# Properties

 Label 336.4.a.k.1.1 Level $336$ Weight $4$ Character 336.1 Self dual yes Analytic conductor $19.825$ Analytic rank $0$ Dimension $1$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$336 = 2^{4} \cdot 3 \cdot 7$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 336.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$19.8246417619$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 84) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Character $$\chi$$ $$=$$ 336.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q+3.00000 q^{3} +6.00000 q^{5} -7.00000 q^{7} +9.00000 q^{9} +O(q^{10})$$ $$q+3.00000 q^{3} +6.00000 q^{5} -7.00000 q^{7} +9.00000 q^{9} -36.0000 q^{11} +62.0000 q^{13} +18.0000 q^{15} +114.000 q^{17} +76.0000 q^{19} -21.0000 q^{21} +24.0000 q^{23} -89.0000 q^{25} +27.0000 q^{27} +54.0000 q^{29} +112.000 q^{31} -108.000 q^{33} -42.0000 q^{35} -178.000 q^{37} +186.000 q^{39} +378.000 q^{41} +172.000 q^{43} +54.0000 q^{45} +192.000 q^{47} +49.0000 q^{49} +342.000 q^{51} -402.000 q^{53} -216.000 q^{55} +228.000 q^{57} -396.000 q^{59} +254.000 q^{61} -63.0000 q^{63} +372.000 q^{65} +1012.00 q^{67} +72.0000 q^{69} -840.000 q^{71} +890.000 q^{73} -267.000 q^{75} +252.000 q^{77} -80.0000 q^{79} +81.0000 q^{81} +108.000 q^{83} +684.000 q^{85} +162.000 q^{87} -1638.00 q^{89} -434.000 q^{91} +336.000 q^{93} +456.000 q^{95} +1010.00 q^{97} -324.000 q^{99} +O(q^{100})$$

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 0 0
$$3$$ 3.00000 0.577350
$$4$$ 0 0
$$5$$ 6.00000 0.536656 0.268328 0.963328i $$-0.413529\pi$$
0.268328 + 0.963328i $$0.413529\pi$$
$$6$$ 0 0
$$7$$ −7.00000 −0.377964
$$8$$ 0 0
$$9$$ 9.00000 0.333333
$$10$$ 0 0
$$11$$ −36.0000 −0.986764 −0.493382 0.869813i $$-0.664240\pi$$
−0.493382 + 0.869813i $$0.664240\pi$$
$$12$$ 0 0
$$13$$ 62.0000 1.32275 0.661373 0.750057i $$-0.269974\pi$$
0.661373 + 0.750057i $$0.269974\pi$$
$$14$$ 0 0
$$15$$ 18.0000 0.309839
$$16$$ 0 0
$$17$$ 114.000 1.62642 0.813208 0.581974i $$-0.197719\pi$$
0.813208 + 0.581974i $$0.197719\pi$$
$$18$$ 0 0
$$19$$ 76.0000 0.917663 0.458831 0.888523i $$-0.348268\pi$$
0.458831 + 0.888523i $$0.348268\pi$$
$$20$$ 0 0
$$21$$ −21.0000 −0.218218
$$22$$ 0 0
$$23$$ 24.0000 0.217580 0.108790 0.994065i $$-0.465302\pi$$
0.108790 + 0.994065i $$0.465302\pi$$
$$24$$ 0 0
$$25$$ −89.0000 −0.712000
$$26$$ 0 0
$$27$$ 27.0000 0.192450
$$28$$ 0 0
$$29$$ 54.0000 0.345778 0.172889 0.984941i $$-0.444690\pi$$
0.172889 + 0.984941i $$0.444690\pi$$
$$30$$ 0 0
$$31$$ 112.000 0.648897 0.324448 0.945903i $$-0.394821\pi$$
0.324448 + 0.945903i $$0.394821\pi$$
$$32$$ 0 0
$$33$$ −108.000 −0.569709
$$34$$ 0 0
$$35$$ −42.0000 −0.202837
$$36$$ 0 0
$$37$$ −178.000 −0.790892 −0.395446 0.918489i $$-0.629410\pi$$
−0.395446 + 0.918489i $$0.629410\pi$$
$$38$$ 0 0
$$39$$ 186.000 0.763688
$$40$$ 0 0
$$41$$ 378.000 1.43985 0.719923 0.694054i $$-0.244177\pi$$
0.719923 + 0.694054i $$0.244177\pi$$
$$42$$ 0 0
$$43$$ 172.000 0.609994 0.304997 0.952353i $$-0.401344\pi$$
0.304997 + 0.952353i $$0.401344\pi$$
$$44$$ 0 0
$$45$$ 54.0000 0.178885
$$46$$ 0 0
$$47$$ 192.000 0.595874 0.297937 0.954586i $$-0.403701\pi$$
0.297937 + 0.954586i $$0.403701\pi$$
$$48$$ 0 0
$$49$$ 49.0000 0.142857
$$50$$ 0 0
$$51$$ 342.000 0.939011
$$52$$ 0 0
$$53$$ −402.000 −1.04187 −0.520933 0.853597i $$-0.674416\pi$$
−0.520933 + 0.853597i $$0.674416\pi$$
$$54$$ 0 0
$$55$$ −216.000 −0.529553
$$56$$ 0 0
$$57$$ 228.000 0.529813
$$58$$ 0 0
$$59$$ −396.000 −0.873810 −0.436905 0.899508i $$-0.643925\pi$$
−0.436905 + 0.899508i $$0.643925\pi$$
$$60$$ 0 0
$$61$$ 254.000 0.533137 0.266569 0.963816i $$-0.414110\pi$$
0.266569 + 0.963816i $$0.414110\pi$$
$$62$$ 0 0
$$63$$ −63.0000 −0.125988
$$64$$ 0 0
$$65$$ 372.000 0.709860
$$66$$ 0 0
$$67$$ 1012.00 1.84531 0.922653 0.385632i $$-0.126016\pi$$
0.922653 + 0.385632i $$0.126016\pi$$
$$68$$ 0 0
$$69$$ 72.0000 0.125620
$$70$$ 0 0
$$71$$ −840.000 −1.40408 −0.702040 0.712138i $$-0.747727\pi$$
−0.702040 + 0.712138i $$0.747727\pi$$
$$72$$ 0 0
$$73$$ 890.000 1.42694 0.713470 0.700686i $$-0.247122\pi$$
0.713470 + 0.700686i $$0.247122\pi$$
$$74$$ 0 0
$$75$$ −267.000 −0.411073
$$76$$ 0 0
$$77$$ 252.000 0.372962
$$78$$ 0 0
$$79$$ −80.0000 −0.113933 −0.0569665 0.998376i $$-0.518143\pi$$
−0.0569665 + 0.998376i $$0.518143\pi$$
$$80$$ 0 0
$$81$$ 81.0000 0.111111
$$82$$ 0 0
$$83$$ 108.000 0.142826 0.0714129 0.997447i $$-0.477249\pi$$
0.0714129 + 0.997447i $$0.477249\pi$$
$$84$$ 0 0
$$85$$ 684.000 0.872826
$$86$$ 0 0
$$87$$ 162.000 0.199635
$$88$$ 0 0
$$89$$ −1638.00 −1.95087 −0.975436 0.220282i $$-0.929302\pi$$
−0.975436 + 0.220282i $$0.929302\pi$$
$$90$$ 0 0
$$91$$ −434.000 −0.499951
$$92$$ 0 0
$$93$$ 336.000 0.374641
$$94$$ 0 0
$$95$$ 456.000 0.492470
$$96$$ 0 0
$$97$$ 1010.00 1.05722 0.528608 0.848866i $$-0.322714\pi$$
0.528608 + 0.848866i $$0.322714\pi$$
$$98$$ 0 0
$$99$$ −324.000 −0.328921
$$100$$ 0 0
$$101$$ 6.00000 0.00591111 0.00295556 0.999996i $$-0.499059\pi$$
0.00295556 + 0.999996i $$0.499059\pi$$
$$102$$ 0 0
$$103$$ 472.000 0.451530 0.225765 0.974182i $$-0.427512\pi$$
0.225765 + 0.974182i $$0.427512\pi$$
$$104$$ 0 0
$$105$$ −126.000 −0.117108
$$106$$ 0 0
$$107$$ 972.000 0.878194 0.439097 0.898440i $$-0.355298\pi$$
0.439097 + 0.898440i $$0.355298\pi$$
$$108$$ 0 0
$$109$$ −1786.00 −1.56943 −0.784715 0.619857i $$-0.787190\pi$$
−0.784715 + 0.619857i $$0.787190\pi$$
$$110$$ 0 0
$$111$$ −534.000 −0.456622
$$112$$ 0 0
$$113$$ −2286.00 −1.90309 −0.951543 0.307515i $$-0.900503\pi$$
−0.951543 + 0.307515i $$0.900503\pi$$
$$114$$ 0 0
$$115$$ 144.000 0.116766
$$116$$ 0 0
$$117$$ 558.000 0.440916
$$118$$ 0 0
$$119$$ −798.000 −0.614727
$$120$$ 0 0
$$121$$ −35.0000 −0.0262960
$$122$$ 0 0
$$123$$ 1134.00 0.831295
$$124$$ 0 0
$$125$$ −1284.00 −0.918756
$$126$$ 0 0
$$127$$ −1328.00 −0.927881 −0.463941 0.885866i $$-0.653565\pi$$
−0.463941 + 0.885866i $$0.653565\pi$$
$$128$$ 0 0
$$129$$ 516.000 0.352180
$$130$$ 0 0
$$131$$ 1212.00 0.808343 0.404171 0.914683i $$-0.367560\pi$$
0.404171 + 0.914683i $$0.367560\pi$$
$$132$$ 0 0
$$133$$ −532.000 −0.346844
$$134$$ 0 0
$$135$$ 162.000 0.103280
$$136$$ 0 0
$$137$$ −1254.00 −0.782018 −0.391009 0.920387i $$-0.627874\pi$$
−0.391009 + 0.920387i $$0.627874\pi$$
$$138$$ 0 0
$$139$$ 340.000 0.207471 0.103735 0.994605i $$-0.466921\pi$$
0.103735 + 0.994605i $$0.466921\pi$$
$$140$$ 0 0
$$141$$ 576.000 0.344028
$$142$$ 0 0
$$143$$ −2232.00 −1.30524
$$144$$ 0 0
$$145$$ 324.000 0.185564
$$146$$ 0 0
$$147$$ 147.000 0.0824786
$$148$$ 0 0
$$149$$ 1038.00 0.570713 0.285357 0.958421i $$-0.407888\pi$$
0.285357 + 0.958421i $$0.407888\pi$$
$$150$$ 0 0
$$151$$ −2936.00 −1.58231 −0.791153 0.611618i $$-0.790519\pi$$
−0.791153 + 0.611618i $$0.790519\pi$$
$$152$$ 0 0
$$153$$ 1026.00 0.542138
$$154$$ 0 0
$$155$$ 672.000 0.348234
$$156$$ 0 0
$$157$$ −1330.00 −0.676086 −0.338043 0.941131i $$-0.609765\pi$$
−0.338043 + 0.941131i $$0.609765\pi$$
$$158$$ 0 0
$$159$$ −1206.00 −0.601522
$$160$$ 0 0
$$161$$ −168.000 −0.0822376
$$162$$ 0 0
$$163$$ 3364.00 1.61650 0.808248 0.588842i $$-0.200416\pi$$
0.808248 + 0.588842i $$0.200416\pi$$
$$164$$ 0 0
$$165$$ −648.000 −0.305738
$$166$$ 0 0
$$167$$ −3048.00 −1.41234 −0.706172 0.708041i $$-0.749579\pi$$
−0.706172 + 0.708041i $$0.749579\pi$$
$$168$$ 0 0
$$169$$ 1647.00 0.749659
$$170$$ 0 0
$$171$$ 684.000 0.305888
$$172$$ 0 0
$$173$$ −2706.00 −1.18921 −0.594605 0.804018i $$-0.702692\pi$$
−0.594605 + 0.804018i $$0.702692\pi$$
$$174$$ 0 0
$$175$$ 623.000 0.269111
$$176$$ 0 0
$$177$$ −1188.00 −0.504495
$$178$$ 0 0
$$179$$ −4716.00 −1.96922 −0.984610 0.174766i $$-0.944083\pi$$
−0.984610 + 0.174766i $$0.944083\pi$$
$$180$$ 0 0
$$181$$ 1910.00 0.784360 0.392180 0.919888i $$-0.371721\pi$$
0.392180 + 0.919888i $$0.371721\pi$$
$$182$$ 0 0
$$183$$ 762.000 0.307807
$$184$$ 0 0
$$185$$ −1068.00 −0.424437
$$186$$ 0 0
$$187$$ −4104.00 −1.60489
$$188$$ 0 0
$$189$$ −189.000 −0.0727393
$$190$$ 0 0
$$191$$ −4080.00 −1.54565 −0.772823 0.634621i $$-0.781156\pi$$
−0.772823 + 0.634621i $$0.781156\pi$$
$$192$$ 0 0
$$193$$ −2686.00 −1.00177 −0.500887 0.865512i $$-0.666993\pi$$
−0.500887 + 0.865512i $$0.666993\pi$$
$$194$$ 0 0
$$195$$ 1116.00 0.409838
$$196$$ 0 0
$$197$$ 510.000 0.184447 0.0922233 0.995738i $$-0.470603\pi$$
0.0922233 + 0.995738i $$0.470603\pi$$
$$198$$ 0 0
$$199$$ −1352.00 −0.481612 −0.240806 0.970573i $$-0.577412\pi$$
−0.240806 + 0.970573i $$0.577412\pi$$
$$200$$ 0 0
$$201$$ 3036.00 1.06539
$$202$$ 0 0
$$203$$ −378.000 −0.130692
$$204$$ 0 0
$$205$$ 2268.00 0.772702
$$206$$ 0 0
$$207$$ 216.000 0.0725268
$$208$$ 0 0
$$209$$ −2736.00 −0.905517
$$210$$ 0 0
$$211$$ 3364.00 1.09757 0.548785 0.835963i $$-0.315091\pi$$
0.548785 + 0.835963i $$0.315091\pi$$
$$212$$ 0 0
$$213$$ −2520.00 −0.810646
$$214$$ 0 0
$$215$$ 1032.00 0.327357
$$216$$ 0 0
$$217$$ −784.000 −0.245260
$$218$$ 0 0
$$219$$ 2670.00 0.823844
$$220$$ 0 0
$$221$$ 7068.00 2.15134
$$222$$ 0 0
$$223$$ 4768.00 1.43179 0.715894 0.698209i $$-0.246019\pi$$
0.715894 + 0.698209i $$0.246019\pi$$
$$224$$ 0 0
$$225$$ −801.000 −0.237333
$$226$$ 0 0
$$227$$ −420.000 −0.122803 −0.0614017 0.998113i $$-0.519557\pi$$
−0.0614017 + 0.998113i $$0.519557\pi$$
$$228$$ 0 0
$$229$$ −1882.00 −0.543083 −0.271542 0.962427i $$-0.587533\pi$$
−0.271542 + 0.962427i $$0.587533\pi$$
$$230$$ 0 0
$$231$$ 756.000 0.215330
$$232$$ 0 0
$$233$$ 5082.00 1.42890 0.714448 0.699688i $$-0.246678\pi$$
0.714448 + 0.699688i $$0.246678\pi$$
$$234$$ 0 0
$$235$$ 1152.00 0.319780
$$236$$ 0 0
$$237$$ −240.000 −0.0657792
$$238$$ 0 0
$$239$$ 5424.00 1.46799 0.733995 0.679155i $$-0.237654\pi$$
0.733995 + 0.679155i $$0.237654\pi$$
$$240$$ 0 0
$$241$$ −2590.00 −0.692268 −0.346134 0.938185i $$-0.612506\pi$$
−0.346134 + 0.938185i $$0.612506\pi$$
$$242$$ 0 0
$$243$$ 243.000 0.0641500
$$244$$ 0 0
$$245$$ 294.000 0.0766652
$$246$$ 0 0
$$247$$ 4712.00 1.21384
$$248$$ 0 0
$$249$$ 324.000 0.0824605
$$250$$ 0 0
$$251$$ 4932.00 1.24026 0.620130 0.784499i $$-0.287080\pi$$
0.620130 + 0.784499i $$0.287080\pi$$
$$252$$ 0 0
$$253$$ −864.000 −0.214700
$$254$$ 0 0
$$255$$ 2052.00 0.503926
$$256$$ 0 0
$$257$$ −3438.00 −0.834461 −0.417231 0.908801i $$-0.636999\pi$$
−0.417231 + 0.908801i $$0.636999\pi$$
$$258$$ 0 0
$$259$$ 1246.00 0.298929
$$260$$ 0 0
$$261$$ 486.000 0.115259
$$262$$ 0 0
$$263$$ −6120.00 −1.43489 −0.717444 0.696617i $$-0.754688\pi$$
−0.717444 + 0.696617i $$0.754688\pi$$
$$264$$ 0 0
$$265$$ −2412.00 −0.559124
$$266$$ 0 0
$$267$$ −4914.00 −1.12634
$$268$$ 0 0
$$269$$ −18.0000 −0.00407985 −0.00203992 0.999998i $$-0.500649\pi$$
−0.00203992 + 0.999998i $$0.500649\pi$$
$$270$$ 0 0
$$271$$ −6896.00 −1.54576 −0.772882 0.634549i $$-0.781186\pi$$
−0.772882 + 0.634549i $$0.781186\pi$$
$$272$$ 0 0
$$273$$ −1302.00 −0.288647
$$274$$ 0 0
$$275$$ 3204.00 0.702576
$$276$$ 0 0
$$277$$ 6254.00 1.35656 0.678279 0.734805i $$-0.262726\pi$$
0.678279 + 0.734805i $$0.262726\pi$$
$$278$$ 0 0
$$279$$ 1008.00 0.216299
$$280$$ 0 0
$$281$$ 1194.00 0.253481 0.126740 0.991936i $$-0.459549\pi$$
0.126740 + 0.991936i $$0.459549\pi$$
$$282$$ 0 0
$$283$$ 7156.00 1.50311 0.751555 0.659671i $$-0.229304\pi$$
0.751555 + 0.659671i $$0.229304\pi$$
$$284$$ 0 0
$$285$$ 1368.00 0.284327
$$286$$ 0 0
$$287$$ −2646.00 −0.544211
$$288$$ 0 0
$$289$$ 8083.00 1.64523
$$290$$ 0 0
$$291$$ 3030.00 0.610384
$$292$$ 0 0
$$293$$ −3738.00 −0.745312 −0.372656 0.927970i $$-0.621553\pi$$
−0.372656 + 0.927970i $$0.621553\pi$$
$$294$$ 0 0
$$295$$ −2376.00 −0.468936
$$296$$ 0 0
$$297$$ −972.000 −0.189903
$$298$$ 0 0
$$299$$ 1488.00 0.287804
$$300$$ 0 0
$$301$$ −1204.00 −0.230556
$$302$$ 0 0
$$303$$ 18.0000 0.00341278
$$304$$ 0 0
$$305$$ 1524.00 0.286111
$$306$$ 0 0
$$307$$ 844.000 0.156904 0.0784522 0.996918i $$-0.475002\pi$$
0.0784522 + 0.996918i $$0.475002\pi$$
$$308$$ 0 0
$$309$$ 1416.00 0.260691
$$310$$ 0 0
$$311$$ −6312.00 −1.15087 −0.575435 0.817847i $$-0.695167\pi$$
−0.575435 + 0.817847i $$0.695167\pi$$
$$312$$ 0 0
$$313$$ 8282.00 1.49561 0.747806 0.663918i $$-0.231108\pi$$
0.747806 + 0.663918i $$0.231108\pi$$
$$314$$ 0 0
$$315$$ −378.000 −0.0676123
$$316$$ 0 0
$$317$$ 9318.00 1.65095 0.825475 0.564439i $$-0.190907\pi$$
0.825475 + 0.564439i $$0.190907\pi$$
$$318$$ 0 0
$$319$$ −1944.00 −0.341201
$$320$$ 0 0
$$321$$ 2916.00 0.507026
$$322$$ 0 0
$$323$$ 8664.00 1.49250
$$324$$ 0 0
$$325$$ −5518.00 −0.941796
$$326$$ 0 0
$$327$$ −5358.00 −0.906110
$$328$$ 0 0
$$329$$ −1344.00 −0.225219
$$330$$ 0 0
$$331$$ −1652.00 −0.274327 −0.137163 0.990548i $$-0.543799\pi$$
−0.137163 + 0.990548i $$0.543799\pi$$
$$332$$ 0 0
$$333$$ −1602.00 −0.263631
$$334$$ 0 0
$$335$$ 6072.00 0.990295
$$336$$ 0 0
$$337$$ −1294.00 −0.209165 −0.104583 0.994516i $$-0.533351\pi$$
−0.104583 + 0.994516i $$0.533351\pi$$
$$338$$ 0 0
$$339$$ −6858.00 −1.09875
$$340$$ 0 0
$$341$$ −4032.00 −0.640308
$$342$$ 0 0
$$343$$ −343.000 −0.0539949
$$344$$ 0 0
$$345$$ 432.000 0.0674148
$$346$$ 0 0
$$347$$ −3636.00 −0.562509 −0.281255 0.959633i $$-0.590751\pi$$
−0.281255 + 0.959633i $$0.590751\pi$$
$$348$$ 0 0
$$349$$ 10478.0 1.60709 0.803545 0.595244i $$-0.202945\pi$$
0.803545 + 0.595244i $$0.202945\pi$$
$$350$$ 0 0
$$351$$ 1674.00 0.254563
$$352$$ 0 0
$$353$$ −7566.00 −1.14079 −0.570393 0.821372i $$-0.693209\pi$$
−0.570393 + 0.821372i $$0.693209\pi$$
$$354$$ 0 0
$$355$$ −5040.00 −0.753508
$$356$$ 0 0
$$357$$ −2394.00 −0.354913
$$358$$ 0 0
$$359$$ 8040.00 1.18199 0.590996 0.806675i $$-0.298735\pi$$
0.590996 + 0.806675i $$0.298735\pi$$
$$360$$ 0 0
$$361$$ −1083.00 −0.157895
$$362$$ 0 0
$$363$$ −105.000 −0.0151820
$$364$$ 0 0
$$365$$ 5340.00 0.765776
$$366$$ 0 0
$$367$$ −7568.00 −1.07642 −0.538210 0.842811i $$-0.680899\pi$$
−0.538210 + 0.842811i $$0.680899\pi$$
$$368$$ 0 0
$$369$$ 3402.00 0.479949
$$370$$ 0 0
$$371$$ 2814.00 0.393789
$$372$$ 0 0
$$373$$ −13522.0 −1.87706 −0.938529 0.345200i $$-0.887811\pi$$
−0.938529 + 0.345200i $$0.887811\pi$$
$$374$$ 0 0
$$375$$ −3852.00 −0.530444
$$376$$ 0 0
$$377$$ 3348.00 0.457376
$$378$$ 0 0
$$379$$ −2468.00 −0.334492 −0.167246 0.985915i $$-0.553487\pi$$
−0.167246 + 0.985915i $$0.553487\pi$$
$$380$$ 0 0
$$381$$ −3984.00 −0.535713
$$382$$ 0 0
$$383$$ 12336.0 1.64580 0.822898 0.568189i $$-0.192356\pi$$
0.822898 + 0.568189i $$0.192356\pi$$
$$384$$ 0 0
$$385$$ 1512.00 0.200152
$$386$$ 0 0
$$387$$ 1548.00 0.203331
$$388$$ 0 0
$$389$$ −3762.00 −0.490337 −0.245168 0.969481i $$-0.578843\pi$$
−0.245168 + 0.969481i $$0.578843\pi$$
$$390$$ 0 0
$$391$$ 2736.00 0.353876
$$392$$ 0 0
$$393$$ 3636.00 0.466697
$$394$$ 0 0
$$395$$ −480.000 −0.0611428
$$396$$ 0 0
$$397$$ −8770.00 −1.10870 −0.554350 0.832284i $$-0.687033\pi$$
−0.554350 + 0.832284i $$0.687033\pi$$
$$398$$ 0 0
$$399$$ −1596.00 −0.200250
$$400$$ 0 0
$$401$$ 6642.00 0.827146 0.413573 0.910471i $$-0.364281\pi$$
0.413573 + 0.910471i $$0.364281\pi$$
$$402$$ 0 0
$$403$$ 6944.00 0.858326
$$404$$ 0 0
$$405$$ 486.000 0.0596285
$$406$$ 0 0
$$407$$ 6408.00 0.780424
$$408$$ 0 0
$$409$$ −1510.00 −0.182554 −0.0912771 0.995826i $$-0.529095\pi$$
−0.0912771 + 0.995826i $$0.529095\pi$$
$$410$$ 0 0
$$411$$ −3762.00 −0.451498
$$412$$ 0 0
$$413$$ 2772.00 0.330269
$$414$$ 0 0
$$415$$ 648.000 0.0766484
$$416$$ 0 0
$$417$$ 1020.00 0.119783
$$418$$ 0 0
$$419$$ 1260.00 0.146909 0.0734547 0.997299i $$-0.476598\pi$$
0.0734547 + 0.997299i $$0.476598\pi$$
$$420$$ 0 0
$$421$$ 3998.00 0.462828 0.231414 0.972855i $$-0.425665\pi$$
0.231414 + 0.972855i $$0.425665\pi$$
$$422$$ 0 0
$$423$$ 1728.00 0.198625
$$424$$ 0 0
$$425$$ −10146.0 −1.15801
$$426$$ 0 0
$$427$$ −1778.00 −0.201507
$$428$$ 0 0
$$429$$ −6696.00 −0.753580
$$430$$ 0 0
$$431$$ 2736.00 0.305774 0.152887 0.988244i $$-0.451143\pi$$
0.152887 + 0.988244i $$0.451143\pi$$
$$432$$ 0 0
$$433$$ 2690.00 0.298552 0.149276 0.988796i $$-0.452306\pi$$
0.149276 + 0.988796i $$0.452306\pi$$
$$434$$ 0 0
$$435$$ 972.000 0.107135
$$436$$ 0 0
$$437$$ 1824.00 0.199665
$$438$$ 0 0
$$439$$ 1240.00 0.134811 0.0674054 0.997726i $$-0.478528\pi$$
0.0674054 + 0.997726i $$0.478528\pi$$
$$440$$ 0 0
$$441$$ 441.000 0.0476190
$$442$$ 0 0
$$443$$ 3900.00 0.418272 0.209136 0.977887i $$-0.432935\pi$$
0.209136 + 0.977887i $$0.432935\pi$$
$$444$$ 0 0
$$445$$ −9828.00 −1.04695
$$446$$ 0 0
$$447$$ 3114.00 0.329501
$$448$$ 0 0
$$449$$ −10878.0 −1.14335 −0.571675 0.820480i $$-0.693706\pi$$
−0.571675 + 0.820480i $$0.693706\pi$$
$$450$$ 0 0
$$451$$ −13608.0 −1.42079
$$452$$ 0 0
$$453$$ −8808.00 −0.913545
$$454$$ 0 0
$$455$$ −2604.00 −0.268302
$$456$$ 0 0
$$457$$ 2330.00 0.238496 0.119248 0.992864i $$-0.461952\pi$$
0.119248 + 0.992864i $$0.461952\pi$$
$$458$$ 0 0
$$459$$ 3078.00 0.313004
$$460$$ 0 0
$$461$$ 15150.0 1.53060 0.765299 0.643675i $$-0.222591\pi$$
0.765299 + 0.643675i $$0.222591\pi$$
$$462$$ 0 0
$$463$$ 2992.00 0.300324 0.150162 0.988661i $$-0.452020\pi$$
0.150162 + 0.988661i $$0.452020\pi$$
$$464$$ 0 0
$$465$$ 2016.00 0.201053
$$466$$ 0 0
$$467$$ −8724.00 −0.864451 −0.432225 0.901766i $$-0.642272\pi$$
−0.432225 + 0.901766i $$0.642272\pi$$
$$468$$ 0 0
$$469$$ −7084.00 −0.697460
$$470$$ 0 0
$$471$$ −3990.00 −0.390339
$$472$$ 0 0
$$473$$ −6192.00 −0.601921
$$474$$ 0 0
$$475$$ −6764.00 −0.653376
$$476$$ 0 0
$$477$$ −3618.00 −0.347289
$$478$$ 0 0
$$479$$ −9744.00 −0.929467 −0.464734 0.885451i $$-0.653850\pi$$
−0.464734 + 0.885451i $$0.653850\pi$$
$$480$$ 0 0
$$481$$ −11036.0 −1.04615
$$482$$ 0 0
$$483$$ −504.000 −0.0474799
$$484$$ 0 0
$$485$$ 6060.00 0.567362
$$486$$ 0 0
$$487$$ −4136.00 −0.384846 −0.192423 0.981312i $$-0.561635\pi$$
−0.192423 + 0.981312i $$0.561635\pi$$
$$488$$ 0 0
$$489$$ 10092.0 0.933284
$$490$$ 0 0
$$491$$ −16212.0 −1.49010 −0.745048 0.667011i $$-0.767574\pi$$
−0.745048 + 0.667011i $$0.767574\pi$$
$$492$$ 0 0
$$493$$ 6156.00 0.562378
$$494$$ 0 0
$$495$$ −1944.00 −0.176518
$$496$$ 0 0
$$497$$ 5880.00 0.530692
$$498$$ 0 0
$$499$$ −2396.00 −0.214949 −0.107475 0.994208i $$-0.534276\pi$$
−0.107475 + 0.994208i $$0.534276\pi$$
$$500$$ 0 0
$$501$$ −9144.00 −0.815417
$$502$$ 0 0
$$503$$ −13128.0 −1.16371 −0.581857 0.813291i $$-0.697674\pi$$
−0.581857 + 0.813291i $$0.697674\pi$$
$$504$$ 0 0
$$505$$ 36.0000 0.00317224
$$506$$ 0 0
$$507$$ 4941.00 0.432816
$$508$$ 0 0
$$509$$ 12798.0 1.11446 0.557231 0.830357i $$-0.311864\pi$$
0.557231 + 0.830357i $$0.311864\pi$$
$$510$$ 0 0
$$511$$ −6230.00 −0.539333
$$512$$ 0 0
$$513$$ 2052.00 0.176604
$$514$$ 0 0
$$515$$ 2832.00 0.242316
$$516$$ 0 0
$$517$$ −6912.00 −0.587987
$$518$$ 0 0
$$519$$ −8118.00 −0.686591
$$520$$ 0 0
$$521$$ 7386.00 0.621087 0.310544 0.950559i $$-0.399489\pi$$
0.310544 + 0.950559i $$0.399489\pi$$
$$522$$ 0 0
$$523$$ −5180.00 −0.433089 −0.216545 0.976273i $$-0.569479\pi$$
−0.216545 + 0.976273i $$0.569479\pi$$
$$524$$ 0 0
$$525$$ 1869.00 0.155371
$$526$$ 0 0
$$527$$ 12768.0 1.05538
$$528$$ 0 0
$$529$$ −11591.0 −0.952659
$$530$$ 0 0
$$531$$ −3564.00 −0.291270
$$532$$ 0 0
$$533$$ 23436.0 1.90455
$$534$$ 0 0
$$535$$ 5832.00 0.471288
$$536$$ 0 0
$$537$$ −14148.0 −1.13693
$$538$$ 0 0
$$539$$ −1764.00 −0.140966
$$540$$ 0 0
$$541$$ 4070.00 0.323444 0.161722 0.986836i $$-0.448295\pi$$
0.161722 + 0.986836i $$0.448295\pi$$
$$542$$ 0 0
$$543$$ 5730.00 0.452851
$$544$$ 0 0
$$545$$ −10716.0 −0.842244
$$546$$ 0 0
$$547$$ −14780.0 −1.15530 −0.577648 0.816286i $$-0.696029\pi$$
−0.577648 + 0.816286i $$0.696029\pi$$
$$548$$ 0 0
$$549$$ 2286.00 0.177712
$$550$$ 0 0
$$551$$ 4104.00 0.317307
$$552$$ 0 0
$$553$$ 560.000 0.0430626
$$554$$ 0 0
$$555$$ −3204.00 −0.245049
$$556$$ 0 0
$$557$$ −6858.00 −0.521693 −0.260846 0.965380i $$-0.584002\pi$$
−0.260846 + 0.965380i $$0.584002\pi$$
$$558$$ 0 0
$$559$$ 10664.0 0.806868
$$560$$ 0 0
$$561$$ −12312.0 −0.926583
$$562$$ 0 0
$$563$$ −6660.00 −0.498553 −0.249277 0.968432i $$-0.580193\pi$$
−0.249277 + 0.968432i $$0.580193\pi$$
$$564$$ 0 0
$$565$$ −13716.0 −1.02130
$$566$$ 0 0
$$567$$ −567.000 −0.0419961
$$568$$ 0 0
$$569$$ −150.000 −0.0110515 −0.00552577 0.999985i $$-0.501759\pi$$
−0.00552577 + 0.999985i $$0.501759\pi$$
$$570$$ 0 0
$$571$$ 8188.00 0.600100 0.300050 0.953923i $$-0.402997\pi$$
0.300050 + 0.953923i $$0.402997\pi$$
$$572$$ 0 0
$$573$$ −12240.0 −0.892379
$$574$$ 0 0
$$575$$ −2136.00 −0.154917
$$576$$ 0 0
$$577$$ −5854.00 −0.422366 −0.211183 0.977447i $$-0.567732\pi$$
−0.211183 + 0.977447i $$0.567732\pi$$
$$578$$ 0 0
$$579$$ −8058.00 −0.578375
$$580$$ 0 0
$$581$$ −756.000 −0.0539831
$$582$$ 0 0
$$583$$ 14472.0 1.02808
$$584$$ 0 0
$$585$$ 3348.00 0.236620
$$586$$ 0 0
$$587$$ −17580.0 −1.23612 −0.618062 0.786130i $$-0.712082\pi$$
−0.618062 + 0.786130i $$0.712082\pi$$
$$588$$ 0 0
$$589$$ 8512.00 0.595468
$$590$$ 0 0
$$591$$ 1530.00 0.106490
$$592$$ 0 0
$$593$$ 17154.0 1.18791 0.593955 0.804498i $$-0.297566\pi$$
0.593955 + 0.804498i $$0.297566\pi$$
$$594$$ 0 0
$$595$$ −4788.00 −0.329897
$$596$$ 0 0
$$597$$ −4056.00 −0.278059
$$598$$ 0 0
$$599$$ 18120.0 1.23600 0.617999 0.786179i $$-0.287943\pi$$
0.617999 + 0.786179i $$0.287943\pi$$
$$600$$ 0 0
$$601$$ 17546.0 1.19088 0.595438 0.803401i $$-0.296979\pi$$
0.595438 + 0.803401i $$0.296979\pi$$
$$602$$ 0 0
$$603$$ 9108.00 0.615102
$$604$$ 0 0
$$605$$ −210.000 −0.0141119
$$606$$ 0 0
$$607$$ 14560.0 0.973595 0.486798 0.873515i $$-0.338165\pi$$
0.486798 + 0.873515i $$0.338165\pi$$
$$608$$ 0 0
$$609$$ −1134.00 −0.0754548
$$610$$ 0 0
$$611$$ 11904.0 0.788190
$$612$$ 0 0
$$613$$ −4498.00 −0.296366 −0.148183 0.988960i $$-0.547343\pi$$
−0.148183 + 0.988960i $$0.547343\pi$$
$$614$$ 0 0
$$615$$ 6804.00 0.446120
$$616$$ 0 0
$$617$$ −5478.00 −0.357433 −0.178716 0.983901i $$-0.557194\pi$$
−0.178716 + 0.983901i $$0.557194\pi$$
$$618$$ 0 0
$$619$$ −6044.00 −0.392454 −0.196227 0.980559i $$-0.562869\pi$$
−0.196227 + 0.980559i $$0.562869\pi$$
$$620$$ 0 0
$$621$$ 648.000 0.0418733
$$622$$ 0 0
$$623$$ 11466.0 0.737360
$$624$$ 0 0
$$625$$ 3421.00 0.218944
$$626$$ 0 0
$$627$$ −8208.00 −0.522801
$$628$$ 0 0
$$629$$ −20292.0 −1.28632
$$630$$ 0 0
$$631$$ 15352.0 0.968547 0.484274 0.874917i $$-0.339084\pi$$
0.484274 + 0.874917i $$0.339084\pi$$
$$632$$ 0 0
$$633$$ 10092.0 0.633682
$$634$$ 0 0
$$635$$ −7968.00 −0.497953
$$636$$ 0 0
$$637$$ 3038.00 0.188964
$$638$$ 0 0
$$639$$ −7560.00 −0.468027
$$640$$ 0 0
$$641$$ −22398.0 −1.38014 −0.690068 0.723744i $$-0.742420\pi$$
−0.690068 + 0.723744i $$0.742420\pi$$
$$642$$ 0 0
$$643$$ −3764.00 −0.230852 −0.115426 0.993316i $$-0.536823\pi$$
−0.115426 + 0.993316i $$0.536823\pi$$
$$644$$ 0 0
$$645$$ 3096.00 0.189000
$$646$$ 0 0
$$647$$ 17688.0 1.07479 0.537393 0.843332i $$-0.319409\pi$$
0.537393 + 0.843332i $$0.319409\pi$$
$$648$$ 0 0
$$649$$ 14256.0 0.862245
$$650$$ 0 0
$$651$$ −2352.00 −0.141601
$$652$$ 0 0
$$653$$ 19878.0 1.19125 0.595625 0.803263i $$-0.296904\pi$$
0.595625 + 0.803263i $$0.296904\pi$$
$$654$$ 0 0
$$655$$ 7272.00 0.433802
$$656$$ 0 0
$$657$$ 8010.00 0.475647
$$658$$ 0 0
$$659$$ 20004.0 1.18247 0.591233 0.806501i $$-0.298641\pi$$
0.591233 + 0.806501i $$0.298641\pi$$
$$660$$ 0 0
$$661$$ −1306.00 −0.0768495 −0.0384247 0.999261i $$-0.512234\pi$$
−0.0384247 + 0.999261i $$0.512234\pi$$
$$662$$ 0 0
$$663$$ 21204.0 1.24207
$$664$$ 0 0
$$665$$ −3192.00 −0.186136
$$666$$ 0 0
$$667$$ 1296.00 0.0752344
$$668$$ 0 0
$$669$$ 14304.0 0.826644
$$670$$ 0 0
$$671$$ −9144.00 −0.526081
$$672$$ 0 0
$$673$$ −13054.0 −0.747689 −0.373845 0.927491i $$-0.621961\pi$$
−0.373845 + 0.927491i $$0.621961\pi$$
$$674$$ 0 0
$$675$$ −2403.00 −0.137024
$$676$$ 0 0
$$677$$ 5046.00 0.286460 0.143230 0.989689i $$-0.454251\pi$$
0.143230 + 0.989689i $$0.454251\pi$$
$$678$$ 0 0
$$679$$ −7070.00 −0.399590
$$680$$ 0 0
$$681$$ −1260.00 −0.0709006
$$682$$ 0 0
$$683$$ −12468.0 −0.698499 −0.349249 0.937030i $$-0.613563\pi$$
−0.349249 + 0.937030i $$0.613563\pi$$
$$684$$ 0 0
$$685$$ −7524.00 −0.419675
$$686$$ 0 0
$$687$$ −5646.00 −0.313549
$$688$$ 0 0
$$689$$ −24924.0 −1.37813
$$690$$ 0 0
$$691$$ 23212.0 1.27790 0.638948 0.769250i $$-0.279370\pi$$
0.638948 + 0.769250i $$0.279370\pi$$
$$692$$ 0 0
$$693$$ 2268.00 0.124321
$$694$$ 0 0
$$695$$ 2040.00 0.111340
$$696$$ 0 0
$$697$$ 43092.0 2.34179
$$698$$ 0 0
$$699$$ 15246.0 0.824974
$$700$$ 0 0
$$701$$ 35958.0 1.93740 0.968698 0.248241i $$-0.0798526\pi$$
0.968698 + 0.248241i $$0.0798526\pi$$
$$702$$ 0 0
$$703$$ −13528.0 −0.725773
$$704$$ 0 0
$$705$$ 3456.00 0.184625
$$706$$ 0 0
$$707$$ −42.0000 −0.00223419
$$708$$ 0 0
$$709$$ 6446.00 0.341445 0.170723 0.985319i $$-0.445390\pi$$
0.170723 + 0.985319i $$0.445390\pi$$
$$710$$ 0 0
$$711$$ −720.000 −0.0379777
$$712$$ 0 0
$$713$$ 2688.00 0.141187
$$714$$ 0 0
$$715$$ −13392.0 −0.700465
$$716$$ 0 0
$$717$$ 16272.0 0.847544
$$718$$ 0 0
$$719$$ 4704.00 0.243991 0.121996 0.992531i $$-0.461071\pi$$
0.121996 + 0.992531i $$0.461071\pi$$
$$720$$ 0 0
$$721$$ −3304.00 −0.170662
$$722$$ 0 0
$$723$$ −7770.00 −0.399681
$$724$$ 0 0
$$725$$ −4806.00 −0.246194
$$726$$ 0 0
$$727$$ 10600.0 0.540760 0.270380 0.962754i $$-0.412851\pi$$
0.270380 + 0.962754i $$0.412851\pi$$
$$728$$ 0 0
$$729$$ 729.000 0.0370370
$$730$$ 0 0
$$731$$ 19608.0 0.992104
$$732$$ 0 0
$$733$$ 12542.0 0.631991 0.315995 0.948761i $$-0.397662\pi$$
0.315995 + 0.948761i $$0.397662\pi$$
$$734$$ 0 0
$$735$$ 882.000 0.0442627
$$736$$ 0 0
$$737$$ −36432.0 −1.82088
$$738$$ 0 0
$$739$$ −23324.0 −1.16101 −0.580506 0.814256i $$-0.697145\pi$$
−0.580506 + 0.814256i $$0.697145\pi$$
$$740$$ 0 0
$$741$$ 14136.0 0.700808
$$742$$ 0 0
$$743$$ 6312.00 0.311662 0.155831 0.987784i $$-0.450194\pi$$
0.155831 + 0.987784i $$0.450194\pi$$
$$744$$ 0 0
$$745$$ 6228.00 0.306277
$$746$$ 0 0
$$747$$ 972.000 0.0476086
$$748$$ 0 0
$$749$$ −6804.00 −0.331926
$$750$$ 0 0
$$751$$ −35840.0 −1.74144 −0.870719 0.491781i $$-0.836346\pi$$
−0.870719 + 0.491781i $$0.836346\pi$$
$$752$$ 0 0
$$753$$ 14796.0 0.716064
$$754$$ 0 0
$$755$$ −17616.0 −0.849155
$$756$$ 0 0
$$757$$ −34594.0 −1.66095 −0.830476 0.557055i $$-0.811931\pi$$
−0.830476 + 0.557055i $$0.811931\pi$$
$$758$$ 0 0
$$759$$ −2592.00 −0.123957
$$760$$ 0 0
$$761$$ 23946.0 1.14066 0.570330 0.821416i $$-0.306815\pi$$
0.570330 + 0.821416i $$0.306815\pi$$
$$762$$ 0 0
$$763$$ 12502.0 0.593188
$$764$$ 0 0
$$765$$ 6156.00 0.290942
$$766$$ 0 0
$$767$$ −24552.0 −1.15583
$$768$$ 0 0
$$769$$ 18770.0 0.880187 0.440093 0.897952i $$-0.354945\pi$$
0.440093 + 0.897952i $$0.354945\pi$$
$$770$$ 0 0
$$771$$ −10314.0 −0.481776
$$772$$ 0 0
$$773$$ 30342.0 1.41181 0.705903 0.708309i $$-0.250541\pi$$
0.705903 + 0.708309i $$0.250541\pi$$
$$774$$ 0 0
$$775$$ −9968.00 −0.462014
$$776$$ 0 0
$$777$$ 3738.00 0.172587
$$778$$ 0 0
$$779$$ 28728.0 1.32129
$$780$$ 0 0
$$781$$ 30240.0 1.38550
$$782$$ 0 0
$$783$$ 1458.00 0.0665449
$$784$$ 0 0
$$785$$ −7980.00 −0.362826
$$786$$ 0 0
$$787$$ 26188.0 1.18615 0.593076 0.805147i $$-0.297913\pi$$
0.593076 + 0.805147i $$0.297913\pi$$
$$788$$ 0 0
$$789$$ −18360.0 −0.828433
$$790$$ 0 0
$$791$$ 16002.0 0.719299
$$792$$ 0 0
$$793$$ 15748.0 0.705205
$$794$$ 0 0
$$795$$ −7236.00 −0.322811
$$796$$ 0 0
$$797$$ −34818.0 −1.54745 −0.773724 0.633522i $$-0.781609\pi$$
−0.773724 + 0.633522i $$0.781609\pi$$
$$798$$ 0 0
$$799$$ 21888.0 0.969139
$$800$$ 0 0
$$801$$ −14742.0 −0.650291
$$802$$ 0 0
$$803$$ −32040.0 −1.40805
$$804$$ 0 0
$$805$$ −1008.00 −0.0441333
$$806$$ 0 0
$$807$$ −54.0000 −0.00235550
$$808$$ 0 0
$$809$$ −21702.0 −0.943142 −0.471571 0.881828i $$-0.656313\pi$$
−0.471571 + 0.881828i $$0.656313\pi$$
$$810$$ 0 0
$$811$$ 20356.0 0.881376 0.440688 0.897660i $$-0.354735\pi$$
0.440688 + 0.897660i $$0.354735\pi$$
$$812$$ 0 0
$$813$$ −20688.0 −0.892448
$$814$$ 0 0
$$815$$ 20184.0 0.867503
$$816$$ 0 0
$$817$$ 13072.0 0.559769
$$818$$ 0 0
$$819$$ −3906.00 −0.166650
$$820$$ 0 0
$$821$$ −19890.0 −0.845513 −0.422756 0.906243i $$-0.638937\pi$$
−0.422756 + 0.906243i $$0.638937\pi$$
$$822$$ 0 0
$$823$$ −4232.00 −0.179245 −0.0896223 0.995976i $$-0.528566\pi$$
−0.0896223 + 0.995976i $$0.528566\pi$$
$$824$$ 0 0
$$825$$ 9612.00 0.405633
$$826$$ 0 0
$$827$$ −9636.00 −0.405171 −0.202586 0.979265i $$-0.564934\pi$$
−0.202586 + 0.979265i $$0.564934\pi$$
$$828$$ 0 0
$$829$$ 35294.0 1.47866 0.739331 0.673342i $$-0.235142\pi$$
0.739331 + 0.673342i $$0.235142\pi$$
$$830$$ 0 0
$$831$$ 18762.0 0.783209
$$832$$ 0 0
$$833$$ 5586.00 0.232345
$$834$$ 0 0
$$835$$ −18288.0 −0.757943
$$836$$ 0 0
$$837$$ 3024.00 0.124880
$$838$$ 0 0
$$839$$ 3768.00 0.155049 0.0775243 0.996990i $$-0.475298\pi$$
0.0775243 + 0.996990i $$0.475298\pi$$
$$840$$ 0 0
$$841$$ −21473.0 −0.880438
$$842$$ 0 0
$$843$$ 3582.00 0.146347
$$844$$ 0 0
$$845$$ 9882.00 0.402309
$$846$$ 0 0
$$847$$ 245.000 0.00993896
$$848$$ 0 0
$$849$$ 21468.0 0.867821
$$850$$ 0 0
$$851$$ −4272.00 −0.172083
$$852$$ 0 0
$$853$$ −39466.0 −1.58416 −0.792081 0.610416i $$-0.791002\pi$$
−0.792081 + 0.610416i $$0.791002\pi$$
$$854$$ 0 0
$$855$$ 4104.00 0.164157
$$856$$ 0 0
$$857$$ −34038.0 −1.35673 −0.678364 0.734726i $$-0.737311\pi$$
−0.678364 + 0.734726i $$0.737311\pi$$
$$858$$ 0 0
$$859$$ 3364.00 0.133618 0.0668092 0.997766i $$-0.478718\pi$$
0.0668092 + 0.997766i $$0.478718\pi$$
$$860$$ 0 0
$$861$$ −7938.00 −0.314200
$$862$$ 0 0
$$863$$ −13104.0 −0.516878 −0.258439 0.966028i $$-0.583208\pi$$
−0.258439 + 0.966028i $$0.583208\pi$$
$$864$$ 0 0
$$865$$ −16236.0 −0.638197
$$866$$ 0 0
$$867$$ 24249.0 0.949872
$$868$$ 0 0
$$869$$ 2880.00 0.112425
$$870$$ 0 0
$$871$$ 62744.0 2.44087
$$872$$ 0 0
$$873$$ 9090.00 0.352405
$$874$$ 0 0
$$875$$ 8988.00 0.347257
$$876$$ 0 0
$$877$$ −40858.0 −1.57318 −0.786589 0.617477i $$-0.788155\pi$$
−0.786589 + 0.617477i $$0.788155\pi$$
$$878$$ 0 0
$$879$$ −11214.0 −0.430306
$$880$$ 0 0
$$881$$ −37374.0 −1.42924 −0.714621 0.699512i $$-0.753401\pi$$
−0.714621 + 0.699512i $$0.753401\pi$$
$$882$$ 0 0
$$883$$ −9788.00 −0.373038 −0.186519 0.982451i $$-0.559721\pi$$
−0.186519 + 0.982451i $$0.559721\pi$$
$$884$$ 0 0
$$885$$ −7128.00 −0.270740
$$886$$ 0 0
$$887$$ −50424.0 −1.90876 −0.954381 0.298591i $$-0.903483\pi$$
−0.954381 + 0.298591i $$0.903483\pi$$
$$888$$ 0 0
$$889$$ 9296.00 0.350706
$$890$$ 0 0
$$891$$ −2916.00 −0.109640
$$892$$ 0 0
$$893$$ 14592.0 0.546811
$$894$$ 0 0
$$895$$ −28296.0 −1.05679
$$896$$ 0 0
$$897$$ 4464.00 0.166163
$$898$$ 0 0
$$899$$ 6048.00 0.224374
$$900$$ 0 0
$$901$$ −45828.0 −1.69451
$$902$$ 0 0
$$903$$ −3612.00 −0.133112
$$904$$ 0 0
$$905$$ 11460.0 0.420932
$$906$$ 0 0
$$907$$ 12412.0 0.454392 0.227196 0.973849i $$-0.427044\pi$$
0.227196 + 0.973849i $$0.427044\pi$$
$$908$$ 0 0
$$909$$ 54.0000 0.00197037
$$910$$ 0 0
$$911$$ −6576.00 −0.239158 −0.119579 0.992825i $$-0.538154\pi$$
−0.119579 + 0.992825i $$0.538154\pi$$
$$912$$ 0 0
$$913$$ −3888.00 −0.140935
$$914$$ 0 0
$$915$$ 4572.00 0.165187
$$916$$ 0 0
$$917$$ −8484.00 −0.305525
$$918$$ 0 0
$$919$$ −8264.00 −0.296631 −0.148316 0.988940i $$-0.547385\pi$$
−0.148316 + 0.988940i $$0.547385\pi$$
$$920$$ 0 0
$$921$$ 2532.00 0.0905887
$$922$$ 0 0
$$923$$ −52080.0 −1.85724
$$924$$ 0 0
$$925$$ 15842.0 0.563115
$$926$$ 0 0
$$927$$ 4248.00 0.150510
$$928$$ 0 0
$$929$$ 39426.0 1.39238 0.696192 0.717855i $$-0.254876\pi$$
0.696192 + 0.717855i $$0.254876\pi$$
$$930$$ 0 0
$$931$$ 3724.00 0.131095
$$932$$ 0 0
$$933$$ −18936.0 −0.664455
$$934$$ 0 0
$$935$$ −24624.0 −0.861274
$$936$$ 0 0
$$937$$ −4678.00 −0.163099 −0.0815494 0.996669i $$-0.525987\pi$$
−0.0815494 + 0.996669i $$0.525987\pi$$
$$938$$ 0 0
$$939$$ 24846.0 0.863492
$$940$$ 0 0
$$941$$ −17346.0 −0.600918 −0.300459 0.953795i $$-0.597140\pi$$
−0.300459 + 0.953795i $$0.597140\pi$$
$$942$$ 0 0
$$943$$ 9072.00 0.313282
$$944$$ 0 0
$$945$$ −1134.00 −0.0390360
$$946$$ 0 0
$$947$$ −19452.0 −0.667482 −0.333741 0.942665i $$-0.608311\pi$$
−0.333741 + 0.942665i $$0.608311\pi$$
$$948$$ 0 0
$$949$$ 55180.0 1.88748
$$950$$ 0 0
$$951$$ 27954.0 0.953176
$$952$$ 0 0
$$953$$ 4458.00 0.151531 0.0757654 0.997126i $$-0.475860\pi$$
0.0757654 + 0.997126i $$0.475860\pi$$
$$954$$ 0 0
$$955$$ −24480.0 −0.829481
$$956$$ 0 0
$$957$$ −5832.00 −0.196992
$$958$$ 0 0
$$959$$ 8778.00 0.295575
$$960$$ 0 0
$$961$$ −17247.0 −0.578933
$$962$$ 0 0
$$963$$ 8748.00 0.292731
$$964$$ 0 0
$$965$$ −16116.0 −0.537609
$$966$$ 0 0
$$967$$ −52520.0 −1.74657 −0.873283 0.487213i $$-0.838013\pi$$
−0.873283 + 0.487213i $$0.838013\pi$$
$$968$$ 0 0
$$969$$ 25992.0 0.861696
$$970$$ 0 0
$$971$$ 10404.0 0.343852 0.171926 0.985110i $$-0.445001\pi$$
0.171926 + 0.985110i $$0.445001\pi$$
$$972$$ 0 0
$$973$$ −2380.00 −0.0784165
$$974$$ 0 0
$$975$$ −16554.0 −0.543746
$$976$$ 0 0
$$977$$ −7566.00 −0.247756 −0.123878 0.992297i $$-0.539533\pi$$
−0.123878 + 0.992297i $$0.539533\pi$$
$$978$$ 0 0
$$979$$ 58968.0 1.92505
$$980$$ 0 0
$$981$$ −16074.0 −0.523143
$$982$$ 0 0
$$983$$ −44376.0 −1.43985 −0.719926 0.694051i $$-0.755824\pi$$
−0.719926 + 0.694051i $$0.755824\pi$$
$$984$$ 0 0
$$985$$ 3060.00 0.0989845
$$986$$ 0 0
$$987$$ −4032.00 −0.130030
$$988$$ 0 0
$$989$$ 4128.00 0.132723
$$990$$ 0 0
$$991$$ 27328.0 0.875986 0.437993 0.898978i $$-0.355689\pi$$
0.437993 + 0.898978i $$0.355689\pi$$
$$992$$ 0 0
$$993$$ −4956.00 −0.158383
$$994$$ 0 0
$$995$$ −8112.00 −0.258460
$$996$$ 0 0
$$997$$ 2774.00 0.0881178 0.0440589 0.999029i $$-0.485971\pi$$
0.0440589 + 0.999029i $$0.485971\pi$$
$$998$$ 0 0
$$999$$ −4806.00 −0.152207
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 336.4.a.k.1.1 1
3.2 odd 2 1008.4.a.h.1.1 1
4.3 odd 2 84.4.a.a.1.1 1
7.6 odd 2 2352.4.a.d.1.1 1
8.3 odd 2 1344.4.a.q.1.1 1
8.5 even 2 1344.4.a.d.1.1 1
12.11 even 2 252.4.a.b.1.1 1
20.3 even 4 2100.4.k.j.1849.1 2
20.7 even 4 2100.4.k.j.1849.2 2
20.19 odd 2 2100.4.a.l.1.1 1
28.3 even 6 588.4.i.c.373.1 2
28.11 odd 6 588.4.i.f.373.1 2
28.19 even 6 588.4.i.c.361.1 2
28.23 odd 6 588.4.i.f.361.1 2
28.27 even 2 588.4.a.d.1.1 1
84.11 even 6 1764.4.k.l.1549.1 2
84.23 even 6 1764.4.k.l.361.1 2
84.47 odd 6 1764.4.k.f.361.1 2
84.59 odd 6 1764.4.k.f.1549.1 2
84.83 odd 2 1764.4.a.j.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
84.4.a.a.1.1 1 4.3 odd 2
252.4.a.b.1.1 1 12.11 even 2
336.4.a.k.1.1 1 1.1 even 1 trivial
588.4.a.d.1.1 1 28.27 even 2
588.4.i.c.361.1 2 28.19 even 6
588.4.i.c.373.1 2 28.3 even 6
588.4.i.f.361.1 2 28.23 odd 6
588.4.i.f.373.1 2 28.11 odd 6
1008.4.a.h.1.1 1 3.2 odd 2
1344.4.a.d.1.1 1 8.5 even 2
1344.4.a.q.1.1 1 8.3 odd 2
1764.4.a.j.1.1 1 84.83 odd 2
1764.4.k.f.361.1 2 84.47 odd 6
1764.4.k.f.1549.1 2 84.59 odd 6
1764.4.k.l.361.1 2 84.23 even 6
1764.4.k.l.1549.1 2 84.11 even 6
2100.4.a.l.1.1 1 20.19 odd 2
2100.4.k.j.1849.1 2 20.3 even 4
2100.4.k.j.1849.2 2 20.7 even 4
2352.4.a.d.1.1 1 7.6 odd 2