# Properties

 Label 342.2.a.b.1.1 Level $342$ Weight $2$ Character 342.1 Self dual yes Analytic conductor $2.731$ Analytic rank $0$ Dimension $1$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$342 = 2 \cdot 3^{2} \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 342.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$2.73088374913$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 114) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Character $$\chi$$ $$=$$ 342.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q-1.00000 q^{2} +1.00000 q^{4} -2.00000 q^{5} -1.00000 q^{8} +O(q^{10})$$ $$q-1.00000 q^{2} +1.00000 q^{4} -2.00000 q^{5} -1.00000 q^{8} +2.00000 q^{10} +4.00000 q^{11} +2.00000 q^{13} +1.00000 q^{16} +6.00000 q^{17} -1.00000 q^{19} -2.00000 q^{20} -4.00000 q^{22} +4.00000 q^{23} -1.00000 q^{25} -2.00000 q^{26} +2.00000 q^{29} +4.00000 q^{31} -1.00000 q^{32} -6.00000 q^{34} +10.0000 q^{37} +1.00000 q^{38} +2.00000 q^{40} -10.0000 q^{41} +4.00000 q^{43} +4.00000 q^{44} -4.00000 q^{46} +4.00000 q^{47} -7.00000 q^{49} +1.00000 q^{50} +2.00000 q^{52} +10.0000 q^{53} -8.00000 q^{55} -2.00000 q^{58} -12.0000 q^{59} +14.0000 q^{61} -4.00000 q^{62} +1.00000 q^{64} -4.00000 q^{65} -12.0000 q^{67} +6.00000 q^{68} -8.00000 q^{71} -6.00000 q^{73} -10.0000 q^{74} -1.00000 q^{76} -4.00000 q^{79} -2.00000 q^{80} +10.0000 q^{82} -12.0000 q^{83} -12.0000 q^{85} -4.00000 q^{86} -4.00000 q^{88} +6.00000 q^{89} +4.00000 q^{92} -4.00000 q^{94} +2.00000 q^{95} +10.0000 q^{97} +7.00000 q^{98} +O(q^{100})$$

## Coefficient data

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

Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ −1.00000 −0.707107
$$3$$ 0 0
$$4$$ 1.00000 0.500000
$$5$$ −2.00000 −0.894427 −0.447214 0.894427i $$-0.647584\pi$$
−0.447214 + 0.894427i $$0.647584\pi$$
$$6$$ 0 0
$$7$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$8$$ −1.00000 −0.353553
$$9$$ 0 0
$$10$$ 2.00000 0.632456
$$11$$ 4.00000 1.20605 0.603023 0.797724i $$-0.293963\pi$$
0.603023 + 0.797724i $$0.293963\pi$$
$$12$$ 0 0
$$13$$ 2.00000 0.554700 0.277350 0.960769i $$-0.410544\pi$$
0.277350 + 0.960769i $$0.410544\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 1.00000 0.250000
$$17$$ 6.00000 1.45521 0.727607 0.685994i $$-0.240633\pi$$
0.727607 + 0.685994i $$0.240633\pi$$
$$18$$ 0 0
$$19$$ −1.00000 −0.229416
$$20$$ −2.00000 −0.447214
$$21$$ 0 0
$$22$$ −4.00000 −0.852803
$$23$$ 4.00000 0.834058 0.417029 0.908893i $$-0.363071\pi$$
0.417029 + 0.908893i $$0.363071\pi$$
$$24$$ 0 0
$$25$$ −1.00000 −0.200000
$$26$$ −2.00000 −0.392232
$$27$$ 0 0
$$28$$ 0 0
$$29$$ 2.00000 0.371391 0.185695 0.982607i $$-0.440546\pi$$
0.185695 + 0.982607i $$0.440546\pi$$
$$30$$ 0 0
$$31$$ 4.00000 0.718421 0.359211 0.933257i $$-0.383046\pi$$
0.359211 + 0.933257i $$0.383046\pi$$
$$32$$ −1.00000 −0.176777
$$33$$ 0 0
$$34$$ −6.00000 −1.02899
$$35$$ 0 0
$$36$$ 0 0
$$37$$ 10.0000 1.64399 0.821995 0.569495i $$-0.192861\pi$$
0.821995 + 0.569495i $$0.192861\pi$$
$$38$$ 1.00000 0.162221
$$39$$ 0 0
$$40$$ 2.00000 0.316228
$$41$$ −10.0000 −1.56174 −0.780869 0.624695i $$-0.785223\pi$$
−0.780869 + 0.624695i $$0.785223\pi$$
$$42$$ 0 0
$$43$$ 4.00000 0.609994 0.304997 0.952353i $$-0.401344\pi$$
0.304997 + 0.952353i $$0.401344\pi$$
$$44$$ 4.00000 0.603023
$$45$$ 0 0
$$46$$ −4.00000 −0.589768
$$47$$ 4.00000 0.583460 0.291730 0.956501i $$-0.405769\pi$$
0.291730 + 0.956501i $$0.405769\pi$$
$$48$$ 0 0
$$49$$ −7.00000 −1.00000
$$50$$ 1.00000 0.141421
$$51$$ 0 0
$$52$$ 2.00000 0.277350
$$53$$ 10.0000 1.37361 0.686803 0.726844i $$-0.259014\pi$$
0.686803 + 0.726844i $$0.259014\pi$$
$$54$$ 0 0
$$55$$ −8.00000 −1.07872
$$56$$ 0 0
$$57$$ 0 0
$$58$$ −2.00000 −0.262613
$$59$$ −12.0000 −1.56227 −0.781133 0.624364i $$-0.785358\pi$$
−0.781133 + 0.624364i $$0.785358\pi$$
$$60$$ 0 0
$$61$$ 14.0000 1.79252 0.896258 0.443533i $$-0.146275\pi$$
0.896258 + 0.443533i $$0.146275\pi$$
$$62$$ −4.00000 −0.508001
$$63$$ 0 0
$$64$$ 1.00000 0.125000
$$65$$ −4.00000 −0.496139
$$66$$ 0 0
$$67$$ −12.0000 −1.46603 −0.733017 0.680211i $$-0.761888\pi$$
−0.733017 + 0.680211i $$0.761888\pi$$
$$68$$ 6.00000 0.727607
$$69$$ 0 0
$$70$$ 0 0
$$71$$ −8.00000 −0.949425 −0.474713 0.880141i $$-0.657448\pi$$
−0.474713 + 0.880141i $$0.657448\pi$$
$$72$$ 0 0
$$73$$ −6.00000 −0.702247 −0.351123 0.936329i $$-0.614200\pi$$
−0.351123 + 0.936329i $$0.614200\pi$$
$$74$$ −10.0000 −1.16248
$$75$$ 0 0
$$76$$ −1.00000 −0.114708
$$77$$ 0 0
$$78$$ 0 0
$$79$$ −4.00000 −0.450035 −0.225018 0.974355i $$-0.572244\pi$$
−0.225018 + 0.974355i $$0.572244\pi$$
$$80$$ −2.00000 −0.223607
$$81$$ 0 0
$$82$$ 10.0000 1.10432
$$83$$ −12.0000 −1.31717 −0.658586 0.752506i $$-0.728845\pi$$
−0.658586 + 0.752506i $$0.728845\pi$$
$$84$$ 0 0
$$85$$ −12.0000 −1.30158
$$86$$ −4.00000 −0.431331
$$87$$ 0 0
$$88$$ −4.00000 −0.426401
$$89$$ 6.00000 0.635999 0.317999 0.948091i $$-0.396989\pi$$
0.317999 + 0.948091i $$0.396989\pi$$
$$90$$ 0 0
$$91$$ 0 0
$$92$$ 4.00000 0.417029
$$93$$ 0 0
$$94$$ −4.00000 −0.412568
$$95$$ 2.00000 0.205196
$$96$$ 0 0
$$97$$ 10.0000 1.01535 0.507673 0.861550i $$-0.330506\pi$$
0.507673 + 0.861550i $$0.330506\pi$$
$$98$$ 7.00000 0.707107
$$99$$ 0 0
$$100$$ −1.00000 −0.100000
$$101$$ −2.00000 −0.199007 −0.0995037 0.995037i $$-0.531726\pi$$
−0.0995037 + 0.995037i $$0.531726\pi$$
$$102$$ 0 0
$$103$$ −12.0000 −1.18240 −0.591198 0.806527i $$-0.701345\pi$$
−0.591198 + 0.806527i $$0.701345\pi$$
$$104$$ −2.00000 −0.196116
$$105$$ 0 0
$$106$$ −10.0000 −0.971286
$$107$$ 4.00000 0.386695 0.193347 0.981130i $$-0.438066\pi$$
0.193347 + 0.981130i $$0.438066\pi$$
$$108$$ 0 0
$$109$$ −6.00000 −0.574696 −0.287348 0.957826i $$-0.592774\pi$$
−0.287348 + 0.957826i $$0.592774\pi$$
$$110$$ 8.00000 0.762770
$$111$$ 0 0
$$112$$ 0 0
$$113$$ −2.00000 −0.188144 −0.0940721 0.995565i $$-0.529988\pi$$
−0.0940721 + 0.995565i $$0.529988\pi$$
$$114$$ 0 0
$$115$$ −8.00000 −0.746004
$$116$$ 2.00000 0.185695
$$117$$ 0 0
$$118$$ 12.0000 1.10469
$$119$$ 0 0
$$120$$ 0 0
$$121$$ 5.00000 0.454545
$$122$$ −14.0000 −1.26750
$$123$$ 0 0
$$124$$ 4.00000 0.359211
$$125$$ 12.0000 1.07331
$$126$$ 0 0
$$127$$ −12.0000 −1.06483 −0.532414 0.846484i $$-0.678715\pi$$
−0.532414 + 0.846484i $$0.678715\pi$$
$$128$$ −1.00000 −0.0883883
$$129$$ 0 0
$$130$$ 4.00000 0.350823
$$131$$ −12.0000 −1.04844 −0.524222 0.851581i $$-0.675644\pi$$
−0.524222 + 0.851581i $$0.675644\pi$$
$$132$$ 0 0
$$133$$ 0 0
$$134$$ 12.0000 1.03664
$$135$$ 0 0
$$136$$ −6.00000 −0.514496
$$137$$ 14.0000 1.19610 0.598050 0.801459i $$-0.295942\pi$$
0.598050 + 0.801459i $$0.295942\pi$$
$$138$$ 0 0
$$139$$ 12.0000 1.01783 0.508913 0.860818i $$-0.330047\pi$$
0.508913 + 0.860818i $$0.330047\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ 8.00000 0.671345
$$143$$ 8.00000 0.668994
$$144$$ 0 0
$$145$$ −4.00000 −0.332182
$$146$$ 6.00000 0.496564
$$147$$ 0 0
$$148$$ 10.0000 0.821995
$$149$$ 6.00000 0.491539 0.245770 0.969328i $$-0.420959\pi$$
0.245770 + 0.969328i $$0.420959\pi$$
$$150$$ 0 0
$$151$$ 20.0000 1.62758 0.813788 0.581161i $$-0.197401\pi$$
0.813788 + 0.581161i $$0.197401\pi$$
$$152$$ 1.00000 0.0811107
$$153$$ 0 0
$$154$$ 0 0
$$155$$ −8.00000 −0.642575
$$156$$ 0 0
$$157$$ 22.0000 1.75579 0.877896 0.478852i $$-0.158947\pi$$
0.877896 + 0.478852i $$0.158947\pi$$
$$158$$ 4.00000 0.318223
$$159$$ 0 0
$$160$$ 2.00000 0.158114
$$161$$ 0 0
$$162$$ 0 0
$$163$$ 20.0000 1.56652 0.783260 0.621694i $$-0.213555\pi$$
0.783260 + 0.621694i $$0.213555\pi$$
$$164$$ −10.0000 −0.780869
$$165$$ 0 0
$$166$$ 12.0000 0.931381
$$167$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$168$$ 0 0
$$169$$ −9.00000 −0.692308
$$170$$ 12.0000 0.920358
$$171$$ 0 0
$$172$$ 4.00000 0.304997
$$173$$ −6.00000 −0.456172 −0.228086 0.973641i $$-0.573247\pi$$
−0.228086 + 0.973641i $$0.573247\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ 4.00000 0.301511
$$177$$ 0 0
$$178$$ −6.00000 −0.449719
$$179$$ −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 $$-0.674182\pi$$
−0.520306 + 0.853980i $$0.674182\pi$$
$$182$$ 0 0
$$183$$ 0 0
$$184$$ −4.00000 −0.294884
$$185$$ −20.0000 −1.47043
$$186$$ 0 0
$$187$$ 24.0000 1.75505
$$188$$ 4.00000 0.291730
$$189$$ 0 0
$$190$$ −2.00000 −0.145095
$$191$$ −4.00000 −0.289430 −0.144715 0.989473i $$-0.546227\pi$$
−0.144715 + 0.989473i $$0.546227\pi$$
$$192$$ 0 0
$$193$$ −6.00000 −0.431889 −0.215945 0.976406i $$-0.569283\pi$$
−0.215945 + 0.976406i $$0.569283\pi$$
$$194$$ −10.0000 −0.717958
$$195$$ 0 0
$$196$$ −7.00000 −0.500000
$$197$$ 22.0000 1.56744 0.783718 0.621117i $$-0.213321\pi$$
0.783718 + 0.621117i $$0.213321\pi$$
$$198$$ 0 0
$$199$$ −16.0000 −1.13421 −0.567105 0.823646i $$-0.691937\pi$$
−0.567105 + 0.823646i $$0.691937\pi$$
$$200$$ 1.00000 0.0707107
$$201$$ 0 0
$$202$$ 2.00000 0.140720
$$203$$ 0 0
$$204$$ 0 0
$$205$$ 20.0000 1.39686
$$206$$ 12.0000 0.836080
$$207$$ 0 0
$$208$$ 2.00000 0.138675
$$209$$ −4.00000 −0.276686
$$210$$ 0 0
$$211$$ 12.0000 0.826114 0.413057 0.910705i $$-0.364461\pi$$
0.413057 + 0.910705i $$0.364461\pi$$
$$212$$ 10.0000 0.686803
$$213$$ 0 0
$$214$$ −4.00000 −0.273434
$$215$$ −8.00000 −0.545595
$$216$$ 0 0
$$217$$ 0 0
$$218$$ 6.00000 0.406371
$$219$$ 0 0
$$220$$ −8.00000 −0.539360
$$221$$ 12.0000 0.807207
$$222$$ 0 0
$$223$$ −28.0000 −1.87502 −0.937509 0.347960i $$-0.886874\pi$$
−0.937509 + 0.347960i $$0.886874\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ 2.00000 0.133038
$$227$$ −28.0000 −1.85843 −0.929213 0.369546i $$-0.879513\pi$$
−0.929213 + 0.369546i $$0.879513\pi$$
$$228$$ 0 0
$$229$$ −10.0000 −0.660819 −0.330409 0.943838i $$-0.607187\pi$$
−0.330409 + 0.943838i $$0.607187\pi$$
$$230$$ 8.00000 0.527504
$$231$$ 0 0
$$232$$ −2.00000 −0.131306
$$233$$ 6.00000 0.393073 0.196537 0.980497i $$-0.437031\pi$$
0.196537 + 0.980497i $$0.437031\pi$$
$$234$$ 0 0
$$235$$ −8.00000 −0.521862
$$236$$ −12.0000 −0.781133
$$237$$ 0 0
$$238$$ 0 0
$$239$$ −12.0000 −0.776215 −0.388108 0.921614i $$-0.626871\pi$$
−0.388108 + 0.921614i $$0.626871\pi$$
$$240$$ 0 0
$$241$$ 10.0000 0.644157 0.322078 0.946713i $$-0.395619\pi$$
0.322078 + 0.946713i $$0.395619\pi$$
$$242$$ −5.00000 −0.321412
$$243$$ 0 0
$$244$$ 14.0000 0.896258
$$245$$ 14.0000 0.894427
$$246$$ 0 0
$$247$$ −2.00000 −0.127257
$$248$$ −4.00000 −0.254000
$$249$$ 0 0
$$250$$ −12.0000 −0.758947
$$251$$ 28.0000 1.76734 0.883672 0.468106i $$-0.155064\pi$$
0.883672 + 0.468106i $$0.155064\pi$$
$$252$$ 0 0
$$253$$ 16.0000 1.00591
$$254$$ 12.0000 0.752947
$$255$$ 0 0
$$256$$ 1.00000 0.0625000
$$257$$ −2.00000 −0.124757 −0.0623783 0.998053i $$-0.519869\pi$$
−0.0623783 + 0.998053i $$0.519869\pi$$
$$258$$ 0 0
$$259$$ 0 0
$$260$$ −4.00000 −0.248069
$$261$$ 0 0
$$262$$ 12.0000 0.741362
$$263$$ 12.0000 0.739952 0.369976 0.929041i $$-0.379366\pi$$
0.369976 + 0.929041i $$0.379366\pi$$
$$264$$ 0 0
$$265$$ −20.0000 −1.22859
$$266$$ 0 0
$$267$$ 0 0
$$268$$ −12.0000 −0.733017
$$269$$ −6.00000 −0.365826 −0.182913 0.983129i $$-0.558553\pi$$
−0.182913 + 0.983129i $$0.558553\pi$$
$$270$$ 0 0
$$271$$ −8.00000 −0.485965 −0.242983 0.970031i $$-0.578126\pi$$
−0.242983 + 0.970031i $$0.578126\pi$$
$$272$$ 6.00000 0.363803
$$273$$ 0 0
$$274$$ −14.0000 −0.845771
$$275$$ −4.00000 −0.241209
$$276$$ 0 0
$$277$$ −26.0000 −1.56219 −0.781094 0.624413i $$-0.785338\pi$$
−0.781094 + 0.624413i $$0.785338\pi$$
$$278$$ −12.0000 −0.719712
$$279$$ 0 0
$$280$$ 0 0
$$281$$ −10.0000 −0.596550 −0.298275 0.954480i $$-0.596411\pi$$
−0.298275 + 0.954480i $$0.596411\pi$$
$$282$$ 0 0
$$283$$ 12.0000 0.713326 0.356663 0.934233i $$-0.383914\pi$$
0.356663 + 0.934233i $$0.383914\pi$$
$$284$$ −8.00000 −0.474713
$$285$$ 0 0
$$286$$ −8.00000 −0.473050
$$287$$ 0 0
$$288$$ 0 0
$$289$$ 19.0000 1.11765
$$290$$ 4.00000 0.234888
$$291$$ 0 0
$$292$$ −6.00000 −0.351123
$$293$$ 18.0000 1.05157 0.525786 0.850617i $$-0.323771\pi$$
0.525786 + 0.850617i $$0.323771\pi$$
$$294$$ 0 0
$$295$$ 24.0000 1.39733
$$296$$ −10.0000 −0.581238
$$297$$ 0 0
$$298$$ −6.00000 −0.347571
$$299$$ 8.00000 0.462652
$$300$$ 0 0
$$301$$ 0 0
$$302$$ −20.0000 −1.15087
$$303$$ 0 0
$$304$$ −1.00000 −0.0573539
$$305$$ −28.0000 −1.60328
$$306$$ 0 0
$$307$$ −12.0000 −0.684876 −0.342438 0.939540i $$-0.611253\pi$$
−0.342438 + 0.939540i $$0.611253\pi$$
$$308$$ 0 0
$$309$$ 0 0
$$310$$ 8.00000 0.454369
$$311$$ −4.00000 −0.226819 −0.113410 0.993548i $$-0.536177\pi$$
−0.113410 + 0.993548i $$0.536177\pi$$
$$312$$ 0 0
$$313$$ −22.0000 −1.24351 −0.621757 0.783210i $$-0.713581\pi$$
−0.621757 + 0.783210i $$0.713581\pi$$
$$314$$ −22.0000 −1.24153
$$315$$ 0 0
$$316$$ −4.00000 −0.225018
$$317$$ −6.00000 −0.336994 −0.168497 0.985702i $$-0.553891\pi$$
−0.168497 + 0.985702i $$0.553891\pi$$
$$318$$ 0 0
$$319$$ 8.00000 0.447914
$$320$$ −2.00000 −0.111803
$$321$$ 0 0
$$322$$ 0 0
$$323$$ −6.00000 −0.333849
$$324$$ 0 0
$$325$$ −2.00000 −0.110940
$$326$$ −20.0000 −1.10770
$$327$$ 0 0
$$328$$ 10.0000 0.552158
$$329$$ 0 0
$$330$$ 0 0
$$331$$ −4.00000 −0.219860 −0.109930 0.993939i $$-0.535063\pi$$
−0.109930 + 0.993939i $$0.535063\pi$$
$$332$$ −12.0000 −0.658586
$$333$$ 0 0
$$334$$ 0 0
$$335$$ 24.0000 1.31126
$$336$$ 0 0
$$337$$ −14.0000 −0.762629 −0.381314 0.924445i $$-0.624528\pi$$
−0.381314 + 0.924445i $$0.624528\pi$$
$$338$$ 9.00000 0.489535
$$339$$ 0 0
$$340$$ −12.0000 −0.650791
$$341$$ 16.0000 0.866449
$$342$$ 0 0
$$343$$ 0 0
$$344$$ −4.00000 −0.215666
$$345$$ 0 0
$$346$$ 6.00000 0.322562
$$347$$ 20.0000 1.07366 0.536828 0.843692i $$-0.319622\pi$$
0.536828 + 0.843692i $$0.319622\pi$$
$$348$$ 0 0
$$349$$ −26.0000 −1.39175 −0.695874 0.718164i $$-0.744983\pi$$
−0.695874 + 0.718164i $$0.744983\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ −4.00000 −0.213201
$$353$$ −18.0000 −0.958043 −0.479022 0.877803i $$-0.659008\pi$$
−0.479022 + 0.877803i $$0.659008\pi$$
$$354$$ 0 0
$$355$$ 16.0000 0.849192
$$356$$ 6.00000 0.317999
$$357$$ 0 0
$$358$$ 12.0000 0.634220
$$359$$ −12.0000 −0.633336 −0.316668 0.948536i $$-0.602564\pi$$
−0.316668 + 0.948536i $$0.602564\pi$$
$$360$$ 0 0
$$361$$ 1.00000 0.0526316
$$362$$ 14.0000 0.735824
$$363$$ 0 0
$$364$$ 0 0
$$365$$ 12.0000 0.628109
$$366$$ 0 0
$$367$$ −8.00000 −0.417597 −0.208798 0.977959i $$-0.566955\pi$$
−0.208798 + 0.977959i $$0.566955\pi$$
$$368$$ 4.00000 0.208514
$$369$$ 0 0
$$370$$ 20.0000 1.03975
$$371$$ 0 0
$$372$$ 0 0
$$373$$ 26.0000 1.34623 0.673114 0.739538i $$-0.264956\pi$$
0.673114 + 0.739538i $$0.264956\pi$$
$$374$$ −24.0000 −1.24101
$$375$$ 0 0
$$376$$ −4.00000 −0.206284
$$377$$ 4.00000 0.206010
$$378$$ 0 0
$$379$$ −36.0000 −1.84920 −0.924598 0.380945i $$-0.875599\pi$$
−0.924598 + 0.380945i $$0.875599\pi$$
$$380$$ 2.00000 0.102598
$$381$$ 0 0
$$382$$ 4.00000 0.204658
$$383$$ −16.0000 −0.817562 −0.408781 0.912633i $$-0.634046\pi$$
−0.408781 + 0.912633i $$0.634046\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 6.00000 0.305392
$$387$$ 0 0
$$388$$ 10.0000 0.507673
$$389$$ −18.0000 −0.912636 −0.456318 0.889817i $$-0.650832\pi$$
−0.456318 + 0.889817i $$0.650832\pi$$
$$390$$ 0 0
$$391$$ 24.0000 1.21373
$$392$$ 7.00000 0.353553
$$393$$ 0 0
$$394$$ −22.0000 −1.10834
$$395$$ 8.00000 0.402524
$$396$$ 0 0
$$397$$ −10.0000 −0.501886 −0.250943 0.968002i $$-0.580741\pi$$
−0.250943 + 0.968002i $$0.580741\pi$$
$$398$$ 16.0000 0.802008
$$399$$ 0 0
$$400$$ −1.00000 −0.0500000
$$401$$ 14.0000 0.699127 0.349563 0.936913i $$-0.386330\pi$$
0.349563 + 0.936913i $$0.386330\pi$$
$$402$$ 0 0
$$403$$ 8.00000 0.398508
$$404$$ −2.00000 −0.0995037
$$405$$ 0 0
$$406$$ 0 0
$$407$$ 40.0000 1.98273
$$408$$ 0 0
$$409$$ −14.0000 −0.692255 −0.346128 0.938187i $$-0.612504\pi$$
−0.346128 + 0.938187i $$0.612504\pi$$
$$410$$ −20.0000 −0.987730
$$411$$ 0 0
$$412$$ −12.0000 −0.591198
$$413$$ 0 0
$$414$$ 0 0
$$415$$ 24.0000 1.17811
$$416$$ −2.00000 −0.0980581
$$417$$ 0 0
$$418$$ 4.00000 0.195646
$$419$$ 12.0000 0.586238 0.293119 0.956076i $$-0.405307\pi$$
0.293119 + 0.956076i $$0.405307\pi$$
$$420$$ 0 0
$$421$$ 26.0000 1.26716 0.633581 0.773676i $$-0.281584\pi$$
0.633581 + 0.773676i $$0.281584\pi$$
$$422$$ −12.0000 −0.584151
$$423$$ 0 0
$$424$$ −10.0000 −0.485643
$$425$$ −6.00000 −0.291043
$$426$$ 0 0
$$427$$ 0 0
$$428$$ 4.00000 0.193347
$$429$$ 0 0
$$430$$ 8.00000 0.385794
$$431$$ 24.0000 1.15604 0.578020 0.816023i $$-0.303826\pi$$
0.578020 + 0.816023i $$0.303826\pi$$
$$432$$ 0 0
$$433$$ 26.0000 1.24948 0.624740 0.780833i $$-0.285205\pi$$
0.624740 + 0.780833i $$0.285205\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ −6.00000 −0.287348
$$437$$ −4.00000 −0.191346
$$438$$ 0 0
$$439$$ −4.00000 −0.190910 −0.0954548 0.995434i $$-0.530431\pi$$
−0.0954548 + 0.995434i $$0.530431\pi$$
$$440$$ 8.00000 0.381385
$$441$$ 0 0
$$442$$ −12.0000 −0.570782
$$443$$ 20.0000 0.950229 0.475114 0.879924i $$-0.342407\pi$$
0.475114 + 0.879924i $$0.342407\pi$$
$$444$$ 0 0
$$445$$ −12.0000 −0.568855
$$446$$ 28.0000 1.32584
$$447$$ 0 0
$$448$$ 0 0
$$449$$ −34.0000 −1.60456 −0.802280 0.596948i $$-0.796380\pi$$
−0.802280 + 0.596948i $$0.796380\pi$$
$$450$$ 0 0
$$451$$ −40.0000 −1.88353
$$452$$ −2.00000 −0.0940721
$$453$$ 0 0
$$454$$ 28.0000 1.31411
$$455$$ 0 0
$$456$$ 0 0
$$457$$ 26.0000 1.21623 0.608114 0.793849i $$-0.291926\pi$$
0.608114 + 0.793849i $$0.291926\pi$$
$$458$$ 10.0000 0.467269
$$459$$ 0 0
$$460$$ −8.00000 −0.373002
$$461$$ −2.00000 −0.0931493 −0.0465746 0.998915i $$-0.514831\pi$$
−0.0465746 + 0.998915i $$0.514831\pi$$
$$462$$ 0 0
$$463$$ −8.00000 −0.371792 −0.185896 0.982569i $$-0.559519\pi$$
−0.185896 + 0.982569i $$0.559519\pi$$
$$464$$ 2.00000 0.0928477
$$465$$ 0 0
$$466$$ −6.00000 −0.277945
$$467$$ 20.0000 0.925490 0.462745 0.886492i $$-0.346865\pi$$
0.462745 + 0.886492i $$0.346865\pi$$
$$468$$ 0 0
$$469$$ 0 0
$$470$$ 8.00000 0.369012
$$471$$ 0 0
$$472$$ 12.0000 0.552345
$$473$$ 16.0000 0.735681
$$474$$ 0 0
$$475$$ 1.00000 0.0458831
$$476$$ 0 0
$$477$$ 0 0
$$478$$ 12.0000 0.548867
$$479$$ −4.00000 −0.182765 −0.0913823 0.995816i $$-0.529129\pi$$
−0.0913823 + 0.995816i $$0.529129\pi$$
$$480$$ 0 0
$$481$$ 20.0000 0.911922
$$482$$ −10.0000 −0.455488
$$483$$ 0 0
$$484$$ 5.00000 0.227273
$$485$$ −20.0000 −0.908153
$$486$$ 0 0
$$487$$ 4.00000 0.181257 0.0906287 0.995885i $$-0.471112\pi$$
0.0906287 + 0.995885i $$0.471112\pi$$
$$488$$ −14.0000 −0.633750
$$489$$ 0 0
$$490$$ −14.0000 −0.632456
$$491$$ −20.0000 −0.902587 −0.451294 0.892375i $$-0.649037\pi$$
−0.451294 + 0.892375i $$0.649037\pi$$
$$492$$ 0 0
$$493$$ 12.0000 0.540453
$$494$$ 2.00000 0.0899843
$$495$$ 0 0
$$496$$ 4.00000 0.179605
$$497$$ 0 0
$$498$$ 0 0
$$499$$ −20.0000 −0.895323 −0.447661 0.894203i $$-0.647743\pi$$
−0.447661 + 0.894203i $$0.647743\pi$$
$$500$$ 12.0000 0.536656
$$501$$ 0 0
$$502$$ −28.0000 −1.24970
$$503$$ −36.0000 −1.60516 −0.802580 0.596544i $$-0.796540\pi$$
−0.802580 + 0.596544i $$0.796540\pi$$
$$504$$ 0 0
$$505$$ 4.00000 0.177998
$$506$$ −16.0000 −0.711287
$$507$$ 0 0
$$508$$ −12.0000 −0.532414
$$509$$ −6.00000 −0.265945 −0.132973 0.991120i $$-0.542452\pi$$
−0.132973 + 0.991120i $$0.542452\pi$$
$$510$$ 0 0
$$511$$ 0 0
$$512$$ −1.00000 −0.0441942
$$513$$ 0 0
$$514$$ 2.00000 0.0882162
$$515$$ 24.0000 1.05757
$$516$$ 0 0
$$517$$ 16.0000 0.703679
$$518$$ 0 0
$$519$$ 0 0
$$520$$ 4.00000 0.175412
$$521$$ −26.0000 −1.13908 −0.569540 0.821963i $$-0.692879\pi$$
−0.569540 + 0.821963i $$0.692879\pi$$
$$522$$ 0 0
$$523$$ −4.00000 −0.174908 −0.0874539 0.996169i $$-0.527873\pi$$
−0.0874539 + 0.996169i $$0.527873\pi$$
$$524$$ −12.0000 −0.524222
$$525$$ 0 0
$$526$$ −12.0000 −0.523225
$$527$$ 24.0000 1.04546
$$528$$ 0 0
$$529$$ −7.00000 −0.304348
$$530$$ 20.0000 0.868744
$$531$$ 0 0
$$532$$ 0 0
$$533$$ −20.0000 −0.866296
$$534$$ 0 0
$$535$$ −8.00000 −0.345870
$$536$$ 12.0000 0.518321
$$537$$ 0 0
$$538$$ 6.00000 0.258678
$$539$$ −28.0000 −1.20605
$$540$$ 0 0
$$541$$ 38.0000 1.63375 0.816874 0.576816i $$-0.195705\pi$$
0.816874 + 0.576816i $$0.195705\pi$$
$$542$$ 8.00000 0.343629
$$543$$ 0 0
$$544$$ −6.00000 −0.257248
$$545$$ 12.0000 0.514024
$$546$$ 0 0
$$547$$ 4.00000 0.171028 0.0855138 0.996337i $$-0.472747\pi$$
0.0855138 + 0.996337i $$0.472747\pi$$
$$548$$ 14.0000 0.598050
$$549$$ 0 0
$$550$$ 4.00000 0.170561
$$551$$ −2.00000 −0.0852029
$$552$$ 0 0
$$553$$ 0 0
$$554$$ 26.0000 1.10463
$$555$$ 0 0
$$556$$ 12.0000 0.508913
$$557$$ −34.0000 −1.44063 −0.720313 0.693649i $$-0.756002\pi$$
−0.720313 + 0.693649i $$0.756002\pi$$
$$558$$ 0 0
$$559$$ 8.00000 0.338364
$$560$$ 0 0
$$561$$ 0 0
$$562$$ 10.0000 0.421825
$$563$$ 12.0000 0.505740 0.252870 0.967500i $$-0.418626\pi$$
0.252870 + 0.967500i $$0.418626\pi$$
$$564$$ 0 0
$$565$$ 4.00000 0.168281
$$566$$ −12.0000 −0.504398
$$567$$ 0 0
$$568$$ 8.00000 0.335673
$$569$$ 38.0000 1.59304 0.796521 0.604610i $$-0.206671\pi$$
0.796521 + 0.604610i $$0.206671\pi$$
$$570$$ 0 0
$$571$$ 20.0000 0.836974 0.418487 0.908223i $$-0.362561\pi$$
0.418487 + 0.908223i $$0.362561\pi$$
$$572$$ 8.00000 0.334497
$$573$$ 0 0
$$574$$ 0 0
$$575$$ −4.00000 −0.166812
$$576$$ 0 0
$$577$$ 2.00000 0.0832611 0.0416305 0.999133i $$-0.486745\pi$$
0.0416305 + 0.999133i $$0.486745\pi$$
$$578$$ −19.0000 −0.790296
$$579$$ 0 0
$$580$$ −4.00000 −0.166091
$$581$$ 0 0
$$582$$ 0 0
$$583$$ 40.0000 1.65663
$$584$$ 6.00000 0.248282
$$585$$ 0 0
$$586$$ −18.0000 −0.743573
$$587$$ −12.0000 −0.495293 −0.247647 0.968850i $$-0.579657\pi$$
−0.247647 + 0.968850i $$0.579657\pi$$
$$588$$ 0 0
$$589$$ −4.00000 −0.164817
$$590$$ −24.0000 −0.988064
$$591$$ 0 0
$$592$$ 10.0000 0.410997
$$593$$ −34.0000 −1.39621 −0.698106 0.715994i $$-0.745974\pi$$
−0.698106 + 0.715994i $$0.745974\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 6.00000 0.245770
$$597$$ 0 0
$$598$$ −8.00000 −0.327144
$$599$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$600$$ 0 0
$$601$$ 10.0000 0.407909 0.203954 0.978980i $$-0.434621\pi$$
0.203954 + 0.978980i $$0.434621\pi$$
$$602$$ 0 0
$$603$$ 0 0
$$604$$ 20.0000 0.813788
$$605$$ −10.0000 −0.406558
$$606$$ 0 0
$$607$$ −4.00000 −0.162355 −0.0811775 0.996700i $$-0.525868\pi$$
−0.0811775 + 0.996700i $$0.525868\pi$$
$$608$$ 1.00000 0.0405554
$$609$$ 0 0
$$610$$ 28.0000 1.13369
$$611$$ 8.00000 0.323645
$$612$$ 0 0
$$613$$ −2.00000 −0.0807792 −0.0403896 0.999184i $$-0.512860\pi$$
−0.0403896 + 0.999184i $$0.512860\pi$$
$$614$$ 12.0000 0.484281
$$615$$ 0 0
$$616$$ 0 0
$$617$$ −42.0000 −1.69086 −0.845428 0.534089i $$-0.820655\pi$$
−0.845428 + 0.534089i $$0.820655\pi$$
$$618$$ 0 0
$$619$$ 4.00000 0.160774 0.0803868 0.996764i $$-0.474384\pi$$
0.0803868 + 0.996764i $$0.474384\pi$$
$$620$$ −8.00000 −0.321288
$$621$$ 0 0
$$622$$ 4.00000 0.160385
$$623$$ 0 0
$$624$$ 0 0
$$625$$ −19.0000 −0.760000
$$626$$ 22.0000 0.879297
$$627$$ 0 0
$$628$$ 22.0000 0.877896
$$629$$ 60.0000 2.39236
$$630$$ 0 0
$$631$$ −40.0000 −1.59237 −0.796187 0.605050i $$-0.793153\pi$$
−0.796187 + 0.605050i $$0.793153\pi$$
$$632$$ 4.00000 0.159111
$$633$$ 0 0
$$634$$ 6.00000 0.238290
$$635$$ 24.0000 0.952411
$$636$$ 0 0
$$637$$ −14.0000 −0.554700
$$638$$ −8.00000 −0.316723
$$639$$ 0 0
$$640$$ 2.00000 0.0790569
$$641$$ −18.0000 −0.710957 −0.355479 0.934684i $$-0.615682\pi$$
−0.355479 + 0.934684i $$0.615682\pi$$
$$642$$ 0 0
$$643$$ 4.00000 0.157745 0.0788723 0.996885i $$-0.474868\pi$$
0.0788723 + 0.996885i $$0.474868\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 6.00000 0.236067
$$647$$ 4.00000 0.157256 0.0786281 0.996904i $$-0.474946\pi$$
0.0786281 + 0.996904i $$0.474946\pi$$
$$648$$ 0 0
$$649$$ −48.0000 −1.88416
$$650$$ 2.00000 0.0784465
$$651$$ 0 0
$$652$$ 20.0000 0.783260
$$653$$ −18.0000 −0.704394 −0.352197 0.935926i $$-0.614565\pi$$
−0.352197 + 0.935926i $$0.614565\pi$$
$$654$$ 0 0
$$655$$ 24.0000 0.937758
$$656$$ −10.0000 −0.390434
$$657$$ 0 0
$$658$$ 0 0
$$659$$ 4.00000 0.155818 0.0779089 0.996960i $$-0.475176\pi$$
0.0779089 + 0.996960i $$0.475176\pi$$
$$660$$ 0 0
$$661$$ −38.0000 −1.47803 −0.739014 0.673690i $$-0.764708\pi$$
−0.739014 + 0.673690i $$0.764708\pi$$
$$662$$ 4.00000 0.155464
$$663$$ 0 0
$$664$$ 12.0000 0.465690
$$665$$ 0 0
$$666$$ 0 0
$$667$$ 8.00000 0.309761
$$668$$ 0 0
$$669$$ 0 0
$$670$$ −24.0000 −0.927201
$$671$$ 56.0000 2.16186
$$672$$ 0 0
$$673$$ 26.0000 1.00223 0.501113 0.865382i $$-0.332924\pi$$
0.501113 + 0.865382i $$0.332924\pi$$
$$674$$ 14.0000 0.539260
$$675$$ 0 0
$$676$$ −9.00000 −0.346154
$$677$$ 26.0000 0.999261 0.499631 0.866239i $$-0.333469\pi$$
0.499631 + 0.866239i $$0.333469\pi$$
$$678$$ 0 0
$$679$$ 0 0
$$680$$ 12.0000 0.460179
$$681$$ 0 0
$$682$$ −16.0000 −0.612672
$$683$$ −4.00000 −0.153056 −0.0765279 0.997067i $$-0.524383\pi$$
−0.0765279 + 0.997067i $$0.524383\pi$$
$$684$$ 0 0
$$685$$ −28.0000 −1.06983
$$686$$ 0 0
$$687$$ 0 0
$$688$$ 4.00000 0.152499
$$689$$ 20.0000 0.761939
$$690$$ 0 0
$$691$$ −36.0000 −1.36950 −0.684752 0.728776i $$-0.740090\pi$$
−0.684752 + 0.728776i $$0.740090\pi$$
$$692$$ −6.00000 −0.228086
$$693$$ 0 0
$$694$$ −20.0000 −0.759190
$$695$$ −24.0000 −0.910372
$$696$$ 0 0
$$697$$ −60.0000 −2.27266
$$698$$ 26.0000 0.984115
$$699$$ 0 0
$$700$$ 0 0
$$701$$ 22.0000 0.830929 0.415464 0.909610i $$-0.363619\pi$$
0.415464 + 0.909610i $$0.363619\pi$$
$$702$$ 0 0
$$703$$ −10.0000 −0.377157
$$704$$ 4.00000 0.150756
$$705$$ 0 0
$$706$$ 18.0000 0.677439
$$707$$ 0 0
$$708$$ 0 0
$$709$$ −42.0000 −1.57734 −0.788672 0.614815i $$-0.789231\pi$$
−0.788672 + 0.614815i $$0.789231\pi$$
$$710$$ −16.0000 −0.600469
$$711$$ 0 0
$$712$$ −6.00000 −0.224860
$$713$$ 16.0000 0.599205
$$714$$ 0 0
$$715$$ −16.0000 −0.598366
$$716$$ −12.0000 −0.448461
$$717$$ 0 0
$$718$$ 12.0000 0.447836
$$719$$ −28.0000 −1.04422 −0.522112 0.852877i $$-0.674856\pi$$
−0.522112 + 0.852877i $$0.674856\pi$$
$$720$$ 0 0
$$721$$ 0 0
$$722$$ −1.00000 −0.0372161
$$723$$ 0 0
$$724$$ −14.0000 −0.520306
$$725$$ −2.00000 −0.0742781
$$726$$ 0 0
$$727$$ −8.00000 −0.296704 −0.148352 0.988935i $$-0.547397\pi$$
−0.148352 + 0.988935i $$0.547397\pi$$
$$728$$ 0 0
$$729$$ 0 0
$$730$$ −12.0000 −0.444140
$$731$$ 24.0000 0.887672
$$732$$ 0 0
$$733$$ 6.00000 0.221615 0.110808 0.993842i $$-0.464656\pi$$
0.110808 + 0.993842i $$0.464656\pi$$
$$734$$ 8.00000 0.295285
$$735$$ 0 0
$$736$$ −4.00000 −0.147442
$$737$$ −48.0000 −1.76810
$$738$$ 0 0
$$739$$ 20.0000 0.735712 0.367856 0.929883i $$-0.380092\pi$$
0.367856 + 0.929883i $$0.380092\pi$$
$$740$$ −20.0000 −0.735215
$$741$$ 0 0
$$742$$ 0 0
$$743$$ −32.0000 −1.17397 −0.586983 0.809599i $$-0.699684\pi$$
−0.586983 + 0.809599i $$0.699684\pi$$
$$744$$ 0 0
$$745$$ −12.0000 −0.439646
$$746$$ −26.0000 −0.951928
$$747$$ 0 0
$$748$$ 24.0000 0.877527
$$749$$ 0 0
$$750$$ 0 0
$$751$$ 44.0000 1.60558 0.802791 0.596260i $$-0.203347\pi$$
0.802791 + 0.596260i $$0.203347\pi$$
$$752$$ 4.00000 0.145865
$$753$$ 0 0
$$754$$ −4.00000 −0.145671
$$755$$ −40.0000 −1.45575
$$756$$ 0 0
$$757$$ −2.00000 −0.0726912 −0.0363456 0.999339i $$-0.511572\pi$$
−0.0363456 + 0.999339i $$0.511572\pi$$
$$758$$ 36.0000 1.30758
$$759$$ 0 0
$$760$$ −2.00000 −0.0725476
$$761$$ 30.0000 1.08750 0.543750 0.839248i $$-0.317004\pi$$
0.543750 + 0.839248i $$0.317004\pi$$
$$762$$ 0 0
$$763$$ 0 0
$$764$$ −4.00000 −0.144715
$$765$$ 0 0
$$766$$ 16.0000 0.578103
$$767$$ −24.0000 −0.866590
$$768$$ 0 0
$$769$$ −14.0000 −0.504853 −0.252426 0.967616i $$-0.581229\pi$$
−0.252426 + 0.967616i $$0.581229\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ −6.00000 −0.215945
$$773$$ −30.0000 −1.07903 −0.539513 0.841978i $$-0.681391\pi$$
−0.539513 + 0.841978i $$0.681391\pi$$
$$774$$ 0 0
$$775$$ −4.00000 −0.143684
$$776$$ −10.0000 −0.358979
$$777$$ 0 0
$$778$$ 18.0000 0.645331
$$779$$ 10.0000 0.358287
$$780$$ 0 0
$$781$$ −32.0000 −1.14505
$$782$$ −24.0000 −0.858238
$$783$$ 0 0
$$784$$ −7.00000 −0.250000
$$785$$ −44.0000 −1.57043
$$786$$ 0 0
$$787$$ −20.0000 −0.712923 −0.356462 0.934310i $$-0.616017\pi$$
−0.356462 + 0.934310i $$0.616017\pi$$
$$788$$ 22.0000 0.783718
$$789$$ 0 0
$$790$$ −8.00000 −0.284627
$$791$$ 0 0
$$792$$ 0 0
$$793$$ 28.0000 0.994309
$$794$$ 10.0000 0.354887
$$795$$ 0 0
$$796$$ −16.0000 −0.567105
$$797$$ −6.00000 −0.212531 −0.106265 0.994338i $$-0.533889\pi$$
−0.106265 + 0.994338i $$0.533889\pi$$
$$798$$ 0 0
$$799$$ 24.0000 0.849059
$$800$$ 1.00000 0.0353553
$$801$$ 0 0
$$802$$ −14.0000 −0.494357
$$803$$ −24.0000 −0.846942
$$804$$ 0 0
$$805$$ 0 0
$$806$$ −8.00000 −0.281788
$$807$$ 0 0
$$808$$ 2.00000 0.0703598
$$809$$ 30.0000 1.05474 0.527372 0.849635i $$-0.323177\pi$$
0.527372 + 0.849635i $$0.323177\pi$$
$$810$$ 0 0
$$811$$ 12.0000 0.421377 0.210688 0.977553i $$-0.432429\pi$$
0.210688 + 0.977553i $$0.432429\pi$$
$$812$$ 0 0
$$813$$ 0 0
$$814$$ −40.0000 −1.40200
$$815$$ −40.0000 −1.40114
$$816$$ 0 0
$$817$$ −4.00000 −0.139942
$$818$$ 14.0000 0.489499
$$819$$ 0 0
$$820$$ 20.0000 0.698430
$$821$$ 30.0000 1.04701 0.523504 0.852023i $$-0.324625\pi$$
0.523504 + 0.852023i $$0.324625\pi$$
$$822$$ 0 0
$$823$$ 8.00000 0.278862 0.139431 0.990232i $$-0.455473\pi$$
0.139431 + 0.990232i $$0.455473\pi$$
$$824$$ 12.0000 0.418040
$$825$$ 0 0
$$826$$ 0 0
$$827$$ 28.0000 0.973655 0.486828 0.873498i $$-0.338154\pi$$
0.486828 + 0.873498i $$0.338154\pi$$
$$828$$ 0 0
$$829$$ 42.0000 1.45872 0.729360 0.684130i $$-0.239818\pi$$
0.729360 + 0.684130i $$0.239818\pi$$
$$830$$ −24.0000 −0.833052
$$831$$ 0 0
$$832$$ 2.00000 0.0693375
$$833$$ −42.0000 −1.45521
$$834$$ 0 0
$$835$$ 0 0
$$836$$ −4.00000 −0.138343
$$837$$ 0 0
$$838$$ −12.0000 −0.414533
$$839$$ 16.0000 0.552381 0.276191 0.961103i $$-0.410928\pi$$
0.276191 + 0.961103i $$0.410928\pi$$
$$840$$ 0 0
$$841$$ −25.0000 −0.862069
$$842$$ −26.0000 −0.896019
$$843$$ 0 0
$$844$$ 12.0000 0.413057
$$845$$ 18.0000 0.619219
$$846$$ 0 0
$$847$$ 0 0
$$848$$ 10.0000 0.343401
$$849$$ 0 0
$$850$$ 6.00000 0.205798
$$851$$ 40.0000 1.37118
$$852$$ 0 0
$$853$$ 6.00000 0.205436 0.102718 0.994711i $$-0.467246\pi$$
0.102718 + 0.994711i $$0.467246\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ −4.00000 −0.136717
$$857$$ −42.0000 −1.43469 −0.717346 0.696717i $$-0.754643\pi$$
−0.717346 + 0.696717i $$0.754643\pi$$
$$858$$ 0 0
$$859$$ 52.0000 1.77422 0.887109 0.461561i $$-0.152710\pi$$
0.887109 + 0.461561i $$0.152710\pi$$
$$860$$ −8.00000 −0.272798
$$861$$ 0 0
$$862$$ −24.0000 −0.817443
$$863$$ 24.0000 0.816970 0.408485 0.912765i $$-0.366057\pi$$
0.408485 + 0.912765i $$0.366057\pi$$
$$864$$ 0 0
$$865$$ 12.0000 0.408012
$$866$$ −26.0000 −0.883516
$$867$$ 0 0
$$868$$ 0 0
$$869$$ −16.0000 −0.542763
$$870$$ 0 0
$$871$$ −24.0000 −0.813209
$$872$$ 6.00000 0.203186
$$873$$ 0 0
$$874$$ 4.00000 0.135302
$$875$$ 0 0
$$876$$ 0 0
$$877$$ 34.0000 1.14810 0.574049 0.818821i $$-0.305372\pi$$
0.574049 + 0.818821i $$0.305372\pi$$
$$878$$ 4.00000 0.134993
$$879$$ 0 0
$$880$$ −8.00000 −0.269680
$$881$$ −18.0000 −0.606435 −0.303218 0.952921i $$-0.598061\pi$$
−0.303218 + 0.952921i $$0.598061\pi$$
$$882$$ 0 0
$$883$$ −20.0000 −0.673054 −0.336527 0.941674i $$-0.609252\pi$$
−0.336527 + 0.941674i $$0.609252\pi$$
$$884$$ 12.0000 0.403604
$$885$$ 0 0
$$886$$ −20.0000 −0.671913
$$887$$ 32.0000 1.07445 0.537227 0.843437i $$-0.319472\pi$$
0.537227 + 0.843437i $$0.319472\pi$$
$$888$$ 0 0
$$889$$ 0 0
$$890$$ 12.0000 0.402241
$$891$$ 0 0
$$892$$ −28.0000 −0.937509
$$893$$ −4.00000 −0.133855
$$894$$ 0 0
$$895$$ 24.0000 0.802232
$$896$$ 0 0
$$897$$ 0 0
$$898$$ 34.0000 1.13459
$$899$$ 8.00000 0.266815
$$900$$ 0 0
$$901$$ 60.0000 1.99889
$$902$$ 40.0000 1.33185
$$903$$ 0 0
$$904$$ 2.00000 0.0665190
$$905$$ 28.0000 0.930751
$$906$$ 0 0
$$907$$ 44.0000 1.46100 0.730498 0.682915i $$-0.239288\pi$$
0.730498 + 0.682915i $$0.239288\pi$$
$$908$$ −28.0000 −0.929213
$$909$$ 0 0
$$910$$ 0 0
$$911$$ −24.0000 −0.795155 −0.397578 0.917568i $$-0.630149\pi$$
−0.397578 + 0.917568i $$0.630149\pi$$
$$912$$ 0 0
$$913$$ −48.0000 −1.58857
$$914$$ −26.0000 −0.860004
$$915$$ 0 0
$$916$$ −10.0000 −0.330409
$$917$$ 0 0
$$918$$ 0 0
$$919$$ 8.00000 0.263896 0.131948 0.991257i $$-0.457877\pi$$
0.131948 + 0.991257i $$0.457877\pi$$
$$920$$ 8.00000 0.263752
$$921$$ 0 0
$$922$$ 2.00000 0.0658665
$$923$$ −16.0000 −0.526646
$$924$$ 0 0
$$925$$ −10.0000 −0.328798
$$926$$ 8.00000 0.262896
$$927$$ 0 0
$$928$$ −2.00000 −0.0656532
$$929$$ 22.0000 0.721797 0.360898 0.932605i $$-0.382470\pi$$
0.360898 + 0.932605i $$0.382470\pi$$
$$930$$ 0 0
$$931$$ 7.00000 0.229416
$$932$$ 6.00000 0.196537
$$933$$ 0 0
$$934$$ −20.0000 −0.654420
$$935$$ −48.0000 −1.56977
$$936$$ 0 0
$$937$$ 26.0000 0.849383 0.424691 0.905338i $$-0.360383\pi$$
0.424691 + 0.905338i $$0.360383\pi$$
$$938$$ 0 0
$$939$$ 0 0
$$940$$ −8.00000 −0.260931
$$941$$ −22.0000 −0.717180 −0.358590 0.933495i $$-0.616742\pi$$
−0.358590 + 0.933495i $$0.616742\pi$$
$$942$$ 0 0
$$943$$ −40.0000 −1.30258
$$944$$ −12.0000 −0.390567
$$945$$ 0 0
$$946$$ −16.0000 −0.520205
$$947$$ −12.0000 −0.389948 −0.194974 0.980808i $$-0.562462\pi$$
−0.194974 + 0.980808i $$0.562462\pi$$
$$948$$ 0 0
$$949$$ −12.0000 −0.389536
$$950$$ −1.00000 −0.0324443
$$951$$ 0 0
$$952$$ 0 0
$$953$$ 22.0000 0.712650 0.356325 0.934362i $$-0.384030\pi$$
0.356325 + 0.934362i $$0.384030\pi$$
$$954$$ 0 0
$$955$$ 8.00000 0.258874
$$956$$ −12.0000 −0.388108
$$957$$ 0 0
$$958$$ 4.00000 0.129234
$$959$$ 0 0
$$960$$ 0 0
$$961$$ −15.0000 −0.483871
$$962$$ −20.0000 −0.644826
$$963$$ 0 0
$$964$$ 10.0000 0.322078
$$965$$ 12.0000 0.386294
$$966$$ 0 0
$$967$$ 48.0000 1.54358 0.771788 0.635880i $$-0.219363\pi$$
0.771788 + 0.635880i $$0.219363\pi$$
$$968$$ −5.00000 −0.160706
$$969$$ 0 0
$$970$$ 20.0000 0.642161
$$971$$ 28.0000 0.898563 0.449281 0.893390i $$-0.351680\pi$$
0.449281 + 0.893390i $$0.351680\pi$$
$$972$$ 0 0
$$973$$ 0 0
$$974$$ −4.00000 −0.128168
$$975$$ 0 0
$$976$$ 14.0000 0.448129
$$977$$ 46.0000 1.47167 0.735835 0.677161i $$-0.236790\pi$$
0.735835 + 0.677161i $$0.236790\pi$$
$$978$$ 0 0
$$979$$ 24.0000 0.767043
$$980$$ 14.0000 0.447214
$$981$$ 0 0
$$982$$ 20.0000 0.638226
$$983$$ 24.0000 0.765481 0.382741 0.923856i $$-0.374980\pi$$
0.382741 + 0.923856i $$0.374980\pi$$
$$984$$ 0 0
$$985$$ −44.0000 −1.40196
$$986$$ −12.0000 −0.382158
$$987$$ 0 0
$$988$$ −2.00000 −0.0636285
$$989$$ 16.0000 0.508770
$$990$$ 0 0
$$991$$ 4.00000 0.127064 0.0635321 0.997980i $$-0.479763\pi$$
0.0635321 + 0.997980i $$0.479763\pi$$
$$992$$ −4.00000 −0.127000
$$993$$ 0 0
$$994$$ 0 0
$$995$$ 32.0000 1.01447
$$996$$ 0 0
$$997$$ −2.00000 −0.0633406 −0.0316703 0.999498i $$-0.510083\pi$$
−0.0316703 + 0.999498i $$0.510083\pi$$
$$998$$ 20.0000 0.633089
$$999$$ 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 342.2.a.b.1.1 1
3.2 odd 2 114.2.a.b.1.1 1
4.3 odd 2 2736.2.a.d.1.1 1
5.4 even 2 8550.2.a.ba.1.1 1
12.11 even 2 912.2.a.k.1.1 1
15.2 even 4 2850.2.d.b.799.2 2
15.8 even 4 2850.2.d.b.799.1 2
15.14 odd 2 2850.2.a.j.1.1 1
19.18 odd 2 6498.2.a.p.1.1 1
21.20 even 2 5586.2.a.y.1.1 1
24.5 odd 2 3648.2.a.x.1.1 1
24.11 even 2 3648.2.a.c.1.1 1
57.56 even 2 2166.2.a.d.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
114.2.a.b.1.1 1 3.2 odd 2
342.2.a.b.1.1 1 1.1 even 1 trivial
912.2.a.k.1.1 1 12.11 even 2
2166.2.a.d.1.1 1 57.56 even 2
2736.2.a.d.1.1 1 4.3 odd 2
2850.2.a.j.1.1 1 15.14 odd 2
2850.2.d.b.799.1 2 15.8 even 4
2850.2.d.b.799.2 2 15.2 even 4
3648.2.a.c.1.1 1 24.11 even 2
3648.2.a.x.1.1 1 24.5 odd 2
5586.2.a.y.1.1 1 21.20 even 2
6498.2.a.p.1.1 1 19.18 odd 2
8550.2.a.ba.1.1 1 5.4 even 2