# Properties

 Label 441.4.a.k.1.1 Level $441$ Weight $4$ Character 441.1 Self dual yes Analytic conductor $26.020$ Analytic rank $0$ Dimension $1$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$441 = 3^{2} \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 441.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$26.0198423125$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 21) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Character $$\chi$$ $$=$$ 441.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q+3.00000 q^{2} +1.00000 q^{4} -3.00000 q^{5} -21.0000 q^{8} +O(q^{10})$$ $$q+3.00000 q^{2} +1.00000 q^{4} -3.00000 q^{5} -21.0000 q^{8} -9.00000 q^{10} +15.0000 q^{11} +64.0000 q^{13} -71.0000 q^{16} +84.0000 q^{17} +16.0000 q^{19} -3.00000 q^{20} +45.0000 q^{22} +84.0000 q^{23} -116.000 q^{25} +192.000 q^{26} +297.000 q^{29} +253.000 q^{31} -45.0000 q^{32} +252.000 q^{34} -316.000 q^{37} +48.0000 q^{38} +63.0000 q^{40} +360.000 q^{41} +26.0000 q^{43} +15.0000 q^{44} +252.000 q^{46} -30.0000 q^{47} -348.000 q^{50} +64.0000 q^{52} -363.000 q^{53} -45.0000 q^{55} +891.000 q^{58} -15.0000 q^{59} +118.000 q^{61} +759.000 q^{62} +433.000 q^{64} -192.000 q^{65} -370.000 q^{67} +84.0000 q^{68} +342.000 q^{71} -362.000 q^{73} -948.000 q^{74} +16.0000 q^{76} +467.000 q^{79} +213.000 q^{80} +1080.00 q^{82} +477.000 q^{83} -252.000 q^{85} +78.0000 q^{86} -315.000 q^{88} +906.000 q^{89} +84.0000 q^{92} -90.0000 q^{94} -48.0000 q^{95} -503.000 q^{97} +O(q^{100})$$

## Coefficient data

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

Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 3.00000 1.06066 0.530330 0.847791i $$-0.322068\pi$$
0.530330 + 0.847791i $$0.322068\pi$$
$$3$$ 0 0
$$4$$ 1.00000 0.125000
$$5$$ −3.00000 −0.268328 −0.134164 0.990959i $$-0.542835\pi$$
−0.134164 + 0.990959i $$0.542835\pi$$
$$6$$ 0 0
$$7$$ 0 0
$$8$$ −21.0000 −0.928078
$$9$$ 0 0
$$10$$ −9.00000 −0.284605
$$11$$ 15.0000 0.411152 0.205576 0.978641i $$-0.434093\pi$$
0.205576 + 0.978641i $$0.434093\pi$$
$$12$$ 0 0
$$13$$ 64.0000 1.36542 0.682708 0.730691i $$-0.260802\pi$$
0.682708 + 0.730691i $$0.260802\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ −71.0000 −1.10938
$$17$$ 84.0000 1.19841 0.599206 0.800595i $$-0.295483\pi$$
0.599206 + 0.800595i $$0.295483\pi$$
$$18$$ 0 0
$$19$$ 16.0000 0.193192 0.0965961 0.995324i $$-0.469204\pi$$
0.0965961 + 0.995324i $$0.469204\pi$$
$$20$$ −3.00000 −0.0335410
$$21$$ 0 0
$$22$$ 45.0000 0.436092
$$23$$ 84.0000 0.761531 0.380765 0.924672i $$-0.375661\pi$$
0.380765 + 0.924672i $$0.375661\pi$$
$$24$$ 0 0
$$25$$ −116.000 −0.928000
$$26$$ 192.000 1.44824
$$27$$ 0 0
$$28$$ 0 0
$$29$$ 297.000 1.90178 0.950888 0.309535i $$-0.100173\pi$$
0.950888 + 0.309535i $$0.100173\pi$$
$$30$$ 0 0
$$31$$ 253.000 1.46581 0.732906 0.680330i $$-0.238164\pi$$
0.732906 + 0.680330i $$0.238164\pi$$
$$32$$ −45.0000 −0.248592
$$33$$ 0 0
$$34$$ 252.000 1.27111
$$35$$ 0 0
$$36$$ 0 0
$$37$$ −316.000 −1.40406 −0.702028 0.712149i $$-0.747722\pi$$
−0.702028 + 0.712149i $$0.747722\pi$$
$$38$$ 48.0000 0.204911
$$39$$ 0 0
$$40$$ 63.0000 0.249029
$$41$$ 360.000 1.37128 0.685641 0.727940i $$-0.259522\pi$$
0.685641 + 0.727940i $$0.259522\pi$$
$$42$$ 0 0
$$43$$ 26.0000 0.0922084 0.0461042 0.998937i $$-0.485319\pi$$
0.0461042 + 0.998937i $$0.485319\pi$$
$$44$$ 15.0000 0.0513940
$$45$$ 0 0
$$46$$ 252.000 0.807725
$$47$$ −30.0000 −0.0931053 −0.0465527 0.998916i $$-0.514824\pi$$
−0.0465527 + 0.998916i $$0.514824\pi$$
$$48$$ 0 0
$$49$$ 0 0
$$50$$ −348.000 −0.984293
$$51$$ 0 0
$$52$$ 64.0000 0.170677
$$53$$ −363.000 −0.940790 −0.470395 0.882456i $$-0.655889\pi$$
−0.470395 + 0.882456i $$0.655889\pi$$
$$54$$ 0 0
$$55$$ −45.0000 −0.110324
$$56$$ 0 0
$$57$$ 0 0
$$58$$ 891.000 2.01714
$$59$$ −15.0000 −0.0330989 −0.0165494 0.999863i $$-0.505268\pi$$
−0.0165494 + 0.999863i $$0.505268\pi$$
$$60$$ 0 0
$$61$$ 118.000 0.247678 0.123839 0.992302i $$-0.460479\pi$$
0.123839 + 0.992302i $$0.460479\pi$$
$$62$$ 759.000 1.55473
$$63$$ 0 0
$$64$$ 433.000 0.845703
$$65$$ −192.000 −0.366380
$$66$$ 0 0
$$67$$ −370.000 −0.674667 −0.337334 0.941385i $$-0.609525\pi$$
−0.337334 + 0.941385i $$0.609525\pi$$
$$68$$ 84.0000 0.149801
$$69$$ 0 0
$$70$$ 0 0
$$71$$ 342.000 0.571661 0.285831 0.958280i $$-0.407731\pi$$
0.285831 + 0.958280i $$0.407731\pi$$
$$72$$ 0 0
$$73$$ −362.000 −0.580396 −0.290198 0.956967i $$-0.593721\pi$$
−0.290198 + 0.956967i $$0.593721\pi$$
$$74$$ −948.000 −1.48923
$$75$$ 0 0
$$76$$ 16.0000 0.0241490
$$77$$ 0 0
$$78$$ 0 0
$$79$$ 467.000 0.665084 0.332542 0.943089i $$-0.392094\pi$$
0.332542 + 0.943089i $$0.392094\pi$$
$$80$$ 213.000 0.297677
$$81$$ 0 0
$$82$$ 1080.00 1.45446
$$83$$ 477.000 0.630814 0.315407 0.948957i $$-0.397859\pi$$
0.315407 + 0.948957i $$0.397859\pi$$
$$84$$ 0 0
$$85$$ −252.000 −0.321568
$$86$$ 78.0000 0.0978018
$$87$$ 0 0
$$88$$ −315.000 −0.381581
$$89$$ 906.000 1.07905 0.539527 0.841968i $$-0.318603\pi$$
0.539527 + 0.841968i $$0.318603\pi$$
$$90$$ 0 0
$$91$$ 0 0
$$92$$ 84.0000 0.0951914
$$93$$ 0 0
$$94$$ −90.0000 −0.0987531
$$95$$ −48.0000 −0.0518389
$$96$$ 0 0
$$97$$ −503.000 −0.526515 −0.263257 0.964726i $$-0.584797\pi$$
−0.263257 + 0.964726i $$0.584797\pi$$
$$98$$ 0 0
$$99$$ 0 0
$$100$$ −116.000 −0.116000
$$101$$ −1086.00 −1.06991 −0.534956 0.844880i $$-0.679672\pi$$
−0.534956 + 0.844880i $$0.679672\pi$$
$$102$$ 0 0
$$103$$ −1736.00 −1.66071 −0.830355 0.557235i $$-0.811862\pi$$
−0.830355 + 0.557235i $$0.811862\pi$$
$$104$$ −1344.00 −1.26721
$$105$$ 0 0
$$106$$ −1089.00 −0.997859
$$107$$ 1353.00 1.22242 0.611212 0.791467i $$-0.290682\pi$$
0.611212 + 0.791467i $$0.290682\pi$$
$$108$$ 0 0
$$109$$ −370.000 −0.325134 −0.162567 0.986698i $$-0.551977\pi$$
−0.162567 + 0.986698i $$0.551977\pi$$
$$110$$ −135.000 −0.117016
$$111$$ 0 0
$$112$$ 0 0
$$113$$ 648.000 0.539458 0.269729 0.962936i $$-0.413066\pi$$
0.269729 + 0.962936i $$0.413066\pi$$
$$114$$ 0 0
$$115$$ −252.000 −0.204340
$$116$$ 297.000 0.237722
$$117$$ 0 0
$$118$$ −45.0000 −0.0351067
$$119$$ 0 0
$$120$$ 0 0
$$121$$ −1106.00 −0.830954
$$122$$ 354.000 0.262702
$$123$$ 0 0
$$124$$ 253.000 0.183226
$$125$$ 723.000 0.517337
$$126$$ 0 0
$$127$$ 377.000 0.263412 0.131706 0.991289i $$-0.457954\pi$$
0.131706 + 0.991289i $$0.457954\pi$$
$$128$$ 1659.00 1.14560
$$129$$ 0 0
$$130$$ −576.000 −0.388604
$$131$$ −651.000 −0.434184 −0.217092 0.976151i $$-0.569657\pi$$
−0.217092 + 0.976151i $$0.569657\pi$$
$$132$$ 0 0
$$133$$ 0 0
$$134$$ −1110.00 −0.715593
$$135$$ 0 0
$$136$$ −1764.00 −1.11222
$$137$$ 1770.00 1.10381 0.551903 0.833909i $$-0.313902\pi$$
0.551903 + 0.833909i $$0.313902\pi$$
$$138$$ 0 0
$$139$$ 1558.00 0.950704 0.475352 0.879796i $$-0.342321\pi$$
0.475352 + 0.879796i $$0.342321\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ 1026.00 0.606338
$$143$$ 960.000 0.561393
$$144$$ 0 0
$$145$$ −891.000 −0.510300
$$146$$ −1086.00 −0.615603
$$147$$ 0 0
$$148$$ −316.000 −0.175507
$$149$$ −2454.00 −1.34926 −0.674629 0.738157i $$-0.735696\pi$$
−0.674629 + 0.738157i $$0.735696\pi$$
$$150$$ 0 0
$$151$$ 1259.00 0.678516 0.339258 0.940693i $$-0.389824\pi$$
0.339258 + 0.940693i $$0.389824\pi$$
$$152$$ −336.000 −0.179297
$$153$$ 0 0
$$154$$ 0 0
$$155$$ −759.000 −0.393318
$$156$$ 0 0
$$157$$ 196.000 0.0996338 0.0498169 0.998758i $$-0.484136\pi$$
0.0498169 + 0.998758i $$0.484136\pi$$
$$158$$ 1401.00 0.705428
$$159$$ 0 0
$$160$$ 135.000 0.0667043
$$161$$ 0 0
$$162$$ 0 0
$$163$$ −1252.00 −0.601621 −0.300810 0.953684i $$-0.597257\pi$$
−0.300810 + 0.953684i $$0.597257\pi$$
$$164$$ 360.000 0.171410
$$165$$ 0 0
$$166$$ 1431.00 0.669079
$$167$$ −2646.00 −1.22607 −0.613035 0.790056i $$-0.710051\pi$$
−0.613035 + 0.790056i $$0.710051\pi$$
$$168$$ 0 0
$$169$$ 1899.00 0.864360
$$170$$ −756.000 −0.341074
$$171$$ 0 0
$$172$$ 26.0000 0.0115261
$$173$$ −786.000 −0.345425 −0.172712 0.984972i $$-0.555253\pi$$
−0.172712 + 0.984972i $$0.555253\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ −1065.00 −0.456122
$$177$$ 0 0
$$178$$ 2718.00 1.14451
$$179$$ −2892.00 −1.20759 −0.603794 0.797140i $$-0.706345\pi$$
−0.603794 + 0.797140i $$0.706345\pi$$
$$180$$ 0 0
$$181$$ −1352.00 −0.555212 −0.277606 0.960695i $$-0.589541\pi$$
−0.277606 + 0.960695i $$0.589541\pi$$
$$182$$ 0 0
$$183$$ 0 0
$$184$$ −1764.00 −0.706760
$$185$$ 948.000 0.376748
$$186$$ 0 0
$$187$$ 1260.00 0.492729
$$188$$ −30.0000 −0.0116382
$$189$$ 0 0
$$190$$ −144.000 −0.0549835
$$191$$ −3912.00 −1.48200 −0.741001 0.671504i $$-0.765649\pi$$
−0.741001 + 0.671504i $$0.765649\pi$$
$$192$$ 0 0
$$193$$ 1493.00 0.556832 0.278416 0.960461i $$-0.410191\pi$$
0.278416 + 0.960461i $$0.410191\pi$$
$$194$$ −1509.00 −0.558453
$$195$$ 0 0
$$196$$ 0 0
$$197$$ 4086.00 1.47774 0.738872 0.673846i $$-0.235359\pi$$
0.738872 + 0.673846i $$0.235359\pi$$
$$198$$ 0 0
$$199$$ 3556.00 1.26672 0.633362 0.773855i $$-0.281674\pi$$
0.633362 + 0.773855i $$0.281674\pi$$
$$200$$ 2436.00 0.861256
$$201$$ 0 0
$$202$$ −3258.00 −1.13481
$$203$$ 0 0
$$204$$ 0 0
$$205$$ −1080.00 −0.367954
$$206$$ −5208.00 −1.76145
$$207$$ 0 0
$$208$$ −4544.00 −1.51476
$$209$$ 240.000 0.0794313
$$210$$ 0 0
$$211$$ 1250.00 0.407837 0.203918 0.978988i $$-0.434632\pi$$
0.203918 + 0.978988i $$0.434632\pi$$
$$212$$ −363.000 −0.117599
$$213$$ 0 0
$$214$$ 4059.00 1.29658
$$215$$ −78.0000 −0.0247421
$$216$$ 0 0
$$217$$ 0 0
$$218$$ −1110.00 −0.344856
$$219$$ 0 0
$$220$$ −45.0000 −0.0137905
$$221$$ 5376.00 1.63633
$$222$$ 0 0
$$223$$ −425.000 −0.127624 −0.0638119 0.997962i $$-0.520326\pi$$
−0.0638119 + 0.997962i $$0.520326\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ 1944.00 0.572181
$$227$$ 3855.00 1.12716 0.563580 0.826061i $$-0.309424\pi$$
0.563580 + 0.826061i $$0.309424\pi$$
$$228$$ 0 0
$$229$$ 2188.00 0.631385 0.315692 0.948862i $$-0.397763\pi$$
0.315692 + 0.948862i $$0.397763\pi$$
$$230$$ −756.000 −0.216735
$$231$$ 0 0
$$232$$ −6237.00 −1.76500
$$233$$ −852.000 −0.239555 −0.119778 0.992801i $$-0.538218\pi$$
−0.119778 + 0.992801i $$0.538218\pi$$
$$234$$ 0 0
$$235$$ 90.0000 0.0249828
$$236$$ −15.0000 −0.00413736
$$237$$ 0 0
$$238$$ 0 0
$$239$$ −5508.00 −1.49072 −0.745362 0.666660i $$-0.767723\pi$$
−0.745362 + 0.666660i $$0.767723\pi$$
$$240$$ 0 0
$$241$$ −791.000 −0.211422 −0.105711 0.994397i $$-0.533712\pi$$
−0.105711 + 0.994397i $$0.533712\pi$$
$$242$$ −3318.00 −0.881360
$$243$$ 0 0
$$244$$ 118.000 0.0309597
$$245$$ 0 0
$$246$$ 0 0
$$247$$ 1024.00 0.263788
$$248$$ −5313.00 −1.36039
$$249$$ 0 0
$$250$$ 2169.00 0.548718
$$251$$ 5265.00 1.32400 0.662000 0.749504i $$-0.269708\pi$$
0.662000 + 0.749504i $$0.269708\pi$$
$$252$$ 0 0
$$253$$ 1260.00 0.313105
$$254$$ 1131.00 0.279391
$$255$$ 0 0
$$256$$ 1513.00 0.369385
$$257$$ −6870.00 −1.66747 −0.833733 0.552168i $$-0.813801\pi$$
−0.833733 + 0.552168i $$0.813801\pi$$
$$258$$ 0 0
$$259$$ 0 0
$$260$$ −192.000 −0.0457974
$$261$$ 0 0
$$262$$ −1953.00 −0.460522
$$263$$ 222.000 0.0520498 0.0260249 0.999661i $$-0.491715\pi$$
0.0260249 + 0.999661i $$0.491715\pi$$
$$264$$ 0 0
$$265$$ 1089.00 0.252441
$$266$$ 0 0
$$267$$ 0 0
$$268$$ −370.000 −0.0843334
$$269$$ 7851.00 1.77949 0.889747 0.456454i $$-0.150881\pi$$
0.889747 + 0.456454i $$0.150881\pi$$
$$270$$ 0 0
$$271$$ −5183.00 −1.16179 −0.580895 0.813979i $$-0.697297\pi$$
−0.580895 + 0.813979i $$0.697297\pi$$
$$272$$ −5964.00 −1.32949
$$273$$ 0 0
$$274$$ 5310.00 1.17076
$$275$$ −1740.00 −0.381549
$$276$$ 0 0
$$277$$ −4960.00 −1.07588 −0.537938 0.842985i $$-0.680796\pi$$
−0.537938 + 0.842985i $$0.680796\pi$$
$$278$$ 4674.00 1.00837
$$279$$ 0 0
$$280$$ 0 0
$$281$$ 774.000 0.164317 0.0821583 0.996619i $$-0.473819\pi$$
0.0821583 + 0.996619i $$0.473819\pi$$
$$282$$ 0 0
$$283$$ −3698.00 −0.776761 −0.388380 0.921499i $$-0.626965\pi$$
−0.388380 + 0.921499i $$0.626965\pi$$
$$284$$ 342.000 0.0714576
$$285$$ 0 0
$$286$$ 2880.00 0.595447
$$287$$ 0 0
$$288$$ 0 0
$$289$$ 2143.00 0.436190
$$290$$ −2673.00 −0.541255
$$291$$ 0 0
$$292$$ −362.000 −0.0725495
$$293$$ −6273.00 −1.25076 −0.625380 0.780321i $$-0.715056\pi$$
−0.625380 + 0.780321i $$0.715056\pi$$
$$294$$ 0 0
$$295$$ 45.0000 0.00888136
$$296$$ 6636.00 1.30307
$$297$$ 0 0
$$298$$ −7362.00 −1.43110
$$299$$ 5376.00 1.03981
$$300$$ 0 0
$$301$$ 0 0
$$302$$ 3777.00 0.719675
$$303$$ 0 0
$$304$$ −1136.00 −0.214323
$$305$$ −354.000 −0.0664590
$$306$$ 0 0
$$307$$ 1684.00 0.313065 0.156533 0.987673i $$-0.449968\pi$$
0.156533 + 0.987673i $$0.449968\pi$$
$$308$$ 0 0
$$309$$ 0 0
$$310$$ −2277.00 −0.417177
$$311$$ −1320.00 −0.240676 −0.120338 0.992733i $$-0.538398\pi$$
−0.120338 + 0.992733i $$0.538398\pi$$
$$312$$ 0 0
$$313$$ 8503.00 1.53552 0.767760 0.640737i $$-0.221371\pi$$
0.767760 + 0.640737i $$0.221371\pi$$
$$314$$ 588.000 0.105678
$$315$$ 0 0
$$316$$ 467.000 0.0831355
$$317$$ 2577.00 0.456589 0.228295 0.973592i $$-0.426685\pi$$
0.228295 + 0.973592i $$0.426685\pi$$
$$318$$ 0 0
$$319$$ 4455.00 0.781919
$$320$$ −1299.00 −0.226926
$$321$$ 0 0
$$322$$ 0 0
$$323$$ 1344.00 0.231524
$$324$$ 0 0
$$325$$ −7424.00 −1.26711
$$326$$ −3756.00 −0.638115
$$327$$ 0 0
$$328$$ −7560.00 −1.27266
$$329$$ 0 0
$$330$$ 0 0
$$331$$ −484.000 −0.0803717 −0.0401859 0.999192i $$-0.512795\pi$$
−0.0401859 + 0.999192i $$0.512795\pi$$
$$332$$ 477.000 0.0788517
$$333$$ 0 0
$$334$$ −7938.00 −1.30044
$$335$$ 1110.00 0.181032
$$336$$ 0 0
$$337$$ −8359.00 −1.35117 −0.675584 0.737283i $$-0.736109\pi$$
−0.675584 + 0.737283i $$0.736109\pi$$
$$338$$ 5697.00 0.916793
$$339$$ 0 0
$$340$$ −252.000 −0.0401959
$$341$$ 3795.00 0.602671
$$342$$ 0 0
$$343$$ 0 0
$$344$$ −546.000 −0.0855766
$$345$$ 0 0
$$346$$ −2358.00 −0.366378
$$347$$ 1860.00 0.287752 0.143876 0.989596i $$-0.454043\pi$$
0.143876 + 0.989596i $$0.454043\pi$$
$$348$$ 0 0
$$349$$ 1918.00 0.294178 0.147089 0.989123i $$-0.453010\pi$$
0.147089 + 0.989123i $$0.453010\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ −675.000 −0.102209
$$353$$ −3048.00 −0.459571 −0.229786 0.973241i $$-0.573803\pi$$
−0.229786 + 0.973241i $$0.573803\pi$$
$$354$$ 0 0
$$355$$ −1026.00 −0.153393
$$356$$ 906.000 0.134882
$$357$$ 0 0
$$358$$ −8676.00 −1.28084
$$359$$ 30.0000 0.00441042 0.00220521 0.999998i $$-0.499298\pi$$
0.00220521 + 0.999998i $$0.499298\pi$$
$$360$$ 0 0
$$361$$ −6603.00 −0.962677
$$362$$ −4056.00 −0.588891
$$363$$ 0 0
$$364$$ 0 0
$$365$$ 1086.00 0.155737
$$366$$ 0 0
$$367$$ 11311.0 1.60880 0.804400 0.594088i $$-0.202487\pi$$
0.804400 + 0.594088i $$0.202487\pi$$
$$368$$ −5964.00 −0.844823
$$369$$ 0 0
$$370$$ 2844.00 0.399601
$$371$$ 0 0
$$372$$ 0 0
$$373$$ 1208.00 0.167689 0.0838443 0.996479i $$-0.473280\pi$$
0.0838443 + 0.996479i $$0.473280\pi$$
$$374$$ 3780.00 0.522618
$$375$$ 0 0
$$376$$ 630.000 0.0864090
$$377$$ 19008.0 2.59672
$$378$$ 0 0
$$379$$ 7640.00 1.03546 0.517731 0.855543i $$-0.326777\pi$$
0.517731 + 0.855543i $$0.326777\pi$$
$$380$$ −48.0000 −0.00647986
$$381$$ 0 0
$$382$$ −11736.0 −1.57190
$$383$$ 12750.0 1.70103 0.850515 0.525951i $$-0.176290\pi$$
0.850515 + 0.525951i $$0.176290\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 4479.00 0.590609
$$387$$ 0 0
$$388$$ −503.000 −0.0658143
$$389$$ −3126.00 −0.407441 −0.203720 0.979029i $$-0.565303\pi$$
−0.203720 + 0.979029i $$0.565303\pi$$
$$390$$ 0 0
$$391$$ 7056.00 0.912627
$$392$$ 0 0
$$393$$ 0 0
$$394$$ 12258.0 1.56738
$$395$$ −1401.00 −0.178461
$$396$$ 0 0
$$397$$ 5932.00 0.749921 0.374960 0.927041i $$-0.377656\pi$$
0.374960 + 0.927041i $$0.377656\pi$$
$$398$$ 10668.0 1.34356
$$399$$ 0 0
$$400$$ 8236.00 1.02950
$$401$$ −1608.00 −0.200249 −0.100124 0.994975i $$-0.531924\pi$$
−0.100124 + 0.994975i $$0.531924\pi$$
$$402$$ 0 0
$$403$$ 16192.0 2.00144
$$404$$ −1086.00 −0.133739
$$405$$ 0 0
$$406$$ 0 0
$$407$$ −4740.00 −0.577280
$$408$$ 0 0
$$409$$ 4465.00 0.539805 0.269902 0.962888i $$-0.413009\pi$$
0.269902 + 0.962888i $$0.413009\pi$$
$$410$$ −3240.00 −0.390274
$$411$$ 0 0
$$412$$ −1736.00 −0.207589
$$413$$ 0 0
$$414$$ 0 0
$$415$$ −1431.00 −0.169265
$$416$$ −2880.00 −0.339432
$$417$$ 0 0
$$418$$ 720.000 0.0842496
$$419$$ −1584.00 −0.184686 −0.0923430 0.995727i $$-0.529436\pi$$
−0.0923430 + 0.995727i $$0.529436\pi$$
$$420$$ 0 0
$$421$$ −1330.00 −0.153967 −0.0769837 0.997032i $$-0.524529\pi$$
−0.0769837 + 0.997032i $$0.524529\pi$$
$$422$$ 3750.00 0.432576
$$423$$ 0 0
$$424$$ 7623.00 0.873126
$$425$$ −9744.00 −1.11213
$$426$$ 0 0
$$427$$ 0 0
$$428$$ 1353.00 0.152803
$$429$$ 0 0
$$430$$ −234.000 −0.0262430
$$431$$ −9588.00 −1.07155 −0.535775 0.844361i $$-0.679980\pi$$
−0.535775 + 0.844361i $$0.679980\pi$$
$$432$$ 0 0
$$433$$ −494.000 −0.0548271 −0.0274135 0.999624i $$-0.508727\pi$$
−0.0274135 + 0.999624i $$0.508727\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ −370.000 −0.0406417
$$437$$ 1344.00 0.147122
$$438$$ 0 0
$$439$$ 16009.0 1.74047 0.870237 0.492634i $$-0.163966\pi$$
0.870237 + 0.492634i $$0.163966\pi$$
$$440$$ 945.000 0.102389
$$441$$ 0 0
$$442$$ 16128.0 1.73559
$$443$$ −7773.00 −0.833649 −0.416824 0.908987i $$-0.636857\pi$$
−0.416824 + 0.908987i $$0.636857\pi$$
$$444$$ 0 0
$$445$$ −2718.00 −0.289541
$$446$$ −1275.00 −0.135365
$$447$$ 0 0
$$448$$ 0 0
$$449$$ −864.000 −0.0908122 −0.0454061 0.998969i $$-0.514458\pi$$
−0.0454061 + 0.998969i $$0.514458\pi$$
$$450$$ 0 0
$$451$$ 5400.00 0.563805
$$452$$ 648.000 0.0674322
$$453$$ 0 0
$$454$$ 11565.0 1.19553
$$455$$ 0 0
$$456$$ 0 0
$$457$$ 2519.00 0.257842 0.128921 0.991655i $$-0.458849\pi$$
0.128921 + 0.991655i $$0.458849\pi$$
$$458$$ 6564.00 0.669685
$$459$$ 0 0
$$460$$ −252.000 −0.0255425
$$461$$ −342.000 −0.0345521 −0.0172761 0.999851i $$-0.505499\pi$$
−0.0172761 + 0.999851i $$0.505499\pi$$
$$462$$ 0 0
$$463$$ −4336.00 −0.435229 −0.217614 0.976035i $$-0.569828\pi$$
−0.217614 + 0.976035i $$0.569828\pi$$
$$464$$ −21087.0 −2.10978
$$465$$ 0 0
$$466$$ −2556.00 −0.254087
$$467$$ 18636.0 1.84662 0.923310 0.384056i $$-0.125473\pi$$
0.923310 + 0.384056i $$0.125473\pi$$
$$468$$ 0 0
$$469$$ 0 0
$$470$$ 270.000 0.0264982
$$471$$ 0 0
$$472$$ 315.000 0.0307183
$$473$$ 390.000 0.0379117
$$474$$ 0 0
$$475$$ −1856.00 −0.179282
$$476$$ 0 0
$$477$$ 0 0
$$478$$ −16524.0 −1.58115
$$479$$ 15078.0 1.43827 0.719135 0.694870i $$-0.244538\pi$$
0.719135 + 0.694870i $$0.244538\pi$$
$$480$$ 0 0
$$481$$ −20224.0 −1.91712
$$482$$ −2373.00 −0.224247
$$483$$ 0 0
$$484$$ −1106.00 −0.103869
$$485$$ 1509.00 0.141279
$$486$$ 0 0
$$487$$ 6221.00 0.578851 0.289425 0.957201i $$-0.406536\pi$$
0.289425 + 0.957201i $$0.406536\pi$$
$$488$$ −2478.00 −0.229864
$$489$$ 0 0
$$490$$ 0 0
$$491$$ 7371.00 0.677492 0.338746 0.940878i $$-0.389997\pi$$
0.338746 + 0.940878i $$0.389997\pi$$
$$492$$ 0 0
$$493$$ 24948.0 2.27911
$$494$$ 3072.00 0.279789
$$495$$ 0 0
$$496$$ −17963.0 −1.62613
$$497$$ 0 0
$$498$$ 0 0
$$499$$ 4274.00 0.383428 0.191714 0.981451i $$-0.438595\pi$$
0.191714 + 0.981451i $$0.438595\pi$$
$$500$$ 723.000 0.0646671
$$501$$ 0 0
$$502$$ 15795.0 1.40431
$$503$$ −2520.00 −0.223382 −0.111691 0.993743i $$-0.535627\pi$$
−0.111691 + 0.993743i $$0.535627\pi$$
$$504$$ 0 0
$$505$$ 3258.00 0.287087
$$506$$ 3780.00 0.332098
$$507$$ 0 0
$$508$$ 377.000 0.0329265
$$509$$ −14277.0 −1.24326 −0.621628 0.783313i $$-0.713528\pi$$
−0.621628 + 0.783313i $$0.713528\pi$$
$$510$$ 0 0
$$511$$ 0 0
$$512$$ −8733.00 −0.753804
$$513$$ 0 0
$$514$$ −20610.0 −1.76862
$$515$$ 5208.00 0.445615
$$516$$ 0 0
$$517$$ −450.000 −0.0382804
$$518$$ 0 0
$$519$$ 0 0
$$520$$ 4032.00 0.340029
$$521$$ −6306.00 −0.530270 −0.265135 0.964211i $$-0.585417\pi$$
−0.265135 + 0.964211i $$0.585417\pi$$
$$522$$ 0 0
$$523$$ −8072.00 −0.674883 −0.337442 0.941346i $$-0.609562\pi$$
−0.337442 + 0.941346i $$0.609562\pi$$
$$524$$ −651.000 −0.0542730
$$525$$ 0 0
$$526$$ 666.000 0.0552072
$$527$$ 21252.0 1.75664
$$528$$ 0 0
$$529$$ −5111.00 −0.420071
$$530$$ 3267.00 0.267754
$$531$$ 0 0
$$532$$ 0 0
$$533$$ 23040.0 1.87237
$$534$$ 0 0
$$535$$ −4059.00 −0.328011
$$536$$ 7770.00 0.626143
$$537$$ 0 0
$$538$$ 23553.0 1.88744
$$539$$ 0 0
$$540$$ 0 0
$$541$$ −22858.0 −1.81653 −0.908264 0.418396i $$-0.862592\pi$$
−0.908264 + 0.418396i $$0.862592\pi$$
$$542$$ −15549.0 −1.23226
$$543$$ 0 0
$$544$$ −3780.00 −0.297916
$$545$$ 1110.00 0.0872425
$$546$$ 0 0
$$547$$ −24724.0 −1.93258 −0.966291 0.257454i $$-0.917116\pi$$
−0.966291 + 0.257454i $$0.917116\pi$$
$$548$$ 1770.00 0.137976
$$549$$ 0 0
$$550$$ −5220.00 −0.404694
$$551$$ 4752.00 0.367408
$$552$$ 0 0
$$553$$ 0 0
$$554$$ −14880.0 −1.14114
$$555$$ 0 0
$$556$$ 1558.00 0.118838
$$557$$ 9843.00 0.748764 0.374382 0.927275i $$-0.377855\pi$$
0.374382 + 0.927275i $$0.377855\pi$$
$$558$$ 0 0
$$559$$ 1664.00 0.125903
$$560$$ 0 0
$$561$$ 0 0
$$562$$ 2322.00 0.174284
$$563$$ −13371.0 −1.00092 −0.500462 0.865758i $$-0.666837\pi$$
−0.500462 + 0.865758i $$0.666837\pi$$
$$564$$ 0 0
$$565$$ −1944.00 −0.144752
$$566$$ −11094.0 −0.823879
$$567$$ 0 0
$$568$$ −7182.00 −0.530546
$$569$$ 5232.00 0.385478 0.192739 0.981250i $$-0.438263\pi$$
0.192739 + 0.981250i $$0.438263\pi$$
$$570$$ 0 0
$$571$$ −14398.0 −1.05523 −0.527616 0.849483i $$-0.676914\pi$$
−0.527616 + 0.849483i $$0.676914\pi$$
$$572$$ 960.000 0.0701742
$$573$$ 0 0
$$574$$ 0 0
$$575$$ −9744.00 −0.706701
$$576$$ 0 0
$$577$$ −19871.0 −1.43369 −0.716846 0.697231i $$-0.754415\pi$$
−0.716846 + 0.697231i $$0.754415\pi$$
$$578$$ 6429.00 0.462649
$$579$$ 0 0
$$580$$ −891.000 −0.0637875
$$581$$ 0 0
$$582$$ 0 0
$$583$$ −5445.00 −0.386808
$$584$$ 7602.00 0.538652
$$585$$ 0 0
$$586$$ −18819.0 −1.32663
$$587$$ −16137.0 −1.13466 −0.567330 0.823491i $$-0.692024\pi$$
−0.567330 + 0.823491i $$0.692024\pi$$
$$588$$ 0 0
$$589$$ 4048.00 0.283183
$$590$$ 135.000 0.00942011
$$591$$ 0 0
$$592$$ 22436.0 1.55762
$$593$$ −21324.0 −1.47668 −0.738340 0.674428i $$-0.764390\pi$$
−0.738340 + 0.674428i $$0.764390\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ −2454.00 −0.168657
$$597$$ 0 0
$$598$$ 16128.0 1.10288
$$599$$ 8646.00 0.589760 0.294880 0.955534i $$-0.404720\pi$$
0.294880 + 0.955534i $$0.404720\pi$$
$$600$$ 0 0
$$601$$ −11195.0 −0.759823 −0.379911 0.925023i $$-0.624046\pi$$
−0.379911 + 0.925023i $$0.624046\pi$$
$$602$$ 0 0
$$603$$ 0 0
$$604$$ 1259.00 0.0848145
$$605$$ 3318.00 0.222968
$$606$$ 0 0
$$607$$ 8971.00 0.599871 0.299935 0.953959i $$-0.403035\pi$$
0.299935 + 0.953959i $$0.403035\pi$$
$$608$$ −720.000 −0.0480261
$$609$$ 0 0
$$610$$ −1062.00 −0.0704904
$$611$$ −1920.00 −0.127127
$$612$$ 0 0
$$613$$ −12772.0 −0.841527 −0.420764 0.907170i $$-0.638238\pi$$
−0.420764 + 0.907170i $$0.638238\pi$$
$$614$$ 5052.00 0.332056
$$615$$ 0 0
$$616$$ 0 0
$$617$$ −12762.0 −0.832705 −0.416352 0.909203i $$-0.636692\pi$$
−0.416352 + 0.909203i $$0.636692\pi$$
$$618$$ 0 0
$$619$$ −12842.0 −0.833867 −0.416933 0.908937i $$-0.636895\pi$$
−0.416933 + 0.908937i $$0.636895\pi$$
$$620$$ −759.000 −0.0491648
$$621$$ 0 0
$$622$$ −3960.00 −0.255276
$$623$$ 0 0
$$624$$ 0 0
$$625$$ 12331.0 0.789184
$$626$$ 25509.0 1.62867
$$627$$ 0 0
$$628$$ 196.000 0.0124542
$$629$$ −26544.0 −1.68264
$$630$$ 0 0
$$631$$ 21365.0 1.34790 0.673952 0.738775i $$-0.264596\pi$$
0.673952 + 0.738775i $$0.264596\pi$$
$$632$$ −9807.00 −0.617249
$$633$$ 0 0
$$634$$ 7731.00 0.484286
$$635$$ −1131.00 −0.0706809
$$636$$ 0 0
$$637$$ 0 0
$$638$$ 13365.0 0.829350
$$639$$ 0 0
$$640$$ −4977.00 −0.307396
$$641$$ −8274.00 −0.509834 −0.254917 0.966963i $$-0.582048\pi$$
−0.254917 + 0.966963i $$0.582048\pi$$
$$642$$ 0 0
$$643$$ −27998.0 −1.71716 −0.858580 0.512680i $$-0.828653\pi$$
−0.858580 + 0.512680i $$0.828653\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 4032.00 0.245568
$$647$$ −17466.0 −1.06130 −0.530649 0.847592i $$-0.678052\pi$$
−0.530649 + 0.847592i $$0.678052\pi$$
$$648$$ 0 0
$$649$$ −225.000 −0.0136087
$$650$$ −22272.0 −1.34397
$$651$$ 0 0
$$652$$ −1252.00 −0.0752026
$$653$$ −2157.00 −0.129265 −0.0646324 0.997909i $$-0.520587\pi$$
−0.0646324 + 0.997909i $$0.520587\pi$$
$$654$$ 0 0
$$655$$ 1953.00 0.116504
$$656$$ −25560.0 −1.52127
$$657$$ 0 0
$$658$$ 0 0
$$659$$ −19944.0 −1.17892 −0.589460 0.807798i $$-0.700659\pi$$
−0.589460 + 0.807798i $$0.700659\pi$$
$$660$$ 0 0
$$661$$ −27506.0 −1.61855 −0.809273 0.587432i $$-0.800139\pi$$
−0.809273 + 0.587432i $$0.800139\pi$$
$$662$$ −1452.00 −0.0852471
$$663$$ 0 0
$$664$$ −10017.0 −0.585444
$$665$$ 0 0
$$666$$ 0 0
$$667$$ 24948.0 1.44826
$$668$$ −2646.00 −0.153259
$$669$$ 0 0
$$670$$ 3330.00 0.192014
$$671$$ 1770.00 0.101833
$$672$$ 0 0
$$673$$ −19123.0 −1.09530 −0.547650 0.836707i $$-0.684478\pi$$
−0.547650 + 0.836707i $$0.684478\pi$$
$$674$$ −25077.0 −1.43313
$$675$$ 0 0
$$676$$ 1899.00 0.108045
$$677$$ 13857.0 0.786658 0.393329 0.919398i $$-0.371323\pi$$
0.393329 + 0.919398i $$0.371323\pi$$
$$678$$ 0 0
$$679$$ 0 0
$$680$$ 5292.00 0.298440
$$681$$ 0 0
$$682$$ 11385.0 0.639229
$$683$$ 22245.0 1.24624 0.623120 0.782127i $$-0.285865\pi$$
0.623120 + 0.782127i $$0.285865\pi$$
$$684$$ 0 0
$$685$$ −5310.00 −0.296182
$$686$$ 0 0
$$687$$ 0 0
$$688$$ −1846.00 −0.102294
$$689$$ −23232.0 −1.28457
$$690$$ 0 0
$$691$$ 640.000 0.0352341 0.0176170 0.999845i $$-0.494392\pi$$
0.0176170 + 0.999845i $$0.494392\pi$$
$$692$$ −786.000 −0.0431781
$$693$$ 0 0
$$694$$ 5580.00 0.305207
$$695$$ −4674.00 −0.255101
$$696$$ 0 0
$$697$$ 30240.0 1.64336
$$698$$ 5754.00 0.312023
$$699$$ 0 0
$$700$$ 0 0
$$701$$ 15561.0 0.838418 0.419209 0.907890i $$-0.362307\pi$$
0.419209 + 0.907890i $$0.362307\pi$$
$$702$$ 0 0
$$703$$ −5056.00 −0.271253
$$704$$ 6495.00 0.347712
$$705$$ 0 0
$$706$$ −9144.00 −0.487449
$$707$$ 0 0
$$708$$ 0 0
$$709$$ 5534.00 0.293136 0.146568 0.989201i $$-0.453177\pi$$
0.146568 + 0.989201i $$0.453177\pi$$
$$710$$ −3078.00 −0.162698
$$711$$ 0 0
$$712$$ −19026.0 −1.00145
$$713$$ 21252.0 1.11626
$$714$$ 0 0
$$715$$ −2880.00 −0.150638
$$716$$ −2892.00 −0.150948
$$717$$ 0 0
$$718$$ 90.0000 0.00467795
$$719$$ −21846.0 −1.13313 −0.566564 0.824018i $$-0.691727\pi$$
−0.566564 + 0.824018i $$0.691727\pi$$
$$720$$ 0 0
$$721$$ 0 0
$$722$$ −19809.0 −1.02107
$$723$$ 0 0
$$724$$ −1352.00 −0.0694015
$$725$$ −34452.0 −1.76485
$$726$$ 0 0
$$727$$ 11089.0 0.565706 0.282853 0.959163i $$-0.408719\pi$$
0.282853 + 0.959163i $$0.408719\pi$$
$$728$$ 0 0
$$729$$ 0 0
$$730$$ 3258.00 0.165184
$$731$$ 2184.00 0.110504
$$732$$ 0 0
$$733$$ −11762.0 −0.592687 −0.296343 0.955081i $$-0.595767\pi$$
−0.296343 + 0.955081i $$0.595767\pi$$
$$734$$ 33933.0 1.70639
$$735$$ 0 0
$$736$$ −3780.00 −0.189311
$$737$$ −5550.00 −0.277391
$$738$$ 0 0
$$739$$ −22726.0 −1.13124 −0.565622 0.824665i $$-0.691364\pi$$
−0.565622 + 0.824665i $$0.691364\pi$$
$$740$$ 948.000 0.0470935
$$741$$ 0 0
$$742$$ 0 0
$$743$$ −6678.00 −0.329734 −0.164867 0.986316i $$-0.552719\pi$$
−0.164867 + 0.986316i $$0.552719\pi$$
$$744$$ 0 0
$$745$$ 7362.00 0.362044
$$746$$ 3624.00 0.177861
$$747$$ 0 0
$$748$$ 1260.00 0.0615911
$$749$$ 0 0
$$750$$ 0 0
$$751$$ −19987.0 −0.971153 −0.485577 0.874194i $$-0.661390\pi$$
−0.485577 + 0.874194i $$0.661390\pi$$
$$752$$ 2130.00 0.103289
$$753$$ 0 0
$$754$$ 57024.0 2.75423
$$755$$ −3777.00 −0.182065
$$756$$ 0 0
$$757$$ 314.000 0.0150760 0.00753799 0.999972i $$-0.497601\pi$$
0.00753799 + 0.999972i $$0.497601\pi$$
$$758$$ 22920.0 1.09827
$$759$$ 0 0
$$760$$ 1008.00 0.0481105
$$761$$ −11496.0 −0.547608 −0.273804 0.961786i $$-0.588282\pi$$
−0.273804 + 0.961786i $$0.588282\pi$$
$$762$$ 0 0
$$763$$ 0 0
$$764$$ −3912.00 −0.185250
$$765$$ 0 0
$$766$$ 38250.0 1.80421
$$767$$ −960.000 −0.0451937
$$768$$ 0 0
$$769$$ −2765.00 −0.129660 −0.0648299 0.997896i $$-0.520650\pi$$
−0.0648299 + 0.997896i $$0.520650\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ 1493.00 0.0696039
$$773$$ −14046.0 −0.653557 −0.326778 0.945101i $$-0.605963\pi$$
−0.326778 + 0.945101i $$0.605963\pi$$
$$774$$ 0 0
$$775$$ −29348.0 −1.36027
$$776$$ 10563.0 0.488646
$$777$$ 0 0
$$778$$ −9378.00 −0.432156
$$779$$ 5760.00 0.264921
$$780$$ 0 0
$$781$$ 5130.00 0.235039
$$782$$ 21168.0 0.967987
$$783$$ 0 0
$$784$$ 0 0
$$785$$ −588.000 −0.0267345
$$786$$ 0 0
$$787$$ 18514.0 0.838568 0.419284 0.907855i $$-0.362281\pi$$
0.419284 + 0.907855i $$0.362281\pi$$
$$788$$ 4086.00 0.184718
$$789$$ 0 0
$$790$$ −4203.00 −0.189286
$$791$$ 0 0
$$792$$ 0 0
$$793$$ 7552.00 0.338183
$$794$$ 17796.0 0.795411
$$795$$ 0 0
$$796$$ 3556.00 0.158341
$$797$$ −27495.0 −1.22199 −0.610993 0.791636i $$-0.709230\pi$$
−0.610993 + 0.791636i $$0.709230\pi$$
$$798$$ 0 0
$$799$$ −2520.00 −0.111578
$$800$$ 5220.00 0.230694
$$801$$ 0 0
$$802$$ −4824.00 −0.212396
$$803$$ −5430.00 −0.238631
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 48576.0 2.12285
$$807$$ 0 0
$$808$$ 22806.0 0.992961
$$809$$ 7944.00 0.345236 0.172618 0.984989i $$-0.444777\pi$$
0.172618 + 0.984989i $$0.444777\pi$$
$$810$$ 0 0
$$811$$ 28942.0 1.25313 0.626567 0.779368i $$-0.284460\pi$$
0.626567 + 0.779368i $$0.284460\pi$$
$$812$$ 0 0
$$813$$ 0 0
$$814$$ −14220.0 −0.612298
$$815$$ 3756.00 0.161432
$$816$$ 0 0
$$817$$ 416.000 0.0178140
$$818$$ 13395.0 0.572549
$$819$$ 0 0
$$820$$ −1080.00 −0.0459942
$$821$$ −8187.00 −0.348025 −0.174012 0.984743i $$-0.555673\pi$$
−0.174012 + 0.984743i $$0.555673\pi$$
$$822$$ 0 0
$$823$$ −280.000 −0.0118593 −0.00592964 0.999982i $$-0.501887\pi$$
−0.00592964 + 0.999982i $$0.501887\pi$$
$$824$$ 36456.0 1.54127
$$825$$ 0 0
$$826$$ 0 0
$$827$$ −25317.0 −1.06452 −0.532260 0.846581i $$-0.678657\pi$$
−0.532260 + 0.846581i $$0.678657\pi$$
$$828$$ 0 0
$$829$$ −15320.0 −0.641840 −0.320920 0.947106i $$-0.603992\pi$$
−0.320920 + 0.947106i $$0.603992\pi$$
$$830$$ −4293.00 −0.179533
$$831$$ 0 0
$$832$$ 27712.0 1.15474
$$833$$ 0 0
$$834$$ 0 0
$$835$$ 7938.00 0.328989
$$836$$ 240.000 0.00992892
$$837$$ 0 0
$$838$$ −4752.00 −0.195889
$$839$$ 34092.0 1.40284 0.701422 0.712746i $$-0.252549\pi$$
0.701422 + 0.712746i $$0.252549\pi$$
$$840$$ 0 0
$$841$$ 63820.0 2.61675
$$842$$ −3990.00 −0.163307
$$843$$ 0 0
$$844$$ 1250.00 0.0509796
$$845$$ −5697.00 −0.231932
$$846$$ 0 0
$$847$$ 0 0
$$848$$ 25773.0 1.04369
$$849$$ 0 0
$$850$$ −29232.0 −1.17959
$$851$$ −26544.0 −1.06923
$$852$$ 0 0
$$853$$ 7378.00 0.296152 0.148076 0.988976i $$-0.452692\pi$$
0.148076 + 0.988976i $$0.452692\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ −28413.0 −1.13451
$$857$$ −15594.0 −0.621565 −0.310782 0.950481i $$-0.600591\pi$$
−0.310782 + 0.950481i $$0.600591\pi$$
$$858$$ 0 0
$$859$$ 30538.0 1.21297 0.606486 0.795094i $$-0.292579\pi$$
0.606486 + 0.795094i $$0.292579\pi$$
$$860$$ −78.0000 −0.00309277
$$861$$ 0 0
$$862$$ −28764.0 −1.13655
$$863$$ 822.000 0.0324232 0.0162116 0.999869i $$-0.494839\pi$$
0.0162116 + 0.999869i $$0.494839\pi$$
$$864$$ 0 0
$$865$$ 2358.00 0.0926872
$$866$$ −1482.00 −0.0581529
$$867$$ 0 0
$$868$$ 0 0
$$869$$ 7005.00 0.273450
$$870$$ 0 0
$$871$$ −23680.0 −0.921201
$$872$$ 7770.00 0.301749
$$873$$ 0 0
$$874$$ 4032.00 0.156046
$$875$$ 0 0
$$876$$ 0 0
$$877$$ −41824.0 −1.61037 −0.805186 0.593022i $$-0.797935\pi$$
−0.805186 + 0.593022i $$0.797935\pi$$
$$878$$ 48027.0 1.84605
$$879$$ 0 0
$$880$$ 3195.00 0.122390
$$881$$ −46098.0 −1.76286 −0.881431 0.472313i $$-0.843419\pi$$
−0.881431 + 0.472313i $$0.843419\pi$$
$$882$$ 0 0
$$883$$ 21008.0 0.800652 0.400326 0.916373i $$-0.368897\pi$$
0.400326 + 0.916373i $$0.368897\pi$$
$$884$$ 5376.00 0.204541
$$885$$ 0 0
$$886$$ −23319.0 −0.884218
$$887$$ 24036.0 0.909865 0.454932 0.890526i $$-0.349663\pi$$
0.454932 + 0.890526i $$0.349663\pi$$
$$888$$ 0 0
$$889$$ 0 0
$$890$$ −8154.00 −0.307104
$$891$$ 0 0
$$892$$ −425.000 −0.0159530
$$893$$ −480.000 −0.0179872
$$894$$ 0 0
$$895$$ 8676.00 0.324030
$$896$$ 0 0
$$897$$ 0 0
$$898$$ −2592.00 −0.0963209
$$899$$ 75141.0 2.78764
$$900$$ 0 0
$$901$$ −30492.0 −1.12745
$$902$$ 16200.0 0.598006
$$903$$ 0 0
$$904$$ −13608.0 −0.500659
$$905$$ 4056.00 0.148979
$$906$$ 0 0
$$907$$ 13292.0 0.486608 0.243304 0.969950i $$-0.421769\pi$$
0.243304 + 0.969950i $$0.421769\pi$$
$$908$$ 3855.00 0.140895
$$909$$ 0 0
$$910$$ 0 0
$$911$$ 9306.00 0.338443 0.169221 0.985578i $$-0.445875\pi$$
0.169221 + 0.985578i $$0.445875\pi$$
$$912$$ 0 0
$$913$$ 7155.00 0.259360
$$914$$ 7557.00 0.273483
$$915$$ 0 0
$$916$$ 2188.00 0.0789231
$$917$$ 0 0
$$918$$ 0 0
$$919$$ 16496.0 0.592114 0.296057 0.955170i $$-0.404328\pi$$
0.296057 + 0.955170i $$0.404328\pi$$
$$920$$ 5292.00 0.189644
$$921$$ 0 0
$$922$$ −1026.00 −0.0366481
$$923$$ 21888.0 0.780555
$$924$$ 0 0
$$925$$ 36656.0 1.30296
$$926$$ −13008.0 −0.461630
$$927$$ 0 0
$$928$$ −13365.0 −0.472767
$$929$$ 14154.0 0.499868 0.249934 0.968263i $$-0.419591\pi$$
0.249934 + 0.968263i $$0.419591\pi$$
$$930$$ 0 0
$$931$$ 0 0
$$932$$ −852.000 −0.0299444
$$933$$ 0 0
$$934$$ 55908.0 1.95864
$$935$$ −3780.00 −0.132213
$$936$$ 0 0
$$937$$ 3781.00 0.131825 0.0659124 0.997825i $$-0.479004\pi$$
0.0659124 + 0.997825i $$0.479004\pi$$
$$938$$ 0 0
$$939$$ 0 0
$$940$$ 90.0000 0.00312285
$$941$$ −25863.0 −0.895972 −0.447986 0.894041i $$-0.647859\pi$$
−0.447986 + 0.894041i $$0.647859\pi$$
$$942$$ 0 0
$$943$$ 30240.0 1.04427
$$944$$ 1065.00 0.0367191
$$945$$ 0 0
$$946$$ 1170.00 0.0402114
$$947$$ 42384.0 1.45438 0.727188 0.686438i $$-0.240827\pi$$
0.727188 + 0.686438i $$0.240827\pi$$
$$948$$ 0 0
$$949$$ −23168.0 −0.792482
$$950$$ −5568.00 −0.190158
$$951$$ 0 0
$$952$$ 0 0
$$953$$ −10530.0 −0.357923 −0.178961 0.983856i $$-0.557274\pi$$
−0.178961 + 0.983856i $$0.557274\pi$$
$$954$$ 0 0
$$955$$ 11736.0 0.397663
$$956$$ −5508.00 −0.186340
$$957$$ 0 0
$$958$$ 45234.0 1.52552
$$959$$ 0 0
$$960$$ 0 0
$$961$$ 34218.0 1.14860
$$962$$ −60672.0 −2.03341
$$963$$ 0 0
$$964$$ −791.000 −0.0264278
$$965$$ −4479.00 −0.149414
$$966$$ 0 0
$$967$$ −38341.0 −1.27504 −0.637520 0.770434i $$-0.720040\pi$$
−0.637520 + 0.770434i $$0.720040\pi$$
$$968$$ 23226.0 0.771190
$$969$$ 0 0
$$970$$ 4527.00 0.149849
$$971$$ 1923.00 0.0635551 0.0317776 0.999495i $$-0.489883\pi$$
0.0317776 + 0.999495i $$0.489883\pi$$
$$972$$ 0 0
$$973$$ 0 0
$$974$$ 18663.0 0.613964
$$975$$ 0 0
$$976$$ −8378.00 −0.274768
$$977$$ −57090.0 −1.86947 −0.934734 0.355347i $$-0.884363\pi$$
−0.934734 + 0.355347i $$0.884363\pi$$
$$978$$ 0 0
$$979$$ 13590.0 0.443655
$$980$$ 0 0
$$981$$ 0 0
$$982$$ 22113.0 0.718589
$$983$$ 5484.00 0.177937 0.0889687 0.996034i $$-0.471643\pi$$
0.0889687 + 0.996034i $$0.471643\pi$$
$$984$$ 0 0
$$985$$ −12258.0 −0.396520
$$986$$ 74844.0 2.41736
$$987$$ 0 0
$$988$$ 1024.00 0.0329735
$$989$$ 2184.00 0.0702196
$$990$$ 0 0
$$991$$ −22465.0 −0.720105 −0.360053 0.932932i $$-0.617241\pi$$
−0.360053 + 0.932932i $$0.617241\pi$$
$$992$$ −11385.0 −0.364389
$$993$$ 0 0
$$994$$ 0 0
$$995$$ −10668.0 −0.339898
$$996$$ 0 0
$$997$$ −29366.0 −0.932829 −0.466415 0.884566i $$-0.654454\pi$$
−0.466415 + 0.884566i $$0.654454\pi$$
$$998$$ 12822.0 0.406687
$$999$$ 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 441.4.a.k.1.1 1
3.2 odd 2 147.4.a.a.1.1 1
7.2 even 3 441.4.e.c.361.1 2
7.3 odd 6 63.4.e.a.37.1 2
7.4 even 3 441.4.e.c.226.1 2
7.5 odd 6 63.4.e.a.46.1 2
7.6 odd 2 441.4.a.l.1.1 1
12.11 even 2 2352.4.a.bd.1.1 1
21.2 odd 6 147.4.e.h.67.1 2
21.5 even 6 21.4.e.a.4.1 2
21.11 odd 6 147.4.e.h.79.1 2
21.17 even 6 21.4.e.a.16.1 yes 2
21.20 even 2 147.4.a.b.1.1 1
84.47 odd 6 336.4.q.e.193.1 2
84.59 odd 6 336.4.q.e.289.1 2
84.83 odd 2 2352.4.a.i.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
21.4.e.a.4.1 2 21.5 even 6
21.4.e.a.16.1 yes 2 21.17 even 6
63.4.e.a.37.1 2 7.3 odd 6
63.4.e.a.46.1 2 7.5 odd 6
147.4.a.a.1.1 1 3.2 odd 2
147.4.a.b.1.1 1 21.20 even 2
147.4.e.h.67.1 2 21.2 odd 6
147.4.e.h.79.1 2 21.11 odd 6
336.4.q.e.193.1 2 84.47 odd 6
336.4.q.e.289.1 2 84.59 odd 6
441.4.a.k.1.1 1 1.1 even 1 trivial
441.4.a.l.1.1 1 7.6 odd 2
441.4.e.c.226.1 2 7.4 even 3
441.4.e.c.361.1 2 7.2 even 3
2352.4.a.i.1.1 1 84.83 odd 2
2352.4.a.bd.1.1 1 12.11 even 2