# Properties

 Label 2352.4.a.bx.1.1 Level $2352$ Weight $4$ Character 2352.1 Self dual yes Analytic conductor $138.772$ Analytic rank $1$ Dimension $2$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$2352 = 2^{4} \cdot 3 \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 2352.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$138.772492334$$ Analytic rank: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{193})$$ Defining polynomial: $$x^{2} - x - 48$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 84) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Root $$7.44622$$ of defining polynomial Character $$\chi$$ $$=$$ 2352.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q+3.00000 q^{3} -12.4462 q^{5} +9.00000 q^{9} +O(q^{10})$$ $$q+3.00000 q^{3} -12.4462 q^{5} +9.00000 q^{9} +51.1236 q^{11} -37.2311 q^{13} -37.3387 q^{15} -22.2151 q^{17} +54.3387 q^{19} -176.924 q^{23} +29.9084 q^{25} +27.0000 q^{27} +61.0916 q^{29} +319.924 q^{31} +153.371 q^{33} -315.080 q^{37} -111.693 q^{39} +206.032 q^{41} -339.661 q^{43} -112.016 q^{45} +142.064 q^{47} -66.6453 q^{51} +310.016 q^{53} -636.295 q^{55} +163.016 q^{57} -281.650 q^{59} +543.849 q^{61} +463.387 q^{65} +479.359 q^{67} -530.773 q^{69} -1105.63 q^{71} -239.350 q^{73} +89.7253 q^{75} -1160.67 q^{79} +81.0000 q^{81} -2.93158 q^{83} +276.494 q^{85} +183.275 q^{87} +1278.74 q^{89} +959.773 q^{93} -676.311 q^{95} -79.0596 q^{97} +460.112 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + 6q^{3} - 11q^{5} + 18q^{9} + O(q^{10})$$ $$2q + 6q^{3} - 11q^{5} + 18q^{9} + 5q^{11} - 5q^{13} - 33q^{15} - 100q^{17} + 67q^{19} - 76q^{23} - 93q^{25} + 54q^{27} + 275q^{29} + 362q^{31} + 15q^{33} - 5q^{37} - 15q^{39} + 162q^{41} - 721q^{43} - 99q^{45} - 216q^{47} - 300q^{51} + 495q^{53} - 703q^{55} + 201q^{57} + 173q^{59} + 532q^{61} + 510q^{65} - 111q^{67} - 228q^{69} - 1600q^{71} - 1215q^{73} - 279q^{75} - 1460q^{79} + 162q^{81} - 1409q^{83} + 164q^{85} + 825q^{87} + 1974q^{89} + 1086q^{93} - 658q^{95} - 561q^{97} + 45q^{99} + O(q^{100})$$

## Coefficient data

For each $$n$$ we display the coefficients of the $$q$$-expansion $$a_n$$, the Satake parameters $$\alpha_p$$, and the Satake angles $$\theta_p = \textrm{Arg}(\alpha_p)$$.

Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 0 0
$$3$$ 3.00000 0.577350
$$4$$ 0 0
$$5$$ −12.4462 −1.11322 −0.556612 0.830773i $$-0.687899\pi$$
−0.556612 + 0.830773i $$0.687899\pi$$
$$6$$ 0 0
$$7$$ 0 0
$$8$$ 0 0
$$9$$ 9.00000 0.333333
$$10$$ 0 0
$$11$$ 51.1236 1.40130 0.700651 0.713504i $$-0.252893\pi$$
0.700651 + 0.713504i $$0.252893\pi$$
$$12$$ 0 0
$$13$$ −37.2311 −0.794312 −0.397156 0.917751i $$-0.630003\pi$$
−0.397156 + 0.917751i $$0.630003\pi$$
$$14$$ 0 0
$$15$$ −37.3387 −0.642720
$$16$$ 0 0
$$17$$ −22.2151 −0.316939 −0.158469 0.987364i $$-0.550656\pi$$
−0.158469 + 0.987364i $$0.550656\pi$$
$$18$$ 0 0
$$19$$ 54.3387 0.656113 0.328056 0.944658i $$-0.393606\pi$$
0.328056 + 0.944658i $$0.393606\pi$$
$$20$$ 0 0
$$21$$ 0 0
$$22$$ 0 0
$$23$$ −176.924 −1.60397 −0.801985 0.597345i $$-0.796222\pi$$
−0.801985 + 0.597345i $$0.796222\pi$$
$$24$$ 0 0
$$25$$ 29.9084 0.239268
$$26$$ 0 0
$$27$$ 27.0000 0.192450
$$28$$ 0 0
$$29$$ 61.0916 0.391187 0.195593 0.980685i $$-0.437337\pi$$
0.195593 + 0.980685i $$0.437337\pi$$
$$30$$ 0 0
$$31$$ 319.924 1.85355 0.926776 0.375614i $$-0.122568\pi$$
0.926776 + 0.375614i $$0.122568\pi$$
$$32$$ 0 0
$$33$$ 153.371 0.809043
$$34$$ 0 0
$$35$$ 0 0
$$36$$ 0 0
$$37$$ −315.080 −1.39997 −0.699984 0.714158i $$-0.746810\pi$$
−0.699984 + 0.714158i $$0.746810\pi$$
$$38$$ 0 0
$$39$$ −111.693 −0.458596
$$40$$ 0 0
$$41$$ 206.032 0.784800 0.392400 0.919795i $$-0.371645\pi$$
0.392400 + 0.919795i $$0.371645\pi$$
$$42$$ 0 0
$$43$$ −339.661 −1.20460 −0.602301 0.798269i $$-0.705749\pi$$
−0.602301 + 0.798269i $$0.705749\pi$$
$$44$$ 0 0
$$45$$ −112.016 −0.371075
$$46$$ 0 0
$$47$$ 142.064 0.440897 0.220449 0.975399i $$-0.429248\pi$$
0.220449 + 0.975399i $$0.429248\pi$$
$$48$$ 0 0
$$49$$ 0 0
$$50$$ 0 0
$$51$$ −66.6453 −0.182985
$$52$$ 0 0
$$53$$ 310.016 0.803471 0.401736 0.915756i $$-0.368407\pi$$
0.401736 + 0.915756i $$0.368407\pi$$
$$54$$ 0 0
$$55$$ −636.295 −1.55996
$$56$$ 0 0
$$57$$ 163.016 0.378807
$$58$$ 0 0
$$59$$ −281.650 −0.621486 −0.310743 0.950494i $$-0.600578\pi$$
−0.310743 + 0.950494i $$0.600578\pi$$
$$60$$ 0 0
$$61$$ 543.849 1.14152 0.570760 0.821117i $$-0.306649\pi$$
0.570760 + 0.821117i $$0.306649\pi$$
$$62$$ 0 0
$$63$$ 0 0
$$64$$ 0 0
$$65$$ 463.387 0.884247
$$66$$ 0 0
$$67$$ 479.359 0.874075 0.437038 0.899443i $$-0.356028\pi$$
0.437038 + 0.899443i $$0.356028\pi$$
$$68$$ 0 0
$$69$$ −530.773 −0.926052
$$70$$ 0 0
$$71$$ −1105.63 −1.84809 −0.924046 0.382280i $$-0.875139\pi$$
−0.924046 + 0.382280i $$0.875139\pi$$
$$72$$ 0 0
$$73$$ −239.350 −0.383751 −0.191876 0.981419i $$-0.561457\pi$$
−0.191876 + 0.981419i $$0.561457\pi$$
$$74$$ 0 0
$$75$$ 89.7253 0.138141
$$76$$ 0 0
$$77$$ 0 0
$$78$$ 0 0
$$79$$ −1160.67 −1.65298 −0.826488 0.562954i $$-0.809665\pi$$
−0.826488 + 0.562954i $$0.809665\pi$$
$$80$$ 0 0
$$81$$ 81.0000 0.111111
$$82$$ 0 0
$$83$$ −2.93158 −0.00387690 −0.00193845 0.999998i $$-0.500617\pi$$
−0.00193845 + 0.999998i $$0.500617\pi$$
$$84$$ 0 0
$$85$$ 276.494 0.352824
$$86$$ 0 0
$$87$$ 183.275 0.225852
$$88$$ 0 0
$$89$$ 1278.74 1.52299 0.761496 0.648169i $$-0.224465\pi$$
0.761496 + 0.648169i $$0.224465\pi$$
$$90$$ 0 0
$$91$$ 0 0
$$92$$ 0 0
$$93$$ 959.773 1.07015
$$94$$ 0 0
$$95$$ −676.311 −0.730401
$$96$$ 0 0
$$97$$ −79.0596 −0.0827555 −0.0413777 0.999144i $$-0.513175\pi$$
−0.0413777 + 0.999144i $$0.513175\pi$$
$$98$$ 0 0
$$99$$ 460.112 0.467101
$$100$$ 0 0
$$101$$ −1372.10 −1.35178 −0.675889 0.737004i $$-0.736240\pi$$
−0.675889 + 0.737004i $$0.736240\pi$$
$$102$$ 0 0
$$103$$ 258.531 0.247318 0.123659 0.992325i $$-0.460537\pi$$
0.123659 + 0.992325i $$0.460537\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ 1262.37 1.14054 0.570270 0.821458i $$-0.306839\pi$$
0.570270 + 0.821458i $$0.306839\pi$$
$$108$$ 0 0
$$109$$ 276.833 0.243264 0.121632 0.992575i $$-0.461187\pi$$
0.121632 + 0.992575i $$0.461187\pi$$
$$110$$ 0 0
$$111$$ −945.240 −0.808272
$$112$$ 0 0
$$113$$ 52.9156 0.0440520 0.0220260 0.999757i $$-0.492988\pi$$
0.0220260 + 0.999757i $$0.492988\pi$$
$$114$$ 0 0
$$115$$ 2202.04 1.78558
$$116$$ 0 0
$$117$$ −335.080 −0.264771
$$118$$ 0 0
$$119$$ 0 0
$$120$$ 0 0
$$121$$ 1282.62 0.963650
$$122$$ 0 0
$$123$$ 618.096 0.453104
$$124$$ 0 0
$$125$$ 1183.53 0.846866
$$126$$ 0 0
$$127$$ −443.700 −0.310016 −0.155008 0.987913i $$-0.549540\pi$$
−0.155008 + 0.987913i $$0.549540\pi$$
$$128$$ 0 0
$$129$$ −1018.98 −0.695477
$$130$$ 0 0
$$131$$ −2153.48 −1.43627 −0.718133 0.695906i $$-0.755003\pi$$
−0.718133 + 0.695906i $$0.755003\pi$$
$$132$$ 0 0
$$133$$ 0 0
$$134$$ 0 0
$$135$$ −336.048 −0.214240
$$136$$ 0 0
$$137$$ 263.108 0.164079 0.0820394 0.996629i $$-0.473857\pi$$
0.0820394 + 0.996629i $$0.473857\pi$$
$$138$$ 0 0
$$139$$ 1165.77 0.711360 0.355680 0.934608i $$-0.384249\pi$$
0.355680 + 0.934608i $$0.384249\pi$$
$$140$$ 0 0
$$141$$ 426.192 0.254552
$$142$$ 0 0
$$143$$ −1903.39 −1.11307
$$144$$ 0 0
$$145$$ −760.359 −0.435479
$$146$$ 0 0
$$147$$ 0 0
$$148$$ 0 0
$$149$$ −1765.19 −0.970539 −0.485270 0.874364i $$-0.661279\pi$$
−0.485270 + 0.874364i $$0.661279\pi$$
$$150$$ 0 0
$$151$$ −2692.31 −1.45098 −0.725488 0.688235i $$-0.758386\pi$$
−0.725488 + 0.688235i $$0.758386\pi$$
$$152$$ 0 0
$$153$$ −199.936 −0.105646
$$154$$ 0 0
$$155$$ −3981.85 −2.06342
$$156$$ 0 0
$$157$$ 1941.38 0.986873 0.493437 0.869782i $$-0.335740\pi$$
0.493437 + 0.869782i $$0.335740\pi$$
$$158$$ 0 0
$$159$$ 930.048 0.463884
$$160$$ 0 0
$$161$$ 0 0
$$162$$ 0 0
$$163$$ −2102.84 −1.01047 −0.505236 0.862981i $$-0.668595\pi$$
−0.505236 + 0.862981i $$0.668595\pi$$
$$164$$ 0 0
$$165$$ −1908.89 −0.900646
$$166$$ 0 0
$$167$$ −2344.22 −1.08623 −0.543116 0.839658i $$-0.682756\pi$$
−0.543116 + 0.839658i $$0.682756\pi$$
$$168$$ 0 0
$$169$$ −810.844 −0.369069
$$170$$ 0 0
$$171$$ 489.048 0.218704
$$172$$ 0 0
$$173$$ −3470.65 −1.52525 −0.762626 0.646840i $$-0.776090\pi$$
−0.762626 + 0.646840i $$0.776090\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ 0 0
$$177$$ −844.949 −0.358815
$$178$$ 0 0
$$179$$ 955.771 0.399093 0.199547 0.979888i $$-0.436053\pi$$
0.199547 + 0.979888i $$0.436053\pi$$
$$180$$ 0 0
$$181$$ −4220.26 −1.73309 −0.866546 0.499098i $$-0.833665\pi$$
−0.866546 + 0.499098i $$0.833665\pi$$
$$182$$ 0 0
$$183$$ 1631.55 0.659057
$$184$$ 0 0
$$185$$ 3921.56 1.55848
$$186$$ 0 0
$$187$$ −1135.72 −0.444127
$$188$$ 0 0
$$189$$ 0 0
$$190$$ 0 0
$$191$$ 3518.03 1.33275 0.666377 0.745615i $$-0.267844\pi$$
0.666377 + 0.745615i $$0.267844\pi$$
$$192$$ 0 0
$$193$$ −5017.67 −1.87140 −0.935699 0.352799i $$-0.885230\pi$$
−0.935699 + 0.352799i $$0.885230\pi$$
$$194$$ 0 0
$$195$$ 1390.16 0.510520
$$196$$ 0 0
$$197$$ −2838.14 −1.02644 −0.513221 0.858257i $$-0.671548\pi$$
−0.513221 + 0.858257i $$0.671548\pi$$
$$198$$ 0 0
$$199$$ 354.901 0.126424 0.0632118 0.998000i $$-0.479866\pi$$
0.0632118 + 0.998000i $$0.479866\pi$$
$$200$$ 0 0
$$201$$ 1438.08 0.504648
$$202$$ 0 0
$$203$$ 0 0
$$204$$ 0 0
$$205$$ −2564.32 −0.873658
$$206$$ 0 0
$$207$$ −1592.32 −0.534656
$$208$$ 0 0
$$209$$ 2777.99 0.919413
$$210$$ 0 0
$$211$$ −752.672 −0.245574 −0.122787 0.992433i $$-0.539183\pi$$
−0.122787 + 0.992433i $$0.539183\pi$$
$$212$$ 0 0
$$213$$ −3316.90 −1.06700
$$214$$ 0 0
$$215$$ 4227.50 1.34099
$$216$$ 0 0
$$217$$ 0 0
$$218$$ 0 0
$$219$$ −718.051 −0.221559
$$220$$ 0 0
$$221$$ 827.093 0.251748
$$222$$ 0 0
$$223$$ −3077.75 −0.924221 −0.462111 0.886822i $$-0.652908\pi$$
−0.462111 + 0.886822i $$0.652908\pi$$
$$224$$ 0 0
$$225$$ 269.176 0.0797558
$$226$$ 0 0
$$227$$ −6217.43 −1.81791 −0.908955 0.416894i $$-0.863119\pi$$
−0.908955 + 0.416894i $$0.863119\pi$$
$$228$$ 0 0
$$229$$ −503.254 −0.145223 −0.0726113 0.997360i $$-0.523133\pi$$
−0.0726113 + 0.997360i $$0.523133\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 0 0
$$233$$ −268.340 −0.0754486 −0.0377243 0.999288i $$-0.512011\pi$$
−0.0377243 + 0.999288i $$0.512011\pi$$
$$234$$ 0 0
$$235$$ −1768.16 −0.490817
$$236$$ 0 0
$$237$$ −3482.00 −0.954346
$$238$$ 0 0
$$239$$ 5189.77 1.40459 0.702297 0.711884i $$-0.252158\pi$$
0.702297 + 0.711884i $$0.252158\pi$$
$$240$$ 0 0
$$241$$ −6170.94 −1.64940 −0.824699 0.565571i $$-0.808656\pi$$
−0.824699 + 0.565571i $$0.808656\pi$$
$$242$$ 0 0
$$243$$ 243.000 0.0641500
$$244$$ 0 0
$$245$$ 0 0
$$246$$ 0 0
$$247$$ −2023.09 −0.521158
$$248$$ 0 0
$$249$$ −8.79474 −0.00223833
$$250$$ 0 0
$$251$$ 1891.91 0.475763 0.237882 0.971294i $$-0.423547\pi$$
0.237882 + 0.971294i $$0.423547\pi$$
$$252$$ 0 0
$$253$$ −9045.01 −2.24765
$$254$$ 0 0
$$255$$ 829.483 0.203703
$$256$$ 0 0
$$257$$ 6539.93 1.58735 0.793676 0.608340i $$-0.208164\pi$$
0.793676 + 0.608340i $$0.208164\pi$$
$$258$$ 0 0
$$259$$ 0 0
$$260$$ 0 0
$$261$$ 549.824 0.130396
$$262$$ 0 0
$$263$$ −5375.90 −1.26043 −0.630214 0.776422i $$-0.717033\pi$$
−0.630214 + 0.776422i $$0.717033\pi$$
$$264$$ 0 0
$$265$$ −3858.53 −0.894443
$$266$$ 0 0
$$267$$ 3836.22 0.879300
$$268$$ 0 0
$$269$$ 3238.88 0.734119 0.367060 0.930197i $$-0.380364\pi$$
0.367060 + 0.930197i $$0.380364\pi$$
$$270$$ 0 0
$$271$$ 1357.46 0.304279 0.152140 0.988359i $$-0.451384\pi$$
0.152140 + 0.988359i $$0.451384\pi$$
$$272$$ 0 0
$$273$$ 0 0
$$274$$ 0 0
$$275$$ 1529.03 0.335286
$$276$$ 0 0
$$277$$ −2561.63 −0.555645 −0.277823 0.960632i $$-0.589613\pi$$
−0.277823 + 0.960632i $$0.589613\pi$$
$$278$$ 0 0
$$279$$ 2879.32 0.617851
$$280$$ 0 0
$$281$$ 1786.17 0.379196 0.189598 0.981862i $$-0.439282\pi$$
0.189598 + 0.981862i $$0.439282\pi$$
$$282$$ 0 0
$$283$$ −7388.28 −1.55190 −0.775950 0.630794i $$-0.782729\pi$$
−0.775950 + 0.630794i $$0.782729\pi$$
$$284$$ 0 0
$$285$$ −2028.93 −0.421697
$$286$$ 0 0
$$287$$ 0 0
$$288$$ 0 0
$$289$$ −4419.49 −0.899550
$$290$$ 0 0
$$291$$ −237.179 −0.0477789
$$292$$ 0 0
$$293$$ −492.981 −0.0982945 −0.0491472 0.998792i $$-0.515650\pi$$
−0.0491472 + 0.998792i $$0.515650\pi$$
$$294$$ 0 0
$$295$$ 3505.48 0.691853
$$296$$ 0 0
$$297$$ 1380.34 0.269681
$$298$$ 0 0
$$299$$ 6587.09 1.27405
$$300$$ 0 0
$$301$$ 0 0
$$302$$ 0 0
$$303$$ −4116.31 −0.780449
$$304$$ 0 0
$$305$$ −6768.86 −1.27077
$$306$$ 0 0
$$307$$ −988.810 −0.183825 −0.0919126 0.995767i $$-0.529298\pi$$
−0.0919126 + 0.995767i $$0.529298\pi$$
$$308$$ 0 0
$$309$$ 775.592 0.142789
$$310$$ 0 0
$$311$$ −9596.57 −1.74975 −0.874874 0.484351i $$-0.839056\pi$$
−0.874874 + 0.484351i $$0.839056\pi$$
$$312$$ 0 0
$$313$$ −965.713 −0.174394 −0.0871970 0.996191i $$-0.527791\pi$$
−0.0871970 + 0.996191i $$0.527791\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ −8985.97 −1.59212 −0.796061 0.605217i $$-0.793086\pi$$
−0.796061 + 0.605217i $$0.793086\pi$$
$$318$$ 0 0
$$319$$ 3123.22 0.548171
$$320$$ 0 0
$$321$$ 3787.10 0.658491
$$322$$ 0 0
$$323$$ −1207.14 −0.207947
$$324$$ 0 0
$$325$$ −1113.52 −0.190053
$$326$$ 0 0
$$327$$ 830.499 0.140449
$$328$$ 0 0
$$329$$ 0 0
$$330$$ 0 0
$$331$$ 3807.30 0.632230 0.316115 0.948721i $$-0.397621\pi$$
0.316115 + 0.948721i $$0.397621\pi$$
$$332$$ 0 0
$$333$$ −2835.72 −0.466656
$$334$$ 0 0
$$335$$ −5966.21 −0.973041
$$336$$ 0 0
$$337$$ −1649.82 −0.266681 −0.133340 0.991070i $$-0.542570\pi$$
−0.133340 + 0.991070i $$0.542570\pi$$
$$338$$ 0 0
$$339$$ 158.747 0.0254334
$$340$$ 0 0
$$341$$ 16355.7 2.59739
$$342$$ 0 0
$$343$$ 0 0
$$344$$ 0 0
$$345$$ 6606.12 1.03090
$$346$$ 0 0
$$347$$ −5709.99 −0.883368 −0.441684 0.897171i $$-0.645619\pi$$
−0.441684 + 0.897171i $$0.645619\pi$$
$$348$$ 0 0
$$349$$ 447.244 0.0685973 0.0342986 0.999412i $$-0.489080\pi$$
0.0342986 + 0.999412i $$0.489080\pi$$
$$350$$ 0 0
$$351$$ −1005.24 −0.152865
$$352$$ 0 0
$$353$$ 10645.7 1.60514 0.802569 0.596560i $$-0.203466\pi$$
0.802569 + 0.596560i $$0.203466\pi$$
$$354$$ 0 0
$$355$$ 13761.0 2.05734
$$356$$ 0 0
$$357$$ 0 0
$$358$$ 0 0
$$359$$ 9097.47 1.33745 0.668727 0.743508i $$-0.266839\pi$$
0.668727 + 0.743508i $$0.266839\pi$$
$$360$$ 0 0
$$361$$ −3906.31 −0.569516
$$362$$ 0 0
$$363$$ 3847.85 0.556363
$$364$$ 0 0
$$365$$ 2979.01 0.427201
$$366$$ 0 0
$$367$$ 5287.83 0.752104 0.376052 0.926598i $$-0.377281\pi$$
0.376052 + 0.926598i $$0.377281\pi$$
$$368$$ 0 0
$$369$$ 1854.29 0.261600
$$370$$ 0 0
$$371$$ 0 0
$$372$$ 0 0
$$373$$ 5895.01 0.818317 0.409159 0.912463i $$-0.365822\pi$$
0.409159 + 0.912463i $$0.365822\pi$$
$$374$$ 0 0
$$375$$ 3550.59 0.488938
$$376$$ 0 0
$$377$$ −2274.51 −0.310724
$$378$$ 0 0
$$379$$ −3842.41 −0.520769 −0.260384 0.965505i $$-0.583849\pi$$
−0.260384 + 0.965505i $$0.583849\pi$$
$$380$$ 0 0
$$381$$ −1331.10 −0.178988
$$382$$ 0 0
$$383$$ 5013.74 0.668903 0.334452 0.942413i $$-0.391449\pi$$
0.334452 + 0.942413i $$0.391449\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 0 0
$$387$$ −3056.95 −0.401534
$$388$$ 0 0
$$389$$ 11182.5 1.45752 0.728758 0.684771i $$-0.240098\pi$$
0.728758 + 0.684771i $$0.240098\pi$$
$$390$$ 0 0
$$391$$ 3930.40 0.508360
$$392$$ 0 0
$$393$$ −6460.45 −0.829228
$$394$$ 0 0
$$395$$ 14445.9 1.84013
$$396$$ 0 0
$$397$$ 3406.80 0.430686 0.215343 0.976539i $$-0.430913\pi$$
0.215343 + 0.976539i $$0.430913\pi$$
$$398$$ 0 0
$$399$$ 0 0
$$400$$ 0 0
$$401$$ 83.0401 0.0103412 0.00517061 0.999987i $$-0.498354\pi$$
0.00517061 + 0.999987i $$0.498354\pi$$
$$402$$ 0 0
$$403$$ −11911.1 −1.47230
$$404$$ 0 0
$$405$$ −1008.14 −0.123692
$$406$$ 0 0
$$407$$ −16108.0 −1.96178
$$408$$ 0 0
$$409$$ −2456.20 −0.296947 −0.148473 0.988916i $$-0.547436\pi$$
−0.148473 + 0.988916i $$0.547436\pi$$
$$410$$ 0 0
$$411$$ 789.323 0.0947309
$$412$$ 0 0
$$413$$ 0 0
$$414$$ 0 0
$$415$$ 36.4871 0.00431586
$$416$$ 0 0
$$417$$ 3497.30 0.410704
$$418$$ 0 0
$$419$$ −3437.96 −0.400848 −0.200424 0.979709i $$-0.564232\pi$$
−0.200424 + 0.979709i $$0.564232\pi$$
$$420$$ 0 0
$$421$$ −5347.62 −0.619067 −0.309533 0.950889i $$-0.600173\pi$$
−0.309533 + 0.950889i $$0.600173\pi$$
$$422$$ 0 0
$$423$$ 1278.58 0.146966
$$424$$ 0 0
$$425$$ −664.419 −0.0758331
$$426$$ 0 0
$$427$$ 0 0
$$428$$ 0 0
$$429$$ −5710.16 −0.642632
$$430$$ 0 0
$$431$$ −851.643 −0.0951791 −0.0475895 0.998867i $$-0.515154\pi$$
−0.0475895 + 0.998867i $$0.515154\pi$$
$$432$$ 0 0
$$433$$ 3433.42 0.381061 0.190531 0.981681i $$-0.438979\pi$$
0.190531 + 0.981681i $$0.438979\pi$$
$$434$$ 0 0
$$435$$ −2281.08 −0.251424
$$436$$ 0 0
$$437$$ −9613.84 −1.05238
$$438$$ 0 0
$$439$$ −9738.39 −1.05874 −0.529371 0.848390i $$-0.677572\pi$$
−0.529371 + 0.848390i $$0.677572\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ 0 0
$$443$$ −9934.96 −1.06552 −0.532759 0.846267i $$-0.678845\pi$$
−0.532759 + 0.846267i $$0.678845\pi$$
$$444$$ 0 0
$$445$$ −15915.5 −1.69543
$$446$$ 0 0
$$447$$ −5295.58 −0.560341
$$448$$ 0 0
$$449$$ −7557.33 −0.794327 −0.397163 0.917748i $$-0.630005\pi$$
−0.397163 + 0.917748i $$0.630005\pi$$
$$450$$ 0 0
$$451$$ 10533.1 1.09974
$$452$$ 0 0
$$453$$ −8076.94 −0.837721
$$454$$ 0 0
$$455$$ 0 0
$$456$$ 0 0
$$457$$ 14011.8 1.43424 0.717118 0.696951i $$-0.245461\pi$$
0.717118 + 0.696951i $$0.245461\pi$$
$$458$$ 0 0
$$459$$ −599.808 −0.0609949
$$460$$ 0 0
$$461$$ 1669.61 0.168680 0.0843399 0.996437i $$-0.473122\pi$$
0.0843399 + 0.996437i $$0.473122\pi$$
$$462$$ 0 0
$$463$$ −14785.4 −1.48409 −0.742046 0.670349i $$-0.766144\pi$$
−0.742046 + 0.670349i $$0.766144\pi$$
$$464$$ 0 0
$$465$$ −11945.6 −1.19132
$$466$$ 0 0
$$467$$ 4603.17 0.456122 0.228061 0.973647i $$-0.426761\pi$$
0.228061 + 0.973647i $$0.426761\pi$$
$$468$$ 0 0
$$469$$ 0 0
$$470$$ 0 0
$$471$$ 5824.14 0.569772
$$472$$ 0 0
$$473$$ −17364.7 −1.68801
$$474$$ 0 0
$$475$$ 1625.18 0.156987
$$476$$ 0 0
$$477$$ 2790.14 0.267824
$$478$$ 0 0
$$479$$ 3477.35 0.331700 0.165850 0.986151i $$-0.446963\pi$$
0.165850 + 0.986151i $$0.446963\pi$$
$$480$$ 0 0
$$481$$ 11730.8 1.11201
$$482$$ 0 0
$$483$$ 0 0
$$484$$ 0 0
$$485$$ 983.993 0.0921254
$$486$$ 0 0
$$487$$ −4344.17 −0.404216 −0.202108 0.979363i $$-0.564779\pi$$
−0.202108 + 0.979363i $$0.564779\pi$$
$$488$$ 0 0
$$489$$ −6308.51 −0.583396
$$490$$ 0 0
$$491$$ −4982.89 −0.457993 −0.228997 0.973427i $$-0.573544\pi$$
−0.228997 + 0.973427i $$0.573544\pi$$
$$492$$ 0 0
$$493$$ −1357.16 −0.123982
$$494$$ 0 0
$$495$$ −5726.66 −0.519988
$$496$$ 0 0
$$497$$ 0 0
$$498$$ 0 0
$$499$$ −15326.2 −1.37494 −0.687468 0.726215i $$-0.741278\pi$$
−0.687468 + 0.726215i $$0.741278\pi$$
$$500$$ 0 0
$$501$$ −7032.65 −0.627137
$$502$$ 0 0
$$503$$ −1516.04 −0.134387 −0.0671936 0.997740i $$-0.521405\pi$$
−0.0671936 + 0.997740i $$0.521405\pi$$
$$504$$ 0 0
$$505$$ 17077.5 1.50483
$$506$$ 0 0
$$507$$ −2432.53 −0.213082
$$508$$ 0 0
$$509$$ −2653.64 −0.231081 −0.115541 0.993303i $$-0.536860\pi$$
−0.115541 + 0.993303i $$0.536860\pi$$
$$510$$ 0 0
$$511$$ 0 0
$$512$$ 0 0
$$513$$ 1467.14 0.126269
$$514$$ 0 0
$$515$$ −3217.73 −0.275321
$$516$$ 0 0
$$517$$ 7262.82 0.617830
$$518$$ 0 0
$$519$$ −10411.9 −0.880604
$$520$$ 0 0
$$521$$ 13132.1 1.10428 0.552139 0.833752i $$-0.313812\pi$$
0.552139 + 0.833752i $$0.313812\pi$$
$$522$$ 0 0
$$523$$ −3086.34 −0.258042 −0.129021 0.991642i $$-0.541184\pi$$
−0.129021 + 0.991642i $$0.541184\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ −7107.16 −0.587462
$$528$$ 0 0
$$529$$ 19135.3 1.57272
$$530$$ 0 0
$$531$$ −2534.85 −0.207162
$$532$$ 0 0
$$533$$ −7670.80 −0.623376
$$534$$ 0 0
$$535$$ −15711.7 −1.26968
$$536$$ 0 0
$$537$$ 2867.31 0.230416
$$538$$ 0 0
$$539$$ 0 0
$$540$$ 0 0
$$541$$ 926.095 0.0735969 0.0367985 0.999323i $$-0.488284\pi$$
0.0367985 + 0.999323i $$0.488284\pi$$
$$542$$ 0 0
$$543$$ −12660.8 −1.00060
$$544$$ 0 0
$$545$$ −3445.52 −0.270807
$$546$$ 0 0
$$547$$ −592.871 −0.0463425 −0.0231712 0.999732i $$-0.507376\pi$$
−0.0231712 + 0.999732i $$0.507376\pi$$
$$548$$ 0 0
$$549$$ 4894.64 0.380507
$$550$$ 0 0
$$551$$ 3319.63 0.256663
$$552$$ 0 0
$$553$$ 0 0
$$554$$ 0 0
$$555$$ 11764.7 0.899788
$$556$$ 0 0
$$557$$ −12245.3 −0.931512 −0.465756 0.884913i $$-0.654217\pi$$
−0.465756 + 0.884913i $$0.654217\pi$$
$$558$$ 0 0
$$559$$ 12646.0 0.956829
$$560$$ 0 0
$$561$$ −3407.15 −0.256417
$$562$$ 0 0
$$563$$ 14594.6 1.09252 0.546261 0.837615i $$-0.316051\pi$$
0.546261 + 0.837615i $$0.316051\pi$$
$$564$$ 0 0
$$565$$ −658.599 −0.0490398
$$566$$ 0 0
$$567$$ 0 0
$$568$$ 0 0
$$569$$ 22911.2 1.68803 0.844015 0.536320i $$-0.180186\pi$$
0.844015 + 0.536320i $$0.180186\pi$$
$$570$$ 0 0
$$571$$ −5904.64 −0.432752 −0.216376 0.976310i $$-0.569424\pi$$
−0.216376 + 0.976310i $$0.569424\pi$$
$$572$$ 0 0
$$573$$ 10554.1 0.769465
$$574$$ 0 0
$$575$$ −5291.53 −0.383778
$$576$$ 0 0
$$577$$ −9513.21 −0.686378 −0.343189 0.939266i $$-0.611507\pi$$
−0.343189 + 0.939266i $$0.611507\pi$$
$$578$$ 0 0
$$579$$ −15053.0 −1.08045
$$580$$ 0 0
$$581$$ 0 0
$$582$$ 0 0
$$583$$ 15849.1 1.12591
$$584$$ 0 0
$$585$$ 4170.48 0.294749
$$586$$ 0 0
$$587$$ −22790.6 −1.60250 −0.801252 0.598327i $$-0.795833\pi$$
−0.801252 + 0.598327i $$0.795833\pi$$
$$588$$ 0 0
$$589$$ 17384.3 1.21614
$$590$$ 0 0
$$591$$ −8514.41 −0.592616
$$592$$ 0 0
$$593$$ 18262.8 1.26469 0.632346 0.774686i $$-0.282092\pi$$
0.632346 + 0.774686i $$0.282092\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 0 0
$$597$$ 1064.70 0.0729907
$$598$$ 0 0
$$599$$ −6958.09 −0.474624 −0.237312 0.971433i $$-0.576266\pi$$
−0.237312 + 0.971433i $$0.576266\pi$$
$$600$$ 0 0
$$601$$ −2305.39 −0.156471 −0.0782353 0.996935i $$-0.524929\pi$$
−0.0782353 + 0.996935i $$0.524929\pi$$
$$602$$ 0 0
$$603$$ 4314.23 0.291358
$$604$$ 0 0
$$605$$ −15963.7 −1.07276
$$606$$ 0 0
$$607$$ −16179.2 −1.08187 −0.540935 0.841064i $$-0.681929\pi$$
−0.540935 + 0.841064i $$0.681929\pi$$
$$608$$ 0 0
$$609$$ 0 0
$$610$$ 0 0
$$611$$ −5289.20 −0.350210
$$612$$ 0 0
$$613$$ 20540.7 1.35340 0.676699 0.736260i $$-0.263410\pi$$
0.676699 + 0.736260i $$0.263410\pi$$
$$614$$ 0 0
$$615$$ −7692.96 −0.504407
$$616$$ 0 0
$$617$$ 6918.19 0.451403 0.225702 0.974196i $$-0.427533\pi$$
0.225702 + 0.974196i $$0.427533\pi$$
$$618$$ 0 0
$$619$$ −8081.62 −0.524762 −0.262381 0.964964i $$-0.584508\pi$$
−0.262381 + 0.964964i $$0.584508\pi$$
$$620$$ 0 0
$$621$$ −4776.96 −0.308684
$$622$$ 0 0
$$623$$ 0 0
$$624$$ 0 0
$$625$$ −18469.0 −1.18202
$$626$$ 0 0
$$627$$ 8333.96 0.530823
$$628$$ 0 0
$$629$$ 6999.54 0.443704
$$630$$ 0 0
$$631$$ 27293.3 1.72191 0.860957 0.508677i $$-0.169865\pi$$
0.860957 + 0.508677i $$0.169865\pi$$
$$632$$ 0 0
$$633$$ −2258.02 −0.141782
$$634$$ 0 0
$$635$$ 5522.39 0.345117
$$636$$ 0 0
$$637$$ 0 0
$$638$$ 0 0
$$639$$ −9950.70 −0.616031
$$640$$ 0 0
$$641$$ 19255.6 1.18651 0.593254 0.805015i $$-0.297843\pi$$
0.593254 + 0.805015i $$0.297843\pi$$
$$642$$ 0 0
$$643$$ −19996.4 −1.22641 −0.613204 0.789925i $$-0.710120\pi$$
−0.613204 + 0.789925i $$0.710120\pi$$
$$644$$ 0 0
$$645$$ 12682.5 0.774222
$$646$$ 0 0
$$647$$ 7064.20 0.429246 0.214623 0.976697i $$-0.431148\pi$$
0.214623 + 0.976697i $$0.431148\pi$$
$$648$$ 0 0
$$649$$ −14398.9 −0.870890
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 0 0
$$653$$ −13821.1 −0.828269 −0.414134 0.910216i $$-0.635916\pi$$
−0.414134 + 0.910216i $$0.635916\pi$$
$$654$$ 0 0
$$655$$ 26802.7 1.59889
$$656$$ 0 0
$$657$$ −2154.15 −0.127917
$$658$$ 0 0
$$659$$ 7802.80 0.461235 0.230617 0.973044i $$-0.425925\pi$$
0.230617 + 0.973044i $$0.425925\pi$$
$$660$$ 0 0
$$661$$ −15817.3 −0.930744 −0.465372 0.885115i $$-0.654079\pi$$
−0.465372 + 0.885115i $$0.654079\pi$$
$$662$$ 0 0
$$663$$ 2481.28 0.145347
$$664$$ 0 0
$$665$$ 0 0
$$666$$ 0 0
$$667$$ −10808.6 −0.627452
$$668$$ 0 0
$$669$$ −9233.25 −0.533599
$$670$$ 0 0
$$671$$ 27803.5 1.59962
$$672$$ 0 0
$$673$$ 2943.30 0.168582 0.0842911 0.996441i $$-0.473137\pi$$
0.0842911 + 0.996441i $$0.473137\pi$$
$$674$$ 0 0
$$675$$ 807.528 0.0460471
$$676$$ 0 0
$$677$$ 3171.48 0.180044 0.0900220 0.995940i $$-0.471306\pi$$
0.0900220 + 0.995940i $$0.471306\pi$$
$$678$$ 0 0
$$679$$ 0 0
$$680$$ 0 0
$$681$$ −18652.3 −1.04957
$$682$$ 0 0
$$683$$ 24453.2 1.36995 0.684975 0.728567i $$-0.259813\pi$$
0.684975 + 0.728567i $$0.259813\pi$$
$$684$$ 0 0
$$685$$ −3274.70 −0.182656
$$686$$ 0 0
$$687$$ −1509.76 −0.0838443
$$688$$ 0 0
$$689$$ −11542.2 −0.638207
$$690$$ 0 0
$$691$$ 8595.89 0.473232 0.236616 0.971603i $$-0.423962\pi$$
0.236616 + 0.971603i $$0.423962\pi$$
$$692$$ 0 0
$$693$$ 0 0
$$694$$ 0 0
$$695$$ −14509.4 −0.791903
$$696$$ 0 0
$$697$$ −4577.02 −0.248733
$$698$$ 0 0
$$699$$ −805.019 −0.0435602
$$700$$ 0 0
$$701$$ −21476.1 −1.15712 −0.578561 0.815639i $$-0.696385\pi$$
−0.578561 + 0.815639i $$0.696385\pi$$
$$702$$ 0 0
$$703$$ −17121.0 −0.918537
$$704$$ 0 0
$$705$$ −5304.48 −0.283373
$$706$$ 0 0
$$707$$ 0 0
$$708$$ 0 0
$$709$$ −13538.9 −0.717158 −0.358579 0.933499i $$-0.616739\pi$$
−0.358579 + 0.933499i $$0.616739\pi$$
$$710$$ 0 0
$$711$$ −10446.0 −0.550992
$$712$$ 0 0
$$713$$ −56602.5 −2.97304
$$714$$ 0 0
$$715$$ 23690.0 1.23910
$$716$$ 0 0
$$717$$ 15569.3 0.810943
$$718$$ 0 0
$$719$$ 13883.7 0.720131 0.360065 0.932927i $$-0.382754\pi$$
0.360065 + 0.932927i $$0.382754\pi$$
$$720$$ 0 0
$$721$$ 0 0
$$722$$ 0 0
$$723$$ −18512.8 −0.952281
$$724$$ 0 0
$$725$$ 1827.15 0.0935983
$$726$$ 0 0
$$727$$ −18292.9 −0.933215 −0.466607 0.884465i $$-0.654524\pi$$
−0.466607 + 0.884465i $$0.654524\pi$$
$$728$$ 0 0
$$729$$ 729.000 0.0370370
$$730$$ 0 0
$$731$$ 7545.61 0.381785
$$732$$ 0 0
$$733$$ −14245.8 −0.717848 −0.358924 0.933367i $$-0.616856\pi$$
−0.358924 + 0.933367i $$0.616856\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ 24506.5 1.22484
$$738$$ 0 0
$$739$$ 1363.89 0.0678913 0.0339456 0.999424i $$-0.489193\pi$$
0.0339456 + 0.999424i $$0.489193\pi$$
$$740$$ 0 0
$$741$$ −6069.27 −0.300891
$$742$$ 0 0
$$743$$ −21789.4 −1.07588 −0.537938 0.842984i $$-0.680797\pi$$
−0.537938 + 0.842984i $$0.680797\pi$$
$$744$$ 0 0
$$745$$ 21970.0 1.08043
$$746$$ 0 0
$$747$$ −26.3842 −0.00129230
$$748$$ 0 0
$$749$$ 0 0
$$750$$ 0 0
$$751$$ 2119.55 0.102987 0.0514937 0.998673i $$-0.483602\pi$$
0.0514937 + 0.998673i $$0.483602\pi$$
$$752$$ 0 0
$$753$$ 5675.74 0.274682
$$754$$ 0 0
$$755$$ 33509.1 1.61526
$$756$$ 0 0
$$757$$ 28202.4 1.35408 0.677038 0.735948i $$-0.263263\pi$$
0.677038 + 0.735948i $$0.263263\pi$$
$$758$$ 0 0
$$759$$ −27135.0 −1.29768
$$760$$ 0 0
$$761$$ −11147.0 −0.530985 −0.265493 0.964113i $$-0.585535\pi$$
−0.265493 + 0.964113i $$0.585535\pi$$
$$762$$ 0 0
$$763$$ 0 0
$$764$$ 0 0
$$765$$ 2488.45 0.117608
$$766$$ 0 0
$$767$$ 10486.1 0.493654
$$768$$ 0 0
$$769$$ 4109.29 0.192698 0.0963491 0.995348i $$-0.469283\pi$$
0.0963491 + 0.995348i $$0.469283\pi$$
$$770$$ 0 0
$$771$$ 19619.8 0.916459
$$772$$ 0 0
$$773$$ −13891.4 −0.646362 −0.323181 0.946337i $$-0.604752\pi$$
−0.323181 + 0.946337i $$0.604752\pi$$
$$774$$ 0 0
$$775$$ 9568.44 0.443495
$$776$$ 0 0
$$777$$ 0 0
$$778$$ 0 0
$$779$$ 11195.5 0.514917
$$780$$ 0 0
$$781$$ −56523.9 −2.58974
$$782$$ 0 0
$$783$$ 1649.47 0.0752839
$$784$$ 0 0
$$785$$ −24162.9 −1.09861
$$786$$ 0 0
$$787$$ −9342.35 −0.423150 −0.211575 0.977362i $$-0.567859\pi$$
−0.211575 + 0.977362i $$0.567859\pi$$
$$788$$ 0 0
$$789$$ −16127.7 −0.727708
$$790$$ 0 0
$$791$$ 0 0
$$792$$ 0 0
$$793$$ −20248.1 −0.906723
$$794$$ 0 0
$$795$$ −11575.6 −0.516407
$$796$$ 0 0
$$797$$ 24324.3 1.08107 0.540534 0.841322i $$-0.318222\pi$$
0.540534 + 0.841322i $$0.318222\pi$$
$$798$$ 0 0
$$799$$ −3155.97 −0.139737
$$800$$ 0 0
$$801$$ 11508.7 0.507664
$$802$$ 0 0
$$803$$ −12236.4 −0.537751
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 0 0
$$807$$ 9716.65 0.423844
$$808$$ 0 0
$$809$$ 34454.9 1.49737 0.748684 0.662927i $$-0.230686\pi$$
0.748684 + 0.662927i $$0.230686\pi$$
$$810$$ 0 0
$$811$$ −8350.13 −0.361545 −0.180772 0.983525i $$-0.557860\pi$$
−0.180772 + 0.983525i $$0.557860\pi$$
$$812$$ 0 0
$$813$$ 4072.37 0.175676
$$814$$ 0 0
$$815$$ 26172.4 1.12488
$$816$$ 0 0
$$817$$ −18456.7 −0.790355
$$818$$ 0 0
$$819$$ 0 0
$$820$$ 0 0
$$821$$ −18834.8 −0.800655 −0.400327 0.916372i $$-0.631104\pi$$
−0.400327 + 0.916372i $$0.631104\pi$$
$$822$$ 0 0
$$823$$ −9656.76 −0.409008 −0.204504 0.978866i $$-0.565558\pi$$
−0.204504 + 0.978866i $$0.565558\pi$$
$$824$$ 0 0
$$825$$ 4587.08 0.193578
$$826$$ 0 0
$$827$$ 20759.6 0.872892 0.436446 0.899731i $$-0.356237\pi$$
0.436446 + 0.899731i $$0.356237\pi$$
$$828$$ 0 0
$$829$$ 15616.1 0.654245 0.327123 0.944982i $$-0.393921\pi$$
0.327123 + 0.944982i $$0.393921\pi$$
$$830$$ 0 0
$$831$$ −7684.90 −0.320802
$$832$$ 0 0
$$833$$ 0 0
$$834$$ 0 0
$$835$$ 29176.6 1.20922
$$836$$ 0 0
$$837$$ 8637.96 0.356716
$$838$$ 0 0
$$839$$ −417.027 −0.0171601 −0.00858007 0.999963i $$-0.502731\pi$$
−0.00858007 + 0.999963i $$0.502731\pi$$
$$840$$ 0 0
$$841$$ −20656.8 −0.846973
$$842$$ 0 0
$$843$$ 5358.52 0.218929
$$844$$ 0 0
$$845$$ 10092.0 0.410856
$$846$$ 0 0
$$847$$ 0 0
$$848$$ 0 0
$$849$$ −22164.8 −0.895990
$$850$$ 0 0
$$851$$ 55745.4 2.24551
$$852$$ 0 0
$$853$$ 24917.4 1.00018 0.500092 0.865972i $$-0.333300\pi$$
0.500092 + 0.865972i $$0.333300\pi$$
$$854$$ 0 0
$$855$$ −6086.80 −0.243467
$$856$$ 0 0
$$857$$ 44523.8 1.77468 0.887342 0.461112i $$-0.152549\pi$$
0.887342 + 0.461112i $$0.152549\pi$$
$$858$$ 0 0
$$859$$ −24146.4 −0.959097 −0.479548 0.877515i $$-0.659200\pi$$
−0.479548 + 0.877515i $$0.659200\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ 0 0
$$863$$ 26442.6 1.04301 0.521505 0.853248i $$-0.325371\pi$$
0.521505 + 0.853248i $$0.325371\pi$$
$$864$$ 0 0
$$865$$ 43196.5 1.69795
$$866$$ 0 0
$$867$$ −13258.5 −0.519355
$$868$$ 0 0
$$869$$ −59337.4 −2.31632
$$870$$ 0 0
$$871$$ −17847.1 −0.694288
$$872$$ 0 0
$$873$$ −711.536 −0.0275852
$$874$$ 0 0
$$875$$ 0 0
$$876$$ 0 0
$$877$$ −22516.1 −0.866950 −0.433475 0.901166i $$-0.642713\pi$$
−0.433475 + 0.901166i $$0.642713\pi$$
$$878$$ 0 0
$$879$$ −1478.94 −0.0567503
$$880$$ 0 0
$$881$$ −10120.6 −0.387027 −0.193514 0.981098i $$-0.561988\pi$$
−0.193514 + 0.981098i $$0.561988\pi$$
$$882$$ 0 0
$$883$$ 20748.5 0.790761 0.395380 0.918517i $$-0.370613\pi$$
0.395380 + 0.918517i $$0.370613\pi$$
$$884$$ 0 0
$$885$$ 10516.4 0.399442
$$886$$ 0 0
$$887$$ 25499.4 0.965259 0.482630 0.875825i $$-0.339682\pi$$
0.482630 + 0.875825i $$0.339682\pi$$
$$888$$ 0 0
$$889$$ 0 0
$$890$$ 0 0
$$891$$ 4141.01 0.155700
$$892$$ 0 0
$$893$$ 7719.57 0.289278
$$894$$ 0 0
$$895$$ −11895.7 −0.444280
$$896$$ 0 0
$$897$$ 19761.3 0.735574
$$898$$ 0 0
$$899$$ 19544.7 0.725085
$$900$$ 0 0
$$901$$ −6887.04 −0.254651
$$902$$ 0 0
$$903$$ 0 0
$$904$$ 0 0
$$905$$ 52526.3 1.92932
$$906$$ 0 0
$$907$$ −37807.0 −1.38408 −0.692040 0.721859i $$-0.743288\pi$$
−0.692040 + 0.721859i $$0.743288\pi$$
$$908$$ 0 0
$$909$$ −12348.9 −0.450593
$$910$$ 0 0
$$911$$ 3230.08 0.117472 0.0587362 0.998274i $$-0.481293\pi$$
0.0587362 + 0.998274i $$0.481293\pi$$
$$912$$ 0 0
$$913$$ −149.873 −0.00543271
$$914$$ 0 0
$$915$$ −20306.6 −0.733678
$$916$$ 0 0
$$917$$ 0 0
$$918$$ 0 0
$$919$$ 35343.1 1.26862 0.634310 0.773079i $$-0.281284\pi$$
0.634310 + 0.773079i $$0.281284\pi$$
$$920$$ 0 0
$$921$$ −2966.43 −0.106132
$$922$$ 0 0
$$923$$ 41164.0 1.46796
$$924$$ 0 0
$$925$$ −9423.55 −0.334967
$$926$$ 0 0
$$927$$ 2326.78 0.0824394
$$928$$ 0 0
$$929$$ −33030.2 −1.16651 −0.583254 0.812289i $$-0.698221\pi$$
−0.583254 + 0.812289i $$0.698221\pi$$
$$930$$ 0 0
$$931$$ 0 0
$$932$$ 0 0
$$933$$ −28789.7 −1.01022
$$934$$ 0 0
$$935$$ 14135.4 0.494413
$$936$$ 0 0
$$937$$ −54695.9 −1.90698 −0.953488 0.301430i $$-0.902536\pi$$
−0.953488 + 0.301430i $$0.902536\pi$$
$$938$$ 0 0
$$939$$ −2897.14 −0.100686
$$940$$ 0 0
$$941$$ 18255.3 0.632418 0.316209 0.948690i $$-0.397590\pi$$
0.316209 + 0.948690i $$0.397590\pi$$
$$942$$ 0 0
$$943$$ −36452.1 −1.25879
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ −6215.75 −0.213289 −0.106645 0.994297i $$-0.534011\pi$$
−0.106645 + 0.994297i $$0.534011\pi$$
$$948$$ 0 0
$$949$$ 8911.27 0.304818
$$950$$ 0 0
$$951$$ −26957.9 −0.919212
$$952$$ 0 0
$$953$$ 8594.53 0.292135 0.146067 0.989275i $$-0.453338\pi$$
0.146067 + 0.989275i $$0.453338\pi$$
$$954$$ 0 0
$$955$$ −43786.2 −1.48365
$$956$$ 0 0
$$957$$ 9369.65 0.316487
$$958$$ 0 0
$$959$$ 0 0
$$960$$ 0 0
$$961$$ 72560.6 2.43566
$$962$$ 0 0
$$963$$ 11361.3 0.380180
$$964$$ 0 0
$$965$$ 62451.1 2.08329
$$966$$ 0 0
$$967$$ 17168.1 0.570929 0.285464 0.958389i $$-0.407852\pi$$
0.285464 + 0.958389i $$0.407852\pi$$
$$968$$ 0 0
$$969$$ −3621.42 −0.120059
$$970$$ 0 0
$$971$$ 11926.2 0.394162 0.197081 0.980387i $$-0.436854\pi$$
0.197081 + 0.980387i $$0.436854\pi$$
$$972$$ 0 0
$$973$$ 0 0
$$974$$ 0 0
$$975$$ −3340.57 −0.109727
$$976$$ 0 0
$$977$$ 14038.2 0.459696 0.229848 0.973227i $$-0.426177\pi$$
0.229848 + 0.973227i $$0.426177\pi$$
$$978$$ 0 0
$$979$$ 65373.8 2.13417
$$980$$ 0 0
$$981$$ 2491.50 0.0810880
$$982$$ 0 0
$$983$$ 20351.4 0.660334 0.330167 0.943923i $$-0.392895\pi$$
0.330167 + 0.943923i $$0.392895\pi$$
$$984$$ 0 0
$$985$$ 35324.1 1.14266
$$986$$ 0 0
$$987$$ 0 0
$$988$$ 0 0
$$989$$ 60094.4 1.93214
$$990$$ 0 0
$$991$$ 32505.7 1.04196 0.520978 0.853570i $$-0.325567\pi$$
0.520978 + 0.853570i $$0.325567\pi$$
$$992$$ 0 0
$$993$$ 11421.9 0.365018
$$994$$ 0 0
$$995$$ −4417.18 −0.140738
$$996$$ 0 0
$$997$$ 24847.1 0.789282 0.394641 0.918835i $$-0.370869\pi$$
0.394641 + 0.918835i $$0.370869\pi$$
$$998$$ 0 0
$$999$$ −8507.16 −0.269424
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 2352.4.a.bx.1.1 2
4.3 odd 2 588.4.a.f.1.1 2
7.3 odd 6 336.4.q.i.289.1 4
7.5 odd 6 336.4.q.i.193.1 4
7.6 odd 2 2352.4.a.bt.1.2 2
12.11 even 2 1764.4.a.y.1.2 2
28.3 even 6 84.4.i.a.37.1 yes 4
28.11 odd 6 588.4.i.j.373.2 4
28.19 even 6 84.4.i.a.25.1 4
28.23 odd 6 588.4.i.j.361.2 4
28.27 even 2 588.4.a.i.1.2 2
84.11 even 6 1764.4.k.q.1549.1 4
84.23 even 6 1764.4.k.q.361.1 4
84.47 odd 6 252.4.k.f.109.2 4
84.59 odd 6 252.4.k.f.37.2 4
84.83 odd 2 1764.4.a.o.1.1 2

By twisted newform
Twist Min Dim Char Parity Ord Type
84.4.i.a.25.1 4 28.19 even 6
84.4.i.a.37.1 yes 4 28.3 even 6
252.4.k.f.37.2 4 84.59 odd 6
252.4.k.f.109.2 4 84.47 odd 6
336.4.q.i.193.1 4 7.5 odd 6
336.4.q.i.289.1 4 7.3 odd 6
588.4.a.f.1.1 2 4.3 odd 2
588.4.a.i.1.2 2 28.27 even 2
588.4.i.j.361.2 4 28.23 odd 6
588.4.i.j.373.2 4 28.11 odd 6
1764.4.a.o.1.1 2 84.83 odd 2
1764.4.a.y.1.2 2 12.11 even 2
1764.4.k.q.361.1 4 84.23 even 6
1764.4.k.q.1549.1 4 84.11 even 6
2352.4.a.bt.1.2 2 7.6 odd 2
2352.4.a.bx.1.1 2 1.1 even 1 trivial