# Properties

 Label 4410.2.a.g.1.1 Level 4410 Weight 2 Character 4410.1 Self dual yes Analytic conductor 35.214 Analytic rank 0 Dimension 1 CM no Inner twists 1

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$4410 = 2 \cdot 3^{2} \cdot 5 \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 4410.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$35.2140272914$$ 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 $$\chi$$ $$=$$ 4410.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q-1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{5} -1.00000 q^{8} +O(q^{10})$$ $$q-1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{5} -1.00000 q^{8} +1.00000 q^{10} +1.00000 q^{11} +7.00000 q^{13} +1.00000 q^{16} +4.00000 q^{17} +1.00000 q^{19} -1.00000 q^{20} -1.00000 q^{22} -1.00000 q^{23} +1.00000 q^{25} -7.00000 q^{26} +8.00000 q^{29} +6.00000 q^{31} -1.00000 q^{32} -4.00000 q^{34} -3.00000 q^{37} -1.00000 q^{38} +1.00000 q^{40} -9.00000 q^{41} -4.00000 q^{43} +1.00000 q^{44} +1.00000 q^{46} +3.00000 q^{47} -1.00000 q^{50} +7.00000 q^{52} +1.00000 q^{53} -1.00000 q^{55} -8.00000 q^{58} -12.0000 q^{59} -4.00000 q^{61} -6.00000 q^{62} +1.00000 q^{64} -7.00000 q^{65} +12.0000 q^{67} +4.00000 q^{68} +14.0000 q^{71} -14.0000 q^{73} +3.00000 q^{74} +1.00000 q^{76} +4.00000 q^{79} -1.00000 q^{80} +9.00000 q^{82} -12.0000 q^{83} -4.00000 q^{85} +4.00000 q^{86} -1.00000 q^{88} +2.00000 q^{89} -1.00000 q^{92} -3.00000 q^{94} -1.00000 q^{95} -16.0000 q^{97} +O(q^{100})$$

## 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
$$5$$ −1.00000 −0.447214
$$6$$ 0 0
$$7$$ 0 0
$$8$$ −1.00000 −0.353553
$$9$$ 0 0
$$10$$ 1.00000 0.316228
$$11$$ 1.00000 0.301511 0.150756 0.988571i $$-0.451829\pi$$
0.150756 + 0.988571i $$0.451829\pi$$
$$12$$ 0 0
$$13$$ 7.00000 1.94145 0.970725 0.240192i $$-0.0772105\pi$$
0.970725 + 0.240192i $$0.0772105\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 1.00000 0.250000
$$17$$ 4.00000 0.970143 0.485071 0.874475i $$-0.338794\pi$$
0.485071 + 0.874475i $$0.338794\pi$$
$$18$$ 0 0
$$19$$ 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 $$-0.533247\pi$$
−0.104257 + 0.994550i $$0.533247\pi$$
$$24$$ 0 0
$$25$$ 1.00000 0.200000
$$26$$ −7.00000 −1.37281
$$27$$ 0 0
$$28$$ 0 0
$$29$$ 8.00000 1.48556 0.742781 0.669534i $$-0.233506\pi$$
0.742781 + 0.669534i $$0.233506\pi$$
$$30$$ 0 0
$$31$$ 6.00000 1.07763 0.538816 0.842424i $$-0.318872\pi$$
0.538816 + 0.842424i $$0.318872\pi$$
$$32$$ −1.00000 −0.176777
$$33$$ 0 0
$$34$$ −4.00000 −0.685994
$$35$$ 0 0
$$36$$ 0 0
$$37$$ −3.00000 −0.493197 −0.246598 0.969118i $$-0.579313\pi$$
−0.246598 + 0.969118i $$0.579313\pi$$
$$38$$ −1.00000 −0.162221
$$39$$ 0 0
$$40$$ 1.00000 0.158114
$$41$$ −9.00000 −1.40556 −0.702782 0.711405i $$-0.748059\pi$$
−0.702782 + 0.711405i $$0.748059\pi$$
$$42$$ 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
$$45$$ 0 0
$$46$$ 1.00000 0.147442
$$47$$ 3.00000 0.437595 0.218797 0.975770i $$-0.429787\pi$$
0.218797 + 0.975770i $$0.429787\pi$$
$$48$$ 0 0
$$49$$ 0 0
$$50$$ −1.00000 −0.141421
$$51$$ 0 0
$$52$$ 7.00000 0.970725
$$53$$ 1.00000 0.137361 0.0686803 0.997639i $$-0.478121\pi$$
0.0686803 + 0.997639i $$0.478121\pi$$
$$54$$ 0 0
$$55$$ −1.00000 −0.134840
$$56$$ 0 0
$$57$$ 0 0
$$58$$ −8.00000 −1.05045
$$59$$ −12.0000 −1.56227 −0.781133 0.624364i $$-0.785358\pi$$
−0.781133 + 0.624364i $$0.785358\pi$$
$$60$$ 0 0
$$61$$ −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
$$66$$ 0 0
$$67$$ 12.0000 1.46603 0.733017 0.680211i $$-0.238112\pi$$
0.733017 + 0.680211i $$0.238112\pi$$
$$68$$ 4.00000 0.485071
$$69$$ 0 0
$$70$$ 0 0
$$71$$ 14.0000 1.66149 0.830747 0.556650i $$-0.187914\pi$$
0.830747 + 0.556650i $$0.187914\pi$$
$$72$$ 0 0
$$73$$ −14.0000 −1.63858 −0.819288 0.573382i $$-0.805631\pi$$
−0.819288 + 0.573382i $$0.805631\pi$$
$$74$$ 3.00000 0.348743
$$75$$ 0 0
$$76$$ 1.00000 0.114708
$$77$$ 0 0
$$78$$ 0 0
$$79$$ 4.00000 0.450035 0.225018 0.974355i $$-0.427756\pi$$
0.225018 + 0.974355i $$0.427756\pi$$
$$80$$ −1.00000 −0.111803
$$81$$ 0 0
$$82$$ 9.00000 0.993884
$$83$$ −12.0000 −1.31717 −0.658586 0.752506i $$-0.728845\pi$$
−0.658586 + 0.752506i $$0.728845\pi$$
$$84$$ 0 0
$$85$$ −4.00000 −0.433861
$$86$$ 4.00000 0.431331
$$87$$ 0 0
$$88$$ −1.00000 −0.106600
$$89$$ 2.00000 0.212000 0.106000 0.994366i $$-0.466196\pi$$
0.106000 + 0.994366i $$0.466196\pi$$
$$90$$ 0 0
$$91$$ 0 0
$$92$$ −1.00000 −0.104257
$$93$$ 0 0
$$94$$ −3.00000 −0.309426
$$95$$ −1.00000 −0.102598
$$96$$ 0 0
$$97$$ −16.0000 −1.62455 −0.812277 0.583272i $$-0.801772\pi$$
−0.812277 + 0.583272i $$0.801772\pi$$
$$98$$ 0 0
$$99$$ 0 0
$$100$$ 1.00000 0.100000
$$101$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$102$$ 0 0
$$103$$ 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 $$-0.164078\pi$$
0.870063 + 0.492941i $$0.164078\pi$$
$$108$$ 0 0
$$109$$ −10.0000 −0.957826 −0.478913 0.877862i $$-0.658969\pi$$
−0.478913 + 0.877862i $$0.658969\pi$$
$$110$$ 1.00000 0.0953463
$$111$$ 0 0
$$112$$ 0 0
$$113$$ 6.00000 0.564433 0.282216 0.959351i $$-0.408930\pi$$
0.282216 + 0.959351i $$0.408930\pi$$
$$114$$ 0 0
$$115$$ 1.00000 0.0932505
$$116$$ 8.00000 0.742781
$$117$$ 0 0
$$118$$ 12.0000 1.10469
$$119$$ 0 0
$$120$$ 0 0
$$121$$ −10.0000 −0.909091
$$122$$ 4.00000 0.362143
$$123$$ 0 0
$$124$$ 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
$$129$$ 0 0
$$130$$ 7.00000 0.613941
$$131$$ 13.0000 1.13582 0.567908 0.823092i $$-0.307753\pi$$
0.567908 + 0.823092i $$0.307753\pi$$
$$132$$ 0 0
$$133$$ 0 0
$$134$$ −12.0000 −1.03664
$$135$$ 0 0
$$136$$ −4.00000 −0.342997
$$137$$ 2.00000 0.170872 0.0854358 0.996344i $$-0.472772\pi$$
0.0854358 + 0.996344i $$0.472772\pi$$
$$138$$ 0 0
$$139$$ −4.00000 −0.339276 −0.169638 0.985506i $$-0.554260\pi$$
−0.169638 + 0.985506i $$0.554260\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ −14.0000 −1.17485
$$143$$ 7.00000 0.585369
$$144$$ 0 0
$$145$$ −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 $$-0.447610\pi$$
0.163846 + 0.986486i $$0.447610\pi$$
$$150$$ 0 0
$$151$$ −2.00000 −0.162758 −0.0813788 0.996683i $$-0.525932\pi$$
−0.0813788 + 0.996683i $$0.525932\pi$$
$$152$$ −1.00000 −0.0811107
$$153$$ 0 0
$$154$$ 0 0
$$155$$ −6.00000 −0.481932
$$156$$ 0 0
$$157$$ 15.0000 1.19713 0.598565 0.801074i $$-0.295738\pi$$
0.598565 + 0.801074i $$0.295738\pi$$
$$158$$ −4.00000 −0.318223
$$159$$ 0 0
$$160$$ 1.00000 0.0790569
$$161$$ 0 0
$$162$$ 0 0
$$163$$ 8.00000 0.626608 0.313304 0.949653i $$-0.398564\pi$$
0.313304 + 0.949653i $$0.398564\pi$$
$$164$$ −9.00000 −0.702782
$$165$$ 0 0
$$166$$ 12.0000 0.931381
$$167$$ 5.00000 0.386912 0.193456 0.981109i $$-0.438030\pi$$
0.193456 + 0.981109i $$0.438030\pi$$
$$168$$ 0 0
$$169$$ 36.0000 2.76923
$$170$$ 4.00000 0.306786
$$171$$ 0 0
$$172$$ −4.00000 −0.304997
$$173$$ 21.0000 1.59660 0.798300 0.602260i $$-0.205733\pi$$
0.798300 + 0.602260i $$0.205733\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ 1.00000 0.0753778
$$177$$ 0 0
$$178$$ −2.00000 −0.149906
$$179$$ −13.0000 −0.971666 −0.485833 0.874052i $$-0.661484\pi$$
−0.485833 + 0.874052i $$0.661484\pi$$
$$180$$ 0 0
$$181$$ −12.0000 −0.891953 −0.445976 0.895045i $$-0.647144\pi$$
−0.445976 + 0.895045i $$0.647144\pi$$
$$182$$ 0 0
$$183$$ 0 0
$$184$$ 1.00000 0.0737210
$$185$$ 3.00000 0.220564
$$186$$ 0 0
$$187$$ 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 $$-0.382167\pi$$
0.361787 + 0.932261i $$0.382167\pi$$
$$192$$ 0 0
$$193$$ 26.0000 1.87152 0.935760 0.352636i $$-0.114715\pi$$
0.935760 + 0.352636i $$0.114715\pi$$
$$194$$ 16.0000 1.14873
$$195$$ 0 0
$$196$$ 0 0
$$197$$ 3.00000 0.213741 0.106871 0.994273i $$-0.465917\pi$$
0.106871 + 0.994273i $$0.465917\pi$$
$$198$$ 0 0
$$199$$ −12.0000 −0.850657 −0.425329 0.905039i $$-0.639842\pi$$
−0.425329 + 0.905039i $$0.639842\pi$$
$$200$$ −1.00000 −0.0707107
$$201$$ 0 0
$$202$$ 0 0
$$203$$ 0 0
$$204$$ 0 0
$$205$$ 9.00000 0.628587
$$206$$ −16.0000 −1.11477
$$207$$ 0 0
$$208$$ 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
$$213$$ 0 0
$$214$$ −18.0000 −1.23045
$$215$$ 4.00000 0.272798
$$216$$ 0 0
$$217$$ 0 0
$$218$$ 10.0000 0.677285
$$219$$ 0 0
$$220$$ −1.00000 −0.0674200
$$221$$ 28.0000 1.88348
$$222$$ 0 0
$$223$$ −4.00000 −0.267860 −0.133930 0.990991i $$-0.542760\pi$$
−0.133930 + 0.990991i $$0.542760\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ −6.00000 −0.399114
$$227$$ 20.0000 1.32745 0.663723 0.747978i $$-0.268975\pi$$
0.663723 + 0.747978i $$0.268975\pi$$
$$228$$ 0 0
$$229$$ −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 $$-0.824403\pi$$
−0.851658 + 0.524097i $$0.824403\pi$$
$$234$$ 0 0
$$235$$ −3.00000 −0.195698
$$236$$ −12.0000 −0.781133
$$237$$ 0 0
$$238$$ 0 0
$$239$$ 6.00000 0.388108 0.194054 0.980991i $$-0.437836\pi$$
0.194054 + 0.980991i $$0.437836\pi$$
$$240$$ 0 0
$$241$$ 7.00000 0.450910 0.225455 0.974254i $$-0.427613\pi$$
0.225455 + 0.974254i $$0.427613\pi$$
$$242$$ 10.0000 0.642824
$$243$$ 0 0
$$244$$ −4.00000 −0.256074
$$245$$ 0 0
$$246$$ 0 0
$$247$$ 7.00000 0.445399
$$248$$ −6.00000 −0.381000
$$249$$ 0 0
$$250$$ 1.00000 0.0632456
$$251$$ 3.00000 0.189358 0.0946792 0.995508i $$-0.469817\pi$$
0.0946792 + 0.995508i $$0.469817\pi$$
$$252$$ 0 0
$$253$$ −1.00000 −0.0628695
$$254$$ −5.00000 −0.313728
$$255$$ 0 0
$$256$$ 1.00000 0.0625000
$$257$$ 8.00000 0.499026 0.249513 0.968371i $$-0.419729\pi$$
0.249513 + 0.968371i $$0.419729\pi$$
$$258$$ 0 0
$$259$$ 0 0
$$260$$ −7.00000 −0.434122
$$261$$ 0 0
$$262$$ −13.0000 −0.803143
$$263$$ 16.0000 0.986602 0.493301 0.869859i $$-0.335790\pi$$
0.493301 + 0.869859i $$0.335790\pi$$
$$264$$ 0 0
$$265$$ −1.00000 −0.0614295
$$266$$ 0 0
$$267$$ 0 0
$$268$$ 12.0000 0.733017
$$269$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$270$$ 0 0
$$271$$ 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
$$276$$ 0 0
$$277$$ −2.00000 −0.120168 −0.0600842 0.998193i $$-0.519137\pi$$
−0.0600842 + 0.998193i $$0.519137\pi$$
$$278$$ 4.00000 0.239904
$$279$$ 0 0
$$280$$ 0 0
$$281$$ −3.00000 −0.178965 −0.0894825 0.995988i $$-0.528521\pi$$
−0.0894825 + 0.995988i $$0.528521\pi$$
$$282$$ 0 0
$$283$$ −2.00000 −0.118888 −0.0594438 0.998232i $$-0.518933\pi$$
−0.0594438 + 0.998232i $$0.518933\pi$$
$$284$$ 14.0000 0.830747
$$285$$ 0 0
$$286$$ −7.00000 −0.413919
$$287$$ 0 0
$$288$$ 0 0
$$289$$ −1.00000 −0.0588235
$$290$$ 8.00000 0.469776
$$291$$ 0 0
$$292$$ −14.0000 −0.819288
$$293$$ −9.00000 −0.525786 −0.262893 0.964825i $$-0.584677\pi$$
−0.262893 + 0.964825i $$0.584677\pi$$
$$294$$ 0 0
$$295$$ 12.0000 0.698667
$$296$$ 3.00000 0.174371
$$297$$ 0 0
$$298$$ −4.00000 −0.231714
$$299$$ −7.00000 −0.404820
$$300$$ 0 0
$$301$$ 0 0
$$302$$ 2.00000 0.115087
$$303$$ 0 0
$$304$$ 1.00000 0.0573539
$$305$$ 4.00000 0.229039
$$306$$ 0 0
$$307$$ 8.00000 0.456584 0.228292 0.973593i $$-0.426686\pi$$
0.228292 + 0.973593i $$0.426686\pi$$
$$308$$ 0 0
$$309$$ 0 0
$$310$$ 6.00000 0.340777
$$311$$ −16.0000 −0.907277 −0.453638 0.891186i $$-0.649874\pi$$
−0.453638 + 0.891186i $$0.649874\pi$$
$$312$$ 0 0
$$313$$ −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 $$-0.590609\pi$$
−0.280828 + 0.959758i $$0.590609\pi$$
$$318$$ 0 0
$$319$$ 8.00000 0.447914
$$320$$ −1.00000 −0.0559017
$$321$$ 0 0
$$322$$ 0 0
$$323$$ 4.00000 0.222566
$$324$$ 0 0
$$325$$ 7.00000 0.388290
$$326$$ −8.00000 −0.443079
$$327$$ 0 0
$$328$$ 9.00000 0.496942
$$329$$ 0 0
$$330$$ 0 0
$$331$$ −9.00000 −0.494685 −0.247342 0.968928i $$-0.579557\pi$$
−0.247342 + 0.968928i $$0.579557\pi$$
$$332$$ −12.0000 −0.658586
$$333$$ 0 0
$$334$$ −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
$$339$$ 0 0
$$340$$ −4.00000 −0.216930
$$341$$ 6.00000 0.324918
$$342$$ 0 0
$$343$$ 0 0
$$344$$ 4.00000 0.215666
$$345$$ 0 0
$$346$$ −21.0000 −1.12897
$$347$$ 34.0000 1.82522 0.912608 0.408836i $$-0.134065\pi$$
0.912608 + 0.408836i $$0.134065\pi$$
$$348$$ 0 0
$$349$$ −28.0000 −1.49881 −0.749403 0.662114i $$-0.769659\pi$$
−0.749403 + 0.662114i $$0.769659\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ −1.00000 −0.0533002
$$353$$ 8.00000 0.425797 0.212899 0.977074i $$-0.431710\pi$$
0.212899 + 0.977074i $$0.431710\pi$$
$$354$$ 0 0
$$355$$ −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 $$-0.898921\pi$$
−0.950004 + 0.312239i $$0.898921\pi$$
$$360$$ 0 0
$$361$$ −18.0000 −0.947368
$$362$$ 12.0000 0.630706
$$363$$ 0 0
$$364$$ 0 0
$$365$$ 14.0000 0.732793
$$366$$ 0 0
$$367$$ 19.0000 0.991792 0.495896 0.868382i $$-0.334840\pi$$
0.495896 + 0.868382i $$0.334840\pi$$
$$368$$ −1.00000 −0.0521286
$$369$$ 0 0
$$370$$ −3.00000 −0.155963
$$371$$ 0 0
$$372$$ 0 0
$$373$$ 26.0000 1.34623 0.673114 0.739538i $$-0.264956\pi$$
0.673114 + 0.739538i $$0.264956\pi$$
$$374$$ −4.00000 −0.206835
$$375$$ 0 0
$$376$$ −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
$$381$$ 0 0
$$382$$ −10.0000 −0.511645
$$383$$ −13.0000 −0.664269 −0.332134 0.943232i $$-0.607769\pi$$
−0.332134 + 0.943232i $$0.607769\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ −26.0000 −1.32337
$$387$$ 0 0
$$388$$ −16.0000 −0.812277
$$389$$ 14.0000 0.709828 0.354914 0.934899i $$-0.384510\pi$$
0.354914 + 0.934899i $$0.384510\pi$$
$$390$$ 0 0
$$391$$ −4.00000 −0.202289
$$392$$ 0 0
$$393$$ 0 0
$$394$$ −3.00000 −0.151138
$$395$$ −4.00000 −0.201262
$$396$$ 0 0
$$397$$ 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 $$-0.360461\pi$$
0.424470 + 0.905442i $$0.360461\pi$$
$$402$$ 0 0
$$403$$ 42.0000 2.09217
$$404$$ 0 0
$$405$$ 0 0
$$406$$ 0 0
$$407$$ −3.00000 −0.148704
$$408$$ 0 0
$$409$$ 10.0000 0.494468 0.247234 0.968956i $$-0.420478\pi$$
0.247234 + 0.968956i $$0.420478\pi$$
$$410$$ −9.00000 −0.444478
$$411$$ 0 0
$$412$$ 16.0000 0.788263
$$413$$ 0 0
$$414$$ 0 0
$$415$$ 12.0000 0.589057
$$416$$ −7.00000 −0.343203
$$417$$ 0 0
$$418$$ −1.00000 −0.0489116
$$419$$ −11.0000 −0.537385 −0.268693 0.963226i $$-0.586592\pi$$
−0.268693 + 0.963226i $$0.586592\pi$$
$$420$$ 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
$$423$$ 0 0
$$424$$ −1.00000 −0.0485643
$$425$$ 4.00000 0.194029
$$426$$ 0 0
$$427$$ 0 0
$$428$$ 18.0000 0.870063
$$429$$ 0 0
$$430$$ −4.00000 −0.192897
$$431$$ 12.0000 0.578020 0.289010 0.957326i $$-0.406674\pi$$
0.289010 + 0.957326i $$0.406674\pi$$
$$432$$ 0 0
$$433$$ 40.0000 1.92228 0.961139 0.276066i $$-0.0890309\pi$$
0.961139 + 0.276066i $$0.0890309\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ −10.0000 −0.478913
$$437$$ −1.00000 −0.0478365
$$438$$ 0 0
$$439$$ 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 $$-0.173431\pi$$
0.855206 + 0.518289i $$0.173431\pi$$
$$444$$ 0 0
$$445$$ −2.00000 −0.0948091
$$446$$ 4.00000 0.189405
$$447$$ 0 0
$$448$$ 0 0
$$449$$ 25.0000 1.17982 0.589911 0.807468i $$-0.299163\pi$$
0.589911 + 0.807468i $$0.299163\pi$$
$$450$$ 0 0
$$451$$ −9.00000 −0.423793
$$452$$ 6.00000 0.282216
$$453$$ 0 0
$$454$$ −20.0000 −0.938647
$$455$$ 0 0
$$456$$ 0 0
$$457$$ 10.0000 0.467780 0.233890 0.972263i $$-0.424854\pi$$
0.233890 + 0.972263i $$0.424854\pi$$
$$458$$ 22.0000 1.02799
$$459$$ 0 0
$$460$$ 1.00000 0.0466252
$$461$$ 28.0000 1.30409 0.652045 0.758180i $$-0.273911\pi$$
0.652045 + 0.758180i $$0.273911\pi$$
$$462$$ 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
$$465$$ 0 0
$$466$$ 26.0000 1.20443
$$467$$ −12.0000 −0.555294 −0.277647 0.960683i $$-0.589555\pi$$
−0.277647 + 0.960683i $$0.589555\pi$$
$$468$$ 0 0
$$469$$ 0 0
$$470$$ 3.00000 0.138380
$$471$$ 0 0
$$472$$ 12.0000 0.552345
$$473$$ −4.00000 −0.183920
$$474$$ 0 0
$$475$$ 1.00000 0.0458831
$$476$$ 0 0
$$477$$ 0 0
$$478$$ −6.00000 −0.274434
$$479$$ −26.0000 −1.18797 −0.593985 0.804476i $$-0.702446\pi$$
−0.593985 + 0.804476i $$0.702446\pi$$
$$480$$ 0 0
$$481$$ −21.0000 −0.957518
$$482$$ −7.00000 −0.318841
$$483$$ 0 0
$$484$$ −10.0000 −0.454545
$$485$$ 16.0000 0.726523
$$486$$ 0 0
$$487$$ −8.00000 −0.362515 −0.181257 0.983436i $$-0.558017\pi$$
−0.181257 + 0.983436i $$0.558017\pi$$
$$488$$ 4.00000 0.181071
$$489$$ 0 0
$$490$$ 0 0
$$491$$ 12.0000 0.541552 0.270776 0.962642i $$-0.412720\pi$$
0.270776 + 0.962642i $$0.412720\pi$$
$$492$$ 0 0
$$493$$ 32.0000 1.44121
$$494$$ −7.00000 −0.314945
$$495$$ 0 0
$$496$$ 6.00000 0.269408
$$497$$ 0 0
$$498$$ 0 0
$$499$$ 24.0000 1.07439 0.537194 0.843459i $$-0.319484\pi$$
0.537194 + 0.843459i $$0.319484\pi$$
$$500$$ −1.00000 −0.0447214
$$501$$ 0 0
$$502$$ −3.00000 −0.133897
$$503$$ −28.0000 −1.24846 −0.624229 0.781241i $$-0.714587\pi$$
−0.624229 + 0.781241i $$0.714587\pi$$
$$504$$ 0 0
$$505$$ 0 0
$$506$$ 1.00000 0.0444554
$$507$$ 0 0
$$508$$ 5.00000 0.221839
$$509$$ −30.0000 −1.32973 −0.664863 0.746965i $$-0.731510\pi$$
−0.664863 + 0.746965i $$0.731510\pi$$
$$510$$ 0 0
$$511$$ 0 0
$$512$$ −1.00000 −0.0441942
$$513$$ 0 0
$$514$$ −8.00000 −0.352865
$$515$$ −16.0000 −0.705044
$$516$$ 0 0
$$517$$ 3.00000 0.131940
$$518$$ 0 0
$$519$$ 0 0
$$520$$ 7.00000 0.306970
$$521$$ 21.0000 0.920027 0.460013 0.887912i $$-0.347845\pi$$
0.460013 + 0.887912i $$0.347845\pi$$
$$522$$ 0 0
$$523$$ −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
$$528$$ 0 0
$$529$$ −22.0000 −0.956522
$$530$$ 1.00000 0.0434372
$$531$$ 0 0
$$532$$ 0 0
$$533$$ −63.0000 −2.72883
$$534$$ 0 0
$$535$$ −18.0000 −0.778208
$$536$$ −12.0000 −0.518321
$$537$$ 0 0
$$538$$ 0 0
$$539$$ 0 0
$$540$$ 0 0
$$541$$ 2.00000 0.0859867 0.0429934 0.999075i $$-0.486311\pi$$
0.0429934 + 0.999075i $$0.486311\pi$$
$$542$$ −16.0000 −0.687259
$$543$$ 0 0
$$544$$ −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
$$549$$ 0 0
$$550$$ −1.00000 −0.0426401
$$551$$ 8.00000 0.340811
$$552$$ 0 0
$$553$$ 0 0
$$554$$ 2.00000 0.0849719
$$555$$ 0 0
$$556$$ −4.00000 −0.169638
$$557$$ −45.0000 −1.90671 −0.953356 0.301849i $$-0.902396\pi$$
−0.953356 + 0.301849i $$0.902396\pi$$
$$558$$ 0 0
$$559$$ −28.0000 −1.18427
$$560$$ 0 0
$$561$$ 0 0
$$562$$ 3.00000 0.126547
$$563$$ −14.0000 −0.590030 −0.295015 0.955493i $$-0.595325\pi$$
−0.295015 + 0.955493i $$0.595325\pi$$
$$564$$ 0 0
$$565$$ −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 $$-0.217467\pi$$
0.775560 + 0.631273i $$0.217467\pi$$
$$570$$ 0 0
$$571$$ −8.00000 −0.334790 −0.167395 0.985890i $$-0.553535\pi$$
−0.167395 + 0.985890i $$0.553535\pi$$
$$572$$ 7.00000 0.292685
$$573$$ 0 0
$$574$$ 0 0
$$575$$ −1.00000 −0.0417029
$$576$$ 0 0
$$577$$ −14.0000 −0.582828 −0.291414 0.956597i $$-0.594126\pi$$
−0.291414 + 0.956597i $$0.594126\pi$$
$$578$$ 1.00000 0.0415945
$$579$$ 0 0
$$580$$ −8.00000 −0.332182
$$581$$ 0 0
$$582$$ 0 0
$$583$$ 1.00000 0.0414158
$$584$$ 14.0000 0.579324
$$585$$ 0 0
$$586$$ 9.00000 0.371787
$$587$$ −42.0000 −1.73353 −0.866763 0.498721i $$-0.833803\pi$$
−0.866763 + 0.498721i $$0.833803\pi$$
$$588$$ 0 0
$$589$$ 6.00000 0.247226
$$590$$ −12.0000 −0.494032
$$591$$ 0 0
$$592$$ −3.00000 −0.123299
$$593$$ −12.0000 −0.492781 −0.246390 0.969171i $$-0.579245\pi$$
−0.246390 + 0.969171i $$0.579245\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 4.00000 0.163846
$$597$$ 0 0
$$598$$ 7.00000 0.286251
$$599$$ −6.00000 −0.245153 −0.122577 0.992459i $$-0.539116\pi$$
−0.122577 + 0.992459i $$0.539116\pi$$
$$600$$ 0 0
$$601$$ 22.0000 0.897399 0.448699 0.893683i $$-0.351887\pi$$
0.448699 + 0.893683i $$0.351887\pi$$
$$602$$ 0 0
$$603$$ 0 0
$$604$$ −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
$$612$$ 0 0
$$613$$ −15.0000 −0.605844 −0.302922 0.953015i $$-0.597962\pi$$
−0.302922 + 0.953015i $$0.597962\pi$$
$$614$$ −8.00000 −0.322854
$$615$$ 0 0
$$616$$ 0 0
$$617$$ −8.00000 −0.322068 −0.161034 0.986949i $$-0.551483\pi$$
−0.161034 + 0.986949i $$0.551483\pi$$
$$618$$ 0 0
$$619$$ −7.00000 −0.281354 −0.140677 0.990056i $$-0.544928\pi$$
−0.140677 + 0.990056i $$0.544928\pi$$
$$620$$ −6.00000 −0.240966
$$621$$ 0 0
$$622$$ 16.0000 0.641542
$$623$$ 0 0
$$624$$ 0 0
$$625$$ 1.00000 0.0400000
$$626$$ 24.0000 0.959233
$$627$$ 0 0
$$628$$ 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
$$633$$ 0 0
$$634$$ 10.0000 0.397151
$$635$$ −5.00000 −0.198419
$$636$$ 0 0
$$637$$ 0 0
$$638$$ −8.00000 −0.316723
$$639$$ 0 0
$$640$$ 1.00000 0.0395285
$$641$$ 23.0000 0.908445 0.454223 0.890888i $$-0.349917\pi$$
0.454223 + 0.890888i $$0.349917\pi$$
$$642$$ 0 0
$$643$$ 26.0000 1.02534 0.512670 0.858586i $$-0.328656\pi$$
0.512670 + 0.858586i $$0.328656\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ −4.00000 −0.157378
$$647$$ 15.0000 0.589711 0.294855 0.955542i $$-0.404729\pi$$
0.294855 + 0.955542i $$0.404729\pi$$
$$648$$ 0 0
$$649$$ −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 $$-0.692062\pi$$
−0.567429 + 0.823422i $$0.692062\pi$$
$$654$$ 0 0
$$655$$ −13.0000 −0.507952
$$656$$ −9.00000 −0.351391
$$657$$ 0 0
$$658$$ 0 0
$$659$$ 12.0000 0.467454 0.233727 0.972302i $$-0.424908\pi$$
0.233727 + 0.972302i $$0.424908\pi$$
$$660$$ 0 0
$$661$$ 8.00000 0.311164 0.155582 0.987823i $$-0.450275\pi$$
0.155582 + 0.987823i $$0.450275\pi$$
$$662$$ 9.00000 0.349795
$$663$$ 0 0
$$664$$ 12.0000 0.465690
$$665$$ 0 0
$$666$$ 0 0
$$667$$ −8.00000 −0.309761
$$668$$ 5.00000 0.193456
$$669$$ 0 0
$$670$$ 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
$$675$$ 0 0
$$676$$ 36.0000 1.38462
$$677$$ −1.00000 −0.0384331 −0.0192166 0.999815i $$-0.506117\pi$$
−0.0192166 + 0.999815i $$0.506117\pi$$
$$678$$ 0 0
$$679$$ 0 0
$$680$$ 4.00000 0.153393
$$681$$ 0 0
$$682$$ −6.00000 −0.229752
$$683$$ −12.0000 −0.459167 −0.229584 0.973289i $$-0.573736\pi$$
−0.229584 + 0.973289i $$0.573736\pi$$
$$684$$ 0 0
$$685$$ −2.00000 −0.0764161
$$686$$ 0 0
$$687$$ 0 0
$$688$$ −4.00000 −0.152499
$$689$$ 7.00000 0.266679
$$690$$ 0 0
$$691$$ 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
$$696$$ 0 0
$$697$$ −36.0000 −1.36360
$$698$$ 28.0000 1.05982
$$699$$ 0 0
$$700$$ 0 0
$$701$$ −6.00000 −0.226617 −0.113308 0.993560i $$-0.536145\pi$$
−0.113308 + 0.993560i $$0.536145\pi$$
$$702$$ 0 0
$$703$$ −3.00000 −0.113147
$$704$$ 1.00000 0.0376889
$$705$$ 0 0
$$706$$ −8.00000 −0.301084
$$707$$ 0 0
$$708$$ 0 0
$$709$$ 4.00000 0.150223 0.0751116 0.997175i $$-0.476069\pi$$
0.0751116 + 0.997175i $$0.476069\pi$$
$$710$$ 14.0000 0.525411
$$711$$ 0 0
$$712$$ −2.00000 −0.0749532
$$713$$ −6.00000 −0.224702
$$714$$ 0 0
$$715$$ −7.00000 −0.261785
$$716$$ −13.0000 −0.485833
$$717$$ 0 0
$$718$$ 36.0000 1.34351
$$719$$ −26.0000 −0.969636 −0.484818 0.874615i $$-0.661114\pi$$
−0.484818 + 0.874615i $$0.661114\pi$$
$$720$$ 0 0
$$721$$ 0 0
$$722$$ 18.0000 0.669891
$$723$$ 0 0
$$724$$ −12.0000 −0.445976
$$725$$ 8.00000 0.297113
$$726$$ 0 0
$$727$$ 17.0000 0.630495 0.315248 0.949009i $$-0.397912\pi$$
0.315248 + 0.949009i $$0.397912\pi$$
$$728$$ 0 0
$$729$$ 0 0
$$730$$ −14.0000 −0.518163
$$731$$ −16.0000 −0.591781
$$732$$ 0 0
$$733$$ 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
$$738$$ 0 0
$$739$$ 41.0000 1.50821 0.754105 0.656754i $$-0.228071\pi$$
0.754105 + 0.656754i $$0.228071\pi$$
$$740$$ 3.00000 0.110282
$$741$$ 0 0
$$742$$ 0 0
$$743$$ 9.00000 0.330178 0.165089 0.986279i $$-0.447209\pi$$
0.165089 + 0.986279i $$0.447209\pi$$
$$744$$ 0 0
$$745$$ −4.00000 −0.146549
$$746$$ −26.0000 −0.951928
$$747$$ 0 0
$$748$$ 4.00000 0.146254
$$749$$ 0 0
$$750$$ 0 0
$$751$$ −26.0000 −0.948753 −0.474377 0.880322i $$-0.657327\pi$$
−0.474377 + 0.880322i $$0.657327\pi$$
$$752$$ 3.00000 0.109399
$$753$$ 0 0
$$754$$ −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
$$759$$ 0 0
$$760$$ 1.00000 0.0362738
$$761$$ 17.0000 0.616250 0.308125 0.951346i $$-0.400299\pi$$
0.308125 + 0.951346i $$0.400299\pi$$
$$762$$ 0 0
$$763$$ 0 0
$$764$$ 10.0000 0.361787
$$765$$ 0 0
$$766$$ 13.0000 0.469709
$$767$$ −84.0000 −3.03306
$$768$$ 0 0
$$769$$ −29.0000 −1.04577 −0.522883 0.852404i $$-0.675144\pi$$
−0.522883 + 0.852404i $$0.675144\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ 26.0000 0.935760
$$773$$ −43.0000 −1.54660 −0.773301 0.634039i $$-0.781396\pi$$
−0.773301 + 0.634039i $$0.781396\pi$$
$$774$$ 0 0
$$775$$ 6.00000 0.215526
$$776$$ 16.0000 0.574367
$$777$$ 0 0
$$778$$ −14.0000 −0.501924
$$779$$ −9.00000 −0.322458
$$780$$ 0 0
$$781$$ 14.0000 0.500959
$$782$$ 4.00000 0.143040
$$783$$ 0 0
$$784$$ 0 0
$$785$$ −15.0000 −0.535373
$$786$$ 0 0
$$787$$ −22.0000 −0.784215 −0.392108 0.919919i $$-0.628254\pi$$
−0.392108 + 0.919919i $$0.628254\pi$$
$$788$$ 3.00000 0.106871
$$789$$ 0 0
$$790$$ 4.00000 0.142314
$$791$$ 0 0
$$792$$ 0 0
$$793$$ −28.0000 −0.994309
$$794$$ −18.0000 −0.638796
$$795$$ 0 0
$$796$$ −12.0000 −0.425329
$$797$$ 6.00000 0.212531 0.106265 0.994338i $$-0.466111\pi$$
0.106265 + 0.994338i $$0.466111\pi$$
$$798$$ 0 0
$$799$$ 12.0000 0.424529
$$800$$ −1.00000 −0.0353553
$$801$$ 0 0
$$802$$ −17.0000 −0.600291
$$803$$ −14.0000 −0.494049
$$804$$ 0 0
$$805$$ 0 0
$$806$$ −42.0000 −1.47939
$$807$$ 0 0
$$808$$ 0 0
$$809$$ −53.0000 −1.86338 −0.931690 0.363253i $$-0.881666\pi$$
−0.931690 + 0.363253i $$0.881666\pi$$
$$810$$ 0 0
$$811$$ 37.0000 1.29925 0.649623 0.760257i $$-0.274927\pi$$
0.649623 + 0.760257i $$0.274927\pi$$
$$812$$ 0 0
$$813$$ 0 0
$$814$$ 3.00000 0.105150
$$815$$ −8.00000 −0.280228
$$816$$ 0 0
$$817$$ −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 $$-0.702177\pi$$
−0.593304 + 0.804978i $$0.702177\pi$$
$$822$$ 0 0
$$823$$ 48.0000 1.67317 0.836587 0.547833i $$-0.184547\pi$$
0.836587 + 0.547833i $$0.184547\pi$$
$$824$$ −16.0000 −0.557386
$$825$$ 0 0
$$826$$ 0 0
$$827$$ −10.0000 −0.347734 −0.173867 0.984769i $$-0.555626\pi$$
−0.173867 + 0.984769i $$0.555626\pi$$
$$828$$ 0 0
$$829$$ 12.0000 0.416777 0.208389 0.978046i $$-0.433178\pi$$
0.208389 + 0.978046i $$0.433178\pi$$
$$830$$ −12.0000 −0.416526
$$831$$ 0 0
$$832$$ 7.00000 0.242681
$$833$$ 0 0
$$834$$ 0 0
$$835$$ −5.00000 −0.173032
$$836$$ 1.00000 0.0345857
$$837$$ 0 0
$$838$$ 11.0000 0.379989
$$839$$ −44.0000 −1.51905 −0.759524 0.650479i $$-0.774568\pi$$
−0.759524 + 0.650479i $$0.774568\pi$$
$$840$$ 0 0
$$841$$ 35.0000 1.20690
$$842$$ 14.0000 0.482472
$$843$$ 0 0
$$844$$ −15.0000 −0.516321
$$845$$ −36.0000 −1.23844
$$846$$ 0 0
$$847$$ 0 0
$$848$$ 1.00000 0.0343401
$$849$$ 0 0
$$850$$ −4.00000 −0.137199
$$851$$ 3.00000 0.102839
$$852$$ 0 0
$$853$$ 1.00000 0.0342393 0.0171197 0.999853i $$-0.494550\pi$$
0.0171197 + 0.999853i $$0.494550\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ −18.0000 −0.615227
$$857$$ 22.0000 0.751506 0.375753 0.926720i $$-0.377384\pi$$
0.375753 + 0.926720i $$0.377384\pi$$
$$858$$ 0 0
$$859$$ −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 $$-0.664314\pi$$
−0.493586 + 0.869697i $$0.664314\pi$$
$$864$$ 0 0
$$865$$ −21.0000 −0.714021
$$866$$ −40.0000 −1.35926
$$867$$ 0 0
$$868$$ 0 0
$$869$$ 4.00000 0.135691
$$870$$ 0 0
$$871$$ 84.0000 2.84623
$$872$$ 10.0000 0.338643
$$873$$ 0 0
$$874$$ 1.00000 0.0338255
$$875$$ 0 0
$$876$$ 0 0
$$877$$ 23.0000 0.776655 0.388327 0.921521i $$-0.373053\pi$$
0.388327 + 0.921521i $$0.373053\pi$$
$$878$$ −16.0000 −0.539974
$$879$$ 0 0
$$880$$ −1.00000 −0.0337100
$$881$$ 25.0000 0.842271 0.421136 0.906998i $$-0.361632\pi$$
0.421136 + 0.906998i $$0.361632\pi$$
$$882$$ 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
$$885$$ 0 0
$$886$$ −36.0000 −1.20944
$$887$$ −20.0000 −0.671534 −0.335767 0.941945i $$-0.608996\pi$$
−0.335767 + 0.941945i $$0.608996\pi$$
$$888$$ 0 0
$$889$$ 0 0
$$890$$ 2.00000 0.0670402
$$891$$ 0 0
$$892$$ −4.00000 −0.133930
$$893$$ 3.00000 0.100391
$$894$$ 0 0
$$895$$ 13.0000 0.434542
$$896$$ 0 0
$$897$$ 0 0
$$898$$ −25.0000 −0.834261
$$899$$ 48.0000 1.60089
$$900$$ 0 0
$$901$$ 4.00000 0.133259
$$902$$ 9.00000 0.299667
$$903$$ 0 0
$$904$$ −6.00000 −0.199557
$$905$$ 12.0000 0.398893
$$906$$ 0 0
$$907$$ 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 $$-0.0894057\pi$$
0.960813 + 0.277198i $$0.0894057\pi$$
$$912$$ 0 0
$$913$$ −12.0000 −0.397142
$$914$$ −10.0000 −0.330771
$$915$$ 0 0
$$916$$ −22.0000 −0.726900
$$917$$ 0 0
$$918$$ 0 0
$$919$$ −44.0000 −1.45143 −0.725713 0.687998i $$-0.758490\pi$$
−0.725713 + 0.687998i $$0.758490\pi$$
$$920$$ −1.00000 −0.0329690
$$921$$ 0 0
$$922$$ −28.0000 −0.922131
$$923$$ 98.0000 3.22571
$$924$$ 0 0
$$925$$ −3.00000 −0.0986394
$$926$$ −33.0000 −1.08445
$$927$$ 0 0
$$928$$ −8.00000 −0.262613
$$929$$ −31.0000 −1.01708 −0.508539 0.861039i $$-0.669814\pi$$
−0.508539 + 0.861039i $$0.669814\pi$$
$$930$$ 0 0
$$931$$ 0 0
$$932$$ −26.0000 −0.851658
$$933$$ 0 0
$$934$$ 12.0000 0.392652
$$935$$ −4.00000 −0.130814
$$936$$ 0 0
$$937$$ 16.0000 0.522697 0.261349 0.965244i $$-0.415833\pi$$
0.261349 + 0.965244i $$0.415833\pi$$
$$938$$ 0 0
$$939$$ 0 0
$$940$$ −3.00000 −0.0978492
$$941$$ 30.0000 0.977972 0.488986 0.872292i $$-0.337367\pi$$
0.488986 + 0.872292i $$0.337367\pi$$
$$942$$ 0 0
$$943$$ 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 $$-0.768698\pi$$
−0.747400 + 0.664375i $$0.768698\pi$$
$$948$$ 0 0
$$949$$ −98.0000 −3.18121
$$950$$ −1.00000 −0.0324443
$$951$$ 0 0
$$952$$ 0 0
$$953$$ −44.0000 −1.42530 −0.712650 0.701520i $$-0.752505\pi$$
−0.712650 + 0.701520i $$0.752505\pi$$
$$954$$ 0 0
$$955$$ −10.0000 −0.323592
$$956$$ 6.00000 0.194054
$$957$$ 0 0
$$958$$ 26.0000 0.840022
$$959$$ 0 0
$$960$$ 0 0
$$961$$ 5.00000 0.161290
$$962$$ 21.0000 0.677067
$$963$$ 0 0
$$964$$ 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
$$969$$ 0 0
$$970$$ −16.0000 −0.513729
$$971$$ −43.0000 −1.37994 −0.689968 0.723840i $$-0.742375\pi$$
−0.689968 + 0.723840i $$0.742375\pi$$
$$972$$ 0 0
$$973$$ 0 0
$$974$$ 8.00000 0.256337
$$975$$ 0 0
$$976$$ −4.00000 −0.128037
$$977$$ −18.0000 −0.575871 −0.287936 0.957650i $$-0.592969\pi$$
−0.287936 + 0.957650i $$0.592969\pi$$
$$978$$ 0 0
$$979$$ 2.00000 0.0639203
$$980$$ 0 0
$$981$$ 0 0
$$982$$ −12.0000 −0.382935
$$983$$ 33.0000 1.05254 0.526268 0.850319i $$-0.323591\pi$$
0.526268 + 0.850319i $$0.323591\pi$$
$$984$$ 0 0
$$985$$ −3.00000 −0.0955879
$$986$$ −32.0000 −1.01909
$$987$$ 0 0
$$988$$ 7.00000 0.222700
$$989$$ 4.00000 0.127193
$$990$$ 0 0
$$991$$ 10.0000 0.317660 0.158830 0.987306i $$-0.449228\pi$$
0.158830 + 0.987306i $$0.449228\pi$$
$$992$$ −6.00000 −0.190500
$$993$$ 0 0
$$994$$ 0 0
$$995$$ 12.0000 0.380426
$$996$$ 0 0
$$997$$ 10.0000 0.316703 0.158352 0.987383i $$-0.449382\pi$$
0.158352 + 0.987383i $$0.449382\pi$$
$$998$$ −24.0000 −0.759707
$$999$$ 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 4410.2.a.g.1.1 1
3.2 odd 2 1470.2.a.r.1.1 1
7.2 even 3 630.2.k.h.361.1 2
7.4 even 3 630.2.k.h.541.1 2
7.6 odd 2 4410.2.a.q.1.1 1
15.14 odd 2 7350.2.a.j.1.1 1
21.2 odd 6 210.2.i.a.151.1 yes 2
21.5 even 6 1470.2.i.i.361.1 2
21.11 odd 6 210.2.i.a.121.1 2
21.17 even 6 1470.2.i.i.961.1 2
21.20 even 2 1470.2.a.k.1.1 1
84.11 even 6 1680.2.bg.k.961.1 2
84.23 even 6 1680.2.bg.k.1201.1 2
105.2 even 12 1050.2.o.j.949.2 4
105.23 even 12 1050.2.o.j.949.1 4
105.32 even 12 1050.2.o.j.499.1 4
105.44 odd 6 1050.2.i.s.151.1 2
105.53 even 12 1050.2.o.j.499.2 4
105.74 odd 6 1050.2.i.s.751.1 2
105.104 even 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 21.11 odd 6
210.2.i.a.151.1 yes 2 21.2 odd 6
630.2.k.h.361.1 2 7.2 even 3
630.2.k.h.541.1 2 7.4 even 3
1050.2.i.s.151.1 2 105.44 odd 6
1050.2.i.s.751.1 2 105.74 odd 6
1050.2.o.j.499.1 4 105.32 even 12
1050.2.o.j.499.2 4 105.53 even 12
1050.2.o.j.949.1 4 105.23 even 12
1050.2.o.j.949.2 4 105.2 even 12
1470.2.a.k.1.1 1 21.20 even 2
1470.2.a.r.1.1 1 3.2 odd 2
1470.2.i.i.361.1 2 21.5 even 6
1470.2.i.i.961.1 2 21.17 even 6
1680.2.bg.k.961.1 2 84.11 even 6
1680.2.bg.k.1201.1 2 84.23 even 6
4410.2.a.g.1.1 1 1.1 even 1 trivial
4410.2.a.q.1.1 1 7.6 odd 2
7350.2.a.j.1.1 1 15.14 odd 2
7350.2.a.ba.1.1 1 105.104 even 2