# Properties

 Label 1728.4.c.j.1727.6 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 1727.6 Root $$-2.18604 - 2.07664i$$ of defining polynomial Character $$\chi$$ $$=$$ 1728.1727 Dual form 1728.4.c.j.1727.7

## $q$-expansion

 $$f(q)$$ $$=$$ $$q-1.49508i q^{5} +26.1852i q^{7} +O(q^{10})$$ $$q-1.49508i q^{5} +26.1852i q^{7} -56.3941 q^{11} +41.3170 q^{13} +51.0410i q^{17} -79.0640i q^{19} +27.3688 q^{23} +122.765 q^{25} +134.567i q^{29} +187.192i q^{31} +39.1491 q^{35} +196.585 q^{37} +298.015i q^{41} -465.576i q^{43} -373.845 q^{47} -342.667 q^{49} +620.093i q^{53} +84.3140i q^{55} +321.152 q^{59} -674.699 q^{61} -61.7724i q^{65} +576.075i q^{67} +223.813 q^{71} +70.1371 q^{73} -1476.69i q^{77} -1052.32i q^{79} -1219.05 q^{83} +76.3107 q^{85} -1340.64i q^{89} +1081.89i q^{91} -118.207 q^{95} -576.059 q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$12q + O(q^{10})$$ $$12q + 72q^{13} - 384q^{25} + 240q^{37} + 288q^{49} - 144q^{61} + 156q^{73} + 168q^{85} + 516q^{97} + O(q^{100})$$

## 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
$$5$$ − 1.49508i − 0.133724i −0.997762 0.0668622i $$-0.978701\pi$$
0.997762 0.0668622i $$-0.0212988\pi$$
$$6$$ 0 0
$$7$$ 26.1852i 1.41387i 0.707279 + 0.706935i $$0.249923\pi$$
−0.707279 + 0.706935i $$0.750077\pi$$
$$8$$ 0 0
$$9$$ 0 0
$$10$$ 0 0
$$11$$ −56.3941 −1.54577 −0.772885 0.634546i $$-0.781187\pi$$
−0.772885 + 0.634546i $$0.781187\pi$$
$$12$$ 0 0
$$13$$ 41.3170 0.881482 0.440741 0.897634i $$-0.354716\pi$$
0.440741 + 0.897634i $$0.354716\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 0 0
$$17$$ 51.0410i 0.728192i 0.931361 + 0.364096i $$0.118622\pi$$
−0.931361 + 0.364096i $$0.881378\pi$$
$$18$$ 0 0
$$19$$ − 79.0640i − 0.954659i −0.878724 0.477330i $$-0.841605\pi$$
0.878724 0.477330i $$-0.158395\pi$$
$$20$$ 0 0
$$21$$ 0 0
$$22$$ 0 0
$$23$$ 27.3688 0.248121 0.124061 0.992275i $$-0.460408\pi$$
0.124061 + 0.992275i $$0.460408\pi$$
$$24$$ 0 0
$$25$$ 122.765 0.982118
$$26$$ 0 0
$$27$$ 0 0
$$28$$ 0 0
$$29$$ 134.567i 0.861674i 0.902430 + 0.430837i $$0.141782\pi$$
−0.902430 + 0.430837i $$0.858218\pi$$
$$30$$ 0 0
$$31$$ 187.192i 1.08454i 0.840204 + 0.542270i $$0.182435\pi$$
−0.840204 + 0.542270i $$0.817565\pi$$
$$32$$ 0 0
$$33$$ 0 0
$$34$$ 0 0
$$35$$ 39.1491 0.189069
$$36$$ 0 0
$$37$$ 196.585 0.873469 0.436734 0.899590i $$-0.356135\pi$$
0.436734 + 0.899590i $$0.356135\pi$$
$$38$$ 0 0
$$39$$ 0 0
$$40$$ 0 0
$$41$$ 298.015i 1.13517i 0.823313 + 0.567587i $$0.192123\pi$$
−0.823313 + 0.567587i $$0.807877\pi$$
$$42$$ 0 0
$$43$$ − 465.576i − 1.65115i −0.564289 0.825577i $$-0.690850\pi$$
0.564289 0.825577i $$-0.309150\pi$$
$$44$$ 0 0
$$45$$ 0 0
$$46$$ 0 0
$$47$$ −373.845 −1.16023 −0.580116 0.814534i $$-0.696993\pi$$
−0.580116 + 0.814534i $$0.696993\pi$$
$$48$$ 0 0
$$49$$ −342.667 −0.999028
$$50$$ 0 0
$$51$$ 0 0
$$52$$ 0 0
$$53$$ 620.093i 1.60710i 0.595237 + 0.803550i $$0.297058\pi$$
−0.595237 + 0.803550i $$0.702942\pi$$
$$54$$ 0 0
$$55$$ 84.3140i 0.206707i
$$56$$ 0 0
$$57$$ 0 0
$$58$$ 0 0
$$59$$ 321.152 0.708652 0.354326 0.935122i $$-0.384710\pi$$
0.354326 + 0.935122i $$0.384710\pi$$
$$60$$ 0 0
$$61$$ −674.699 −1.41617 −0.708085 0.706127i $$-0.750441\pi$$
−0.708085 + 0.706127i $$0.750441\pi$$
$$62$$ 0 0
$$63$$ 0 0
$$64$$ 0 0
$$65$$ − 61.7724i − 0.117876i
$$66$$ 0 0
$$67$$ 576.075i 1.05043i 0.850970 + 0.525215i $$0.176015\pi$$
−0.850970 + 0.525215i $$0.823985\pi$$
$$68$$ 0 0
$$69$$ 0 0
$$70$$ 0 0
$$71$$ 223.813 0.374110 0.187055 0.982349i $$-0.440106\pi$$
0.187055 + 0.982349i $$0.440106\pi$$
$$72$$ 0 0
$$73$$ 70.1371 0.112451 0.0562255 0.998418i $$-0.482093\pi$$
0.0562255 + 0.998418i $$0.482093\pi$$
$$74$$ 0 0
$$75$$ 0 0
$$76$$ 0 0
$$77$$ − 1476.69i − 2.18552i
$$78$$ 0 0
$$79$$ − 1052.32i − 1.49868i −0.662187 0.749338i $$-0.730372\pi$$
0.662187 0.749338i $$-0.269628\pi$$
$$80$$ 0 0
$$81$$ 0 0
$$82$$ 0 0
$$83$$ −1219.05 −1.61214 −0.806070 0.591820i $$-0.798409\pi$$
−0.806070 + 0.591820i $$0.798409\pi$$
$$84$$ 0 0
$$85$$ 76.3107 0.0973771
$$86$$ 0 0
$$87$$ 0 0
$$88$$ 0 0
$$89$$ − 1340.64i − 1.59672i −0.602183 0.798358i $$-0.705702\pi$$
0.602183 0.798358i $$-0.294298\pi$$
$$90$$ 0 0
$$91$$ 1081.89i 1.24630i
$$92$$ 0 0
$$93$$ 0 0
$$94$$ 0 0
$$95$$ −118.207 −0.127661
$$96$$ 0 0
$$97$$ −576.059 −0.602989 −0.301494 0.953468i $$-0.597485\pi$$
−0.301494 + 0.953468i $$0.597485\pi$$
$$98$$ 0 0
$$99$$ 0 0
$$100$$ 0 0
$$101$$ 116.079i 0.114359i 0.998364 + 0.0571797i $$0.0182108\pi$$
−0.998364 + 0.0571797i $$0.981789\pi$$
$$102$$ 0 0
$$103$$ − 165.074i − 0.157915i −0.996878 0.0789573i $$-0.974841\pi$$
0.996878 0.0789573i $$-0.0251591\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ −936.718 −0.846318 −0.423159 0.906056i $$-0.639079\pi$$
−0.423159 + 0.906056i $$0.639079\pi$$
$$108$$ 0 0
$$109$$ −346.957 −0.304885 −0.152443 0.988312i $$-0.548714\pi$$
−0.152443 + 0.988312i $$0.548714\pi$$
$$110$$ 0 0
$$111$$ 0 0
$$112$$ 0 0
$$113$$ 1462.22i 1.21729i 0.793443 + 0.608645i $$0.208287\pi$$
−0.793443 + 0.608645i $$0.791713\pi$$
$$114$$ 0 0
$$115$$ − 40.9187i − 0.0331799i
$$116$$ 0 0
$$117$$ 0 0
$$118$$ 0 0
$$119$$ −1336.52 −1.02957
$$120$$ 0 0
$$121$$ 1849.30 1.38941
$$122$$ 0 0
$$123$$ 0 0
$$124$$ 0 0
$$125$$ − 370.429i − 0.265058i
$$126$$ 0 0
$$127$$ 265.004i 0.185160i 0.995705 + 0.0925800i $$0.0295114\pi$$
−0.995705 + 0.0925800i $$0.970489\pi$$
$$128$$ 0 0
$$129$$ 0 0
$$130$$ 0 0
$$131$$ −1151.86 −0.768231 −0.384115 0.923285i $$-0.625493\pi$$
−0.384115 + 0.923285i $$0.625493\pi$$
$$132$$ 0 0
$$133$$ 2070.31 1.34976
$$134$$ 0 0
$$135$$ 0 0
$$136$$ 0 0
$$137$$ − 2348.67i − 1.46467i −0.680943 0.732337i $$-0.738430\pi$$
0.680943 0.732337i $$-0.261570\pi$$
$$138$$ 0 0
$$139$$ 215.240i 0.131341i 0.997841 + 0.0656706i $$0.0209187\pi$$
−0.997841 + 0.0656706i $$0.979081\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ 0 0
$$143$$ −2330.04 −1.36257
$$144$$ 0 0
$$145$$ 201.190 0.115227
$$146$$ 0 0
$$147$$ 0 0
$$148$$ 0 0
$$149$$ 219.954i 0.120935i 0.998170 + 0.0604674i $$0.0192591\pi$$
−0.998170 + 0.0604674i $$0.980741\pi$$
$$150$$ 0 0
$$151$$ − 1148.02i − 0.618703i −0.950948 0.309351i $$-0.899888\pi$$
0.950948 0.309351i $$-0.100112\pi$$
$$152$$ 0 0
$$153$$ 0 0
$$154$$ 0 0
$$155$$ 279.869 0.145030
$$156$$ 0 0
$$157$$ 276.320 0.140463 0.0702316 0.997531i $$-0.477626\pi$$
0.0702316 + 0.997531i $$0.477626\pi$$
$$158$$ 0 0
$$159$$ 0 0
$$160$$ 0 0
$$161$$ 716.659i 0.350811i
$$162$$ 0 0
$$163$$ − 1419.15i − 0.681941i −0.940074 0.340971i $$-0.889244\pi$$
0.940074 0.340971i $$-0.110756\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ 0 0
$$167$$ −3508.69 −1.62581 −0.812906 0.582394i $$-0.802116\pi$$
−0.812906 + 0.582394i $$0.802116\pi$$
$$168$$ 0 0
$$169$$ −489.908 −0.222990
$$170$$ 0 0
$$171$$ 0 0
$$172$$ 0 0
$$173$$ − 330.965i − 0.145450i −0.997352 0.0727248i $$-0.976831\pi$$
0.997352 0.0727248i $$-0.0231695\pi$$
$$174$$ 0 0
$$175$$ 3214.62i 1.38859i
$$176$$ 0 0
$$177$$ 0 0
$$178$$ 0 0
$$179$$ −1136.75 −0.474662 −0.237331 0.971429i $$-0.576273\pi$$
−0.237331 + 0.971429i $$0.576273\pi$$
$$180$$ 0 0
$$181$$ −2056.92 −0.844692 −0.422346 0.906435i $$-0.638793\pi$$
−0.422346 + 0.906435i $$0.638793\pi$$
$$182$$ 0 0
$$183$$ 0 0
$$184$$ 0 0
$$185$$ − 293.911i − 0.116804i
$$186$$ 0 0
$$187$$ − 2878.42i − 1.12562i
$$188$$ 0 0
$$189$$ 0 0
$$190$$ 0 0
$$191$$ 1983.23 0.751316 0.375658 0.926758i $$-0.377417\pi$$
0.375658 + 0.926758i $$0.377417\pi$$
$$192$$ 0 0
$$193$$ −189.908 −0.0708283 −0.0354141 0.999373i $$-0.511275\pi$$
−0.0354141 + 0.999373i $$0.511275\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ 0 0
$$197$$ 2160.42i 0.781337i 0.920531 + 0.390669i $$0.127756\pi$$
−0.920531 + 0.390669i $$0.872244\pi$$
$$198$$ 0 0
$$199$$ 2656.23i 0.946205i 0.881007 + 0.473103i $$0.156866\pi$$
−0.881007 + 0.473103i $$0.843134\pi$$
$$200$$ 0 0
$$201$$ 0 0
$$202$$ 0 0
$$203$$ −3523.68 −1.21829
$$204$$ 0 0
$$205$$ 445.558 0.151801
$$206$$ 0 0
$$207$$ 0 0
$$208$$ 0 0
$$209$$ 4458.75i 1.47568i
$$210$$ 0 0
$$211$$ − 1001.20i − 0.326662i −0.986571 0.163331i $$-0.947776\pi$$
0.986571 0.163331i $$-0.0522239\pi$$
$$212$$ 0 0
$$213$$ 0 0
$$214$$ 0 0
$$215$$ −696.075 −0.220800
$$216$$ 0 0
$$217$$ −4901.68 −1.53340
$$218$$ 0 0
$$219$$ 0 0
$$220$$ 0 0
$$221$$ 2108.86i 0.641888i
$$222$$ 0 0
$$223$$ 3193.62i 0.959015i 0.877538 + 0.479507i $$0.159185\pi$$
−0.877538 + 0.479507i $$0.840815\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ 0 0
$$227$$ −1714.72 −0.501366 −0.250683 0.968069i $$-0.580655\pi$$
−0.250683 + 0.968069i $$0.580655\pi$$
$$228$$ 0 0
$$229$$ 407.497 0.117590 0.0587951 0.998270i $$-0.481274\pi$$
0.0587951 + 0.998270i $$0.481274\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 0 0
$$233$$ 3210.54i 0.902702i 0.892346 + 0.451351i $$0.149058\pi$$
−0.892346 + 0.451351i $$0.850942\pi$$
$$234$$ 0 0
$$235$$ 558.930i 0.155151i
$$236$$ 0 0
$$237$$ 0 0
$$238$$ 0 0
$$239$$ −1561.19 −0.422532 −0.211266 0.977429i $$-0.567759\pi$$
−0.211266 + 0.977429i $$0.567759\pi$$
$$240$$ 0 0
$$241$$ 1460.89 0.390475 0.195238 0.980756i $$-0.437452\pi$$
0.195238 + 0.980756i $$0.437452\pi$$
$$242$$ 0 0
$$243$$ 0 0
$$244$$ 0 0
$$245$$ 512.316i 0.133594i
$$246$$ 0 0
$$247$$ − 3266.69i − 0.841515i
$$248$$ 0 0
$$249$$ 0 0
$$250$$ 0 0
$$251$$ −6868.44 −1.72722 −0.863609 0.504162i $$-0.831802\pi$$
−0.863609 + 0.504162i $$0.831802\pi$$
$$252$$ 0 0
$$253$$ −1543.44 −0.383539
$$254$$ 0 0
$$255$$ 0 0
$$256$$ 0 0
$$257$$ − 4450.84i − 1.08029i −0.841570 0.540147i $$-0.818368\pi$$
0.841570 0.540147i $$-0.181632\pi$$
$$258$$ 0 0
$$259$$ 5147.62i 1.23497i
$$260$$ 0 0
$$261$$ 0 0
$$262$$ 0 0
$$263$$ −6525.31 −1.52991 −0.764957 0.644081i $$-0.777240\pi$$
−0.764957 + 0.644081i $$0.777240\pi$$
$$264$$ 0 0
$$265$$ 927.091 0.214909
$$266$$ 0 0
$$267$$ 0 0
$$268$$ 0 0
$$269$$ − 3222.22i − 0.730343i −0.930940 0.365171i $$-0.881010\pi$$
0.930940 0.365171i $$-0.118990\pi$$
$$270$$ 0 0
$$271$$ − 7368.93i − 1.65177i −0.563837 0.825886i $$-0.690675\pi$$
0.563837 0.825886i $$-0.309325\pi$$
$$272$$ 0 0
$$273$$ 0 0
$$274$$ 0 0
$$275$$ −6923.21 −1.51813
$$276$$ 0 0
$$277$$ −9078.61 −1.96924 −0.984622 0.174699i $$-0.944105\pi$$
−0.984622 + 0.174699i $$0.944105\pi$$
$$278$$ 0 0
$$279$$ 0 0
$$280$$ 0 0
$$281$$ 2613.34i 0.554801i 0.960754 + 0.277400i $$0.0894729\pi$$
−0.960754 + 0.277400i $$0.910527\pi$$
$$282$$ 0 0
$$283$$ 927.970i 0.194919i 0.995239 + 0.0974596i $$0.0310717\pi$$
−0.995239 + 0.0974596i $$0.968928\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ 0 0
$$287$$ −7803.60 −1.60499
$$288$$ 0 0
$$289$$ 2307.81 0.469736
$$290$$ 0 0
$$291$$ 0 0
$$292$$ 0 0
$$293$$ 5351.73i 1.06707i 0.845778 + 0.533535i $$0.179137\pi$$
−0.845778 + 0.533535i $$0.820863\pi$$
$$294$$ 0 0
$$295$$ − 480.150i − 0.0947641i
$$296$$ 0 0
$$297$$ 0 0
$$298$$ 0 0
$$299$$ 1130.80 0.218714
$$300$$ 0 0
$$301$$ 12191.2 2.33452
$$302$$ 0 0
$$303$$ 0 0
$$304$$ 0 0
$$305$$ 1008.73i 0.189377i
$$306$$ 0 0
$$307$$ − 1892.38i − 0.351804i −0.984408 0.175902i $$-0.943716\pi$$
0.984408 0.175902i $$-0.0562842\pi$$
$$308$$ 0 0
$$309$$ 0 0
$$310$$ 0 0
$$311$$ −5645.22 −1.02930 −0.514648 0.857402i $$-0.672077\pi$$
−0.514648 + 0.857402i $$0.672077\pi$$
$$312$$ 0 0
$$313$$ −818.001 −0.147719 −0.0738597 0.997269i $$-0.523532\pi$$
−0.0738597 + 0.997269i $$0.523532\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ − 1946.72i − 0.344917i −0.985017 0.172458i $$-0.944829\pi$$
0.985017 0.172458i $$-0.0551710\pi$$
$$318$$ 0 0
$$319$$ − 7588.81i − 1.33195i
$$320$$ 0 0
$$321$$ 0 0
$$322$$ 0 0
$$323$$ 4035.51 0.695176
$$324$$ 0 0
$$325$$ 5072.27 0.865719
$$326$$ 0 0
$$327$$ 0 0
$$328$$ 0 0
$$329$$ − 9789.22i − 1.64042i
$$330$$ 0 0
$$331$$ 3404.83i 0.565396i 0.959209 + 0.282698i $$0.0912295\pi$$
−0.959209 + 0.282698i $$0.908771\pi$$
$$332$$ 0 0
$$333$$ 0 0
$$334$$ 0 0
$$335$$ 861.281 0.140468
$$336$$ 0 0
$$337$$ 7072.15 1.14316 0.571579 0.820547i $$-0.306331\pi$$
0.571579 + 0.820547i $$0.306331\pi$$
$$338$$ 0 0
$$339$$ 0 0
$$340$$ 0 0
$$341$$ − 10556.6i − 1.67645i
$$342$$ 0 0
$$343$$ 8.73194i 0.00137458i
$$344$$ 0 0
$$345$$ 0 0
$$346$$ 0 0
$$347$$ 5783.02 0.894665 0.447332 0.894368i $$-0.352374\pi$$
0.447332 + 0.894368i $$0.352374\pi$$
$$348$$ 0 0
$$349$$ −1748.60 −0.268196 −0.134098 0.990968i $$-0.542814\pi$$
−0.134098 + 0.990968i $$0.542814\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ 0 0
$$353$$ − 9552.31i − 1.44028i −0.693830 0.720139i $$-0.744078\pi$$
0.693830 0.720139i $$-0.255922\pi$$
$$354$$ 0 0
$$355$$ − 334.620i − 0.0500276i
$$356$$ 0 0
$$357$$ 0 0
$$358$$ 0 0
$$359$$ −921.392 −0.135457 −0.0677287 0.997704i $$-0.521575\pi$$
−0.0677287 + 0.997704i $$0.521575\pi$$
$$360$$ 0 0
$$361$$ 607.881 0.0886254
$$362$$ 0 0
$$363$$ 0 0
$$364$$ 0 0
$$365$$ − 104.861i − 0.0150375i
$$366$$ 0 0
$$367$$ 8490.66i 1.20765i 0.797115 + 0.603827i $$0.206358\pi$$
−0.797115 + 0.603827i $$0.793642\pi$$
$$368$$ 0 0
$$369$$ 0 0
$$370$$ 0 0
$$371$$ −16237.3 −2.27223
$$372$$ 0 0
$$373$$ −2824.92 −0.392142 −0.196071 0.980590i $$-0.562818\pi$$
−0.196071 + 0.980590i $$0.562818\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ 0 0
$$377$$ 5559.92i 0.759550i
$$378$$ 0 0
$$379$$ 5322.35i 0.721348i 0.932692 + 0.360674i $$0.117453\pi$$
−0.932692 + 0.360674i $$0.882547\pi$$
$$380$$ 0 0
$$381$$ 0 0
$$382$$ 0 0
$$383$$ 8320.49 1.11007 0.555035 0.831827i $$-0.312705\pi$$
0.555035 + 0.831827i $$0.312705\pi$$
$$384$$ 0 0
$$385$$ −2207.78 −0.292257
$$386$$ 0 0
$$387$$ 0 0
$$388$$ 0 0
$$389$$ − 8358.52i − 1.08944i −0.838617 0.544722i $$-0.816635\pi$$
0.838617 0.544722i $$-0.183365\pi$$
$$390$$ 0 0
$$391$$ 1396.93i 0.180680i
$$392$$ 0 0
$$393$$ 0 0
$$394$$ 0 0
$$395$$ −1573.31 −0.200410
$$396$$ 0 0
$$397$$ −7283.17 −0.920735 −0.460367 0.887728i $$-0.652282\pi$$
−0.460367 + 0.887728i $$0.652282\pi$$
$$398$$ 0 0
$$399$$ 0 0
$$400$$ 0 0
$$401$$ 1549.41i 0.192952i 0.995335 + 0.0964759i $$0.0307571\pi$$
−0.995335 + 0.0964759i $$0.969243\pi$$
$$402$$ 0 0
$$403$$ 7734.22i 0.956003i
$$404$$ 0 0
$$405$$ 0 0
$$406$$ 0 0
$$407$$ −11086.2 −1.35018
$$408$$ 0 0
$$409$$ 3291.67 0.397952 0.198976 0.980004i $$-0.436238\pi$$
0.198976 + 0.980004i $$0.436238\pi$$
$$410$$ 0 0
$$411$$ 0 0
$$412$$ 0 0
$$413$$ 8409.45i 1.00194i
$$414$$ 0 0
$$415$$ 1822.58i 0.215583i
$$416$$ 0 0
$$417$$ 0 0
$$418$$ 0 0
$$419$$ −5299.78 −0.617927 −0.308964 0.951074i $$-0.599982\pi$$
−0.308964 + 0.951074i $$0.599982\pi$$
$$420$$ 0 0
$$421$$ 10681.4 1.23653 0.618265 0.785970i $$-0.287836\pi$$
0.618265 + 0.785970i $$0.287836\pi$$
$$422$$ 0 0
$$423$$ 0 0
$$424$$ 0 0
$$425$$ 6266.04i 0.715170i
$$426$$ 0 0
$$427$$ − 17667.2i − 2.00228i
$$428$$ 0 0
$$429$$ 0 0
$$430$$ 0 0
$$431$$ 9866.13 1.10263 0.551316 0.834296i $$-0.314126\pi$$
0.551316 + 0.834296i $$0.314126\pi$$
$$432$$ 0 0
$$433$$ −12560.2 −1.39400 −0.697002 0.717069i $$-0.745483\pi$$
−0.697002 + 0.717069i $$0.745483\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 0 0
$$437$$ − 2163.89i − 0.236871i
$$438$$ 0 0
$$439$$ 929.228i 0.101024i 0.998723 + 0.0505121i $$0.0160854\pi$$
−0.998723 + 0.0505121i $$0.983915\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ 0 0
$$443$$ 4373.85 0.469092 0.234546 0.972105i $$-0.424640\pi$$
0.234546 + 0.972105i $$0.424640\pi$$
$$444$$ 0 0
$$445$$ −2004.37 −0.213520
$$446$$ 0 0
$$447$$ 0 0
$$448$$ 0 0
$$449$$ − 14149.5i − 1.48721i −0.668618 0.743606i $$-0.733114\pi$$
0.668618 0.743606i $$-0.266886\pi$$
$$450$$ 0 0
$$451$$ − 16806.3i − 1.75472i
$$452$$ 0 0
$$453$$ 0 0
$$454$$ 0 0
$$455$$ 1617.52 0.166661
$$456$$ 0 0
$$457$$ −12582.4 −1.28792 −0.643960 0.765059i $$-0.722710\pi$$
−0.643960 + 0.765059i $$0.722710\pi$$
$$458$$ 0 0
$$459$$ 0 0
$$460$$ 0 0
$$461$$ 10602.2i 1.07114i 0.844492 + 0.535568i $$0.179902\pi$$
−0.844492 + 0.535568i $$0.820098\pi$$
$$462$$ 0 0
$$463$$ 5080.88i 0.509997i 0.966941 + 0.254999i $$0.0820750\pi$$
−0.966941 + 0.254999i $$0.917925\pi$$
$$464$$ 0 0
$$465$$ 0 0
$$466$$ 0 0
$$467$$ −6903.92 −0.684102 −0.342051 0.939681i $$-0.611121\pi$$
−0.342051 + 0.939681i $$0.611121\pi$$
$$468$$ 0 0
$$469$$ −15084.7 −1.48517
$$470$$ 0 0
$$471$$ 0 0
$$472$$ 0 0
$$473$$ 26255.8i 2.55231i
$$474$$ 0 0
$$475$$ − 9706.27i − 0.937588i
$$476$$ 0 0
$$477$$ 0 0
$$478$$ 0 0
$$479$$ −5114.56 −0.487871 −0.243936 0.969791i $$-0.578439\pi$$
−0.243936 + 0.969791i $$0.578439\pi$$
$$480$$ 0 0
$$481$$ 8122.29 0.769947
$$482$$ 0 0
$$483$$ 0 0
$$484$$ 0 0
$$485$$ 861.257i 0.0806344i
$$486$$ 0 0
$$487$$ 15594.2i 1.45101i 0.688218 + 0.725504i $$0.258393\pi$$
−0.688218 + 0.725504i $$0.741607\pi$$
$$488$$ 0 0
$$489$$ 0 0
$$490$$ 0 0
$$491$$ 10521.0 0.967023 0.483511 0.875338i $$-0.339361\pi$$
0.483511 + 0.875338i $$0.339361\pi$$
$$492$$ 0 0
$$493$$ −6868.46 −0.627464
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 0 0
$$497$$ 5860.61i 0.528942i
$$498$$ 0 0
$$499$$ 8984.43i 0.806009i 0.915198 + 0.403004i $$0.132034\pi$$
−0.915198 + 0.403004i $$0.867966\pi$$
$$500$$ 0 0
$$501$$ 0 0
$$502$$ 0 0
$$503$$ −4299.12 −0.381090 −0.190545 0.981678i $$-0.561026\pi$$
−0.190545 + 0.981678i $$0.561026\pi$$
$$504$$ 0 0
$$505$$ 173.548 0.0152926
$$506$$ 0 0
$$507$$ 0 0
$$508$$ 0 0
$$509$$ 2984.83i 0.259922i 0.991519 + 0.129961i $$0.0414853\pi$$
−0.991519 + 0.129961i $$0.958515\pi$$
$$510$$ 0 0
$$511$$ 1836.56i 0.158991i
$$512$$ 0 0
$$513$$ 0 0
$$514$$ 0 0
$$515$$ −246.799 −0.0211170
$$516$$ 0 0
$$517$$ 21082.7 1.79345
$$518$$ 0 0
$$519$$ 0 0
$$520$$ 0 0
$$521$$ − 12255.7i − 1.03058i −0.857016 0.515290i $$-0.827684\pi$$
0.857016 0.515290i $$-0.172316\pi$$
$$522$$ 0 0
$$523$$ 15148.5i 1.26654i 0.773932 + 0.633269i $$0.218287\pi$$
−0.773932 + 0.633269i $$0.781713\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ −9554.49 −0.789754
$$528$$ 0 0
$$529$$ −11417.9 −0.938436
$$530$$ 0 0
$$531$$ 0 0
$$532$$ 0 0
$$533$$ 12313.1i 1.00064i
$$534$$ 0 0
$$535$$ 1400.47i 0.113173i
$$536$$ 0 0
$$537$$ 0 0
$$538$$ 0 0
$$539$$ 19324.4 1.54427
$$540$$ 0 0
$$541$$ −5723.84 −0.454875 −0.227437 0.973793i $$-0.573035\pi$$
−0.227437 + 0.973793i $$0.573035\pi$$
$$542$$ 0 0
$$543$$ 0 0
$$544$$ 0 0
$$545$$ 518.730i 0.0407706i
$$546$$ 0 0
$$547$$ − 8367.43i − 0.654051i −0.945016 0.327025i $$-0.893954\pi$$
0.945016 0.327025i $$-0.106046\pi$$
$$548$$ 0 0
$$549$$ 0 0
$$550$$ 0 0
$$551$$ 10639.4 0.822605
$$552$$ 0 0
$$553$$ 27555.3 2.11893
$$554$$ 0 0
$$555$$ 0 0
$$556$$ 0 0
$$557$$ 15824.2i 1.20375i 0.798589 + 0.601877i $$0.205580\pi$$
−0.798589 + 0.601877i $$0.794420\pi$$
$$558$$ 0 0
$$559$$ − 19236.2i − 1.45546i
$$560$$ 0 0
$$561$$ 0 0
$$562$$ 0 0
$$563$$ 2781.72 0.208233 0.104117 0.994565i $$-0.466798\pi$$
0.104117 + 0.994565i $$0.466798\pi$$
$$564$$ 0 0
$$565$$ 2186.14 0.162781
$$566$$ 0 0
$$567$$ 0 0
$$568$$ 0 0
$$569$$ − 4687.63i − 0.345370i −0.984977 0.172685i $$-0.944756\pi$$
0.984977 0.172685i $$-0.0552443\pi$$
$$570$$ 0 0
$$571$$ 15169.4i 1.11177i 0.831259 + 0.555885i $$0.187621\pi$$
−0.831259 + 0.555885i $$0.812379\pi$$
$$572$$ 0 0
$$573$$ 0 0
$$574$$ 0 0
$$575$$ 3359.92 0.243684
$$576$$ 0 0
$$577$$ −14452.3 −1.04273 −0.521367 0.853332i $$-0.674578\pi$$
−0.521367 + 0.853332i $$0.674578\pi$$
$$578$$ 0 0
$$579$$ 0 0
$$580$$ 0 0
$$581$$ − 31921.0i − 2.27936i
$$582$$ 0 0
$$583$$ − 34969.6i − 2.48421i
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 0 0
$$587$$ 7405.55 0.520715 0.260358 0.965512i $$-0.416160\pi$$
0.260358 + 0.965512i $$0.416160\pi$$
$$588$$ 0 0
$$589$$ 14800.2 1.03537
$$590$$ 0 0
$$591$$ 0 0
$$592$$ 0 0
$$593$$ 19888.1i 1.37725i 0.725120 + 0.688623i $$0.241784\pi$$
−0.725120 + 0.688623i $$0.758216\pi$$
$$594$$ 0 0
$$595$$ 1998.21i 0.137679i
$$596$$ 0 0
$$597$$ 0 0
$$598$$ 0 0
$$599$$ −2335.49 −0.159308 −0.0796539 0.996823i $$-0.525382\pi$$
−0.0796539 + 0.996823i $$0.525382\pi$$
$$600$$ 0 0
$$601$$ 24547.9 1.66611 0.833054 0.553192i $$-0.186590\pi$$
0.833054 + 0.553192i $$0.186590\pi$$
$$602$$ 0 0
$$603$$ 0 0
$$604$$ 0 0
$$605$$ − 2764.86i − 0.185798i
$$606$$ 0 0
$$607$$ 18328.9i 1.22561i 0.790233 + 0.612806i $$0.209959\pi$$
−0.790233 + 0.612806i $$0.790041\pi$$
$$608$$ 0 0
$$609$$ 0 0
$$610$$ 0 0
$$611$$ −15446.1 −1.02272
$$612$$ 0 0
$$613$$ −6450.01 −0.424981 −0.212491 0.977163i $$-0.568157\pi$$
−0.212491 + 0.977163i $$0.568157\pi$$
$$614$$ 0 0
$$615$$ 0 0
$$616$$ 0 0
$$617$$ − 6724.96i − 0.438795i −0.975636 0.219398i $$-0.929591\pi$$
0.975636 0.219398i $$-0.0704092\pi$$
$$618$$ 0 0
$$619$$ 3136.55i 0.203665i 0.994802 + 0.101832i $$0.0324705\pi$$
−0.994802 + 0.101832i $$0.967529\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ 0 0
$$623$$ 35105.0 2.25755
$$624$$ 0 0
$$625$$ 14791.8 0.946673
$$626$$ 0 0
$$627$$ 0 0
$$628$$ 0 0
$$629$$ 10033.9i 0.636053i
$$630$$ 0 0
$$631$$ 6061.57i 0.382420i 0.981549 + 0.191210i $$0.0612412\pi$$
−0.981549 + 0.191210i $$0.938759\pi$$
$$632$$ 0 0
$$633$$ 0 0
$$634$$ 0 0
$$635$$ 396.204 0.0247604
$$636$$ 0 0
$$637$$ −14157.9 −0.880625
$$638$$ 0 0
$$639$$ 0 0
$$640$$ 0 0
$$641$$ − 18603.3i − 1.14631i −0.819446 0.573157i $$-0.805719\pi$$
0.819446 0.573157i $$-0.194281\pi$$
$$642$$ 0 0
$$643$$ 21602.4i 1.32491i 0.749103 + 0.662453i $$0.230485\pi$$
−0.749103 + 0.662453i $$0.769515\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 0 0
$$647$$ 672.754 0.0408790 0.0204395 0.999791i $$-0.493493\pi$$
0.0204395 + 0.999791i $$0.493493\pi$$
$$648$$ 0 0
$$649$$ −18111.1 −1.09541
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 0 0
$$653$$ − 3322.88i − 0.199134i −0.995031 0.0995668i $$-0.968254\pi$$
0.995031 0.0995668i $$-0.0317457\pi$$
$$654$$ 0 0
$$655$$ 1722.12i 0.102731i
$$656$$ 0 0
$$657$$ 0 0
$$658$$ 0 0
$$659$$ 25093.7 1.48332 0.741662 0.670773i $$-0.234038\pi$$
0.741662 + 0.670773i $$0.234038\pi$$
$$660$$ 0 0
$$661$$ 28966.3 1.70447 0.852237 0.523157i $$-0.175246\pi$$
0.852237 + 0.523157i $$0.175246\pi$$
$$662$$ 0 0
$$663$$ 0 0
$$664$$ 0 0
$$665$$ − 3095.29i − 0.180496i
$$666$$ 0 0
$$667$$ 3682.95i 0.213800i
$$668$$ 0 0
$$669$$ 0 0
$$670$$ 0 0
$$671$$ 38049.1 2.18907
$$672$$ 0 0
$$673$$ −4710.36 −0.269793 −0.134897 0.990860i $$-0.543070\pi$$
−0.134897 + 0.990860i $$0.543070\pi$$
$$674$$ 0 0
$$675$$ 0 0
$$676$$ 0 0
$$677$$ 4784.53i 0.271617i 0.990735 + 0.135808i $$0.0433631\pi$$
−0.990735 + 0.135808i $$0.956637\pi$$
$$678$$ 0 0
$$679$$ − 15084.2i − 0.852548i
$$680$$ 0 0
$$681$$ 0 0
$$682$$ 0 0
$$683$$ 18019.8 1.00953 0.504764 0.863258i $$-0.331580\pi$$
0.504764 + 0.863258i $$0.331580\pi$$
$$684$$ 0 0
$$685$$ −3511.46 −0.195863
$$686$$ 0 0
$$687$$ 0 0
$$688$$ 0 0
$$689$$ 25620.4i 1.41663i
$$690$$ 0 0
$$691$$ − 16956.5i − 0.933511i −0.884386 0.466756i $$-0.845423\pi$$
0.884386 0.466756i $$-0.154577\pi$$
$$692$$ 0 0
$$693$$ 0 0
$$694$$ 0 0
$$695$$ 321.802 0.0175635
$$696$$ 0 0
$$697$$ −15211.0 −0.826625
$$698$$ 0 0
$$699$$ 0 0
$$700$$ 0 0
$$701$$ − 30620.5i − 1.64982i −0.565266 0.824909i $$-0.691226\pi$$
0.565266 0.824909i $$-0.308774\pi$$
$$702$$ 0 0
$$703$$ − 15542.8i − 0.833865i
$$704$$ 0 0
$$705$$ 0 0
$$706$$ 0 0
$$707$$ −3039.56 −0.161689
$$708$$ 0 0
$$709$$ −24192.8 −1.28149 −0.640747 0.767752i $$-0.721375\pi$$
−0.640747 + 0.767752i $$0.721375\pi$$
$$710$$ 0 0
$$711$$ 0 0
$$712$$ 0 0
$$713$$ 5123.23i 0.269098i
$$714$$ 0 0
$$715$$ 3483.60i 0.182209i
$$716$$ 0 0
$$717$$ 0 0
$$718$$ 0 0
$$719$$ 37146.6 1.92675 0.963377 0.268151i $$-0.0864124\pi$$
0.963377 + 0.268151i $$0.0864124\pi$$
$$720$$ 0 0
$$721$$ 4322.49 0.223271
$$722$$ 0 0
$$723$$ 0 0
$$724$$ 0 0
$$725$$ 16520.1i 0.846265i
$$726$$ 0 0
$$727$$ 14614.3i 0.745551i 0.927922 + 0.372775i $$0.121594\pi$$
−0.927922 + 0.372775i $$0.878406\pi$$
$$728$$ 0 0
$$729$$ 0 0
$$730$$ 0 0
$$731$$ 23763.5 1.20236
$$732$$ 0 0
$$733$$ 8044.73 0.405374 0.202687 0.979244i $$-0.435033\pi$$
0.202687 + 0.979244i $$0.435033\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ − 32487.3i − 1.62372i
$$738$$ 0 0
$$739$$ 7025.01i 0.349688i 0.984596 + 0.174844i $$0.0559420\pi$$
−0.984596 + 0.174844i $$0.944058\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ 0 0
$$743$$ −27063.4 −1.33629 −0.668143 0.744033i $$-0.732911\pi$$
−0.668143 + 0.744033i $$0.732911\pi$$
$$744$$ 0 0
$$745$$ 328.849 0.0161719
$$746$$ 0 0
$$747$$ 0 0
$$748$$ 0 0
$$749$$ − 24528.2i − 1.19658i
$$750$$ 0 0
$$751$$ − 11434.2i − 0.555579i −0.960642 0.277789i $$-0.910398\pi$$
0.960642 0.277789i $$-0.0896017\pi$$
$$752$$ 0 0
$$753$$ 0 0
$$754$$ 0 0
$$755$$ −1716.38 −0.0827357
$$756$$ 0 0
$$757$$ −22087.2 −1.06046 −0.530232 0.847853i $$-0.677895\pi$$
−0.530232 + 0.847853i $$0.677895\pi$$
$$758$$ 0 0
$$759$$ 0 0
$$760$$ 0 0
$$761$$ − 35524.0i − 1.69218i −0.533043 0.846088i $$-0.678952\pi$$
0.533043 0.846088i $$-0.321048\pi$$
$$762$$ 0 0
$$763$$ − 9085.16i − 0.431068i
$$764$$ 0 0
$$765$$ 0 0
$$766$$ 0 0
$$767$$ 13269.0 0.624664
$$768$$ 0 0
$$769$$ 21213.0 0.994745 0.497372 0.867537i $$-0.334298\pi$$
0.497372 + 0.867537i $$0.334298\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ 0 0
$$773$$ 14370.7i 0.668663i 0.942456 + 0.334332i $$0.108511\pi$$
−0.942456 + 0.334332i $$0.891489\pi$$
$$774$$ 0 0
$$775$$ 22980.6i 1.06515i
$$776$$ 0 0
$$777$$ 0 0
$$778$$ 0 0
$$779$$ 23562.3 1.08371
$$780$$ 0 0
$$781$$ −12621.8 −0.578287
$$782$$ 0 0
$$783$$ 0 0
$$784$$ 0 0
$$785$$ − 413.122i − 0.0187834i
$$786$$ 0 0
$$787$$ − 41642.9i − 1.88616i −0.332564 0.943081i $$-0.607914\pi$$
0.332564 0.943081i $$-0.392086\pi$$
$$788$$ 0 0
$$789$$ 0 0
$$790$$ 0 0
$$791$$ −38288.5 −1.72109
$$792$$ 0 0
$$793$$ −27876.5 −1.24833
$$794$$ 0 0
$$795$$ 0 0
$$796$$ 0 0
$$797$$ − 23527.1i − 1.04564i −0.852445 0.522818i $$-0.824881\pi$$
0.852445 0.522818i $$-0.175119\pi$$
$$798$$ 0 0
$$799$$ − 19081.4i − 0.844872i
$$800$$ 0 0
$$801$$ 0 0
$$802$$ 0 0
$$803$$ −3955.32 −0.173823
$$804$$ 0 0
$$805$$ 1071.47 0.0469120
$$806$$ 0 0
$$807$$ 0 0
$$808$$ 0 0
$$809$$ 7508.22i 0.326298i 0.986601 + 0.163149i $$0.0521651\pi$$
−0.986601 + 0.163149i $$0.947835\pi$$
$$810$$ 0 0
$$811$$ − 15834.0i − 0.685581i −0.939412 0.342791i $$-0.888628\pi$$
0.939412 0.342791i $$-0.111372\pi$$
$$812$$ 0 0
$$813$$ 0 0
$$814$$ 0 0
$$815$$ −2121.75 −0.0911923
$$816$$ 0 0
$$817$$ −36810.3 −1.57629
$$818$$ 0 0
$$819$$ 0 0
$$820$$ 0 0
$$821$$ − 13279.1i − 0.564486i −0.959343 0.282243i $$-0.908922\pi$$
0.959343 0.282243i $$-0.0910784\pi$$
$$822$$ 0 0
$$823$$ 27997.4i 1.18582i 0.805270 + 0.592909i $$0.202021\pi$$
−0.805270 + 0.592909i $$0.797979\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ 0 0
$$827$$ 35177.8 1.47914 0.739571 0.673078i $$-0.235028\pi$$
0.739571 + 0.673078i $$0.235028\pi$$
$$828$$ 0 0
$$829$$ 13390.3 0.560996 0.280498 0.959855i $$-0.409500\pi$$
0.280498 + 0.959855i $$0.409500\pi$$
$$830$$ 0 0
$$831$$ 0 0
$$832$$ 0 0
$$833$$ − 17490.1i − 0.727484i
$$834$$ 0 0
$$835$$ 5245.79i 0.217411i
$$836$$ 0 0
$$837$$ 0 0
$$838$$ 0 0
$$839$$ 8466.38 0.348381 0.174191 0.984712i $$-0.444269\pi$$
0.174191 + 0.984712i $$0.444269\pi$$
$$840$$ 0 0
$$841$$ 6280.62 0.257518
$$842$$ 0 0
$$843$$ 0 0
$$844$$ 0 0
$$845$$ 732.454i 0.0298192i
$$846$$ 0 0
$$847$$ 48424.4i 1.96444i
$$848$$ 0 0
$$849$$ 0 0
$$850$$ 0 0
$$851$$ 5380.29 0.216726
$$852$$ 0 0
$$853$$ 16448.2 0.660229 0.330114 0.943941i $$-0.392913\pi$$
0.330114 + 0.943941i $$0.392913\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ 0 0
$$857$$ 13593.7i 0.541832i 0.962603 + 0.270916i $$0.0873266\pi$$
−0.962603 + 0.270916i $$0.912673\pi$$
$$858$$ 0 0
$$859$$ − 15107.9i − 0.600086i −0.953926 0.300043i $$-0.902999\pi$$
0.953926 0.300043i $$-0.0970011\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ 0 0
$$863$$ 43717.5 1.72440 0.862202 0.506565i $$-0.169085\pi$$
0.862202 + 0.506565i $$0.169085\pi$$
$$864$$ 0 0
$$865$$ −494.820 −0.0194502
$$866$$ 0 0
$$867$$ 0 0
$$868$$ 0 0
$$869$$ 59344.8i 2.31661i
$$870$$ 0 0
$$871$$ 23801.7i 0.925935i
$$872$$ 0 0
$$873$$ 0 0
$$874$$ 0 0
$$875$$ 9699.78 0.374757
$$876$$ 0 0
$$877$$ 23868.3 0.919012 0.459506 0.888175i $$-0.348026\pi$$
0.459506 + 0.888175i $$0.348026\pi$$
$$878$$ 0 0
$$879$$ 0 0
$$880$$ 0 0
$$881$$ 48039.2i 1.83710i 0.395310 + 0.918548i $$0.370637\pi$$
−0.395310 + 0.918548i $$0.629363\pi$$
$$882$$ 0 0
$$883$$ 10091.3i 0.384595i 0.981337 + 0.192298i $$0.0615939\pi$$
−0.981337 + 0.192298i $$0.938406\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ 0 0
$$887$$ 37951.2 1.43661 0.718307 0.695726i $$-0.244917\pi$$
0.718307 + 0.695726i $$0.244917\pi$$
$$888$$ 0 0
$$889$$ −6939.20 −0.261792
$$890$$ 0 0
$$891$$ 0 0
$$892$$ 0 0
$$893$$ 29557.7i 1.10763i
$$894$$ 0 0
$$895$$ 1699.53i 0.0634739i
$$896$$ 0 0
$$897$$ 0 0
$$898$$ 0 0
$$899$$ −25190.0 −0.934520
$$900$$ 0 0
$$901$$ −31650.2 −1.17028
$$902$$ 0 0
$$903$$ 0 0
$$904$$ 0 0
$$905$$ 3075.26i 0.112956i
$$906$$ 0 0
$$907$$ − 23697.4i − 0.867540i −0.901024 0.433770i $$-0.857183\pi$$
0.901024 0.433770i $$-0.142817\pi$$
$$908$$ 0 0
$$909$$ 0 0
$$910$$ 0 0
$$911$$ −11833.0 −0.430347 −0.215174 0.976576i $$-0.569032\pi$$
−0.215174 + 0.976576i $$0.569032\pi$$
$$912$$ 0 0
$$913$$ 68747.0 2.49200
$$914$$ 0 0
$$915$$ 0 0
$$916$$ 0 0
$$917$$ − 30161.6i − 1.08618i
$$918$$ 0 0
$$919$$ − 51576.2i − 1.85130i −0.378386 0.925648i $$-0.623521\pi$$
0.378386 0.925648i $$-0.376479\pi$$
$$920$$ 0 0
$$921$$ 0 0
$$922$$ 0 0
$$923$$ 9247.29 0.329771
$$924$$ 0 0
$$925$$ 24133.7 0.857849
$$926$$ 0 0
$$927$$ 0 0
$$928$$ 0 0
$$929$$ 46577.3i 1.64494i 0.568807 + 0.822471i $$0.307405\pi$$
−0.568807 + 0.822471i $$0.692595\pi$$
$$930$$ 0 0
$$931$$ 27092.6i 0.953731i
$$932$$ 0 0
$$933$$ 0 0
$$934$$ 0 0
$$935$$ −4303.47 −0.150523
$$936$$ 0 0
$$937$$ 11409.2 0.397783 0.198891 0.980022i $$-0.436266\pi$$
0.198891 + 0.980022i $$0.436266\pi$$
$$938$$ 0 0
$$939$$ 0 0
$$940$$ 0 0
$$941$$ 32444.9i 1.12399i 0.827141 + 0.561994i $$0.189966\pi$$
−0.827141 + 0.561994i $$0.810034\pi$$
$$942$$ 0 0
$$943$$ 8156.32i 0.281661i
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ −22327.6 −0.766154 −0.383077 0.923716i $$-0.625136\pi$$
−0.383077 + 0.923716i $$0.625136\pi$$
$$948$$ 0 0
$$949$$ 2897.85 0.0991236
$$950$$ 0 0
$$951$$ 0 0
$$952$$ 0 0
$$953$$ 15314.3i 0.520546i 0.965535 + 0.260273i $$0.0838126\pi$$
−0.965535 + 0.260273i $$0.916187\pi$$
$$954$$ 0 0
$$955$$ − 2965.09i − 0.100469i
$$956$$ 0 0
$$957$$ 0 0
$$958$$ 0 0
$$959$$ 61500.4 2.07086
$$960$$ 0 0
$$961$$ −5250.01 −0.176228
$$962$$ 0 0
$$963$$ 0 0
$$964$$ 0 0
$$965$$ 283.928i 0.00947147i
$$966$$ 0 0
$$967$$ − 16913.6i − 0.562465i −0.959640 0.281232i $$-0.909257\pi$$
0.959640 0.281232i $$-0.0907431\pi$$
$$968$$ 0 0
$$969$$ 0 0
$$970$$ 0 0
$$971$$ 56451.5 1.86572 0.932861 0.360237i $$-0.117304\pi$$
0.932861 + 0.360237i $$0.117304\pi$$
$$972$$ 0 0
$$973$$ −5636.12 −0.185699
$$974$$ 0 0
$$975$$ 0 0
$$976$$ 0 0
$$977$$ − 7397.50i − 0.242238i −0.992638 0.121119i $$-0.961352\pi$$
0.992638 0.121119i $$-0.0386483\pi$$
$$978$$ 0 0
$$979$$ 75604.3i 2.46816i
$$980$$ 0 0
$$981$$ 0 0
$$982$$ 0 0
$$983$$ −18261.7 −0.592532 −0.296266 0.955105i $$-0.595742\pi$$
−0.296266 + 0.955105i $$0.595742\pi$$
$$984$$ 0 0
$$985$$ 3230.01 0.104484
$$986$$ 0 0
$$987$$ 0 0
$$988$$ 0 0
$$989$$ − 12742.3i − 0.409687i
$$990$$ 0 0
$$991$$ − 7020.98i − 0.225054i −0.993649 0.112527i $$-0.964105\pi$$
0.993649 0.112527i $$-0.0358945\pi$$
$$992$$ 0 0
$$993$$ 0 0
$$994$$ 0 0
$$995$$ 3971.28 0.126531
$$996$$ 0 0
$$997$$ −6983.34 −0.221830 −0.110915 0.993830i $$-0.535378\pi$$
−0.110915 + 0.993830i $$0.535378\pi$$
$$998$$ 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.1727.6 12
3.2 odd 2 inner 1728.4.c.j.1727.8 12
4.3 odd 2 inner 1728.4.c.j.1727.5 12
8.3 odd 2 108.4.b.b.107.7 yes 12
8.5 even 2 108.4.b.b.107.5 12
12.11 even 2 inner 1728.4.c.j.1727.7 12
24.5 odd 2 108.4.b.b.107.8 yes 12
24.11 even 2 108.4.b.b.107.6 yes 12

By twisted newform
Twist Min Dim Char Parity Ord Type
108.4.b.b.107.5 12 8.5 even 2
108.4.b.b.107.6 yes 12 24.11 even 2
108.4.b.b.107.7 yes 12 8.3 odd 2
108.4.b.b.107.8 yes 12 24.5 odd 2
1728.4.c.j.1727.5 12 4.3 odd 2 inner
1728.4.c.j.1727.6 12 1.1 even 1 trivial
1728.4.c.j.1727.7 12 12.11 even 2 inner
1728.4.c.j.1727.8 12 3.2 odd 2 inner