# Properties

 Label 9408.2.a.ej.1.3 Level $9408$ Weight $2$ Character 9408.1 Self dual yes Analytic conductor $75.123$ Analytic rank $0$ Dimension $3$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$9408 = 2^{6} \cdot 3 \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 9408.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$75.1232582216$$ Analytic rank: $$0$$ Dimension: $$3$$ Coefficient field: 3.3.621.1 Defining polynomial: $$x^{3} - 6 x - 3$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 672) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.3 Root $$-0.523976$$ of defining polynomial Character $$\chi$$ $$=$$ 9408.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q+1.00000 q^{3} +3.20147 q^{5} +1.00000 q^{9} +O(q^{10})$$ $$q+1.00000 q^{3} +3.20147 q^{5} +1.00000 q^{9} +4.24943 q^{11} +3.15352 q^{13} +3.20147 q^{15} -4.40294 q^{17} +3.15352 q^{19} -4.40294 q^{23} +5.24943 q^{25} +1.00000 q^{27} -7.20147 q^{29} -2.04795 q^{31} +4.24943 q^{33} +9.65237 q^{37} +3.15352 q^{39} -10.4989 q^{41} +0.750575 q^{43} +3.20147 q^{45} +2.40294 q^{47} -4.40294 q^{51} +3.29738 q^{53} +13.6044 q^{55} +3.15352 q^{57} +8.24943 q^{59} +8.09591 q^{61} +10.0959 q^{65} +5.15352 q^{67} -4.40294 q^{69} -6.40294 q^{71} +15.2494 q^{73} +5.24943 q^{75} +16.4509 q^{79} +1.00000 q^{81} +14.5565 q^{83} -14.0959 q^{85} -7.20147 q^{87} +2.40294 q^{89} -2.04795 q^{93} +10.0959 q^{95} +4.24943 q^{97} +4.24943 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3q + 3q^{3} + 3q^{9} + O(q^{10})$$ $$3q + 3q^{3} + 3q^{9} + 3q^{13} + 6q^{17} + 3q^{19} + 6q^{23} + 3q^{25} + 3q^{27} - 12q^{29} - 3q^{31} - 3q^{37} + 3q^{39} - 6q^{41} + 15q^{43} - 12q^{47} + 6q^{51} - 6q^{53} + 12q^{55} + 3q^{57} + 12q^{59} + 18q^{61} + 24q^{65} + 9q^{67} + 6q^{69} + 33q^{73} + 3q^{75} + 27q^{79} + 3q^{81} + 18q^{83} - 36q^{85} - 12q^{87} - 12q^{89} - 3q^{93} + 24q^{95} + O(q^{100})$$

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 0 0
$$3$$ 1.00000 0.577350
$$4$$ 0 0
$$5$$ 3.20147 1.43174 0.715871 0.698233i $$-0.246030\pi$$
0.715871 + 0.698233i $$0.246030\pi$$
$$6$$ 0 0
$$7$$ 0 0
$$8$$ 0 0
$$9$$ 1.00000 0.333333
$$10$$ 0 0
$$11$$ 4.24943 1.28125 0.640625 0.767854i $$-0.278675\pi$$
0.640625 + 0.767854i $$0.278675\pi$$
$$12$$ 0 0
$$13$$ 3.15352 0.874629 0.437314 0.899309i $$-0.355930\pi$$
0.437314 + 0.899309i $$0.355930\pi$$
$$14$$ 0 0
$$15$$ 3.20147 0.826617
$$16$$ 0 0
$$17$$ −4.40294 −1.06787 −0.533935 0.845525i $$-0.679287\pi$$
−0.533935 + 0.845525i $$0.679287\pi$$
$$18$$ 0 0
$$19$$ 3.15352 0.723467 0.361734 0.932282i $$-0.382185\pi$$
0.361734 + 0.932282i $$0.382185\pi$$
$$20$$ 0 0
$$21$$ 0 0
$$22$$ 0 0
$$23$$ −4.40294 −0.918077 −0.459039 0.888416i $$-0.651806\pi$$
−0.459039 + 0.888416i $$0.651806\pi$$
$$24$$ 0 0
$$25$$ 5.24943 1.04989
$$26$$ 0 0
$$27$$ 1.00000 0.192450
$$28$$ 0 0
$$29$$ −7.20147 −1.33728 −0.668640 0.743586i $$-0.733123\pi$$
−0.668640 + 0.743586i $$0.733123\pi$$
$$30$$ 0 0
$$31$$ −2.04795 −0.367823 −0.183912 0.982943i $$-0.558876\pi$$
−0.183912 + 0.982943i $$0.558876\pi$$
$$32$$ 0 0
$$33$$ 4.24943 0.739730
$$34$$ 0 0
$$35$$ 0 0
$$36$$ 0 0
$$37$$ 9.65237 1.58684 0.793420 0.608675i $$-0.208299\pi$$
0.793420 + 0.608675i $$0.208299\pi$$
$$38$$ 0 0
$$39$$ 3.15352 0.504967
$$40$$ 0 0
$$41$$ −10.4989 −1.63964 −0.819822 0.572618i $$-0.805928\pi$$
−0.819822 + 0.572618i $$0.805928\pi$$
$$42$$ 0 0
$$43$$ 0.750575 0.114462 0.0572308 0.998361i $$-0.481773\pi$$
0.0572308 + 0.998361i $$0.481773\pi$$
$$44$$ 0 0
$$45$$ 3.20147 0.477247
$$46$$ 0 0
$$47$$ 2.40294 0.350506 0.175253 0.984523i $$-0.443926\pi$$
0.175253 + 0.984523i $$0.443926\pi$$
$$48$$ 0 0
$$49$$ 0 0
$$50$$ 0 0
$$51$$ −4.40294 −0.616536
$$52$$ 0 0
$$53$$ 3.29738 0.452930 0.226465 0.974019i $$-0.427283\pi$$
0.226465 + 0.974019i $$0.427283\pi$$
$$54$$ 0 0
$$55$$ 13.6044 1.83442
$$56$$ 0 0
$$57$$ 3.15352 0.417694
$$58$$ 0 0
$$59$$ 8.24943 1.07398 0.536992 0.843587i $$-0.319561\pi$$
0.536992 + 0.843587i $$0.319561\pi$$
$$60$$ 0 0
$$61$$ 8.09591 1.03657 0.518287 0.855207i $$-0.326570\pi$$
0.518287 + 0.855207i $$0.326570\pi$$
$$62$$ 0 0
$$63$$ 0 0
$$64$$ 0 0
$$65$$ 10.0959 1.25224
$$66$$ 0 0
$$67$$ 5.15352 0.629603 0.314801 0.949158i $$-0.398062\pi$$
0.314801 + 0.949158i $$0.398062\pi$$
$$68$$ 0 0
$$69$$ −4.40294 −0.530052
$$70$$ 0 0
$$71$$ −6.40294 −0.759890 −0.379945 0.925009i $$-0.624057\pi$$
−0.379945 + 0.925009i $$0.624057\pi$$
$$72$$ 0 0
$$73$$ 15.2494 1.78481 0.892405 0.451235i $$-0.149016\pi$$
0.892405 + 0.451235i $$0.149016\pi$$
$$74$$ 0 0
$$75$$ 5.24943 0.606151
$$76$$ 0 0
$$77$$ 0 0
$$78$$ 0 0
$$79$$ 16.4509 1.85087 0.925435 0.378906i $$-0.123700\pi$$
0.925435 + 0.378906i $$0.123700\pi$$
$$80$$ 0 0
$$81$$ 1.00000 0.111111
$$82$$ 0 0
$$83$$ 14.5565 1.59778 0.798890 0.601477i $$-0.205421\pi$$
0.798890 + 0.601477i $$0.205421\pi$$
$$84$$ 0 0
$$85$$ −14.0959 −1.52892
$$86$$ 0 0
$$87$$ −7.20147 −0.772079
$$88$$ 0 0
$$89$$ 2.40294 0.254712 0.127356 0.991857i $$-0.459351\pi$$
0.127356 + 0.991857i $$0.459351\pi$$
$$90$$ 0 0
$$91$$ 0 0
$$92$$ 0 0
$$93$$ −2.04795 −0.212363
$$94$$ 0 0
$$95$$ 10.0959 1.03582
$$96$$ 0 0
$$97$$ 4.24943 0.431464 0.215732 0.976453i $$-0.430786\pi$$
0.215732 + 0.976453i $$0.430786\pi$$
$$98$$ 0 0
$$99$$ 4.24943 0.427083
$$100$$ 0 0
$$101$$ −3.90409 −0.388472 −0.194236 0.980955i $$-0.562223\pi$$
−0.194236 + 0.980955i $$0.562223\pi$$
$$102$$ 0 0
$$103$$ −9.24943 −0.911373 −0.455686 0.890140i $$-0.650606\pi$$
−0.455686 + 0.890140i $$0.650606\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ 16.5565 1.60057 0.800287 0.599617i $$-0.204681\pi$$
0.800287 + 0.599617i $$0.204681\pi$$
$$108$$ 0 0
$$109$$ −20.0553 −1.92095 −0.960475 0.278365i $$-0.910208\pi$$
−0.960475 + 0.278365i $$0.910208\pi$$
$$110$$ 0 0
$$111$$ 9.65237 0.916162
$$112$$ 0 0
$$113$$ 8.00000 0.752577 0.376288 0.926503i $$-0.377200\pi$$
0.376288 + 0.926503i $$0.377200\pi$$
$$114$$ 0 0
$$115$$ −14.0959 −1.31445
$$116$$ 0 0
$$117$$ 3.15352 0.291543
$$118$$ 0 0
$$119$$ 0 0
$$120$$ 0 0
$$121$$ 7.05761 0.641601
$$122$$ 0 0
$$123$$ −10.4989 −0.946649
$$124$$ 0 0
$$125$$ 0.798528 0.0714225
$$126$$ 0 0
$$127$$ −12.4509 −1.10484 −0.552419 0.833566i $$-0.686295\pi$$
−0.552419 + 0.833566i $$0.686295\pi$$
$$128$$ 0 0
$$129$$ 0.750575 0.0660844
$$130$$ 0 0
$$131$$ 4.24943 0.371274 0.185637 0.982618i $$-0.440565\pi$$
0.185637 + 0.982618i $$0.440565\pi$$
$$132$$ 0 0
$$133$$ 0 0
$$134$$ 0 0
$$135$$ 3.20147 0.275539
$$136$$ 0 0
$$137$$ −10.4029 −0.888784 −0.444392 0.895833i $$-0.646580\pi$$
−0.444392 + 0.895833i $$0.646580\pi$$
$$138$$ 0 0
$$139$$ 11.6524 0.988341 0.494171 0.869365i $$-0.335472\pi$$
0.494171 + 0.869365i $$0.335472\pi$$
$$140$$ 0 0
$$141$$ 2.40294 0.202364
$$142$$ 0 0
$$143$$ 13.4006 1.12062
$$144$$ 0 0
$$145$$ −23.0553 −1.91464
$$146$$ 0 0
$$147$$ 0 0
$$148$$ 0 0
$$149$$ −10.4029 −0.852242 −0.426121 0.904666i $$-0.640120\pi$$
−0.426121 + 0.904666i $$0.640120\pi$$
$$150$$ 0 0
$$151$$ 20.0074 1.62818 0.814088 0.580742i $$-0.197237\pi$$
0.814088 + 0.580742i $$0.197237\pi$$
$$152$$ 0 0
$$153$$ −4.40294 −0.355957
$$154$$ 0 0
$$155$$ −6.55646 −0.526628
$$156$$ 0 0
$$157$$ −14.4989 −1.15713 −0.578567 0.815635i $$-0.696388\pi$$
−0.578567 + 0.815635i $$0.696388\pi$$
$$158$$ 0 0
$$159$$ 3.29738 0.261499
$$160$$ 0 0
$$161$$ 0 0
$$162$$ 0 0
$$163$$ −10.7100 −0.838871 −0.419435 0.907785i $$-0.637772\pi$$
−0.419435 + 0.907785i $$0.637772\pi$$
$$164$$ 0 0
$$165$$ 13.6044 1.05910
$$166$$ 0 0
$$167$$ −17.3047 −1.33908 −0.669540 0.742776i $$-0.733509\pi$$
−0.669540 + 0.742776i $$0.733509\pi$$
$$168$$ 0 0
$$169$$ −3.05531 −0.235024
$$170$$ 0 0
$$171$$ 3.15352 0.241156
$$172$$ 0 0
$$173$$ 3.90409 0.296823 0.148411 0.988926i $$-0.452584\pi$$
0.148411 + 0.988926i $$0.452584\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ 0 0
$$177$$ 8.24943 0.620065
$$178$$ 0 0
$$179$$ 20.8059 1.55511 0.777553 0.628818i $$-0.216461\pi$$
0.777553 + 0.628818i $$0.216461\pi$$
$$180$$ 0 0
$$181$$ 13.2494 0.984822 0.492411 0.870363i $$-0.336116\pi$$
0.492411 + 0.870363i $$0.336116\pi$$
$$182$$ 0 0
$$183$$ 8.09591 0.598467
$$184$$ 0 0
$$185$$ 30.9018 2.27195
$$186$$ 0 0
$$187$$ −18.7100 −1.36821
$$188$$ 0 0
$$189$$ 0 0
$$190$$ 0 0
$$191$$ −4.19181 −0.303309 −0.151654 0.988434i $$-0.548460\pi$$
−0.151654 + 0.988434i $$0.548460\pi$$
$$192$$ 0 0
$$193$$ −18.3047 −1.31760 −0.658802 0.752316i $$-0.728936\pi$$
−0.658802 + 0.752316i $$0.728936\pi$$
$$194$$ 0 0
$$195$$ 10.0959 0.722983
$$196$$ 0 0
$$197$$ −11.9041 −0.848132 −0.424066 0.905631i $$-0.639397\pi$$
−0.424066 + 0.905631i $$0.639397\pi$$
$$198$$ 0 0
$$199$$ −0.805889 −0.0571280 −0.0285640 0.999592i $$-0.509093\pi$$
−0.0285640 + 0.999592i $$0.509093\pi$$
$$200$$ 0 0
$$201$$ 5.15352 0.363501
$$202$$ 0 0
$$203$$ 0 0
$$204$$ 0 0
$$205$$ −33.6118 −2.34755
$$206$$ 0 0
$$207$$ −4.40294 −0.306026
$$208$$ 0 0
$$209$$ 13.4006 0.926942
$$210$$ 0 0
$$211$$ 8.49885 0.585085 0.292542 0.956253i $$-0.405499\pi$$
0.292542 + 0.956253i $$0.405499\pi$$
$$212$$ 0 0
$$213$$ −6.40294 −0.438723
$$214$$ 0 0
$$215$$ 2.40294 0.163879
$$216$$ 0 0
$$217$$ 0 0
$$218$$ 0 0
$$219$$ 15.2494 1.03046
$$220$$ 0 0
$$221$$ −13.8848 −0.933991
$$222$$ 0 0
$$223$$ 16.7985 1.12491 0.562456 0.826827i $$-0.309856\pi$$
0.562456 + 0.826827i $$0.309856\pi$$
$$224$$ 0 0
$$225$$ 5.24943 0.349962
$$226$$ 0 0
$$227$$ −16.8635 −1.11927 −0.559635 0.828739i $$-0.689059\pi$$
−0.559635 + 0.828739i $$0.689059\pi$$
$$228$$ 0 0
$$229$$ −20.5542 −1.35826 −0.679129 0.734019i $$-0.737642\pi$$
−0.679129 + 0.734019i $$0.737642\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 0 0
$$233$$ 9.90409 0.648839 0.324419 0.945913i $$-0.394831\pi$$
0.324419 + 0.945913i $$0.394831\pi$$
$$234$$ 0 0
$$235$$ 7.69296 0.501833
$$236$$ 0 0
$$237$$ 16.4509 1.06860
$$238$$ 0 0
$$239$$ −4.40294 −0.284803 −0.142401 0.989809i $$-0.545482\pi$$
−0.142401 + 0.989809i $$0.545482\pi$$
$$240$$ 0 0
$$241$$ 20.8635 1.34394 0.671968 0.740580i $$-0.265449\pi$$
0.671968 + 0.740580i $$0.265449\pi$$
$$242$$ 0 0
$$243$$ 1.00000 0.0641500
$$244$$ 0 0
$$245$$ 0 0
$$246$$ 0 0
$$247$$ 9.94469 0.632765
$$248$$ 0 0
$$249$$ 14.5565 0.922478
$$250$$ 0 0
$$251$$ 3.44354 0.217354 0.108677 0.994077i $$-0.465339\pi$$
0.108677 + 0.994077i $$0.465339\pi$$
$$252$$ 0 0
$$253$$ −18.7100 −1.17629
$$254$$ 0 0
$$255$$ −14.0959 −0.882720
$$256$$ 0 0
$$257$$ −5.59706 −0.349135 −0.174567 0.984645i $$-0.555853\pi$$
−0.174567 + 0.984645i $$0.555853\pi$$
$$258$$ 0 0
$$259$$ 0 0
$$260$$ 0 0
$$261$$ −7.20147 −0.445760
$$262$$ 0 0
$$263$$ 1.59706 0.0984787 0.0492393 0.998787i $$-0.484320\pi$$
0.0492393 + 0.998787i $$0.484320\pi$$
$$264$$ 0 0
$$265$$ 10.5565 0.648478
$$266$$ 0 0
$$267$$ 2.40294 0.147058
$$268$$ 0 0
$$269$$ −17.7003 −1.07921 −0.539604 0.841919i $$-0.681426\pi$$
−0.539604 + 0.841919i $$0.681426\pi$$
$$270$$ 0 0
$$271$$ 2.39558 0.145521 0.0727607 0.997349i $$-0.476819\pi$$
0.0727607 + 0.997349i $$0.476819\pi$$
$$272$$ 0 0
$$273$$ 0 0
$$274$$ 0 0
$$275$$ 22.3070 1.34517
$$276$$ 0 0
$$277$$ 5.24943 0.315407 0.157704 0.987486i $$-0.449591\pi$$
0.157704 + 0.987486i $$0.449591\pi$$
$$278$$ 0 0
$$279$$ −2.04795 −0.122608
$$280$$ 0 0
$$281$$ 20.0959 1.19882 0.599411 0.800442i $$-0.295402\pi$$
0.599411 + 0.800442i $$0.295402\pi$$
$$282$$ 0 0
$$283$$ −23.2494 −1.38203 −0.691017 0.722838i $$-0.742837\pi$$
−0.691017 + 0.722838i $$0.742837\pi$$
$$284$$ 0 0
$$285$$ 10.0959 0.598030
$$286$$ 0 0
$$287$$ 0 0
$$288$$ 0 0
$$289$$ 2.38592 0.140348
$$290$$ 0 0
$$291$$ 4.24943 0.249106
$$292$$ 0 0
$$293$$ −11.2015 −0.654397 −0.327199 0.944956i $$-0.606105\pi$$
−0.327199 + 0.944956i $$0.606105\pi$$
$$294$$ 0 0
$$295$$ 26.4103 1.53767
$$296$$ 0 0
$$297$$ 4.24943 0.246577
$$298$$ 0 0
$$299$$ −13.8848 −0.802977
$$300$$ 0 0
$$301$$ 0 0
$$302$$ 0 0
$$303$$ −3.90409 −0.224284
$$304$$ 0 0
$$305$$ 25.9188 1.48411
$$306$$ 0 0
$$307$$ −28.9571 −1.65267 −0.826335 0.563179i $$-0.809578\pi$$
−0.826335 + 0.563179i $$0.809578\pi$$
$$308$$ 0 0
$$309$$ −9.24943 −0.526181
$$310$$ 0 0
$$311$$ −27.3047 −1.54831 −0.774155 0.632996i $$-0.781825\pi$$
−0.774155 + 0.632996i $$0.781825\pi$$
$$312$$ 0 0
$$313$$ −5.49885 −0.310813 −0.155407 0.987851i $$-0.549669\pi$$
−0.155407 + 0.987851i $$0.549669\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ 9.60442 0.539438 0.269719 0.962939i $$-0.413069\pi$$
0.269719 + 0.962939i $$0.413069\pi$$
$$318$$ 0 0
$$319$$ −30.6021 −1.71339
$$320$$ 0 0
$$321$$ 16.5565 0.924092
$$322$$ 0 0
$$323$$ −13.8848 −0.772569
$$324$$ 0 0
$$325$$ 16.5542 0.918260
$$326$$ 0 0
$$327$$ −20.0553 −1.10906
$$328$$ 0 0
$$329$$ 0 0
$$330$$ 0 0
$$331$$ 15.5565 0.855061 0.427530 0.904001i $$-0.359384\pi$$
0.427530 + 0.904001i $$0.359384\pi$$
$$332$$ 0 0
$$333$$ 9.65237 0.528947
$$334$$ 0 0
$$335$$ 16.4989 0.901428
$$336$$ 0 0
$$337$$ 17.9977 0.980397 0.490199 0.871611i $$-0.336924\pi$$
0.490199 + 0.871611i $$0.336924\pi$$
$$338$$ 0 0
$$339$$ 8.00000 0.434500
$$340$$ 0 0
$$341$$ −8.70262 −0.471273
$$342$$ 0 0
$$343$$ 0 0
$$344$$ 0 0
$$345$$ −14.0959 −0.758898
$$346$$ 0 0
$$347$$ 5.69296 0.305614 0.152807 0.988256i $$-0.451169\pi$$
0.152807 + 0.988256i $$0.451169\pi$$
$$348$$ 0 0
$$349$$ 18.1918 0.973785 0.486893 0.873462i $$-0.338130\pi$$
0.486893 + 0.873462i $$0.338130\pi$$
$$350$$ 0 0
$$351$$ 3.15352 0.168322
$$352$$ 0 0
$$353$$ −20.9977 −1.11759 −0.558797 0.829304i $$-0.688737\pi$$
−0.558797 + 0.829304i $$0.688737\pi$$
$$354$$ 0 0
$$355$$ −20.4989 −1.08797
$$356$$ 0 0
$$357$$ 0 0
$$358$$ 0 0
$$359$$ 24.6907 1.30312 0.651562 0.758596i $$-0.274114\pi$$
0.651562 + 0.758596i $$0.274114\pi$$
$$360$$ 0 0
$$361$$ −9.05531 −0.476595
$$362$$ 0 0
$$363$$ 7.05761 0.370429
$$364$$ 0 0
$$365$$ 48.8206 2.55539
$$366$$ 0 0
$$367$$ 5.85614 0.305688 0.152844 0.988250i $$-0.451157\pi$$
0.152844 + 0.988250i $$0.451157\pi$$
$$368$$ 0 0
$$369$$ −10.4989 −0.546548
$$370$$ 0 0
$$371$$ 0 0
$$372$$ 0 0
$$373$$ −0.635347 −0.0328970 −0.0164485 0.999865i $$-0.505236\pi$$
−0.0164485 + 0.999865i $$0.505236\pi$$
$$374$$ 0 0
$$375$$ 0.798528 0.0412358
$$376$$ 0 0
$$377$$ −22.7100 −1.16962
$$378$$ 0 0
$$379$$ 10.7506 0.552220 0.276110 0.961126i $$-0.410955\pi$$
0.276110 + 0.961126i $$0.410955\pi$$
$$380$$ 0 0
$$381$$ −12.4509 −0.637879
$$382$$ 0 0
$$383$$ −14.9977 −0.766347 −0.383173 0.923676i $$-0.625169\pi$$
−0.383173 + 0.923676i $$0.625169\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 0 0
$$387$$ 0.750575 0.0381539
$$388$$ 0 0
$$389$$ −6.21113 −0.314917 −0.157458 0.987526i $$-0.550330\pi$$
−0.157458 + 0.987526i $$0.550330\pi$$
$$390$$ 0 0
$$391$$ 19.3859 0.980388
$$392$$ 0 0
$$393$$ 4.24943 0.214355
$$394$$ 0 0
$$395$$ 52.6671 2.64997
$$396$$ 0 0
$$397$$ −1.24943 −0.0627068 −0.0313534 0.999508i $$-0.509982\pi$$
−0.0313534 + 0.999508i $$0.509982\pi$$
$$398$$ 0 0
$$399$$ 0 0
$$400$$ 0 0
$$401$$ 2.11523 0.105629 0.0528147 0.998604i $$-0.483181\pi$$
0.0528147 + 0.998604i $$0.483181\pi$$
$$402$$ 0 0
$$403$$ −6.45826 −0.321709
$$404$$ 0 0
$$405$$ 3.20147 0.159082
$$406$$ 0 0
$$407$$ 41.0170 2.03314
$$408$$ 0 0
$$409$$ −16.1129 −0.796733 −0.398367 0.917226i $$-0.630423\pi$$
−0.398367 + 0.917226i $$0.630423\pi$$
$$410$$ 0 0
$$411$$ −10.4029 −0.513139
$$412$$ 0 0
$$413$$ 0 0
$$414$$ 0 0
$$415$$ 46.6021 2.28761
$$416$$ 0 0
$$417$$ 11.6524 0.570619
$$418$$ 0 0
$$419$$ −20.1106 −0.982469 −0.491234 0.871027i $$-0.663454\pi$$
−0.491234 + 0.871027i $$0.663454\pi$$
$$420$$ 0 0
$$421$$ −4.96171 −0.241819 −0.120909 0.992664i $$-0.538581\pi$$
−0.120909 + 0.992664i $$0.538581\pi$$
$$422$$ 0 0
$$423$$ 2.40294 0.116835
$$424$$ 0 0
$$425$$ −23.1129 −1.12114
$$426$$ 0 0
$$427$$ 0 0
$$428$$ 0 0
$$429$$ 13.4006 0.646989
$$430$$ 0 0
$$431$$ 20.8059 1.00218 0.501092 0.865394i $$-0.332932\pi$$
0.501092 + 0.865394i $$0.332932\pi$$
$$432$$ 0 0
$$433$$ −8.05531 −0.387114 −0.193557 0.981089i $$-0.562002\pi$$
−0.193557 + 0.981089i $$0.562002\pi$$
$$434$$ 0 0
$$435$$ −23.0553 −1.10542
$$436$$ 0 0
$$437$$ −13.8848 −0.664199
$$438$$ 0 0
$$439$$ 16.1992 0.773144 0.386572 0.922259i $$-0.373659\pi$$
0.386572 + 0.922259i $$0.373659\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ 0 0
$$443$$ 15.4435 0.733745 0.366872 0.930271i $$-0.380429\pi$$
0.366872 + 0.930271i $$0.380429\pi$$
$$444$$ 0 0
$$445$$ 7.69296 0.364681
$$446$$ 0 0
$$447$$ −10.4029 −0.492042
$$448$$ 0 0
$$449$$ 31.5159 1.48733 0.743663 0.668555i $$-0.233087\pi$$
0.743663 + 0.668555i $$0.233087\pi$$
$$450$$ 0 0
$$451$$ −44.6141 −2.10079
$$452$$ 0 0
$$453$$ 20.0074 0.940028
$$454$$ 0 0
$$455$$ 0 0
$$456$$ 0 0
$$457$$ −11.1152 −0.519948 −0.259974 0.965616i $$-0.583714\pi$$
−0.259974 + 0.965616i $$0.583714\pi$$
$$458$$ 0 0
$$459$$ −4.40294 −0.205512
$$460$$ 0 0
$$461$$ −13.5971 −0.633278 −0.316639 0.948546i $$-0.602554\pi$$
−0.316639 + 0.948546i $$0.602554\pi$$
$$462$$ 0 0
$$463$$ −2.55876 −0.118916 −0.0594579 0.998231i $$-0.518937\pi$$
−0.0594579 + 0.998231i $$0.518937\pi$$
$$464$$ 0 0
$$465$$ −6.55646 −0.304049
$$466$$ 0 0
$$467$$ −0.614078 −0.0284161 −0.0142081 0.999899i $$-0.504523\pi$$
−0.0142081 + 0.999899i $$0.504523\pi$$
$$468$$ 0 0
$$469$$ 0 0
$$470$$ 0 0
$$471$$ −14.4989 −0.668072
$$472$$ 0 0
$$473$$ 3.18951 0.146654
$$474$$ 0 0
$$475$$ 16.5542 0.759557
$$476$$ 0 0
$$477$$ 3.29738 0.150977
$$478$$ 0 0
$$479$$ −20.0959 −0.918205 −0.459103 0.888383i $$-0.651829\pi$$
−0.459103 + 0.888383i $$0.651829\pi$$
$$480$$ 0 0
$$481$$ 30.4389 1.38790
$$482$$ 0 0
$$483$$ 0 0
$$484$$ 0 0
$$485$$ 13.6044 0.617745
$$486$$ 0 0
$$487$$ −23.3720 −1.05909 −0.529544 0.848283i $$-0.677637\pi$$
−0.529544 + 0.848283i $$0.677637\pi$$
$$488$$ 0 0
$$489$$ −10.7100 −0.484322
$$490$$ 0 0
$$491$$ 22.7483 1.02662 0.513308 0.858205i $$-0.328420\pi$$
0.513308 + 0.858205i $$0.328420\pi$$
$$492$$ 0 0
$$493$$ 31.7077 1.42804
$$494$$ 0 0
$$495$$ 13.6044 0.611473
$$496$$ 0 0
$$497$$ 0 0
$$498$$ 0 0
$$499$$ −8.53944 −0.382278 −0.191139 0.981563i $$-0.561218\pi$$
−0.191139 + 0.981563i $$0.561218\pi$$
$$500$$ 0 0
$$501$$ −17.3047 −0.773119
$$502$$ 0 0
$$503$$ −7.59706 −0.338736 −0.169368 0.985553i $$-0.554173\pi$$
−0.169368 + 0.985553i $$0.554173\pi$$
$$504$$ 0 0
$$505$$ −12.4989 −0.556192
$$506$$ 0 0
$$507$$ −3.05531 −0.135691
$$508$$ 0 0
$$509$$ 4.99034 0.221193 0.110596 0.993865i $$-0.464724\pi$$
0.110596 + 0.993865i $$0.464724\pi$$
$$510$$ 0 0
$$511$$ 0 0
$$512$$ 0 0
$$513$$ 3.15352 0.139231
$$514$$ 0 0
$$515$$ −29.6118 −1.30485
$$516$$ 0 0
$$517$$ 10.2111 0.449085
$$518$$ 0 0
$$519$$ 3.90409 0.171371
$$520$$ 0 0
$$521$$ −12.4989 −0.547585 −0.273792 0.961789i $$-0.588278\pi$$
−0.273792 + 0.961789i $$0.588278\pi$$
$$522$$ 0 0
$$523$$ 44.2618 1.93544 0.967718 0.252036i $$-0.0811003\pi$$
0.967718 + 0.252036i $$0.0811003\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ 9.01702 0.392788
$$528$$ 0 0
$$529$$ −3.61408 −0.157134
$$530$$ 0 0
$$531$$ 8.24943 0.357995
$$532$$ 0 0
$$533$$ −33.1083 −1.43408
$$534$$ 0 0
$$535$$ 53.0051 2.29161
$$536$$ 0 0
$$537$$ 20.8059 0.897840
$$538$$ 0 0
$$539$$ 0 0
$$540$$ 0 0
$$541$$ −25.9594 −1.11608 −0.558041 0.829813i $$-0.688447\pi$$
−0.558041 + 0.829813i $$0.688447\pi$$
$$542$$ 0 0
$$543$$ 13.2494 0.568587
$$544$$ 0 0
$$545$$ −64.2065 −2.75031
$$546$$ 0 0
$$547$$ 22.9018 0.979210 0.489605 0.871944i $$-0.337141\pi$$
0.489605 + 0.871944i $$0.337141\pi$$
$$548$$ 0 0
$$549$$ 8.09591 0.345525
$$550$$ 0 0
$$551$$ −22.7100 −0.967478
$$552$$ 0 0
$$553$$ 0 0
$$554$$ 0 0
$$555$$ 30.9018 1.31171
$$556$$ 0 0
$$557$$ 6.50621 0.275677 0.137839 0.990455i $$-0.455984\pi$$
0.137839 + 0.990455i $$0.455984\pi$$
$$558$$ 0 0
$$559$$ 2.36695 0.100111
$$560$$ 0 0
$$561$$ −18.7100 −0.789936
$$562$$ 0 0
$$563$$ −44.2448 −1.86470 −0.932349 0.361561i $$-0.882244\pi$$
−0.932349 + 0.361561i $$0.882244\pi$$
$$564$$ 0 0
$$565$$ 25.6118 1.07750
$$566$$ 0 0
$$567$$ 0 0
$$568$$ 0 0
$$569$$ −13.0170 −0.545702 −0.272851 0.962056i $$-0.587967\pi$$
−0.272851 + 0.962056i $$0.587967\pi$$
$$570$$ 0 0
$$571$$ 30.1705 1.26260 0.631299 0.775540i $$-0.282522\pi$$
0.631299 + 0.775540i $$0.282522\pi$$
$$572$$ 0 0
$$573$$ −4.19181 −0.175115
$$574$$ 0 0
$$575$$ −23.1129 −0.963876
$$576$$ 0 0
$$577$$ −29.8059 −1.24084 −0.620418 0.784272i $$-0.713037\pi$$
−0.620418 + 0.784272i $$0.713037\pi$$
$$578$$ 0 0
$$579$$ −18.3047 −0.760719
$$580$$ 0 0
$$581$$ 0 0
$$582$$ 0 0
$$583$$ 14.0120 0.580316
$$584$$ 0 0
$$585$$ 10.0959 0.417414
$$586$$ 0 0
$$587$$ 18.2494 0.753234 0.376617 0.926369i $$-0.377087\pi$$
0.376617 + 0.926369i $$0.377087\pi$$
$$588$$ 0 0
$$589$$ −6.45826 −0.266108
$$590$$ 0 0
$$591$$ −11.9041 −0.489669
$$592$$ 0 0
$$593$$ −23.7889 −0.976892 −0.488446 0.872594i $$-0.662436\pi$$
−0.488446 + 0.872594i $$0.662436\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 0 0
$$597$$ −0.805889 −0.0329829
$$598$$ 0 0
$$599$$ 15.5011 0.633360 0.316680 0.948532i $$-0.397432\pi$$
0.316680 + 0.948532i $$0.397432\pi$$
$$600$$ 0 0
$$601$$ 15.8059 0.644736 0.322368 0.946614i $$-0.395521\pi$$
0.322368 + 0.946614i $$0.395521\pi$$
$$602$$ 0 0
$$603$$ 5.15352 0.209868
$$604$$ 0 0
$$605$$ 22.5948 0.918607
$$606$$ 0 0
$$607$$ −5.95205 −0.241586 −0.120793 0.992678i $$-0.538544\pi$$
−0.120793 + 0.992678i $$0.538544\pi$$
$$608$$ 0 0
$$609$$ 0 0
$$610$$ 0 0
$$611$$ 7.57773 0.306562
$$612$$ 0 0
$$613$$ 28.4029 1.14718 0.573592 0.819141i $$-0.305549\pi$$
0.573592 + 0.819141i $$0.305549\pi$$
$$614$$ 0 0
$$615$$ −33.6118 −1.35536
$$616$$ 0 0
$$617$$ −10.1918 −0.410307 −0.205153 0.978730i $$-0.565769\pi$$
−0.205153 + 0.978730i $$0.565769\pi$$
$$618$$ 0 0
$$619$$ 20.8612 0.838483 0.419241 0.907875i $$-0.362296\pi$$
0.419241 + 0.907875i $$0.362296\pi$$
$$620$$ 0 0
$$621$$ −4.40294 −0.176684
$$622$$ 0 0
$$623$$ 0 0
$$624$$ 0 0
$$625$$ −23.6907 −0.947626
$$626$$ 0 0
$$627$$ 13.4006 0.535170
$$628$$ 0 0
$$629$$ −42.4989 −1.69454
$$630$$ 0 0
$$631$$ −41.6191 −1.65683 −0.828416 0.560113i $$-0.810758\pi$$
−0.828416 + 0.560113i $$0.810758\pi$$
$$632$$ 0 0
$$633$$ 8.49885 0.337799
$$634$$ 0 0
$$635$$ −39.8612 −1.58184
$$636$$ 0 0
$$637$$ 0 0
$$638$$ 0 0
$$639$$ −6.40294 −0.253297
$$640$$ 0 0
$$641$$ −48.8013 −1.92754 −0.963768 0.266744i $$-0.914052\pi$$
−0.963768 + 0.266744i $$0.914052\pi$$
$$642$$ 0 0
$$643$$ 11.8442 0.467089 0.233544 0.972346i $$-0.424968\pi$$
0.233544 + 0.972346i $$0.424968\pi$$
$$644$$ 0 0
$$645$$ 2.40294 0.0946159
$$646$$ 0 0
$$647$$ −33.5159 −1.31764 −0.658822 0.752298i $$-0.728945\pi$$
−0.658822 + 0.752298i $$0.728945\pi$$
$$648$$ 0 0
$$649$$ 35.0553 1.37604
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 0 0
$$653$$ 46.5062 1.81993 0.909964 0.414686i $$-0.136109\pi$$
0.909964 + 0.414686i $$0.136109\pi$$
$$654$$ 0 0
$$655$$ 13.6044 0.531569
$$656$$ 0 0
$$657$$ 15.2494 0.594937
$$658$$ 0 0
$$659$$ −10.1152 −0.394033 −0.197017 0.980400i $$-0.563125\pi$$
−0.197017 + 0.980400i $$0.563125\pi$$
$$660$$ 0 0
$$661$$ −33.2641 −1.29383 −0.646913 0.762564i $$-0.723940\pi$$
−0.646913 + 0.762564i $$0.723940\pi$$
$$662$$ 0 0
$$663$$ −13.8848 −0.539240
$$664$$ 0 0
$$665$$ 0 0
$$666$$ 0 0
$$667$$ 31.7077 1.22773
$$668$$ 0 0
$$669$$ 16.7985 0.649469
$$670$$ 0 0
$$671$$ 34.4029 1.32811
$$672$$ 0 0
$$673$$ −50.4966 −1.94650 −0.973249 0.229751i $$-0.926209\pi$$
−0.973249 + 0.229751i $$0.926209\pi$$
$$674$$ 0 0
$$675$$ 5.24943 0.202050
$$676$$ 0 0
$$677$$ −28.7985 −1.10682 −0.553409 0.832910i $$-0.686673\pi$$
−0.553409 + 0.832910i $$0.686673\pi$$
$$678$$ 0 0
$$679$$ 0 0
$$680$$ 0 0
$$681$$ −16.8635 −0.646211
$$682$$ 0 0
$$683$$ −13.2471 −0.506887 −0.253444 0.967350i $$-0.581563\pi$$
−0.253444 + 0.967350i $$0.581563\pi$$
$$684$$ 0 0
$$685$$ −33.3047 −1.27251
$$686$$ 0 0
$$687$$ −20.5542 −0.784190
$$688$$ 0 0
$$689$$ 10.3983 0.396145
$$690$$ 0 0
$$691$$ −39.6671 −1.50901 −0.754504 0.656296i $$-0.772122\pi$$
−0.754504 + 0.656296i $$0.772122\pi$$
$$692$$ 0 0
$$693$$ 0 0
$$694$$ 0 0
$$695$$ 37.3047 1.41505
$$696$$ 0 0
$$697$$ 46.2259 1.75093
$$698$$ 0 0
$$699$$ 9.90409 0.374607
$$700$$ 0 0
$$701$$ 2.10787 0.0796130 0.0398065 0.999207i $$-0.487326\pi$$
0.0398065 + 0.999207i $$0.487326\pi$$
$$702$$ 0 0
$$703$$ 30.4389 1.14803
$$704$$ 0 0
$$705$$ 7.69296 0.289734
$$706$$ 0 0
$$707$$ 0 0
$$708$$ 0 0
$$709$$ 14.9977 0.563250 0.281625 0.959524i $$-0.409127\pi$$
0.281625 + 0.959524i $$0.409127\pi$$
$$710$$ 0 0
$$711$$ 16.4509 0.616957
$$712$$ 0 0
$$713$$ 9.01702 0.337690
$$714$$ 0 0
$$715$$ 42.9018 1.60444
$$716$$ 0 0
$$717$$ −4.40294 −0.164431
$$718$$ 0 0
$$719$$ −22.8059 −0.850516 −0.425258 0.905072i $$-0.639817\pi$$
−0.425258 + 0.905072i $$0.639817\pi$$
$$720$$ 0 0
$$721$$ 0 0
$$722$$ 0 0
$$723$$ 20.8635 0.775922
$$724$$ 0 0
$$725$$ −37.8036 −1.40399
$$726$$ 0 0
$$727$$ −16.9691 −0.629348 −0.314674 0.949200i $$-0.601895\pi$$
−0.314674 + 0.949200i $$0.601895\pi$$
$$728$$ 0 0
$$729$$ 1.00000 0.0370370
$$730$$ 0 0
$$731$$ −3.30474 −0.122230
$$732$$ 0 0
$$733$$ −1.34533 −0.0496909 −0.0248455 0.999691i $$-0.507909\pi$$
−0.0248455 + 0.999691i $$0.507909\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ 21.8995 0.806678
$$738$$ 0 0
$$739$$ 45.2641 1.66507 0.832534 0.553973i $$-0.186889\pi$$
0.832534 + 0.553973i $$0.186889\pi$$
$$740$$ 0 0
$$741$$ 9.94469 0.365327
$$742$$ 0 0
$$743$$ 27.8036 1.02001 0.510007 0.860170i $$-0.329643\pi$$
0.510007 + 0.860170i $$0.329643\pi$$
$$744$$ 0 0
$$745$$ −33.3047 −1.22019
$$746$$ 0 0
$$747$$ 14.5565 0.532593
$$748$$ 0 0
$$749$$ 0 0
$$750$$ 0 0
$$751$$ −30.0627 −1.09700 −0.548501 0.836150i $$-0.684801\pi$$
−0.548501 + 0.836150i $$0.684801\pi$$
$$752$$ 0 0
$$753$$ 3.44354 0.125489
$$754$$ 0 0
$$755$$ 64.0530 2.33113
$$756$$ 0 0
$$757$$ 17.3241 0.629654 0.314827 0.949149i $$-0.398054\pi$$
0.314827 + 0.949149i $$0.398054\pi$$
$$758$$ 0 0
$$759$$ −18.7100 −0.679129
$$760$$ 0 0
$$761$$ 12.2111 0.442653 0.221327 0.975200i $$-0.428961\pi$$
0.221327 + 0.975200i $$0.428961\pi$$
$$762$$ 0 0
$$763$$ 0 0
$$764$$ 0 0
$$765$$ −14.0959 −0.509639
$$766$$ 0 0
$$767$$ 26.0147 0.939337
$$768$$ 0 0
$$769$$ 11.1152 0.400825 0.200413 0.979712i $$-0.435772\pi$$
0.200413 + 0.979712i $$0.435772\pi$$
$$770$$ 0 0
$$771$$ −5.59706 −0.201573
$$772$$ 0 0
$$773$$ 0.287717 0.0103485 0.00517423 0.999987i $$-0.498353\pi$$
0.00517423 + 0.999987i $$0.498353\pi$$
$$774$$ 0 0
$$775$$ −10.7506 −0.386172
$$776$$ 0 0
$$777$$ 0 0
$$778$$ 0 0
$$779$$ −33.1083 −1.18623
$$780$$ 0 0
$$781$$ −27.2088 −0.973609
$$782$$ 0 0
$$783$$ −7.20147 −0.257360
$$784$$ 0 0
$$785$$ −46.4177 −1.65672
$$786$$ 0 0
$$787$$ −11.5011 −0.409972 −0.204986 0.978765i $$-0.565715\pi$$
−0.204986 + 0.978765i $$0.565715\pi$$
$$788$$ 0 0
$$789$$ 1.59706 0.0568567
$$790$$ 0 0
$$791$$ 0 0
$$792$$ 0 0
$$793$$ 25.5306 0.906618
$$794$$ 0 0
$$795$$ 10.5565 0.374399
$$796$$ 0 0
$$797$$ 18.6980 0.662318 0.331159 0.943575i $$-0.392560\pi$$
0.331159 + 0.943575i $$0.392560\pi$$
$$798$$ 0 0
$$799$$ −10.5800 −0.374295
$$800$$ 0 0
$$801$$ 2.40294 0.0849039
$$802$$ 0 0
$$803$$ 64.8013 2.28679
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 0 0
$$807$$ −17.7003 −0.623081
$$808$$ 0 0
$$809$$ −53.8036 −1.89163 −0.945817 0.324701i $$-0.894736\pi$$
−0.945817 + 0.324701i $$0.894736\pi$$
$$810$$ 0 0
$$811$$ −34.7866 −1.22152 −0.610761 0.791815i $$-0.709136\pi$$
−0.610761 + 0.791815i $$0.709136\pi$$
$$812$$ 0 0
$$813$$ 2.39558 0.0840168
$$814$$ 0 0
$$815$$ −34.2877 −1.20105
$$816$$ 0 0
$$817$$ 2.36695 0.0828092
$$818$$ 0 0
$$819$$ 0 0
$$820$$ 0 0
$$821$$ −37.8110 −1.31961 −0.659806 0.751436i $$-0.729361\pi$$
−0.659806 + 0.751436i $$0.729361\pi$$
$$822$$ 0 0
$$823$$ 54.4177 1.89688 0.948440 0.316956i $$-0.102661\pi$$
0.948440 + 0.316956i $$0.102661\pi$$
$$824$$ 0 0
$$825$$ 22.3070 0.776631
$$826$$ 0 0
$$827$$ −7.86120 −0.273361 −0.136680 0.990615i $$-0.543643\pi$$
−0.136680 + 0.990615i $$0.543643\pi$$
$$828$$ 0 0
$$829$$ −6.45826 −0.224305 −0.112152 0.993691i $$-0.535774\pi$$
−0.112152 + 0.993691i $$0.535774\pi$$
$$830$$ 0 0
$$831$$ 5.24943 0.182101
$$832$$ 0 0
$$833$$ 0 0
$$834$$ 0 0
$$835$$ −55.4006 −1.91722
$$836$$ 0 0
$$837$$ −2.04795 −0.0707876
$$838$$ 0 0
$$839$$ 28.3218 0.977776 0.488888 0.872347i $$-0.337403\pi$$
0.488888 + 0.872347i $$0.337403\pi$$
$$840$$ 0 0
$$841$$ 22.8612 0.788317
$$842$$ 0 0
$$843$$ 20.0959 0.692140
$$844$$ 0 0
$$845$$ −9.78150 −0.336494
$$846$$ 0 0
$$847$$ 0 0
$$848$$ 0 0
$$849$$ −23.2494 −0.797918
$$850$$ 0 0
$$851$$ −42.4989 −1.45684
$$852$$ 0 0
$$853$$ −8.75057 −0.299614 −0.149807 0.988715i $$-0.547865\pi$$
−0.149807 + 0.988715i $$0.547865\pi$$
$$854$$ 0 0
$$855$$ 10.0959 0.345273
$$856$$ 0 0
$$857$$ −24.6907 −0.843417 −0.421708 0.906731i $$-0.638569\pi$$
−0.421708 + 0.906731i $$0.638569\pi$$
$$858$$ 0 0
$$859$$ −8.49885 −0.289977 −0.144989 0.989433i $$-0.546315\pi$$
−0.144989 + 0.989433i $$0.546315\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ 0 0
$$863$$ −18.6141 −0.633631 −0.316815 0.948487i $$-0.602614\pi$$
−0.316815 + 0.948487i $$0.602614\pi$$
$$864$$ 0 0
$$865$$ 12.4989 0.424974
$$866$$ 0 0
$$867$$ 2.38592 0.0810302
$$868$$ 0 0
$$869$$ 69.9069 2.37143
$$870$$ 0 0
$$871$$ 16.2517 0.550669
$$872$$ 0 0
$$873$$ 4.24943 0.143821
$$874$$ 0 0
$$875$$ 0 0
$$876$$ 0 0
$$877$$ 44.1106 1.48951 0.744755 0.667338i $$-0.232566\pi$$
0.744755 + 0.667338i $$0.232566\pi$$
$$878$$ 0 0
$$879$$ −11.2015 −0.377816
$$880$$ 0 0
$$881$$ −11.2900 −0.380370 −0.190185 0.981748i $$-0.560909\pi$$
−0.190185 + 0.981748i $$0.560909\pi$$
$$882$$ 0 0
$$883$$ −42.2471 −1.42173 −0.710864 0.703329i $$-0.751696\pi$$
−0.710864 + 0.703329i $$0.751696\pi$$
$$884$$ 0 0
$$885$$ 26.4103 0.887773
$$886$$ 0 0
$$887$$ 3.98068 0.133658 0.0668290 0.997764i $$-0.478712\pi$$
0.0668290 + 0.997764i $$0.478712\pi$$
$$888$$ 0 0
$$889$$ 0 0
$$890$$ 0 0
$$891$$ 4.24943 0.142361
$$892$$ 0 0
$$893$$ 7.57773 0.253579
$$894$$ 0 0
$$895$$ 66.6095 2.22651
$$896$$ 0 0
$$897$$ −13.8848 −0.463599
$$898$$ 0 0
$$899$$ 14.7483 0.491883
$$900$$ 0 0
$$901$$ −14.5182 −0.483670
$$902$$ 0 0
$$903$$ 0 0
$$904$$ 0 0
$$905$$ 42.4177 1.41001
$$906$$ 0 0
$$907$$ −12.5735 −0.417496 −0.208748 0.977969i $$-0.566939\pi$$
−0.208748 + 0.977969i $$0.566939\pi$$
$$908$$ 0 0
$$909$$ −3.90409 −0.129491
$$910$$ 0 0
$$911$$ −7.09821 −0.235174 −0.117587 0.993063i $$-0.537516\pi$$
−0.117587 + 0.993063i $$0.537516\pi$$
$$912$$ 0 0
$$913$$ 61.8566 2.04715
$$914$$ 0 0
$$915$$ 25.9188 0.856850
$$916$$ 0 0
$$917$$ 0 0
$$918$$ 0 0
$$919$$ 37.4412 1.23507 0.617536 0.786542i $$-0.288131\pi$$
0.617536 + 0.786542i $$0.288131\pi$$
$$920$$ 0 0
$$921$$ −28.9571 −0.954169
$$922$$ 0 0
$$923$$ −20.1918 −0.664622
$$924$$ 0 0
$$925$$ 50.6694 1.66600
$$926$$ 0 0
$$927$$ −9.24943 −0.303791
$$928$$ 0 0
$$929$$ 7.88477 0.258691 0.129345 0.991600i $$-0.458712\pi$$
0.129345 + 0.991600i $$0.458712\pi$$
$$930$$ 0 0
$$931$$ 0 0
$$932$$ 0 0
$$933$$ −27.3047 −0.893917
$$934$$ 0 0
$$935$$ −59.8995 −1.95892
$$936$$ 0 0
$$937$$ 36.9954 1.20859 0.604294 0.796762i $$-0.293455\pi$$
0.604294 + 0.796762i $$0.293455\pi$$
$$938$$ 0 0
$$939$$ −5.49885 −0.179448
$$940$$ 0 0
$$941$$ 12.7026 0.414094 0.207047 0.978331i $$-0.433615\pi$$
0.207047 + 0.978331i $$0.433615\pi$$
$$942$$ 0 0
$$943$$ 46.2259 1.50532
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ −23.1129 −0.751069 −0.375535 0.926808i $$-0.622541\pi$$
−0.375535 + 0.926808i $$0.622541\pi$$
$$948$$ 0 0
$$949$$ 48.0894 1.56105
$$950$$ 0 0
$$951$$ 9.60442 0.311445
$$952$$ 0 0
$$953$$ −43.4200 −1.40651 −0.703255 0.710937i $$-0.748271\pi$$
−0.703255 + 0.710937i $$0.748271\pi$$
$$954$$ 0 0
$$955$$ −13.4200 −0.434260
$$956$$ 0 0
$$957$$ −30.6021 −0.989226
$$958$$ 0 0
$$959$$ 0 0
$$960$$ 0 0
$$961$$ −26.8059 −0.864706
$$962$$ 0 0
$$963$$ 16.5565 0.533525
$$964$$ 0 0
$$965$$ −58.6021 −1.88647
$$966$$ 0 0
$$967$$ 11.6450 0.374478 0.187239 0.982314i $$-0.440046\pi$$
0.187239 + 0.982314i $$0.440046\pi$$
$$968$$ 0 0
$$969$$ −13.8848 −0.446043
$$970$$ 0 0
$$971$$ 33.3624 1.07065 0.535324 0.844647i $$-0.320189\pi$$
0.535324 + 0.844647i $$0.320189\pi$$
$$972$$ 0 0
$$973$$ 0 0
$$974$$ 0 0
$$975$$ 16.5542 0.530158
$$976$$ 0 0
$$977$$ 19.1895 0.613927 0.306963 0.951721i $$-0.400687\pi$$
0.306963 + 0.951721i $$0.400687\pi$$
$$978$$ 0 0
$$979$$ 10.2111 0.326349
$$980$$ 0 0
$$981$$ −20.0553 −0.640317
$$982$$ 0 0
$$983$$ −6.07658 −0.193813 −0.0969065 0.995293i $$-0.530895\pi$$
−0.0969065 + 0.995293i $$0.530895\pi$$
$$984$$ 0 0
$$985$$ −38.1106 −1.21431
$$986$$ 0 0
$$987$$ 0 0
$$988$$ 0 0
$$989$$ −3.30474 −0.105085
$$990$$ 0 0
$$991$$ −31.6597 −1.00570 −0.502852 0.864372i $$-0.667716\pi$$
−0.502852 + 0.864372i $$0.667716\pi$$
$$992$$ 0 0
$$993$$ 15.5565 0.493669
$$994$$ 0 0
$$995$$ −2.58003 −0.0817925
$$996$$ 0 0
$$997$$ −15.3601 −0.486458 −0.243229 0.969969i $$-0.578207\pi$$
−0.243229 + 0.969969i $$0.578207\pi$$
$$998$$ 0 0
$$999$$ 9.65237 0.305387
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 9408.2.a.ej.1.3 3
4.3 odd 2 9408.2.a.eh.1.3 3
7.2 even 3 1344.2.q.y.193.1 6
7.4 even 3 1344.2.q.y.961.1 6
7.6 odd 2 9408.2.a.eg.1.1 3
8.3 odd 2 4704.2.a.bu.1.1 3
8.5 even 2 4704.2.a.bs.1.1 3
28.11 odd 6 1344.2.q.z.961.1 6
28.23 odd 6 1344.2.q.z.193.1 6
28.27 even 2 9408.2.a.ei.1.1 3
56.11 odd 6 672.2.q.k.289.3 yes 6
56.13 odd 2 4704.2.a.bv.1.3 3
56.27 even 2 4704.2.a.bt.1.3 3
56.37 even 6 672.2.q.l.193.3 yes 6
56.51 odd 6 672.2.q.k.193.3 6
56.53 even 6 672.2.q.l.289.3 yes 6
168.11 even 6 2016.2.s.u.289.1 6
168.53 odd 6 2016.2.s.v.289.1 6
168.107 even 6 2016.2.s.u.865.1 6
168.149 odd 6 2016.2.s.v.865.1 6

By twisted newform
Twist Min Dim Char Parity Ord Type
672.2.q.k.193.3 6 56.51 odd 6
672.2.q.k.289.3 yes 6 56.11 odd 6
672.2.q.l.193.3 yes 6 56.37 even 6
672.2.q.l.289.3 yes 6 56.53 even 6
1344.2.q.y.193.1 6 7.2 even 3
1344.2.q.y.961.1 6 7.4 even 3
1344.2.q.z.193.1 6 28.23 odd 6
1344.2.q.z.961.1 6 28.11 odd 6
2016.2.s.u.289.1 6 168.11 even 6
2016.2.s.u.865.1 6 168.107 even 6
2016.2.s.v.289.1 6 168.53 odd 6
2016.2.s.v.865.1 6 168.149 odd 6
4704.2.a.bs.1.1 3 8.5 even 2
4704.2.a.bt.1.3 3 56.27 even 2
4704.2.a.bu.1.1 3 8.3 odd 2
4704.2.a.bv.1.3 3 56.13 odd 2
9408.2.a.eg.1.1 3 7.6 odd 2
9408.2.a.eh.1.3 3 4.3 odd 2
9408.2.a.ei.1.1 3 28.27 even 2
9408.2.a.ej.1.3 3 1.1 even 1 trivial