# Properties

 Label 7581.2.a.d.1.1 Level $7581$ Weight $2$ Character 7581.1 Self dual yes Analytic conductor $60.535$ Analytic rank $0$ Dimension $1$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$7581 = 3 \cdot 7 \cdot 19^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 7581.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$60.5345897723$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 21) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Character $$\chi$$ $$=$$ 7581.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q+1.00000 q^{2} -1.00000 q^{3} -1.00000 q^{4} -2.00000 q^{5} -1.00000 q^{6} -1.00000 q^{7} -3.00000 q^{8} +1.00000 q^{9} +O(q^{10})$$ $$q+1.00000 q^{2} -1.00000 q^{3} -1.00000 q^{4} -2.00000 q^{5} -1.00000 q^{6} -1.00000 q^{7} -3.00000 q^{8} +1.00000 q^{9} -2.00000 q^{10} +4.00000 q^{11} +1.00000 q^{12} +2.00000 q^{13} -1.00000 q^{14} +2.00000 q^{15} -1.00000 q^{16} -6.00000 q^{17} +1.00000 q^{18} +2.00000 q^{20} +1.00000 q^{21} +4.00000 q^{22} +3.00000 q^{24} -1.00000 q^{25} +2.00000 q^{26} -1.00000 q^{27} +1.00000 q^{28} +2.00000 q^{29} +2.00000 q^{30} +5.00000 q^{32} -4.00000 q^{33} -6.00000 q^{34} +2.00000 q^{35} -1.00000 q^{36} -6.00000 q^{37} -2.00000 q^{39} +6.00000 q^{40} -2.00000 q^{41} +1.00000 q^{42} -4.00000 q^{43} -4.00000 q^{44} -2.00000 q^{45} +1.00000 q^{48} +1.00000 q^{49} -1.00000 q^{50} +6.00000 q^{51} -2.00000 q^{52} -6.00000 q^{53} -1.00000 q^{54} -8.00000 q^{55} +3.00000 q^{56} +2.00000 q^{58} -12.0000 q^{59} -2.00000 q^{60} -2.00000 q^{61} -1.00000 q^{63} +7.00000 q^{64} -4.00000 q^{65} -4.00000 q^{66} -4.00000 q^{67} +6.00000 q^{68} +2.00000 q^{70} -3.00000 q^{72} -6.00000 q^{73} -6.00000 q^{74} +1.00000 q^{75} -4.00000 q^{77} -2.00000 q^{78} +16.0000 q^{79} +2.00000 q^{80} +1.00000 q^{81} -2.00000 q^{82} -12.0000 q^{83} -1.00000 q^{84} +12.0000 q^{85} -4.00000 q^{86} -2.00000 q^{87} -12.0000 q^{88} +14.0000 q^{89} -2.00000 q^{90} -2.00000 q^{91} -5.00000 q^{96} -18.0000 q^{97} +1.00000 q^{98} +4.00000 q^{99} +O(q^{100})$$

## 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 $$0.384973\pi$$
$$3$$ −1.00000 −0.577350
$$4$$ −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
$$7$$ −1.00000 −0.377964
$$8$$ −3.00000 −1.06066
$$9$$ 1.00000 0.333333
$$10$$ −2.00000 −0.632456
$$11$$ 4.00000 1.20605 0.603023 0.797724i $$-0.293963\pi$$
0.603023 + 0.797724i $$0.293963\pi$$
$$12$$ 1.00000 0.288675
$$13$$ 2.00000 0.554700 0.277350 0.960769i $$-0.410544\pi$$
0.277350 + 0.960769i $$0.410544\pi$$
$$14$$ −1.00000 −0.267261
$$15$$ 2.00000 0.516398
$$16$$ −1.00000 −0.250000
$$17$$ −6.00000 −1.45521 −0.727607 0.685994i $$-0.759367\pi$$
−0.727607 + 0.685994i $$0.759367\pi$$
$$18$$ 1.00000 0.235702
$$19$$ 0 0
$$20$$ 2.00000 0.447214
$$21$$ 1.00000 0.218218
$$22$$ 4.00000 0.852803
$$23$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$24$$ 3.00000 0.612372
$$25$$ −1.00000 −0.200000
$$26$$ 2.00000 0.392232
$$27$$ −1.00000 −0.192450
$$28$$ 1.00000 0.188982
$$29$$ 2.00000 0.371391 0.185695 0.982607i $$-0.440546\pi$$
0.185695 + 0.982607i $$0.440546\pi$$
$$30$$ 2.00000 0.365148
$$31$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$32$$ 5.00000 0.883883
$$33$$ −4.00000 −0.696311
$$34$$ −6.00000 −1.02899
$$35$$ 2.00000 0.338062
$$36$$ −1.00000 −0.166667
$$37$$ −6.00000 −0.986394 −0.493197 0.869918i $$-0.664172\pi$$
−0.493197 + 0.869918i $$0.664172\pi$$
$$38$$ 0 0
$$39$$ −2.00000 −0.320256
$$40$$ 6.00000 0.948683
$$41$$ −2.00000 −0.312348 −0.156174 0.987730i $$-0.549916\pi$$
−0.156174 + 0.987730i $$0.549916\pi$$
$$42$$ 1.00000 0.154303
$$43$$ −4.00000 −0.609994 −0.304997 0.952353i $$-0.598656\pi$$
−0.304997 + 0.952353i $$0.598656\pi$$
$$44$$ −4.00000 −0.603023
$$45$$ −2.00000 −0.298142
$$46$$ 0 0
$$47$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$48$$ 1.00000 0.144338
$$49$$ 1.00000 0.142857
$$50$$ −1.00000 −0.141421
$$51$$ 6.00000 0.840168
$$52$$ −2.00000 −0.277350
$$53$$ −6.00000 −0.824163 −0.412082 0.911147i $$-0.635198\pi$$
−0.412082 + 0.911147i $$0.635198\pi$$
$$54$$ −1.00000 −0.136083
$$55$$ −8.00000 −1.07872
$$56$$ 3.00000 0.400892
$$57$$ 0 0
$$58$$ 2.00000 0.262613
$$59$$ −12.0000 −1.56227 −0.781133 0.624364i $$-0.785358\pi$$
−0.781133 + 0.624364i $$0.785358\pi$$
$$60$$ −2.00000 −0.258199
$$61$$ −2.00000 −0.256074 −0.128037 0.991769i $$-0.540868\pi$$
−0.128037 + 0.991769i $$0.540868\pi$$
$$62$$ 0 0
$$63$$ −1.00000 −0.125988
$$64$$ 7.00000 0.875000
$$65$$ −4.00000 −0.496139
$$66$$ −4.00000 −0.492366
$$67$$ −4.00000 −0.488678 −0.244339 0.969690i $$-0.578571\pi$$
−0.244339 + 0.969690i $$0.578571\pi$$
$$68$$ 6.00000 0.727607
$$69$$ 0 0
$$70$$ 2.00000 0.239046
$$71$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$72$$ −3.00000 −0.353553
$$73$$ −6.00000 −0.702247 −0.351123 0.936329i $$-0.614200\pi$$
−0.351123 + 0.936329i $$0.614200\pi$$
$$74$$ −6.00000 −0.697486
$$75$$ 1.00000 0.115470
$$76$$ 0 0
$$77$$ −4.00000 −0.455842
$$78$$ −2.00000 −0.226455
$$79$$ 16.0000 1.80014 0.900070 0.435745i $$-0.143515\pi$$
0.900070 + 0.435745i $$0.143515\pi$$
$$80$$ 2.00000 0.223607
$$81$$ 1.00000 0.111111
$$82$$ −2.00000 −0.220863
$$83$$ −12.0000 −1.31717 −0.658586 0.752506i $$-0.728845\pi$$
−0.658586 + 0.752506i $$0.728845\pi$$
$$84$$ −1.00000 −0.109109
$$85$$ 12.0000 1.30158
$$86$$ −4.00000 −0.431331
$$87$$ −2.00000 −0.214423
$$88$$ −12.0000 −1.27920
$$89$$ 14.0000 1.48400 0.741999 0.670402i $$-0.233878\pi$$
0.741999 + 0.670402i $$0.233878\pi$$
$$90$$ −2.00000 −0.210819
$$91$$ −2.00000 −0.209657
$$92$$ 0 0
$$93$$ 0 0
$$94$$ 0 0
$$95$$ 0 0
$$96$$ −5.00000 −0.510310
$$97$$ −18.0000 −1.82762 −0.913812 0.406138i $$-0.866875\pi$$
−0.913812 + 0.406138i $$0.866875\pi$$
$$98$$ 1.00000 0.101015
$$99$$ 4.00000 0.402015
$$100$$ 1.00000 0.100000
$$101$$ 14.0000 1.39305 0.696526 0.717532i $$-0.254728\pi$$
0.696526 + 0.717532i $$0.254728\pi$$
$$102$$ 6.00000 0.594089
$$103$$ −8.00000 −0.788263 −0.394132 0.919054i $$-0.628955\pi$$
−0.394132 + 0.919054i $$0.628955\pi$$
$$104$$ −6.00000 −0.588348
$$105$$ −2.00000 −0.195180
$$106$$ −6.00000 −0.582772
$$107$$ −4.00000 −0.386695 −0.193347 0.981130i $$-0.561934\pi$$
−0.193347 + 0.981130i $$0.561934\pi$$
$$108$$ 1.00000 0.0962250
$$109$$ 18.0000 1.72409 0.862044 0.506834i $$-0.169184\pi$$
0.862044 + 0.506834i $$0.169184\pi$$
$$110$$ −8.00000 −0.762770
$$111$$ 6.00000 0.569495
$$112$$ 1.00000 0.0944911
$$113$$ 14.0000 1.31701 0.658505 0.752577i $$-0.271189\pi$$
0.658505 + 0.752577i $$0.271189\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ −2.00000 −0.185695
$$117$$ 2.00000 0.184900
$$118$$ −12.0000 −1.10469
$$119$$ 6.00000 0.550019
$$120$$ −6.00000 −0.547723
$$121$$ 5.00000 0.454545
$$122$$ −2.00000 −0.181071
$$123$$ 2.00000 0.180334
$$124$$ 0 0
$$125$$ 12.0000 1.07331
$$126$$ −1.00000 −0.0890871
$$127$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$128$$ −3.00000 −0.265165
$$129$$ 4.00000 0.352180
$$130$$ −4.00000 −0.350823
$$131$$ 4.00000 0.349482 0.174741 0.984614i $$-0.444091\pi$$
0.174741 + 0.984614i $$0.444091\pi$$
$$132$$ 4.00000 0.348155
$$133$$ 0 0
$$134$$ −4.00000 −0.345547
$$135$$ 2.00000 0.172133
$$136$$ 18.0000 1.54349
$$137$$ −6.00000 −0.512615 −0.256307 0.966595i $$-0.582506\pi$$
−0.256307 + 0.966595i $$0.582506\pi$$
$$138$$ 0 0
$$139$$ 12.0000 1.01783 0.508913 0.860818i $$-0.330047\pi$$
0.508913 + 0.860818i $$0.330047\pi$$
$$140$$ −2.00000 −0.169031
$$141$$ 0 0
$$142$$ 0 0
$$143$$ 8.00000 0.668994
$$144$$ −1.00000 −0.0833333
$$145$$ −4.00000 −0.332182
$$146$$ −6.00000 −0.496564
$$147$$ −1.00000 −0.0824786
$$148$$ 6.00000 0.493197
$$149$$ 6.00000 0.491539 0.245770 0.969328i $$-0.420959\pi$$
0.245770 + 0.969328i $$0.420959\pi$$
$$150$$ 1.00000 0.0816497
$$151$$ −8.00000 −0.651031 −0.325515 0.945537i $$-0.605538\pi$$
−0.325515 + 0.945537i $$0.605538\pi$$
$$152$$ 0 0
$$153$$ −6.00000 −0.485071
$$154$$ −4.00000 −0.322329
$$155$$ 0 0
$$156$$ 2.00000 0.160128
$$157$$ −2.00000 −0.159617 −0.0798087 0.996810i $$-0.525431\pi$$
−0.0798087 + 0.996810i $$0.525431\pi$$
$$158$$ 16.0000 1.27289
$$159$$ 6.00000 0.475831
$$160$$ −10.0000 −0.790569
$$161$$ 0 0
$$162$$ 1.00000 0.0785674
$$163$$ 4.00000 0.313304 0.156652 0.987654i $$-0.449930\pi$$
0.156652 + 0.987654i $$0.449930\pi$$
$$164$$ 2.00000 0.156174
$$165$$ 8.00000 0.622799
$$166$$ −12.0000 −0.931381
$$167$$ 8.00000 0.619059 0.309529 0.950890i $$-0.399829\pi$$
0.309529 + 0.950890i $$0.399829\pi$$
$$168$$ −3.00000 −0.231455
$$169$$ −9.00000 −0.692308
$$170$$ 12.0000 0.920358
$$171$$ 0 0
$$172$$ 4.00000 0.304997
$$173$$ 10.0000 0.760286 0.380143 0.924928i $$-0.375875\pi$$
0.380143 + 0.924928i $$0.375875\pi$$
$$174$$ −2.00000 −0.151620
$$175$$ 1.00000 0.0755929
$$176$$ −4.00000 −0.301511
$$177$$ 12.0000 0.901975
$$178$$ 14.0000 1.04934
$$179$$ 4.00000 0.298974 0.149487 0.988764i $$-0.452238\pi$$
0.149487 + 0.988764i $$0.452238\pi$$
$$180$$ 2.00000 0.149071
$$181$$ 26.0000 1.93256 0.966282 0.257485i $$-0.0828937\pi$$
0.966282 + 0.257485i $$0.0828937\pi$$
$$182$$ −2.00000 −0.148250
$$183$$ 2.00000 0.147844
$$184$$ 0 0
$$185$$ 12.0000 0.882258
$$186$$ 0 0
$$187$$ −24.0000 −1.75505
$$188$$ 0 0
$$189$$ 1.00000 0.0727393
$$190$$ 0 0
$$191$$ −8.00000 −0.578860 −0.289430 0.957199i $$-0.593466\pi$$
−0.289430 + 0.957199i $$0.593466\pi$$
$$192$$ −7.00000 −0.505181
$$193$$ −2.00000 −0.143963 −0.0719816 0.997406i $$-0.522932\pi$$
−0.0719816 + 0.997406i $$0.522932\pi$$
$$194$$ −18.0000 −1.29232
$$195$$ 4.00000 0.286446
$$196$$ −1.00000 −0.0714286
$$197$$ 22.0000 1.56744 0.783718 0.621117i $$-0.213321\pi$$
0.783718 + 0.621117i $$0.213321\pi$$
$$198$$ 4.00000 0.284268
$$199$$ 24.0000 1.70131 0.850657 0.525720i $$-0.176204\pi$$
0.850657 + 0.525720i $$0.176204\pi$$
$$200$$ 3.00000 0.212132
$$201$$ 4.00000 0.282138
$$202$$ 14.0000 0.985037
$$203$$ −2.00000 −0.140372
$$204$$ −6.00000 −0.420084
$$205$$ 4.00000 0.279372
$$206$$ −8.00000 −0.557386
$$207$$ 0 0
$$208$$ −2.00000 −0.138675
$$209$$ 0 0
$$210$$ −2.00000 −0.138013
$$211$$ −4.00000 −0.275371 −0.137686 0.990476i $$-0.543966\pi$$
−0.137686 + 0.990476i $$0.543966\pi$$
$$212$$ 6.00000 0.412082
$$213$$ 0 0
$$214$$ −4.00000 −0.273434
$$215$$ 8.00000 0.545595
$$216$$ 3.00000 0.204124
$$217$$ 0 0
$$218$$ 18.0000 1.21911
$$219$$ 6.00000 0.405442
$$220$$ 8.00000 0.539360
$$221$$ −12.0000 −0.807207
$$222$$ 6.00000 0.402694
$$223$$ −16.0000 −1.07144 −0.535720 0.844396i $$-0.679960\pi$$
−0.535720 + 0.844396i $$0.679960\pi$$
$$224$$ −5.00000 −0.334077
$$225$$ −1.00000 −0.0666667
$$226$$ 14.0000 0.931266
$$227$$ 12.0000 0.796468 0.398234 0.917284i $$-0.369623\pi$$
0.398234 + 0.917284i $$0.369623\pi$$
$$228$$ 0 0
$$229$$ −10.0000 −0.660819 −0.330409 0.943838i $$-0.607187\pi$$
−0.330409 + 0.943838i $$0.607187\pi$$
$$230$$ 0 0
$$231$$ 4.00000 0.263181
$$232$$ −6.00000 −0.393919
$$233$$ −6.00000 −0.393073 −0.196537 0.980497i $$-0.562969\pi$$
−0.196537 + 0.980497i $$0.562969\pi$$
$$234$$ 2.00000 0.130744
$$235$$ 0 0
$$236$$ 12.0000 0.781133
$$237$$ −16.0000 −1.03931
$$238$$ 6.00000 0.388922
$$239$$ 24.0000 1.55243 0.776215 0.630468i $$-0.217137\pi$$
0.776215 + 0.630468i $$0.217137\pi$$
$$240$$ −2.00000 −0.129099
$$241$$ −2.00000 −0.128831 −0.0644157 0.997923i $$-0.520518\pi$$
−0.0644157 + 0.997923i $$0.520518\pi$$
$$242$$ 5.00000 0.321412
$$243$$ −1.00000 −0.0641500
$$244$$ 2.00000 0.128037
$$245$$ −2.00000 −0.127775
$$246$$ 2.00000 0.127515
$$247$$ 0 0
$$248$$ 0 0
$$249$$ 12.0000 0.760469
$$250$$ 12.0000 0.758947
$$251$$ −20.0000 −1.26239 −0.631194 0.775625i $$-0.717435\pi$$
−0.631194 + 0.775625i $$0.717435\pi$$
$$252$$ 1.00000 0.0629941
$$253$$ 0 0
$$254$$ 0 0
$$255$$ −12.0000 −0.751469
$$256$$ −17.0000 −1.06250
$$257$$ −26.0000 −1.62184 −0.810918 0.585160i $$-0.801032\pi$$
−0.810918 + 0.585160i $$0.801032\pi$$
$$258$$ 4.00000 0.249029
$$259$$ 6.00000 0.372822
$$260$$ 4.00000 0.248069
$$261$$ 2.00000 0.123797
$$262$$ 4.00000 0.247121
$$263$$ 16.0000 0.986602 0.493301 0.869859i $$-0.335790\pi$$
0.493301 + 0.869859i $$0.335790\pi$$
$$264$$ 12.0000 0.738549
$$265$$ 12.0000 0.737154
$$266$$ 0 0
$$267$$ −14.0000 −0.856786
$$268$$ 4.00000 0.244339
$$269$$ −6.00000 −0.365826 −0.182913 0.983129i $$-0.558553\pi$$
−0.182913 + 0.983129i $$0.558553\pi$$
$$270$$ 2.00000 0.121716
$$271$$ 16.0000 0.971931 0.485965 0.873978i $$-0.338468\pi$$
0.485965 + 0.873978i $$0.338468\pi$$
$$272$$ 6.00000 0.363803
$$273$$ 2.00000 0.121046
$$274$$ −6.00000 −0.362473
$$275$$ −4.00000 −0.241209
$$276$$ 0 0
$$277$$ 22.0000 1.32185 0.660926 0.750451i $$-0.270164\pi$$
0.660926 + 0.750451i $$0.270164\pi$$
$$278$$ 12.0000 0.719712
$$279$$ 0 0
$$280$$ −6.00000 −0.358569
$$281$$ 22.0000 1.31241 0.656205 0.754583i $$-0.272161\pi$$
0.656205 + 0.754583i $$0.272161\pi$$
$$282$$ 0 0
$$283$$ −20.0000 −1.18888 −0.594438 0.804141i $$-0.702626\pi$$
−0.594438 + 0.804141i $$0.702626\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ 8.00000 0.473050
$$287$$ 2.00000 0.118056
$$288$$ 5.00000 0.294628
$$289$$ 19.0000 1.11765
$$290$$ −4.00000 −0.234888
$$291$$ 18.0000 1.05518
$$292$$ 6.00000 0.351123
$$293$$ −14.0000 −0.817889 −0.408944 0.912559i $$-0.634103\pi$$
−0.408944 + 0.912559i $$0.634103\pi$$
$$294$$ −1.00000 −0.0583212
$$295$$ 24.0000 1.39733
$$296$$ 18.0000 1.04623
$$297$$ −4.00000 −0.232104
$$298$$ 6.00000 0.347571
$$299$$ 0 0
$$300$$ −1.00000 −0.0577350
$$301$$ 4.00000 0.230556
$$302$$ −8.00000 −0.460348
$$303$$ −14.0000 −0.804279
$$304$$ 0 0
$$305$$ 4.00000 0.229039
$$306$$ −6.00000 −0.342997
$$307$$ −4.00000 −0.228292 −0.114146 0.993464i $$-0.536413\pi$$
−0.114146 + 0.993464i $$0.536413\pi$$
$$308$$ 4.00000 0.227921
$$309$$ 8.00000 0.455104
$$310$$ 0 0
$$311$$ −24.0000 −1.36092 −0.680458 0.732787i $$-0.738219\pi$$
−0.680458 + 0.732787i $$0.738219\pi$$
$$312$$ 6.00000 0.339683
$$313$$ 26.0000 1.46961 0.734803 0.678280i $$-0.237274\pi$$
0.734803 + 0.678280i $$0.237274\pi$$
$$314$$ −2.00000 −0.112867
$$315$$ 2.00000 0.112687
$$316$$ −16.0000 −0.900070
$$317$$ 18.0000 1.01098 0.505490 0.862832i $$-0.331312\pi$$
0.505490 + 0.862832i $$0.331312\pi$$
$$318$$ 6.00000 0.336463
$$319$$ 8.00000 0.447914
$$320$$ −14.0000 −0.782624
$$321$$ 4.00000 0.223258
$$322$$ 0 0
$$323$$ 0 0
$$324$$ −1.00000 −0.0555556
$$325$$ −2.00000 −0.110940
$$326$$ 4.00000 0.221540
$$327$$ −18.0000 −0.995402
$$328$$ 6.00000 0.331295
$$329$$ 0 0
$$330$$ 8.00000 0.440386
$$331$$ 4.00000 0.219860 0.109930 0.993939i $$-0.464937\pi$$
0.109930 + 0.993939i $$0.464937\pi$$
$$332$$ 12.0000 0.658586
$$333$$ −6.00000 −0.328798
$$334$$ 8.00000 0.437741
$$335$$ 8.00000 0.437087
$$336$$ −1.00000 −0.0545545
$$337$$ 14.0000 0.762629 0.381314 0.924445i $$-0.375472\pi$$
0.381314 + 0.924445i $$0.375472\pi$$
$$338$$ −9.00000 −0.489535
$$339$$ −14.0000 −0.760376
$$340$$ −12.0000 −0.650791
$$341$$ 0 0
$$342$$ 0 0
$$343$$ −1.00000 −0.0539949
$$344$$ 12.0000 0.646997
$$345$$ 0 0
$$346$$ 10.0000 0.537603
$$347$$ −28.0000 −1.50312 −0.751559 0.659665i $$-0.770698\pi$$
−0.751559 + 0.659665i $$0.770698\pi$$
$$348$$ 2.00000 0.107211
$$349$$ −2.00000 −0.107058 −0.0535288 0.998566i $$-0.517047\pi$$
−0.0535288 + 0.998566i $$0.517047\pi$$
$$350$$ 1.00000 0.0534522
$$351$$ −2.00000 −0.106752
$$352$$ 20.0000 1.06600
$$353$$ 10.0000 0.532246 0.266123 0.963939i $$-0.414257\pi$$
0.266123 + 0.963939i $$0.414257\pi$$
$$354$$ 12.0000 0.637793
$$355$$ 0 0
$$356$$ −14.0000 −0.741999
$$357$$ −6.00000 −0.317554
$$358$$ 4.00000 0.211407
$$359$$ 32.0000 1.68890 0.844448 0.535638i $$-0.179929\pi$$
0.844448 + 0.535638i $$0.179929\pi$$
$$360$$ 6.00000 0.316228
$$361$$ 0 0
$$362$$ 26.0000 1.36653
$$363$$ −5.00000 −0.262432
$$364$$ 2.00000 0.104828
$$365$$ 12.0000 0.628109
$$366$$ 2.00000 0.104542
$$367$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$368$$ 0 0
$$369$$ −2.00000 −0.104116
$$370$$ 12.0000 0.623850
$$371$$ 6.00000 0.311504
$$372$$ 0 0
$$373$$ 10.0000 0.517780 0.258890 0.965907i $$-0.416643\pi$$
0.258890 + 0.965907i $$0.416643\pi$$
$$374$$ −24.0000 −1.24101
$$375$$ −12.0000 −0.619677
$$376$$ 0 0
$$377$$ 4.00000 0.206010
$$378$$ 1.00000 0.0514344
$$379$$ −12.0000 −0.616399 −0.308199 0.951322i $$-0.599726\pi$$
−0.308199 + 0.951322i $$0.599726\pi$$
$$380$$ 0 0
$$381$$ 0 0
$$382$$ −8.00000 −0.409316
$$383$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$384$$ 3.00000 0.153093
$$385$$ 8.00000 0.407718
$$386$$ −2.00000 −0.101797
$$387$$ −4.00000 −0.203331
$$388$$ 18.0000 0.913812
$$389$$ 6.00000 0.304212 0.152106 0.988364i $$-0.451394\pi$$
0.152106 + 0.988364i $$0.451394\pi$$
$$390$$ 4.00000 0.202548
$$391$$ 0 0
$$392$$ −3.00000 −0.151523
$$393$$ −4.00000 −0.201773
$$394$$ 22.0000 1.10834
$$395$$ −32.0000 −1.61009
$$396$$ −4.00000 −0.201008
$$397$$ −18.0000 −0.903394 −0.451697 0.892171i $$-0.649181\pi$$
−0.451697 + 0.892171i $$0.649181\pi$$
$$398$$ 24.0000 1.20301
$$399$$ 0 0
$$400$$ 1.00000 0.0500000
$$401$$ 30.0000 1.49813 0.749064 0.662497i $$-0.230503\pi$$
0.749064 + 0.662497i $$0.230503\pi$$
$$402$$ 4.00000 0.199502
$$403$$ 0 0
$$404$$ −14.0000 −0.696526
$$405$$ −2.00000 −0.0993808
$$406$$ −2.00000 −0.0992583
$$407$$ −24.0000 −1.18964
$$408$$ −18.0000 −0.891133
$$409$$ 22.0000 1.08783 0.543915 0.839140i $$-0.316941\pi$$
0.543915 + 0.839140i $$0.316941\pi$$
$$410$$ 4.00000 0.197546
$$411$$ 6.00000 0.295958
$$412$$ 8.00000 0.394132
$$413$$ 12.0000 0.590481
$$414$$ 0 0
$$415$$ 24.0000 1.17811
$$416$$ 10.0000 0.490290
$$417$$ −12.0000 −0.587643
$$418$$ 0 0
$$419$$ −12.0000 −0.586238 −0.293119 0.956076i $$-0.594693\pi$$
−0.293119 + 0.956076i $$0.594693\pi$$
$$420$$ 2.00000 0.0975900
$$421$$ −38.0000 −1.85201 −0.926003 0.377515i $$-0.876779\pi$$
−0.926003 + 0.377515i $$0.876779\pi$$
$$422$$ −4.00000 −0.194717
$$423$$ 0 0
$$424$$ 18.0000 0.874157
$$425$$ 6.00000 0.291043
$$426$$ 0 0
$$427$$ 2.00000 0.0967868
$$428$$ 4.00000 0.193347
$$429$$ −8.00000 −0.386244
$$430$$ 8.00000 0.385794
$$431$$ 24.0000 1.15604 0.578020 0.816023i $$-0.303826\pi$$
0.578020 + 0.816023i $$0.303826\pi$$
$$432$$ 1.00000 0.0481125
$$433$$ 14.0000 0.672797 0.336399 0.941720i $$-0.390791\pi$$
0.336399 + 0.941720i $$0.390791\pi$$
$$434$$ 0 0
$$435$$ 4.00000 0.191785
$$436$$ −18.0000 −0.862044
$$437$$ 0 0
$$438$$ 6.00000 0.286691
$$439$$ 24.0000 1.14546 0.572729 0.819745i $$-0.305885\pi$$
0.572729 + 0.819745i $$0.305885\pi$$
$$440$$ 24.0000 1.14416
$$441$$ 1.00000 0.0476190
$$442$$ −12.0000 −0.570782
$$443$$ 36.0000 1.71041 0.855206 0.518289i $$-0.173431\pi$$
0.855206 + 0.518289i $$0.173431\pi$$
$$444$$ −6.00000 −0.284747
$$445$$ −28.0000 −1.32733
$$446$$ −16.0000 −0.757622
$$447$$ −6.00000 −0.283790
$$448$$ −7.00000 −0.330719
$$449$$ 30.0000 1.41579 0.707894 0.706319i $$-0.249646\pi$$
0.707894 + 0.706319i $$0.249646\pi$$
$$450$$ −1.00000 −0.0471405
$$451$$ −8.00000 −0.376705
$$452$$ −14.0000 −0.658505
$$453$$ 8.00000 0.375873
$$454$$ 12.0000 0.563188
$$455$$ 4.00000 0.187523
$$456$$ 0 0
$$457$$ 10.0000 0.467780 0.233890 0.972263i $$-0.424854\pi$$
0.233890 + 0.972263i $$0.424854\pi$$
$$458$$ −10.0000 −0.467269
$$459$$ 6.00000 0.280056
$$460$$ 0 0
$$461$$ −10.0000 −0.465746 −0.232873 0.972507i $$-0.574813\pi$$
−0.232873 + 0.972507i $$0.574813\pi$$
$$462$$ 4.00000 0.186097
$$463$$ 16.0000 0.743583 0.371792 0.928316i $$-0.378744\pi$$
0.371792 + 0.928316i $$0.378744\pi$$
$$464$$ −2.00000 −0.0928477
$$465$$ 0 0
$$466$$ −6.00000 −0.277945
$$467$$ 36.0000 1.66588 0.832941 0.553362i $$-0.186655\pi$$
0.832941 + 0.553362i $$0.186655\pi$$
$$468$$ −2.00000 −0.0924500
$$469$$ 4.00000 0.184703
$$470$$ 0 0
$$471$$ 2.00000 0.0921551
$$472$$ 36.0000 1.65703
$$473$$ −16.0000 −0.735681
$$474$$ −16.0000 −0.734904
$$475$$ 0 0
$$476$$ −6.00000 −0.275010
$$477$$ −6.00000 −0.274721
$$478$$ 24.0000 1.09773
$$479$$ −16.0000 −0.731059 −0.365529 0.930800i $$-0.619112\pi$$
−0.365529 + 0.930800i $$0.619112\pi$$
$$480$$ 10.0000 0.456435
$$481$$ −12.0000 −0.547153
$$482$$ −2.00000 −0.0910975
$$483$$ 0 0
$$484$$ −5.00000 −0.227273
$$485$$ 36.0000 1.63468
$$486$$ −1.00000 −0.0453609
$$487$$ 8.00000 0.362515 0.181257 0.983436i $$-0.441983\pi$$
0.181257 + 0.983436i $$0.441983\pi$$
$$488$$ 6.00000 0.271607
$$489$$ −4.00000 −0.180886
$$490$$ −2.00000 −0.0903508
$$491$$ 20.0000 0.902587 0.451294 0.892375i $$-0.350963\pi$$
0.451294 + 0.892375i $$0.350963\pi$$
$$492$$ −2.00000 −0.0901670
$$493$$ −12.0000 −0.540453
$$494$$ 0 0
$$495$$ −8.00000 −0.359573
$$496$$ 0 0
$$497$$ 0 0
$$498$$ 12.0000 0.537733
$$499$$ 4.00000 0.179065 0.0895323 0.995984i $$-0.471463\pi$$
0.0895323 + 0.995984i $$0.471463\pi$$
$$500$$ −12.0000 −0.536656
$$501$$ −8.00000 −0.357414
$$502$$ −20.0000 −0.892644
$$503$$ 24.0000 1.07011 0.535054 0.844818i $$-0.320291\pi$$
0.535054 + 0.844818i $$0.320291\pi$$
$$504$$ 3.00000 0.133631
$$505$$ −28.0000 −1.24598
$$506$$ 0 0
$$507$$ 9.00000 0.399704
$$508$$ 0 0
$$509$$ 10.0000 0.443242 0.221621 0.975133i $$-0.428865\pi$$
0.221621 + 0.975133i $$0.428865\pi$$
$$510$$ −12.0000 −0.531369
$$511$$ 6.00000 0.265424
$$512$$ −11.0000 −0.486136
$$513$$ 0 0
$$514$$ −26.0000 −1.14681
$$515$$ 16.0000 0.705044
$$516$$ −4.00000 −0.176090
$$517$$ 0 0
$$518$$ 6.00000 0.263625
$$519$$ −10.0000 −0.438951
$$520$$ 12.0000 0.526235
$$521$$ −18.0000 −0.788594 −0.394297 0.918983i $$-0.629012\pi$$
−0.394297 + 0.918983i $$0.629012\pi$$
$$522$$ 2.00000 0.0875376
$$523$$ 20.0000 0.874539 0.437269 0.899331i $$-0.355946\pi$$
0.437269 + 0.899331i $$0.355946\pi$$
$$524$$ −4.00000 −0.174741
$$525$$ −1.00000 −0.0436436
$$526$$ 16.0000 0.697633
$$527$$ 0 0
$$528$$ 4.00000 0.174078
$$529$$ −23.0000 −1.00000
$$530$$ 12.0000 0.521247
$$531$$ −12.0000 −0.520756
$$532$$ 0 0
$$533$$ −4.00000 −0.173259
$$534$$ −14.0000 −0.605839
$$535$$ 8.00000 0.345870
$$536$$ 12.0000 0.518321
$$537$$ −4.00000 −0.172613
$$538$$ −6.00000 −0.258678
$$539$$ 4.00000 0.172292
$$540$$ −2.00000 −0.0860663
$$541$$ −34.0000 −1.46177 −0.730887 0.682498i $$-0.760893\pi$$
−0.730887 + 0.682498i $$0.760893\pi$$
$$542$$ 16.0000 0.687259
$$543$$ −26.0000 −1.11577
$$544$$ −30.0000 −1.28624
$$545$$ −36.0000 −1.54207
$$546$$ 2.00000 0.0855921
$$547$$ −4.00000 −0.171028 −0.0855138 0.996337i $$-0.527253\pi$$
−0.0855138 + 0.996337i $$0.527253\pi$$
$$548$$ 6.00000 0.256307
$$549$$ −2.00000 −0.0853579
$$550$$ −4.00000 −0.170561
$$551$$ 0 0
$$552$$ 0 0
$$553$$ −16.0000 −0.680389
$$554$$ 22.0000 0.934690
$$555$$ −12.0000 −0.509372
$$556$$ −12.0000 −0.508913
$$557$$ −2.00000 −0.0847427 −0.0423714 0.999102i $$-0.513491\pi$$
−0.0423714 + 0.999102i $$0.513491\pi$$
$$558$$ 0 0
$$559$$ −8.00000 −0.338364
$$560$$ −2.00000 −0.0845154
$$561$$ 24.0000 1.01328
$$562$$ 22.0000 0.928014
$$563$$ −4.00000 −0.168580 −0.0842900 0.996441i $$-0.526862\pi$$
−0.0842900 + 0.996441i $$0.526862\pi$$
$$564$$ 0 0
$$565$$ −28.0000 −1.17797
$$566$$ −20.0000 −0.840663
$$567$$ −1.00000 −0.0419961
$$568$$ 0 0
$$569$$ −10.0000 −0.419222 −0.209611 0.977785i $$-0.567220\pi$$
−0.209611 + 0.977785i $$0.567220\pi$$
$$570$$ 0 0
$$571$$ −4.00000 −0.167395 −0.0836974 0.996491i $$-0.526673\pi$$
−0.0836974 + 0.996491i $$0.526673\pi$$
$$572$$ −8.00000 −0.334497
$$573$$ 8.00000 0.334205
$$574$$ 2.00000 0.0834784
$$575$$ 0 0
$$576$$ 7.00000 0.291667
$$577$$ 34.0000 1.41544 0.707719 0.706494i $$-0.249724\pi$$
0.707719 + 0.706494i $$0.249724\pi$$
$$578$$ 19.0000 0.790296
$$579$$ 2.00000 0.0831172
$$580$$ 4.00000 0.166091
$$581$$ 12.0000 0.497844
$$582$$ 18.0000 0.746124
$$583$$ −24.0000 −0.993978
$$584$$ 18.0000 0.744845
$$585$$ −4.00000 −0.165380
$$586$$ −14.0000 −0.578335
$$587$$ 28.0000 1.15568 0.577842 0.816149i $$-0.303895\pi$$
0.577842 + 0.816149i $$0.303895\pi$$
$$588$$ 1.00000 0.0412393
$$589$$ 0 0
$$590$$ 24.0000 0.988064
$$591$$ −22.0000 −0.904959
$$592$$ 6.00000 0.246598
$$593$$ −6.00000 −0.246390 −0.123195 0.992382i $$-0.539314\pi$$
−0.123195 + 0.992382i $$0.539314\pi$$
$$594$$ −4.00000 −0.164122
$$595$$ −12.0000 −0.491952
$$596$$ −6.00000 −0.245770
$$597$$ −24.0000 −0.982255
$$598$$ 0 0
$$599$$ −48.0000 −1.96123 −0.980613 0.195952i $$-0.937220\pi$$
−0.980613 + 0.195952i $$0.937220\pi$$
$$600$$ −3.00000 −0.122474
$$601$$ 6.00000 0.244745 0.122373 0.992484i $$-0.460950\pi$$
0.122373 + 0.992484i $$0.460950\pi$$
$$602$$ 4.00000 0.163028
$$603$$ −4.00000 −0.162893
$$604$$ 8.00000 0.325515
$$605$$ −10.0000 −0.406558
$$606$$ −14.0000 −0.568711
$$607$$ 16.0000 0.649420 0.324710 0.945814i $$-0.394733\pi$$
0.324710 + 0.945814i $$0.394733\pi$$
$$608$$ 0 0
$$609$$ 2.00000 0.0810441
$$610$$ 4.00000 0.161955
$$611$$ 0 0
$$612$$ 6.00000 0.242536
$$613$$ −26.0000 −1.05013 −0.525065 0.851062i $$-0.675959\pi$$
−0.525065 + 0.851062i $$0.675959\pi$$
$$614$$ −4.00000 −0.161427
$$615$$ −4.00000 −0.161296
$$616$$ 12.0000 0.483494
$$617$$ −6.00000 −0.241551 −0.120775 0.992680i $$-0.538538\pi$$
−0.120775 + 0.992680i $$0.538538\pi$$
$$618$$ 8.00000 0.321807
$$619$$ −20.0000 −0.803868 −0.401934 0.915669i $$-0.631662\pi$$
−0.401934 + 0.915669i $$0.631662\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ −24.0000 −0.962312
$$623$$ −14.0000 −0.560898
$$624$$ 2.00000 0.0800641
$$625$$ −19.0000 −0.760000
$$626$$ 26.0000 1.03917
$$627$$ 0 0
$$628$$ 2.00000 0.0798087
$$629$$ 36.0000 1.43541
$$630$$ 2.00000 0.0796819
$$631$$ −40.0000 −1.59237 −0.796187 0.605050i $$-0.793153\pi$$
−0.796187 + 0.605050i $$0.793153\pi$$
$$632$$ −48.0000 −1.90934
$$633$$ 4.00000 0.158986
$$634$$ 18.0000 0.714871
$$635$$ 0 0
$$636$$ −6.00000 −0.237915
$$637$$ 2.00000 0.0792429
$$638$$ 8.00000 0.316723
$$639$$ 0 0
$$640$$ 6.00000 0.237171
$$641$$ −18.0000 −0.710957 −0.355479 0.934684i $$-0.615682\pi$$
−0.355479 + 0.934684i $$0.615682\pi$$
$$642$$ 4.00000 0.157867
$$643$$ 20.0000 0.788723 0.394362 0.918955i $$-0.370966\pi$$
0.394362 + 0.918955i $$0.370966\pi$$
$$644$$ 0 0
$$645$$ −8.00000 −0.315000
$$646$$ 0 0
$$647$$ −40.0000 −1.57256 −0.786281 0.617869i $$-0.787996\pi$$
−0.786281 + 0.617869i $$0.787996\pi$$
$$648$$ −3.00000 −0.117851
$$649$$ −48.0000 −1.88416
$$650$$ −2.00000 −0.0784465
$$651$$ 0 0
$$652$$ −4.00000 −0.156652
$$653$$ −18.0000 −0.704394 −0.352197 0.935926i $$-0.614565\pi$$
−0.352197 + 0.935926i $$0.614565\pi$$
$$654$$ −18.0000 −0.703856
$$655$$ −8.00000 −0.312586
$$656$$ 2.00000 0.0780869
$$657$$ −6.00000 −0.234082
$$658$$ 0 0
$$659$$ −12.0000 −0.467454 −0.233727 0.972302i $$-0.575092\pi$$
−0.233727 + 0.972302i $$0.575092\pi$$
$$660$$ −8.00000 −0.311400
$$661$$ −22.0000 −0.855701 −0.427850 0.903850i $$-0.640729\pi$$
−0.427850 + 0.903850i $$0.640729\pi$$
$$662$$ 4.00000 0.155464
$$663$$ 12.0000 0.466041
$$664$$ 36.0000 1.39707
$$665$$ 0 0
$$666$$ −6.00000 −0.232495
$$667$$ 0 0
$$668$$ −8.00000 −0.309529
$$669$$ 16.0000 0.618596
$$670$$ 8.00000 0.309067
$$671$$ −8.00000 −0.308837
$$672$$ 5.00000 0.192879
$$673$$ −34.0000 −1.31060 −0.655302 0.755367i $$-0.727459\pi$$
−0.655302 + 0.755367i $$0.727459\pi$$
$$674$$ 14.0000 0.539260
$$675$$ 1.00000 0.0384900
$$676$$ 9.00000 0.346154
$$677$$ 18.0000 0.691796 0.345898 0.938272i $$-0.387574\pi$$
0.345898 + 0.938272i $$0.387574\pi$$
$$678$$ −14.0000 −0.537667
$$679$$ 18.0000 0.690777
$$680$$ −36.0000 −1.38054
$$681$$ −12.0000 −0.459841
$$682$$ 0 0
$$683$$ 12.0000 0.459167 0.229584 0.973289i $$-0.426264\pi$$
0.229584 + 0.973289i $$0.426264\pi$$
$$684$$ 0 0
$$685$$ 12.0000 0.458496
$$686$$ −1.00000 −0.0381802
$$687$$ 10.0000 0.381524
$$688$$ 4.00000 0.152499
$$689$$ −12.0000 −0.457164
$$690$$ 0 0
$$691$$ 20.0000 0.760836 0.380418 0.924815i $$-0.375780\pi$$
0.380418 + 0.924815i $$0.375780\pi$$
$$692$$ −10.0000 −0.380143
$$693$$ −4.00000 −0.151947
$$694$$ −28.0000 −1.06287
$$695$$ −24.0000 −0.910372
$$696$$ 6.00000 0.227429
$$697$$ 12.0000 0.454532
$$698$$ −2.00000 −0.0757011
$$699$$ 6.00000 0.226941
$$700$$ −1.00000 −0.0377964
$$701$$ 30.0000 1.13308 0.566542 0.824033i $$-0.308281\pi$$
0.566542 + 0.824033i $$0.308281\pi$$
$$702$$ −2.00000 −0.0754851
$$703$$ 0 0
$$704$$ 28.0000 1.05529
$$705$$ 0 0
$$706$$ 10.0000 0.376355
$$707$$ −14.0000 −0.526524
$$708$$ −12.0000 −0.450988
$$709$$ 6.00000 0.225335 0.112667 0.993633i $$-0.464061\pi$$
0.112667 + 0.993633i $$0.464061\pi$$
$$710$$ 0 0
$$711$$ 16.0000 0.600047
$$712$$ −42.0000 −1.57402
$$713$$ 0 0
$$714$$ −6.00000 −0.224544
$$715$$ −16.0000 −0.598366
$$716$$ −4.00000 −0.149487
$$717$$ −24.0000 −0.896296
$$718$$ 32.0000 1.19423
$$719$$ −48.0000 −1.79010 −0.895049 0.445968i $$-0.852860\pi$$
−0.895049 + 0.445968i $$0.852860\pi$$
$$720$$ 2.00000 0.0745356
$$721$$ 8.00000 0.297936
$$722$$ 0 0
$$723$$ 2.00000 0.0743808
$$724$$ −26.0000 −0.966282
$$725$$ −2.00000 −0.0742781
$$726$$ −5.00000 −0.185567
$$727$$ −40.0000 −1.48352 −0.741759 0.670667i $$-0.766008\pi$$
−0.741759 + 0.670667i $$0.766008\pi$$
$$728$$ 6.00000 0.222375
$$729$$ 1.00000 0.0370370
$$730$$ 12.0000 0.444140
$$731$$ 24.0000 0.887672
$$732$$ −2.00000 −0.0739221
$$733$$ −18.0000 −0.664845 −0.332423 0.943131i $$-0.607866\pi$$
−0.332423 + 0.943131i $$0.607866\pi$$
$$734$$ 0 0
$$735$$ 2.00000 0.0737711
$$736$$ 0 0
$$737$$ −16.0000 −0.589368
$$738$$ −2.00000 −0.0736210
$$739$$ 36.0000 1.32428 0.662141 0.749380i $$-0.269648\pi$$
0.662141 + 0.749380i $$0.269648\pi$$
$$740$$ −12.0000 −0.441129
$$741$$ 0 0
$$742$$ 6.00000 0.220267
$$743$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$744$$ 0 0
$$745$$ −12.0000 −0.439646
$$746$$ 10.0000 0.366126
$$747$$ −12.0000 −0.439057
$$748$$ 24.0000 0.877527
$$749$$ 4.00000 0.146157
$$750$$ −12.0000 −0.438178
$$751$$ 32.0000 1.16770 0.583848 0.811863i $$-0.301546\pi$$
0.583848 + 0.811863i $$0.301546\pi$$
$$752$$ 0 0
$$753$$ 20.0000 0.728841
$$754$$ 4.00000 0.145671
$$755$$ 16.0000 0.582300
$$756$$ −1.00000 −0.0363696
$$757$$ −10.0000 −0.363456 −0.181728 0.983349i $$-0.558169\pi$$
−0.181728 + 0.983349i $$0.558169\pi$$
$$758$$ −12.0000 −0.435860
$$759$$ 0 0
$$760$$ 0 0
$$761$$ 18.0000 0.652499 0.326250 0.945284i $$-0.394215\pi$$
0.326250 + 0.945284i $$0.394215\pi$$
$$762$$ 0 0
$$763$$ −18.0000 −0.651644
$$764$$ 8.00000 0.289430
$$765$$ 12.0000 0.433861
$$766$$ 0 0
$$767$$ −24.0000 −0.866590
$$768$$ 17.0000 0.613435
$$769$$ 2.00000 0.0721218 0.0360609 0.999350i $$-0.488519\pi$$
0.0360609 + 0.999350i $$0.488519\pi$$
$$770$$ 8.00000 0.288300
$$771$$ 26.0000 0.936367
$$772$$ 2.00000 0.0719816
$$773$$ −14.0000 −0.503545 −0.251773 0.967786i $$-0.581013\pi$$
−0.251773 + 0.967786i $$0.581013\pi$$
$$774$$ −4.00000 −0.143777
$$775$$ 0 0
$$776$$ 54.0000 1.93849
$$777$$ −6.00000 −0.215249
$$778$$ 6.00000 0.215110
$$779$$ 0 0
$$780$$ −4.00000 −0.143223
$$781$$ 0 0
$$782$$ 0 0
$$783$$ −2.00000 −0.0714742
$$784$$ −1.00000 −0.0357143
$$785$$ 4.00000 0.142766
$$786$$ −4.00000 −0.142675
$$787$$ 44.0000 1.56843 0.784215 0.620489i $$-0.213066\pi$$
0.784215 + 0.620489i $$0.213066\pi$$
$$788$$ −22.0000 −0.783718
$$789$$ −16.0000 −0.569615
$$790$$ −32.0000 −1.13851
$$791$$ −14.0000 −0.497783
$$792$$ −12.0000 −0.426401
$$793$$ −4.00000 −0.142044
$$794$$ −18.0000 −0.638796
$$795$$ −12.0000 −0.425596
$$796$$ −24.0000 −0.850657
$$797$$ 26.0000 0.920967 0.460484 0.887668i $$-0.347676\pi$$
0.460484 + 0.887668i $$0.347676\pi$$
$$798$$ 0 0
$$799$$ 0 0
$$800$$ −5.00000 −0.176777
$$801$$ 14.0000 0.494666
$$802$$ 30.0000 1.05934
$$803$$ −24.0000 −0.846942
$$804$$ −4.00000 −0.141069
$$805$$ 0 0
$$806$$ 0 0
$$807$$ 6.00000 0.211210
$$808$$ −42.0000 −1.47755
$$809$$ 42.0000 1.47664 0.738321 0.674450i $$-0.235619\pi$$
0.738321 + 0.674450i $$0.235619\pi$$
$$810$$ −2.00000 −0.0702728
$$811$$ −44.0000 −1.54505 −0.772524 0.634985i $$-0.781006\pi$$
−0.772524 + 0.634985i $$0.781006\pi$$
$$812$$ 2.00000 0.0701862
$$813$$ −16.0000 −0.561144
$$814$$ −24.0000 −0.841200
$$815$$ −8.00000 −0.280228
$$816$$ −6.00000 −0.210042
$$817$$ 0 0
$$818$$ 22.0000 0.769212
$$819$$ −2.00000 −0.0698857
$$820$$ −4.00000 −0.139686
$$821$$ 38.0000 1.32621 0.663105 0.748527i $$-0.269238\pi$$
0.663105 + 0.748527i $$0.269238\pi$$
$$822$$ 6.00000 0.209274
$$823$$ 24.0000 0.836587 0.418294 0.908312i $$-0.362628\pi$$
0.418294 + 0.908312i $$0.362628\pi$$
$$824$$ 24.0000 0.836080
$$825$$ 4.00000 0.139262
$$826$$ 12.0000 0.417533
$$827$$ 12.0000 0.417281 0.208640 0.977992i $$-0.433096\pi$$
0.208640 + 0.977992i $$0.433096\pi$$
$$828$$ 0 0
$$829$$ −14.0000 −0.486240 −0.243120 0.969996i $$-0.578171\pi$$
−0.243120 + 0.969996i $$0.578171\pi$$
$$830$$ 24.0000 0.833052
$$831$$ −22.0000 −0.763172
$$832$$ 14.0000 0.485363
$$833$$ −6.00000 −0.207888
$$834$$ −12.0000 −0.415526
$$835$$ −16.0000 −0.553703
$$836$$ 0 0
$$837$$ 0 0
$$838$$ −12.0000 −0.414533
$$839$$ 8.00000 0.276191 0.138095 0.990419i $$-0.455902\pi$$
0.138095 + 0.990419i $$0.455902\pi$$
$$840$$ 6.00000 0.207020
$$841$$ −25.0000 −0.862069
$$842$$ −38.0000 −1.30957
$$843$$ −22.0000 −0.757720
$$844$$ 4.00000 0.137686
$$845$$ 18.0000 0.619219
$$846$$ 0 0
$$847$$ −5.00000 −0.171802
$$848$$ 6.00000 0.206041
$$849$$ 20.0000 0.686398
$$850$$ 6.00000 0.205798
$$851$$ 0 0
$$852$$ 0 0
$$853$$ −10.0000 −0.342393 −0.171197 0.985237i $$-0.554763\pi$$
−0.171197 + 0.985237i $$0.554763\pi$$
$$854$$ 2.00000 0.0684386
$$855$$ 0 0
$$856$$ 12.0000 0.410152
$$857$$ 14.0000 0.478231 0.239115 0.970991i $$-0.423143\pi$$
0.239115 + 0.970991i $$0.423143\pi$$
$$858$$ −8.00000 −0.273115
$$859$$ 44.0000 1.50126 0.750630 0.660722i $$-0.229750\pi$$
0.750630 + 0.660722i $$0.229750\pi$$
$$860$$ −8.00000 −0.272798
$$861$$ −2.00000 −0.0681598
$$862$$ 24.0000 0.817443
$$863$$ 24.0000 0.816970 0.408485 0.912765i $$-0.366057\pi$$
0.408485 + 0.912765i $$0.366057\pi$$
$$864$$ −5.00000 −0.170103
$$865$$ −20.0000 −0.680020
$$866$$ 14.0000 0.475739
$$867$$ −19.0000 −0.645274
$$868$$ 0 0
$$869$$ 64.0000 2.17105
$$870$$ 4.00000 0.135613
$$871$$ −8.00000 −0.271070
$$872$$ −54.0000 −1.82867
$$873$$ −18.0000 −0.609208
$$874$$ 0 0
$$875$$ −12.0000 −0.405674
$$876$$ −6.00000 −0.202721
$$877$$ −46.0000 −1.55331 −0.776655 0.629926i $$-0.783085\pi$$
−0.776655 + 0.629926i $$0.783085\pi$$
$$878$$ 24.0000 0.809961
$$879$$ 14.0000 0.472208
$$880$$ 8.00000 0.269680
$$881$$ −6.00000 −0.202145 −0.101073 0.994879i $$-0.532227\pi$$
−0.101073 + 0.994879i $$0.532227\pi$$
$$882$$ 1.00000 0.0336718
$$883$$ −28.0000 −0.942275 −0.471138 0.882060i $$-0.656156\pi$$
−0.471138 + 0.882060i $$0.656156\pi$$
$$884$$ 12.0000 0.403604
$$885$$ −24.0000 −0.806751
$$886$$ 36.0000 1.20944
$$887$$ −8.00000 −0.268614 −0.134307 0.990940i $$-0.542881\pi$$
−0.134307 + 0.990940i $$0.542881\pi$$
$$888$$ −18.0000 −0.604040
$$889$$ 0 0
$$890$$ −28.0000 −0.938562
$$891$$ 4.00000 0.134005
$$892$$ 16.0000 0.535720
$$893$$ 0 0
$$894$$ −6.00000 −0.200670
$$895$$ −8.00000 −0.267411
$$896$$ 3.00000 0.100223
$$897$$ 0 0
$$898$$ 30.0000 1.00111
$$899$$ 0 0
$$900$$ 1.00000 0.0333333
$$901$$ 36.0000 1.19933
$$902$$ −8.00000 −0.266371
$$903$$ −4.00000 −0.133112
$$904$$ −42.0000 −1.39690
$$905$$ −52.0000 −1.72854
$$906$$ 8.00000 0.265782
$$907$$ 4.00000 0.132818 0.0664089 0.997792i $$-0.478846\pi$$
0.0664089 + 0.997792i $$0.478846\pi$$
$$908$$ −12.0000 −0.398234
$$909$$ 14.0000 0.464351
$$910$$ 4.00000 0.132599
$$911$$ 24.0000 0.795155 0.397578 0.917568i $$-0.369851\pi$$
0.397578 + 0.917568i $$0.369851\pi$$
$$912$$ 0 0
$$913$$ −48.0000 −1.58857
$$914$$ 10.0000 0.330771
$$915$$ −4.00000 −0.132236
$$916$$ 10.0000 0.330409
$$917$$ −4.00000 −0.132092
$$918$$ 6.00000 0.198030
$$919$$ 8.00000 0.263896 0.131948 0.991257i $$-0.457877\pi$$
0.131948 + 0.991257i $$0.457877\pi$$
$$920$$ 0 0
$$921$$ 4.00000 0.131804
$$922$$ −10.0000 −0.329332
$$923$$ 0 0
$$924$$ −4.00000 −0.131590
$$925$$ 6.00000 0.197279
$$926$$ 16.0000 0.525793
$$927$$ −8.00000 −0.262754
$$928$$ 10.0000 0.328266
$$929$$ 26.0000 0.853032 0.426516 0.904480i $$-0.359741\pi$$
0.426516 + 0.904480i $$0.359741\pi$$
$$930$$ 0 0
$$931$$ 0 0
$$932$$ 6.00000 0.196537
$$933$$ 24.0000 0.785725
$$934$$ 36.0000 1.17796
$$935$$ 48.0000 1.56977
$$936$$ −6.00000 −0.196116
$$937$$ 42.0000 1.37208 0.686040 0.727564i $$-0.259347\pi$$
0.686040 + 0.727564i $$0.259347\pi$$
$$938$$ 4.00000 0.130605
$$939$$ −26.0000 −0.848478
$$940$$ 0 0
$$941$$ −38.0000 −1.23876 −0.619382 0.785090i $$-0.712617\pi$$
−0.619382 + 0.785090i $$0.712617\pi$$
$$942$$ 2.00000 0.0651635
$$943$$ 0 0
$$944$$ 12.0000 0.390567
$$945$$ −2.00000 −0.0650600
$$946$$ −16.0000 −0.520205
$$947$$ 44.0000 1.42981 0.714904 0.699223i $$-0.246470\pi$$
0.714904 + 0.699223i $$0.246470\pi$$
$$948$$ 16.0000 0.519656
$$949$$ −12.0000 −0.389536
$$950$$ 0 0
$$951$$ −18.0000 −0.583690
$$952$$ −18.0000 −0.583383
$$953$$ −26.0000 −0.842223 −0.421111 0.907009i $$-0.638360\pi$$
−0.421111 + 0.907009i $$0.638360\pi$$
$$954$$ −6.00000 −0.194257
$$955$$ 16.0000 0.517748
$$956$$ −24.0000 −0.776215
$$957$$ −8.00000 −0.258603
$$958$$ −16.0000 −0.516937
$$959$$ 6.00000 0.193750
$$960$$ 14.0000 0.451848
$$961$$ −31.0000 −1.00000
$$962$$ −12.0000 −0.386896
$$963$$ −4.00000 −0.128898
$$964$$ 2.00000 0.0644157
$$965$$ 4.00000 0.128765
$$966$$ 0 0
$$967$$ 40.0000 1.28631 0.643157 0.765735i $$-0.277624\pi$$
0.643157 + 0.765735i $$0.277624\pi$$
$$968$$ −15.0000 −0.482118
$$969$$ 0 0
$$970$$ 36.0000 1.15589
$$971$$ −12.0000 −0.385098 −0.192549 0.981287i $$-0.561675\pi$$
−0.192549 + 0.981287i $$0.561675\pi$$
$$972$$ 1.00000 0.0320750
$$973$$ −12.0000 −0.384702
$$974$$ 8.00000 0.256337
$$975$$ 2.00000 0.0640513
$$976$$ 2.00000 0.0640184
$$977$$ 30.0000 0.959785 0.479893 0.877327i $$-0.340676\pi$$
0.479893 + 0.877327i $$0.340676\pi$$
$$978$$ −4.00000 −0.127906
$$979$$ 56.0000 1.78977
$$980$$ 2.00000 0.0638877
$$981$$ 18.0000 0.574696
$$982$$ 20.0000 0.638226
$$983$$ −24.0000 −0.765481 −0.382741 0.923856i $$-0.625020\pi$$
−0.382741 + 0.923856i $$0.625020\pi$$
$$984$$ −6.00000 −0.191273
$$985$$ −44.0000 −1.40196
$$986$$ −12.0000 −0.382158
$$987$$ 0 0
$$988$$ 0 0
$$989$$ 0 0
$$990$$ −8.00000 −0.254257
$$991$$ 16.0000 0.508257 0.254128 0.967170i $$-0.418211\pi$$
0.254128 + 0.967170i $$0.418211\pi$$
$$992$$ 0 0
$$993$$ −4.00000 −0.126936
$$994$$ 0 0
$$995$$ −48.0000 −1.52170
$$996$$ −12.0000 −0.380235
$$997$$ −26.0000 −0.823428 −0.411714 0.911313i $$-0.635070\pi$$
−0.411714 + 0.911313i $$0.635070\pi$$
$$998$$ 4.00000 0.126618
$$999$$ 6.00000 0.189832
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 7581.2.a.d.1.1 1
19.18 odd 2 21.2.a.a.1.1 1
57.56 even 2 63.2.a.a.1.1 1
76.75 even 2 336.2.a.a.1.1 1
95.18 even 4 525.2.d.a.274.2 2
95.37 even 4 525.2.d.a.274.1 2
95.94 odd 2 525.2.a.d.1.1 1
133.18 odd 6 147.2.e.b.79.1 2
133.37 odd 6 147.2.e.b.67.1 2
133.75 even 6 147.2.e.c.67.1 2
133.94 even 6 147.2.e.c.79.1 2
133.132 even 2 147.2.a.a.1.1 1
152.37 odd 2 1344.2.a.g.1.1 1
152.75 even 2 1344.2.a.s.1.1 1
171.56 even 6 567.2.f.b.190.1 2
171.94 odd 6 567.2.f.g.379.1 2
171.113 even 6 567.2.f.b.379.1 2
171.151 odd 6 567.2.f.g.190.1 2
209.208 even 2 2541.2.a.j.1.1 1
228.227 odd 2 1008.2.a.l.1.1 1
247.246 odd 2 3549.2.a.c.1.1 1
285.113 odd 4 1575.2.d.a.1324.1 2
285.227 odd 4 1575.2.d.a.1324.2 2
285.284 even 2 1575.2.a.c.1.1 1
304.37 odd 4 5376.2.c.r.2689.1 2
304.75 even 4 5376.2.c.l.2689.2 2
304.189 odd 4 5376.2.c.r.2689.2 2
304.227 even 4 5376.2.c.l.2689.1 2
323.322 odd 2 6069.2.a.b.1.1 1
380.379 even 2 8400.2.a.bn.1.1 1
399.170 even 6 441.2.e.a.361.1 2
399.227 odd 6 441.2.e.b.226.1 2
399.284 even 6 441.2.e.a.226.1 2
399.341 odd 6 441.2.e.b.361.1 2
399.398 odd 2 441.2.a.f.1.1 1
456.227 odd 2 4032.2.a.k.1.1 1
456.341 even 2 4032.2.a.h.1.1 1
532.75 odd 6 2352.2.q.e.1537.1 2
532.151 even 6 2352.2.q.x.961.1 2
532.227 odd 6 2352.2.q.e.961.1 2
532.303 even 6 2352.2.q.x.1537.1 2
532.531 odd 2 2352.2.a.v.1.1 1
627.626 odd 2 7623.2.a.g.1.1 1
665.664 even 2 3675.2.a.n.1.1 1
1064.531 odd 2 9408.2.a.m.1.1 1
1064.797 even 2 9408.2.a.bv.1.1 1
1596.1595 even 2 7056.2.a.p.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
21.2.a.a.1.1 1 19.18 odd 2
63.2.a.a.1.1 1 57.56 even 2
147.2.a.a.1.1 1 133.132 even 2
147.2.e.b.67.1 2 133.37 odd 6
147.2.e.b.79.1 2 133.18 odd 6
147.2.e.c.67.1 2 133.75 even 6
147.2.e.c.79.1 2 133.94 even 6
336.2.a.a.1.1 1 76.75 even 2
441.2.a.f.1.1 1 399.398 odd 2
441.2.e.a.226.1 2 399.284 even 6
441.2.e.a.361.1 2 399.170 even 6
441.2.e.b.226.1 2 399.227 odd 6
441.2.e.b.361.1 2 399.341 odd 6
525.2.a.d.1.1 1 95.94 odd 2
525.2.d.a.274.1 2 95.37 even 4
525.2.d.a.274.2 2 95.18 even 4
567.2.f.b.190.1 2 171.56 even 6
567.2.f.b.379.1 2 171.113 even 6
567.2.f.g.190.1 2 171.151 odd 6
567.2.f.g.379.1 2 171.94 odd 6
1008.2.a.l.1.1 1 228.227 odd 2
1344.2.a.g.1.1 1 152.37 odd 2
1344.2.a.s.1.1 1 152.75 even 2
1575.2.a.c.1.1 1 285.284 even 2
1575.2.d.a.1324.1 2 285.113 odd 4
1575.2.d.a.1324.2 2 285.227 odd 4
2352.2.a.v.1.1 1 532.531 odd 2
2352.2.q.e.961.1 2 532.227 odd 6
2352.2.q.e.1537.1 2 532.75 odd 6
2352.2.q.x.961.1 2 532.151 even 6
2352.2.q.x.1537.1 2 532.303 even 6
2541.2.a.j.1.1 1 209.208 even 2
3549.2.a.c.1.1 1 247.246 odd 2
3675.2.a.n.1.1 1 665.664 even 2
4032.2.a.h.1.1 1 456.341 even 2
4032.2.a.k.1.1 1 456.227 odd 2
5376.2.c.l.2689.1 2 304.227 even 4
5376.2.c.l.2689.2 2 304.75 even 4
5376.2.c.r.2689.1 2 304.37 odd 4
5376.2.c.r.2689.2 2 304.189 odd 4
6069.2.a.b.1.1 1 323.322 odd 2
7056.2.a.p.1.1 1 1596.1595 even 2
7581.2.a.d.1.1 1 1.1 even 1 trivial
7623.2.a.g.1.1 1 627.626 odd 2
8400.2.a.bn.1.1 1 380.379 even 2
9408.2.a.m.1.1 1 1064.531 odd 2
9408.2.a.bv.1.1 1 1064.797 even 2