# Properties

 Label 147.4.a.a.1.1 Level $147$ Weight $4$ Character 147.1 Self dual yes Analytic conductor $8.673$ Analytic rank $1$ Dimension $1$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$147 = 3 \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 147.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$8.67328077084$$ Analytic rank: $$1$$ 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 $$\chi$$ $$=$$ 147.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q-3.00000 q^{2} -3.00000 q^{3} +1.00000 q^{4} +3.00000 q^{5} +9.00000 q^{6} +21.0000 q^{8} +9.00000 q^{9} +O(q^{10})$$ $$q-3.00000 q^{2} -3.00000 q^{3} +1.00000 q^{4} +3.00000 q^{5} +9.00000 q^{6} +21.0000 q^{8} +9.00000 q^{9} -9.00000 q^{10} -15.0000 q^{11} -3.00000 q^{12} +64.0000 q^{13} -9.00000 q^{15} -71.0000 q^{16} -84.0000 q^{17} -27.0000 q^{18} +16.0000 q^{19} +3.00000 q^{20} +45.0000 q^{22} -84.0000 q^{23} -63.0000 q^{24} -116.000 q^{25} -192.000 q^{26} -27.0000 q^{27} -297.000 q^{29} +27.0000 q^{30} +253.000 q^{31} +45.0000 q^{32} +45.0000 q^{33} +252.000 q^{34} +9.00000 q^{36} -316.000 q^{37} -48.0000 q^{38} -192.000 q^{39} +63.0000 q^{40} -360.000 q^{41} +26.0000 q^{43} -15.0000 q^{44} +27.0000 q^{45} +252.000 q^{46} +30.0000 q^{47} +213.000 q^{48} +348.000 q^{50} +252.000 q^{51} +64.0000 q^{52} +363.000 q^{53} +81.0000 q^{54} -45.0000 q^{55} -48.0000 q^{57} +891.000 q^{58} +15.0000 q^{59} -9.00000 q^{60} +118.000 q^{61} -759.000 q^{62} +433.000 q^{64} +192.000 q^{65} -135.000 q^{66} -370.000 q^{67} -84.0000 q^{68} +252.000 q^{69} -342.000 q^{71} +189.000 q^{72} -362.000 q^{73} +948.000 q^{74} +348.000 q^{75} +16.0000 q^{76} +576.000 q^{78} +467.000 q^{79} -213.000 q^{80} +81.0000 q^{81} +1080.00 q^{82} -477.000 q^{83} -252.000 q^{85} -78.0000 q^{86} +891.000 q^{87} -315.000 q^{88} -906.000 q^{89} -81.0000 q^{90} -84.0000 q^{92} -759.000 q^{93} -90.0000 q^{94} +48.0000 q^{95} -135.000 q^{96} -503.000 q^{97} -135.000 q^{99} +O(q^{100})$$

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ −3.00000 −1.06066 −0.530330 0.847791i $$-0.677932\pi$$
−0.530330 + 0.847791i $$0.677932\pi$$
$$3$$ −3.00000 −0.577350
$$4$$ 1.00000 0.125000
$$5$$ 3.00000 0.268328 0.134164 0.990959i $$-0.457165\pi$$
0.134164 + 0.990959i $$0.457165\pi$$
$$6$$ 9.00000 0.612372
$$7$$ 0 0
$$8$$ 21.0000 0.928078
$$9$$ 9.00000 0.333333
$$10$$ −9.00000 −0.284605
$$11$$ −15.0000 −0.411152 −0.205576 0.978641i $$-0.565907\pi$$
−0.205576 + 0.978641i $$0.565907\pi$$
$$12$$ −3.00000 −0.0721688
$$13$$ 64.0000 1.36542 0.682708 0.730691i $$-0.260802\pi$$
0.682708 + 0.730691i $$0.260802\pi$$
$$14$$ 0 0
$$15$$ −9.00000 −0.154919
$$16$$ −71.0000 −1.10938
$$17$$ −84.0000 −1.19841 −0.599206 0.800595i $$-0.704517\pi$$
−0.599206 + 0.800595i $$0.704517\pi$$
$$18$$ −27.0000 −0.353553
$$19$$ 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 $$-0.624339\pi$$
−0.380765 + 0.924672i $$0.624339\pi$$
$$24$$ −63.0000 −0.535826
$$25$$ −116.000 −0.928000
$$26$$ −192.000 −1.44824
$$27$$ −27.0000 −0.192450
$$28$$ 0 0
$$29$$ −297.000 −1.90178 −0.950888 0.309535i $$-0.899827\pi$$
−0.950888 + 0.309535i $$0.899827\pi$$
$$30$$ 27.0000 0.164317
$$31$$ 253.000 1.46581 0.732906 0.680330i $$-0.238164\pi$$
0.732906 + 0.680330i $$0.238164\pi$$
$$32$$ 45.0000 0.248592
$$33$$ 45.0000 0.237379
$$34$$ 252.000 1.27111
$$35$$ 0 0
$$36$$ 9.00000 0.0416667
$$37$$ −316.000 −1.40406 −0.702028 0.712149i $$-0.747722\pi$$
−0.702028 + 0.712149i $$0.747722\pi$$
$$38$$ −48.0000 −0.204911
$$39$$ −192.000 −0.788323
$$40$$ 63.0000 0.249029
$$41$$ −360.000 −1.37128 −0.685641 0.727940i $$-0.740478\pi$$
−0.685641 + 0.727940i $$0.740478\pi$$
$$42$$ 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
$$45$$ 27.0000 0.0894427
$$46$$ 252.000 0.807725
$$47$$ 30.0000 0.0931053 0.0465527 0.998916i $$-0.485176\pi$$
0.0465527 + 0.998916i $$0.485176\pi$$
$$48$$ 213.000 0.640498
$$49$$ 0 0
$$50$$ 348.000 0.984293
$$51$$ 252.000 0.691903
$$52$$ 64.0000 0.170677
$$53$$ 363.000 0.940790 0.470395 0.882456i $$-0.344111\pi$$
0.470395 + 0.882456i $$0.344111\pi$$
$$54$$ 81.0000 0.204124
$$55$$ −45.0000 −0.110324
$$56$$ 0 0
$$57$$ −48.0000 −0.111540
$$58$$ 891.000 2.01714
$$59$$ 15.0000 0.0330989 0.0165494 0.999863i $$-0.494732\pi$$
0.0165494 + 0.999863i $$0.494732\pi$$
$$60$$ −9.00000 −0.0193649
$$61$$ 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
$$66$$ −135.000 −0.251778
$$67$$ −370.000 −0.674667 −0.337334 0.941385i $$-0.609525\pi$$
−0.337334 + 0.941385i $$0.609525\pi$$
$$68$$ −84.0000 −0.149801
$$69$$ 252.000 0.439670
$$70$$ 0 0
$$71$$ −342.000 −0.571661 −0.285831 0.958280i $$-0.592269\pi$$
−0.285831 + 0.958280i $$0.592269\pi$$
$$72$$ 189.000 0.309359
$$73$$ −362.000 −0.580396 −0.290198 0.956967i $$-0.593721\pi$$
−0.290198 + 0.956967i $$0.593721\pi$$
$$74$$ 948.000 1.48923
$$75$$ 348.000 0.535781
$$76$$ 16.0000 0.0241490
$$77$$ 0 0
$$78$$ 576.000 0.836143
$$79$$ 467.000 0.665084 0.332542 0.943089i $$-0.392094\pi$$
0.332542 + 0.943089i $$0.392094\pi$$
$$80$$ −213.000 −0.297677
$$81$$ 81.0000 0.111111
$$82$$ 1080.00 1.45446
$$83$$ −477.000 −0.630814 −0.315407 0.948957i $$-0.602141\pi$$
−0.315407 + 0.948957i $$0.602141\pi$$
$$84$$ 0 0
$$85$$ −252.000 −0.321568
$$86$$ −78.0000 −0.0978018
$$87$$ 891.000 1.09799
$$88$$ −315.000 −0.381581
$$89$$ −906.000 −1.07905 −0.539527 0.841968i $$-0.681397\pi$$
−0.539527 + 0.841968i $$0.681397\pi$$
$$90$$ −81.0000 −0.0948683
$$91$$ 0 0
$$92$$ −84.0000 −0.0951914
$$93$$ −759.000 −0.846286
$$94$$ −90.0000 −0.0987531
$$95$$ 48.0000 0.0518389
$$96$$ −135.000 −0.143525
$$97$$ −503.000 −0.526515 −0.263257 0.964726i $$-0.584797\pi$$
−0.263257 + 0.964726i $$0.584797\pi$$
$$98$$ 0 0
$$99$$ −135.000 −0.137051
$$100$$ −116.000 −0.116000
$$101$$ 1086.00 1.06991 0.534956 0.844880i $$-0.320328\pi$$
0.534956 + 0.844880i $$0.320328\pi$$
$$102$$ −756.000 −0.733874
$$103$$ −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 $$-0.709318\pi$$
−0.611212 + 0.791467i $$0.709318\pi$$
$$108$$ −27.0000 −0.0240563
$$109$$ −370.000 −0.325134 −0.162567 0.986698i $$-0.551977\pi$$
−0.162567 + 0.986698i $$0.551977\pi$$
$$110$$ 135.000 0.117016
$$111$$ 948.000 0.810632
$$112$$ 0 0
$$113$$ −648.000 −0.539458 −0.269729 0.962936i $$-0.586934\pi$$
−0.269729 + 0.962936i $$0.586934\pi$$
$$114$$ 144.000 0.118306
$$115$$ −252.000 −0.204340
$$116$$ −297.000 −0.237722
$$117$$ 576.000 0.455139
$$118$$ −45.0000 −0.0351067
$$119$$ 0 0
$$120$$ −189.000 −0.143777
$$121$$ −1106.00 −0.830954
$$122$$ −354.000 −0.262702
$$123$$ 1080.00 0.791710
$$124$$ 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
$$129$$ −78.0000 −0.0532366
$$130$$ −576.000 −0.388604
$$131$$ 651.000 0.434184 0.217092 0.976151i $$-0.430343\pi$$
0.217092 + 0.976151i $$0.430343\pi$$
$$132$$ 45.0000 0.0296723
$$133$$ 0 0
$$134$$ 1110.00 0.715593
$$135$$ −81.0000 −0.0516398
$$136$$ −1764.00 −1.11222
$$137$$ −1770.00 −1.10381 −0.551903 0.833909i $$-0.686098\pi$$
−0.551903 + 0.833909i $$0.686098\pi$$
$$138$$ −756.000 −0.466341
$$139$$ 1558.00 0.950704 0.475352 0.879796i $$-0.342321\pi$$
0.475352 + 0.879796i $$0.342321\pi$$
$$140$$ 0 0
$$141$$ −90.0000 −0.0537544
$$142$$ 1026.00 0.606338
$$143$$ −960.000 −0.561393
$$144$$ −639.000 −0.369792
$$145$$ −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 $$-0.264304\pi$$
0.674629 + 0.738157i $$0.264304\pi$$
$$150$$ −1044.00 −0.568282
$$151$$ 1259.00 0.678516 0.339258 0.940693i $$-0.389824\pi$$
0.339258 + 0.940693i $$0.389824\pi$$
$$152$$ 336.000 0.179297
$$153$$ −756.000 −0.399470
$$154$$ 0 0
$$155$$ 759.000 0.393318
$$156$$ −192.000 −0.0985404
$$157$$ 196.000 0.0996338 0.0498169 0.998758i $$-0.484136\pi$$
0.0498169 + 0.998758i $$0.484136\pi$$
$$158$$ −1401.00 −0.705428
$$159$$ −1089.00 −0.543166
$$160$$ 135.000 0.0667043
$$161$$ 0 0
$$162$$ −243.000 −0.117851
$$163$$ −1252.00 −0.601621 −0.300810 0.953684i $$-0.597257\pi$$
−0.300810 + 0.953684i $$0.597257\pi$$
$$164$$ −360.000 −0.171410
$$165$$ 135.000 0.0636954
$$166$$ 1431.00 0.669079
$$167$$ 2646.00 1.22607 0.613035 0.790056i $$-0.289949\pi$$
0.613035 + 0.790056i $$0.289949\pi$$
$$168$$ 0 0
$$169$$ 1899.00 0.864360
$$170$$ 756.000 0.341074
$$171$$ 144.000 0.0643974
$$172$$ 26.0000 0.0115261
$$173$$ 786.000 0.345425 0.172712 0.984972i $$-0.444747\pi$$
0.172712 + 0.984972i $$0.444747\pi$$
$$174$$ −2673.00 −1.16460
$$175$$ 0 0
$$176$$ 1065.00 0.456122
$$177$$ −45.0000 −0.0191096
$$178$$ 2718.00 1.14451
$$179$$ 2892.00 1.20759 0.603794 0.797140i $$-0.293655\pi$$
0.603794 + 0.797140i $$0.293655\pi$$
$$180$$ 27.0000 0.0111803
$$181$$ −1352.00 −0.555212 −0.277606 0.960695i $$-0.589541\pi$$
−0.277606 + 0.960695i $$0.589541\pi$$
$$182$$ 0 0
$$183$$ −354.000 −0.142997
$$184$$ −1764.00 −0.706760
$$185$$ −948.000 −0.376748
$$186$$ 2277.00 0.897622
$$187$$ 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 $$-0.234351\pi$$
0.741001 + 0.671504i $$0.234351\pi$$
$$192$$ −1299.00 −0.488267
$$193$$ 1493.00 0.556832 0.278416 0.960461i $$-0.410191\pi$$
0.278416 + 0.960461i $$0.410191\pi$$
$$194$$ 1509.00 0.558453
$$195$$ −576.000 −0.211529
$$196$$ 0 0
$$197$$ −4086.00 −1.47774 −0.738872 0.673846i $$-0.764641\pi$$
−0.738872 + 0.673846i $$0.764641\pi$$
$$198$$ 405.000 0.145364
$$199$$ 3556.00 1.26672 0.633362 0.773855i $$-0.281674\pi$$
0.633362 + 0.773855i $$0.281674\pi$$
$$200$$ −2436.00 −0.861256
$$201$$ 1110.00 0.389519
$$202$$ −3258.00 −1.13481
$$203$$ 0 0
$$204$$ 252.000 0.0864879
$$205$$ −1080.00 −0.367954
$$206$$ 5208.00 1.76145
$$207$$ −756.000 −0.253844
$$208$$ −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
$$213$$ 1026.00 0.330049
$$214$$ 4059.00 1.29658
$$215$$ 78.0000 0.0247421
$$216$$ −567.000 −0.178609
$$217$$ 0 0
$$218$$ 1110.00 0.344856
$$219$$ 1086.00 0.335092
$$220$$ −45.0000 −0.0137905
$$221$$ −5376.00 −1.63633
$$222$$ −2844.00 −0.859805
$$223$$ −425.000 −0.127624 −0.0638119 0.997962i $$-0.520326\pi$$
−0.0638119 + 0.997962i $$0.520326\pi$$
$$224$$ 0 0
$$225$$ −1044.00 −0.309333
$$226$$ 1944.00 0.572181
$$227$$ −3855.00 −1.12716 −0.563580 0.826061i $$-0.690576\pi$$
−0.563580 + 0.826061i $$0.690576\pi$$
$$228$$ −48.0000 −0.0139424
$$229$$ 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 $$-0.461782\pi$$
0.119778 + 0.992801i $$0.461782\pi$$
$$234$$ −1728.00 −0.482747
$$235$$ 90.0000 0.0249828
$$236$$ 15.0000 0.00413736
$$237$$ −1401.00 −0.383986
$$238$$ 0 0
$$239$$ 5508.00 1.49072 0.745362 0.666660i $$-0.232277\pi$$
0.745362 + 0.666660i $$0.232277\pi$$
$$240$$ 639.000 0.171864
$$241$$ −791.000 −0.211422 −0.105711 0.994397i $$-0.533712\pi$$
−0.105711 + 0.994397i $$0.533712\pi$$
$$242$$ 3318.00 0.881360
$$243$$ −243.000 −0.0641500
$$244$$ 118.000 0.0309597
$$245$$ 0 0
$$246$$ −3240.00 −0.839735
$$247$$ 1024.00 0.263788
$$248$$ 5313.00 1.36039
$$249$$ 1431.00 0.364201
$$250$$ 2169.00 0.548718
$$251$$ −5265.00 −1.32400 −0.662000 0.749504i $$-0.730292\pi$$
−0.662000 + 0.749504i $$0.730292\pi$$
$$252$$ 0 0
$$253$$ 1260.00 0.313105
$$254$$ −1131.00 −0.279391
$$255$$ 756.000 0.185657
$$256$$ 1513.00 0.369385
$$257$$ 6870.00 1.66747 0.833733 0.552168i $$-0.186199\pi$$
0.833733 + 0.552168i $$0.186199\pi$$
$$258$$ 234.000 0.0564659
$$259$$ 0 0
$$260$$ 192.000 0.0457974
$$261$$ −2673.00 −0.633925
$$262$$ −1953.00 −0.460522
$$263$$ −222.000 −0.0520498 −0.0260249 0.999661i $$-0.508285\pi$$
−0.0260249 + 0.999661i $$0.508285\pi$$
$$264$$ 945.000 0.220306
$$265$$ 1089.00 0.252441
$$266$$ 0 0
$$267$$ 2718.00 0.622992
$$268$$ −370.000 −0.0843334
$$269$$ −7851.00 −1.77949 −0.889747 0.456454i $$-0.849119\pi$$
−0.889747 + 0.456454i $$0.849119\pi$$
$$270$$ 243.000 0.0547723
$$271$$ −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
$$276$$ 252.000 0.0549588
$$277$$ −4960.00 −1.07588 −0.537938 0.842985i $$-0.680796\pi$$
−0.537938 + 0.842985i $$0.680796\pi$$
$$278$$ −4674.00 −1.00837
$$279$$ 2277.00 0.488604
$$280$$ 0 0
$$281$$ −774.000 −0.164317 −0.0821583 0.996619i $$-0.526181\pi$$
−0.0821583 + 0.996619i $$0.526181\pi$$
$$282$$ 270.000 0.0570151
$$283$$ −3698.00 −0.776761 −0.388380 0.921499i $$-0.626965\pi$$
−0.388380 + 0.921499i $$0.626965\pi$$
$$284$$ −342.000 −0.0714576
$$285$$ −144.000 −0.0299292
$$286$$ 2880.00 0.595447
$$287$$ 0 0
$$288$$ 405.000 0.0828641
$$289$$ 2143.00 0.436190
$$290$$ 2673.00 0.541255
$$291$$ 1509.00 0.303983
$$292$$ −362.000 −0.0725495
$$293$$ 6273.00 1.25076 0.625380 0.780321i $$-0.284944\pi$$
0.625380 + 0.780321i $$0.284944\pi$$
$$294$$ 0 0
$$295$$ 45.0000 0.00888136
$$296$$ −6636.00 −1.30307
$$297$$ 405.000 0.0791262
$$298$$ −7362.00 −1.43110
$$299$$ −5376.00 −1.03981
$$300$$ 348.000 0.0669726
$$301$$ 0 0
$$302$$ −3777.00 −0.719675
$$303$$ −3258.00 −0.617714
$$304$$ −1136.00 −0.214323
$$305$$ 354.000 0.0664590
$$306$$ 2268.00 0.423702
$$307$$ 1684.00 0.313065 0.156533 0.987673i $$-0.449968\pi$$
0.156533 + 0.987673i $$0.449968\pi$$
$$308$$ 0 0
$$309$$ 5208.00 0.958812
$$310$$ −2277.00 −0.417177
$$311$$ 1320.00 0.240676 0.120338 0.992733i $$-0.461602\pi$$
0.120338 + 0.992733i $$0.461602\pi$$
$$312$$ −4032.00 −0.731625
$$313$$ 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 $$-0.573315\pi$$
−0.228295 + 0.973592i $$0.573315\pi$$
$$318$$ 3267.00 0.576114
$$319$$ 4455.00 0.781919
$$320$$ 1299.00 0.226926
$$321$$ 4059.00 0.705767
$$322$$ 0 0
$$323$$ −1344.00 −0.231524
$$324$$ 81.0000 0.0138889
$$325$$ −7424.00 −1.26711
$$326$$ 3756.00 0.638115
$$327$$ 1110.00 0.187716
$$328$$ −7560.00 −1.27266
$$329$$ 0 0
$$330$$ −405.000 −0.0675591
$$331$$ −484.000 −0.0803717 −0.0401859 0.999192i $$-0.512795\pi$$
−0.0401859 + 0.999192i $$0.512795\pi$$
$$332$$ −477.000 −0.0788517
$$333$$ −2844.00 −0.468019
$$334$$ −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
$$339$$ 1944.00 0.311456
$$340$$ −252.000 −0.0401959
$$341$$ −3795.00 −0.602671
$$342$$ −432.000 −0.0683038
$$343$$ 0 0
$$344$$ 546.000 0.0855766
$$345$$ 756.000 0.117976
$$346$$ −2358.00 −0.366378
$$347$$ −1860.00 −0.287752 −0.143876 0.989596i $$-0.545957\pi$$
−0.143876 + 0.989596i $$0.545957\pi$$
$$348$$ 891.000 0.137249
$$349$$ 1918.00 0.294178 0.147089 0.989123i $$-0.453010\pi$$
0.147089 + 0.989123i $$0.453010\pi$$
$$350$$ 0 0
$$351$$ −1728.00 −0.262774
$$352$$ −675.000 −0.102209
$$353$$ 3048.00 0.459571 0.229786 0.973241i $$-0.426197\pi$$
0.229786 + 0.973241i $$0.426197\pi$$
$$354$$ 135.000 0.0202688
$$355$$ −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 $$-0.500702\pi$$
−0.00220521 + 0.999998i $$0.500702\pi$$
$$360$$ 567.000 0.0830098
$$361$$ −6603.00 −0.962677
$$362$$ 4056.00 0.588891
$$363$$ 3318.00 0.479752
$$364$$ 0 0
$$365$$ −1086.00 −0.155737
$$366$$ 1062.00 0.151671
$$367$$ 11311.0 1.60880 0.804400 0.594088i $$-0.202487\pi$$
0.804400 + 0.594088i $$0.202487\pi$$
$$368$$ 5964.00 0.844823
$$369$$ −3240.00 −0.457094
$$370$$ 2844.00 0.399601
$$371$$ 0 0
$$372$$ −759.000 −0.105786
$$373$$ 1208.00 0.167689 0.0838443 0.996479i $$-0.473280\pi$$
0.0838443 + 0.996479i $$0.473280\pi$$
$$374$$ −3780.00 −0.522618
$$375$$ 2169.00 0.298684
$$376$$ 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
$$381$$ −1131.00 −0.152081
$$382$$ −11736.0 −1.57190
$$383$$ −12750.0 −1.70103 −0.850515 0.525951i $$-0.823710\pi$$
−0.850515 + 0.525951i $$0.823710\pi$$
$$384$$ 4977.00 0.661410
$$385$$ 0 0
$$386$$ −4479.00 −0.590609
$$387$$ 234.000 0.0307361
$$388$$ −503.000 −0.0658143
$$389$$ 3126.00 0.407441 0.203720 0.979029i $$-0.434697\pi$$
0.203720 + 0.979029i $$0.434697\pi$$
$$390$$ 1728.00 0.224361
$$391$$ 7056.00 0.912627
$$392$$ 0 0
$$393$$ −1953.00 −0.250676
$$394$$ 12258.0 1.56738
$$395$$ 1401.00 0.178461
$$396$$ −135.000 −0.0171313
$$397$$ 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 $$-0.468076\pi$$
0.100124 + 0.994975i $$0.468076\pi$$
$$402$$ −3330.00 −0.413148
$$403$$ 16192.0 2.00144
$$404$$ 1086.00 0.133739
$$405$$ 243.000 0.0298142
$$406$$ 0 0
$$407$$ 4740.00 0.577280
$$408$$ 5292.00 0.642140
$$409$$ 4465.00 0.539805 0.269902 0.962888i $$-0.413009\pi$$
0.269902 + 0.962888i $$0.413009\pi$$
$$410$$ 3240.00 0.390274
$$411$$ 5310.00 0.637282
$$412$$ −1736.00 −0.207589
$$413$$ 0 0
$$414$$ 2268.00 0.269242
$$415$$ −1431.00 −0.169265
$$416$$ 2880.00 0.339432
$$417$$ −4674.00 −0.548889
$$418$$ 720.000 0.0842496
$$419$$ 1584.00 0.184686 0.0923430 0.995727i $$-0.470564\pi$$
0.0923430 + 0.995727i $$0.470564\pi$$
$$420$$ 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
$$423$$ 270.000 0.0310351
$$424$$ 7623.00 0.873126
$$425$$ 9744.00 1.11213
$$426$$ −3078.00 −0.350069
$$427$$ 0 0
$$428$$ −1353.00 −0.152803
$$429$$ 2880.00 0.324121
$$430$$ −234.000 −0.0262430
$$431$$ 9588.00 1.07155 0.535775 0.844361i $$-0.320020\pi$$
0.535775 + 0.844361i $$0.320020\pi$$
$$432$$ 1917.00 0.213499
$$433$$ −494.000 −0.0548271 −0.0274135 0.999624i $$-0.508727\pi$$
−0.0274135 + 0.999624i $$0.508727\pi$$
$$434$$ 0 0
$$435$$ 2673.00 0.294622
$$436$$ −370.000 −0.0406417
$$437$$ −1344.00 −0.147122
$$438$$ −3258.00 −0.355418
$$439$$ 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 $$-0.363143\pi$$
0.416824 + 0.908987i $$0.363143\pi$$
$$444$$ 948.000 0.101329
$$445$$ −2718.00 −0.289541
$$446$$ 1275.00 0.135365
$$447$$ −7362.00 −0.778995
$$448$$ 0 0
$$449$$ 864.000 0.0908122 0.0454061 0.998969i $$-0.485542\pi$$
0.0454061 + 0.998969i $$0.485542\pi$$
$$450$$ 3132.00 0.328098
$$451$$ 5400.00 0.563805
$$452$$ −648.000 −0.0674322
$$453$$ −3777.00 −0.391742
$$454$$ 11565.0 1.19553
$$455$$ 0 0
$$456$$ −1008.00 −0.103517
$$457$$ 2519.00 0.257842 0.128921 0.991655i $$-0.458849\pi$$
0.128921 + 0.991655i $$0.458849\pi$$
$$458$$ −6564.00 −0.669685
$$459$$ 2268.00 0.230634
$$460$$ −252.000 −0.0255425
$$461$$ 342.000 0.0345521 0.0172761 0.999851i $$-0.494501\pi$$
0.0172761 + 0.999851i $$0.494501\pi$$
$$462$$ 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
$$465$$ −2277.00 −0.227082
$$466$$ −2556.00 −0.254087
$$467$$ −18636.0 −1.84662 −0.923310 0.384056i $$-0.874527\pi$$
−0.923310 + 0.384056i $$0.874527\pi$$
$$468$$ 576.000 0.0568923
$$469$$ 0 0
$$470$$ −270.000 −0.0264982
$$471$$ −588.000 −0.0575236
$$472$$ 315.000 0.0307183
$$473$$ −390.000 −0.0379117
$$474$$ 4203.00 0.407279
$$475$$ −1856.00 −0.179282
$$476$$ 0 0
$$477$$ 3267.00 0.313597
$$478$$ −16524.0 −1.58115
$$479$$ −15078.0 −1.43827 −0.719135 0.694870i $$-0.755462\pi$$
−0.719135 + 0.694870i $$0.755462\pi$$
$$480$$ −405.000 −0.0385117
$$481$$ −20224.0 −1.91712
$$482$$ 2373.00 0.224247
$$483$$ 0 0
$$484$$ −1106.00 −0.103869
$$485$$ −1509.00 −0.141279
$$486$$ 729.000 0.0680414
$$487$$ 6221.00 0.578851 0.289425 0.957201i $$-0.406536\pi$$
0.289425 + 0.957201i $$0.406536\pi$$
$$488$$ 2478.00 0.229864
$$489$$ 3756.00 0.347346
$$490$$ 0 0
$$491$$ −7371.00 −0.677492 −0.338746 0.940878i $$-0.610003\pi$$
−0.338746 + 0.940878i $$0.610003\pi$$
$$492$$ 1080.00 0.0989637
$$493$$ 24948.0 2.27911
$$494$$ −3072.00 −0.279789
$$495$$ −405.000 −0.0367745
$$496$$ −17963.0 −1.62613
$$497$$ 0 0
$$498$$ −4293.00 −0.386293
$$499$$ 4274.00 0.383428 0.191714 0.981451i $$-0.438595\pi$$
0.191714 + 0.981451i $$0.438595\pi$$
$$500$$ −723.000 −0.0646671
$$501$$ −7938.00 −0.707872
$$502$$ 15795.0 1.40431
$$503$$ 2520.00 0.223382 0.111691 0.993743i $$-0.464373\pi$$
0.111691 + 0.993743i $$0.464373\pi$$
$$504$$ 0 0
$$505$$ 3258.00 0.287087
$$506$$ −3780.00 −0.332098
$$507$$ −5697.00 −0.499039
$$508$$ 377.000 0.0329265
$$509$$ 14277.0 1.24326 0.621628 0.783313i $$-0.286472\pi$$
0.621628 + 0.783313i $$0.286472\pi$$
$$510$$ −2268.00 −0.196919
$$511$$ 0 0
$$512$$ 8733.00 0.753804
$$513$$ −432.000 −0.0371799
$$514$$ −20610.0 −1.76862
$$515$$ −5208.00 −0.445615
$$516$$ −78.0000 −0.00665457
$$517$$ −450.000 −0.0382804
$$518$$ 0 0
$$519$$ −2358.00 −0.199431
$$520$$ 4032.00 0.340029
$$521$$ 6306.00 0.530270 0.265135 0.964211i $$-0.414583\pi$$
0.265135 + 0.964211i $$0.414583\pi$$
$$522$$ 8019.00 0.672379
$$523$$ −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
$$528$$ −3195.00 −0.263342
$$529$$ −5111.00 −0.420071
$$530$$ −3267.00 −0.267754
$$531$$ 135.000 0.0110330
$$532$$ 0 0
$$533$$ −23040.0 −1.87237
$$534$$ −8154.00 −0.660783
$$535$$ −4059.00 −0.328011
$$536$$ −7770.00 −0.626143
$$537$$ −8676.00 −0.697201
$$538$$ 23553.0 1.88744
$$539$$ 0 0
$$540$$ −81.0000 −0.00645497
$$541$$ −22858.0 −1.81653 −0.908264 0.418396i $$-0.862592\pi$$
−0.908264 + 0.418396i $$0.862592\pi$$
$$542$$ 15549.0 1.23226
$$543$$ 4056.00 0.320552
$$544$$ −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
$$549$$ 1062.00 0.0825593
$$550$$ −5220.00 −0.404694
$$551$$ −4752.00 −0.367408
$$552$$ 5292.00 0.408048
$$553$$ 0 0
$$554$$ 14880.0 1.14114
$$555$$ 2844.00 0.217515
$$556$$ 1558.00 0.118838
$$557$$ −9843.00 −0.748764 −0.374382 0.927275i $$-0.622145\pi$$
−0.374382 + 0.927275i $$0.622145\pi$$
$$558$$ −6831.00 −0.518242
$$559$$ 1664.00 0.125903
$$560$$ 0 0
$$561$$ −3780.00 −0.284477
$$562$$ 2322.00 0.174284
$$563$$ 13371.0 1.00092 0.500462 0.865758i $$-0.333163\pi$$
0.500462 + 0.865758i $$0.333163\pi$$
$$564$$ −90.0000 −0.00671930
$$565$$ −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 $$-0.561737\pi$$
−0.192739 + 0.981250i $$0.561737\pi$$
$$570$$ 432.000 0.0317447
$$571$$ −14398.0 −1.05523 −0.527616 0.849483i $$-0.676914\pi$$
−0.527616 + 0.849483i $$0.676914\pi$$
$$572$$ −960.000 −0.0701742
$$573$$ −11736.0 −0.855634
$$574$$ 0 0
$$575$$ 9744.00 0.706701
$$576$$ 3897.00 0.281901
$$577$$ −19871.0 −1.43369 −0.716846 0.697231i $$-0.754415\pi$$
−0.716846 + 0.697231i $$0.754415\pi$$
$$578$$ −6429.00 −0.462649
$$579$$ −4479.00 −0.321487
$$580$$ −891.000 −0.0637875
$$581$$ 0 0
$$582$$ −4527.00 −0.322423
$$583$$ −5445.00 −0.386808
$$584$$ −7602.00 −0.538652
$$585$$ 1728.00 0.122127
$$586$$ −18819.0 −1.32663
$$587$$ 16137.0 1.13466 0.567330 0.823491i $$-0.307976\pi$$
0.567330 + 0.823491i $$0.307976\pi$$
$$588$$ 0 0
$$589$$ 4048.00 0.283183
$$590$$ −135.000 −0.00942011
$$591$$ 12258.0 0.853176
$$592$$ 22436.0 1.55762
$$593$$ 21324.0 1.47668 0.738340 0.674428i $$-0.235610\pi$$
0.738340 + 0.674428i $$0.235610\pi$$
$$594$$ −1215.00 −0.0839260
$$595$$ 0 0
$$596$$ 2454.00 0.168657
$$597$$ −10668.0 −0.731344
$$598$$ 16128.0 1.10288
$$599$$ −8646.00 −0.589760 −0.294880 0.955534i $$-0.595280\pi$$
−0.294880 + 0.955534i $$0.595280\pi$$
$$600$$ 7308.00 0.497246
$$601$$ −11195.0 −0.759823 −0.379911 0.925023i $$-0.624046\pi$$
−0.379911 + 0.925023i $$0.624046\pi$$
$$602$$ 0 0
$$603$$ −3330.00 −0.224889
$$604$$ 1259.00 0.0848145
$$605$$ −3318.00 −0.222968
$$606$$ 9774.00 0.655184
$$607$$ 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
$$612$$ −756.000 −0.0499338
$$613$$ −12772.0 −0.841527 −0.420764 0.907170i $$-0.638238\pi$$
−0.420764 + 0.907170i $$0.638238\pi$$
$$614$$ −5052.00 −0.332056
$$615$$ 3240.00 0.212438
$$616$$ 0 0
$$617$$ 12762.0 0.832705 0.416352 0.909203i $$-0.363308\pi$$
0.416352 + 0.909203i $$0.363308\pi$$
$$618$$ −15624.0 −1.01697
$$619$$ −12842.0 −0.833867 −0.416933 0.908937i $$-0.636895\pi$$
−0.416933 + 0.908937i $$0.636895\pi$$
$$620$$ 759.000 0.0491648
$$621$$ 2268.00 0.146557
$$622$$ −3960.00 −0.255276
$$623$$ 0 0
$$624$$ 13632.0 0.874546
$$625$$ 12331.0 0.789184
$$626$$ −25509.0 −1.62867
$$627$$ 720.000 0.0458597
$$628$$ 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
$$633$$ −3750.00 −0.235465
$$634$$ 7731.00 0.484286
$$635$$ 1131.00 0.0706809
$$636$$ −1089.00 −0.0678957
$$637$$ 0 0
$$638$$ −13365.0 −0.829350
$$639$$ −3078.00 −0.190554
$$640$$ −4977.00 −0.307396
$$641$$ 8274.00 0.509834 0.254917 0.966963i $$-0.417952\pi$$
0.254917 + 0.966963i $$0.417952\pi$$
$$642$$ −12177.0 −0.748579
$$643$$ −27998.0 −1.71716 −0.858580 0.512680i $$-0.828653\pi$$
−0.858580 + 0.512680i $$0.828653\pi$$
$$644$$ 0 0
$$645$$ −234.000 −0.0142849
$$646$$ 4032.00 0.245568
$$647$$ 17466.0 1.06130 0.530649 0.847592i $$-0.321948\pi$$
0.530649 + 0.847592i $$0.321948\pi$$
$$648$$ 1701.00 0.103120
$$649$$ −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 $$-0.479413\pi$$
0.0646324 + 0.997909i $$0.479413\pi$$
$$654$$ −3330.00 −0.199103
$$655$$ 1953.00 0.116504
$$656$$ 25560.0 1.52127
$$657$$ −3258.00 −0.193465
$$658$$ 0 0
$$659$$ 19944.0 1.17892 0.589460 0.807798i $$-0.299341\pi$$
0.589460 + 0.807798i $$0.299341\pi$$
$$660$$ 135.000 0.00796192
$$661$$ −27506.0 −1.61855 −0.809273 0.587432i $$-0.800139\pi$$
−0.809273 + 0.587432i $$0.800139\pi$$
$$662$$ 1452.00 0.0852471
$$663$$ 16128.0 0.944735
$$664$$ −10017.0 −0.585444
$$665$$ 0 0
$$666$$ 8532.00 0.496409
$$667$$ 24948.0 1.44826
$$668$$ 2646.00 0.153259
$$669$$ 1275.00 0.0736836
$$670$$ 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
$$675$$ 3132.00 0.178594
$$676$$ 1899.00 0.108045
$$677$$ −13857.0 −0.786658 −0.393329 0.919398i $$-0.628677\pi$$
−0.393329 + 0.919398i $$0.628677\pi$$
$$678$$ −5832.00 −0.330349
$$679$$ 0 0
$$680$$ −5292.00 −0.298440
$$681$$ 11565.0 0.650766
$$682$$ 11385.0 0.639229
$$683$$ −22245.0 −1.24624 −0.623120 0.782127i $$-0.714135\pi$$
−0.623120 + 0.782127i $$0.714135\pi$$
$$684$$ 144.000 0.00804967
$$685$$ −5310.00 −0.296182
$$686$$ 0 0
$$687$$ −6564.00 −0.364530
$$688$$ −1846.00 −0.102294
$$689$$ 23232.0 1.28457
$$690$$ −2268.00 −0.125132
$$691$$ 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
$$696$$ 18711.0 1.01902
$$697$$ 30240.0 1.64336
$$698$$ −5754.00 −0.312023
$$699$$ −2556.00 −0.138307
$$700$$ 0 0
$$701$$ −15561.0 −0.838418 −0.419209 0.907890i $$-0.637693\pi$$
−0.419209 + 0.907890i $$0.637693\pi$$
$$702$$ 5184.00 0.278714
$$703$$ −5056.00 −0.271253
$$704$$ −6495.00 −0.347712
$$705$$ −270.000 −0.0144238
$$706$$ −9144.00 −0.487449
$$707$$ 0 0
$$708$$ −45.0000 −0.00238871
$$709$$ 5534.00 0.293136 0.146568 0.989201i $$-0.453177\pi$$
0.146568 + 0.989201i $$0.453177\pi$$
$$710$$ 3078.00 0.162698
$$711$$ 4203.00 0.221695
$$712$$ −19026.0 −1.00145
$$713$$ −21252.0 −1.11626
$$714$$ 0 0
$$715$$ −2880.00 −0.150638
$$716$$ 2892.00 0.150948
$$717$$ −16524.0 −0.860670
$$718$$ 90.0000 0.00467795
$$719$$ 21846.0 1.13313 0.566564 0.824018i $$-0.308273\pi$$
0.566564 + 0.824018i $$0.308273\pi$$
$$720$$ −1917.00 −0.0992255
$$721$$ 0 0
$$722$$ 19809.0 1.02107
$$723$$ 2373.00 0.122065
$$724$$ −1352.00 −0.0694015
$$725$$ 34452.0 1.76485
$$726$$ −9954.00 −0.508853
$$727$$ 11089.0 0.565706 0.282853 0.959163i $$-0.408719\pi$$
0.282853 + 0.959163i $$0.408719\pi$$
$$728$$ 0 0
$$729$$ 729.000 0.0370370
$$730$$ 3258.00 0.165184
$$731$$ −2184.00 −0.110504
$$732$$ −354.000 −0.0178746
$$733$$ −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
$$738$$ 9720.00 0.484821
$$739$$ −22726.0 −1.13124 −0.565622 0.824665i $$-0.691364\pi$$
−0.565622 + 0.824665i $$0.691364\pi$$
$$740$$ −948.000 −0.0470935
$$741$$ −3072.00 −0.152298
$$742$$ 0 0
$$743$$ 6678.00 0.329734 0.164867 0.986316i $$-0.447281\pi$$
0.164867 + 0.986316i $$0.447281\pi$$
$$744$$ −15939.0 −0.785419
$$745$$ 7362.00 0.362044
$$746$$ −3624.00 −0.177861
$$747$$ −4293.00 −0.210271
$$748$$ 1260.00 0.0615911
$$749$$ 0 0
$$750$$ −6507.00 −0.316803
$$751$$ −19987.0 −0.971153 −0.485577 0.874194i $$-0.661390\pi$$
−0.485577 + 0.874194i $$0.661390\pi$$
$$752$$ −2130.00 −0.103289
$$753$$ 15795.0 0.764411
$$754$$ 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
$$759$$ −3780.00 −0.180771
$$760$$ 1008.00 0.0481105
$$761$$ 11496.0 0.547608 0.273804 0.961786i $$-0.411718\pi$$
0.273804 + 0.961786i $$0.411718\pi$$
$$762$$ 3393.00 0.161306
$$763$$ 0 0
$$764$$ 3912.00 0.185250
$$765$$ −2268.00 −0.107189
$$766$$ 38250.0 1.80421
$$767$$ 960.000 0.0451937
$$768$$ −4539.00 −0.213264
$$769$$ −2765.00 −0.129660 −0.0648299 0.997896i $$-0.520650\pi$$
−0.0648299 + 0.997896i $$0.520650\pi$$
$$770$$ 0 0
$$771$$ −20610.0 −0.962712
$$772$$ 1493.00 0.0696039
$$773$$ 14046.0 0.653557 0.326778 0.945101i $$-0.394037\pi$$
0.326778 + 0.945101i $$0.394037\pi$$
$$774$$ −702.000 −0.0326006
$$775$$ −29348.0 −1.36027
$$776$$ −10563.0 −0.488646
$$777$$ 0 0
$$778$$ −9378.00 −0.432156
$$779$$ −5760.00 −0.264921
$$780$$ −576.000 −0.0264412
$$781$$ 5130.00 0.235039
$$782$$ −21168.0 −0.967987
$$783$$ 8019.00 0.365997
$$784$$ 0 0
$$785$$ 588.000 0.0267345
$$786$$ 5859.00 0.265882
$$787$$ 18514.0 0.838568 0.419284 0.907855i $$-0.362281\pi$$
0.419284 + 0.907855i $$0.362281\pi$$
$$788$$ −4086.00 −0.184718
$$789$$ 666.000 0.0300510
$$790$$ −4203.00 −0.189286
$$791$$ 0 0
$$792$$ −2835.00 −0.127194
$$793$$ 7552.00 0.338183
$$794$$ −17796.0 −0.795411
$$795$$ −3267.00 −0.145747
$$796$$ 3556.00 0.158341
$$797$$ 27495.0 1.22199 0.610993 0.791636i $$-0.290770\pi$$
0.610993 + 0.791636i $$0.290770\pi$$
$$798$$ 0 0
$$799$$ −2520.00 −0.111578
$$800$$ −5220.00 −0.230694
$$801$$ −8154.00 −0.359685
$$802$$ −4824.00 −0.212396
$$803$$ 5430.00 0.238631
$$804$$ 1110.00 0.0486899
$$805$$ 0 0
$$806$$ −48576.0 −2.12285
$$807$$ 23553.0 1.02739
$$808$$ 22806.0 0.992961
$$809$$ −7944.00 −0.345236 −0.172618 0.984989i $$-0.555223\pi$$
−0.172618 + 0.984989i $$0.555223\pi$$
$$810$$ −729.000 −0.0316228
$$811$$ 28942.0 1.25313 0.626567 0.779368i $$-0.284460\pi$$
0.626567 + 0.779368i $$0.284460\pi$$
$$812$$ 0 0
$$813$$ 15549.0 0.670759
$$814$$ −14220.0 −0.612298
$$815$$ −3756.00 −0.161432
$$816$$ −17892.0 −0.767580
$$817$$ 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 $$-0.444327\pi$$
0.174012 + 0.984743i $$0.444327\pi$$
$$822$$ −15930.0 −0.675940
$$823$$ −280.000 −0.0118593 −0.00592964 0.999982i $$-0.501887\pi$$
−0.00592964 + 0.999982i $$0.501887\pi$$
$$824$$ −36456.0 −1.54127
$$825$$ −5220.00 −0.220287
$$826$$ 0 0
$$827$$ 25317.0 1.06452 0.532260 0.846581i $$-0.321343\pi$$
0.532260 + 0.846581i $$0.321343\pi$$
$$828$$ −756.000 −0.0317305
$$829$$ −15320.0 −0.641840 −0.320920 0.947106i $$-0.603992\pi$$
−0.320920 + 0.947106i $$0.603992\pi$$
$$830$$ 4293.00 0.179533
$$831$$ 14880.0 0.621157
$$832$$ 27712.0 1.15474
$$833$$ 0 0
$$834$$ 14022.0 0.582185
$$835$$ 7938.00 0.328989
$$836$$ −240.000 −0.00992892
$$837$$ −6831.00 −0.282095
$$838$$ −4752.00 −0.195889
$$839$$ −34092.0 −1.40284 −0.701422 0.712746i $$-0.747451\pi$$
−0.701422 + 0.712746i $$0.747451\pi$$
$$840$$ 0 0
$$841$$ 63820.0 2.61675
$$842$$ 3990.00 0.163307
$$843$$ 2322.00 0.0948682
$$844$$ 1250.00 0.0509796
$$845$$ 5697.00 0.231932
$$846$$ −810.000 −0.0329177
$$847$$ 0 0
$$848$$ −25773.0 −1.04369
$$849$$ 11094.0 0.448463
$$850$$ −29232.0 −1.17959
$$851$$ 26544.0 1.06923
$$852$$ 1026.00 0.0412561
$$853$$ 7378.00 0.296152 0.148076 0.988976i $$-0.452692\pi$$
0.148076 + 0.988976i $$0.452692\pi$$
$$854$$ 0 0
$$855$$ 432.000 0.0172796
$$856$$ −28413.0 −1.13451
$$857$$ 15594.0 0.621565 0.310782 0.950481i $$-0.399409\pi$$
0.310782 + 0.950481i $$0.399409\pi$$
$$858$$ −8640.00 −0.343782
$$859$$ 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 $$-0.505161\pi$$
−0.0162116 + 0.999869i $$0.505161\pi$$
$$864$$ −1215.00 −0.0478416
$$865$$ 2358.00 0.0926872
$$866$$ 1482.00 0.0581529
$$867$$ −6429.00 −0.251834
$$868$$ 0 0
$$869$$ −7005.00 −0.273450
$$870$$ −8019.00 −0.312494
$$871$$ −23680.0 −0.921201
$$872$$ −7770.00 −0.301749
$$873$$ −4527.00 −0.175505
$$874$$ 4032.00 0.156046
$$875$$ 0 0
$$876$$ 1086.00 0.0418865
$$877$$ −41824.0 −1.61037 −0.805186 0.593022i $$-0.797935\pi$$
−0.805186 + 0.593022i $$0.797935\pi$$
$$878$$ −48027.0 −1.84605
$$879$$ −18819.0 −0.722126
$$880$$ 3195.00 0.122390
$$881$$ 46098.0 1.76286 0.881431 0.472313i $$-0.156581\pi$$
0.881431 + 0.472313i $$0.156581\pi$$
$$882$$ 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
$$885$$ −135.000 −0.00512766
$$886$$ −23319.0 −0.884218
$$887$$ −24036.0 −0.909865 −0.454932 0.890526i $$-0.650337\pi$$
−0.454932 + 0.890526i $$0.650337\pi$$
$$888$$ 19908.0 0.752330
$$889$$ 0 0
$$890$$ 8154.00 0.307104
$$891$$ −1215.00 −0.0456835
$$892$$ −425.000 −0.0159530
$$893$$ 480.000 0.0179872
$$894$$ 22086.0 0.826249
$$895$$ 8676.00 0.324030
$$896$$ 0 0
$$897$$ 16128.0 0.600332
$$898$$ −2592.00 −0.0963209
$$899$$ −75141.0 −2.78764
$$900$$ −1044.00 −0.0386667
$$901$$ −30492.0 −1.12745
$$902$$ −16200.0 −0.598006
$$903$$ 0 0
$$904$$ −13608.0 −0.500659
$$905$$ −4056.00 −0.148979
$$906$$ 11331.0 0.415505
$$907$$ 13292.0 0.486608 0.243304 0.969950i $$-0.421769\pi$$
0.243304 + 0.969950i $$0.421769\pi$$
$$908$$ −3855.00 −0.140895
$$909$$ 9774.00 0.356637
$$910$$ 0 0
$$911$$ −9306.00 −0.338443 −0.169221 0.985578i $$-0.554125\pi$$
−0.169221 + 0.985578i $$0.554125\pi$$
$$912$$ 3408.00 0.123739
$$913$$ 7155.00 0.259360
$$914$$ −7557.00 −0.273483
$$915$$ −1062.00 −0.0383701
$$916$$ 2188.00 0.0789231
$$917$$ 0 0
$$918$$ −6804.00 −0.244625
$$919$$ 16496.0 0.592114 0.296057 0.955170i $$-0.404328\pi$$
0.296057 + 0.955170i $$0.404328\pi$$
$$920$$ −5292.00 −0.189644
$$921$$ −5052.00 −0.180748
$$922$$ −1026.00 −0.0366481
$$923$$ −21888.0 −0.780555
$$924$$ 0 0
$$925$$ 36656.0 1.30296
$$926$$ 13008.0 0.461630
$$927$$ −15624.0 −0.553570
$$928$$ −13365.0 −0.472767
$$929$$ −14154.0 −0.499868 −0.249934 0.968263i $$-0.580409\pi$$
−0.249934 + 0.968263i $$0.580409\pi$$
$$930$$ 6831.00 0.240857
$$931$$ 0 0
$$932$$ 852.000 0.0299444
$$933$$ −3960.00 −0.138955
$$934$$ 55908.0 1.95864
$$935$$ 3780.00 0.132213
$$936$$ 12096.0 0.422404
$$937$$ 3781.00 0.131825 0.0659124 0.997825i $$-0.479004\pi$$
0.0659124 + 0.997825i $$0.479004\pi$$
$$938$$ 0 0
$$939$$ −25509.0 −0.886533
$$940$$ 90.0000 0.00312285
$$941$$ 25863.0 0.895972 0.447986 0.894041i $$-0.352141\pi$$
0.447986 + 0.894041i $$0.352141\pi$$
$$942$$ 1764.00 0.0610130
$$943$$ 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 $$-0.759173\pi$$
−0.727188 + 0.686438i $$0.759173\pi$$
$$948$$ −1401.00 −0.0479983
$$949$$ −23168.0 −0.792482
$$950$$ 5568.00 0.190158
$$951$$ 7731.00 0.263612
$$952$$ 0 0
$$953$$ 10530.0 0.357923 0.178961 0.983856i $$-0.442726\pi$$
0.178961 + 0.983856i $$0.442726\pi$$
$$954$$ −9801.00 −0.332620
$$955$$ 11736.0 0.397663
$$956$$ 5508.00 0.186340
$$957$$ −13365.0 −0.451441
$$958$$ 45234.0 1.52552
$$959$$ 0 0
$$960$$ −3897.00 −0.131016
$$961$$ 34218.0 1.14860
$$962$$ 60672.0 2.03341
$$963$$ −12177.0 −0.407475
$$964$$ −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
$$969$$ 4032.00 0.133670
$$970$$ 4527.00 0.149849
$$971$$ −1923.00 −0.0635551 −0.0317776 0.999495i $$-0.510117\pi$$
−0.0317776 + 0.999495i $$0.510117\pi$$
$$972$$ −243.000 −0.00801875
$$973$$ 0 0
$$974$$ −18663.0 −0.613964
$$975$$ 22272.0 0.731564
$$976$$ −8378.00 −0.274768
$$977$$ 57090.0 1.86947 0.934734 0.355347i $$-0.115637\pi$$
0.934734 + 0.355347i $$0.115637\pi$$
$$978$$ −11268.0 −0.368416
$$979$$ 13590.0 0.443655
$$980$$ 0 0
$$981$$ −3330.00 −0.108378
$$982$$ 22113.0 0.718589
$$983$$ −5484.00 −0.177937 −0.0889687 0.996034i $$-0.528357\pi$$
−0.0889687 + 0.996034i $$0.528357\pi$$
$$984$$ 22680.0 0.734768
$$985$$ −12258.0 −0.396520
$$986$$ −74844.0 −2.41736
$$987$$ 0 0
$$988$$ 1024.00 0.0329735
$$989$$ −2184.00 −0.0702196
$$990$$ 1215.00 0.0390053
$$991$$ −22465.0 −0.720105 −0.360053 0.932932i $$-0.617241\pi$$
−0.360053 + 0.932932i $$0.617241\pi$$
$$992$$ 11385.0 0.364389
$$993$$ 1452.00 0.0464026
$$994$$ 0 0
$$995$$ 10668.0 0.339898
$$996$$ 1431.00 0.0455251
$$997$$ −29366.0 −0.932829 −0.466415 0.884566i $$-0.654454\pi$$
−0.466415 + 0.884566i $$0.654454\pi$$
$$998$$ −12822.0 −0.406687
$$999$$ 8532.00 0.270211
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 147.4.a.a.1.1 1
3.2 odd 2 441.4.a.k.1.1 1
4.3 odd 2 2352.4.a.bd.1.1 1
7.2 even 3 147.4.e.h.67.1 2
7.3 odd 6 21.4.e.a.16.1 yes 2
7.4 even 3 147.4.e.h.79.1 2
7.5 odd 6 21.4.e.a.4.1 2
7.6 odd 2 147.4.a.b.1.1 1
21.2 odd 6 441.4.e.c.361.1 2
21.5 even 6 63.4.e.a.46.1 2
21.11 odd 6 441.4.e.c.226.1 2
21.17 even 6 63.4.e.a.37.1 2
21.20 even 2 441.4.a.l.1.1 1
28.3 even 6 336.4.q.e.289.1 2
28.19 even 6 336.4.q.e.193.1 2
28.27 even 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 7.5 odd 6
21.4.e.a.16.1 yes 2 7.3 odd 6
63.4.e.a.37.1 2 21.17 even 6
63.4.e.a.46.1 2 21.5 even 6
147.4.a.a.1.1 1 1.1 even 1 trivial
147.4.a.b.1.1 1 7.6 odd 2
147.4.e.h.67.1 2 7.2 even 3
147.4.e.h.79.1 2 7.4 even 3
336.4.q.e.193.1 2 28.19 even 6
336.4.q.e.289.1 2 28.3 even 6
441.4.a.k.1.1 1 3.2 odd 2
441.4.a.l.1.1 1 21.20 even 2
441.4.e.c.226.1 2 21.11 odd 6
441.4.e.c.361.1 2 21.2 odd 6
2352.4.a.i.1.1 1 28.27 even 2
2352.4.a.bd.1.1 1 4.3 odd 2