# Properties

 Label 1728.4.c.j.1727.9 Level $1728$ Weight $4$ Character 1728.1727 Analytic conductor $101.955$ Analytic rank $0$ Dimension $12$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1728 = 2^{6} \cdot 3^{3}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 1728.c (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$101.955300490$$ Analytic rank: $$0$$ Dimension: $$12$$ Coefficient field: $$\mathbb{Q}[x]/(x^{12} + \cdots)$$ Defining polynomial: $$x^{12} + 3 x^{10} - 12 x^{9} + 73 x^{8} - 12 x^{7} + 589 x^{6} + 84 x^{5} + 2452 x^{4} + 852 x^{3} + 6854 x^{2} - 888 x + 9496$$ Coefficient ring: $$\Z[a_1, \ldots, a_{19}]$$ Coefficient ring index: $$2^{32}\cdot 3^{12}$$ Twist minimal: no (minimal twist has level 108) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## Embedding invariants

 Embedding label 1727.9 Root $$-1.29835 + 1.36719i$$ of defining polynomial Character $$\chi$$ $$=$$ 1728.1727 Dual form 1728.4.c.j.1727.4

## $q$-expansion

 $$f(q)$$ $$=$$ $$q+5.83890i q^{5} -8.83113i q^{7} +O(q^{10})$$ $$q+5.83890i q^{5} -8.83113i q^{7} +23.6146 q^{11} -54.6531 q^{13} -117.211i q^{17} -109.576i q^{19} +33.5763 q^{23} +90.9072 q^{25} -40.0490i q^{29} +292.510i q^{31} +51.5641 q^{35} -283.265 q^{37} +367.472i q^{41} +323.337i q^{43} -66.2249 q^{47} +265.011 q^{49} -158.506i q^{53} +137.883i q^{55} -848.630 q^{59} +348.716 q^{61} -319.114i q^{65} -194.285i q^{67} -939.761 q^{71} -473.826 q^{73} -208.544i q^{77} -273.221i q^{79} +338.366 q^{83} +684.386 q^{85} -739.884i q^{89} +482.648i q^{91} +639.805 q^{95} -448.629 q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$12q + O(q^{10})$$ $$12q + 72q^{13} - 384q^{25} + 240q^{37} + 288q^{49} - 144q^{61} + 156q^{73} + 168q^{85} + 516q^{97} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1728\mathbb{Z}\right)^\times$$.

 $$n$$ $$325$$ $$703$$ $$1217$$ $$\chi(n)$$ $$1$$ $$-1$$ $$-1$$

## 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
$$3$$ 0 0
$$4$$ 0 0
$$5$$ 5.83890i 0.522247i 0.965305 + 0.261124i $$0.0840930\pi$$
−0.965305 + 0.261124i $$0.915907\pi$$
$$6$$ 0 0
$$7$$ − 8.83113i − 0.476836i −0.971163 0.238418i $$-0.923371\pi$$
0.971163 0.238418i $$-0.0766288\pi$$
$$8$$ 0 0
$$9$$ 0 0
$$10$$ 0 0
$$11$$ 23.6146 0.647279 0.323639 0.946180i $$-0.395094\pi$$
0.323639 + 0.946180i $$0.395094\pi$$
$$12$$ 0 0
$$13$$ −54.6531 −1.16600 −0.583001 0.812471i $$-0.698122\pi$$
−0.583001 + 0.812471i $$0.698122\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 0 0
$$17$$ − 117.211i − 1.67223i −0.548553 0.836116i $$-0.684821\pi$$
0.548553 0.836116i $$-0.315179\pi$$
$$18$$ 0 0
$$19$$ − 109.576i − 1.32308i −0.749910 0.661539i $$-0.769903\pi$$
0.749910 0.661539i $$-0.230097\pi$$
$$20$$ 0 0
$$21$$ 0 0
$$22$$ 0 0
$$23$$ 33.5763 0.304397 0.152199 0.988350i $$-0.451365\pi$$
0.152199 + 0.988350i $$0.451365\pi$$
$$24$$ 0 0
$$25$$ 90.9072 0.727258
$$26$$ 0 0
$$27$$ 0 0
$$28$$ 0 0
$$29$$ − 40.0490i − 0.256445i −0.991745 0.128223i $$-0.959073\pi$$
0.991745 0.128223i $$-0.0409272\pi$$
$$30$$ 0 0
$$31$$ 292.510i 1.69472i 0.531020 + 0.847359i $$0.321809\pi$$
−0.531020 + 0.847359i $$0.678191\pi$$
$$32$$ 0 0
$$33$$ 0 0
$$34$$ 0 0
$$35$$ 51.5641 0.249026
$$36$$ 0 0
$$37$$ −283.265 −1.25861 −0.629304 0.777159i $$-0.716660\pi$$
−0.629304 + 0.777159i $$0.716660\pi$$
$$38$$ 0 0
$$39$$ 0 0
$$40$$ 0 0
$$41$$ 367.472i 1.39974i 0.714269 + 0.699871i $$0.246759\pi$$
−0.714269 + 0.699871i $$0.753241\pi$$
$$42$$ 0 0
$$43$$ 323.337i 1.14671i 0.819308 + 0.573354i $$0.194358\pi$$
−0.819308 + 0.573354i $$0.805642\pi$$
$$44$$ 0 0
$$45$$ 0 0
$$46$$ 0 0
$$47$$ −66.2249 −0.205530 −0.102765 0.994706i $$-0.532769\pi$$
−0.102765 + 0.994706i $$0.532769\pi$$
$$48$$ 0 0
$$49$$ 265.011 0.772627
$$50$$ 0 0
$$51$$ 0 0
$$52$$ 0 0
$$53$$ − 158.506i − 0.410801i −0.978678 0.205401i $$-0.934150\pi$$
0.978678 0.205401i $$-0.0658498\pi$$
$$54$$ 0 0
$$55$$ 137.883i 0.338040i
$$56$$ 0 0
$$57$$ 0 0
$$58$$ 0 0
$$59$$ −848.630 −1.87258 −0.936290 0.351228i $$-0.885764\pi$$
−0.936290 + 0.351228i $$0.885764\pi$$
$$60$$ 0 0
$$61$$ 348.716 0.731943 0.365972 0.930626i $$-0.380737\pi$$
0.365972 + 0.930626i $$0.380737\pi$$
$$62$$ 0 0
$$63$$ 0 0
$$64$$ 0 0
$$65$$ − 319.114i − 0.608942i
$$66$$ 0 0
$$67$$ − 194.285i − 0.354264i −0.984187 0.177132i $$-0.943318\pi$$
0.984187 0.177132i $$-0.0566819\pi$$
$$68$$ 0 0
$$69$$ 0 0
$$70$$ 0 0
$$71$$ −939.761 −1.57083 −0.785417 0.618968i $$-0.787551\pi$$
−0.785417 + 0.618968i $$0.787551\pi$$
$$72$$ 0 0
$$73$$ −473.826 −0.759686 −0.379843 0.925051i $$-0.624022\pi$$
−0.379843 + 0.925051i $$0.624022\pi$$
$$74$$ 0 0
$$75$$ 0 0
$$76$$ 0 0
$$77$$ − 208.544i − 0.308646i
$$78$$ 0 0
$$79$$ − 273.221i − 0.389112i −0.980891 0.194556i $$-0.937673\pi$$
0.980891 0.194556i $$-0.0623265\pi$$
$$80$$ 0 0
$$81$$ 0 0
$$82$$ 0 0
$$83$$ 338.366 0.447476 0.223738 0.974649i $$-0.428174\pi$$
0.223738 + 0.974649i $$0.428174\pi$$
$$84$$ 0 0
$$85$$ 684.386 0.873318
$$86$$ 0 0
$$87$$ 0 0
$$88$$ 0 0
$$89$$ − 739.884i − 0.881208i −0.897702 0.440604i $$-0.854764\pi$$
0.897702 0.440604i $$-0.145236\pi$$
$$90$$ 0 0
$$91$$ 482.648i 0.555992i
$$92$$ 0 0
$$93$$ 0 0
$$94$$ 0 0
$$95$$ 639.805 0.690974
$$96$$ 0 0
$$97$$ −448.629 −0.469602 −0.234801 0.972044i $$-0.575444\pi$$
−0.234801 + 0.972044i $$0.575444\pi$$
$$98$$ 0 0
$$99$$ 0 0
$$100$$ 0 0
$$101$$ 1152.87i 1.13580i 0.823099 + 0.567898i $$0.192243\pi$$
−0.823099 + 0.567898i $$0.807757\pi$$
$$102$$ 0 0
$$103$$ − 1389.90i − 1.32962i −0.747013 0.664809i $$-0.768513\pi$$
0.747013 0.664809i $$-0.231487\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ 245.479 0.221788 0.110894 0.993832i $$-0.464629\pi$$
0.110894 + 0.993832i $$0.464629\pi$$
$$108$$ 0 0
$$109$$ 644.998 0.566785 0.283393 0.959004i $$-0.408540\pi$$
0.283393 + 0.959004i $$0.408540\pi$$
$$110$$ 0 0
$$111$$ 0 0
$$112$$ 0 0
$$113$$ 825.442i 0.687177i 0.939120 + 0.343589i $$0.111643\pi$$
−0.939120 + 0.343589i $$0.888357\pi$$
$$114$$ 0 0
$$115$$ 196.049i 0.158971i
$$116$$ 0 0
$$117$$ 0 0
$$118$$ 0 0
$$119$$ −1035.11 −0.797380
$$120$$ 0 0
$$121$$ −773.351 −0.581030
$$122$$ 0 0
$$123$$ 0 0
$$124$$ 0 0
$$125$$ 1260.66i 0.902056i
$$126$$ 0 0
$$127$$ − 754.649i − 0.527278i −0.964621 0.263639i $$-0.915077\pi$$
0.964621 0.263639i $$-0.0849228\pi$$
$$128$$ 0 0
$$129$$ 0 0
$$130$$ 0 0
$$131$$ −2026.43 −1.35153 −0.675765 0.737117i $$-0.736186\pi$$
−0.675765 + 0.737117i $$0.736186\pi$$
$$132$$ 0 0
$$133$$ −967.681 −0.630892
$$134$$ 0 0
$$135$$ 0 0
$$136$$ 0 0
$$137$$ 1882.88i 1.17420i 0.809515 + 0.587099i $$0.199730\pi$$
−0.809515 + 0.587099i $$0.800270\pi$$
$$138$$ 0 0
$$139$$ 93.7277i 0.0571934i 0.999591 + 0.0285967i $$0.00910385\pi$$
−0.999591 + 0.0285967i $$0.990896\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ 0 0
$$143$$ −1290.61 −0.754729
$$144$$ 0 0
$$145$$ 233.842 0.133928
$$146$$ 0 0
$$147$$ 0 0
$$148$$ 0 0
$$149$$ − 2764.69i − 1.52008i −0.649874 0.760042i $$-0.725178\pi$$
0.649874 0.760042i $$-0.274822\pi$$
$$150$$ 0 0
$$151$$ 3694.11i 1.99088i 0.0954051 + 0.995439i $$0.469585\pi$$
−0.0954051 + 0.995439i $$0.530415\pi$$
$$152$$ 0 0
$$153$$ 0 0
$$154$$ 0 0
$$155$$ −1707.94 −0.885062
$$156$$ 0 0
$$157$$ −812.401 −0.412972 −0.206486 0.978450i $$-0.566203\pi$$
−0.206486 + 0.978450i $$0.566203\pi$$
$$158$$ 0 0
$$159$$ 0 0
$$160$$ 0 0
$$161$$ − 296.516i − 0.145148i
$$162$$ 0 0
$$163$$ − 1034.18i − 0.496953i −0.968638 0.248477i $$-0.920070\pi$$
0.968638 0.248477i $$-0.0799299\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ 0 0
$$167$$ −1453.29 −0.673409 −0.336704 0.941610i $$-0.609312\pi$$
−0.336704 + 0.941610i $$0.609312\pi$$
$$168$$ 0 0
$$169$$ 789.957 0.359562
$$170$$ 0 0
$$171$$ 0 0
$$172$$ 0 0
$$173$$ − 634.902i − 0.279022i −0.990221 0.139511i $$-0.955447\pi$$
0.990221 0.139511i $$-0.0445530\pi$$
$$174$$ 0 0
$$175$$ − 802.813i − 0.346783i
$$176$$ 0 0
$$177$$ 0 0
$$178$$ 0 0
$$179$$ −2909.35 −1.21483 −0.607417 0.794383i $$-0.707794\pi$$
−0.607417 + 0.794383i $$0.707794\pi$$
$$180$$ 0 0
$$181$$ −1354.46 −0.556222 −0.278111 0.960549i $$-0.589708\pi$$
−0.278111 + 0.960549i $$0.589708\pi$$
$$182$$ 0 0
$$183$$ 0 0
$$184$$ 0 0
$$185$$ − 1653.96i − 0.657305i
$$186$$ 0 0
$$187$$ − 2767.90i − 1.08240i
$$188$$ 0 0
$$189$$ 0 0
$$190$$ 0 0
$$191$$ −3488.40 −1.32153 −0.660765 0.750593i $$-0.729768\pi$$
−0.660765 + 0.750593i $$0.729768\pi$$
$$192$$ 0 0
$$193$$ 2912.99 1.08643 0.543217 0.839592i $$-0.317206\pi$$
0.543217 + 0.839592i $$0.317206\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ 0 0
$$197$$ − 2432.66i − 0.879795i −0.898048 0.439897i $$-0.855015\pi$$
0.898048 0.439897i $$-0.144985\pi$$
$$198$$ 0 0
$$199$$ − 563.190i − 0.200621i −0.994956 0.100310i $$-0.968016\pi$$
0.994956 0.100310i $$-0.0319836\pi$$
$$200$$ 0 0
$$201$$ 0 0
$$202$$ 0 0
$$203$$ −353.678 −0.122282
$$204$$ 0 0
$$205$$ −2145.63 −0.731012
$$206$$ 0 0
$$207$$ 0 0
$$208$$ 0 0
$$209$$ − 2587.60i − 0.856401i
$$210$$ 0 0
$$211$$ − 1447.62i − 0.472314i −0.971715 0.236157i $$-0.924112\pi$$
0.971715 0.236157i $$-0.0758879\pi$$
$$212$$ 0 0
$$213$$ 0 0
$$214$$ 0 0
$$215$$ −1887.93 −0.598865
$$216$$ 0 0
$$217$$ 2583.19 0.808103
$$218$$ 0 0
$$219$$ 0 0
$$220$$ 0 0
$$221$$ 6405.96i 1.94983i
$$222$$ 0 0
$$223$$ − 5049.28i − 1.51625i −0.652107 0.758127i $$-0.726115\pi$$
0.652107 0.758127i $$-0.273885\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ 0 0
$$227$$ −5716.54 −1.67145 −0.835727 0.549146i $$-0.814953\pi$$
−0.835727 + 0.549146i $$0.814953\pi$$
$$228$$ 0 0
$$229$$ 2582.55 0.745240 0.372620 0.927984i $$-0.378460\pi$$
0.372620 + 0.927984i $$0.378460\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 0 0
$$233$$ − 587.204i − 0.165103i −0.996587 0.0825516i $$-0.973693\pi$$
0.996587 0.0825516i $$-0.0263069\pi$$
$$234$$ 0 0
$$235$$ − 386.681i − 0.107337i
$$236$$ 0 0
$$237$$ 0 0
$$238$$ 0 0
$$239$$ −5583.00 −1.51102 −0.755512 0.655135i $$-0.772612\pi$$
−0.755512 + 0.655135i $$0.772612\pi$$
$$240$$ 0 0
$$241$$ −2056.95 −0.549791 −0.274895 0.961474i $$-0.588643\pi$$
−0.274895 + 0.961474i $$0.588643\pi$$
$$242$$ 0 0
$$243$$ 0 0
$$244$$ 0 0
$$245$$ 1547.37i 0.403503i
$$246$$ 0 0
$$247$$ 5988.67i 1.54271i
$$248$$ 0 0
$$249$$ 0 0
$$250$$ 0 0
$$251$$ −1256.79 −0.316047 −0.158023 0.987435i $$-0.550512\pi$$
−0.158023 + 0.987435i $$0.550512\pi$$
$$252$$ 0 0
$$253$$ 792.890 0.197030
$$254$$ 0 0
$$255$$ 0 0
$$256$$ 0 0
$$257$$ 643.773i 0.156255i 0.996943 + 0.0781273i $$0.0248941\pi$$
−0.996943 + 0.0781273i $$0.975106\pi$$
$$258$$ 0 0
$$259$$ 2501.55i 0.600150i
$$260$$ 0 0
$$261$$ 0 0
$$262$$ 0 0
$$263$$ −917.108 −0.215024 −0.107512 0.994204i $$-0.534288\pi$$
−0.107512 + 0.994204i $$0.534288\pi$$
$$264$$ 0 0
$$265$$ 925.501 0.214540
$$266$$ 0 0
$$267$$ 0 0
$$268$$ 0 0
$$269$$ − 6577.00i − 1.49073i −0.666656 0.745366i $$-0.732275\pi$$
0.666656 0.745366i $$-0.267725\pi$$
$$270$$ 0 0
$$271$$ 4656.21i 1.04371i 0.853035 + 0.521854i $$0.174759\pi$$
−0.853035 + 0.521854i $$0.825241\pi$$
$$272$$ 0 0
$$273$$ 0 0
$$274$$ 0 0
$$275$$ 2146.74 0.470739
$$276$$ 0 0
$$277$$ 2881.42 0.625010 0.312505 0.949916i $$-0.398832\pi$$
0.312505 + 0.949916i $$0.398832\pi$$
$$278$$ 0 0
$$279$$ 0 0
$$280$$ 0 0
$$281$$ 2604.93i 0.553015i 0.961012 + 0.276507i $$0.0891771\pi$$
−0.961012 + 0.276507i $$0.910823\pi$$
$$282$$ 0 0
$$283$$ 5786.02i 1.21535i 0.794187 + 0.607674i $$0.207897\pi$$
−0.794187 + 0.607674i $$0.792103\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ 0 0
$$287$$ 3245.19 0.667448
$$288$$ 0 0
$$289$$ −8825.50 −1.79636
$$290$$ 0 0
$$291$$ 0 0
$$292$$ 0 0
$$293$$ 3279.55i 0.653902i 0.945041 + 0.326951i $$0.106021\pi$$
−0.945041 + 0.326951i $$0.893979\pi$$
$$294$$ 0 0
$$295$$ − 4955.07i − 0.977950i
$$296$$ 0 0
$$297$$ 0 0
$$298$$ 0 0
$$299$$ −1835.05 −0.354928
$$300$$ 0 0
$$301$$ 2855.43 0.546792
$$302$$ 0 0
$$303$$ 0 0
$$304$$ 0 0
$$305$$ 2036.12i 0.382255i
$$306$$ 0 0
$$307$$ 3014.47i 0.560407i 0.959941 + 0.280203i $$0.0904019\pi$$
−0.959941 + 0.280203i $$0.909598\pi$$
$$308$$ 0 0
$$309$$ 0 0
$$310$$ 0 0
$$311$$ −6230.80 −1.13606 −0.568032 0.823006i $$-0.692295\pi$$
−0.568032 + 0.823006i $$0.692295\pi$$
$$312$$ 0 0
$$313$$ −3922.49 −0.708346 −0.354173 0.935180i $$-0.615238\pi$$
−0.354173 + 0.935180i $$0.615238\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ − 6882.19i − 1.21938i −0.792641 0.609689i $$-0.791295\pi$$
0.792641 0.609689i $$-0.208705\pi$$
$$318$$ 0 0
$$319$$ − 945.741i − 0.165992i
$$320$$ 0 0
$$321$$ 0 0
$$322$$ 0 0
$$323$$ −12843.6 −2.21249
$$324$$ 0 0
$$325$$ −4968.36 −0.847984
$$326$$ 0 0
$$327$$ 0 0
$$328$$ 0 0
$$329$$ 584.841i 0.0980040i
$$330$$ 0 0
$$331$$ 9489.08i 1.57573i 0.615847 + 0.787866i $$0.288814\pi$$
−0.615847 + 0.787866i $$0.711186\pi$$
$$332$$ 0 0
$$333$$ 0 0
$$334$$ 0 0
$$335$$ 1134.41 0.185013
$$336$$ 0 0
$$337$$ 4578.52 0.740083 0.370041 0.929015i $$-0.379344\pi$$
0.370041 + 0.929015i $$0.379344\pi$$
$$338$$ 0 0
$$339$$ 0 0
$$340$$ 0 0
$$341$$ 6907.50i 1.09696i
$$342$$ 0 0
$$343$$ − 5369.42i − 0.845253i
$$344$$ 0 0
$$345$$ 0 0
$$346$$ 0 0
$$347$$ 942.483 0.145807 0.0729037 0.997339i $$-0.476773\pi$$
0.0729037 + 0.997339i $$0.476773\pi$$
$$348$$ 0 0
$$349$$ 2124.84 0.325903 0.162951 0.986634i $$-0.447899\pi$$
0.162951 + 0.986634i $$0.447899\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ 0 0
$$353$$ 3881.60i 0.585260i 0.956226 + 0.292630i $$0.0945304\pi$$
−0.956226 + 0.292630i $$0.905470\pi$$
$$354$$ 0 0
$$355$$ − 5487.18i − 0.820363i
$$356$$ 0 0
$$357$$ 0 0
$$358$$ 0 0
$$359$$ −8050.44 −1.18353 −0.591763 0.806112i $$-0.701568\pi$$
−0.591763 + 0.806112i $$0.701568\pi$$
$$360$$ 0 0
$$361$$ −5147.94 −0.750537
$$362$$ 0 0
$$363$$ 0 0
$$364$$ 0 0
$$365$$ − 2766.62i − 0.396744i
$$366$$ 0 0
$$367$$ 6890.96i 0.980123i 0.871688 + 0.490062i $$0.163026\pi$$
−0.871688 + 0.490062i $$0.836974\pi$$
$$368$$ 0 0
$$369$$ 0 0
$$370$$ 0 0
$$371$$ −1399.79 −0.195885
$$372$$ 0 0
$$373$$ 10512.5 1.45930 0.729649 0.683822i $$-0.239684\pi$$
0.729649 + 0.683822i $$0.239684\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ 0 0
$$377$$ 2188.80i 0.299016i
$$378$$ 0 0
$$379$$ − 3139.69i − 0.425528i −0.977104 0.212764i $$-0.931753\pi$$
0.977104 0.212764i $$-0.0682466\pi$$
$$380$$ 0 0
$$381$$ 0 0
$$382$$ 0 0
$$383$$ −5117.53 −0.682751 −0.341375 0.939927i $$-0.610893\pi$$
−0.341375 + 0.939927i $$0.610893\pi$$
$$384$$ 0 0
$$385$$ 1217.67 0.161190
$$386$$ 0 0
$$387$$ 0 0
$$388$$ 0 0
$$389$$ 787.004i 0.102578i 0.998684 + 0.0512888i $$0.0163329\pi$$
−0.998684 + 0.0512888i $$0.983667\pi$$
$$390$$ 0 0
$$391$$ − 3935.52i − 0.509023i
$$392$$ 0 0
$$393$$ 0 0
$$394$$ 0 0
$$395$$ 1595.31 0.203212
$$396$$ 0 0
$$397$$ −11160.6 −1.41092 −0.705458 0.708752i $$-0.749259\pi$$
−0.705458 + 0.708752i $$0.749259\pi$$
$$398$$ 0 0
$$399$$ 0 0
$$400$$ 0 0
$$401$$ − 13224.6i − 1.64689i −0.567394 0.823447i $$-0.692048\pi$$
0.567394 0.823447i $$-0.307952\pi$$
$$402$$ 0 0
$$403$$ − 15986.5i − 1.97605i
$$404$$ 0 0
$$405$$ 0 0
$$406$$ 0 0
$$407$$ −6689.20 −0.814671
$$408$$ 0 0
$$409$$ 8153.09 0.985683 0.492841 0.870119i $$-0.335958\pi$$
0.492841 + 0.870119i $$0.335958\pi$$
$$410$$ 0 0
$$411$$ 0 0
$$412$$ 0 0
$$413$$ 7494.36i 0.892914i
$$414$$ 0 0
$$415$$ 1975.69i 0.233693i
$$416$$ 0 0
$$417$$ 0 0
$$418$$ 0 0
$$419$$ −11682.5 −1.36212 −0.681061 0.732226i $$-0.738481\pi$$
−0.681061 + 0.732226i $$0.738481\pi$$
$$420$$ 0 0
$$421$$ −9595.77 −1.11085 −0.555426 0.831566i $$-0.687445\pi$$
−0.555426 + 0.831566i $$0.687445\pi$$
$$422$$ 0 0
$$423$$ 0 0
$$424$$ 0 0
$$425$$ − 10655.4i − 1.21614i
$$426$$ 0 0
$$427$$ − 3079.56i − 0.349017i
$$428$$ 0 0
$$429$$ 0 0
$$430$$ 0 0
$$431$$ 4294.48 0.479948 0.239974 0.970779i $$-0.422861\pi$$
0.239974 + 0.970779i $$0.422861\pi$$
$$432$$ 0 0
$$433$$ 168.392 0.0186892 0.00934460 0.999956i $$-0.497025\pi$$
0.00934460 + 0.999956i $$0.497025\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 0 0
$$437$$ − 3679.16i − 0.402742i
$$438$$ 0 0
$$439$$ 1459.53i 0.158677i 0.996848 + 0.0793387i $$0.0252809\pi$$
−0.996848 + 0.0793387i $$0.974719\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ 0 0
$$443$$ 333.411 0.0357581 0.0178790 0.999840i $$-0.494309\pi$$
0.0178790 + 0.999840i $$0.494309\pi$$
$$444$$ 0 0
$$445$$ 4320.11 0.460209
$$446$$ 0 0
$$447$$ 0 0
$$448$$ 0 0
$$449$$ 11454.3i 1.20393i 0.798523 + 0.601964i $$0.205615\pi$$
−0.798523 + 0.601964i $$0.794385\pi$$
$$450$$ 0 0
$$451$$ 8677.70i 0.906024i
$$452$$ 0 0
$$453$$ 0 0
$$454$$ 0 0
$$455$$ −2818.14 −0.290365
$$456$$ 0 0
$$457$$ 9680.26 0.990861 0.495431 0.868648i $$-0.335010\pi$$
0.495431 + 0.868648i $$0.335010\pi$$
$$458$$ 0 0
$$459$$ 0 0
$$460$$ 0 0
$$461$$ 14806.9i 1.49593i 0.663737 + 0.747966i $$0.268970\pi$$
−0.663737 + 0.747966i $$0.731030\pi$$
$$462$$ 0 0
$$463$$ − 3658.08i − 0.367183i −0.983003 0.183591i $$-0.941228\pi$$
0.983003 0.183591i $$-0.0587723\pi$$
$$464$$ 0 0
$$465$$ 0 0
$$466$$ 0 0
$$467$$ 9852.37 0.976260 0.488130 0.872771i $$-0.337679\pi$$
0.488130 + 0.872771i $$0.337679\pi$$
$$468$$ 0 0
$$469$$ −1715.75 −0.168926
$$470$$ 0 0
$$471$$ 0 0
$$472$$ 0 0
$$473$$ 7635.47i 0.742240i
$$474$$ 0 0
$$475$$ − 9961.26i − 0.962219i
$$476$$ 0 0
$$477$$ 0 0
$$478$$ 0 0
$$479$$ 11107.9 1.05957 0.529784 0.848133i $$-0.322273\pi$$
0.529784 + 0.848133i $$0.322273\pi$$
$$480$$ 0 0
$$481$$ 15481.3 1.46754
$$482$$ 0 0
$$483$$ 0 0
$$484$$ 0 0
$$485$$ − 2619.50i − 0.245248i
$$486$$ 0 0
$$487$$ 11703.7i 1.08900i 0.838761 + 0.544500i $$0.183281\pi$$
−0.838761 + 0.544500i $$0.816719\pi$$
$$488$$ 0 0
$$489$$ 0 0
$$490$$ 0 0
$$491$$ 2851.25 0.262067 0.131034 0.991378i $$-0.458170\pi$$
0.131034 + 0.991378i $$0.458170\pi$$
$$492$$ 0 0
$$493$$ −4694.20 −0.428836
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 0 0
$$497$$ 8299.15i 0.749030i
$$498$$ 0 0
$$499$$ 14936.0i 1.33994i 0.742389 + 0.669969i $$0.233693\pi$$
−0.742389 + 0.669969i $$0.766307\pi$$
$$500$$ 0 0
$$501$$ 0 0
$$502$$ 0 0
$$503$$ −9047.91 −0.802041 −0.401020 0.916069i $$-0.631344\pi$$
−0.401020 + 0.916069i $$0.631344\pi$$
$$504$$ 0 0
$$505$$ −6731.52 −0.593166
$$506$$ 0 0
$$507$$ 0 0
$$508$$ 0 0
$$509$$ − 15173.5i − 1.32132i −0.750683 0.660662i $$-0.770276\pi$$
0.750683 0.660662i $$-0.229724\pi$$
$$510$$ 0 0
$$511$$ 4184.41i 0.362246i
$$512$$ 0 0
$$513$$ 0 0
$$514$$ 0 0
$$515$$ 8115.47 0.694389
$$516$$ 0 0
$$517$$ −1563.87 −0.133035
$$518$$ 0 0
$$519$$ 0 0
$$520$$ 0 0
$$521$$ − 7375.96i − 0.620243i −0.950697 0.310121i $$-0.899630\pi$$
0.950697 0.310121i $$-0.100370\pi$$
$$522$$ 0 0
$$523$$ − 8514.96i − 0.711918i −0.934502 0.355959i $$-0.884154\pi$$
0.934502 0.355959i $$-0.115846\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ 34285.4 2.83396
$$528$$ 0 0
$$529$$ −11039.6 −0.907342
$$530$$ 0 0
$$531$$ 0 0
$$532$$ 0 0
$$533$$ − 20083.5i − 1.63210i
$$534$$ 0 0
$$535$$ 1433.33i 0.115828i
$$536$$ 0 0
$$537$$ 0 0
$$538$$ 0 0
$$539$$ 6258.13 0.500105
$$540$$ 0 0
$$541$$ 2214.98 0.176025 0.0880125 0.996119i $$-0.471948\pi$$
0.0880125 + 0.996119i $$0.471948\pi$$
$$542$$ 0 0
$$543$$ 0 0
$$544$$ 0 0
$$545$$ 3766.08i 0.296002i
$$546$$ 0 0
$$547$$ 3906.55i 0.305360i 0.988276 + 0.152680i $$0.0487904\pi$$
−0.988276 + 0.152680i $$0.951210\pi$$
$$548$$ 0 0
$$549$$ 0 0
$$550$$ 0 0
$$551$$ −4388.41 −0.339297
$$552$$ 0 0
$$553$$ −2412.85 −0.185542
$$554$$ 0 0
$$555$$ 0 0
$$556$$ 0 0
$$557$$ − 2978.14i − 0.226549i −0.993564 0.113275i $$-0.963866\pi$$
0.993564 0.113275i $$-0.0361340\pi$$
$$558$$ 0 0
$$559$$ − 17671.4i − 1.33706i
$$560$$ 0 0
$$561$$ 0 0
$$562$$ 0 0
$$563$$ −8786.44 −0.657734 −0.328867 0.944376i $$-0.606667\pi$$
−0.328867 + 0.944376i $$0.606667\pi$$
$$564$$ 0 0
$$565$$ −4819.67 −0.358876
$$566$$ 0 0
$$567$$ 0 0
$$568$$ 0 0
$$569$$ − 19271.0i − 1.41983i −0.704288 0.709914i $$-0.748734\pi$$
0.704288 0.709914i $$-0.251266\pi$$
$$570$$ 0 0
$$571$$ − 15535.7i − 1.13861i −0.822125 0.569307i $$-0.807212\pi$$
0.822125 0.569307i $$-0.192788\pi$$
$$572$$ 0 0
$$573$$ 0 0
$$574$$ 0 0
$$575$$ 3052.33 0.221375
$$576$$ 0 0
$$577$$ 3783.58 0.272985 0.136493 0.990641i $$-0.456417\pi$$
0.136493 + 0.990641i $$0.456417\pi$$
$$578$$ 0 0
$$579$$ 0 0
$$580$$ 0 0
$$581$$ − 2988.15i − 0.213373i
$$582$$ 0 0
$$583$$ − 3743.06i − 0.265903i
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 0 0
$$587$$ 15515.4 1.09095 0.545477 0.838126i $$-0.316349\pi$$
0.545477 + 0.838126i $$0.316349\pi$$
$$588$$ 0 0
$$589$$ 32052.1 2.24225
$$590$$ 0 0
$$591$$ 0 0
$$592$$ 0 0
$$593$$ − 14098.1i − 0.976292i −0.872762 0.488146i $$-0.837673\pi$$
0.872762 0.488146i $$-0.162327\pi$$
$$594$$ 0 0
$$595$$ − 6043.90i − 0.416430i
$$596$$ 0 0
$$597$$ 0 0
$$598$$ 0 0
$$599$$ −572.539 −0.0390539 −0.0195270 0.999809i $$-0.506216\pi$$
−0.0195270 + 0.999809i $$0.506216\pi$$
$$600$$ 0 0
$$601$$ −23463.4 −1.59250 −0.796249 0.604969i $$-0.793185\pi$$
−0.796249 + 0.604969i $$0.793185\pi$$
$$602$$ 0 0
$$603$$ 0 0
$$604$$ 0 0
$$605$$ − 4515.52i − 0.303441i
$$606$$ 0 0
$$607$$ 1980.51i 0.132433i 0.997805 + 0.0662163i $$0.0210927\pi$$
−0.997805 + 0.0662163i $$0.978907\pi$$
$$608$$ 0 0
$$609$$ 0 0
$$610$$ 0 0
$$611$$ 3619.39 0.239648
$$612$$ 0 0
$$613$$ −2479.01 −0.163338 −0.0816691 0.996660i $$-0.526025\pi$$
−0.0816691 + 0.996660i $$0.526025\pi$$
$$614$$ 0 0
$$615$$ 0 0
$$616$$ 0 0
$$617$$ 20002.8i 1.30516i 0.757722 + 0.652578i $$0.226312\pi$$
−0.757722 + 0.652578i $$0.773688\pi$$
$$618$$ 0 0
$$619$$ 22292.4i 1.44751i 0.690058 + 0.723754i $$0.257585\pi$$
−0.690058 + 0.723754i $$0.742415\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ 0 0
$$623$$ −6534.01 −0.420192
$$624$$ 0 0
$$625$$ 4002.52 0.256161
$$626$$ 0 0
$$627$$ 0 0
$$628$$ 0 0
$$629$$ 33201.9i 2.10469i
$$630$$ 0 0
$$631$$ − 24203.8i − 1.52700i −0.645807 0.763501i $$-0.723479\pi$$
0.645807 0.763501i $$-0.276521\pi$$
$$632$$ 0 0
$$633$$ 0 0
$$634$$ 0 0
$$635$$ 4406.32 0.275370
$$636$$ 0 0
$$637$$ −14483.7 −0.900885
$$638$$ 0 0
$$639$$ 0 0
$$640$$ 0 0
$$641$$ 4423.87i 0.272593i 0.990668 + 0.136297i $$0.0435200\pi$$
−0.990668 + 0.136297i $$0.956480\pi$$
$$642$$ 0 0
$$643$$ 11961.5i 0.733619i 0.930296 + 0.366810i $$0.119550\pi$$
−0.930296 + 0.366810i $$0.880450\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 0 0
$$647$$ 6287.16 0.382030 0.191015 0.981587i $$-0.438822\pi$$
0.191015 + 0.981587i $$0.438822\pi$$
$$648$$ 0 0
$$649$$ −20040.1 −1.21208
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 0 0
$$653$$ 18793.2i 1.12624i 0.826376 + 0.563119i $$0.190399\pi$$
−0.826376 + 0.563119i $$0.809601\pi$$
$$654$$ 0 0
$$655$$ − 11832.2i − 0.705833i
$$656$$ 0 0
$$657$$ 0 0
$$658$$ 0 0
$$659$$ −20827.2 −1.23113 −0.615563 0.788088i $$-0.711071\pi$$
−0.615563 + 0.788088i $$0.711071\pi$$
$$660$$ 0 0
$$661$$ 6038.16 0.355306 0.177653 0.984093i $$-0.443150\pi$$
0.177653 + 0.984093i $$0.443150\pi$$
$$662$$ 0 0
$$663$$ 0 0
$$664$$ 0 0
$$665$$ − 5650.20i − 0.329482i
$$666$$ 0 0
$$667$$ − 1344.70i − 0.0780612i
$$668$$ 0 0
$$669$$ 0 0
$$670$$ 0 0
$$671$$ 8234.79 0.473771
$$672$$ 0 0
$$673$$ −19811.5 −1.13474 −0.567368 0.823464i $$-0.692038\pi$$
−0.567368 + 0.823464i $$0.692038\pi$$
$$674$$ 0 0
$$675$$ 0 0
$$676$$ 0 0
$$677$$ 22962.5i 1.30357i 0.758402 + 0.651787i $$0.225980\pi$$
−0.758402 + 0.651787i $$0.774020\pi$$
$$678$$ 0 0
$$679$$ 3961.90i 0.223923i
$$680$$ 0 0
$$681$$ 0 0
$$682$$ 0 0
$$683$$ 33097.3 1.85422 0.927109 0.374791i $$-0.122286\pi$$
0.927109 + 0.374791i $$0.122286\pi$$
$$684$$ 0 0
$$685$$ −10993.9 −0.613222
$$686$$ 0 0
$$687$$ 0 0
$$688$$ 0 0
$$689$$ 8662.84i 0.478995i
$$690$$ 0 0
$$691$$ 21151.0i 1.16443i 0.813034 + 0.582216i $$0.197814\pi$$
−0.813034 + 0.582216i $$0.802186\pi$$
$$692$$ 0 0
$$693$$ 0 0
$$694$$ 0 0
$$695$$ −547.267 −0.0298691
$$696$$ 0 0
$$697$$ 43071.9 2.34069
$$698$$ 0 0
$$699$$ 0 0
$$700$$ 0 0
$$701$$ − 18752.3i − 1.01036i −0.863013 0.505182i $$-0.831425\pi$$
0.863013 0.505182i $$-0.168575\pi$$
$$702$$ 0 0
$$703$$ 31039.1i 1.66524i
$$704$$ 0 0
$$705$$ 0 0
$$706$$ 0 0
$$707$$ 10181.2 0.541588
$$708$$ 0 0
$$709$$ 25964.1 1.37532 0.687660 0.726033i $$-0.258638\pi$$
0.687660 + 0.726033i $$0.258638\pi$$
$$710$$ 0 0
$$711$$ 0 0
$$712$$ 0 0
$$713$$ 9821.38i 0.515868i
$$714$$ 0 0
$$715$$ − 7535.75i − 0.394155i
$$716$$ 0 0
$$717$$ 0 0
$$718$$ 0 0
$$719$$ 11741.1 0.608998 0.304499 0.952513i $$-0.401511\pi$$
0.304499 + 0.952513i $$0.401511\pi$$
$$720$$ 0 0
$$721$$ −12274.4 −0.634010
$$722$$ 0 0
$$723$$ 0 0
$$724$$ 0 0
$$725$$ − 3640.74i − 0.186502i
$$726$$ 0 0
$$727$$ 5637.87i 0.287616i 0.989606 + 0.143808i $$0.0459348\pi$$
−0.989606 + 0.143808i $$0.954065\pi$$
$$728$$ 0 0
$$729$$ 0 0
$$730$$ 0 0
$$731$$ 37898.8 1.91756
$$732$$ 0 0
$$733$$ 23807.3 1.19965 0.599825 0.800131i $$-0.295237\pi$$
0.599825 + 0.800131i $$0.295237\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ − 4587.96i − 0.229307i
$$738$$ 0 0
$$739$$ 3556.31i 0.177024i 0.996075 + 0.0885122i $$0.0282112\pi$$
−0.996075 + 0.0885122i $$0.971789\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ 0 0
$$743$$ 24054.6 1.18772 0.593860 0.804568i $$-0.297603\pi$$
0.593860 + 0.804568i $$0.297603\pi$$
$$744$$ 0 0
$$745$$ 16142.8 0.793860
$$746$$ 0 0
$$747$$ 0 0
$$748$$ 0 0
$$749$$ − 2167.86i − 0.105757i
$$750$$ 0 0
$$751$$ 15373.0i 0.746960i 0.927638 + 0.373480i $$0.121836\pi$$
−0.927638 + 0.373480i $$0.878164\pi$$
$$752$$ 0 0
$$753$$ 0 0
$$754$$ 0 0
$$755$$ −21569.6 −1.03973
$$756$$ 0 0
$$757$$ −26000.8 −1.24837 −0.624184 0.781277i $$-0.714569\pi$$
−0.624184 + 0.781277i $$0.714569\pi$$
$$758$$ 0 0
$$759$$ 0 0
$$760$$ 0 0
$$761$$ − 9461.72i − 0.450706i −0.974277 0.225353i $$-0.927646\pi$$
0.974277 0.225353i $$-0.0723535\pi$$
$$762$$ 0 0
$$763$$ − 5696.06i − 0.270264i
$$764$$ 0 0
$$765$$ 0 0
$$766$$ 0 0
$$767$$ 46380.2 2.18343
$$768$$ 0 0
$$769$$ 11196.2 0.525028 0.262514 0.964928i $$-0.415448\pi$$
0.262514 + 0.964928i $$0.415448\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ 0 0
$$773$$ − 39575.0i − 1.84141i −0.390255 0.920707i $$-0.627613\pi$$
0.390255 0.920707i $$-0.372387\pi$$
$$774$$ 0 0
$$775$$ 26591.2i 1.23250i
$$776$$ 0 0
$$777$$ 0 0
$$778$$ 0 0
$$779$$ 40266.2 1.85197
$$780$$ 0 0
$$781$$ −22192.1 −1.01677
$$782$$ 0 0
$$783$$ 0 0
$$784$$ 0 0
$$785$$ − 4743.53i − 0.215674i
$$786$$ 0 0
$$787$$ − 25069.2i − 1.13548i −0.823209 0.567739i $$-0.807818\pi$$
0.823209 0.567739i $$-0.192182\pi$$
$$788$$ 0 0
$$789$$ 0 0
$$790$$ 0 0
$$791$$ 7289.58 0.327671
$$792$$ 0 0
$$793$$ −19058.4 −0.853448
$$794$$ 0 0
$$795$$ 0 0
$$796$$ 0 0
$$797$$ 5639.39i 0.250637i 0.992117 + 0.125318i $$0.0399952\pi$$
−0.992117 + 0.125318i $$0.960005\pi$$
$$798$$ 0 0
$$799$$ 7762.31i 0.343693i
$$800$$ 0 0
$$801$$ 0 0
$$802$$ 0 0
$$803$$ −11189.2 −0.491729
$$804$$ 0 0
$$805$$ 1731.33 0.0758030
$$806$$ 0 0
$$807$$ 0 0
$$808$$ 0 0
$$809$$ 34032.0i 1.47899i 0.673163 + 0.739494i $$0.264935\pi$$
−0.673163 + 0.739494i $$0.735065\pi$$
$$810$$ 0 0
$$811$$ 20119.5i 0.871137i 0.900156 + 0.435569i $$0.143453\pi$$
−0.900156 + 0.435569i $$0.856547\pi$$
$$812$$ 0 0
$$813$$ 0 0
$$814$$ 0 0
$$815$$ 6038.49 0.259532
$$816$$ 0 0
$$817$$ 35430.0 1.51718
$$818$$ 0 0
$$819$$ 0 0
$$820$$ 0 0
$$821$$ − 9057.81i − 0.385042i −0.981293 0.192521i $$-0.938334\pi$$
0.981293 0.192521i $$-0.0616664\pi$$
$$822$$ 0 0
$$823$$ − 26277.7i − 1.11298i −0.830853 0.556491i $$-0.812147\pi$$
0.830853 0.556491i $$-0.187853\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ 0 0
$$827$$ 21737.7 0.914018 0.457009 0.889462i $$-0.348921\pi$$
0.457009 + 0.889462i $$0.348921\pi$$
$$828$$ 0 0
$$829$$ −27552.6 −1.15433 −0.577167 0.816626i $$-0.695842\pi$$
−0.577167 + 0.816626i $$0.695842\pi$$
$$830$$ 0 0
$$831$$ 0 0
$$832$$ 0 0
$$833$$ − 31062.3i − 1.29201i
$$834$$ 0 0
$$835$$ − 8485.65i − 0.351686i
$$836$$ 0 0
$$837$$ 0 0
$$838$$ 0 0
$$839$$ −12231.7 −0.503320 −0.251660 0.967816i $$-0.580977\pi$$
−0.251660 + 0.967816i $$0.580977\pi$$
$$840$$ 0 0
$$841$$ 22785.1 0.934236
$$842$$ 0 0
$$843$$ 0 0
$$844$$ 0 0
$$845$$ 4612.48i 0.187780i
$$846$$ 0 0
$$847$$ 6829.56i 0.277056i
$$848$$ 0 0
$$849$$ 0 0
$$850$$ 0 0
$$851$$ −9511.00 −0.383117
$$852$$ 0 0
$$853$$ 2257.34 0.0906093 0.0453046 0.998973i $$-0.485574\pi$$
0.0453046 + 0.998973i $$0.485574\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ 0 0
$$857$$ 24908.2i 0.992820i 0.868088 + 0.496410i $$0.165349\pi$$
−0.868088 + 0.496410i $$0.834651\pi$$
$$858$$ 0 0
$$859$$ 11850.7i 0.470710i 0.971909 + 0.235355i $$0.0756253\pi$$
−0.971909 + 0.235355i $$0.924375\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ 0 0
$$863$$ 43145.0 1.70182 0.850910 0.525311i $$-0.176051\pi$$
0.850910 + 0.525311i $$0.176051\pi$$
$$864$$ 0 0
$$865$$ 3707.13 0.145718
$$866$$ 0 0
$$867$$ 0 0
$$868$$ 0 0
$$869$$ − 6452.01i − 0.251864i
$$870$$ 0 0
$$871$$ 10618.3i 0.413072i
$$872$$ 0 0
$$873$$ 0 0
$$874$$ 0 0
$$875$$ 11133.1 0.430133
$$876$$ 0 0
$$877$$ 14333.4 0.551886 0.275943 0.961174i $$-0.411010\pi$$
0.275943 + 0.961174i $$0.411010\pi$$
$$878$$ 0 0
$$879$$ 0 0
$$880$$ 0 0
$$881$$ 5066.51i 0.193752i 0.995296 + 0.0968758i $$0.0308850\pi$$
−0.995296 + 0.0968758i $$0.969115\pi$$
$$882$$ 0 0
$$883$$ − 7046.42i − 0.268551i −0.990944 0.134276i $$-0.957129\pi$$
0.990944 0.134276i $$-0.0428708\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ 0 0
$$887$$ −52122.7 −1.97307 −0.986534 0.163559i $$-0.947703\pi$$
−0.986534 + 0.163559i $$0.947703\pi$$
$$888$$ 0 0
$$889$$ −6664.41 −0.251425
$$890$$ 0 0
$$891$$ 0 0
$$892$$ 0 0
$$893$$ 7256.67i 0.271932i
$$894$$ 0 0
$$895$$ − 16987.4i − 0.634444i
$$896$$ 0 0
$$897$$ 0 0
$$898$$ 0 0
$$899$$ 11714.7 0.434602
$$900$$ 0 0
$$901$$ −18578.7 −0.686955
$$902$$ 0 0
$$903$$ 0 0
$$904$$ 0 0
$$905$$ − 7908.56i − 0.290486i
$$906$$ 0 0
$$907$$ − 41205.1i − 1.50848i −0.656599 0.754240i $$-0.728006\pi$$
0.656599 0.754240i $$-0.271994\pi$$
$$908$$ 0 0
$$909$$ 0 0
$$910$$ 0 0
$$911$$ 34305.8 1.24764 0.623821 0.781567i $$-0.285580\pi$$
0.623821 + 0.781567i $$0.285580\pi$$
$$912$$ 0 0
$$913$$ 7990.37 0.289642
$$914$$ 0 0
$$915$$ 0 0
$$916$$ 0 0
$$917$$ 17895.7i 0.644458i
$$918$$ 0 0
$$919$$ − 43537.1i − 1.56274i −0.624068 0.781370i $$-0.714521\pi$$
0.624068 0.781370i $$-0.285479\pi$$
$$920$$ 0 0
$$921$$ 0 0
$$922$$ 0 0
$$923$$ 51360.8 1.83160
$$924$$ 0 0
$$925$$ −25750.9 −0.915333
$$926$$ 0 0
$$927$$ 0 0
$$928$$ 0 0
$$929$$ − 8764.21i − 0.309520i −0.987952 0.154760i $$-0.950539\pi$$
0.987952 0.154760i $$-0.0494605\pi$$
$$930$$ 0 0
$$931$$ − 29038.9i − 1.02225i
$$932$$ 0 0
$$933$$ 0 0
$$934$$ 0 0
$$935$$ 16161.5 0.565281
$$936$$ 0 0
$$937$$ −22745.0 −0.793006 −0.396503 0.918033i $$-0.629776\pi$$
−0.396503 + 0.918033i $$0.629776\pi$$
$$938$$ 0 0
$$939$$ 0 0
$$940$$ 0 0
$$941$$ 24740.5i 0.857084i 0.903522 + 0.428542i $$0.140973\pi$$
−0.903522 + 0.428542i $$0.859027\pi$$
$$942$$ 0 0
$$943$$ 12338.3i 0.426078i
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ −1886.69 −0.0647405 −0.0323703 0.999476i $$-0.510306\pi$$
−0.0323703 + 0.999476i $$0.510306\pi$$
$$948$$ 0 0
$$949$$ 25896.0 0.885796
$$950$$ 0 0
$$951$$ 0 0
$$952$$ 0 0
$$953$$ − 36292.1i − 1.23360i −0.787122 0.616798i $$-0.788430\pi$$
0.787122 0.616798i $$-0.211570\pi$$
$$954$$ 0 0
$$955$$ − 20368.5i − 0.690165i
$$956$$ 0 0
$$957$$ 0 0
$$958$$ 0 0
$$959$$ 16627.9 0.559900
$$960$$ 0 0
$$961$$ −55770.8 −1.87207
$$962$$ 0 0
$$963$$ 0 0
$$964$$ 0 0
$$965$$ 17008.7i 0.567387i
$$966$$ 0 0
$$967$$ 12599.4i 0.418996i 0.977809 + 0.209498i $$0.0671830\pi$$
−0.977809 + 0.209498i $$0.932817\pi$$
$$968$$ 0 0
$$969$$ 0 0
$$970$$ 0 0
$$971$$ −41974.3 −1.38725 −0.693625 0.720336i $$-0.743988\pi$$
−0.693625 + 0.720336i $$0.743988\pi$$
$$972$$ 0 0
$$973$$ 827.721 0.0272719
$$974$$ 0 0
$$975$$ 0 0
$$976$$ 0 0
$$977$$ − 33488.8i − 1.09662i −0.836274 0.548312i $$-0.815271\pi$$
0.836274 0.548312i $$-0.184729\pi$$
$$978$$ 0 0
$$979$$ − 17472.1i − 0.570387i
$$980$$ 0 0
$$981$$ 0 0
$$982$$ 0 0
$$983$$ −3461.54 −0.112315 −0.0561576 0.998422i $$-0.517885\pi$$
−0.0561576 + 0.998422i $$0.517885\pi$$
$$984$$ 0 0
$$985$$ 14204.0 0.459470
$$986$$ 0 0
$$987$$ 0 0
$$988$$ 0 0
$$989$$ 10856.5i 0.349055i
$$990$$ 0 0
$$991$$ − 20067.6i − 0.643256i −0.946866 0.321628i $$-0.895770\pi$$
0.946866 0.321628i $$-0.104230\pi$$
$$992$$ 0 0
$$993$$ 0 0
$$994$$ 0 0
$$995$$ 3288.41 0.104774
$$996$$ 0 0
$$997$$ −821.726 −0.0261026 −0.0130513 0.999915i $$-0.504154\pi$$
−0.0130513 + 0.999915i $$0.504154\pi$$
$$998$$ 0 0
$$999$$ 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 1728.4.c.j.1727.9 12
3.2 odd 2 inner 1728.4.c.j.1727.3 12
4.3 odd 2 inner 1728.4.c.j.1727.10 12
8.3 odd 2 108.4.b.b.107.3 12
8.5 even 2 108.4.b.b.107.9 yes 12
12.11 even 2 inner 1728.4.c.j.1727.4 12
24.5 odd 2 108.4.b.b.107.4 yes 12
24.11 even 2 108.4.b.b.107.10 yes 12

By twisted newform
Twist Min Dim Char Parity Ord Type
108.4.b.b.107.3 12 8.3 odd 2
108.4.b.b.107.4 yes 12 24.5 odd 2
108.4.b.b.107.9 yes 12 8.5 even 2
108.4.b.b.107.10 yes 12 24.11 even 2
1728.4.c.j.1727.3 12 3.2 odd 2 inner
1728.4.c.j.1727.4 12 12.11 even 2 inner
1728.4.c.j.1727.9 12 1.1 even 1 trivial
1728.4.c.j.1727.10 12 4.3 odd 2 inner