# Properties

 Label 5225.2.a.c.1.1 Level $5225$ Weight $2$ Character 5225.1 Self dual yes Analytic conductor $41.722$ Analytic rank $1$ Dimension $1$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [5225,2,Mod(1,5225)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(5225, 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("5225.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$5225 = 5^{2} \cdot 11 \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 5225.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: $$41.7218350561$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 1045) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Character $$\chi$$ $$=$$ 5225.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+1.00000 q^{2} +2.00000 q^{3} -1.00000 q^{4} +2.00000 q^{6} +2.00000 q^{7} -3.00000 q^{8} +1.00000 q^{9} +O(q^{10})$$ $$q+1.00000 q^{2} +2.00000 q^{3} -1.00000 q^{4} +2.00000 q^{6} +2.00000 q^{7} -3.00000 q^{8} +1.00000 q^{9} -1.00000 q^{11} -2.00000 q^{12} -6.00000 q^{13} +2.00000 q^{14} -1.00000 q^{16} +1.00000 q^{18} -1.00000 q^{19} +4.00000 q^{21} -1.00000 q^{22} -6.00000 q^{24} -6.00000 q^{26} -4.00000 q^{27} -2.00000 q^{28} +6.00000 q^{29} +4.00000 q^{31} +5.00000 q^{32} -2.00000 q^{33} -1.00000 q^{36} -1.00000 q^{38} -12.0000 q^{39} -6.00000 q^{41} +4.00000 q^{42} +6.00000 q^{43} +1.00000 q^{44} +4.00000 q^{47} -2.00000 q^{48} -3.00000 q^{49} +6.00000 q^{52} -12.0000 q^{53} -4.00000 q^{54} -6.00000 q^{56} -2.00000 q^{57} +6.00000 q^{58} -8.00000 q^{59} -14.0000 q^{61} +4.00000 q^{62} +2.00000 q^{63} +7.00000 q^{64} -2.00000 q^{66} -6.00000 q^{67} -8.00000 q^{71} -3.00000 q^{72} -16.0000 q^{73} +1.00000 q^{76} -2.00000 q^{77} -12.0000 q^{78} -8.00000 q^{79} -11.0000 q^{81} -6.00000 q^{82} -6.00000 q^{83} -4.00000 q^{84} +6.00000 q^{86} +12.0000 q^{87} +3.00000 q^{88} -2.00000 q^{89} -12.0000 q^{91} +8.00000 q^{93} +4.00000 q^{94} +10.0000 q^{96} -4.00000 q^{97} -3.00000 q^{98} -1.00000 q^{99} +O(q^{100})$$

## Coefficient data

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

Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 1.00000 0.707107 0.353553 0.935414i $$-0.384973\pi$$
0.353553 + 0.935414i $$0.384973\pi$$
$$3$$ 2.00000 1.15470 0.577350 0.816497i $$-0.304087\pi$$
0.577350 + 0.816497i $$0.304087\pi$$
$$4$$ −1.00000 −0.500000
$$5$$ 0 0
$$6$$ 2.00000 0.816497
$$7$$ 2.00000 0.755929 0.377964 0.925820i $$-0.376624\pi$$
0.377964 + 0.925820i $$0.376624\pi$$
$$8$$ −3.00000 −1.06066
$$9$$ 1.00000 0.333333
$$10$$ 0 0
$$11$$ −1.00000 −0.301511
$$12$$ −2.00000 −0.577350
$$13$$ −6.00000 −1.66410 −0.832050 0.554700i $$-0.812833\pi$$
−0.832050 + 0.554700i $$0.812833\pi$$
$$14$$ 2.00000 0.534522
$$15$$ 0 0
$$16$$ −1.00000 −0.250000
$$17$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$18$$ 1.00000 0.235702
$$19$$ −1.00000 −0.229416
$$20$$ 0 0
$$21$$ 4.00000 0.872872
$$22$$ −1.00000 −0.213201
$$23$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$24$$ −6.00000 −1.22474
$$25$$ 0 0
$$26$$ −6.00000 −1.17670
$$27$$ −4.00000 −0.769800
$$28$$ −2.00000 −0.377964
$$29$$ 6.00000 1.11417 0.557086 0.830455i $$-0.311919\pi$$
0.557086 + 0.830455i $$0.311919\pi$$
$$30$$ 0 0
$$31$$ 4.00000 0.718421 0.359211 0.933257i $$-0.383046\pi$$
0.359211 + 0.933257i $$0.383046\pi$$
$$32$$ 5.00000 0.883883
$$33$$ −2.00000 −0.348155
$$34$$ 0 0
$$35$$ 0 0
$$36$$ −1.00000 −0.166667
$$37$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$38$$ −1.00000 −0.162221
$$39$$ −12.0000 −1.92154
$$40$$ 0 0
$$41$$ −6.00000 −0.937043 −0.468521 0.883452i $$-0.655213\pi$$
−0.468521 + 0.883452i $$0.655213\pi$$
$$42$$ 4.00000 0.617213
$$43$$ 6.00000 0.914991 0.457496 0.889212i $$-0.348747\pi$$
0.457496 + 0.889212i $$0.348747\pi$$
$$44$$ 1.00000 0.150756
$$45$$ 0 0
$$46$$ 0 0
$$47$$ 4.00000 0.583460 0.291730 0.956501i $$-0.405769\pi$$
0.291730 + 0.956501i $$0.405769\pi$$
$$48$$ −2.00000 −0.288675
$$49$$ −3.00000 −0.428571
$$50$$ 0 0
$$51$$ 0 0
$$52$$ 6.00000 0.832050
$$53$$ −12.0000 −1.64833 −0.824163 0.566352i $$-0.808354\pi$$
−0.824163 + 0.566352i $$0.808354\pi$$
$$54$$ −4.00000 −0.544331
$$55$$ 0 0
$$56$$ −6.00000 −0.801784
$$57$$ −2.00000 −0.264906
$$58$$ 6.00000 0.787839
$$59$$ −8.00000 −1.04151 −0.520756 0.853706i $$-0.674350\pi$$
−0.520756 + 0.853706i $$0.674350\pi$$
$$60$$ 0 0
$$61$$ −14.0000 −1.79252 −0.896258 0.443533i $$-0.853725\pi$$
−0.896258 + 0.443533i $$0.853725\pi$$
$$62$$ 4.00000 0.508001
$$63$$ 2.00000 0.251976
$$64$$ 7.00000 0.875000
$$65$$ 0 0
$$66$$ −2.00000 −0.246183
$$67$$ −6.00000 −0.733017 −0.366508 0.930415i $$-0.619447\pi$$
−0.366508 + 0.930415i $$0.619447\pi$$
$$68$$ 0 0
$$69$$ 0 0
$$70$$ 0 0
$$71$$ −8.00000 −0.949425 −0.474713 0.880141i $$-0.657448\pi$$
−0.474713 + 0.880141i $$0.657448\pi$$
$$72$$ −3.00000 −0.353553
$$73$$ −16.0000 −1.87266 −0.936329 0.351123i $$-0.885800\pi$$
−0.936329 + 0.351123i $$0.885800\pi$$
$$74$$ 0 0
$$75$$ 0 0
$$76$$ 1.00000 0.114708
$$77$$ −2.00000 −0.227921
$$78$$ −12.0000 −1.35873
$$79$$ −8.00000 −0.900070 −0.450035 0.893011i $$-0.648589\pi$$
−0.450035 + 0.893011i $$0.648589\pi$$
$$80$$ 0 0
$$81$$ −11.0000 −1.22222
$$82$$ −6.00000 −0.662589
$$83$$ −6.00000 −0.658586 −0.329293 0.944228i $$-0.606810\pi$$
−0.329293 + 0.944228i $$0.606810\pi$$
$$84$$ −4.00000 −0.436436
$$85$$ 0 0
$$86$$ 6.00000 0.646997
$$87$$ 12.0000 1.28654
$$88$$ 3.00000 0.319801
$$89$$ −2.00000 −0.212000 −0.106000 0.994366i $$-0.533804\pi$$
−0.106000 + 0.994366i $$0.533804\pi$$
$$90$$ 0 0
$$91$$ −12.0000 −1.25794
$$92$$ 0 0
$$93$$ 8.00000 0.829561
$$94$$ 4.00000 0.412568
$$95$$ 0 0
$$96$$ 10.0000 1.02062
$$97$$ −4.00000 −0.406138 −0.203069 0.979164i $$-0.565092\pi$$
−0.203069 + 0.979164i $$0.565092\pi$$
$$98$$ −3.00000 −0.303046
$$99$$ −1.00000 −0.100504
$$100$$ 0 0
$$101$$ −18.0000 −1.79107 −0.895533 0.444994i $$-0.853206\pi$$
−0.895533 + 0.444994i $$0.853206\pi$$
$$102$$ 0 0
$$103$$ 10.0000 0.985329 0.492665 0.870219i $$-0.336023\pi$$
0.492665 + 0.870219i $$0.336023\pi$$
$$104$$ 18.0000 1.76505
$$105$$ 0 0
$$106$$ −12.0000 −1.16554
$$107$$ 20.0000 1.93347 0.966736 0.255774i $$-0.0823304\pi$$
0.966736 + 0.255774i $$0.0823304\pi$$
$$108$$ 4.00000 0.384900
$$109$$ −2.00000 −0.191565 −0.0957826 0.995402i $$-0.530535\pi$$
−0.0957826 + 0.995402i $$0.530535\pi$$
$$110$$ 0 0
$$111$$ 0 0
$$112$$ −2.00000 −0.188982
$$113$$ −4.00000 −0.376288 −0.188144 0.982141i $$-0.560247\pi$$
−0.188144 + 0.982141i $$0.560247\pi$$
$$114$$ −2.00000 −0.187317
$$115$$ 0 0
$$116$$ −6.00000 −0.557086
$$117$$ −6.00000 −0.554700
$$118$$ −8.00000 −0.736460
$$119$$ 0 0
$$120$$ 0 0
$$121$$ 1.00000 0.0909091
$$122$$ −14.0000 −1.26750
$$123$$ −12.0000 −1.08200
$$124$$ −4.00000 −0.359211
$$125$$ 0 0
$$126$$ 2.00000 0.178174
$$127$$ −4.00000 −0.354943 −0.177471 0.984126i $$-0.556792\pi$$
−0.177471 + 0.984126i $$0.556792\pi$$
$$128$$ −3.00000 −0.265165
$$129$$ 12.0000 1.05654
$$130$$ 0 0
$$131$$ 4.00000 0.349482 0.174741 0.984614i $$-0.444091\pi$$
0.174741 + 0.984614i $$0.444091\pi$$
$$132$$ 2.00000 0.174078
$$133$$ −2.00000 −0.173422
$$134$$ −6.00000 −0.518321
$$135$$ 0 0
$$136$$ 0 0
$$137$$ −2.00000 −0.170872 −0.0854358 0.996344i $$-0.527228\pi$$
−0.0854358 + 0.996344i $$0.527228\pi$$
$$138$$ 0 0
$$139$$ 12.0000 1.01783 0.508913 0.860818i $$-0.330047\pi$$
0.508913 + 0.860818i $$0.330047\pi$$
$$140$$ 0 0
$$141$$ 8.00000 0.673722
$$142$$ −8.00000 −0.671345
$$143$$ 6.00000 0.501745
$$144$$ −1.00000 −0.0833333
$$145$$ 0 0
$$146$$ −16.0000 −1.32417
$$147$$ −6.00000 −0.494872
$$148$$ 0 0
$$149$$ 6.00000 0.491539 0.245770 0.969328i $$-0.420959\pi$$
0.245770 + 0.969328i $$0.420959\pi$$
$$150$$ 0 0
$$151$$ −8.00000 −0.651031 −0.325515 0.945537i $$-0.605538\pi$$
−0.325515 + 0.945537i $$0.605538\pi$$
$$152$$ 3.00000 0.243332
$$153$$ 0 0
$$154$$ −2.00000 −0.161165
$$155$$ 0 0
$$156$$ 12.0000 0.960769
$$157$$ 10.0000 0.798087 0.399043 0.916932i $$-0.369342\pi$$
0.399043 + 0.916932i $$0.369342\pi$$
$$158$$ −8.00000 −0.636446
$$159$$ −24.0000 −1.90332
$$160$$ 0 0
$$161$$ 0 0
$$162$$ −11.0000 −0.864242
$$163$$ −4.00000 −0.313304 −0.156652 0.987654i $$-0.550070\pi$$
−0.156652 + 0.987654i $$0.550070\pi$$
$$164$$ 6.00000 0.468521
$$165$$ 0 0
$$166$$ −6.00000 −0.465690
$$167$$ 8.00000 0.619059 0.309529 0.950890i $$-0.399829\pi$$
0.309529 + 0.950890i $$0.399829\pi$$
$$168$$ −12.0000 −0.925820
$$169$$ 23.0000 1.76923
$$170$$ 0 0
$$171$$ −1.00000 −0.0764719
$$172$$ −6.00000 −0.457496
$$173$$ −6.00000 −0.456172 −0.228086 0.973641i $$-0.573247\pi$$
−0.228086 + 0.973641i $$0.573247\pi$$
$$174$$ 12.0000 0.909718
$$175$$ 0 0
$$176$$ 1.00000 0.0753778
$$177$$ −16.0000 −1.20263
$$178$$ −2.00000 −0.149906
$$179$$ −12.0000 −0.896922 −0.448461 0.893802i $$-0.648028\pi$$
−0.448461 + 0.893802i $$0.648028\pi$$
$$180$$ 0 0
$$181$$ 14.0000 1.04061 0.520306 0.853980i $$-0.325818\pi$$
0.520306 + 0.853980i $$0.325818\pi$$
$$182$$ −12.0000 −0.889499
$$183$$ −28.0000 −2.06982
$$184$$ 0 0
$$185$$ 0 0
$$186$$ 8.00000 0.586588
$$187$$ 0 0
$$188$$ −4.00000 −0.291730
$$189$$ −8.00000 −0.581914
$$190$$ 0 0
$$191$$ −8.00000 −0.578860 −0.289430 0.957199i $$-0.593466\pi$$
−0.289430 + 0.957199i $$0.593466\pi$$
$$192$$ 14.0000 1.01036
$$193$$ 6.00000 0.431889 0.215945 0.976406i $$-0.430717\pi$$
0.215945 + 0.976406i $$0.430717\pi$$
$$194$$ −4.00000 −0.287183
$$195$$ 0 0
$$196$$ 3.00000 0.214286
$$197$$ 24.0000 1.70993 0.854965 0.518686i $$-0.173579\pi$$
0.854965 + 0.518686i $$0.173579\pi$$
$$198$$ −1.00000 −0.0710669
$$199$$ 16.0000 1.13421 0.567105 0.823646i $$-0.308063\pi$$
0.567105 + 0.823646i $$0.308063\pi$$
$$200$$ 0 0
$$201$$ −12.0000 −0.846415
$$202$$ −18.0000 −1.26648
$$203$$ 12.0000 0.842235
$$204$$ 0 0
$$205$$ 0 0
$$206$$ 10.0000 0.696733
$$207$$ 0 0
$$208$$ 6.00000 0.416025
$$209$$ 1.00000 0.0691714
$$210$$ 0 0
$$211$$ 28.0000 1.92760 0.963800 0.266627i $$-0.0859092\pi$$
0.963800 + 0.266627i $$0.0859092\pi$$
$$212$$ 12.0000 0.824163
$$213$$ −16.0000 −1.09630
$$214$$ 20.0000 1.36717
$$215$$ 0 0
$$216$$ 12.0000 0.816497
$$217$$ 8.00000 0.543075
$$218$$ −2.00000 −0.135457
$$219$$ −32.0000 −2.16236
$$220$$ 0 0
$$221$$ 0 0
$$222$$ 0 0
$$223$$ 22.0000 1.47323 0.736614 0.676313i $$-0.236423\pi$$
0.736614 + 0.676313i $$0.236423\pi$$
$$224$$ 10.0000 0.668153
$$225$$ 0 0
$$226$$ −4.00000 −0.266076
$$227$$ 8.00000 0.530979 0.265489 0.964114i $$-0.414466\pi$$
0.265489 + 0.964114i $$0.414466\pi$$
$$228$$ 2.00000 0.132453
$$229$$ −6.00000 −0.396491 −0.198246 0.980152i $$-0.563524\pi$$
−0.198246 + 0.980152i $$0.563524\pi$$
$$230$$ 0 0
$$231$$ −4.00000 −0.263181
$$232$$ −18.0000 −1.18176
$$233$$ 24.0000 1.57229 0.786146 0.618041i $$-0.212073\pi$$
0.786146 + 0.618041i $$0.212073\pi$$
$$234$$ −6.00000 −0.392232
$$235$$ 0 0
$$236$$ 8.00000 0.520756
$$237$$ −16.0000 −1.03931
$$238$$ 0 0
$$239$$ −24.0000 −1.55243 −0.776215 0.630468i $$-0.782863\pi$$
−0.776215 + 0.630468i $$0.782863\pi$$
$$240$$ 0 0
$$241$$ 14.0000 0.901819 0.450910 0.892570i $$-0.351100\pi$$
0.450910 + 0.892570i $$0.351100\pi$$
$$242$$ 1.00000 0.0642824
$$243$$ −10.0000 −0.641500
$$244$$ 14.0000 0.896258
$$245$$ 0 0
$$246$$ −12.0000 −0.765092
$$247$$ 6.00000 0.381771
$$248$$ −12.0000 −0.762001
$$249$$ −12.0000 −0.760469
$$250$$ 0 0
$$251$$ −4.00000 −0.252478 −0.126239 0.992000i $$-0.540291\pi$$
−0.126239 + 0.992000i $$0.540291\pi$$
$$252$$ −2.00000 −0.125988
$$253$$ 0 0
$$254$$ −4.00000 −0.250982
$$255$$ 0 0
$$256$$ −17.0000 −1.06250
$$257$$ −20.0000 −1.24757 −0.623783 0.781598i $$-0.714405\pi$$
−0.623783 + 0.781598i $$0.714405\pi$$
$$258$$ 12.0000 0.747087
$$259$$ 0 0
$$260$$ 0 0
$$261$$ 6.00000 0.371391
$$262$$ 4.00000 0.247121
$$263$$ −10.0000 −0.616626 −0.308313 0.951285i $$-0.599764\pi$$
−0.308313 + 0.951285i $$0.599764\pi$$
$$264$$ 6.00000 0.369274
$$265$$ 0 0
$$266$$ −2.00000 −0.122628
$$267$$ −4.00000 −0.244796
$$268$$ 6.00000 0.366508
$$269$$ −10.0000 −0.609711 −0.304855 0.952399i $$-0.598608\pi$$
−0.304855 + 0.952399i $$0.598608\pi$$
$$270$$ 0 0
$$271$$ 20.0000 1.21491 0.607457 0.794353i $$-0.292190\pi$$
0.607457 + 0.794353i $$0.292190\pi$$
$$272$$ 0 0
$$273$$ −24.0000 −1.45255
$$274$$ −2.00000 −0.120824
$$275$$ 0 0
$$276$$ 0 0
$$277$$ −8.00000 −0.480673 −0.240337 0.970690i $$-0.577258\pi$$
−0.240337 + 0.970690i $$0.577258\pi$$
$$278$$ 12.0000 0.719712
$$279$$ 4.00000 0.239474
$$280$$ 0 0
$$281$$ 6.00000 0.357930 0.178965 0.983855i $$-0.442725\pi$$
0.178965 + 0.983855i $$0.442725\pi$$
$$282$$ 8.00000 0.476393
$$283$$ −6.00000 −0.356663 −0.178331 0.983970i $$-0.557070\pi$$
−0.178331 + 0.983970i $$0.557070\pi$$
$$284$$ 8.00000 0.474713
$$285$$ 0 0
$$286$$ 6.00000 0.354787
$$287$$ −12.0000 −0.708338
$$288$$ 5.00000 0.294628
$$289$$ −17.0000 −1.00000
$$290$$ 0 0
$$291$$ −8.00000 −0.468968
$$292$$ 16.0000 0.936329
$$293$$ 14.0000 0.817889 0.408944 0.912559i $$-0.365897\pi$$
0.408944 + 0.912559i $$0.365897\pi$$
$$294$$ −6.00000 −0.349927
$$295$$ 0 0
$$296$$ 0 0
$$297$$ 4.00000 0.232104
$$298$$ 6.00000 0.347571
$$299$$ 0 0
$$300$$ 0 0
$$301$$ 12.0000 0.691669
$$302$$ −8.00000 −0.460348
$$303$$ −36.0000 −2.06815
$$304$$ 1.00000 0.0573539
$$305$$ 0 0
$$306$$ 0 0
$$307$$ 12.0000 0.684876 0.342438 0.939540i $$-0.388747\pi$$
0.342438 + 0.939540i $$0.388747\pi$$
$$308$$ 2.00000 0.113961
$$309$$ 20.0000 1.13776
$$310$$ 0 0
$$311$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$312$$ 36.0000 2.03810
$$313$$ −26.0000 −1.46961 −0.734803 0.678280i $$-0.762726\pi$$
−0.734803 + 0.678280i $$0.762726\pi$$
$$314$$ 10.0000 0.564333
$$315$$ 0 0
$$316$$ 8.00000 0.450035
$$317$$ 20.0000 1.12331 0.561656 0.827371i $$-0.310164\pi$$
0.561656 + 0.827371i $$0.310164\pi$$
$$318$$ −24.0000 −1.34585
$$319$$ −6.00000 −0.335936
$$320$$ 0 0
$$321$$ 40.0000 2.23258
$$322$$ 0 0
$$323$$ 0 0
$$324$$ 11.0000 0.611111
$$325$$ 0 0
$$326$$ −4.00000 −0.221540
$$327$$ −4.00000 −0.221201
$$328$$ 18.0000 0.993884
$$329$$ 8.00000 0.441054
$$330$$ 0 0
$$331$$ −28.0000 −1.53902 −0.769510 0.638635i $$-0.779499\pi$$
−0.769510 + 0.638635i $$0.779499\pi$$
$$332$$ 6.00000 0.329293
$$333$$ 0 0
$$334$$ 8.00000 0.437741
$$335$$ 0 0
$$336$$ −4.00000 −0.218218
$$337$$ −10.0000 −0.544735 −0.272367 0.962193i $$-0.587807\pi$$
−0.272367 + 0.962193i $$0.587807\pi$$
$$338$$ 23.0000 1.25104
$$339$$ −8.00000 −0.434500
$$340$$ 0 0
$$341$$ −4.00000 −0.216612
$$342$$ −1.00000 −0.0540738
$$343$$ −20.0000 −1.07990
$$344$$ −18.0000 −0.970495
$$345$$ 0 0
$$346$$ −6.00000 −0.322562
$$347$$ 26.0000 1.39575 0.697877 0.716218i $$-0.254128\pi$$
0.697877 + 0.716218i $$0.254128\pi$$
$$348$$ −12.0000 −0.643268
$$349$$ −30.0000 −1.60586 −0.802932 0.596071i $$-0.796728\pi$$
−0.802932 + 0.596071i $$0.796728\pi$$
$$350$$ 0 0
$$351$$ 24.0000 1.28103
$$352$$ −5.00000 −0.266501
$$353$$ 18.0000 0.958043 0.479022 0.877803i $$-0.340992\pi$$
0.479022 + 0.877803i $$0.340992\pi$$
$$354$$ −16.0000 −0.850390
$$355$$ 0 0
$$356$$ 2.00000 0.106000
$$357$$ 0 0
$$358$$ −12.0000 −0.634220
$$359$$ 12.0000 0.633336 0.316668 0.948536i $$-0.397436\pi$$
0.316668 + 0.948536i $$0.397436\pi$$
$$360$$ 0 0
$$361$$ 1.00000 0.0526316
$$362$$ 14.0000 0.735824
$$363$$ 2.00000 0.104973
$$364$$ 12.0000 0.628971
$$365$$ 0 0
$$366$$ −28.0000 −1.46358
$$367$$ 28.0000 1.46159 0.730794 0.682598i $$-0.239150\pi$$
0.730794 + 0.682598i $$0.239150\pi$$
$$368$$ 0 0
$$369$$ −6.00000 −0.312348
$$370$$ 0 0
$$371$$ −24.0000 −1.24602
$$372$$ −8.00000 −0.414781
$$373$$ −14.0000 −0.724893 −0.362446 0.932005i $$-0.618058\pi$$
−0.362446 + 0.932005i $$0.618058\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ −12.0000 −0.618853
$$377$$ −36.0000 −1.85409
$$378$$ −8.00000 −0.411476
$$379$$ 4.00000 0.205466 0.102733 0.994709i $$-0.467241\pi$$
0.102733 + 0.994709i $$0.467241\pi$$
$$380$$ 0 0
$$381$$ −8.00000 −0.409852
$$382$$ −8.00000 −0.409316
$$383$$ 30.0000 1.53293 0.766464 0.642287i $$-0.222014\pi$$
0.766464 + 0.642287i $$0.222014\pi$$
$$384$$ −6.00000 −0.306186
$$385$$ 0 0
$$386$$ 6.00000 0.305392
$$387$$ 6.00000 0.304997
$$388$$ 4.00000 0.203069
$$389$$ −26.0000 −1.31825 −0.659126 0.752032i $$-0.729074\pi$$
−0.659126 + 0.752032i $$0.729074\pi$$
$$390$$ 0 0
$$391$$ 0 0
$$392$$ 9.00000 0.454569
$$393$$ 8.00000 0.403547
$$394$$ 24.0000 1.20910
$$395$$ 0 0
$$396$$ 1.00000 0.0502519
$$397$$ 18.0000 0.903394 0.451697 0.892171i $$-0.350819\pi$$
0.451697 + 0.892171i $$0.350819\pi$$
$$398$$ 16.0000 0.802008
$$399$$ −4.00000 −0.200250
$$400$$ 0 0
$$401$$ −22.0000 −1.09863 −0.549314 0.835616i $$-0.685111\pi$$
−0.549314 + 0.835616i $$0.685111\pi$$
$$402$$ −12.0000 −0.598506
$$403$$ −24.0000 −1.19553
$$404$$ 18.0000 0.895533
$$405$$ 0 0
$$406$$ 12.0000 0.595550
$$407$$ 0 0
$$408$$ 0 0
$$409$$ −22.0000 −1.08783 −0.543915 0.839140i $$-0.683059\pi$$
−0.543915 + 0.839140i $$0.683059\pi$$
$$410$$ 0 0
$$411$$ −4.00000 −0.197305
$$412$$ −10.0000 −0.492665
$$413$$ −16.0000 −0.787309
$$414$$ 0 0
$$415$$ 0 0
$$416$$ −30.0000 −1.47087
$$417$$ 24.0000 1.17529
$$418$$ 1.00000 0.0489116
$$419$$ −20.0000 −0.977064 −0.488532 0.872546i $$-0.662467\pi$$
−0.488532 + 0.872546i $$0.662467\pi$$
$$420$$ 0 0
$$421$$ −14.0000 −0.682318 −0.341159 0.940006i $$-0.610819\pi$$
−0.341159 + 0.940006i $$0.610819\pi$$
$$422$$ 28.0000 1.36302
$$423$$ 4.00000 0.194487
$$424$$ 36.0000 1.74831
$$425$$ 0 0
$$426$$ −16.0000 −0.775203
$$427$$ −28.0000 −1.35501
$$428$$ −20.0000 −0.966736
$$429$$ 12.0000 0.579365
$$430$$ 0 0
$$431$$ −8.00000 −0.385346 −0.192673 0.981263i $$-0.561716\pi$$
−0.192673 + 0.981263i $$0.561716\pi$$
$$432$$ 4.00000 0.192450
$$433$$ 4.00000 0.192228 0.0961139 0.995370i $$-0.469359\pi$$
0.0961139 + 0.995370i $$0.469359\pi$$
$$434$$ 8.00000 0.384012
$$435$$ 0 0
$$436$$ 2.00000 0.0957826
$$437$$ 0 0
$$438$$ −32.0000 −1.52902
$$439$$ −16.0000 −0.763638 −0.381819 0.924237i $$-0.624702\pi$$
−0.381819 + 0.924237i $$0.624702\pi$$
$$440$$ 0 0
$$441$$ −3.00000 −0.142857
$$442$$ 0 0
$$443$$ −8.00000 −0.380091 −0.190046 0.981775i $$-0.560864\pi$$
−0.190046 + 0.981775i $$0.560864\pi$$
$$444$$ 0 0
$$445$$ 0 0
$$446$$ 22.0000 1.04173
$$447$$ 12.0000 0.567581
$$448$$ 14.0000 0.661438
$$449$$ −14.0000 −0.660701 −0.330350 0.943858i $$-0.607167\pi$$
−0.330350 + 0.943858i $$0.607167\pi$$
$$450$$ 0 0
$$451$$ 6.00000 0.282529
$$452$$ 4.00000 0.188144
$$453$$ −16.0000 −0.751746
$$454$$ 8.00000 0.375459
$$455$$ 0 0
$$456$$ 6.00000 0.280976
$$457$$ −32.0000 −1.49690 −0.748448 0.663193i $$-0.769201\pi$$
−0.748448 + 0.663193i $$0.769201\pi$$
$$458$$ −6.00000 −0.280362
$$459$$ 0 0
$$460$$ 0 0
$$461$$ 30.0000 1.39724 0.698620 0.715493i $$-0.253798\pi$$
0.698620 + 0.715493i $$0.253798\pi$$
$$462$$ −4.00000 −0.186097
$$463$$ −8.00000 −0.371792 −0.185896 0.982569i $$-0.559519\pi$$
−0.185896 + 0.982569i $$0.559519\pi$$
$$464$$ −6.00000 −0.278543
$$465$$ 0 0
$$466$$ 24.0000 1.11178
$$467$$ 36.0000 1.66588 0.832941 0.553362i $$-0.186655\pi$$
0.832941 + 0.553362i $$0.186655\pi$$
$$468$$ 6.00000 0.277350
$$469$$ −12.0000 −0.554109
$$470$$ 0 0
$$471$$ 20.0000 0.921551
$$472$$ 24.0000 1.10469
$$473$$ −6.00000 −0.275880
$$474$$ −16.0000 −0.734904
$$475$$ 0 0
$$476$$ 0 0
$$477$$ −12.0000 −0.549442
$$478$$ −24.0000 −1.09773
$$479$$ 28.0000 1.27935 0.639676 0.768644i $$-0.279068\pi$$
0.639676 + 0.768644i $$0.279068\pi$$
$$480$$ 0 0
$$481$$ 0 0
$$482$$ 14.0000 0.637683
$$483$$ 0 0
$$484$$ −1.00000 −0.0454545
$$485$$ 0 0
$$486$$ −10.0000 −0.453609
$$487$$ −2.00000 −0.0906287 −0.0453143 0.998973i $$-0.514429\pi$$
−0.0453143 + 0.998973i $$0.514429\pi$$
$$488$$ 42.0000 1.90125
$$489$$ −8.00000 −0.361773
$$490$$ 0 0
$$491$$ −16.0000 −0.722070 −0.361035 0.932552i $$-0.617576\pi$$
−0.361035 + 0.932552i $$0.617576\pi$$
$$492$$ 12.0000 0.541002
$$493$$ 0 0
$$494$$ 6.00000 0.269953
$$495$$ 0 0
$$496$$ −4.00000 −0.179605
$$497$$ −16.0000 −0.717698
$$498$$ −12.0000 −0.537733
$$499$$ 4.00000 0.179065 0.0895323 0.995984i $$-0.471463\pi$$
0.0895323 + 0.995984i $$0.471463\pi$$
$$500$$ 0 0
$$501$$ 16.0000 0.714827
$$502$$ −4.00000 −0.178529
$$503$$ 10.0000 0.445878 0.222939 0.974832i $$-0.428435\pi$$
0.222939 + 0.974832i $$0.428435\pi$$
$$504$$ −6.00000 −0.267261
$$505$$ 0 0
$$506$$ 0 0
$$507$$ 46.0000 2.04293
$$508$$ 4.00000 0.177471
$$509$$ −6.00000 −0.265945 −0.132973 0.991120i $$-0.542452\pi$$
−0.132973 + 0.991120i $$0.542452\pi$$
$$510$$ 0 0
$$511$$ −32.0000 −1.41560
$$512$$ −11.0000 −0.486136
$$513$$ 4.00000 0.176604
$$514$$ −20.0000 −0.882162
$$515$$ 0 0
$$516$$ −12.0000 −0.528271
$$517$$ −4.00000 −0.175920
$$518$$ 0 0
$$519$$ −12.0000 −0.526742
$$520$$ 0 0
$$521$$ −6.00000 −0.262865 −0.131432 0.991325i $$-0.541958\pi$$
−0.131432 + 0.991325i $$0.541958\pi$$
$$522$$ 6.00000 0.262613
$$523$$ 28.0000 1.22435 0.612177 0.790721i $$-0.290294\pi$$
0.612177 + 0.790721i $$0.290294\pi$$
$$524$$ −4.00000 −0.174741
$$525$$ 0 0
$$526$$ −10.0000 −0.436021
$$527$$ 0 0
$$528$$ 2.00000 0.0870388
$$529$$ −23.0000 −1.00000
$$530$$ 0 0
$$531$$ −8.00000 −0.347170
$$532$$ 2.00000 0.0867110
$$533$$ 36.0000 1.55933
$$534$$ −4.00000 −0.173097
$$535$$ 0 0
$$536$$ 18.0000 0.777482
$$537$$ −24.0000 −1.03568
$$538$$ −10.0000 −0.431131
$$539$$ 3.00000 0.129219
$$540$$ 0 0
$$541$$ 22.0000 0.945854 0.472927 0.881102i $$-0.343197\pi$$
0.472927 + 0.881102i $$0.343197\pi$$
$$542$$ 20.0000 0.859074
$$543$$ 28.0000 1.20160
$$544$$ 0 0
$$545$$ 0 0
$$546$$ −24.0000 −1.02711
$$547$$ 8.00000 0.342055 0.171028 0.985266i $$-0.445291\pi$$
0.171028 + 0.985266i $$0.445291\pi$$
$$548$$ 2.00000 0.0854358
$$549$$ −14.0000 −0.597505
$$550$$ 0 0
$$551$$ −6.00000 −0.255609
$$552$$ 0 0
$$553$$ −16.0000 −0.680389
$$554$$ −8.00000 −0.339887
$$555$$ 0 0
$$556$$ −12.0000 −0.508913
$$557$$ −28.0000 −1.18640 −0.593199 0.805056i $$-0.702135\pi$$
−0.593199 + 0.805056i $$0.702135\pi$$
$$558$$ 4.00000 0.169334
$$559$$ −36.0000 −1.52264
$$560$$ 0 0
$$561$$ 0 0
$$562$$ 6.00000 0.253095
$$563$$ 36.0000 1.51722 0.758610 0.651546i $$-0.225879\pi$$
0.758610 + 0.651546i $$0.225879\pi$$
$$564$$ −8.00000 −0.336861
$$565$$ 0 0
$$566$$ −6.00000 −0.252199
$$567$$ −22.0000 −0.923913
$$568$$ 24.0000 1.00702
$$569$$ 46.0000 1.92842 0.964210 0.265139i $$-0.0854179\pi$$
0.964210 + 0.265139i $$0.0854179\pi$$
$$570$$ 0 0
$$571$$ 12.0000 0.502184 0.251092 0.967963i $$-0.419210\pi$$
0.251092 + 0.967963i $$0.419210\pi$$
$$572$$ −6.00000 −0.250873
$$573$$ −16.0000 −0.668410
$$574$$ −12.0000 −0.500870
$$575$$ 0 0
$$576$$ 7.00000 0.291667
$$577$$ −14.0000 −0.582828 −0.291414 0.956597i $$-0.594126\pi$$
−0.291414 + 0.956597i $$0.594126\pi$$
$$578$$ −17.0000 −0.707107
$$579$$ 12.0000 0.498703
$$580$$ 0 0
$$581$$ −12.0000 −0.497844
$$582$$ −8.00000 −0.331611
$$583$$ 12.0000 0.496989
$$584$$ 48.0000 1.98625
$$585$$ 0 0
$$586$$ 14.0000 0.578335
$$587$$ 36.0000 1.48588 0.742940 0.669359i $$-0.233431\pi$$
0.742940 + 0.669359i $$0.233431\pi$$
$$588$$ 6.00000 0.247436
$$589$$ −4.00000 −0.164817
$$590$$ 0 0
$$591$$ 48.0000 1.97446
$$592$$ 0 0
$$593$$ 4.00000 0.164260 0.0821302 0.996622i $$-0.473828\pi$$
0.0821302 + 0.996622i $$0.473828\pi$$
$$594$$ 4.00000 0.164122
$$595$$ 0 0
$$596$$ −6.00000 −0.245770
$$597$$ 32.0000 1.30967
$$598$$ 0 0
$$599$$ 40.0000 1.63436 0.817178 0.576386i $$-0.195537\pi$$
0.817178 + 0.576386i $$0.195537\pi$$
$$600$$ 0 0
$$601$$ −46.0000 −1.87638 −0.938190 0.346122i $$-0.887498\pi$$
−0.938190 + 0.346122i $$0.887498\pi$$
$$602$$ 12.0000 0.489083
$$603$$ −6.00000 −0.244339
$$604$$ 8.00000 0.325515
$$605$$ 0 0
$$606$$ −36.0000 −1.46240
$$607$$ −40.0000 −1.62355 −0.811775 0.583970i $$-0.801498\pi$$
−0.811775 + 0.583970i $$0.801498\pi$$
$$608$$ −5.00000 −0.202777
$$609$$ 24.0000 0.972529
$$610$$ 0 0
$$611$$ −24.0000 −0.970936
$$612$$ 0 0
$$613$$ 36.0000 1.45403 0.727013 0.686624i $$-0.240908\pi$$
0.727013 + 0.686624i $$0.240908\pi$$
$$614$$ 12.0000 0.484281
$$615$$ 0 0
$$616$$ 6.00000 0.241747
$$617$$ −26.0000 −1.04672 −0.523360 0.852111i $$-0.675322\pi$$
−0.523360 + 0.852111i $$0.675322\pi$$
$$618$$ 20.0000 0.804518
$$619$$ 4.00000 0.160774 0.0803868 0.996764i $$-0.474384\pi$$
0.0803868 + 0.996764i $$0.474384\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ 0 0
$$623$$ −4.00000 −0.160257
$$624$$ 12.0000 0.480384
$$625$$ 0 0
$$626$$ −26.0000 −1.03917
$$627$$ 2.00000 0.0798723
$$628$$ −10.0000 −0.399043
$$629$$ 0 0
$$630$$ 0 0
$$631$$ 24.0000 0.955425 0.477712 0.878516i $$-0.341466\pi$$
0.477712 + 0.878516i $$0.341466\pi$$
$$632$$ 24.0000 0.954669
$$633$$ 56.0000 2.22580
$$634$$ 20.0000 0.794301
$$635$$ 0 0
$$636$$ 24.0000 0.951662
$$637$$ 18.0000 0.713186
$$638$$ −6.00000 −0.237542
$$639$$ −8.00000 −0.316475
$$640$$ 0 0
$$641$$ −26.0000 −1.02694 −0.513469 0.858108i $$-0.671640\pi$$
−0.513469 + 0.858108i $$0.671640\pi$$
$$642$$ 40.0000 1.57867
$$643$$ 40.0000 1.57745 0.788723 0.614749i $$-0.210743\pi$$
0.788723 + 0.614749i $$0.210743\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 0 0
$$647$$ 12.0000 0.471769 0.235884 0.971781i $$-0.424201\pi$$
0.235884 + 0.971781i $$0.424201\pi$$
$$648$$ 33.0000 1.29636
$$649$$ 8.00000 0.314027
$$650$$ 0 0
$$651$$ 16.0000 0.627089
$$652$$ 4.00000 0.156652
$$653$$ −34.0000 −1.33052 −0.665261 0.746611i $$-0.731680\pi$$
−0.665261 + 0.746611i $$0.731680\pi$$
$$654$$ −4.00000 −0.156412
$$655$$ 0 0
$$656$$ 6.00000 0.234261
$$657$$ −16.0000 −0.624219
$$658$$ 8.00000 0.311872
$$659$$ −44.0000 −1.71400 −0.856998 0.515319i $$-0.827673\pi$$
−0.856998 + 0.515319i $$0.827673\pi$$
$$660$$ 0 0
$$661$$ 18.0000 0.700119 0.350059 0.936727i $$-0.386161\pi$$
0.350059 + 0.936727i $$0.386161\pi$$
$$662$$ −28.0000 −1.08825
$$663$$ 0 0
$$664$$ 18.0000 0.698535
$$665$$ 0 0
$$666$$ 0 0
$$667$$ 0 0
$$668$$ −8.00000 −0.309529
$$669$$ 44.0000 1.70114
$$670$$ 0 0
$$671$$ 14.0000 0.540464
$$672$$ 20.0000 0.771517
$$673$$ −14.0000 −0.539660 −0.269830 0.962908i $$-0.586968\pi$$
−0.269830 + 0.962908i $$0.586968\pi$$
$$674$$ −10.0000 −0.385186
$$675$$ 0 0
$$676$$ −23.0000 −0.884615
$$677$$ 6.00000 0.230599 0.115299 0.993331i $$-0.463217\pi$$
0.115299 + 0.993331i $$0.463217\pi$$
$$678$$ −8.00000 −0.307238
$$679$$ −8.00000 −0.307012
$$680$$ 0 0
$$681$$ 16.0000 0.613121
$$682$$ −4.00000 −0.153168
$$683$$ 18.0000 0.688751 0.344375 0.938832i $$-0.388091\pi$$
0.344375 + 0.938832i $$0.388091\pi$$
$$684$$ 1.00000 0.0382360
$$685$$ 0 0
$$686$$ −20.0000 −0.763604
$$687$$ −12.0000 −0.457829
$$688$$ −6.00000 −0.228748
$$689$$ 72.0000 2.74298
$$690$$ 0 0
$$691$$ 28.0000 1.06517 0.532585 0.846376i $$-0.321221\pi$$
0.532585 + 0.846376i $$0.321221\pi$$
$$692$$ 6.00000 0.228086
$$693$$ −2.00000 −0.0759737
$$694$$ 26.0000 0.986947
$$695$$ 0 0
$$696$$ −36.0000 −1.36458
$$697$$ 0 0
$$698$$ −30.0000 −1.13552
$$699$$ 48.0000 1.81553
$$700$$ 0 0
$$701$$ 18.0000 0.679851 0.339925 0.940452i $$-0.389598\pi$$
0.339925 + 0.940452i $$0.389598\pi$$
$$702$$ 24.0000 0.905822
$$703$$ 0 0
$$704$$ −7.00000 −0.263822
$$705$$ 0 0
$$706$$ 18.0000 0.677439
$$707$$ −36.0000 −1.35392
$$708$$ 16.0000 0.601317
$$709$$ −38.0000 −1.42712 −0.713560 0.700594i $$-0.752918\pi$$
−0.713560 + 0.700594i $$0.752918\pi$$
$$710$$ 0 0
$$711$$ −8.00000 −0.300023
$$712$$ 6.00000 0.224860
$$713$$ 0 0
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 12.0000 0.448461
$$717$$ −48.0000 −1.79259
$$718$$ 12.0000 0.447836
$$719$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$720$$ 0 0
$$721$$ 20.0000 0.744839
$$722$$ 1.00000 0.0372161
$$723$$ 28.0000 1.04133
$$724$$ −14.0000 −0.520306
$$725$$ 0 0
$$726$$ 2.00000 0.0742270
$$727$$ 32.0000 1.18681 0.593407 0.804902i $$-0.297782\pi$$
0.593407 + 0.804902i $$0.297782\pi$$
$$728$$ 36.0000 1.33425
$$729$$ 13.0000 0.481481
$$730$$ 0 0
$$731$$ 0 0
$$732$$ 28.0000 1.03491
$$733$$ −24.0000 −0.886460 −0.443230 0.896408i $$-0.646168\pi$$
−0.443230 + 0.896408i $$0.646168\pi$$
$$734$$ 28.0000 1.03350
$$735$$ 0 0
$$736$$ 0 0
$$737$$ 6.00000 0.221013
$$738$$ −6.00000 −0.220863
$$739$$ 24.0000 0.882854 0.441427 0.897297i $$-0.354472\pi$$
0.441427 + 0.897297i $$0.354472\pi$$
$$740$$ 0 0
$$741$$ 12.0000 0.440831
$$742$$ −24.0000 −0.881068
$$743$$ −40.0000 −1.46746 −0.733729 0.679442i $$-0.762222\pi$$
−0.733729 + 0.679442i $$0.762222\pi$$
$$744$$ −24.0000 −0.879883
$$745$$ 0 0
$$746$$ −14.0000 −0.512576
$$747$$ −6.00000 −0.219529
$$748$$ 0 0
$$749$$ 40.0000 1.46157
$$750$$ 0 0
$$751$$ 4.00000 0.145962 0.0729810 0.997333i $$-0.476749\pi$$
0.0729810 + 0.997333i $$0.476749\pi$$
$$752$$ −4.00000 −0.145865
$$753$$ −8.00000 −0.291536
$$754$$ −36.0000 −1.31104
$$755$$ 0 0
$$756$$ 8.00000 0.290957
$$757$$ −2.00000 −0.0726912 −0.0363456 0.999339i $$-0.511572\pi$$
−0.0363456 + 0.999339i $$0.511572\pi$$
$$758$$ 4.00000 0.145287
$$759$$ 0 0
$$760$$ 0 0
$$761$$ −14.0000 −0.507500 −0.253750 0.967270i $$-0.581664\pi$$
−0.253750 + 0.967270i $$0.581664\pi$$
$$762$$ −8.00000 −0.289809
$$763$$ −4.00000 −0.144810
$$764$$ 8.00000 0.289430
$$765$$ 0 0
$$766$$ 30.0000 1.08394
$$767$$ 48.0000 1.73318
$$768$$ −34.0000 −1.22687
$$769$$ −10.0000 −0.360609 −0.180305 0.983611i $$-0.557708\pi$$
−0.180305 + 0.983611i $$0.557708\pi$$
$$770$$ 0 0
$$771$$ −40.0000 −1.44056
$$772$$ −6.00000 −0.215945
$$773$$ −24.0000 −0.863220 −0.431610 0.902060i $$-0.642054\pi$$
−0.431610 + 0.902060i $$0.642054\pi$$
$$774$$ 6.00000 0.215666
$$775$$ 0 0
$$776$$ 12.0000 0.430775
$$777$$ 0 0
$$778$$ −26.0000 −0.932145
$$779$$ 6.00000 0.214972
$$780$$ 0 0
$$781$$ 8.00000 0.286263
$$782$$ 0 0
$$783$$ −24.0000 −0.857690
$$784$$ 3.00000 0.107143
$$785$$ 0 0
$$786$$ 8.00000 0.285351
$$787$$ 20.0000 0.712923 0.356462 0.934310i $$-0.383983\pi$$
0.356462 + 0.934310i $$0.383983\pi$$
$$788$$ −24.0000 −0.854965
$$789$$ −20.0000 −0.712019
$$790$$ 0 0
$$791$$ −8.00000 −0.284447
$$792$$ 3.00000 0.106600
$$793$$ 84.0000 2.98293
$$794$$ 18.0000 0.638796
$$795$$ 0 0
$$796$$ −16.0000 −0.567105
$$797$$ −48.0000 −1.70025 −0.850124 0.526583i $$-0.823473\pi$$
−0.850124 + 0.526583i $$0.823473\pi$$
$$798$$ −4.00000 −0.141598
$$799$$ 0 0
$$800$$ 0 0
$$801$$ −2.00000 −0.0706665
$$802$$ −22.0000 −0.776847
$$803$$ 16.0000 0.564628
$$804$$ 12.0000 0.423207
$$805$$ 0 0
$$806$$ −24.0000 −0.845364
$$807$$ −20.0000 −0.704033
$$808$$ 54.0000 1.89971
$$809$$ 46.0000 1.61727 0.808637 0.588308i $$-0.200206\pi$$
0.808637 + 0.588308i $$0.200206\pi$$
$$810$$ 0 0
$$811$$ −28.0000 −0.983213 −0.491606 0.870817i $$-0.663590\pi$$
−0.491606 + 0.870817i $$0.663590\pi$$
$$812$$ −12.0000 −0.421117
$$813$$ 40.0000 1.40286
$$814$$ 0 0
$$815$$ 0 0
$$816$$ 0 0
$$817$$ −6.00000 −0.209913
$$818$$ −22.0000 −0.769212
$$819$$ −12.0000 −0.419314
$$820$$ 0 0
$$821$$ 2.00000 0.0698005 0.0349002 0.999391i $$-0.488889\pi$$
0.0349002 + 0.999391i $$0.488889\pi$$
$$822$$ −4.00000 −0.139516
$$823$$ −24.0000 −0.836587 −0.418294 0.908312i $$-0.637372\pi$$
−0.418294 + 0.908312i $$0.637372\pi$$
$$824$$ −30.0000 −1.04510
$$825$$ 0 0
$$826$$ −16.0000 −0.556711
$$827$$ −48.0000 −1.66912 −0.834562 0.550914i $$-0.814279\pi$$
−0.834562 + 0.550914i $$0.814279\pi$$
$$828$$ 0 0
$$829$$ −50.0000 −1.73657 −0.868286 0.496064i $$-0.834778\pi$$
−0.868286 + 0.496064i $$0.834778\pi$$
$$830$$ 0 0
$$831$$ −16.0000 −0.555034
$$832$$ −42.0000 −1.45609
$$833$$ 0 0
$$834$$ 24.0000 0.831052
$$835$$ 0 0
$$836$$ −1.00000 −0.0345857
$$837$$ −16.0000 −0.553041
$$838$$ −20.0000 −0.690889
$$839$$ −36.0000 −1.24286 −0.621429 0.783470i $$-0.713448\pi$$
−0.621429 + 0.783470i $$0.713448\pi$$
$$840$$ 0 0
$$841$$ 7.00000 0.241379
$$842$$ −14.0000 −0.482472
$$843$$ 12.0000 0.413302
$$844$$ −28.0000 −0.963800
$$845$$ 0 0
$$846$$ 4.00000 0.137523
$$847$$ 2.00000 0.0687208
$$848$$ 12.0000 0.412082
$$849$$ −12.0000 −0.411839
$$850$$ 0 0
$$851$$ 0 0
$$852$$ 16.0000 0.548151
$$853$$ 24.0000 0.821744 0.410872 0.911693i $$-0.365224\pi$$
0.410872 + 0.911693i $$0.365224\pi$$
$$854$$ −28.0000 −0.958140
$$855$$ 0 0
$$856$$ −60.0000 −2.05076
$$857$$ −6.00000 −0.204956 −0.102478 0.994735i $$-0.532677\pi$$
−0.102478 + 0.994735i $$0.532677\pi$$
$$858$$ 12.0000 0.409673
$$859$$ −4.00000 −0.136478 −0.0682391 0.997669i $$-0.521738\pi$$
−0.0682391 + 0.997669i $$0.521738\pi$$
$$860$$ 0 0
$$861$$ −24.0000 −0.817918
$$862$$ −8.00000 −0.272481
$$863$$ −10.0000 −0.340404 −0.170202 0.985409i $$-0.554442\pi$$
−0.170202 + 0.985409i $$0.554442\pi$$
$$864$$ −20.0000 −0.680414
$$865$$ 0 0
$$866$$ 4.00000 0.135926
$$867$$ −34.0000 −1.15470
$$868$$ −8.00000 −0.271538
$$869$$ 8.00000 0.271381
$$870$$ 0 0
$$871$$ 36.0000 1.21981
$$872$$ 6.00000 0.203186
$$873$$ −4.00000 −0.135379
$$874$$ 0 0
$$875$$ 0 0
$$876$$ 32.0000 1.08118
$$877$$ −38.0000 −1.28317 −0.641584 0.767052i $$-0.721723\pi$$
−0.641584 + 0.767052i $$0.721723\pi$$
$$878$$ −16.0000 −0.539974
$$879$$ 28.0000 0.944417
$$880$$ 0 0
$$881$$ 14.0000 0.471672 0.235836 0.971793i $$-0.424217\pi$$
0.235836 + 0.971793i $$0.424217\pi$$
$$882$$ −3.00000 −0.101015
$$883$$ 12.0000 0.403832 0.201916 0.979403i $$-0.435283\pi$$
0.201916 + 0.979403i $$0.435283\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ −8.00000 −0.268765
$$887$$ −12.0000 −0.402921 −0.201460 0.979497i $$-0.564569\pi$$
−0.201460 + 0.979497i $$0.564569\pi$$
$$888$$ 0 0
$$889$$ −8.00000 −0.268311
$$890$$ 0 0
$$891$$ 11.0000 0.368514
$$892$$ −22.0000 −0.736614
$$893$$ −4.00000 −0.133855
$$894$$ 12.0000 0.401340
$$895$$ 0 0
$$896$$ −6.00000 −0.200446
$$897$$ 0 0
$$898$$ −14.0000 −0.467186
$$899$$ 24.0000 0.800445
$$900$$ 0 0
$$901$$ 0 0
$$902$$ 6.00000 0.199778
$$903$$ 24.0000 0.798670
$$904$$ 12.0000 0.399114
$$905$$ 0 0
$$906$$ −16.0000 −0.531564
$$907$$ −50.0000 −1.66022 −0.830111 0.557598i $$-0.811723\pi$$
−0.830111 + 0.557598i $$0.811723\pi$$
$$908$$ −8.00000 −0.265489
$$909$$ −18.0000 −0.597022
$$910$$ 0 0
$$911$$ 32.0000 1.06021 0.530104 0.847933i $$-0.322153\pi$$
0.530104 + 0.847933i $$0.322153\pi$$
$$912$$ 2.00000 0.0662266
$$913$$ 6.00000 0.198571
$$914$$ −32.0000 −1.05847
$$915$$ 0 0
$$916$$ 6.00000 0.198246
$$917$$ 8.00000 0.264183
$$918$$ 0 0
$$919$$ 36.0000 1.18753 0.593765 0.804638i $$-0.297641\pi$$
0.593765 + 0.804638i $$0.297641\pi$$
$$920$$ 0 0
$$921$$ 24.0000 0.790827
$$922$$ 30.0000 0.987997
$$923$$ 48.0000 1.57994
$$924$$ 4.00000 0.131590
$$925$$ 0 0
$$926$$ −8.00000 −0.262896
$$927$$ 10.0000 0.328443
$$928$$ 30.0000 0.984798
$$929$$ 18.0000 0.590561 0.295280 0.955411i $$-0.404587\pi$$
0.295280 + 0.955411i $$0.404587\pi$$
$$930$$ 0 0
$$931$$ 3.00000 0.0983210
$$932$$ −24.0000 −0.786146
$$933$$ 0 0
$$934$$ 36.0000 1.17796
$$935$$ 0 0
$$936$$ 18.0000 0.588348
$$937$$ 12.0000 0.392023 0.196011 0.980602i $$-0.437201\pi$$
0.196011 + 0.980602i $$0.437201\pi$$
$$938$$ −12.0000 −0.391814
$$939$$ −52.0000 −1.69696
$$940$$ 0 0
$$941$$ −6.00000 −0.195594 −0.0977972 0.995206i $$-0.531180\pi$$
−0.0977972 + 0.995206i $$0.531180\pi$$
$$942$$ 20.0000 0.651635
$$943$$ 0 0
$$944$$ 8.00000 0.260378
$$945$$ 0 0
$$946$$ −6.00000 −0.195077
$$947$$ −12.0000 −0.389948 −0.194974 0.980808i $$-0.562462\pi$$
−0.194974 + 0.980808i $$0.562462\pi$$
$$948$$ 16.0000 0.519656
$$949$$ 96.0000 3.11629
$$950$$ 0 0
$$951$$ 40.0000 1.29709
$$952$$ 0 0
$$953$$ −10.0000 −0.323932 −0.161966 0.986796i $$-0.551783\pi$$
−0.161966 + 0.986796i $$0.551783\pi$$
$$954$$ −12.0000 −0.388514
$$955$$ 0 0
$$956$$ 24.0000 0.776215
$$957$$ −12.0000 −0.387905
$$958$$ 28.0000 0.904639
$$959$$ −4.00000 −0.129167
$$960$$ 0 0
$$961$$ −15.0000 −0.483871
$$962$$ 0 0
$$963$$ 20.0000 0.644491
$$964$$ −14.0000 −0.450910
$$965$$ 0 0
$$966$$ 0 0
$$967$$ −14.0000 −0.450210 −0.225105 0.974335i $$-0.572272\pi$$
−0.225105 + 0.974335i $$0.572272\pi$$
$$968$$ −3.00000 −0.0964237
$$969$$ 0 0
$$970$$ 0 0
$$971$$ 40.0000 1.28366 0.641831 0.766846i $$-0.278175\pi$$
0.641831 + 0.766846i $$0.278175\pi$$
$$972$$ 10.0000 0.320750
$$973$$ 24.0000 0.769405
$$974$$ −2.00000 −0.0640841
$$975$$ 0 0
$$976$$ 14.0000 0.448129
$$977$$ −32.0000 −1.02377 −0.511885 0.859054i $$-0.671053\pi$$
−0.511885 + 0.859054i $$0.671053\pi$$
$$978$$ −8.00000 −0.255812
$$979$$ 2.00000 0.0639203
$$980$$ 0 0
$$981$$ −2.00000 −0.0638551
$$982$$ −16.0000 −0.510581
$$983$$ 38.0000 1.21201 0.606006 0.795460i $$-0.292771\pi$$
0.606006 + 0.795460i $$0.292771\pi$$
$$984$$ 36.0000 1.14764
$$985$$ 0 0
$$986$$ 0 0
$$987$$ 16.0000 0.509286
$$988$$ −6.00000 −0.190885
$$989$$ 0 0
$$990$$ 0 0
$$991$$ 32.0000 1.01651 0.508257 0.861206i $$-0.330290\pi$$
0.508257 + 0.861206i $$0.330290\pi$$
$$992$$ 20.0000 0.635001
$$993$$ −56.0000 −1.77711
$$994$$ −16.0000 −0.507489
$$995$$ 0 0
$$996$$ 12.0000 0.380235
$$997$$ −12.0000 −0.380044 −0.190022 0.981780i $$-0.560856\pi$$
−0.190022 + 0.981780i $$0.560856\pi$$
$$998$$ 4.00000 0.126618
$$999$$ 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 5225.2.a.c.1.1 1
5.4 even 2 1045.2.a.a.1.1 1
15.14 odd 2 9405.2.a.j.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
1045.2.a.a.1.1 1 5.4 even 2
5225.2.a.c.1.1 1 1.1 even 1 trivial
9405.2.a.j.1.1 1 15.14 odd 2