# Properties

 Label 135.4.a.b.1.1 Level $135$ Weight $4$ Character 135.1 Self dual yes Analytic conductor $7.965$ Analytic rank $0$ Dimension $1$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$135 = 3^{3} \cdot 5$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 135.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$7.96525785077$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: yes Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Character $$\chi$$ $$=$$ 135.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q-1.00000 q^{2} -7.00000 q^{4} -5.00000 q^{5} -6.00000 q^{7} +15.0000 q^{8} +O(q^{10})$$ $$q-1.00000 q^{2} -7.00000 q^{4} -5.00000 q^{5} -6.00000 q^{7} +15.0000 q^{8} +5.00000 q^{10} +47.0000 q^{11} -5.00000 q^{13} +6.00000 q^{14} +41.0000 q^{16} +131.000 q^{17} -56.0000 q^{19} +35.0000 q^{20} -47.0000 q^{22} -3.00000 q^{23} +25.0000 q^{25} +5.00000 q^{26} +42.0000 q^{28} +157.000 q^{29} +225.000 q^{31} -161.000 q^{32} -131.000 q^{34} +30.0000 q^{35} -70.0000 q^{37} +56.0000 q^{38} -75.0000 q^{40} -140.000 q^{41} +397.000 q^{43} -329.000 q^{44} +3.00000 q^{46} +347.000 q^{47} -307.000 q^{49} -25.0000 q^{50} +35.0000 q^{52} -4.00000 q^{53} -235.000 q^{55} -90.0000 q^{56} -157.000 q^{58} -748.000 q^{59} -338.000 q^{61} -225.000 q^{62} -167.000 q^{64} +25.0000 q^{65} +492.000 q^{67} -917.000 q^{68} -30.0000 q^{70} -32.0000 q^{71} +970.000 q^{73} +70.0000 q^{74} +392.000 q^{76} -282.000 q^{77} -1257.00 q^{79} -205.000 q^{80} +140.000 q^{82} +102.000 q^{83} -655.000 q^{85} -397.000 q^{86} +705.000 q^{88} +1488.00 q^{89} +30.0000 q^{91} +21.0000 q^{92} -347.000 q^{94} +280.000 q^{95} +974.000 q^{97} +307.000 q^{98} +O(q^{100})$$

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ −1.00000 −0.353553 −0.176777 0.984251i $$-0.556567\pi$$
−0.176777 + 0.984251i $$0.556567\pi$$
$$3$$ 0 0
$$4$$ −7.00000 −0.875000
$$5$$ −5.00000 −0.447214
$$6$$ 0 0
$$7$$ −6.00000 −0.323970 −0.161985 0.986793i $$-0.551790\pi$$
−0.161985 + 0.986793i $$0.551790\pi$$
$$8$$ 15.0000 0.662913
$$9$$ 0 0
$$10$$ 5.00000 0.158114
$$11$$ 47.0000 1.28828 0.644138 0.764909i $$-0.277216\pi$$
0.644138 + 0.764909i $$0.277216\pi$$
$$12$$ 0 0
$$13$$ −5.00000 −0.106673 −0.0533366 0.998577i $$-0.516986\pi$$
−0.0533366 + 0.998577i $$0.516986\pi$$
$$14$$ 6.00000 0.114541
$$15$$ 0 0
$$16$$ 41.0000 0.640625
$$17$$ 131.000 1.86895 0.934475 0.356027i $$-0.115869\pi$$
0.934475 + 0.356027i $$0.115869\pi$$
$$18$$ 0 0
$$19$$ −56.0000 −0.676173 −0.338086 0.941115i $$-0.609780\pi$$
−0.338086 + 0.941115i $$0.609780\pi$$
$$20$$ 35.0000 0.391312
$$21$$ 0 0
$$22$$ −47.0000 −0.455474
$$23$$ −3.00000 −0.0271975 −0.0135988 0.999908i $$-0.504329\pi$$
−0.0135988 + 0.999908i $$0.504329\pi$$
$$24$$ 0 0
$$25$$ 25.0000 0.200000
$$26$$ 5.00000 0.0377146
$$27$$ 0 0
$$28$$ 42.0000 0.283473
$$29$$ 157.000 1.00532 0.502658 0.864485i $$-0.332355\pi$$
0.502658 + 0.864485i $$0.332355\pi$$
$$30$$ 0 0
$$31$$ 225.000 1.30359 0.651793 0.758397i $$-0.274017\pi$$
0.651793 + 0.758397i $$0.274017\pi$$
$$32$$ −161.000 −0.889408
$$33$$ 0 0
$$34$$ −131.000 −0.660774
$$35$$ 30.0000 0.144884
$$36$$ 0 0
$$37$$ −70.0000 −0.311025 −0.155513 0.987834i $$-0.549703\pi$$
−0.155513 + 0.987834i $$0.549703\pi$$
$$38$$ 56.0000 0.239063
$$39$$ 0 0
$$40$$ −75.0000 −0.296464
$$41$$ −140.000 −0.533276 −0.266638 0.963797i $$-0.585913\pi$$
−0.266638 + 0.963797i $$0.585913\pi$$
$$42$$ 0 0
$$43$$ 397.000 1.40795 0.703976 0.710224i $$-0.251406\pi$$
0.703976 + 0.710224i $$0.251406\pi$$
$$44$$ −329.000 −1.12724
$$45$$ 0 0
$$46$$ 3.00000 0.00961578
$$47$$ 347.000 1.07692 0.538459 0.842652i $$-0.319007\pi$$
0.538459 + 0.842652i $$0.319007\pi$$
$$48$$ 0 0
$$49$$ −307.000 −0.895044
$$50$$ −25.0000 −0.0707107
$$51$$ 0 0
$$52$$ 35.0000 0.0933390
$$53$$ −4.00000 −0.0103668 −0.00518342 0.999987i $$-0.501650\pi$$
−0.00518342 + 0.999987i $$0.501650\pi$$
$$54$$ 0 0
$$55$$ −235.000 −0.576134
$$56$$ −90.0000 −0.214763
$$57$$ 0 0
$$58$$ −157.000 −0.355433
$$59$$ −748.000 −1.65053 −0.825265 0.564745i $$-0.808974\pi$$
−0.825265 + 0.564745i $$0.808974\pi$$
$$60$$ 0 0
$$61$$ −338.000 −0.709450 −0.354725 0.934971i $$-0.615426\pi$$
−0.354725 + 0.934971i $$0.615426\pi$$
$$62$$ −225.000 −0.460888
$$63$$ 0 0
$$64$$ −167.000 −0.326172
$$65$$ 25.0000 0.0477057
$$66$$ 0 0
$$67$$ 492.000 0.897125 0.448562 0.893751i $$-0.351936\pi$$
0.448562 + 0.893751i $$0.351936\pi$$
$$68$$ −917.000 −1.63533
$$69$$ 0 0
$$70$$ −30.0000 −0.0512241
$$71$$ −32.0000 −0.0534888 −0.0267444 0.999642i $$-0.508514\pi$$
−0.0267444 + 0.999642i $$0.508514\pi$$
$$72$$ 0 0
$$73$$ 970.000 1.55520 0.777602 0.628757i $$-0.216436\pi$$
0.777602 + 0.628757i $$0.216436\pi$$
$$74$$ 70.0000 0.109964
$$75$$ 0 0
$$76$$ 392.000 0.591651
$$77$$ −282.000 −0.417362
$$78$$ 0 0
$$79$$ −1257.00 −1.79017 −0.895086 0.445894i $$-0.852886\pi$$
−0.895086 + 0.445894i $$0.852886\pi$$
$$80$$ −205.000 −0.286496
$$81$$ 0 0
$$82$$ 140.000 0.188542
$$83$$ 102.000 0.134891 0.0674455 0.997723i $$-0.478515\pi$$
0.0674455 + 0.997723i $$0.478515\pi$$
$$84$$ 0 0
$$85$$ −655.000 −0.835820
$$86$$ −397.000 −0.497786
$$87$$ 0 0
$$88$$ 705.000 0.854014
$$89$$ 1488.00 1.77222 0.886111 0.463474i $$-0.153397\pi$$
0.886111 + 0.463474i $$0.153397\pi$$
$$90$$ 0 0
$$91$$ 30.0000 0.0345588
$$92$$ 21.0000 0.0237978
$$93$$ 0 0
$$94$$ −347.000 −0.380748
$$95$$ 280.000 0.302394
$$96$$ 0 0
$$97$$ 974.000 1.01953 0.509767 0.860313i $$-0.329732\pi$$
0.509767 + 0.860313i $$0.329732\pi$$
$$98$$ 307.000 0.316446
$$99$$ 0 0
$$100$$ −175.000 −0.175000
$$101$$ 1335.00 1.31522 0.657611 0.753357i $$-0.271567\pi$$
0.657611 + 0.753357i $$0.271567\pi$$
$$102$$ 0 0
$$103$$ 686.000 0.656248 0.328124 0.944635i $$-0.393584\pi$$
0.328124 + 0.944635i $$0.393584\pi$$
$$104$$ −75.0000 −0.0707150
$$105$$ 0 0
$$106$$ 4.00000 0.00366523
$$107$$ 1098.00 0.992034 0.496017 0.868313i $$-0.334795\pi$$
0.496017 + 0.868313i $$0.334795\pi$$
$$108$$ 0 0
$$109$$ −700.000 −0.615118 −0.307559 0.951529i $$-0.599512\pi$$
−0.307559 + 0.951529i $$0.599512\pi$$
$$110$$ 235.000 0.203694
$$111$$ 0 0
$$112$$ −246.000 −0.207543
$$113$$ 1055.00 0.878284 0.439142 0.898418i $$-0.355283\pi$$
0.439142 + 0.898418i $$0.355283\pi$$
$$114$$ 0 0
$$115$$ 15.0000 0.0121631
$$116$$ −1099.00 −0.879652
$$117$$ 0 0
$$118$$ 748.000 0.583551
$$119$$ −786.000 −0.605483
$$120$$ 0 0
$$121$$ 878.000 0.659654
$$122$$ 338.000 0.250829
$$123$$ 0 0
$$124$$ −1575.00 −1.14064
$$125$$ −125.000 −0.0894427
$$126$$ 0 0
$$127$$ −1646.00 −1.15007 −0.575035 0.818129i $$-0.695012\pi$$
−0.575035 + 0.818129i $$0.695012\pi$$
$$128$$ 1455.00 1.00473
$$129$$ 0 0
$$130$$ −25.0000 −0.0168665
$$131$$ 1833.00 1.22252 0.611259 0.791430i $$-0.290663\pi$$
0.611259 + 0.791430i $$0.290663\pi$$
$$132$$ 0 0
$$133$$ 336.000 0.219059
$$134$$ −492.000 −0.317182
$$135$$ 0 0
$$136$$ 1965.00 1.23895
$$137$$ −1098.00 −0.684733 −0.342367 0.939566i $$-0.611229\pi$$
−0.342367 + 0.939566i $$0.611229\pi$$
$$138$$ 0 0
$$139$$ −1042.00 −0.635837 −0.317918 0.948118i $$-0.602984\pi$$
−0.317918 + 0.948118i $$0.602984\pi$$
$$140$$ −210.000 −0.126773
$$141$$ 0 0
$$142$$ 32.0000 0.0189111
$$143$$ −235.000 −0.137424
$$144$$ 0 0
$$145$$ −785.000 −0.449591
$$146$$ −970.000 −0.549848
$$147$$ 0 0
$$148$$ 490.000 0.272147
$$149$$ 2941.00 1.61702 0.808510 0.588482i $$-0.200274\pi$$
0.808510 + 0.588482i $$0.200274\pi$$
$$150$$ 0 0
$$151$$ 511.000 0.275395 0.137697 0.990474i $$-0.456030\pi$$
0.137697 + 0.990474i $$0.456030\pi$$
$$152$$ −840.000 −0.448243
$$153$$ 0 0
$$154$$ 282.000 0.147560
$$155$$ −1125.00 −0.582982
$$156$$ 0 0
$$157$$ −571.000 −0.290260 −0.145130 0.989413i $$-0.546360\pi$$
−0.145130 + 0.989413i $$0.546360\pi$$
$$158$$ 1257.00 0.632921
$$159$$ 0 0
$$160$$ 805.000 0.397755
$$161$$ 18.0000 0.00881117
$$162$$ 0 0
$$163$$ 713.000 0.342616 0.171308 0.985217i $$-0.445201\pi$$
0.171308 + 0.985217i $$0.445201\pi$$
$$164$$ 980.000 0.466617
$$165$$ 0 0
$$166$$ −102.000 −0.0476912
$$167$$ −1596.00 −0.739534 −0.369767 0.929125i $$-0.620563\pi$$
−0.369767 + 0.929125i $$0.620563\pi$$
$$168$$ 0 0
$$169$$ −2172.00 −0.988621
$$170$$ 655.000 0.295507
$$171$$ 0 0
$$172$$ −2779.00 −1.23196
$$173$$ −4134.00 −1.81678 −0.908388 0.418129i $$-0.862686\pi$$
−0.908388 + 0.418129i $$0.862686\pi$$
$$174$$ 0 0
$$175$$ −150.000 −0.0647939
$$176$$ 1927.00 0.825302
$$177$$ 0 0
$$178$$ −1488.00 −0.626575
$$179$$ −1828.00 −0.763302 −0.381651 0.924306i $$-0.624644\pi$$
−0.381651 + 0.924306i $$0.624644\pi$$
$$180$$ 0 0
$$181$$ −520.000 −0.213543 −0.106772 0.994284i $$-0.534051\pi$$
−0.106772 + 0.994284i $$0.534051\pi$$
$$182$$ −30.0000 −0.0122184
$$183$$ 0 0
$$184$$ −45.0000 −0.0180296
$$185$$ 350.000 0.139095
$$186$$ 0 0
$$187$$ 6157.00 2.40772
$$188$$ −2429.00 −0.942303
$$189$$ 0 0
$$190$$ −280.000 −0.106912
$$191$$ −4826.00 −1.82826 −0.914129 0.405424i $$-0.867124\pi$$
−0.914129 + 0.405424i $$0.867124\pi$$
$$192$$ 0 0
$$193$$ 1670.00 0.622846 0.311423 0.950271i $$-0.399194\pi$$
0.311423 + 0.950271i $$0.399194\pi$$
$$194$$ −974.000 −0.360459
$$195$$ 0 0
$$196$$ 2149.00 0.783163
$$197$$ 1380.00 0.499091 0.249546 0.968363i $$-0.419719\pi$$
0.249546 + 0.968363i $$0.419719\pi$$
$$198$$ 0 0
$$199$$ 4357.00 1.55206 0.776029 0.630697i $$-0.217231\pi$$
0.776029 + 0.630697i $$0.217231\pi$$
$$200$$ 375.000 0.132583
$$201$$ 0 0
$$202$$ −1335.00 −0.465001
$$203$$ −942.000 −0.325692
$$204$$ 0 0
$$205$$ 700.000 0.238488
$$206$$ −686.000 −0.232019
$$207$$ 0 0
$$208$$ −205.000 −0.0683375
$$209$$ −2632.00 −0.871097
$$210$$ 0 0
$$211$$ −4162.00 −1.35793 −0.678967 0.734169i $$-0.737572\pi$$
−0.678967 + 0.734169i $$0.737572\pi$$
$$212$$ 28.0000 0.00907098
$$213$$ 0 0
$$214$$ −1098.00 −0.350737
$$215$$ −1985.00 −0.629655
$$216$$ 0 0
$$217$$ −1350.00 −0.422322
$$218$$ 700.000 0.217477
$$219$$ 0 0
$$220$$ 1645.00 0.504118
$$221$$ −655.000 −0.199367
$$222$$ 0 0
$$223$$ −5956.00 −1.78853 −0.894267 0.447533i $$-0.852303\pi$$
−0.894267 + 0.447533i $$0.852303\pi$$
$$224$$ 966.000 0.288141
$$225$$ 0 0
$$226$$ −1055.00 −0.310520
$$227$$ −4940.00 −1.44440 −0.722201 0.691683i $$-0.756870\pi$$
−0.722201 + 0.691683i $$0.756870\pi$$
$$228$$ 0 0
$$229$$ 4344.00 1.25354 0.626768 0.779206i $$-0.284378\pi$$
0.626768 + 0.779206i $$0.284378\pi$$
$$230$$ −15.0000 −0.00430031
$$231$$ 0 0
$$232$$ 2355.00 0.666437
$$233$$ 5202.00 1.46264 0.731318 0.682036i $$-0.238905\pi$$
0.731318 + 0.682036i $$0.238905\pi$$
$$234$$ 0 0
$$235$$ −1735.00 −0.481612
$$236$$ 5236.00 1.44421
$$237$$ 0 0
$$238$$ 786.000 0.214071
$$239$$ −1546.00 −0.418420 −0.209210 0.977871i $$-0.567089\pi$$
−0.209210 + 0.977871i $$0.567089\pi$$
$$240$$ 0 0
$$241$$ −3659.00 −0.977995 −0.488998 0.872285i $$-0.662637\pi$$
−0.488998 + 0.872285i $$0.662637\pi$$
$$242$$ −878.000 −0.233223
$$243$$ 0 0
$$244$$ 2366.00 0.620769
$$245$$ 1535.00 0.400276
$$246$$ 0 0
$$247$$ 280.000 0.0721294
$$248$$ 3375.00 0.864164
$$249$$ 0 0
$$250$$ 125.000 0.0316228
$$251$$ −1221.00 −0.307047 −0.153524 0.988145i $$-0.549062\pi$$
−0.153524 + 0.988145i $$0.549062\pi$$
$$252$$ 0 0
$$253$$ −141.000 −0.0350379
$$254$$ 1646.00 0.406611
$$255$$ 0 0
$$256$$ −119.000 −0.0290527
$$257$$ 6255.00 1.51820 0.759098 0.650977i $$-0.225640\pi$$
0.759098 + 0.650977i $$0.225640\pi$$
$$258$$ 0 0
$$259$$ 420.000 0.100763
$$260$$ −175.000 −0.0417425
$$261$$ 0 0
$$262$$ −1833.00 −0.432226
$$263$$ −836.000 −0.196007 −0.0980037 0.995186i $$-0.531246\pi$$
−0.0980037 + 0.995186i $$0.531246\pi$$
$$264$$ 0 0
$$265$$ 20.0000 0.00463619
$$266$$ −336.000 −0.0774492
$$267$$ 0 0
$$268$$ −3444.00 −0.784984
$$269$$ 2231.00 0.505675 0.252837 0.967509i $$-0.418636\pi$$
0.252837 + 0.967509i $$0.418636\pi$$
$$270$$ 0 0
$$271$$ −4832.00 −1.08311 −0.541556 0.840665i $$-0.682164\pi$$
−0.541556 + 0.840665i $$0.682164\pi$$
$$272$$ 5371.00 1.19730
$$273$$ 0 0
$$274$$ 1098.00 0.242090
$$275$$ 1175.00 0.257655
$$276$$ 0 0
$$277$$ 6450.00 1.39907 0.699536 0.714597i $$-0.253390\pi$$
0.699536 + 0.714597i $$0.253390\pi$$
$$278$$ 1042.00 0.224802
$$279$$ 0 0
$$280$$ 450.000 0.0960452
$$281$$ 1050.00 0.222910 0.111455 0.993769i $$-0.464449\pi$$
0.111455 + 0.993769i $$0.464449\pi$$
$$282$$ 0 0
$$283$$ −1584.00 −0.332717 −0.166359 0.986065i $$-0.553201\pi$$
−0.166359 + 0.986065i $$0.553201\pi$$
$$284$$ 224.000 0.0468027
$$285$$ 0 0
$$286$$ 235.000 0.0485869
$$287$$ 840.000 0.172765
$$288$$ 0 0
$$289$$ 12248.0 2.49298
$$290$$ 785.000 0.158954
$$291$$ 0 0
$$292$$ −6790.00 −1.36080
$$293$$ −6594.00 −1.31476 −0.657382 0.753558i $$-0.728336\pi$$
−0.657382 + 0.753558i $$0.728336\pi$$
$$294$$ 0 0
$$295$$ 3740.00 0.738140
$$296$$ −1050.00 −0.206182
$$297$$ 0 0
$$298$$ −2941.00 −0.571703
$$299$$ 15.0000 0.00290125
$$300$$ 0 0
$$301$$ −2382.00 −0.456134
$$302$$ −511.000 −0.0973667
$$303$$ 0 0
$$304$$ −2296.00 −0.433173
$$305$$ 1690.00 0.317276
$$306$$ 0 0
$$307$$ −4343.00 −0.807388 −0.403694 0.914894i $$-0.632274\pi$$
−0.403694 + 0.914894i $$0.632274\pi$$
$$308$$ 1974.00 0.365192
$$309$$ 0 0
$$310$$ 1125.00 0.206115
$$311$$ 2124.00 0.387270 0.193635 0.981074i $$-0.437972\pi$$
0.193635 + 0.981074i $$0.437972\pi$$
$$312$$ 0 0
$$313$$ −7516.00 −1.35728 −0.678641 0.734470i $$-0.737431\pi$$
−0.678641 + 0.734470i $$0.737431\pi$$
$$314$$ 571.000 0.102622
$$315$$ 0 0
$$316$$ 8799.00 1.56640
$$317$$ 6880.00 1.21899 0.609494 0.792791i $$-0.291373\pi$$
0.609494 + 0.792791i $$0.291373\pi$$
$$318$$ 0 0
$$319$$ 7379.00 1.29512
$$320$$ 835.000 0.145868
$$321$$ 0 0
$$322$$ −18.0000 −0.00311522
$$323$$ −7336.00 −1.26373
$$324$$ 0 0
$$325$$ −125.000 −0.0213346
$$326$$ −713.000 −0.121133
$$327$$ 0 0
$$328$$ −2100.00 −0.353516
$$329$$ −2082.00 −0.348889
$$330$$ 0 0
$$331$$ −4986.00 −0.827962 −0.413981 0.910286i $$-0.635862\pi$$
−0.413981 + 0.910286i $$0.635862\pi$$
$$332$$ −714.000 −0.118030
$$333$$ 0 0
$$334$$ 1596.00 0.261465
$$335$$ −2460.00 −0.401206
$$336$$ 0 0
$$337$$ 904.000 0.146125 0.0730623 0.997327i $$-0.476723\pi$$
0.0730623 + 0.997327i $$0.476723\pi$$
$$338$$ 2172.00 0.349530
$$339$$ 0 0
$$340$$ 4585.00 0.731343
$$341$$ 10575.0 1.67938
$$342$$ 0 0
$$343$$ 3900.00 0.613936
$$344$$ 5955.00 0.933349
$$345$$ 0 0
$$346$$ 4134.00 0.642327
$$347$$ −8860.00 −1.37069 −0.685345 0.728218i $$-0.740349\pi$$
−0.685345 + 0.728218i $$0.740349\pi$$
$$348$$ 0 0
$$349$$ −4454.00 −0.683144 −0.341572 0.939856i $$-0.610959\pi$$
−0.341572 + 0.939856i $$0.610959\pi$$
$$350$$ 150.000 0.0229081
$$351$$ 0 0
$$352$$ −7567.00 −1.14580
$$353$$ 8781.00 1.32398 0.661991 0.749512i $$-0.269712\pi$$
0.661991 + 0.749512i $$0.269712\pi$$
$$354$$ 0 0
$$355$$ 160.000 0.0239209
$$356$$ −10416.0 −1.55069
$$357$$ 0 0
$$358$$ 1828.00 0.269868
$$359$$ −2928.00 −0.430457 −0.215228 0.976564i $$-0.569050\pi$$
−0.215228 + 0.976564i $$0.569050\pi$$
$$360$$ 0 0
$$361$$ −3723.00 −0.542790
$$362$$ 520.000 0.0754989
$$363$$ 0 0
$$364$$ −210.000 −0.0302390
$$365$$ −4850.00 −0.695508
$$366$$ 0 0
$$367$$ 9102.00 1.29461 0.647303 0.762233i $$-0.275897\pi$$
0.647303 + 0.762233i $$0.275897\pi$$
$$368$$ −123.000 −0.0174234
$$369$$ 0 0
$$370$$ −350.000 −0.0491774
$$371$$ 24.0000 0.00335854
$$372$$ 0 0
$$373$$ −8183.00 −1.13592 −0.567962 0.823055i $$-0.692268\pi$$
−0.567962 + 0.823055i $$0.692268\pi$$
$$374$$ −6157.00 −0.851259
$$375$$ 0 0
$$376$$ 5205.00 0.713903
$$377$$ −785.000 −0.107240
$$378$$ 0 0
$$379$$ 6136.00 0.831623 0.415812 0.909451i $$-0.363498\pi$$
0.415812 + 0.909451i $$0.363498\pi$$
$$380$$ −1960.00 −0.264594
$$381$$ 0 0
$$382$$ 4826.00 0.646386
$$383$$ −5643.00 −0.752856 −0.376428 0.926446i $$-0.622848\pi$$
−0.376428 + 0.926446i $$0.622848\pi$$
$$384$$ 0 0
$$385$$ 1410.00 0.186650
$$386$$ −1670.00 −0.220209
$$387$$ 0 0
$$388$$ −6818.00 −0.892092
$$389$$ −8991.00 −1.17188 −0.585941 0.810354i $$-0.699275\pi$$
−0.585941 + 0.810354i $$0.699275\pi$$
$$390$$ 0 0
$$391$$ −393.000 −0.0508309
$$392$$ −4605.00 −0.593336
$$393$$ 0 0
$$394$$ −1380.00 −0.176455
$$395$$ 6285.00 0.800589
$$396$$ 0 0
$$397$$ −12449.0 −1.57380 −0.786898 0.617082i $$-0.788314\pi$$
−0.786898 + 0.617082i $$0.788314\pi$$
$$398$$ −4357.00 −0.548735
$$399$$ 0 0
$$400$$ 1025.00 0.128125
$$401$$ −8076.00 −1.00573 −0.502863 0.864366i $$-0.667720\pi$$
−0.502863 + 0.864366i $$0.667720\pi$$
$$402$$ 0 0
$$403$$ −1125.00 −0.139058
$$404$$ −9345.00 −1.15082
$$405$$ 0 0
$$406$$ 942.000 0.115149
$$407$$ −3290.00 −0.400686
$$408$$ 0 0
$$409$$ −2833.00 −0.342501 −0.171250 0.985228i $$-0.554781\pi$$
−0.171250 + 0.985228i $$0.554781\pi$$
$$410$$ −700.000 −0.0843184
$$411$$ 0 0
$$412$$ −4802.00 −0.574217
$$413$$ 4488.00 0.534722
$$414$$ 0 0
$$415$$ −510.000 −0.0603251
$$416$$ 805.000 0.0948759
$$417$$ 0 0
$$418$$ 2632.00 0.307979
$$419$$ 4777.00 0.556973 0.278487 0.960440i $$-0.410167\pi$$
0.278487 + 0.960440i $$0.410167\pi$$
$$420$$ 0 0
$$421$$ −6464.00 −0.748304 −0.374152 0.927367i $$-0.622066\pi$$
−0.374152 + 0.927367i $$0.622066\pi$$
$$422$$ 4162.00 0.480102
$$423$$ 0 0
$$424$$ −60.0000 −0.00687231
$$425$$ 3275.00 0.373790
$$426$$ 0 0
$$427$$ 2028.00 0.229840
$$428$$ −7686.00 −0.868030
$$429$$ 0 0
$$430$$ 1985.00 0.222617
$$431$$ −10680.0 −1.19359 −0.596795 0.802394i $$-0.703560\pi$$
−0.596795 + 0.802394i $$0.703560\pi$$
$$432$$ 0 0
$$433$$ 11566.0 1.28366 0.641832 0.766845i $$-0.278175\pi$$
0.641832 + 0.766845i $$0.278175\pi$$
$$434$$ 1350.00 0.149314
$$435$$ 0 0
$$436$$ 4900.00 0.538228
$$437$$ 168.000 0.0183902
$$438$$ 0 0
$$439$$ −1448.00 −0.157424 −0.0787122 0.996897i $$-0.525081\pi$$
−0.0787122 + 0.996897i $$0.525081\pi$$
$$440$$ −3525.00 −0.381927
$$441$$ 0 0
$$442$$ 655.000 0.0704868
$$443$$ 2376.00 0.254824 0.127412 0.991850i $$-0.459333\pi$$
0.127412 + 0.991850i $$0.459333\pi$$
$$444$$ 0 0
$$445$$ −7440.00 −0.792561
$$446$$ 5956.00 0.632343
$$447$$ 0 0
$$448$$ 1002.00 0.105670
$$449$$ 14894.0 1.56546 0.782730 0.622362i $$-0.213827\pi$$
0.782730 + 0.622362i $$0.213827\pi$$
$$450$$ 0 0
$$451$$ −6580.00 −0.687007
$$452$$ −7385.00 −0.768498
$$453$$ 0 0
$$454$$ 4940.00 0.510673
$$455$$ −150.000 −0.0154552
$$456$$ 0 0
$$457$$ 16204.0 1.65862 0.829312 0.558786i $$-0.188733\pi$$
0.829312 + 0.558786i $$0.188733\pi$$
$$458$$ −4344.00 −0.443192
$$459$$ 0 0
$$460$$ −105.000 −0.0106427
$$461$$ 5082.00 0.513432 0.256716 0.966487i $$-0.417359\pi$$
0.256716 + 0.966487i $$0.417359\pi$$
$$462$$ 0 0
$$463$$ −10326.0 −1.03648 −0.518240 0.855235i $$-0.673412\pi$$
−0.518240 + 0.855235i $$0.673412\pi$$
$$464$$ 6437.00 0.644031
$$465$$ 0 0
$$466$$ −5202.00 −0.517120
$$467$$ −4184.00 −0.414588 −0.207294 0.978279i $$-0.566466\pi$$
−0.207294 + 0.978279i $$0.566466\pi$$
$$468$$ 0 0
$$469$$ −2952.00 −0.290641
$$470$$ 1735.00 0.170276
$$471$$ 0 0
$$472$$ −11220.0 −1.09416
$$473$$ 18659.0 1.81383
$$474$$ 0 0
$$475$$ −1400.00 −0.135235
$$476$$ 5502.00 0.529798
$$477$$ 0 0
$$478$$ 1546.00 0.147934
$$479$$ 15576.0 1.48577 0.742887 0.669417i $$-0.233456\pi$$
0.742887 + 0.669417i $$0.233456\pi$$
$$480$$ 0 0
$$481$$ 350.000 0.0331780
$$482$$ 3659.00 0.345774
$$483$$ 0 0
$$484$$ −6146.00 −0.577198
$$485$$ −4870.00 −0.455949
$$486$$ 0 0
$$487$$ 10220.0 0.950949 0.475475 0.879729i $$-0.342276\pi$$
0.475475 + 0.879729i $$0.342276\pi$$
$$488$$ −5070.00 −0.470304
$$489$$ 0 0
$$490$$ −1535.00 −0.141519
$$491$$ 2692.00 0.247430 0.123715 0.992318i $$-0.460519\pi$$
0.123715 + 0.992318i $$0.460519\pi$$
$$492$$ 0 0
$$493$$ 20567.0 1.87889
$$494$$ −280.000 −0.0255016
$$495$$ 0 0
$$496$$ 9225.00 0.835110
$$497$$ 192.000 0.0173287
$$498$$ 0 0
$$499$$ 5764.00 0.517098 0.258549 0.965998i $$-0.416756\pi$$
0.258549 + 0.965998i $$0.416756\pi$$
$$500$$ 875.000 0.0782624
$$501$$ 0 0
$$502$$ 1221.00 0.108558
$$503$$ 2437.00 0.216025 0.108012 0.994150i $$-0.465551\pi$$
0.108012 + 0.994150i $$0.465551\pi$$
$$504$$ 0 0
$$505$$ −6675.00 −0.588185
$$506$$ 141.000 0.0123878
$$507$$ 0 0
$$508$$ 11522.0 1.00631
$$509$$ 5849.00 0.509337 0.254668 0.967028i $$-0.418034\pi$$
0.254668 + 0.967028i $$0.418034\pi$$
$$510$$ 0 0
$$511$$ −5820.00 −0.503839
$$512$$ −11521.0 −0.994455
$$513$$ 0 0
$$514$$ −6255.00 −0.536763
$$515$$ −3430.00 −0.293483
$$516$$ 0 0
$$517$$ 16309.0 1.38737
$$518$$ −420.000 −0.0356250
$$519$$ 0 0
$$520$$ 375.000 0.0316247
$$521$$ 17032.0 1.43222 0.716109 0.697989i $$-0.245921\pi$$
0.716109 + 0.697989i $$0.245921\pi$$
$$522$$ 0 0
$$523$$ 4147.00 0.346722 0.173361 0.984858i $$-0.444537\pi$$
0.173361 + 0.984858i $$0.444537\pi$$
$$524$$ −12831.0 −1.06970
$$525$$ 0 0
$$526$$ 836.000 0.0692991
$$527$$ 29475.0 2.43634
$$528$$ 0 0
$$529$$ −12158.0 −0.999260
$$530$$ −20.0000 −0.00163914
$$531$$ 0 0
$$532$$ −2352.00 −0.191677
$$533$$ 700.000 0.0568862
$$534$$ 0 0
$$535$$ −5490.00 −0.443651
$$536$$ 7380.00 0.594715
$$537$$ 0 0
$$538$$ −2231.00 −0.178783
$$539$$ −14429.0 −1.15306
$$540$$ 0 0
$$541$$ −3942.00 −0.313271 −0.156636 0.987656i $$-0.550065\pi$$
−0.156636 + 0.987656i $$0.550065\pi$$
$$542$$ 4832.00 0.382938
$$543$$ 0 0
$$544$$ −21091.0 −1.66226
$$545$$ 3500.00 0.275089
$$546$$ 0 0
$$547$$ −13751.0 −1.07486 −0.537432 0.843307i $$-0.680605\pi$$
−0.537432 + 0.843307i $$0.680605\pi$$
$$548$$ 7686.00 0.599142
$$549$$ 0 0
$$550$$ −1175.00 −0.0910949
$$551$$ −8792.00 −0.679767
$$552$$ 0 0
$$553$$ 7542.00 0.579961
$$554$$ −6450.00 −0.494647
$$555$$ 0 0
$$556$$ 7294.00 0.556357
$$557$$ −7944.00 −0.604305 −0.302153 0.953260i $$-0.597705\pi$$
−0.302153 + 0.953260i $$0.597705\pi$$
$$558$$ 0 0
$$559$$ −1985.00 −0.150191
$$560$$ 1230.00 0.0928160
$$561$$ 0 0
$$562$$ −1050.00 −0.0788106
$$563$$ 6702.00 0.501697 0.250849 0.968026i $$-0.419290\pi$$
0.250849 + 0.968026i $$0.419290\pi$$
$$564$$ 0 0
$$565$$ −5275.00 −0.392780
$$566$$ 1584.00 0.117633
$$567$$ 0 0
$$568$$ −480.000 −0.0354584
$$569$$ 2760.00 0.203348 0.101674 0.994818i $$-0.467580\pi$$
0.101674 + 0.994818i $$0.467580\pi$$
$$570$$ 0 0
$$571$$ 8930.00 0.654481 0.327241 0.944941i $$-0.393881\pi$$
0.327241 + 0.944941i $$0.393881\pi$$
$$572$$ 1645.00 0.120246
$$573$$ 0 0
$$574$$ −840.000 −0.0610817
$$575$$ −75.0000 −0.00543951
$$576$$ 0 0
$$577$$ 6944.00 0.501010 0.250505 0.968115i $$-0.419403\pi$$
0.250505 + 0.968115i $$0.419403\pi$$
$$578$$ −12248.0 −0.881401
$$579$$ 0 0
$$580$$ 5495.00 0.393392
$$581$$ −612.000 −0.0437006
$$582$$ 0 0
$$583$$ −188.000 −0.0133553
$$584$$ 14550.0 1.03096
$$585$$ 0 0
$$586$$ 6594.00 0.464839
$$587$$ 4206.00 0.295741 0.147871 0.989007i $$-0.452758\pi$$
0.147871 + 0.989007i $$0.452758\pi$$
$$588$$ 0 0
$$589$$ −12600.0 −0.881450
$$590$$ −3740.00 −0.260972
$$591$$ 0 0
$$592$$ −2870.00 −0.199250
$$593$$ 6571.00 0.455040 0.227520 0.973773i $$-0.426938\pi$$
0.227520 + 0.973773i $$0.426938\pi$$
$$594$$ 0 0
$$595$$ 3930.00 0.270780
$$596$$ −20587.0 −1.41489
$$597$$ 0 0
$$598$$ −15.0000 −0.00102575
$$599$$ 9490.00 0.647330 0.323665 0.946172i $$-0.395085\pi$$
0.323665 + 0.946172i $$0.395085\pi$$
$$600$$ 0 0
$$601$$ 11861.0 0.805025 0.402513 0.915414i $$-0.368137\pi$$
0.402513 + 0.915414i $$0.368137\pi$$
$$602$$ 2382.00 0.161268
$$603$$ 0 0
$$604$$ −3577.00 −0.240970
$$605$$ −4390.00 −0.295006
$$606$$ 0 0
$$607$$ −518.000 −0.0346375 −0.0173188 0.999850i $$-0.505513\pi$$
−0.0173188 + 0.999850i $$0.505513\pi$$
$$608$$ 9016.00 0.601393
$$609$$ 0 0
$$610$$ −1690.00 −0.112174
$$611$$ −1735.00 −0.114878
$$612$$ 0 0
$$613$$ 15163.0 0.999067 0.499533 0.866295i $$-0.333505\pi$$
0.499533 + 0.866295i $$0.333505\pi$$
$$614$$ 4343.00 0.285455
$$615$$ 0 0
$$616$$ −4230.00 −0.276675
$$617$$ −19011.0 −1.24044 −0.620222 0.784426i $$-0.712958\pi$$
−0.620222 + 0.784426i $$0.712958\pi$$
$$618$$ 0 0
$$619$$ −7906.00 −0.513359 −0.256679 0.966497i $$-0.582628\pi$$
−0.256679 + 0.966497i $$0.582628\pi$$
$$620$$ 7875.00 0.510109
$$621$$ 0 0
$$622$$ −2124.00 −0.136921
$$623$$ −8928.00 −0.574146
$$624$$ 0 0
$$625$$ 625.000 0.0400000
$$626$$ 7516.00 0.479872
$$627$$ 0 0
$$628$$ 3997.00 0.253977
$$629$$ −9170.00 −0.581291
$$630$$ 0 0
$$631$$ −3416.00 −0.215513 −0.107757 0.994177i $$-0.534367\pi$$
−0.107757 + 0.994177i $$0.534367\pi$$
$$632$$ −18855.0 −1.18673
$$633$$ 0 0
$$634$$ −6880.00 −0.430977
$$635$$ 8230.00 0.514327
$$636$$ 0 0
$$637$$ 1535.00 0.0954771
$$638$$ −7379.00 −0.457896
$$639$$ 0 0
$$640$$ −7275.00 −0.449328
$$641$$ −4830.00 −0.297619 −0.148809 0.988866i $$-0.547544\pi$$
−0.148809 + 0.988866i $$0.547544\pi$$
$$642$$ 0 0
$$643$$ 12549.0 0.769649 0.384824 0.922990i $$-0.374262\pi$$
0.384824 + 0.922990i $$0.374262\pi$$
$$644$$ −126.000 −0.00770978
$$645$$ 0 0
$$646$$ 7336.00 0.446797
$$647$$ −8164.00 −0.496074 −0.248037 0.968751i $$-0.579785\pi$$
−0.248037 + 0.968751i $$0.579785\pi$$
$$648$$ 0 0
$$649$$ −35156.0 −2.12634
$$650$$ 125.000 0.00754293
$$651$$ 0 0
$$652$$ −4991.00 −0.299789
$$653$$ −23768.0 −1.42437 −0.712185 0.701992i $$-0.752294\pi$$
−0.712185 + 0.701992i $$0.752294\pi$$
$$654$$ 0 0
$$655$$ −9165.00 −0.546727
$$656$$ −5740.00 −0.341630
$$657$$ 0 0
$$658$$ 2082.00 0.123351
$$659$$ −21472.0 −1.26924 −0.634621 0.772824i $$-0.718844\pi$$
−0.634621 + 0.772824i $$0.718844\pi$$
$$660$$ 0 0
$$661$$ 12982.0 0.763905 0.381953 0.924182i $$-0.375252\pi$$
0.381953 + 0.924182i $$0.375252\pi$$
$$662$$ 4986.00 0.292729
$$663$$ 0 0
$$664$$ 1530.00 0.0894210
$$665$$ −1680.00 −0.0979663
$$666$$ 0 0
$$667$$ −471.000 −0.0273421
$$668$$ 11172.0 0.647092
$$669$$ 0 0
$$670$$ 2460.00 0.141848
$$671$$ −15886.0 −0.913968
$$672$$ 0 0
$$673$$ −6006.00 −0.344003 −0.172002 0.985097i $$-0.555023\pi$$
−0.172002 + 0.985097i $$0.555023\pi$$
$$674$$ −904.000 −0.0516629
$$675$$ 0 0
$$676$$ 15204.0 0.865043
$$677$$ −1164.00 −0.0660800 −0.0330400 0.999454i $$-0.510519\pi$$
−0.0330400 + 0.999454i $$0.510519\pi$$
$$678$$ 0 0
$$679$$ −5844.00 −0.330298
$$680$$ −9825.00 −0.554076
$$681$$ 0 0
$$682$$ −10575.0 −0.593750
$$683$$ −26496.0 −1.48439 −0.742197 0.670182i $$-0.766216\pi$$
−0.742197 + 0.670182i $$0.766216\pi$$
$$684$$ 0 0
$$685$$ 5490.00 0.306222
$$686$$ −3900.00 −0.217059
$$687$$ 0 0
$$688$$ 16277.0 0.901969
$$689$$ 20.0000 0.00110586
$$690$$ 0 0
$$691$$ 17110.0 0.941961 0.470981 0.882144i $$-0.343900\pi$$
0.470981 + 0.882144i $$0.343900\pi$$
$$692$$ 28938.0 1.58968
$$693$$ 0 0
$$694$$ 8860.00 0.484612
$$695$$ 5210.00 0.284355
$$696$$ 0 0
$$697$$ −18340.0 −0.996667
$$698$$ 4454.00 0.241528
$$699$$ 0 0
$$700$$ 1050.00 0.0566947
$$701$$ −30251.0 −1.62991 −0.814953 0.579527i $$-0.803237\pi$$
−0.814953 + 0.579527i $$0.803237\pi$$
$$702$$ 0 0
$$703$$ 3920.00 0.210307
$$704$$ −7849.00 −0.420199
$$705$$ 0 0
$$706$$ −8781.00 −0.468098
$$707$$ −8010.00 −0.426092
$$708$$ 0 0
$$709$$ 18820.0 0.996897 0.498448 0.866919i $$-0.333903\pi$$
0.498448 + 0.866919i $$0.333903\pi$$
$$710$$ −160.000 −0.00845731
$$711$$ 0 0
$$712$$ 22320.0 1.17483
$$713$$ −675.000 −0.0354543
$$714$$ 0 0
$$715$$ 1175.00 0.0614581
$$716$$ 12796.0 0.667890
$$717$$ 0 0
$$718$$ 2928.00 0.152189
$$719$$ −31890.0 −1.65410 −0.827049 0.562130i $$-0.809982\pi$$
−0.827049 + 0.562130i $$0.809982\pi$$
$$720$$ 0 0
$$721$$ −4116.00 −0.212605
$$722$$ 3723.00 0.191905
$$723$$ 0 0
$$724$$ 3640.00 0.186850
$$725$$ 3925.00 0.201063
$$726$$ 0 0
$$727$$ −11452.0 −0.584224 −0.292112 0.956384i $$-0.594358\pi$$
−0.292112 + 0.956384i $$0.594358\pi$$
$$728$$ 450.000 0.0229095
$$729$$ 0 0
$$730$$ 4850.00 0.245899
$$731$$ 52007.0 2.63139
$$732$$ 0 0
$$733$$ 7094.00 0.357466 0.178733 0.983898i $$-0.442800\pi$$
0.178733 + 0.983898i $$0.442800\pi$$
$$734$$ −9102.00 −0.457712
$$735$$ 0 0
$$736$$ 483.000 0.0241897
$$737$$ 23124.0 1.15574
$$738$$ 0 0
$$739$$ −3200.00 −0.159288 −0.0796440 0.996823i $$-0.525378\pi$$
−0.0796440 + 0.996823i $$0.525378\pi$$
$$740$$ −2450.00 −0.121708
$$741$$ 0 0
$$742$$ −24.0000 −0.00118742
$$743$$ 20831.0 1.02855 0.514277 0.857624i $$-0.328060\pi$$
0.514277 + 0.857624i $$0.328060\pi$$
$$744$$ 0 0
$$745$$ −14705.0 −0.723154
$$746$$ 8183.00 0.401610
$$747$$ 0 0
$$748$$ −43099.0 −2.10676
$$749$$ −6588.00 −0.321389
$$750$$ 0 0
$$751$$ −15605.0 −0.758235 −0.379118 0.925349i $$-0.623772\pi$$
−0.379118 + 0.925349i $$0.623772\pi$$
$$752$$ 14227.0 0.689901
$$753$$ 0 0
$$754$$ 785.000 0.0379151
$$755$$ −2555.00 −0.123160
$$756$$ 0 0
$$757$$ 21349.0 1.02502 0.512512 0.858680i $$-0.328715\pi$$
0.512512 + 0.858680i $$0.328715\pi$$
$$758$$ −6136.00 −0.294023
$$759$$ 0 0
$$760$$ 4200.00 0.200461
$$761$$ −3702.00 −0.176343 −0.0881717 0.996105i $$-0.528102\pi$$
−0.0881717 + 0.996105i $$0.528102\pi$$
$$762$$ 0 0
$$763$$ 4200.00 0.199279
$$764$$ 33782.0 1.59972
$$765$$ 0 0
$$766$$ 5643.00 0.266175
$$767$$ 3740.00 0.176067
$$768$$ 0 0
$$769$$ −1393.00 −0.0653223 −0.0326612 0.999466i $$-0.510398\pi$$
−0.0326612 + 0.999466i $$0.510398\pi$$
$$770$$ −1410.00 −0.0659907
$$771$$ 0 0
$$772$$ −11690.0 −0.544990
$$773$$ 6906.00 0.321334 0.160667 0.987009i $$-0.448635\pi$$
0.160667 + 0.987009i $$0.448635\pi$$
$$774$$ 0 0
$$775$$ 5625.00 0.260717
$$776$$ 14610.0 0.675861
$$777$$ 0 0
$$778$$ 8991.00 0.414323
$$779$$ 7840.00 0.360587
$$780$$ 0 0
$$781$$ −1504.00 −0.0689083
$$782$$ 393.000 0.0179714
$$783$$ 0 0
$$784$$ −12587.0 −0.573387
$$785$$ 2855.00 0.129808
$$786$$ 0 0
$$787$$ 30493.0 1.38114 0.690571 0.723265i $$-0.257359\pi$$
0.690571 + 0.723265i $$0.257359\pi$$
$$788$$ −9660.00 −0.436705
$$789$$ 0 0
$$790$$ −6285.00 −0.283051
$$791$$ −6330.00 −0.284537
$$792$$ 0 0
$$793$$ 1690.00 0.0756793
$$794$$ 12449.0 0.556421
$$795$$ 0 0
$$796$$ −30499.0 −1.35805
$$797$$ −33488.0 −1.48834 −0.744169 0.667991i $$-0.767154\pi$$
−0.744169 + 0.667991i $$0.767154\pi$$
$$798$$ 0 0
$$799$$ 45457.0 2.01271
$$800$$ −4025.00 −0.177882
$$801$$ 0 0
$$802$$ 8076.00 0.355578
$$803$$ 45590.0 2.00353
$$804$$ 0 0
$$805$$ −90.0000 −0.00394048
$$806$$ 1125.00 0.0491643
$$807$$ 0 0
$$808$$ 20025.0 0.871878
$$809$$ 15304.0 0.665093 0.332546 0.943087i $$-0.392092\pi$$
0.332546 + 0.943087i $$0.392092\pi$$
$$810$$ 0 0
$$811$$ −40122.0 −1.73721 −0.868603 0.495509i $$-0.834982\pi$$
−0.868603 + 0.495509i $$0.834982\pi$$
$$812$$ 6594.00 0.284980
$$813$$ 0 0
$$814$$ 3290.00 0.141664
$$815$$ −3565.00 −0.153223
$$816$$ 0 0
$$817$$ −22232.0 −0.952019
$$818$$ 2833.00 0.121092
$$819$$ 0 0
$$820$$ −4900.00 −0.208677
$$821$$ −25098.0 −1.06690 −0.533451 0.845831i $$-0.679105\pi$$
−0.533451 + 0.845831i $$0.679105\pi$$
$$822$$ 0 0
$$823$$ −43492.0 −1.84208 −0.921042 0.389462i $$-0.872661\pi$$
−0.921042 + 0.389462i $$0.872661\pi$$
$$824$$ 10290.0 0.435035
$$825$$ 0 0
$$826$$ −4488.00 −0.189053
$$827$$ −11206.0 −0.471186 −0.235593 0.971852i $$-0.575703\pi$$
−0.235593 + 0.971852i $$0.575703\pi$$
$$828$$ 0 0
$$829$$ −23964.0 −1.00399 −0.501993 0.864872i $$-0.667400\pi$$
−0.501993 + 0.864872i $$0.667400\pi$$
$$830$$ 510.000 0.0213281
$$831$$ 0 0
$$832$$ 835.000 0.0347938
$$833$$ −40217.0 −1.67279
$$834$$ 0 0
$$835$$ 7980.00 0.330730
$$836$$ 18424.0 0.762210
$$837$$ 0 0
$$838$$ −4777.00 −0.196920
$$839$$ 34606.0 1.42399 0.711997 0.702182i $$-0.247791\pi$$
0.711997 + 0.702182i $$0.247791\pi$$
$$840$$ 0 0
$$841$$ 260.000 0.0106605
$$842$$ 6464.00 0.264566
$$843$$ 0 0
$$844$$ 29134.0 1.18819
$$845$$ 10860.0 0.442125
$$846$$ 0 0
$$847$$ −5268.00 −0.213708
$$848$$ −164.000 −0.00664125
$$849$$ 0 0
$$850$$ −3275.00 −0.132155
$$851$$ 210.000 0.00845912
$$852$$ 0 0
$$853$$ −18477.0 −0.741665 −0.370833 0.928700i $$-0.620928\pi$$
−0.370833 + 0.928700i $$0.620928\pi$$
$$854$$ −2028.00 −0.0812608
$$855$$ 0 0
$$856$$ 16470.0 0.657632
$$857$$ 41342.0 1.64786 0.823930 0.566692i $$-0.191777\pi$$
0.823930 + 0.566692i $$0.191777\pi$$
$$858$$ 0 0
$$859$$ 21898.0 0.869791 0.434895 0.900481i $$-0.356785\pi$$
0.434895 + 0.900481i $$0.356785\pi$$
$$860$$ 13895.0 0.550948
$$861$$ 0 0
$$862$$ 10680.0 0.421998
$$863$$ 18487.0 0.729206 0.364603 0.931163i $$-0.381205\pi$$
0.364603 + 0.931163i $$0.381205\pi$$
$$864$$ 0 0
$$865$$ 20670.0 0.812487
$$866$$ −11566.0 −0.453844
$$867$$ 0 0
$$868$$ 9450.00 0.369532
$$869$$ −59079.0 −2.30623
$$870$$ 0 0
$$871$$ −2460.00 −0.0956991
$$872$$ −10500.0 −0.407769
$$873$$ 0 0
$$874$$ −168.000 −0.00650193
$$875$$ 750.000 0.0289767
$$876$$ 0 0
$$877$$ 7593.00 0.292357 0.146179 0.989258i $$-0.453303\pi$$
0.146179 + 0.989258i $$0.453303\pi$$
$$878$$ 1448.00 0.0556579
$$879$$ 0 0
$$880$$ −9635.00 −0.369086
$$881$$ 3038.00 0.116178 0.0580890 0.998311i $$-0.481499\pi$$
0.0580890 + 0.998311i $$0.481499\pi$$
$$882$$ 0 0
$$883$$ −16732.0 −0.637686 −0.318843 0.947808i $$-0.603294\pi$$
−0.318843 + 0.947808i $$0.603294\pi$$
$$884$$ 4585.00 0.174446
$$885$$ 0 0
$$886$$ −2376.00 −0.0900940
$$887$$ 8031.00 0.304007 0.152004 0.988380i $$-0.451427\pi$$
0.152004 + 0.988380i $$0.451427\pi$$
$$888$$ 0 0
$$889$$ 9876.00 0.372588
$$890$$ 7440.00 0.280213
$$891$$ 0 0
$$892$$ 41692.0 1.56497
$$893$$ −19432.0 −0.728183
$$894$$ 0 0
$$895$$ 9140.00 0.341359
$$896$$ −8730.00 −0.325501
$$897$$ 0 0
$$898$$ −14894.0 −0.553474
$$899$$ 35325.0 1.31052
$$900$$ 0 0
$$901$$ −524.000 −0.0193751
$$902$$ 6580.00 0.242894
$$903$$ 0 0
$$904$$ 15825.0 0.582225
$$905$$ 2600.00 0.0954994
$$906$$ 0 0
$$907$$ −38487.0 −1.40897 −0.704487 0.709717i $$-0.748823\pi$$
−0.704487 + 0.709717i $$0.748823\pi$$
$$908$$ 34580.0 1.26385
$$909$$ 0 0
$$910$$ 150.000 0.00546423
$$911$$ 5120.00 0.186205 0.0931027 0.995657i $$-0.470321\pi$$
0.0931027 + 0.995657i $$0.470321\pi$$
$$912$$ 0 0
$$913$$ 4794.00 0.173777
$$914$$ −16204.0 −0.586412
$$915$$ 0 0
$$916$$ −30408.0 −1.09684
$$917$$ −10998.0 −0.396059
$$918$$ 0 0
$$919$$ 28075.0 1.00774 0.503868 0.863781i $$-0.331910\pi$$
0.503868 + 0.863781i $$0.331910\pi$$
$$920$$ 225.000 0.00806308
$$921$$ 0 0
$$922$$ −5082.00 −0.181526
$$923$$ 160.000 0.00570581
$$924$$ 0 0
$$925$$ −1750.00 −0.0622050
$$926$$ 10326.0 0.366451
$$927$$ 0 0
$$928$$ −25277.0 −0.894136
$$929$$ 12856.0 0.454028 0.227014 0.973892i $$-0.427104\pi$$
0.227014 + 0.973892i $$0.427104\pi$$
$$930$$ 0 0
$$931$$ 17192.0 0.605204
$$932$$ −36414.0 −1.27981
$$933$$ 0 0
$$934$$ 4184.00 0.146579
$$935$$ −30785.0 −1.07677
$$936$$ 0 0
$$937$$ −1374.00 −0.0479046 −0.0239523 0.999713i $$-0.507625\pi$$
−0.0239523 + 0.999713i $$0.507625\pi$$
$$938$$ 2952.00 0.102757
$$939$$ 0 0
$$940$$ 12145.0 0.421411
$$941$$ −8543.00 −0.295955 −0.147978 0.988991i $$-0.547276\pi$$
−0.147978 + 0.988991i $$0.547276\pi$$
$$942$$ 0 0
$$943$$ 420.000 0.0145038
$$944$$ −30668.0 −1.05737
$$945$$ 0 0
$$946$$ −18659.0 −0.641286
$$947$$ −13506.0 −0.463449 −0.231724 0.972781i $$-0.574437\pi$$
−0.231724 + 0.972781i $$0.574437\pi$$
$$948$$ 0 0
$$949$$ −4850.00 −0.165898
$$950$$ 1400.00 0.0478126
$$951$$ 0 0
$$952$$ −11790.0 −0.401382
$$953$$ −21775.0 −0.740148 −0.370074 0.929002i $$-0.620668\pi$$
−0.370074 + 0.929002i $$0.620668\pi$$
$$954$$ 0 0
$$955$$ 24130.0 0.817621
$$956$$ 10822.0 0.366118
$$957$$ 0 0
$$958$$ −15576.0 −0.525300
$$959$$ 6588.00 0.221833
$$960$$ 0 0
$$961$$ 20834.0 0.699339
$$962$$ −350.000 −0.0117302
$$963$$ 0 0
$$964$$ 25613.0 0.855746
$$965$$ −8350.00 −0.278545
$$966$$ 0 0
$$967$$ 3854.00 0.128166 0.0640829 0.997945i $$-0.479588\pi$$
0.0640829 + 0.997945i $$0.479588\pi$$
$$968$$ 13170.0 0.437293
$$969$$ 0 0
$$970$$ 4870.00 0.161202
$$971$$ −12933.0 −0.427435 −0.213718 0.976895i $$-0.568557\pi$$
−0.213718 + 0.976895i $$0.568557\pi$$
$$972$$ 0 0
$$973$$ 6252.00 0.205992
$$974$$ −10220.0 −0.336211
$$975$$ 0 0
$$976$$ −13858.0 −0.454492
$$977$$ −17521.0 −0.573743 −0.286871 0.957969i $$-0.592615\pi$$
−0.286871 + 0.957969i $$0.592615\pi$$
$$978$$ 0 0
$$979$$ 69936.0 2.28311
$$980$$ −10745.0 −0.350241
$$981$$ 0 0
$$982$$ −2692.00 −0.0874798
$$983$$ 12573.0 0.407952 0.203976 0.978976i $$-0.434614\pi$$
0.203976 + 0.978976i $$0.434614\pi$$
$$984$$ 0 0
$$985$$ −6900.00 −0.223200
$$986$$ −20567.0 −0.664287
$$987$$ 0 0
$$988$$ −1960.00 −0.0631133
$$989$$ −1191.00 −0.0382928
$$990$$ 0 0
$$991$$ 8945.00 0.286728 0.143364 0.989670i $$-0.454208\pi$$
0.143364 + 0.989670i $$0.454208\pi$$
$$992$$ −36225.0 −1.15942
$$993$$ 0 0
$$994$$ −192.000 −0.00612663
$$995$$ −21785.0 −0.694101
$$996$$ 0 0
$$997$$ −58179.0 −1.84809 −0.924046 0.382282i $$-0.875138\pi$$
−0.924046 + 0.382282i $$0.875138\pi$$
$$998$$ −5764.00 −0.182822
$$999$$ 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 135.4.a.b.1.1 1
3.2 odd 2 135.4.a.c.1.1 yes 1
4.3 odd 2 2160.4.a.f.1.1 1
5.2 odd 4 675.4.b.f.649.1 2
5.3 odd 4 675.4.b.f.649.2 2
5.4 even 2 675.4.a.h.1.1 1
9.2 odd 6 405.4.e.f.271.1 2
9.4 even 3 405.4.e.h.136.1 2
9.5 odd 6 405.4.e.f.136.1 2
9.7 even 3 405.4.e.h.271.1 2
12.11 even 2 2160.4.a.p.1.1 1
15.2 even 4 675.4.b.e.649.2 2
15.8 even 4 675.4.b.e.649.1 2
15.14 odd 2 675.4.a.c.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
135.4.a.b.1.1 1 1.1 even 1 trivial
135.4.a.c.1.1 yes 1 3.2 odd 2
405.4.e.f.136.1 2 9.5 odd 6
405.4.e.f.271.1 2 9.2 odd 6
405.4.e.h.136.1 2 9.4 even 3
405.4.e.h.271.1 2 9.7 even 3
675.4.a.c.1.1 1 15.14 odd 2
675.4.a.h.1.1 1 5.4 even 2
675.4.b.e.649.1 2 15.8 even 4
675.4.b.e.649.2 2 15.2 even 4
675.4.b.f.649.1 2 5.2 odd 4
675.4.b.f.649.2 2 5.3 odd 4
2160.4.a.f.1.1 1 4.3 odd 2
2160.4.a.p.1.1 1 12.11 even 2