# Properties

 Label 1470.2.a.r.1.1 Level $1470$ Weight $2$ Character 1470.1 Self dual yes Analytic conductor $11.738$ Analytic rank $0$ Dimension $1$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1470 = 2 \cdot 3 \cdot 5 \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1470.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$11.7380090971$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 210) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Character $$\chi$$ $$=$$ 1470.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} +1.00000 q^{6} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$$ $$q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} +1.00000 q^{6} +1.00000 q^{8} +1.00000 q^{9} +1.00000 q^{10} -1.00000 q^{11} +1.00000 q^{12} +7.00000 q^{13} +1.00000 q^{15} +1.00000 q^{16} -4.00000 q^{17} +1.00000 q^{18} +1.00000 q^{19} +1.00000 q^{20} -1.00000 q^{22} +1.00000 q^{23} +1.00000 q^{24} +1.00000 q^{25} +7.00000 q^{26} +1.00000 q^{27} -8.00000 q^{29} +1.00000 q^{30} +6.00000 q^{31} +1.00000 q^{32} -1.00000 q^{33} -4.00000 q^{34} +1.00000 q^{36} -3.00000 q^{37} +1.00000 q^{38} +7.00000 q^{39} +1.00000 q^{40} +9.00000 q^{41} -4.00000 q^{43} -1.00000 q^{44} +1.00000 q^{45} +1.00000 q^{46} -3.00000 q^{47} +1.00000 q^{48} +1.00000 q^{50} -4.00000 q^{51} +7.00000 q^{52} -1.00000 q^{53} +1.00000 q^{54} -1.00000 q^{55} +1.00000 q^{57} -8.00000 q^{58} +12.0000 q^{59} +1.00000 q^{60} -4.00000 q^{61} +6.00000 q^{62} +1.00000 q^{64} +7.00000 q^{65} -1.00000 q^{66} +12.0000 q^{67} -4.00000 q^{68} +1.00000 q^{69} -14.0000 q^{71} +1.00000 q^{72} -14.0000 q^{73} -3.00000 q^{74} +1.00000 q^{75} +1.00000 q^{76} +7.00000 q^{78} +4.00000 q^{79} +1.00000 q^{80} +1.00000 q^{81} +9.00000 q^{82} +12.0000 q^{83} -4.00000 q^{85} -4.00000 q^{86} -8.00000 q^{87} -1.00000 q^{88} -2.00000 q^{89} +1.00000 q^{90} +1.00000 q^{92} +6.00000 q^{93} -3.00000 q^{94} +1.00000 q^{95} +1.00000 q^{96} -16.0000 q^{97} -1.00000 q^{99} +O(q^{100})$$

## 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
$$3$$ 1.00000 0.577350
$$4$$ 1.00000 0.500000
$$5$$ 1.00000 0.447214
$$6$$ 1.00000 0.408248
$$7$$ 0 0
$$8$$ 1.00000 0.353553
$$9$$ 1.00000 0.333333
$$10$$ 1.00000 0.316228
$$11$$ −1.00000 −0.301511 −0.150756 0.988571i $$-0.548171\pi$$
−0.150756 + 0.988571i $$0.548171\pi$$
$$12$$ 1.00000 0.288675
$$13$$ 7.00000 1.94145 0.970725 0.240192i $$-0.0772105\pi$$
0.970725 + 0.240192i $$0.0772105\pi$$
$$14$$ 0 0
$$15$$ 1.00000 0.258199
$$16$$ 1.00000 0.250000
$$17$$ −4.00000 −0.970143 −0.485071 0.874475i $$-0.661206\pi$$
−0.485071 + 0.874475i $$0.661206\pi$$
$$18$$ 1.00000 0.235702
$$19$$ 1.00000 0.229416 0.114708 0.993399i $$-0.463407\pi$$
0.114708 + 0.993399i $$0.463407\pi$$
$$20$$ 1.00000 0.223607
$$21$$ 0 0
$$22$$ −1.00000 −0.213201
$$23$$ 1.00000 0.208514 0.104257 0.994550i $$-0.466753\pi$$
0.104257 + 0.994550i $$0.466753\pi$$
$$24$$ 1.00000 0.204124
$$25$$ 1.00000 0.200000
$$26$$ 7.00000 1.37281
$$27$$ 1.00000 0.192450
$$28$$ 0 0
$$29$$ −8.00000 −1.48556 −0.742781 0.669534i $$-0.766494\pi$$
−0.742781 + 0.669534i $$0.766494\pi$$
$$30$$ 1.00000 0.182574
$$31$$ 6.00000 1.07763 0.538816 0.842424i $$-0.318872\pi$$
0.538816 + 0.842424i $$0.318872\pi$$
$$32$$ 1.00000 0.176777
$$33$$ −1.00000 −0.174078
$$34$$ −4.00000 −0.685994
$$35$$ 0 0
$$36$$ 1.00000 0.166667
$$37$$ −3.00000 −0.493197 −0.246598 0.969118i $$-0.579313\pi$$
−0.246598 + 0.969118i $$0.579313\pi$$
$$38$$ 1.00000 0.162221
$$39$$ 7.00000 1.12090
$$40$$ 1.00000 0.158114
$$41$$ 9.00000 1.40556 0.702782 0.711405i $$-0.251941\pi$$
0.702782 + 0.711405i $$0.251941\pi$$
$$42$$ 0 0
$$43$$ −4.00000 −0.609994 −0.304997 0.952353i $$-0.598656\pi$$
−0.304997 + 0.952353i $$0.598656\pi$$
$$44$$ −1.00000 −0.150756
$$45$$ 1.00000 0.149071
$$46$$ 1.00000 0.147442
$$47$$ −3.00000 −0.437595 −0.218797 0.975770i $$-0.570213\pi$$
−0.218797 + 0.975770i $$0.570213\pi$$
$$48$$ 1.00000 0.144338
$$49$$ 0 0
$$50$$ 1.00000 0.141421
$$51$$ −4.00000 −0.560112
$$52$$ 7.00000 0.970725
$$53$$ −1.00000 −0.137361 −0.0686803 0.997639i $$-0.521879\pi$$
−0.0686803 + 0.997639i $$0.521879\pi$$
$$54$$ 1.00000 0.136083
$$55$$ −1.00000 −0.134840
$$56$$ 0 0
$$57$$ 1.00000 0.132453
$$58$$ −8.00000 −1.05045
$$59$$ 12.0000 1.56227 0.781133 0.624364i $$-0.214642\pi$$
0.781133 + 0.624364i $$0.214642\pi$$
$$60$$ 1.00000 0.129099
$$61$$ −4.00000 −0.512148 −0.256074 0.966657i $$-0.582429\pi$$
−0.256074 + 0.966657i $$0.582429\pi$$
$$62$$ 6.00000 0.762001
$$63$$ 0 0
$$64$$ 1.00000 0.125000
$$65$$ 7.00000 0.868243
$$66$$ −1.00000 −0.123091
$$67$$ 12.0000 1.46603 0.733017 0.680211i $$-0.238112\pi$$
0.733017 + 0.680211i $$0.238112\pi$$
$$68$$ −4.00000 −0.485071
$$69$$ 1.00000 0.120386
$$70$$ 0 0
$$71$$ −14.0000 −1.66149 −0.830747 0.556650i $$-0.812086\pi$$
−0.830747 + 0.556650i $$0.812086\pi$$
$$72$$ 1.00000 0.117851
$$73$$ −14.0000 −1.63858 −0.819288 0.573382i $$-0.805631\pi$$
−0.819288 + 0.573382i $$0.805631\pi$$
$$74$$ −3.00000 −0.348743
$$75$$ 1.00000 0.115470
$$76$$ 1.00000 0.114708
$$77$$ 0 0
$$78$$ 7.00000 0.792594
$$79$$ 4.00000 0.450035 0.225018 0.974355i $$-0.427756\pi$$
0.225018 + 0.974355i $$0.427756\pi$$
$$80$$ 1.00000 0.111803
$$81$$ 1.00000 0.111111
$$82$$ 9.00000 0.993884
$$83$$ 12.0000 1.31717 0.658586 0.752506i $$-0.271155\pi$$
0.658586 + 0.752506i $$0.271155\pi$$
$$84$$ 0 0
$$85$$ −4.00000 −0.433861
$$86$$ −4.00000 −0.431331
$$87$$ −8.00000 −0.857690
$$88$$ −1.00000 −0.106600
$$89$$ −2.00000 −0.212000 −0.106000 0.994366i $$-0.533804\pi$$
−0.106000 + 0.994366i $$0.533804\pi$$
$$90$$ 1.00000 0.105409
$$91$$ 0 0
$$92$$ 1.00000 0.104257
$$93$$ 6.00000 0.622171
$$94$$ −3.00000 −0.309426
$$95$$ 1.00000 0.102598
$$96$$ 1.00000 0.102062
$$97$$ −16.0000 −1.62455 −0.812277 0.583272i $$-0.801772\pi$$
−0.812277 + 0.583272i $$0.801772\pi$$
$$98$$ 0 0
$$99$$ −1.00000 −0.100504
$$100$$ 1.00000 0.100000
$$101$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$102$$ −4.00000 −0.396059
$$103$$ 16.0000 1.57653 0.788263 0.615338i $$-0.210980\pi$$
0.788263 + 0.615338i $$0.210980\pi$$
$$104$$ 7.00000 0.686406
$$105$$ 0 0
$$106$$ −1.00000 −0.0971286
$$107$$ −18.0000 −1.74013 −0.870063 0.492941i $$-0.835922\pi$$
−0.870063 + 0.492941i $$0.835922\pi$$
$$108$$ 1.00000 0.0962250
$$109$$ −10.0000 −0.957826 −0.478913 0.877862i $$-0.658969\pi$$
−0.478913 + 0.877862i $$0.658969\pi$$
$$110$$ −1.00000 −0.0953463
$$111$$ −3.00000 −0.284747
$$112$$ 0 0
$$113$$ −6.00000 −0.564433 −0.282216 0.959351i $$-0.591070\pi$$
−0.282216 + 0.959351i $$0.591070\pi$$
$$114$$ 1.00000 0.0936586
$$115$$ 1.00000 0.0932505
$$116$$ −8.00000 −0.742781
$$117$$ 7.00000 0.647150
$$118$$ 12.0000 1.10469
$$119$$ 0 0
$$120$$ 1.00000 0.0912871
$$121$$ −10.0000 −0.909091
$$122$$ −4.00000 −0.362143
$$123$$ 9.00000 0.811503
$$124$$ 6.00000 0.538816
$$125$$ 1.00000 0.0894427
$$126$$ 0 0
$$127$$ 5.00000 0.443678 0.221839 0.975083i $$-0.428794\pi$$
0.221839 + 0.975083i $$0.428794\pi$$
$$128$$ 1.00000 0.0883883
$$129$$ −4.00000 −0.352180
$$130$$ 7.00000 0.613941
$$131$$ −13.0000 −1.13582 −0.567908 0.823092i $$-0.692247\pi$$
−0.567908 + 0.823092i $$0.692247\pi$$
$$132$$ −1.00000 −0.0870388
$$133$$ 0 0
$$134$$ 12.0000 1.03664
$$135$$ 1.00000 0.0860663
$$136$$ −4.00000 −0.342997
$$137$$ −2.00000 −0.170872 −0.0854358 0.996344i $$-0.527228\pi$$
−0.0854358 + 0.996344i $$0.527228\pi$$
$$138$$ 1.00000 0.0851257
$$139$$ −4.00000 −0.339276 −0.169638 0.985506i $$-0.554260\pi$$
−0.169638 + 0.985506i $$0.554260\pi$$
$$140$$ 0 0
$$141$$ −3.00000 −0.252646
$$142$$ −14.0000 −1.17485
$$143$$ −7.00000 −0.585369
$$144$$ 1.00000 0.0833333
$$145$$ −8.00000 −0.664364
$$146$$ −14.0000 −1.15865
$$147$$ 0 0
$$148$$ −3.00000 −0.246598
$$149$$ −4.00000 −0.327693 −0.163846 0.986486i $$-0.552390\pi$$
−0.163846 + 0.986486i $$0.552390\pi$$
$$150$$ 1.00000 0.0816497
$$151$$ −2.00000 −0.162758 −0.0813788 0.996683i $$-0.525932\pi$$
−0.0813788 + 0.996683i $$0.525932\pi$$
$$152$$ 1.00000 0.0811107
$$153$$ −4.00000 −0.323381
$$154$$ 0 0
$$155$$ 6.00000 0.481932
$$156$$ 7.00000 0.560449
$$157$$ 15.0000 1.19713 0.598565 0.801074i $$-0.295738\pi$$
0.598565 + 0.801074i $$0.295738\pi$$
$$158$$ 4.00000 0.318223
$$159$$ −1.00000 −0.0793052
$$160$$ 1.00000 0.0790569
$$161$$ 0 0
$$162$$ 1.00000 0.0785674
$$163$$ 8.00000 0.626608 0.313304 0.949653i $$-0.398564\pi$$
0.313304 + 0.949653i $$0.398564\pi$$
$$164$$ 9.00000 0.702782
$$165$$ −1.00000 −0.0778499
$$166$$ 12.0000 0.931381
$$167$$ −5.00000 −0.386912 −0.193456 0.981109i $$-0.561970\pi$$
−0.193456 + 0.981109i $$0.561970\pi$$
$$168$$ 0 0
$$169$$ 36.0000 2.76923
$$170$$ −4.00000 −0.306786
$$171$$ 1.00000 0.0764719
$$172$$ −4.00000 −0.304997
$$173$$ −21.0000 −1.59660 −0.798300 0.602260i $$-0.794267\pi$$
−0.798300 + 0.602260i $$0.794267\pi$$
$$174$$ −8.00000 −0.606478
$$175$$ 0 0
$$176$$ −1.00000 −0.0753778
$$177$$ 12.0000 0.901975
$$178$$ −2.00000 −0.149906
$$179$$ 13.0000 0.971666 0.485833 0.874052i $$-0.338516\pi$$
0.485833 + 0.874052i $$0.338516\pi$$
$$180$$ 1.00000 0.0745356
$$181$$ −12.0000 −0.891953 −0.445976 0.895045i $$-0.647144\pi$$
−0.445976 + 0.895045i $$0.647144\pi$$
$$182$$ 0 0
$$183$$ −4.00000 −0.295689
$$184$$ 1.00000 0.0737210
$$185$$ −3.00000 −0.220564
$$186$$ 6.00000 0.439941
$$187$$ 4.00000 0.292509
$$188$$ −3.00000 −0.218797
$$189$$ 0 0
$$190$$ 1.00000 0.0725476
$$191$$ −10.0000 −0.723575 −0.361787 0.932261i $$-0.617833\pi$$
−0.361787 + 0.932261i $$0.617833\pi$$
$$192$$ 1.00000 0.0721688
$$193$$ 26.0000 1.87152 0.935760 0.352636i $$-0.114715\pi$$
0.935760 + 0.352636i $$0.114715\pi$$
$$194$$ −16.0000 −1.14873
$$195$$ 7.00000 0.501280
$$196$$ 0 0
$$197$$ −3.00000 −0.213741 −0.106871 0.994273i $$-0.534083\pi$$
−0.106871 + 0.994273i $$0.534083\pi$$
$$198$$ −1.00000 −0.0710669
$$199$$ −12.0000 −0.850657 −0.425329 0.905039i $$-0.639842\pi$$
−0.425329 + 0.905039i $$0.639842\pi$$
$$200$$ 1.00000 0.0707107
$$201$$ 12.0000 0.846415
$$202$$ 0 0
$$203$$ 0 0
$$204$$ −4.00000 −0.280056
$$205$$ 9.00000 0.628587
$$206$$ 16.0000 1.11477
$$207$$ 1.00000 0.0695048
$$208$$ 7.00000 0.485363
$$209$$ −1.00000 −0.0691714
$$210$$ 0 0
$$211$$ −15.0000 −1.03264 −0.516321 0.856395i $$-0.672699\pi$$
−0.516321 + 0.856395i $$0.672699\pi$$
$$212$$ −1.00000 −0.0686803
$$213$$ −14.0000 −0.959264
$$214$$ −18.0000 −1.23045
$$215$$ −4.00000 −0.272798
$$216$$ 1.00000 0.0680414
$$217$$ 0 0
$$218$$ −10.0000 −0.677285
$$219$$ −14.0000 −0.946032
$$220$$ −1.00000 −0.0674200
$$221$$ −28.0000 −1.88348
$$222$$ −3.00000 −0.201347
$$223$$ −4.00000 −0.267860 −0.133930 0.990991i $$-0.542760\pi$$
−0.133930 + 0.990991i $$0.542760\pi$$
$$224$$ 0 0
$$225$$ 1.00000 0.0666667
$$226$$ −6.00000 −0.399114
$$227$$ −20.0000 −1.32745 −0.663723 0.747978i $$-0.731025\pi$$
−0.663723 + 0.747978i $$0.731025\pi$$
$$228$$ 1.00000 0.0662266
$$229$$ −22.0000 −1.45380 −0.726900 0.686743i $$-0.759040\pi$$
−0.726900 + 0.686743i $$0.759040\pi$$
$$230$$ 1.00000 0.0659380
$$231$$ 0 0
$$232$$ −8.00000 −0.525226
$$233$$ 26.0000 1.70332 0.851658 0.524097i $$-0.175597\pi$$
0.851658 + 0.524097i $$0.175597\pi$$
$$234$$ 7.00000 0.457604
$$235$$ −3.00000 −0.195698
$$236$$ 12.0000 0.781133
$$237$$ 4.00000 0.259828
$$238$$ 0 0
$$239$$ −6.00000 −0.388108 −0.194054 0.980991i $$-0.562164\pi$$
−0.194054 + 0.980991i $$0.562164\pi$$
$$240$$ 1.00000 0.0645497
$$241$$ 7.00000 0.450910 0.225455 0.974254i $$-0.427613\pi$$
0.225455 + 0.974254i $$0.427613\pi$$
$$242$$ −10.0000 −0.642824
$$243$$ 1.00000 0.0641500
$$244$$ −4.00000 −0.256074
$$245$$ 0 0
$$246$$ 9.00000 0.573819
$$247$$ 7.00000 0.445399
$$248$$ 6.00000 0.381000
$$249$$ 12.0000 0.760469
$$250$$ 1.00000 0.0632456
$$251$$ −3.00000 −0.189358 −0.0946792 0.995508i $$-0.530183\pi$$
−0.0946792 + 0.995508i $$0.530183\pi$$
$$252$$ 0 0
$$253$$ −1.00000 −0.0628695
$$254$$ 5.00000 0.313728
$$255$$ −4.00000 −0.250490
$$256$$ 1.00000 0.0625000
$$257$$ −8.00000 −0.499026 −0.249513 0.968371i $$-0.580271\pi$$
−0.249513 + 0.968371i $$0.580271\pi$$
$$258$$ −4.00000 −0.249029
$$259$$ 0 0
$$260$$ 7.00000 0.434122
$$261$$ −8.00000 −0.495188
$$262$$ −13.0000 −0.803143
$$263$$ −16.0000 −0.986602 −0.493301 0.869859i $$-0.664210\pi$$
−0.493301 + 0.869859i $$0.664210\pi$$
$$264$$ −1.00000 −0.0615457
$$265$$ −1.00000 −0.0614295
$$266$$ 0 0
$$267$$ −2.00000 −0.122398
$$268$$ 12.0000 0.733017
$$269$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$270$$ 1.00000 0.0608581
$$271$$ 16.0000 0.971931 0.485965 0.873978i $$-0.338468\pi$$
0.485965 + 0.873978i $$0.338468\pi$$
$$272$$ −4.00000 −0.242536
$$273$$ 0 0
$$274$$ −2.00000 −0.120824
$$275$$ −1.00000 −0.0603023
$$276$$ 1.00000 0.0601929
$$277$$ −2.00000 −0.120168 −0.0600842 0.998193i $$-0.519137\pi$$
−0.0600842 + 0.998193i $$0.519137\pi$$
$$278$$ −4.00000 −0.239904
$$279$$ 6.00000 0.359211
$$280$$ 0 0
$$281$$ 3.00000 0.178965 0.0894825 0.995988i $$-0.471479\pi$$
0.0894825 + 0.995988i $$0.471479\pi$$
$$282$$ −3.00000 −0.178647
$$283$$ −2.00000 −0.118888 −0.0594438 0.998232i $$-0.518933\pi$$
−0.0594438 + 0.998232i $$0.518933\pi$$
$$284$$ −14.0000 −0.830747
$$285$$ 1.00000 0.0592349
$$286$$ −7.00000 −0.413919
$$287$$ 0 0
$$288$$ 1.00000 0.0589256
$$289$$ −1.00000 −0.0588235
$$290$$ −8.00000 −0.469776
$$291$$ −16.0000 −0.937937
$$292$$ −14.0000 −0.819288
$$293$$ 9.00000 0.525786 0.262893 0.964825i $$-0.415323\pi$$
0.262893 + 0.964825i $$0.415323\pi$$
$$294$$ 0 0
$$295$$ 12.0000 0.698667
$$296$$ −3.00000 −0.174371
$$297$$ −1.00000 −0.0580259
$$298$$ −4.00000 −0.231714
$$299$$ 7.00000 0.404820
$$300$$ 1.00000 0.0577350
$$301$$ 0 0
$$302$$ −2.00000 −0.115087
$$303$$ 0 0
$$304$$ 1.00000 0.0573539
$$305$$ −4.00000 −0.229039
$$306$$ −4.00000 −0.228665
$$307$$ 8.00000 0.456584 0.228292 0.973593i $$-0.426686\pi$$
0.228292 + 0.973593i $$0.426686\pi$$
$$308$$ 0 0
$$309$$ 16.0000 0.910208
$$310$$ 6.00000 0.340777
$$311$$ 16.0000 0.907277 0.453638 0.891186i $$-0.350126\pi$$
0.453638 + 0.891186i $$0.350126\pi$$
$$312$$ 7.00000 0.396297
$$313$$ −24.0000 −1.35656 −0.678280 0.734803i $$-0.737274\pi$$
−0.678280 + 0.734803i $$0.737274\pi$$
$$314$$ 15.0000 0.846499
$$315$$ 0 0
$$316$$ 4.00000 0.225018
$$317$$ 10.0000 0.561656 0.280828 0.959758i $$-0.409391\pi$$
0.280828 + 0.959758i $$0.409391\pi$$
$$318$$ −1.00000 −0.0560772
$$319$$ 8.00000 0.447914
$$320$$ 1.00000 0.0559017
$$321$$ −18.0000 −1.00466
$$322$$ 0 0
$$323$$ −4.00000 −0.222566
$$324$$ 1.00000 0.0555556
$$325$$ 7.00000 0.388290
$$326$$ 8.00000 0.443079
$$327$$ −10.0000 −0.553001
$$328$$ 9.00000 0.496942
$$329$$ 0 0
$$330$$ −1.00000 −0.0550482
$$331$$ −9.00000 −0.494685 −0.247342 0.968928i $$-0.579557\pi$$
−0.247342 + 0.968928i $$0.579557\pi$$
$$332$$ 12.0000 0.658586
$$333$$ −3.00000 −0.164399
$$334$$ −5.00000 −0.273588
$$335$$ 12.0000 0.655630
$$336$$ 0 0
$$337$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$338$$ 36.0000 1.95814
$$339$$ −6.00000 −0.325875
$$340$$ −4.00000 −0.216930
$$341$$ −6.00000 −0.324918
$$342$$ 1.00000 0.0540738
$$343$$ 0 0
$$344$$ −4.00000 −0.215666
$$345$$ 1.00000 0.0538382
$$346$$ −21.0000 −1.12897
$$347$$ −34.0000 −1.82522 −0.912608 0.408836i $$-0.865935\pi$$
−0.912608 + 0.408836i $$0.865935\pi$$
$$348$$ −8.00000 −0.428845
$$349$$ −28.0000 −1.49881 −0.749403 0.662114i $$-0.769659\pi$$
−0.749403 + 0.662114i $$0.769659\pi$$
$$350$$ 0 0
$$351$$ 7.00000 0.373632
$$352$$ −1.00000 −0.0533002
$$353$$ −8.00000 −0.425797 −0.212899 0.977074i $$-0.568290\pi$$
−0.212899 + 0.977074i $$0.568290\pi$$
$$354$$ 12.0000 0.637793
$$355$$ −14.0000 −0.743043
$$356$$ −2.00000 −0.106000
$$357$$ 0 0
$$358$$ 13.0000 0.687071
$$359$$ 36.0000 1.90001 0.950004 0.312239i $$-0.101079\pi$$
0.950004 + 0.312239i $$0.101079\pi$$
$$360$$ 1.00000 0.0527046
$$361$$ −18.0000 −0.947368
$$362$$ −12.0000 −0.630706
$$363$$ −10.0000 −0.524864
$$364$$ 0 0
$$365$$ −14.0000 −0.732793
$$366$$ −4.00000 −0.209083
$$367$$ 19.0000 0.991792 0.495896 0.868382i $$-0.334840\pi$$
0.495896 + 0.868382i $$0.334840\pi$$
$$368$$ 1.00000 0.0521286
$$369$$ 9.00000 0.468521
$$370$$ −3.00000 −0.155963
$$371$$ 0 0
$$372$$ 6.00000 0.311086
$$373$$ 26.0000 1.34623 0.673114 0.739538i $$-0.264956\pi$$
0.673114 + 0.739538i $$0.264956\pi$$
$$374$$ 4.00000 0.206835
$$375$$ 1.00000 0.0516398
$$376$$ −3.00000 −0.154713
$$377$$ −56.0000 −2.88415
$$378$$ 0 0
$$379$$ 1.00000 0.0513665 0.0256833 0.999670i $$-0.491824\pi$$
0.0256833 + 0.999670i $$0.491824\pi$$
$$380$$ 1.00000 0.0512989
$$381$$ 5.00000 0.256158
$$382$$ −10.0000 −0.511645
$$383$$ 13.0000 0.664269 0.332134 0.943232i $$-0.392231\pi$$
0.332134 + 0.943232i $$0.392231\pi$$
$$384$$ 1.00000 0.0510310
$$385$$ 0 0
$$386$$ 26.0000 1.32337
$$387$$ −4.00000 −0.203331
$$388$$ −16.0000 −0.812277
$$389$$ −14.0000 −0.709828 −0.354914 0.934899i $$-0.615490\pi$$
−0.354914 + 0.934899i $$0.615490\pi$$
$$390$$ 7.00000 0.354459
$$391$$ −4.00000 −0.202289
$$392$$ 0 0
$$393$$ −13.0000 −0.655763
$$394$$ −3.00000 −0.151138
$$395$$ 4.00000 0.201262
$$396$$ −1.00000 −0.0502519
$$397$$ 18.0000 0.903394 0.451697 0.892171i $$-0.350819\pi$$
0.451697 + 0.892171i $$0.350819\pi$$
$$398$$ −12.0000 −0.601506
$$399$$ 0 0
$$400$$ 1.00000 0.0500000
$$401$$ −17.0000 −0.848939 −0.424470 0.905442i $$-0.639539\pi$$
−0.424470 + 0.905442i $$0.639539\pi$$
$$402$$ 12.0000 0.598506
$$403$$ 42.0000 2.09217
$$404$$ 0 0
$$405$$ 1.00000 0.0496904
$$406$$ 0 0
$$407$$ 3.00000 0.148704
$$408$$ −4.00000 −0.198030
$$409$$ 10.0000 0.494468 0.247234 0.968956i $$-0.420478\pi$$
0.247234 + 0.968956i $$0.420478\pi$$
$$410$$ 9.00000 0.444478
$$411$$ −2.00000 −0.0986527
$$412$$ 16.0000 0.788263
$$413$$ 0 0
$$414$$ 1.00000 0.0491473
$$415$$ 12.0000 0.589057
$$416$$ 7.00000 0.343203
$$417$$ −4.00000 −0.195881
$$418$$ −1.00000 −0.0489116
$$419$$ 11.0000 0.537385 0.268693 0.963226i $$-0.413408\pi$$
0.268693 + 0.963226i $$0.413408\pi$$
$$420$$ 0 0
$$421$$ −14.0000 −0.682318 −0.341159 0.940006i $$-0.610819\pi$$
−0.341159 + 0.940006i $$0.610819\pi$$
$$422$$ −15.0000 −0.730189
$$423$$ −3.00000 −0.145865
$$424$$ −1.00000 −0.0485643
$$425$$ −4.00000 −0.194029
$$426$$ −14.0000 −0.678302
$$427$$ 0 0
$$428$$ −18.0000 −0.870063
$$429$$ −7.00000 −0.337963
$$430$$ −4.00000 −0.192897
$$431$$ −12.0000 −0.578020 −0.289010 0.957326i $$-0.593326\pi$$
−0.289010 + 0.957326i $$0.593326\pi$$
$$432$$ 1.00000 0.0481125
$$433$$ 40.0000 1.92228 0.961139 0.276066i $$-0.0890309\pi$$
0.961139 + 0.276066i $$0.0890309\pi$$
$$434$$ 0 0
$$435$$ −8.00000 −0.383571
$$436$$ −10.0000 −0.478913
$$437$$ 1.00000 0.0478365
$$438$$ −14.0000 −0.668946
$$439$$ 16.0000 0.763638 0.381819 0.924237i $$-0.375298\pi$$
0.381819 + 0.924237i $$0.375298\pi$$
$$440$$ −1.00000 −0.0476731
$$441$$ 0 0
$$442$$ −28.0000 −1.33182
$$443$$ −36.0000 −1.71041 −0.855206 0.518289i $$-0.826569\pi$$
−0.855206 + 0.518289i $$0.826569\pi$$
$$444$$ −3.00000 −0.142374
$$445$$ −2.00000 −0.0948091
$$446$$ −4.00000 −0.189405
$$447$$ −4.00000 −0.189194
$$448$$ 0 0
$$449$$ −25.0000 −1.17982 −0.589911 0.807468i $$-0.700837\pi$$
−0.589911 + 0.807468i $$0.700837\pi$$
$$450$$ 1.00000 0.0471405
$$451$$ −9.00000 −0.423793
$$452$$ −6.00000 −0.282216
$$453$$ −2.00000 −0.0939682
$$454$$ −20.0000 −0.938647
$$455$$ 0 0
$$456$$ 1.00000 0.0468293
$$457$$ 10.0000 0.467780 0.233890 0.972263i $$-0.424854\pi$$
0.233890 + 0.972263i $$0.424854\pi$$
$$458$$ −22.0000 −1.02799
$$459$$ −4.00000 −0.186704
$$460$$ 1.00000 0.0466252
$$461$$ −28.0000 −1.30409 −0.652045 0.758180i $$-0.726089\pi$$
−0.652045 + 0.758180i $$0.726089\pi$$
$$462$$ 0 0
$$463$$ 33.0000 1.53364 0.766820 0.641862i $$-0.221838\pi$$
0.766820 + 0.641862i $$0.221838\pi$$
$$464$$ −8.00000 −0.371391
$$465$$ 6.00000 0.278243
$$466$$ 26.0000 1.20443
$$467$$ 12.0000 0.555294 0.277647 0.960683i $$-0.410445\pi$$
0.277647 + 0.960683i $$0.410445\pi$$
$$468$$ 7.00000 0.323575
$$469$$ 0 0
$$470$$ −3.00000 −0.138380
$$471$$ 15.0000 0.691164
$$472$$ 12.0000 0.552345
$$473$$ 4.00000 0.183920
$$474$$ 4.00000 0.183726
$$475$$ 1.00000 0.0458831
$$476$$ 0 0
$$477$$ −1.00000 −0.0457869
$$478$$ −6.00000 −0.274434
$$479$$ 26.0000 1.18797 0.593985 0.804476i $$-0.297554\pi$$
0.593985 + 0.804476i $$0.297554\pi$$
$$480$$ 1.00000 0.0456435
$$481$$ −21.0000 −0.957518
$$482$$ 7.00000 0.318841
$$483$$ 0 0
$$484$$ −10.0000 −0.454545
$$485$$ −16.0000 −0.726523
$$486$$ 1.00000 0.0453609
$$487$$ −8.00000 −0.362515 −0.181257 0.983436i $$-0.558017\pi$$
−0.181257 + 0.983436i $$0.558017\pi$$
$$488$$ −4.00000 −0.181071
$$489$$ 8.00000 0.361773
$$490$$ 0 0
$$491$$ −12.0000 −0.541552 −0.270776 0.962642i $$-0.587280\pi$$
−0.270776 + 0.962642i $$0.587280\pi$$
$$492$$ 9.00000 0.405751
$$493$$ 32.0000 1.44121
$$494$$ 7.00000 0.314945
$$495$$ −1.00000 −0.0449467
$$496$$ 6.00000 0.269408
$$497$$ 0 0
$$498$$ 12.0000 0.537733
$$499$$ 24.0000 1.07439 0.537194 0.843459i $$-0.319484\pi$$
0.537194 + 0.843459i $$0.319484\pi$$
$$500$$ 1.00000 0.0447214
$$501$$ −5.00000 −0.223384
$$502$$ −3.00000 −0.133897
$$503$$ 28.0000 1.24846 0.624229 0.781241i $$-0.285413\pi$$
0.624229 + 0.781241i $$0.285413\pi$$
$$504$$ 0 0
$$505$$ 0 0
$$506$$ −1.00000 −0.0444554
$$507$$ 36.0000 1.59882
$$508$$ 5.00000 0.221839
$$509$$ 30.0000 1.32973 0.664863 0.746965i $$-0.268490\pi$$
0.664863 + 0.746965i $$0.268490\pi$$
$$510$$ −4.00000 −0.177123
$$511$$ 0 0
$$512$$ 1.00000 0.0441942
$$513$$ 1.00000 0.0441511
$$514$$ −8.00000 −0.352865
$$515$$ 16.0000 0.705044
$$516$$ −4.00000 −0.176090
$$517$$ 3.00000 0.131940
$$518$$ 0 0
$$519$$ −21.0000 −0.921798
$$520$$ 7.00000 0.306970
$$521$$ −21.0000 −0.920027 −0.460013 0.887912i $$-0.652155\pi$$
−0.460013 + 0.887912i $$0.652155\pi$$
$$522$$ −8.00000 −0.350150
$$523$$ −14.0000 −0.612177 −0.306089 0.952003i $$-0.599020\pi$$
−0.306089 + 0.952003i $$0.599020\pi$$
$$524$$ −13.0000 −0.567908
$$525$$ 0 0
$$526$$ −16.0000 −0.697633
$$527$$ −24.0000 −1.04546
$$528$$ −1.00000 −0.0435194
$$529$$ −22.0000 −0.956522
$$530$$ −1.00000 −0.0434372
$$531$$ 12.0000 0.520756
$$532$$ 0 0
$$533$$ 63.0000 2.72883
$$534$$ −2.00000 −0.0865485
$$535$$ −18.0000 −0.778208
$$536$$ 12.0000 0.518321
$$537$$ 13.0000 0.560991
$$538$$ 0 0
$$539$$ 0 0
$$540$$ 1.00000 0.0430331
$$541$$ 2.00000 0.0859867 0.0429934 0.999075i $$-0.486311\pi$$
0.0429934 + 0.999075i $$0.486311\pi$$
$$542$$ 16.0000 0.687259
$$543$$ −12.0000 −0.514969
$$544$$ −4.00000 −0.171499
$$545$$ −10.0000 −0.428353
$$546$$ 0 0
$$547$$ 36.0000 1.53925 0.769624 0.638497i $$-0.220443\pi$$
0.769624 + 0.638497i $$0.220443\pi$$
$$548$$ −2.00000 −0.0854358
$$549$$ −4.00000 −0.170716
$$550$$ −1.00000 −0.0426401
$$551$$ −8.00000 −0.340811
$$552$$ 1.00000 0.0425628
$$553$$ 0 0
$$554$$ −2.00000 −0.0849719
$$555$$ −3.00000 −0.127343
$$556$$ −4.00000 −0.169638
$$557$$ 45.0000 1.90671 0.953356 0.301849i $$-0.0976040\pi$$
0.953356 + 0.301849i $$0.0976040\pi$$
$$558$$ 6.00000 0.254000
$$559$$ −28.0000 −1.18427
$$560$$ 0 0
$$561$$ 4.00000 0.168880
$$562$$ 3.00000 0.126547
$$563$$ 14.0000 0.590030 0.295015 0.955493i $$-0.404675\pi$$
0.295015 + 0.955493i $$0.404675\pi$$
$$564$$ −3.00000 −0.126323
$$565$$ −6.00000 −0.252422
$$566$$ −2.00000 −0.0840663
$$567$$ 0 0
$$568$$ −14.0000 −0.587427
$$569$$ −37.0000 −1.55112 −0.775560 0.631273i $$-0.782533\pi$$
−0.775560 + 0.631273i $$0.782533\pi$$
$$570$$ 1.00000 0.0418854
$$571$$ −8.00000 −0.334790 −0.167395 0.985890i $$-0.553535\pi$$
−0.167395 + 0.985890i $$0.553535\pi$$
$$572$$ −7.00000 −0.292685
$$573$$ −10.0000 −0.417756
$$574$$ 0 0
$$575$$ 1.00000 0.0417029
$$576$$ 1.00000 0.0416667
$$577$$ −14.0000 −0.582828 −0.291414 0.956597i $$-0.594126\pi$$
−0.291414 + 0.956597i $$0.594126\pi$$
$$578$$ −1.00000 −0.0415945
$$579$$ 26.0000 1.08052
$$580$$ −8.00000 −0.332182
$$581$$ 0 0
$$582$$ −16.0000 −0.663221
$$583$$ 1.00000 0.0414158
$$584$$ −14.0000 −0.579324
$$585$$ 7.00000 0.289414
$$586$$ 9.00000 0.371787
$$587$$ 42.0000 1.73353 0.866763 0.498721i $$-0.166197\pi$$
0.866763 + 0.498721i $$0.166197\pi$$
$$588$$ 0 0
$$589$$ 6.00000 0.247226
$$590$$ 12.0000 0.494032
$$591$$ −3.00000 −0.123404
$$592$$ −3.00000 −0.123299
$$593$$ 12.0000 0.492781 0.246390 0.969171i $$-0.420755\pi$$
0.246390 + 0.969171i $$0.420755\pi$$
$$594$$ −1.00000 −0.0410305
$$595$$ 0 0
$$596$$ −4.00000 −0.163846
$$597$$ −12.0000 −0.491127
$$598$$ 7.00000 0.286251
$$599$$ 6.00000 0.245153 0.122577 0.992459i $$-0.460884\pi$$
0.122577 + 0.992459i $$0.460884\pi$$
$$600$$ 1.00000 0.0408248
$$601$$ 22.0000 0.897399 0.448699 0.893683i $$-0.351887\pi$$
0.448699 + 0.893683i $$0.351887\pi$$
$$602$$ 0 0
$$603$$ 12.0000 0.488678
$$604$$ −2.00000 −0.0813788
$$605$$ −10.0000 −0.406558
$$606$$ 0 0
$$607$$ 25.0000 1.01472 0.507359 0.861735i $$-0.330622\pi$$
0.507359 + 0.861735i $$0.330622\pi$$
$$608$$ 1.00000 0.0405554
$$609$$ 0 0
$$610$$ −4.00000 −0.161955
$$611$$ −21.0000 −0.849569
$$612$$ −4.00000 −0.161690
$$613$$ −15.0000 −0.605844 −0.302922 0.953015i $$-0.597962\pi$$
−0.302922 + 0.953015i $$0.597962\pi$$
$$614$$ 8.00000 0.322854
$$615$$ 9.00000 0.362915
$$616$$ 0 0
$$617$$ 8.00000 0.322068 0.161034 0.986949i $$-0.448517\pi$$
0.161034 + 0.986949i $$0.448517\pi$$
$$618$$ 16.0000 0.643614
$$619$$ −7.00000 −0.281354 −0.140677 0.990056i $$-0.544928\pi$$
−0.140677 + 0.990056i $$0.544928\pi$$
$$620$$ 6.00000 0.240966
$$621$$ 1.00000 0.0401286
$$622$$ 16.0000 0.641542
$$623$$ 0 0
$$624$$ 7.00000 0.280224
$$625$$ 1.00000 0.0400000
$$626$$ −24.0000 −0.959233
$$627$$ −1.00000 −0.0399362
$$628$$ 15.0000 0.598565
$$629$$ 12.0000 0.478471
$$630$$ 0 0
$$631$$ 14.0000 0.557331 0.278666 0.960388i $$-0.410108\pi$$
0.278666 + 0.960388i $$0.410108\pi$$
$$632$$ 4.00000 0.159111
$$633$$ −15.0000 −0.596196
$$634$$ 10.0000 0.397151
$$635$$ 5.00000 0.198419
$$636$$ −1.00000 −0.0396526
$$637$$ 0 0
$$638$$ 8.00000 0.316723
$$639$$ −14.0000 −0.553831
$$640$$ 1.00000 0.0395285
$$641$$ −23.0000 −0.908445 −0.454223 0.890888i $$-0.650083\pi$$
−0.454223 + 0.890888i $$0.650083\pi$$
$$642$$ −18.0000 −0.710403
$$643$$ 26.0000 1.02534 0.512670 0.858586i $$-0.328656\pi$$
0.512670 + 0.858586i $$0.328656\pi$$
$$644$$ 0 0
$$645$$ −4.00000 −0.157500
$$646$$ −4.00000 −0.157378
$$647$$ −15.0000 −0.589711 −0.294855 0.955542i $$-0.595271\pi$$
−0.294855 + 0.955542i $$0.595271\pi$$
$$648$$ 1.00000 0.0392837
$$649$$ −12.0000 −0.471041
$$650$$ 7.00000 0.274563
$$651$$ 0 0
$$652$$ 8.00000 0.313304
$$653$$ 29.0000 1.13486 0.567429 0.823422i $$-0.307938\pi$$
0.567429 + 0.823422i $$0.307938\pi$$
$$654$$ −10.0000 −0.391031
$$655$$ −13.0000 −0.507952
$$656$$ 9.00000 0.351391
$$657$$ −14.0000 −0.546192
$$658$$ 0 0
$$659$$ −12.0000 −0.467454 −0.233727 0.972302i $$-0.575092\pi$$
−0.233727 + 0.972302i $$0.575092\pi$$
$$660$$ −1.00000 −0.0389249
$$661$$ 8.00000 0.311164 0.155582 0.987823i $$-0.450275\pi$$
0.155582 + 0.987823i $$0.450275\pi$$
$$662$$ −9.00000 −0.349795
$$663$$ −28.0000 −1.08743
$$664$$ 12.0000 0.465690
$$665$$ 0 0
$$666$$ −3.00000 −0.116248
$$667$$ −8.00000 −0.309761
$$668$$ −5.00000 −0.193456
$$669$$ −4.00000 −0.154649
$$670$$ 12.0000 0.463600
$$671$$ 4.00000 0.154418
$$672$$ 0 0
$$673$$ −12.0000 −0.462566 −0.231283 0.972887i $$-0.574292\pi$$
−0.231283 + 0.972887i $$0.574292\pi$$
$$674$$ 0 0
$$675$$ 1.00000 0.0384900
$$676$$ 36.0000 1.38462
$$677$$ 1.00000 0.0384331 0.0192166 0.999815i $$-0.493883\pi$$
0.0192166 + 0.999815i $$0.493883\pi$$
$$678$$ −6.00000 −0.230429
$$679$$ 0 0
$$680$$ −4.00000 −0.153393
$$681$$ −20.0000 −0.766402
$$682$$ −6.00000 −0.229752
$$683$$ 12.0000 0.459167 0.229584 0.973289i $$-0.426264\pi$$
0.229584 + 0.973289i $$0.426264\pi$$
$$684$$ 1.00000 0.0382360
$$685$$ −2.00000 −0.0764161
$$686$$ 0 0
$$687$$ −22.0000 −0.839352
$$688$$ −4.00000 −0.152499
$$689$$ −7.00000 −0.266679
$$690$$ 1.00000 0.0380693
$$691$$ 12.0000 0.456502 0.228251 0.973602i $$-0.426699\pi$$
0.228251 + 0.973602i $$0.426699\pi$$
$$692$$ −21.0000 −0.798300
$$693$$ 0 0
$$694$$ −34.0000 −1.29062
$$695$$ −4.00000 −0.151729
$$696$$ −8.00000 −0.303239
$$697$$ −36.0000 −1.36360
$$698$$ −28.0000 −1.05982
$$699$$ 26.0000 0.983410
$$700$$ 0 0
$$701$$ 6.00000 0.226617 0.113308 0.993560i $$-0.463855\pi$$
0.113308 + 0.993560i $$0.463855\pi$$
$$702$$ 7.00000 0.264198
$$703$$ −3.00000 −0.113147
$$704$$ −1.00000 −0.0376889
$$705$$ −3.00000 −0.112987
$$706$$ −8.00000 −0.301084
$$707$$ 0 0
$$708$$ 12.0000 0.450988
$$709$$ 4.00000 0.150223 0.0751116 0.997175i $$-0.476069\pi$$
0.0751116 + 0.997175i $$0.476069\pi$$
$$710$$ −14.0000 −0.525411
$$711$$ 4.00000 0.150012
$$712$$ −2.00000 −0.0749532
$$713$$ 6.00000 0.224702
$$714$$ 0 0
$$715$$ −7.00000 −0.261785
$$716$$ 13.0000 0.485833
$$717$$ −6.00000 −0.224074
$$718$$ 36.0000 1.34351
$$719$$ 26.0000 0.969636 0.484818 0.874615i $$-0.338886\pi$$
0.484818 + 0.874615i $$0.338886\pi$$
$$720$$ 1.00000 0.0372678
$$721$$ 0 0
$$722$$ −18.0000 −0.669891
$$723$$ 7.00000 0.260333
$$724$$ −12.0000 −0.445976
$$725$$ −8.00000 −0.297113
$$726$$ −10.0000 −0.371135
$$727$$ 17.0000 0.630495 0.315248 0.949009i $$-0.397912\pi$$
0.315248 + 0.949009i $$0.397912\pi$$
$$728$$ 0 0
$$729$$ 1.00000 0.0370370
$$730$$ −14.0000 −0.518163
$$731$$ 16.0000 0.591781
$$732$$ −4.00000 −0.147844
$$733$$ 37.0000 1.36663 0.683313 0.730125i $$-0.260538\pi$$
0.683313 + 0.730125i $$0.260538\pi$$
$$734$$ 19.0000 0.701303
$$735$$ 0 0
$$736$$ 1.00000 0.0368605
$$737$$ −12.0000 −0.442026
$$738$$ 9.00000 0.331295
$$739$$ 41.0000 1.50821 0.754105 0.656754i $$-0.228071\pi$$
0.754105 + 0.656754i $$0.228071\pi$$
$$740$$ −3.00000 −0.110282
$$741$$ 7.00000 0.257151
$$742$$ 0 0
$$743$$ −9.00000 −0.330178 −0.165089 0.986279i $$-0.552791\pi$$
−0.165089 + 0.986279i $$0.552791\pi$$
$$744$$ 6.00000 0.219971
$$745$$ −4.00000 −0.146549
$$746$$ 26.0000 0.951928
$$747$$ 12.0000 0.439057
$$748$$ 4.00000 0.146254
$$749$$ 0 0
$$750$$ 1.00000 0.0365148
$$751$$ −26.0000 −0.948753 −0.474377 0.880322i $$-0.657327\pi$$
−0.474377 + 0.880322i $$0.657327\pi$$
$$752$$ −3.00000 −0.109399
$$753$$ −3.00000 −0.109326
$$754$$ −56.0000 −2.03940
$$755$$ −2.00000 −0.0727875
$$756$$ 0 0
$$757$$ −2.00000 −0.0726912 −0.0363456 0.999339i $$-0.511572\pi$$
−0.0363456 + 0.999339i $$0.511572\pi$$
$$758$$ 1.00000 0.0363216
$$759$$ −1.00000 −0.0362977
$$760$$ 1.00000 0.0362738
$$761$$ −17.0000 −0.616250 −0.308125 0.951346i $$-0.599701\pi$$
−0.308125 + 0.951346i $$0.599701\pi$$
$$762$$ 5.00000 0.181131
$$763$$ 0 0
$$764$$ −10.0000 −0.361787
$$765$$ −4.00000 −0.144620
$$766$$ 13.0000 0.469709
$$767$$ 84.0000 3.03306
$$768$$ 1.00000 0.0360844
$$769$$ −29.0000 −1.04577 −0.522883 0.852404i $$-0.675144\pi$$
−0.522883 + 0.852404i $$0.675144\pi$$
$$770$$ 0 0
$$771$$ −8.00000 −0.288113
$$772$$ 26.0000 0.935760
$$773$$ 43.0000 1.54660 0.773301 0.634039i $$-0.218604\pi$$
0.773301 + 0.634039i $$0.218604\pi$$
$$774$$ −4.00000 −0.143777
$$775$$ 6.00000 0.215526
$$776$$ −16.0000 −0.574367
$$777$$ 0 0
$$778$$ −14.0000 −0.501924
$$779$$ 9.00000 0.322458
$$780$$ 7.00000 0.250640
$$781$$ 14.0000 0.500959
$$782$$ −4.00000 −0.143040
$$783$$ −8.00000 −0.285897
$$784$$ 0 0
$$785$$ 15.0000 0.535373
$$786$$ −13.0000 −0.463695
$$787$$ −22.0000 −0.784215 −0.392108 0.919919i $$-0.628254\pi$$
−0.392108 + 0.919919i $$0.628254\pi$$
$$788$$ −3.00000 −0.106871
$$789$$ −16.0000 −0.569615
$$790$$ 4.00000 0.142314
$$791$$ 0 0
$$792$$ −1.00000 −0.0355335
$$793$$ −28.0000 −0.994309
$$794$$ 18.0000 0.638796
$$795$$ −1.00000 −0.0354663
$$796$$ −12.0000 −0.425329
$$797$$ −6.00000 −0.212531 −0.106265 0.994338i $$-0.533889\pi$$
−0.106265 + 0.994338i $$0.533889\pi$$
$$798$$ 0 0
$$799$$ 12.0000 0.424529
$$800$$ 1.00000 0.0353553
$$801$$ −2.00000 −0.0706665
$$802$$ −17.0000 −0.600291
$$803$$ 14.0000 0.494049
$$804$$ 12.0000 0.423207
$$805$$ 0 0
$$806$$ 42.0000 1.47939
$$807$$ 0 0
$$808$$ 0 0
$$809$$ 53.0000 1.86338 0.931690 0.363253i $$-0.118334\pi$$
0.931690 + 0.363253i $$0.118334\pi$$
$$810$$ 1.00000 0.0351364
$$811$$ 37.0000 1.29925 0.649623 0.760257i $$-0.274927\pi$$
0.649623 + 0.760257i $$0.274927\pi$$
$$812$$ 0 0
$$813$$ 16.0000 0.561144
$$814$$ 3.00000 0.105150
$$815$$ 8.00000 0.280228
$$816$$ −4.00000 −0.140028
$$817$$ −4.00000 −0.139942
$$818$$ 10.0000 0.349642
$$819$$ 0 0
$$820$$ 9.00000 0.314294
$$821$$ 34.0000 1.18661 0.593304 0.804978i $$-0.297823\pi$$
0.593304 + 0.804978i $$0.297823\pi$$
$$822$$ −2.00000 −0.0697580
$$823$$ 48.0000 1.67317 0.836587 0.547833i $$-0.184547\pi$$
0.836587 + 0.547833i $$0.184547\pi$$
$$824$$ 16.0000 0.557386
$$825$$ −1.00000 −0.0348155
$$826$$ 0 0
$$827$$ 10.0000 0.347734 0.173867 0.984769i $$-0.444374\pi$$
0.173867 + 0.984769i $$0.444374\pi$$
$$828$$ 1.00000 0.0347524
$$829$$ 12.0000 0.416777 0.208389 0.978046i $$-0.433178\pi$$
0.208389 + 0.978046i $$0.433178\pi$$
$$830$$ 12.0000 0.416526
$$831$$ −2.00000 −0.0693792
$$832$$ 7.00000 0.242681
$$833$$ 0 0
$$834$$ −4.00000 −0.138509
$$835$$ −5.00000 −0.173032
$$836$$ −1.00000 −0.0345857
$$837$$ 6.00000 0.207390
$$838$$ 11.0000 0.379989
$$839$$ 44.0000 1.51905 0.759524 0.650479i $$-0.225432\pi$$
0.759524 + 0.650479i $$0.225432\pi$$
$$840$$ 0 0
$$841$$ 35.0000 1.20690
$$842$$ −14.0000 −0.482472
$$843$$ 3.00000 0.103325
$$844$$ −15.0000 −0.516321
$$845$$ 36.0000 1.23844
$$846$$ −3.00000 −0.103142
$$847$$ 0 0
$$848$$ −1.00000 −0.0343401
$$849$$ −2.00000 −0.0686398
$$850$$ −4.00000 −0.137199
$$851$$ −3.00000 −0.102839
$$852$$ −14.0000 −0.479632
$$853$$ 1.00000 0.0342393 0.0171197 0.999853i $$-0.494550\pi$$
0.0171197 + 0.999853i $$0.494550\pi$$
$$854$$ 0 0
$$855$$ 1.00000 0.0341993
$$856$$ −18.0000 −0.615227
$$857$$ −22.0000 −0.751506 −0.375753 0.926720i $$-0.622616\pi$$
−0.375753 + 0.926720i $$0.622616\pi$$
$$858$$ −7.00000 −0.238976
$$859$$ −20.0000 −0.682391 −0.341196 0.939992i $$-0.610832\pi$$
−0.341196 + 0.939992i $$0.610832\pi$$
$$860$$ −4.00000 −0.136399
$$861$$ 0 0
$$862$$ −12.0000 −0.408722
$$863$$ 29.0000 0.987171 0.493586 0.869697i $$-0.335686\pi$$
0.493586 + 0.869697i $$0.335686\pi$$
$$864$$ 1.00000 0.0340207
$$865$$ −21.0000 −0.714021
$$866$$ 40.0000 1.35926
$$867$$ −1.00000 −0.0339618
$$868$$ 0 0
$$869$$ −4.00000 −0.135691
$$870$$ −8.00000 −0.271225
$$871$$ 84.0000 2.84623
$$872$$ −10.0000 −0.338643
$$873$$ −16.0000 −0.541518
$$874$$ 1.00000 0.0338255
$$875$$ 0 0
$$876$$ −14.0000 −0.473016
$$877$$ 23.0000 0.776655 0.388327 0.921521i $$-0.373053\pi$$
0.388327 + 0.921521i $$0.373053\pi$$
$$878$$ 16.0000 0.539974
$$879$$ 9.00000 0.303562
$$880$$ −1.00000 −0.0337100
$$881$$ −25.0000 −0.842271 −0.421136 0.906998i $$-0.638368\pi$$
−0.421136 + 0.906998i $$0.638368\pi$$
$$882$$ 0 0
$$883$$ −58.0000 −1.95186 −0.975928 0.218094i $$-0.930016\pi$$
−0.975928 + 0.218094i $$0.930016\pi$$
$$884$$ −28.0000 −0.941742
$$885$$ 12.0000 0.403376
$$886$$ −36.0000 −1.20944
$$887$$ 20.0000 0.671534 0.335767 0.941945i $$-0.391004\pi$$
0.335767 + 0.941945i $$0.391004\pi$$
$$888$$ −3.00000 −0.100673
$$889$$ 0 0
$$890$$ −2.00000 −0.0670402
$$891$$ −1.00000 −0.0335013
$$892$$ −4.00000 −0.133930
$$893$$ −3.00000 −0.100391
$$894$$ −4.00000 −0.133780
$$895$$ 13.0000 0.434542
$$896$$ 0 0
$$897$$ 7.00000 0.233723
$$898$$ −25.0000 −0.834261
$$899$$ −48.0000 −1.60089
$$900$$ 1.00000 0.0333333
$$901$$ 4.00000 0.133259
$$902$$ −9.00000 −0.299667
$$903$$ 0 0
$$904$$ −6.00000 −0.199557
$$905$$ −12.0000 −0.398893
$$906$$ −2.00000 −0.0664455
$$907$$ 46.0000 1.52740 0.763702 0.645568i $$-0.223379\pi$$
0.763702 + 0.645568i $$0.223379\pi$$
$$908$$ −20.0000 −0.663723
$$909$$ 0 0
$$910$$ 0 0
$$911$$ −58.0000 −1.92163 −0.960813 0.277198i $$-0.910594\pi$$
−0.960813 + 0.277198i $$0.910594\pi$$
$$912$$ 1.00000 0.0331133
$$913$$ −12.0000 −0.397142
$$914$$ 10.0000 0.330771
$$915$$ −4.00000 −0.132236
$$916$$ −22.0000 −0.726900
$$917$$ 0 0
$$918$$ −4.00000 −0.132020
$$919$$ −44.0000 −1.45143 −0.725713 0.687998i $$-0.758490\pi$$
−0.725713 + 0.687998i $$0.758490\pi$$
$$920$$ 1.00000 0.0329690
$$921$$ 8.00000 0.263609
$$922$$ −28.0000 −0.922131
$$923$$ −98.0000 −3.22571
$$924$$ 0 0
$$925$$ −3.00000 −0.0986394
$$926$$ 33.0000 1.08445
$$927$$ 16.0000 0.525509
$$928$$ −8.00000 −0.262613
$$929$$ 31.0000 1.01708 0.508539 0.861039i $$-0.330186\pi$$
0.508539 + 0.861039i $$0.330186\pi$$
$$930$$ 6.00000 0.196748
$$931$$ 0 0
$$932$$ 26.0000 0.851658
$$933$$ 16.0000 0.523816
$$934$$ 12.0000 0.392652
$$935$$ 4.00000 0.130814
$$936$$ 7.00000 0.228802
$$937$$ 16.0000 0.522697 0.261349 0.965244i $$-0.415833\pi$$
0.261349 + 0.965244i $$0.415833\pi$$
$$938$$ 0 0
$$939$$ −24.0000 −0.783210
$$940$$ −3.00000 −0.0978492
$$941$$ −30.0000 −0.977972 −0.488986 0.872292i $$-0.662633\pi$$
−0.488986 + 0.872292i $$0.662633\pi$$
$$942$$ 15.0000 0.488726
$$943$$ 9.00000 0.293080
$$944$$ 12.0000 0.390567
$$945$$ 0 0
$$946$$ 4.00000 0.130051
$$947$$ 46.0000 1.49480 0.747400 0.664375i $$-0.231302\pi$$
0.747400 + 0.664375i $$0.231302\pi$$
$$948$$ 4.00000 0.129914
$$949$$ −98.0000 −3.18121
$$950$$ 1.00000 0.0324443
$$951$$ 10.0000 0.324272
$$952$$ 0 0
$$953$$ 44.0000 1.42530 0.712650 0.701520i $$-0.247495\pi$$
0.712650 + 0.701520i $$0.247495\pi$$
$$954$$ −1.00000 −0.0323762
$$955$$ −10.0000 −0.323592
$$956$$ −6.00000 −0.194054
$$957$$ 8.00000 0.258603
$$958$$ 26.0000 0.840022
$$959$$ 0 0
$$960$$ 1.00000 0.0322749
$$961$$ 5.00000 0.161290
$$962$$ −21.0000 −0.677067
$$963$$ −18.0000 −0.580042
$$964$$ 7.00000 0.225455
$$965$$ 26.0000 0.836970
$$966$$ 0 0
$$967$$ −20.0000 −0.643157 −0.321578 0.946883i $$-0.604213\pi$$
−0.321578 + 0.946883i $$0.604213\pi$$
$$968$$ −10.0000 −0.321412
$$969$$ −4.00000 −0.128499
$$970$$ −16.0000 −0.513729
$$971$$ 43.0000 1.37994 0.689968 0.723840i $$-0.257625\pi$$
0.689968 + 0.723840i $$0.257625\pi$$
$$972$$ 1.00000 0.0320750
$$973$$ 0 0
$$974$$ −8.00000 −0.256337
$$975$$ 7.00000 0.224179
$$976$$ −4.00000 −0.128037
$$977$$ 18.0000 0.575871 0.287936 0.957650i $$-0.407031\pi$$
0.287936 + 0.957650i $$0.407031\pi$$
$$978$$ 8.00000 0.255812
$$979$$ 2.00000 0.0639203
$$980$$ 0 0
$$981$$ −10.0000 −0.319275
$$982$$ −12.0000 −0.382935
$$983$$ −33.0000 −1.05254 −0.526268 0.850319i $$-0.676409\pi$$
−0.526268 + 0.850319i $$0.676409\pi$$
$$984$$ 9.00000 0.286910
$$985$$ −3.00000 −0.0955879
$$986$$ 32.0000 1.01909
$$987$$ 0 0
$$988$$ 7.00000 0.222700
$$989$$ −4.00000 −0.127193
$$990$$ −1.00000 −0.0317821
$$991$$ 10.0000 0.317660 0.158830 0.987306i $$-0.449228\pi$$
0.158830 + 0.987306i $$0.449228\pi$$
$$992$$ 6.00000 0.190500
$$993$$ −9.00000 −0.285606
$$994$$ 0 0
$$995$$ −12.0000 −0.380426
$$996$$ 12.0000 0.380235
$$997$$ 10.0000 0.316703 0.158352 0.987383i $$-0.449382\pi$$
0.158352 + 0.987383i $$0.449382\pi$$
$$998$$ 24.0000 0.759707
$$999$$ −3.00000 −0.0949158
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 1470.2.a.r.1.1 1
3.2 odd 2 4410.2.a.g.1.1 1
5.4 even 2 7350.2.a.j.1.1 1
7.2 even 3 210.2.i.a.151.1 yes 2
7.3 odd 6 1470.2.i.i.961.1 2
7.4 even 3 210.2.i.a.121.1 2
7.5 odd 6 1470.2.i.i.361.1 2
7.6 odd 2 1470.2.a.k.1.1 1
21.2 odd 6 630.2.k.h.361.1 2
21.11 odd 6 630.2.k.h.541.1 2
21.20 even 2 4410.2.a.q.1.1 1
28.11 odd 6 1680.2.bg.k.961.1 2
28.23 odd 6 1680.2.bg.k.1201.1 2
35.2 odd 12 1050.2.o.j.949.2 4
35.4 even 6 1050.2.i.s.751.1 2
35.9 even 6 1050.2.i.s.151.1 2
35.18 odd 12 1050.2.o.j.499.2 4
35.23 odd 12 1050.2.o.j.949.1 4
35.32 odd 12 1050.2.o.j.499.1 4
35.34 odd 2 7350.2.a.ba.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
210.2.i.a.121.1 2 7.4 even 3
210.2.i.a.151.1 yes 2 7.2 even 3
630.2.k.h.361.1 2 21.2 odd 6
630.2.k.h.541.1 2 21.11 odd 6
1050.2.i.s.151.1 2 35.9 even 6
1050.2.i.s.751.1 2 35.4 even 6
1050.2.o.j.499.1 4 35.32 odd 12
1050.2.o.j.499.2 4 35.18 odd 12
1050.2.o.j.949.1 4 35.23 odd 12
1050.2.o.j.949.2 4 35.2 odd 12
1470.2.a.k.1.1 1 7.6 odd 2
1470.2.a.r.1.1 1 1.1 even 1 trivial
1470.2.i.i.361.1 2 7.5 odd 6
1470.2.i.i.961.1 2 7.3 odd 6
1680.2.bg.k.961.1 2 28.11 odd 6
1680.2.bg.k.1201.1 2 28.23 odd 6
4410.2.a.g.1.1 1 3.2 odd 2
4410.2.a.q.1.1 1 21.20 even 2
7350.2.a.j.1.1 1 5.4 even 2
7350.2.a.ba.1.1 1 35.34 odd 2