$999$$ 0 0
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 1425.2.a.j.1.1 1
3.2 odd 2 4275.2.a.b.1.1 1
5.2 odd 4 1425.2.c.b.799.2 2
5.3 odd 4 1425.2.c.b.799.1 2
5.4 even 2 57.2.a.a.1.1 1
15.14 odd 2 171.2.a.d.1.1 1
20.19 odd 2 912.2.a.g.1.1 1
35.34 odd 2 2793.2.a.b.1.1 1
40.19 odd 2 3648.2.a.r.1.1 1
40.29 even 2 3648.2.a.bh.1.1 1
55.54 odd 2 6897.2.a.f.1.1 1
60.59 even 2 2736.2.a.v.1.1 1
65.64 even 2 9633.2.a.o.1.1 1
95.94 odd 2 1083.2.a.e.1.1 1
105.104 even 2 8379.2.a.p.1.1 1
285.284 even 2 3249.2.a.b.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
57.2.a.a.1.1 1 5.4 even 2
171.2.a.d.1.1 1 15.14 odd 2
912.2.a.g.1.1 1 20.19 odd 2
1083.2.a.e.1.1 1 95.94 odd 2
1425.2.a.j.1.1 1 1.1 even 1 trivial
1425.2.c.b.799.1 2 5.3 odd 4
1425.2.c.b.799.2 2 5.2 odd 4
2736.2.a.v.1.1 1 60.59 even 2
2793.2.a.b.1.1 1 35.34 odd 2
3249.2.a.b.1.1 1 285.284 even 2
3648.2.a.r.1.1 1 40.19 odd 2
3648.2.a.bh.1.1 1 40.29 even 2
4275.2.a.b.1.1 1 3.2 odd 2
6897.2.a.f.1.1 1 55.54 odd 2
8379.2.a.p.1.1 1 105.104 even 2
9633.2.a.o.1.1 1 65.64 even 2