# Properties

 Label 2352.4.a.cq.1.2 Level $2352$ Weight $4$ Character 2352.1 Self dual yes Analytic conductor $138.772$ Analytic rank $0$ Dimension $4$ CM no Inner twists $1$

# Learn more about

## Newspace parameters

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

## Newform invariants

 Self dual: yes Analytic conductor: $$138.772492334$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: 4.4.136768.1 Defining polynomial: $$x^{4} - 2 x^{3} - 23 x^{2} + 18 x + 119$$ Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$2^{4}\cdot 7$$ Twist minimal: no (minimal twist has level 588) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.2 Root $$2.89590$$ of defining polynomial Character $$\chi$$ $$=$$ 2352.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q+3.00000 q^{3} -8.16940 q^{5} +9.00000 q^{9} +O(q^{10})$$ $$q+3.00000 q^{3} -8.16940 q^{5} +9.00000 q^{9} -37.8189 q^{11} -39.9319 q^{13} -24.5082 q^{15} -9.93596 q^{17} -90.4458 q^{19} -118.595 q^{23} -58.2608 q^{25} +27.0000 q^{27} -78.4061 q^{29} +92.0110 q^{31} -113.457 q^{33} +332.435 q^{37} -119.796 q^{39} -71.7451 q^{41} +115.947 q^{43} -73.5246 q^{45} +307.927 q^{47} -29.8079 q^{51} -403.150 q^{53} +308.958 q^{55} -271.337 q^{57} +593.710 q^{59} -333.170 q^{61} +326.220 q^{65} +743.511 q^{67} -355.784 q^{69} +728.272 q^{71} +801.749 q^{73} -174.782 q^{75} -1067.91 q^{79} +81.0000 q^{81} +906.756 q^{83} +81.1709 q^{85} -235.218 q^{87} -1113.27 q^{89} +276.033 q^{93} +738.889 q^{95} -1480.94 q^{97} -340.370 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + 12q^{3} + 36q^{9} + O(q^{10})$$ $$4q + 12q^{3} + 36q^{9} - 48q^{17} + 192q^{19} - 192q^{23} + 324q^{25} + 108q^{27} + 96q^{29} + 48q^{31} + 256q^{37} - 1008q^{41} + 112q^{43} + 864q^{47} - 144q^{51} - 648q^{53} + 2352q^{55} + 576q^{57} + 336q^{59} - 960q^{61} - 360q^{65} - 720q^{67} - 576q^{69} + 1344q^{71} - 672q^{73} + 972q^{75} + 1984q^{79} + 324q^{81} + 3120q^{83} + 680q^{85} + 288q^{87} - 2160q^{89} + 144q^{93} + 3744q^{95} - 2016q^{97} + O(q^{100})$$

## Coefficient data

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

Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 0 0
$$3$$ 3.00000 0.577350
$$4$$ 0 0
$$5$$ −8.16940 −0.730694 −0.365347 0.930871i $$-0.619050\pi$$
−0.365347 + 0.930871i $$0.619050\pi$$
$$6$$ 0 0
$$7$$ 0 0
$$8$$ 0 0
$$9$$ 9.00000 0.333333
$$10$$ 0 0
$$11$$ −37.8189 −1.03662 −0.518310 0.855193i $$-0.673439\pi$$
−0.518310 + 0.855193i $$0.673439\pi$$
$$12$$ 0 0
$$13$$ −39.9319 −0.851933 −0.425966 0.904739i $$-0.640066\pi$$
−0.425966 + 0.904739i $$0.640066\pi$$
$$14$$ 0 0
$$15$$ −24.5082 −0.421866
$$16$$ 0 0
$$17$$ −9.93596 −0.141754 −0.0708772 0.997485i $$-0.522580\pi$$
−0.0708772 + 0.997485i $$0.522580\pi$$
$$18$$ 0 0
$$19$$ −90.4458 −1.09209 −0.546045 0.837756i $$-0.683867\pi$$
−0.546045 + 0.837756i $$0.683867\pi$$
$$20$$ 0 0
$$21$$ 0 0
$$22$$ 0 0
$$23$$ −118.595 −1.07516 −0.537580 0.843213i $$-0.680661\pi$$
−0.537580 + 0.843213i $$0.680661\pi$$
$$24$$ 0 0
$$25$$ −58.2608 −0.466087
$$26$$ 0 0
$$27$$ 27.0000 0.192450
$$28$$ 0 0
$$29$$ −78.4061 −0.502057 −0.251028 0.967980i $$-0.580769\pi$$
−0.251028 + 0.967980i $$0.580769\pi$$
$$30$$ 0 0
$$31$$ 92.0110 0.533086 0.266543 0.963823i $$-0.414119\pi$$
0.266543 + 0.963823i $$0.414119\pi$$
$$32$$ 0 0
$$33$$ −113.457 −0.598493
$$34$$ 0 0
$$35$$ 0 0
$$36$$ 0 0
$$37$$ 332.435 1.47708 0.738541 0.674209i $$-0.235515\pi$$
0.738541 + 0.674209i $$0.235515\pi$$
$$38$$ 0 0
$$39$$ −119.796 −0.491864
$$40$$ 0 0
$$41$$ −71.7451 −0.273285 −0.136643 0.990620i $$-0.543631\pi$$
−0.136643 + 0.990620i $$0.543631\pi$$
$$42$$ 0 0
$$43$$ 115.947 0.411202 0.205601 0.978636i $$-0.434085\pi$$
0.205601 + 0.978636i $$0.434085\pi$$
$$44$$ 0 0
$$45$$ −73.5246 −0.243565
$$46$$ 0 0
$$47$$ 307.927 0.955656 0.477828 0.878453i $$-0.341424\pi$$
0.477828 + 0.878453i $$0.341424\pi$$
$$48$$ 0 0
$$49$$ 0 0
$$50$$ 0 0
$$51$$ −29.8079 −0.0818419
$$52$$ 0 0
$$53$$ −403.150 −1.04485 −0.522424 0.852686i $$-0.674972\pi$$
−0.522424 + 0.852686i $$0.674972\pi$$
$$54$$ 0 0
$$55$$ 308.958 0.757451
$$56$$ 0 0
$$57$$ −271.337 −0.630518
$$58$$ 0 0
$$59$$ 593.710 1.31008 0.655038 0.755596i $$-0.272653\pi$$
0.655038 + 0.755596i $$0.272653\pi$$
$$60$$ 0 0
$$61$$ −333.170 −0.699313 −0.349657 0.936878i $$-0.613702\pi$$
−0.349657 + 0.936878i $$0.613702\pi$$
$$62$$ 0 0
$$63$$ 0 0
$$64$$ 0 0
$$65$$ 326.220 0.622502
$$66$$ 0 0
$$67$$ 743.511 1.35574 0.677869 0.735183i $$-0.262904\pi$$
0.677869 + 0.735183i $$0.262904\pi$$
$$68$$ 0 0
$$69$$ −355.784 −0.620744
$$70$$ 0 0
$$71$$ 728.272 1.21732 0.608662 0.793429i $$-0.291706\pi$$
0.608662 + 0.793429i $$0.291706\pi$$
$$72$$ 0 0
$$73$$ 801.749 1.28545 0.642723 0.766098i $$-0.277804\pi$$
0.642723 + 0.766098i $$0.277804\pi$$
$$74$$ 0 0
$$75$$ −174.782 −0.269095
$$76$$ 0 0
$$77$$ 0 0
$$78$$ 0 0
$$79$$ −1067.91 −1.52087 −0.760435 0.649414i $$-0.775014\pi$$
−0.760435 + 0.649414i $$0.775014\pi$$
$$80$$ 0 0
$$81$$ 81.0000 0.111111
$$82$$ 0 0
$$83$$ 906.756 1.19915 0.599575 0.800319i $$-0.295336\pi$$
0.599575 + 0.800319i $$0.295336\pi$$
$$84$$ 0 0
$$85$$ 81.1709 0.103579
$$86$$ 0 0
$$87$$ −235.218 −0.289863
$$88$$ 0 0
$$89$$ −1113.27 −1.32591 −0.662957 0.748658i $$-0.730699\pi$$
−0.662957 + 0.748658i $$0.730699\pi$$
$$90$$ 0 0
$$91$$ 0 0
$$92$$ 0 0
$$93$$ 276.033 0.307777
$$94$$ 0 0
$$95$$ 738.889 0.797983
$$96$$ 0 0
$$97$$ −1480.94 −1.55017 −0.775084 0.631858i $$-0.782293\pi$$
−0.775084 + 0.631858i $$0.782293\pi$$
$$98$$ 0 0
$$99$$ −340.370 −0.345540
$$100$$ 0 0
$$101$$ 556.316 0.548075 0.274037 0.961719i $$-0.411641\pi$$
0.274037 + 0.961719i $$0.411641\pi$$
$$102$$ 0 0
$$103$$ 552.435 0.528476 0.264238 0.964458i $$-0.414880\pi$$
0.264238 + 0.964458i $$0.414880\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ −533.804 −0.482288 −0.241144 0.970489i $$-0.577523\pi$$
−0.241144 + 0.970489i $$0.577523\pi$$
$$108$$ 0 0
$$109$$ 1094.63 0.961894 0.480947 0.876750i $$-0.340293\pi$$
0.480947 + 0.876750i $$0.340293\pi$$
$$110$$ 0 0
$$111$$ 997.306 0.852794
$$112$$ 0 0
$$113$$ −1425.18 −1.18646 −0.593228 0.805035i $$-0.702147\pi$$
−0.593228 + 0.805035i $$0.702147\pi$$
$$114$$ 0 0
$$115$$ 968.847 0.785612
$$116$$ 0 0
$$117$$ −359.387 −0.283978
$$118$$ 0 0
$$119$$ 0 0
$$120$$ 0 0
$$121$$ 99.2659 0.0745800
$$122$$ 0 0
$$123$$ −215.235 −0.157781
$$124$$ 0 0
$$125$$ 1497.13 1.07126
$$126$$ 0 0
$$127$$ 786.485 0.549522 0.274761 0.961513i $$-0.411401\pi$$
0.274761 + 0.961513i $$0.411401\pi$$
$$128$$ 0 0
$$129$$ 347.840 0.237407
$$130$$ 0 0
$$131$$ −27.7303 −0.0184947 −0.00924735 0.999957i $$-0.502944\pi$$
−0.00924735 + 0.999957i $$0.502944\pi$$
$$132$$ 0 0
$$133$$ 0 0
$$134$$ 0 0
$$135$$ −220.574 −0.140622
$$136$$ 0 0
$$137$$ −2362.98 −1.47360 −0.736798 0.676113i $$-0.763663\pi$$
−0.736798 + 0.676113i $$0.763663\pi$$
$$138$$ 0 0
$$139$$ 2513.28 1.53362 0.766811 0.641873i $$-0.221842\pi$$
0.766811 + 0.641873i $$0.221842\pi$$
$$140$$ 0 0
$$141$$ 923.782 0.551748
$$142$$ 0 0
$$143$$ 1510.18 0.883130
$$144$$ 0 0
$$145$$ 640.531 0.366850
$$146$$ 0 0
$$147$$ 0 0
$$148$$ 0 0
$$149$$ 2265.83 1.24580 0.622900 0.782302i $$-0.285955\pi$$
0.622900 + 0.782302i $$0.285955\pi$$
$$150$$ 0 0
$$151$$ −283.146 −0.152597 −0.0762984 0.997085i $$-0.524310\pi$$
−0.0762984 + 0.997085i $$0.524310\pi$$
$$152$$ 0 0
$$153$$ −89.4237 −0.0472515
$$154$$ 0 0
$$155$$ −751.675 −0.389522
$$156$$ 0 0
$$157$$ −192.581 −0.0978956 −0.0489478 0.998801i $$-0.515587\pi$$
−0.0489478 + 0.998801i $$0.515587\pi$$
$$158$$ 0 0
$$159$$ −1209.45 −0.603243
$$160$$ 0 0
$$161$$ 0 0
$$162$$ 0 0
$$163$$ 842.833 0.405005 0.202502 0.979282i $$-0.435093\pi$$
0.202502 + 0.979282i $$0.435093\pi$$
$$164$$ 0 0
$$165$$ 926.873 0.437315
$$166$$ 0 0
$$167$$ −3859.21 −1.78823 −0.894116 0.447836i $$-0.852195\pi$$
−0.894116 + 0.447836i $$0.852195\pi$$
$$168$$ 0 0
$$169$$ −602.441 −0.274211
$$170$$ 0 0
$$171$$ −814.012 −0.364030
$$172$$ 0 0
$$173$$ −1510.66 −0.663891 −0.331946 0.943298i $$-0.607705\pi$$
−0.331946 + 0.943298i $$0.607705\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ 0 0
$$177$$ 1781.13 0.756372
$$178$$ 0 0
$$179$$ 2464.63 1.02913 0.514566 0.857451i $$-0.327953\pi$$
0.514566 + 0.857451i $$0.327953\pi$$
$$180$$ 0 0
$$181$$ 3297.36 1.35409 0.677047 0.735940i $$-0.263259\pi$$
0.677047 + 0.735940i $$0.263259\pi$$
$$182$$ 0 0
$$183$$ −999.511 −0.403749
$$184$$ 0 0
$$185$$ −2715.80 −1.07929
$$186$$ 0 0
$$187$$ 375.767 0.146945
$$188$$ 0 0
$$189$$ 0 0
$$190$$ 0 0
$$191$$ 4729.71 1.79178 0.895889 0.444278i $$-0.146540\pi$$
0.895889 + 0.444278i $$0.146540\pi$$
$$192$$ 0 0
$$193$$ 4553.18 1.69816 0.849080 0.528264i $$-0.177157\pi$$
0.849080 + 0.528264i $$0.177157\pi$$
$$194$$ 0 0
$$195$$ 978.660 0.359402
$$196$$ 0 0
$$197$$ −3109.06 −1.12442 −0.562212 0.826993i $$-0.690050\pi$$
−0.562212 + 0.826993i $$0.690050\pi$$
$$198$$ 0 0
$$199$$ −443.512 −0.157989 −0.0789943 0.996875i $$-0.525171\pi$$
−0.0789943 + 0.996875i $$0.525171\pi$$
$$200$$ 0 0
$$201$$ 2230.53 0.782735
$$202$$ 0 0
$$203$$ 0 0
$$204$$ 0 0
$$205$$ 586.114 0.199688
$$206$$ 0 0
$$207$$ −1067.35 −0.358387
$$208$$ 0 0
$$209$$ 3420.56 1.13208
$$210$$ 0 0
$$211$$ −4653.28 −1.51822 −0.759112 0.650960i $$-0.774367\pi$$
−0.759112 + 0.650960i $$0.774367\pi$$
$$212$$ 0 0
$$213$$ 2184.82 0.702823
$$214$$ 0 0
$$215$$ −947.214 −0.300463
$$216$$ 0 0
$$217$$ 0 0
$$218$$ 0 0
$$219$$ 2405.25 0.742153
$$220$$ 0 0
$$221$$ 396.762 0.120765
$$222$$ 0 0
$$223$$ 2778.90 0.834481 0.417240 0.908796i $$-0.362997\pi$$
0.417240 + 0.908796i $$0.362997\pi$$
$$224$$ 0 0
$$225$$ −524.347 −0.155362
$$226$$ 0 0
$$227$$ 3217.22 0.940681 0.470340 0.882485i $$-0.344131\pi$$
0.470340 + 0.882485i $$0.344131\pi$$
$$228$$ 0 0
$$229$$ 3728.01 1.07578 0.537891 0.843014i $$-0.319221\pi$$
0.537891 + 0.843014i $$0.319221\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 0 0
$$233$$ −3603.51 −1.01319 −0.506596 0.862183i $$-0.669097\pi$$
−0.506596 + 0.862183i $$0.669097\pi$$
$$234$$ 0 0
$$235$$ −2515.58 −0.698292
$$236$$ 0 0
$$237$$ −3203.72 −0.878075
$$238$$ 0 0
$$239$$ −2348.76 −0.635684 −0.317842 0.948144i $$-0.602958\pi$$
−0.317842 + 0.948144i $$0.602958\pi$$
$$240$$ 0 0
$$241$$ 5093.39 1.36139 0.680693 0.732569i $$-0.261679\pi$$
0.680693 + 0.732569i $$0.261679\pi$$
$$242$$ 0 0
$$243$$ 243.000 0.0641500
$$244$$ 0 0
$$245$$ 0 0
$$246$$ 0 0
$$247$$ 3611.68 0.930386
$$248$$ 0 0
$$249$$ 2720.27 0.692329
$$250$$ 0 0
$$251$$ −5939.02 −1.49350 −0.746749 0.665106i $$-0.768386\pi$$
−0.746749 + 0.665106i $$0.768386\pi$$
$$252$$ 0 0
$$253$$ 4485.11 1.11453
$$254$$ 0 0
$$255$$ 243.513 0.0598014
$$256$$ 0 0
$$257$$ 1515.89 0.367932 0.183966 0.982933i $$-0.441106\pi$$
0.183966 + 0.982933i $$0.441106\pi$$
$$258$$ 0 0
$$259$$ 0 0
$$260$$ 0 0
$$261$$ −705.655 −0.167352
$$262$$ 0 0
$$263$$ 6433.92 1.50849 0.754244 0.656594i $$-0.228003\pi$$
0.754244 + 0.656594i $$0.228003\pi$$
$$264$$ 0 0
$$265$$ 3293.50 0.763464
$$266$$ 0 0
$$267$$ −3339.81 −0.765516
$$268$$ 0 0
$$269$$ 6949.79 1.57523 0.787614 0.616169i $$-0.211316\pi$$
0.787614 + 0.616169i $$0.211316\pi$$
$$270$$ 0 0
$$271$$ −961.994 −0.215635 −0.107817 0.994171i $$-0.534386\pi$$
−0.107817 + 0.994171i $$0.534386\pi$$
$$272$$ 0 0
$$273$$ 0 0
$$274$$ 0 0
$$275$$ 2203.36 0.483154
$$276$$ 0 0
$$277$$ 760.010 0.164854 0.0824271 0.996597i $$-0.473733\pi$$
0.0824271 + 0.996597i $$0.473733\pi$$
$$278$$ 0 0
$$279$$ 828.099 0.177695
$$280$$ 0 0
$$281$$ −4412.07 −0.936662 −0.468331 0.883553i $$-0.655145\pi$$
−0.468331 + 0.883553i $$0.655145\pi$$
$$282$$ 0 0
$$283$$ −2602.15 −0.546578 −0.273289 0.961932i $$-0.588112\pi$$
−0.273289 + 0.961932i $$0.588112\pi$$
$$284$$ 0 0
$$285$$ 2216.67 0.460716
$$286$$ 0 0
$$287$$ 0 0
$$288$$ 0 0
$$289$$ −4814.28 −0.979906
$$290$$ 0 0
$$291$$ −4442.81 −0.894990
$$292$$ 0 0
$$293$$ 9332.18 1.86072 0.930361 0.366644i $$-0.119493\pi$$
0.930361 + 0.366644i $$0.119493\pi$$
$$294$$ 0 0
$$295$$ −4850.26 −0.957264
$$296$$ 0 0
$$297$$ −1021.11 −0.199498
$$298$$ 0 0
$$299$$ 4735.71 0.915964
$$300$$ 0 0
$$301$$ 0 0
$$302$$ 0 0
$$303$$ 1668.95 0.316431
$$304$$ 0 0
$$305$$ 2721.80 0.510984
$$306$$ 0 0
$$307$$ 4895.51 0.910103 0.455051 0.890465i $$-0.349621\pi$$
0.455051 + 0.890465i $$0.349621\pi$$
$$308$$ 0 0
$$309$$ 1657.30 0.305116
$$310$$ 0 0
$$311$$ 7505.28 1.36844 0.684221 0.729275i $$-0.260142\pi$$
0.684221 + 0.729275i $$0.260142\pi$$
$$312$$ 0 0
$$313$$ 6349.27 1.14659 0.573294 0.819350i $$-0.305665\pi$$
0.573294 + 0.819350i $$0.305665\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ −7999.60 −1.41736 −0.708679 0.705531i $$-0.750708\pi$$
−0.708679 + 0.705531i $$0.750708\pi$$
$$318$$ 0 0
$$319$$ 2965.23 0.520442
$$320$$ 0 0
$$321$$ −1601.41 −0.278449
$$322$$ 0 0
$$323$$ 898.666 0.154808
$$324$$ 0 0
$$325$$ 2326.47 0.397074
$$326$$ 0 0
$$327$$ 3283.89 0.555350
$$328$$ 0 0
$$329$$ 0 0
$$330$$ 0 0
$$331$$ 2121.56 0.352300 0.176150 0.984363i $$-0.443636\pi$$
0.176150 + 0.984363i $$0.443636\pi$$
$$332$$ 0 0
$$333$$ 2991.92 0.492361
$$334$$ 0 0
$$335$$ −6074.05 −0.990629
$$336$$ 0 0
$$337$$ −9114.19 −1.47324 −0.736620 0.676307i $$-0.763579\pi$$
−0.736620 + 0.676307i $$0.763579\pi$$
$$338$$ 0 0
$$339$$ −4275.53 −0.685001
$$340$$ 0 0
$$341$$ −3479.75 −0.552607
$$342$$ 0 0
$$343$$ 0 0
$$344$$ 0 0
$$345$$ 2906.54 0.453574
$$346$$ 0 0
$$347$$ 1673.07 0.258833 0.129416 0.991590i $$-0.458690\pi$$
0.129416 + 0.991590i $$0.458690\pi$$
$$348$$ 0 0
$$349$$ 3467.56 0.531845 0.265923 0.963994i $$-0.414323\pi$$
0.265923 + 0.963994i $$0.414323\pi$$
$$350$$ 0 0
$$351$$ −1078.16 −0.163955
$$352$$ 0 0
$$353$$ −3984.24 −0.600736 −0.300368 0.953823i $$-0.597109\pi$$
−0.300368 + 0.953823i $$0.597109\pi$$
$$354$$ 0 0
$$355$$ −5949.55 −0.889491
$$356$$ 0 0
$$357$$ 0 0
$$358$$ 0 0
$$359$$ −2170.93 −0.319157 −0.159579 0.987185i $$-0.551014\pi$$
−0.159579 + 0.987185i $$0.551014\pi$$
$$360$$ 0 0
$$361$$ 1321.45 0.192659
$$362$$ 0 0
$$363$$ 297.798 0.0430588
$$364$$ 0 0
$$365$$ −6549.81 −0.939268
$$366$$ 0 0
$$367$$ 3592.12 0.510918 0.255459 0.966820i $$-0.417773\pi$$
0.255459 + 0.966820i $$0.417773\pi$$
$$368$$ 0 0
$$369$$ −645.705 −0.0910951
$$370$$ 0 0
$$371$$ 0 0
$$372$$ 0 0
$$373$$ 7439.80 1.03276 0.516378 0.856361i $$-0.327280\pi$$
0.516378 + 0.856361i $$0.327280\pi$$
$$374$$ 0 0
$$375$$ 4491.40 0.618492
$$376$$ 0 0
$$377$$ 3130.91 0.427719
$$378$$ 0 0
$$379$$ −11243.9 −1.52390 −0.761951 0.647635i $$-0.775758\pi$$
−0.761951 + 0.647635i $$0.775758\pi$$
$$380$$ 0 0
$$381$$ 2359.46 0.317267
$$382$$ 0 0
$$383$$ 3541.25 0.472452 0.236226 0.971698i $$-0.424089\pi$$
0.236226 + 0.971698i $$0.424089\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 0 0
$$387$$ 1043.52 0.137067
$$388$$ 0 0
$$389$$ −10558.2 −1.37615 −0.688074 0.725640i $$-0.741544\pi$$
−0.688074 + 0.725640i $$0.741544\pi$$
$$390$$ 0 0
$$391$$ 1178.35 0.152409
$$392$$ 0 0
$$393$$ −83.1908 −0.0106779
$$394$$ 0 0
$$395$$ 8724.15 1.11129
$$396$$ 0 0
$$397$$ 540.574 0.0683391 0.0341696 0.999416i $$-0.489121\pi$$
0.0341696 + 0.999416i $$0.489121\pi$$
$$398$$ 0 0
$$399$$ 0 0
$$400$$ 0 0
$$401$$ 2167.03 0.269866 0.134933 0.990855i $$-0.456918\pi$$
0.134933 + 0.990855i $$0.456918\pi$$
$$402$$ 0 0
$$403$$ −3674.17 −0.454153
$$404$$ 0 0
$$405$$ −661.722 −0.0811882
$$406$$ 0 0
$$407$$ −12572.3 −1.53117
$$408$$ 0 0
$$409$$ 957.690 0.115782 0.0578908 0.998323i $$-0.481562\pi$$
0.0578908 + 0.998323i $$0.481562\pi$$
$$410$$ 0 0
$$411$$ −7088.93 −0.850781
$$412$$ 0 0
$$413$$ 0 0
$$414$$ 0 0
$$415$$ −7407.66 −0.876211
$$416$$ 0 0
$$417$$ 7539.84 0.885437
$$418$$ 0 0
$$419$$ 6464.42 0.753718 0.376859 0.926271i $$-0.377004\pi$$
0.376859 + 0.926271i $$0.377004\pi$$
$$420$$ 0 0
$$421$$ −6201.23 −0.717885 −0.358943 0.933360i $$-0.616863\pi$$
−0.358943 + 0.933360i $$0.616863\pi$$
$$422$$ 0 0
$$423$$ 2771.35 0.318552
$$424$$ 0 0
$$425$$ 578.877 0.0660698
$$426$$ 0 0
$$427$$ 0 0
$$428$$ 0 0
$$429$$ 4530.54 0.509875
$$430$$ 0 0
$$431$$ −1415.59 −0.158206 −0.0791029 0.996866i $$-0.525206\pi$$
−0.0791029 + 0.996866i $$0.525206\pi$$
$$432$$ 0 0
$$433$$ −8905.73 −0.988411 −0.494206 0.869345i $$-0.664541\pi$$
−0.494206 + 0.869345i $$0.664541\pi$$
$$434$$ 0 0
$$435$$ 1921.59 0.211801
$$436$$ 0 0
$$437$$ 10726.4 1.17417
$$438$$ 0 0
$$439$$ 10876.5 1.18248 0.591238 0.806497i $$-0.298639\pi$$
0.591238 + 0.806497i $$0.298639\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ 0 0
$$443$$ −11403.0 −1.22296 −0.611479 0.791260i $$-0.709425\pi$$
−0.611479 + 0.791260i $$0.709425\pi$$
$$444$$ 0 0
$$445$$ 9094.74 0.968837
$$446$$ 0 0
$$447$$ 6797.49 0.719262
$$448$$ 0 0
$$449$$ −12689.0 −1.33370 −0.666852 0.745190i $$-0.732359\pi$$
−0.666852 + 0.745190i $$0.732359\pi$$
$$450$$ 0 0
$$451$$ 2713.32 0.283293
$$452$$ 0 0
$$453$$ −849.439 −0.0881019
$$454$$ 0 0
$$455$$ 0 0
$$456$$ 0 0
$$457$$ 2270.05 0.232360 0.116180 0.993228i $$-0.462935\pi$$
0.116180 + 0.993228i $$0.462935\pi$$
$$458$$ 0 0
$$459$$ −268.271 −0.0272806
$$460$$ 0 0
$$461$$ −10731.2 −1.08417 −0.542085 0.840324i $$-0.682365\pi$$
−0.542085 + 0.840324i $$0.682365\pi$$
$$462$$ 0 0
$$463$$ 3307.74 0.332017 0.166008 0.986124i $$-0.446912\pi$$
0.166008 + 0.986124i $$0.446912\pi$$
$$464$$ 0 0
$$465$$ −2255.02 −0.224891
$$466$$ 0 0
$$467$$ −9246.69 −0.916244 −0.458122 0.888889i $$-0.651478\pi$$
−0.458122 + 0.888889i $$0.651478\pi$$
$$468$$ 0 0
$$469$$ 0 0
$$470$$ 0 0
$$471$$ −577.742 −0.0565201
$$472$$ 0 0
$$473$$ −4384.96 −0.426260
$$474$$ 0 0
$$475$$ 5269.45 0.509008
$$476$$ 0 0
$$477$$ −3628.35 −0.348283
$$478$$ 0 0
$$479$$ 15223.3 1.45213 0.726066 0.687625i $$-0.241347\pi$$
0.726066 + 0.687625i $$0.241347\pi$$
$$480$$ 0 0
$$481$$ −13274.8 −1.25837
$$482$$ 0 0
$$483$$ 0 0
$$484$$ 0 0
$$485$$ 12098.4 1.13270
$$486$$ 0 0
$$487$$ 7758.72 0.721933 0.360966 0.932579i $$-0.382447\pi$$
0.360966 + 0.932579i $$0.382447\pi$$
$$488$$ 0 0
$$489$$ 2528.50 0.233830
$$490$$ 0 0
$$491$$ 8342.30 0.766767 0.383384 0.923589i $$-0.374759\pi$$
0.383384 + 0.923589i $$0.374759\pi$$
$$492$$ 0 0
$$493$$ 779.040 0.0711688
$$494$$ 0 0
$$495$$ 2780.62 0.252484
$$496$$ 0 0
$$497$$ 0 0
$$498$$ 0 0
$$499$$ 2586.95 0.232079 0.116040 0.993245i $$-0.462980\pi$$
0.116040 + 0.993245i $$0.462980\pi$$
$$500$$ 0 0
$$501$$ −11577.6 −1.03244
$$502$$ 0 0
$$503$$ 409.682 0.0363157 0.0181578 0.999835i $$-0.494220\pi$$
0.0181578 + 0.999835i $$0.494220\pi$$
$$504$$ 0 0
$$505$$ −4544.77 −0.400475
$$506$$ 0 0
$$507$$ −1807.32 −0.158316
$$508$$ 0 0
$$509$$ −4390.51 −0.382330 −0.191165 0.981558i $$-0.561226\pi$$
−0.191165 + 0.981558i $$0.561226\pi$$
$$510$$ 0 0
$$511$$ 0 0
$$512$$ 0 0
$$513$$ −2442.04 −0.210173
$$514$$ 0 0
$$515$$ −4513.06 −0.386154
$$516$$ 0 0
$$517$$ −11645.5 −0.990652
$$518$$ 0 0
$$519$$ −4531.98 −0.383298
$$520$$ 0 0
$$521$$ 19429.5 1.63382 0.816912 0.576762i $$-0.195684\pi$$
0.816912 + 0.576762i $$0.195684\pi$$
$$522$$ 0 0
$$523$$ 17948.7 1.50065 0.750326 0.661067i $$-0.229896\pi$$
0.750326 + 0.661067i $$0.229896\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ −914.217 −0.0755672
$$528$$ 0 0
$$529$$ 1897.67 0.155968
$$530$$ 0 0
$$531$$ 5343.39 0.436692
$$532$$ 0 0
$$533$$ 2864.92 0.232821
$$534$$ 0 0
$$535$$ 4360.86 0.352405
$$536$$ 0 0
$$537$$ 7393.88 0.594170
$$538$$ 0 0
$$539$$ 0 0
$$540$$ 0 0
$$541$$ 23094.0 1.83528 0.917641 0.397411i $$-0.130091\pi$$
0.917641 + 0.397411i $$0.130091\pi$$
$$542$$ 0 0
$$543$$ 9892.09 0.781787
$$544$$ 0 0
$$545$$ −8942.47 −0.702850
$$546$$ 0 0
$$547$$ 2266.68 0.177178 0.0885889 0.996068i $$-0.471764\pi$$
0.0885889 + 0.996068i $$0.471764\pi$$
$$548$$ 0 0
$$549$$ −2998.53 −0.233104
$$550$$ 0 0
$$551$$ 7091.50 0.548291
$$552$$ 0 0
$$553$$ 0 0
$$554$$ 0 0
$$555$$ −8147.40 −0.623131
$$556$$ 0 0
$$557$$ −11038.5 −0.839709 −0.419854 0.907592i $$-0.637919\pi$$
−0.419854 + 0.907592i $$0.637919\pi$$
$$558$$ 0 0
$$559$$ −4629.97 −0.350316
$$560$$ 0 0
$$561$$ 1127.30 0.0848390
$$562$$ 0 0
$$563$$ −7059.36 −0.528448 −0.264224 0.964461i $$-0.585116\pi$$
−0.264224 + 0.964461i $$0.585116\pi$$
$$564$$ 0 0
$$565$$ 11642.9 0.866936
$$566$$ 0 0
$$567$$ 0 0
$$568$$ 0 0
$$569$$ −1352.81 −0.0996706 −0.0498353 0.998757i $$-0.515870\pi$$
−0.0498353 + 0.998757i $$0.515870\pi$$
$$570$$ 0 0
$$571$$ 20839.2 1.52731 0.763655 0.645625i $$-0.223403\pi$$
0.763655 + 0.645625i $$0.223403\pi$$
$$572$$ 0 0
$$573$$ 14189.1 1.03448
$$574$$ 0 0
$$575$$ 6909.42 0.501117
$$576$$ 0 0
$$577$$ −10014.6 −0.722556 −0.361278 0.932458i $$-0.617659\pi$$
−0.361278 + 0.932458i $$0.617659\pi$$
$$578$$ 0 0
$$579$$ 13659.5 0.980433
$$580$$ 0 0
$$581$$ 0 0
$$582$$ 0 0
$$583$$ 15246.7 1.08311
$$584$$ 0 0
$$585$$ 2935.98 0.207501
$$586$$ 0 0
$$587$$ 20864.8 1.46709 0.733546 0.679640i $$-0.237864\pi$$
0.733546 + 0.679640i $$0.237864\pi$$
$$588$$ 0 0
$$589$$ −8322.01 −0.582177
$$590$$ 0 0
$$591$$ −9327.19 −0.649187
$$592$$ 0 0
$$593$$ −17547.7 −1.21517 −0.607586 0.794254i $$-0.707862\pi$$
−0.607586 + 0.794254i $$0.707862\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 0 0
$$597$$ −1330.54 −0.0912147
$$598$$ 0 0
$$599$$ 130.682 0.00891405 0.00445703 0.999990i $$-0.498581\pi$$
0.00445703 + 0.999990i $$0.498581\pi$$
$$600$$ 0 0
$$601$$ 5964.47 0.404818 0.202409 0.979301i $$-0.435123\pi$$
0.202409 + 0.979301i $$0.435123\pi$$
$$602$$ 0 0
$$603$$ 6691.60 0.451912
$$604$$ 0 0
$$605$$ −810.944 −0.0544951
$$606$$ 0 0
$$607$$ −9911.63 −0.662769 −0.331384 0.943496i $$-0.607516\pi$$
−0.331384 + 0.943496i $$0.607516\pi$$
$$608$$ 0 0
$$609$$ 0 0
$$610$$ 0 0
$$611$$ −12296.1 −0.814155
$$612$$ 0 0
$$613$$ −6482.53 −0.427124 −0.213562 0.976930i $$-0.568506\pi$$
−0.213562 + 0.976930i $$0.568506\pi$$
$$614$$ 0 0
$$615$$ 1758.34 0.115290
$$616$$ 0 0
$$617$$ 7189.36 0.469097 0.234548 0.972104i $$-0.424639\pi$$
0.234548 + 0.972104i $$0.424639\pi$$
$$618$$ 0 0
$$619$$ 1861.39 0.120865 0.0604327 0.998172i $$-0.480752\pi$$
0.0604327 + 0.998172i $$0.480752\pi$$
$$620$$ 0 0
$$621$$ −3202.05 −0.206915
$$622$$ 0 0
$$623$$ 0 0
$$624$$ 0 0
$$625$$ −4948.07 −0.316677
$$626$$ 0 0
$$627$$ 10261.7 0.653607
$$628$$ 0 0
$$629$$ −3303.06 −0.209383
$$630$$ 0 0
$$631$$ 29032.0 1.83161 0.915803 0.401627i $$-0.131555\pi$$
0.915803 + 0.401627i $$0.131555\pi$$
$$632$$ 0 0
$$633$$ −13959.9 −0.876547
$$634$$ 0 0
$$635$$ −6425.12 −0.401532
$$636$$ 0 0
$$637$$ 0 0
$$638$$ 0 0
$$639$$ 6554.45 0.405775
$$640$$ 0 0
$$641$$ 24457.7 1.50705 0.753527 0.657417i $$-0.228351\pi$$
0.753527 + 0.657417i $$0.228351\pi$$
$$642$$ 0 0
$$643$$ 10968.2 0.672695 0.336348 0.941738i $$-0.390808\pi$$
0.336348 + 0.941738i $$0.390808\pi$$
$$644$$ 0 0
$$645$$ −2841.64 −0.173472
$$646$$ 0 0
$$647$$ 5484.74 0.333272 0.166636 0.986018i $$-0.446709\pi$$
0.166636 + 0.986018i $$0.446709\pi$$
$$648$$ 0 0
$$649$$ −22453.4 −1.35805
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 0 0
$$653$$ −10330.0 −0.619055 −0.309527 0.950891i $$-0.600171\pi$$
−0.309527 + 0.950891i $$0.600171\pi$$
$$654$$ 0 0
$$655$$ 226.540 0.0135140
$$656$$ 0 0
$$657$$ 7215.74 0.428482
$$658$$ 0 0
$$659$$ −26822.2 −1.58550 −0.792751 0.609546i $$-0.791352\pi$$
−0.792751 + 0.609546i $$0.791352\pi$$
$$660$$ 0 0
$$661$$ −2498.76 −0.147035 −0.0735177 0.997294i $$-0.523423\pi$$
−0.0735177 + 0.997294i $$0.523423\pi$$
$$662$$ 0 0
$$663$$ 1190.29 0.0697238
$$664$$ 0 0
$$665$$ 0 0
$$666$$ 0 0
$$667$$ 9298.54 0.539791
$$668$$ 0 0
$$669$$ 8336.71 0.481788
$$670$$ 0 0
$$671$$ 12600.1 0.724922
$$672$$ 0 0
$$673$$ 10092.1 0.578044 0.289022 0.957322i $$-0.406670\pi$$
0.289022 + 0.957322i $$0.406670\pi$$
$$674$$ 0 0
$$675$$ −1573.04 −0.0896984
$$676$$ 0 0
$$677$$ 26366.4 1.49681 0.748406 0.663241i $$-0.230820\pi$$
0.748406 + 0.663241i $$0.230820\pi$$
$$678$$ 0 0
$$679$$ 0 0
$$680$$ 0 0
$$681$$ 9651.67 0.543102
$$682$$ 0 0
$$683$$ −22570.8 −1.26449 −0.632245 0.774769i $$-0.717866\pi$$
−0.632245 + 0.774769i $$0.717866\pi$$
$$684$$ 0 0
$$685$$ 19304.1 1.07675
$$686$$ 0 0
$$687$$ 11184.0 0.621103
$$688$$ 0 0
$$689$$ 16098.6 0.890140
$$690$$ 0 0
$$691$$ −11931.5 −0.656867 −0.328434 0.944527i $$-0.606521\pi$$
−0.328434 + 0.944527i $$0.606521\pi$$
$$692$$ 0 0
$$693$$ 0 0
$$694$$ 0 0
$$695$$ −20532.0 −1.12061
$$696$$ 0 0
$$697$$ 712.856 0.0387394
$$698$$ 0 0
$$699$$ −10810.5 −0.584967
$$700$$ 0 0
$$701$$ 16592.6 0.893997 0.446999 0.894535i $$-0.352493\pi$$
0.446999 + 0.894535i $$0.352493\pi$$
$$702$$ 0 0
$$703$$ −30067.4 −1.61311
$$704$$ 0 0
$$705$$ −7546.75 −0.403159
$$706$$ 0 0
$$707$$ 0 0
$$708$$ 0 0
$$709$$ 18714.0 0.991283 0.495641 0.868527i $$-0.334933\pi$$
0.495641 + 0.868527i $$0.334933\pi$$
$$710$$ 0 0
$$711$$ −9611.15 −0.506957
$$712$$ 0 0
$$713$$ −10912.0 −0.573152
$$714$$ 0 0
$$715$$ −12337.3 −0.645298
$$716$$ 0 0
$$717$$ −7046.27 −0.367012
$$718$$ 0 0
$$719$$ −24854.8 −1.28919 −0.644594 0.764525i $$-0.722974\pi$$
−0.644594 + 0.764525i $$0.722974\pi$$
$$720$$ 0 0
$$721$$ 0 0
$$722$$ 0 0
$$723$$ 15280.2 0.785997
$$724$$ 0 0
$$725$$ 4568.00 0.234002
$$726$$ 0 0
$$727$$ −22506.0 −1.14814 −0.574071 0.818805i $$-0.694637\pi$$
−0.574071 + 0.818805i $$0.694637\pi$$
$$728$$ 0 0
$$729$$ 729.000 0.0370370
$$730$$ 0 0
$$731$$ −1152.04 −0.0582897
$$732$$ 0 0
$$733$$ 23792.0 1.19888 0.599440 0.800420i $$-0.295390\pi$$
0.599440 + 0.800420i $$0.295390\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ −28118.8 −1.40538
$$738$$ 0 0
$$739$$ −4803.24 −0.239093 −0.119547 0.992829i $$-0.538144\pi$$
−0.119547 + 0.992829i $$0.538144\pi$$
$$740$$ 0 0
$$741$$ 10835.0 0.537159
$$742$$ 0 0
$$743$$ −5076.10 −0.250638 −0.125319 0.992116i $$-0.539995\pi$$
−0.125319 + 0.992116i $$0.539995\pi$$
$$744$$ 0 0
$$745$$ −18510.5 −0.910298
$$746$$ 0 0
$$747$$ 8160.80 0.399716
$$748$$ 0 0
$$749$$ 0 0
$$750$$ 0 0
$$751$$ 9158.72 0.445015 0.222508 0.974931i $$-0.428576\pi$$
0.222508 + 0.974931i $$0.428576\pi$$
$$752$$ 0 0
$$753$$ −17817.1 −0.862271
$$754$$ 0 0
$$755$$ 2313.14 0.111502
$$756$$ 0 0
$$757$$ 33682.2 1.61717 0.808587 0.588377i $$-0.200233\pi$$
0.808587 + 0.588377i $$0.200233\pi$$
$$758$$ 0 0
$$759$$ 13455.3 0.643475
$$760$$ 0 0
$$761$$ −13346.3 −0.635745 −0.317873 0.948133i $$-0.602968\pi$$
−0.317873 + 0.948133i $$0.602968\pi$$
$$762$$ 0 0
$$763$$ 0 0
$$764$$ 0 0
$$765$$ 730.538 0.0345264
$$766$$ 0 0
$$767$$ −23708.0 −1.11610
$$768$$ 0 0
$$769$$ 15530.6 0.728281 0.364140 0.931344i $$-0.381363\pi$$
0.364140 + 0.931344i $$0.381363\pi$$
$$770$$ 0 0
$$771$$ 4547.66 0.212425
$$772$$ 0 0
$$773$$ 11015.2 0.512536 0.256268 0.966606i $$-0.417507\pi$$
0.256268 + 0.966606i $$0.417507\pi$$
$$774$$ 0 0
$$775$$ −5360.63 −0.248464
$$776$$ 0 0
$$777$$ 0 0
$$778$$ 0 0
$$779$$ 6489.04 0.298452
$$780$$ 0 0
$$781$$ −27542.4 −1.26190
$$782$$ 0 0
$$783$$ −2116.96 −0.0966209
$$784$$ 0 0
$$785$$ 1573.27 0.0715317
$$786$$ 0 0
$$787$$ 21192.6 0.959892 0.479946 0.877298i $$-0.340656\pi$$
0.479946 + 0.877298i $$0.340656\pi$$
$$788$$ 0 0
$$789$$ 19301.8 0.870927
$$790$$ 0 0
$$791$$ 0 0
$$792$$ 0 0
$$793$$ 13304.1 0.595768
$$794$$ 0 0
$$795$$ 9880.49 0.440786
$$796$$ 0 0
$$797$$ −18189.9 −0.808432 −0.404216 0.914663i $$-0.632456\pi$$
−0.404216 + 0.914663i $$0.632456\pi$$
$$798$$ 0 0
$$799$$ −3059.56 −0.135468
$$800$$ 0 0
$$801$$ −10019.4 −0.441971
$$802$$ 0 0
$$803$$ −30321.2 −1.33252
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 0 0
$$807$$ 20849.4 0.909458
$$808$$ 0 0
$$809$$ 45516.5 1.97809 0.989045 0.147611i $$-0.0471584\pi$$
0.989045 + 0.147611i $$0.0471584\pi$$
$$810$$ 0 0
$$811$$ −42099.9 −1.82285 −0.911423 0.411472i $$-0.865015\pi$$
−0.911423 + 0.411472i $$0.865015\pi$$
$$812$$ 0 0
$$813$$ −2885.98 −0.124497
$$814$$ 0 0
$$815$$ −6885.45 −0.295935
$$816$$ 0 0
$$817$$ −10486.9 −0.449069
$$818$$ 0 0
$$819$$ 0 0
$$820$$ 0 0
$$821$$ −26555.9 −1.12888 −0.564439 0.825475i $$-0.690907\pi$$
−0.564439 + 0.825475i $$0.690907\pi$$
$$822$$ 0 0
$$823$$ 1554.25 0.0658295 0.0329147 0.999458i $$-0.489521\pi$$
0.0329147 + 0.999458i $$0.489521\pi$$
$$824$$ 0 0
$$825$$ 6610.07 0.278949
$$826$$ 0 0
$$827$$ 10548.3 0.443531 0.221766 0.975100i $$-0.428818\pi$$
0.221766 + 0.975100i $$0.428818\pi$$
$$828$$ 0 0
$$829$$ −4331.26 −0.181461 −0.0907303 0.995875i $$-0.528920\pi$$
−0.0907303 + 0.995875i $$0.528920\pi$$
$$830$$ 0 0
$$831$$ 2280.03 0.0951786
$$832$$ 0 0
$$833$$ 0 0
$$834$$ 0 0
$$835$$ 31527.5 1.30665
$$836$$ 0 0
$$837$$ 2484.30 0.102592
$$838$$ 0 0
$$839$$ −23557.7 −0.969369 −0.484685 0.874689i $$-0.661066\pi$$
−0.484685 + 0.874689i $$0.661066\pi$$
$$840$$ 0 0
$$841$$ −18241.5 −0.747939
$$842$$ 0 0
$$843$$ −13236.2 −0.540782
$$844$$ 0 0
$$845$$ 4921.59 0.200364
$$846$$ 0 0
$$847$$ 0 0
$$848$$ 0 0
$$849$$ −7806.45 −0.315567
$$850$$ 0 0
$$851$$ −39425.0 −1.58810
$$852$$ 0 0
$$853$$ 11493.6 0.461350 0.230675 0.973031i $$-0.425907\pi$$
0.230675 + 0.973031i $$0.425907\pi$$
$$854$$ 0 0
$$855$$ 6650.00 0.265994
$$856$$ 0 0
$$857$$ 2534.23 0.101012 0.0505062 0.998724i $$-0.483917\pi$$
0.0505062 + 0.998724i $$0.483917\pi$$
$$858$$ 0 0
$$859$$ 3127.09 0.124208 0.0621041 0.998070i $$-0.480219\pi$$
0.0621041 + 0.998070i $$0.480219\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ 0 0
$$863$$ −15309.9 −0.603887 −0.301943 0.953326i $$-0.597635\pi$$
−0.301943 + 0.953326i $$0.597635\pi$$
$$864$$ 0 0
$$865$$ 12341.2 0.485101
$$866$$ 0 0
$$867$$ −14442.8 −0.565749
$$868$$ 0 0
$$869$$ 40387.0 1.57656
$$870$$ 0 0
$$871$$ −29689.8 −1.15500
$$872$$ 0 0
$$873$$ −13328.4 −0.516723
$$874$$ 0 0
$$875$$ 0 0
$$876$$ 0 0
$$877$$ −42420.7 −1.63335 −0.816673 0.577100i $$-0.804184\pi$$
−0.816673 + 0.577100i $$0.804184\pi$$
$$878$$ 0 0
$$879$$ 27996.5 1.07429
$$880$$ 0 0
$$881$$ −10060.8 −0.384740 −0.192370 0.981322i $$-0.561617\pi$$
−0.192370 + 0.981322i $$0.561617\pi$$
$$882$$ 0 0
$$883$$ −17437.8 −0.664585 −0.332293 0.943176i $$-0.607822\pi$$
−0.332293 + 0.943176i $$0.607822\pi$$
$$884$$ 0 0
$$885$$ −14550.8 −0.552677
$$886$$ 0 0
$$887$$ −16016.7 −0.606299 −0.303149 0.952943i $$-0.598038\pi$$
−0.303149 + 0.952943i $$0.598038\pi$$
$$888$$ 0 0
$$889$$ 0 0
$$890$$ 0 0
$$891$$ −3063.33 −0.115180
$$892$$ 0 0
$$893$$ −27850.8 −1.04366
$$894$$ 0 0
$$895$$ −20134.5 −0.751981
$$896$$ 0 0
$$897$$ 14207.1 0.528832
$$898$$ 0 0
$$899$$ −7214.22 −0.267639
$$900$$ 0 0
$$901$$ 4005.69 0.148112
$$902$$ 0 0
$$903$$ 0 0
$$904$$ 0 0
$$905$$ −26937.5 −0.989428
$$906$$ 0 0
$$907$$ 10131.0 0.370887 0.185443 0.982655i $$-0.440628\pi$$
0.185443 + 0.982655i $$0.440628\pi$$
$$908$$ 0 0
$$909$$ 5006.85 0.182692
$$910$$ 0 0
$$911$$ −1320.58 −0.0480273 −0.0240137 0.999712i $$-0.507645\pi$$
−0.0240137 + 0.999712i $$0.507645\pi$$
$$912$$ 0 0
$$913$$ −34292.5 −1.24306
$$914$$ 0 0
$$915$$ 8165.41 0.295017
$$916$$ 0 0
$$917$$ 0 0
$$918$$ 0 0
$$919$$ 7285.76 0.261518 0.130759 0.991414i $$-0.458259\pi$$
0.130759 + 0.991414i $$0.458259\pi$$
$$920$$ 0 0
$$921$$ 14686.5 0.525448
$$922$$ 0 0
$$923$$ −29081.3 −1.03708
$$924$$ 0 0
$$925$$ −19368.0 −0.688448
$$926$$ 0 0
$$927$$ 4971.91 0.176159
$$928$$ 0 0
$$929$$ −23708.0 −0.837280 −0.418640 0.908152i $$-0.637493\pi$$
−0.418640 + 0.908152i $$0.637493\pi$$
$$930$$ 0 0
$$931$$ 0 0
$$932$$ 0 0
$$933$$ 22515.8 0.790070
$$934$$ 0 0
$$935$$ −3069.79 −0.107372
$$936$$ 0 0
$$937$$ 8853.60 0.308682 0.154341 0.988018i $$-0.450675\pi$$
0.154341 + 0.988018i $$0.450675\pi$$
$$938$$ 0 0
$$939$$ 19047.8 0.661983
$$940$$ 0 0
$$941$$ 53971.6 1.86974 0.934869 0.354994i $$-0.115517\pi$$
0.934869 + 0.354994i $$0.115517\pi$$
$$942$$ 0 0
$$943$$ 8508.57 0.293825
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ −8019.80 −0.275194 −0.137597 0.990488i $$-0.543938\pi$$
−0.137597 + 0.990488i $$0.543938\pi$$
$$948$$ 0 0
$$949$$ −32015.4 −1.09511
$$950$$ 0 0
$$951$$ −23998.8 −0.818312
$$952$$ 0 0
$$953$$ 42628.0 1.44896 0.724479 0.689297i $$-0.242080\pi$$
0.724479 + 0.689297i $$0.242080\pi$$
$$954$$ 0 0
$$955$$ −38638.9 −1.30924
$$956$$ 0 0
$$957$$ 8895.69 0.300477
$$958$$ 0 0
$$959$$ 0 0
$$960$$ 0 0
$$961$$ −21325.0 −0.715820
$$962$$ 0 0
$$963$$ −4804.24 −0.160763
$$964$$ 0 0
$$965$$ −37196.8 −1.24084
$$966$$ 0 0
$$967$$ 43161.7 1.43535 0.717676 0.696377i $$-0.245206\pi$$
0.717676 + 0.696377i $$0.245206\pi$$
$$968$$ 0 0
$$969$$ 2696.00 0.0893787
$$970$$ 0 0
$$971$$ −39775.4 −1.31458 −0.657288 0.753640i $$-0.728296\pi$$
−0.657288 + 0.753640i $$0.728296\pi$$
$$972$$ 0 0
$$973$$ 0 0
$$974$$ 0 0
$$975$$ 6979.40 0.229251
$$976$$ 0 0
$$977$$ −26658.1 −0.872945 −0.436472 0.899718i $$-0.643772\pi$$
−0.436472 + 0.899718i $$0.643772\pi$$
$$978$$ 0 0
$$979$$ 42102.6 1.37447
$$980$$ 0 0
$$981$$ 9851.66 0.320631
$$982$$ 0 0
$$983$$ −2942.43 −0.0954719 −0.0477359 0.998860i $$-0.515201\pi$$
−0.0477359 + 0.998860i $$0.515201\pi$$
$$984$$ 0 0
$$985$$ 25399.2 0.821610
$$986$$ 0 0
$$987$$ 0 0
$$988$$ 0 0
$$989$$ −13750.6 −0.442108
$$990$$ 0 0
$$991$$ −44969.8 −1.44149 −0.720743 0.693203i $$-0.756199\pi$$
−0.720743 + 0.693203i $$0.756199\pi$$
$$992$$ 0 0
$$993$$ 6364.68 0.203401
$$994$$ 0 0
$$995$$ 3623.23 0.115441
$$996$$ 0 0
$$997$$ −27116.4 −0.861370 −0.430685 0.902502i $$-0.641728\pi$$
−0.430685 + 0.902502i $$0.641728\pi$$
$$998$$ 0 0
$$999$$ 8975.75 0.284265
Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000

## Twists

By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 2352.4.a.cq.1.2 4
4.3 odd 2 588.4.a.j.1.2 4
7.6 odd 2 2352.4.a.cl.1.3 4
12.11 even 2 1764.4.a.bc.1.3 4
28.3 even 6 588.4.i.k.373.2 8
28.11 odd 6 588.4.i.l.373.3 8
28.19 even 6 588.4.i.k.361.2 8
28.23 odd 6 588.4.i.l.361.3 8
28.27 even 2 588.4.a.k.1.3 yes 4
84.11 even 6 1764.4.k.bb.1549.2 8
84.23 even 6 1764.4.k.bb.361.2 8
84.47 odd 6 1764.4.k.bd.361.3 8
84.59 odd 6 1764.4.k.bd.1549.3 8
84.83 odd 2 1764.4.a.ba.1.2 4

By twisted newform
Twist Min Dim Char Parity Ord Type
588.4.a.j.1.2 4 4.3 odd 2
588.4.a.k.1.3 yes 4 28.27 even 2
588.4.i.k.361.2 8 28.19 even 6
588.4.i.k.373.2 8 28.3 even 6
588.4.i.l.361.3 8 28.23 odd 6
588.4.i.l.373.3 8 28.11 odd 6
1764.4.a.ba.1.2 4 84.83 odd 2
1764.4.a.bc.1.3 4 12.11 even 2
1764.4.k.bb.361.2 8 84.23 even 6
1764.4.k.bb.1549.2 8 84.11 even 6
1764.4.k.bd.361.3 8 84.47 odd 6
1764.4.k.bd.1549.3 8 84.59 odd 6
2352.4.a.cl.1.3 4 7.6 odd 2
2352.4.a.cq.1.2 4 1.1 even 1 trivial