# Properties

 Label 1728.4.a.bh.1.1 Level $1728$ Weight $4$ Character 1728.1 Self dual yes Analytic conductor $101.955$ Analytic rank $1$ Dimension $2$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1728 = 2^{6} \cdot 3^{3}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 1728.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$101.955300490$$ Analytic rank: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{33})$$ Defining polynomial: $$x^{2} - x - 8$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2\cdot 3$$ Twist minimal: no (minimal twist has level 216) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Root $$3.37228$$ of defining polynomial Character $$\chi$$ $$=$$ 1728.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q-21.2337 q^{5} +29.2337 q^{7} +O(q^{10})$$ $$q-21.2337 q^{5} +29.2337 q^{7} -1.00000 q^{11} +52.9348 q^{13} -96.9348 q^{17} -126.467 q^{19} -22.9348 q^{23} +325.870 q^{25} +133.533 q^{29} +101.832 q^{31} -620.739 q^{35} -105.065 q^{37} +16.3369 q^{41} +201.272 q^{43} -251.870 q^{47} +511.609 q^{49} +148.038 q^{53} +21.2337 q^{55} +73.6085 q^{59} +607.478 q^{61} -1124.00 q^{65} -761.945 q^{67} +701.196 q^{71} -287.000 q^{73} -29.2337 q^{77} -128.522 q^{79} -160.478 q^{83} +2058.28 q^{85} +430.206 q^{89} +1547.48 q^{91} +2685.37 q^{95} -31.1305 q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 8q^{5} + 24q^{7} + O(q^{10})$$ $$2q - 8q^{5} + 24q^{7} - 2q^{11} - 32q^{13} - 56q^{17} - 184q^{19} + 92q^{23} + 376q^{25} + 336q^{29} + 376q^{31} - 690q^{35} - 348q^{37} - 312q^{41} - 80q^{43} - 228q^{47} + 196q^{49} - 152q^{53} + 8q^{55} - 680q^{59} + 112q^{61} - 2248q^{65} - 352q^{67} + 1816q^{71} - 574q^{73} - 24q^{77} - 1360q^{79} + 782q^{83} + 2600q^{85} + 240q^{89} + 1992q^{91} + 1924q^{95} - 338q^{97} + O(q^{100})$$

## 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
$$3$$ 0 0
$$4$$ 0 0
$$5$$ −21.2337 −1.89920 −0.949599 0.313466i $$-0.898510\pi$$
−0.949599 + 0.313466i $$0.898510\pi$$
$$6$$ 0 0
$$7$$ 29.2337 1.57847 0.789235 0.614091i $$-0.210477\pi$$
0.789235 + 0.614091i $$0.210477\pi$$
$$8$$ 0 0
$$9$$ 0 0
$$10$$ 0 0
$$11$$ −1.00000 −0.0274101 −0.0137051 0.999906i $$-0.504363\pi$$
−0.0137051 + 0.999906i $$0.504363\pi$$
$$12$$ 0 0
$$13$$ 52.9348 1.12934 0.564671 0.825316i $$-0.309003\pi$$
0.564671 + 0.825316i $$0.309003\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 0 0
$$17$$ −96.9348 −1.38295 −0.691474 0.722401i $$-0.743039\pi$$
−0.691474 + 0.722401i $$0.743039\pi$$
$$18$$ 0 0
$$19$$ −126.467 −1.52703 −0.763516 0.645789i $$-0.776529\pi$$
−0.763516 + 0.645789i $$0.776529\pi$$
$$20$$ 0 0
$$21$$ 0 0
$$22$$ 0 0
$$23$$ −22.9348 −0.207923 −0.103961 0.994581i $$-0.533152\pi$$
−0.103961 + 0.994581i $$0.533152\pi$$
$$24$$ 0 0
$$25$$ 325.870 2.60696
$$26$$ 0 0
$$27$$ 0 0
$$28$$ 0 0
$$29$$ 133.533 0.855048 0.427524 0.904004i $$-0.359386\pi$$
0.427524 + 0.904004i $$0.359386\pi$$
$$30$$ 0 0
$$31$$ 101.832 0.589983 0.294992 0.955500i $$-0.404683\pi$$
0.294992 + 0.955500i $$0.404683\pi$$
$$32$$ 0 0
$$33$$ 0 0
$$34$$ 0 0
$$35$$ −620.739 −2.99783
$$36$$ 0 0
$$37$$ −105.065 −0.466828 −0.233414 0.972378i $$-0.574990\pi$$
−0.233414 + 0.972378i $$0.574990\pi$$
$$38$$ 0 0
$$39$$ 0 0
$$40$$ 0 0
$$41$$ 16.3369 0.0622291 0.0311145 0.999516i $$-0.490094\pi$$
0.0311145 + 0.999516i $$0.490094\pi$$
$$42$$ 0 0
$$43$$ 201.272 0.713805 0.356903 0.934142i $$-0.383833\pi$$
0.356903 + 0.934142i $$0.383833\pi$$
$$44$$ 0 0
$$45$$ 0 0
$$46$$ 0 0
$$47$$ −251.870 −0.781680 −0.390840 0.920459i $$-0.627815\pi$$
−0.390840 + 0.920459i $$0.627815\pi$$
$$48$$ 0 0
$$49$$ 511.609 1.49157
$$50$$ 0 0
$$51$$ 0 0
$$52$$ 0 0
$$53$$ 148.038 0.383671 0.191836 0.981427i $$-0.438556\pi$$
0.191836 + 0.981427i $$0.438556\pi$$
$$54$$ 0 0
$$55$$ 21.2337 0.0520573
$$56$$ 0 0
$$57$$ 0 0
$$58$$ 0 0
$$59$$ 73.6085 0.162424 0.0812120 0.996697i $$-0.474121\pi$$
0.0812120 + 0.996697i $$0.474121\pi$$
$$60$$ 0 0
$$61$$ 607.478 1.27508 0.637538 0.770419i $$-0.279953\pi$$
0.637538 + 0.770419i $$0.279953\pi$$
$$62$$ 0 0
$$63$$ 0 0
$$64$$ 0 0
$$65$$ −1124.00 −2.14485
$$66$$ 0 0
$$67$$ −761.945 −1.38935 −0.694675 0.719324i $$-0.744452\pi$$
−0.694675 + 0.719324i $$0.744452\pi$$
$$68$$ 0 0
$$69$$ 0 0
$$70$$ 0 0
$$71$$ 701.196 1.17207 0.586033 0.810287i $$-0.300689\pi$$
0.586033 + 0.810287i $$0.300689\pi$$
$$72$$ 0 0
$$73$$ −287.000 −0.460148 −0.230074 0.973173i $$-0.573897\pi$$
−0.230074 + 0.973173i $$0.573897\pi$$
$$74$$ 0 0
$$75$$ 0 0
$$76$$ 0 0
$$77$$ −29.2337 −0.0432661
$$78$$ 0 0
$$79$$ −128.522 −0.183036 −0.0915181 0.995803i $$-0.529172\pi$$
−0.0915181 + 0.995803i $$0.529172\pi$$
$$80$$ 0 0
$$81$$ 0 0
$$82$$ 0 0
$$83$$ −160.478 −0.212226 −0.106113 0.994354i $$-0.533841\pi$$
−0.106113 + 0.994354i $$0.533841\pi$$
$$84$$ 0 0
$$85$$ 2058.28 2.62649
$$86$$ 0 0
$$87$$ 0 0
$$88$$ 0 0
$$89$$ 430.206 0.512380 0.256190 0.966626i $$-0.417533\pi$$
0.256190 + 0.966626i $$0.417533\pi$$
$$90$$ 0 0
$$91$$ 1547.48 1.78263
$$92$$ 0 0
$$93$$ 0 0
$$94$$ 0 0
$$95$$ 2685.37 2.90014
$$96$$ 0 0
$$97$$ −31.1305 −0.0325858 −0.0162929 0.999867i $$-0.505186\pi$$
−0.0162929 + 0.999867i $$0.505186\pi$$
$$98$$ 0 0
$$99$$ 0 0
$$100$$ 0 0
$$101$$ 983.103 0.968539 0.484269 0.874919i $$-0.339085\pi$$
0.484269 + 0.874919i $$0.339085\pi$$
$$102$$ 0 0
$$103$$ −952.152 −0.910857 −0.455429 0.890272i $$-0.650514\pi$$
−0.455429 + 0.890272i $$0.650514\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ 272.087 0.245828 0.122914 0.992417i $$-0.460776\pi$$
0.122914 + 0.992417i $$0.460776\pi$$
$$108$$ 0 0
$$109$$ −1355.76 −1.19136 −0.595680 0.803222i $$-0.703117\pi$$
−0.595680 + 0.803222i $$0.703117\pi$$
$$110$$ 0 0
$$111$$ 0 0
$$112$$ 0 0
$$113$$ −1938.36 −1.61368 −0.806838 0.590773i $$-0.798823\pi$$
−0.806838 + 0.590773i $$0.798823\pi$$
$$114$$ 0 0
$$115$$ 486.989 0.394887
$$116$$ 0 0
$$117$$ 0 0
$$118$$ 0 0
$$119$$ −2833.76 −2.18294
$$120$$ 0 0
$$121$$ −1330.00 −0.999249
$$122$$ 0 0
$$123$$ 0 0
$$124$$ 0 0
$$125$$ −4265.20 −3.05193
$$126$$ 0 0
$$127$$ 232.375 0.162362 0.0811808 0.996699i $$-0.474131\pi$$
0.0811808 + 0.996699i $$0.474131\pi$$
$$128$$ 0 0
$$129$$ 0 0
$$130$$ 0 0
$$131$$ 2795.30 1.86433 0.932163 0.362038i $$-0.117919\pi$$
0.932163 + 0.362038i $$0.117919\pi$$
$$132$$ 0 0
$$133$$ −3697.11 −2.41038
$$134$$ 0 0
$$135$$ 0 0
$$136$$ 0 0
$$137$$ −1261.53 −0.786715 −0.393358 0.919386i $$-0.628687\pi$$
−0.393358 + 0.919386i $$0.628687\pi$$
$$138$$ 0 0
$$139$$ 307.369 0.187559 0.0937794 0.995593i $$-0.470105\pi$$
0.0937794 + 0.995593i $$0.470105\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ 0 0
$$143$$ −52.9348 −0.0309554
$$144$$ 0 0
$$145$$ −2835.39 −1.62391
$$146$$ 0 0
$$147$$ 0 0
$$148$$ 0 0
$$149$$ −1597.76 −0.878478 −0.439239 0.898370i $$-0.644752\pi$$
−0.439239 + 0.898370i $$0.644752\pi$$
$$150$$ 0 0
$$151$$ −3415.41 −1.84067 −0.920337 0.391126i $$-0.872086\pi$$
−0.920337 + 0.391126i $$0.872086\pi$$
$$152$$ 0 0
$$153$$ 0 0
$$154$$ 0 0
$$155$$ −2162.26 −1.12050
$$156$$ 0 0
$$157$$ −476.826 −0.242387 −0.121194 0.992629i $$-0.538672\pi$$
−0.121194 + 0.992629i $$0.538672\pi$$
$$158$$ 0 0
$$159$$ 0 0
$$160$$ 0 0
$$161$$ −670.467 −0.328200
$$162$$ 0 0
$$163$$ −3304.04 −1.58768 −0.793842 0.608124i $$-0.791922\pi$$
−0.793842 + 0.608124i $$0.791922\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ 0 0
$$167$$ 2091.65 0.969203 0.484601 0.874735i $$-0.338965\pi$$
0.484601 + 0.874735i $$0.338965\pi$$
$$168$$ 0 0
$$169$$ 605.088 0.275416
$$170$$ 0 0
$$171$$ 0 0
$$172$$ 0 0
$$173$$ −2924.56 −1.28526 −0.642631 0.766176i $$-0.722157\pi$$
−0.642631 + 0.766176i $$0.722157\pi$$
$$174$$ 0 0
$$175$$ 9526.37 4.11500
$$176$$ 0 0
$$177$$ 0 0
$$178$$ 0 0
$$179$$ 231.913 0.0968381 0.0484191 0.998827i $$-0.484582\pi$$
0.0484191 + 0.998827i $$0.484582\pi$$
$$180$$ 0 0
$$181$$ −2291.74 −0.941125 −0.470562 0.882367i $$-0.655949\pi$$
−0.470562 + 0.882367i $$0.655949\pi$$
$$182$$ 0 0
$$183$$ 0 0
$$184$$ 0 0
$$185$$ 2230.92 0.886598
$$186$$ 0 0
$$187$$ 96.9348 0.0379068
$$188$$ 0 0
$$189$$ 0 0
$$190$$ 0 0
$$191$$ −4217.80 −1.59785 −0.798925 0.601430i $$-0.794598\pi$$
−0.798925 + 0.601430i $$0.794598\pi$$
$$192$$ 0 0
$$193$$ −1188.35 −0.443208 −0.221604 0.975137i $$-0.571129\pi$$
−0.221604 + 0.975137i $$0.571129\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ 0 0
$$197$$ 3130.25 1.13209 0.566044 0.824375i $$-0.308473\pi$$
0.566044 + 0.824375i $$0.308473\pi$$
$$198$$ 0 0
$$199$$ −659.146 −0.234802 −0.117401 0.993085i $$-0.537456\pi$$
−0.117401 + 0.993085i $$0.537456\pi$$
$$200$$ 0 0
$$201$$ 0 0
$$202$$ 0 0
$$203$$ 3903.65 1.34967
$$204$$ 0 0
$$205$$ −346.892 −0.118185
$$206$$ 0 0
$$207$$ 0 0
$$208$$ 0 0
$$209$$ 126.467 0.0418561
$$210$$ 0 0
$$211$$ −3613.95 −1.17912 −0.589560 0.807725i $$-0.700699\pi$$
−0.589560 + 0.807725i $$0.700699\pi$$
$$212$$ 0 0
$$213$$ 0 0
$$214$$ 0 0
$$215$$ −4273.74 −1.35566
$$216$$ 0 0
$$217$$ 2976.91 0.931272
$$218$$ 0 0
$$219$$ 0 0
$$220$$ 0 0
$$221$$ −5131.22 −1.56182
$$222$$ 0 0
$$223$$ −2054.13 −0.616837 −0.308419 0.951251i $$-0.599800\pi$$
−0.308419 + 0.951251i $$0.599800\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ 0 0
$$227$$ −2153.61 −0.629692 −0.314846 0.949143i $$-0.601953\pi$$
−0.314846 + 0.949143i $$0.601953\pi$$
$$228$$ 0 0
$$229$$ 1819.87 0.525154 0.262577 0.964911i $$-0.415428\pi$$
0.262577 + 0.964911i $$0.415428\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 0 0
$$233$$ −3989.68 −1.12177 −0.560886 0.827893i $$-0.689539\pi$$
−0.560886 + 0.827893i $$0.689539\pi$$
$$234$$ 0 0
$$235$$ 5348.12 1.48457
$$236$$ 0 0
$$237$$ 0 0
$$238$$ 0 0
$$239$$ −5704.32 −1.54386 −0.771929 0.635709i $$-0.780708\pi$$
−0.771929 + 0.635709i $$0.780708\pi$$
$$240$$ 0 0
$$241$$ 6223.95 1.66357 0.831785 0.555099i $$-0.187319\pi$$
0.831785 + 0.555099i $$0.187319\pi$$
$$242$$ 0 0
$$243$$ 0 0
$$244$$ 0 0
$$245$$ −10863.3 −2.83279
$$246$$ 0 0
$$247$$ −6694.52 −1.72454
$$248$$ 0 0
$$249$$ 0 0
$$250$$ 0 0
$$251$$ −4622.52 −1.16243 −0.581217 0.813749i $$-0.697423\pi$$
−0.581217 + 0.813749i $$0.697423\pi$$
$$252$$ 0 0
$$253$$ 22.9348 0.00569919
$$254$$ 0 0
$$255$$ 0 0
$$256$$ 0 0
$$257$$ −4905.09 −1.19055 −0.595274 0.803523i $$-0.702957\pi$$
−0.595274 + 0.803523i $$0.702957\pi$$
$$258$$ 0 0
$$259$$ −3071.44 −0.736874
$$260$$ 0 0
$$261$$ 0 0
$$262$$ 0 0
$$263$$ 4432.85 1.03932 0.519660 0.854373i $$-0.326059\pi$$
0.519660 + 0.854373i $$0.326059\pi$$
$$264$$ 0 0
$$265$$ −3143.39 −0.728668
$$266$$ 0 0
$$267$$ 0 0
$$268$$ 0 0
$$269$$ −4454.81 −1.00972 −0.504860 0.863201i $$-0.668456\pi$$
−0.504860 + 0.863201i $$0.668456\pi$$
$$270$$ 0 0
$$271$$ 3256.23 0.729897 0.364948 0.931028i $$-0.381087\pi$$
0.364948 + 0.931028i $$0.381087\pi$$
$$272$$ 0 0
$$273$$ 0 0
$$274$$ 0 0
$$275$$ −325.870 −0.0714570
$$276$$ 0 0
$$277$$ 3421.61 0.742182 0.371091 0.928596i $$-0.378984\pi$$
0.371091 + 0.928596i $$0.378984\pi$$
$$278$$ 0 0
$$279$$ 0 0
$$280$$ 0 0
$$281$$ −1715.11 −0.364109 −0.182055 0.983288i $$-0.558275\pi$$
−0.182055 + 0.983288i $$0.558275\pi$$
$$282$$ 0 0
$$283$$ 4487.60 0.942615 0.471307 0.881969i $$-0.343782\pi$$
0.471307 + 0.881969i $$0.343782\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ 0 0
$$287$$ 477.587 0.0982268
$$288$$ 0 0
$$289$$ 4483.35 0.912548
$$290$$ 0 0
$$291$$ 0 0
$$292$$ 0 0
$$293$$ −1685.31 −0.336031 −0.168016 0.985784i $$-0.553736\pi$$
−0.168016 + 0.985784i $$0.553736\pi$$
$$294$$ 0 0
$$295$$ −1562.98 −0.308475
$$296$$ 0 0
$$297$$ 0 0
$$298$$ 0 0
$$299$$ −1214.05 −0.234816
$$300$$ 0 0
$$301$$ 5883.91 1.12672
$$302$$ 0 0
$$303$$ 0 0
$$304$$ 0 0
$$305$$ −12899.0 −2.42162
$$306$$ 0 0
$$307$$ 8079.07 1.50194 0.750972 0.660334i $$-0.229585\pi$$
0.750972 + 0.660334i $$0.229585\pi$$
$$308$$ 0 0
$$309$$ 0 0
$$310$$ 0 0
$$311$$ 2557.63 0.466334 0.233167 0.972437i $$-0.425091\pi$$
0.233167 + 0.972437i $$0.425091\pi$$
$$312$$ 0 0
$$313$$ −4081.43 −0.737049 −0.368524 0.929618i $$-0.620137\pi$$
−0.368524 + 0.929618i $$0.620137\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ 7701.93 1.36462 0.682308 0.731065i $$-0.260976\pi$$
0.682308 + 0.731065i $$0.260976\pi$$
$$318$$ 0 0
$$319$$ −133.533 −0.0234370
$$320$$ 0 0
$$321$$ 0 0
$$322$$ 0 0
$$323$$ 12259.1 2.11181
$$324$$ 0 0
$$325$$ 17249.8 2.94415
$$326$$ 0 0
$$327$$ 0 0
$$328$$ 0 0
$$329$$ −7363.07 −1.23386
$$330$$ 0 0
$$331$$ 357.314 0.0593346 0.0296673 0.999560i $$-0.490555\pi$$
0.0296673 + 0.999560i $$0.490555\pi$$
$$332$$ 0 0
$$333$$ 0 0
$$334$$ 0 0
$$335$$ 16178.9 2.63865
$$336$$ 0 0
$$337$$ −6659.09 −1.07639 −0.538195 0.842820i $$-0.680894\pi$$
−0.538195 + 0.842820i $$0.680894\pi$$
$$338$$ 0 0
$$339$$ 0 0
$$340$$ 0 0
$$341$$ −101.832 −0.0161715
$$342$$ 0 0
$$343$$ 4929.05 0.775929
$$344$$ 0 0
$$345$$ 0 0
$$346$$ 0 0
$$347$$ −7894.04 −1.22125 −0.610626 0.791919i $$-0.709082\pi$$
−0.610626 + 0.791919i $$0.709082\pi$$
$$348$$ 0 0
$$349$$ 1254.46 0.192405 0.0962027 0.995362i $$-0.469330\pi$$
0.0962027 + 0.995362i $$0.469330\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ 0 0
$$353$$ 8604.92 1.29743 0.648716 0.761030i $$-0.275306\pi$$
0.648716 + 0.761030i $$0.275306\pi$$
$$354$$ 0 0
$$355$$ −14889.0 −2.22598
$$356$$ 0 0
$$357$$ 0 0
$$358$$ 0 0
$$359$$ −3875.61 −0.569768 −0.284884 0.958562i $$-0.591955\pi$$
−0.284884 + 0.958562i $$0.591955\pi$$
$$360$$ 0 0
$$361$$ 9135.00 1.33183
$$362$$ 0 0
$$363$$ 0 0
$$364$$ 0 0
$$365$$ 6094.07 0.873913
$$366$$ 0 0
$$367$$ −11890.1 −1.69117 −0.845583 0.533843i $$-0.820747\pi$$
−0.845583 + 0.533843i $$0.820747\pi$$
$$368$$ 0 0
$$369$$ 0 0
$$370$$ 0 0
$$371$$ 4327.70 0.605614
$$372$$ 0 0
$$373$$ −12384.5 −1.71916 −0.859578 0.511005i $$-0.829273\pi$$
−0.859578 + 0.511005i $$0.829273\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ 0 0
$$377$$ 7068.52 0.965642
$$378$$ 0 0
$$379$$ −2062.80 −0.279576 −0.139788 0.990181i $$-0.544642\pi$$
−0.139788 + 0.990181i $$0.544642\pi$$
$$380$$ 0 0
$$381$$ 0 0
$$382$$ 0 0
$$383$$ 11905.2 1.58833 0.794164 0.607704i $$-0.207909\pi$$
0.794164 + 0.607704i $$0.207909\pi$$
$$384$$ 0 0
$$385$$ 620.739 0.0821709
$$386$$ 0 0
$$387$$ 0 0
$$388$$ 0 0
$$389$$ −12716.6 −1.65747 −0.828735 0.559641i $$-0.810939\pi$$
−0.828735 + 0.559641i $$0.810939\pi$$
$$390$$ 0 0
$$391$$ 2223.17 0.287547
$$392$$ 0 0
$$393$$ 0 0
$$394$$ 0 0
$$395$$ 2729.00 0.347622
$$396$$ 0 0
$$397$$ 4531.59 0.572881 0.286441 0.958098i $$-0.407528\pi$$
0.286441 + 0.958098i $$0.407528\pi$$
$$398$$ 0 0
$$399$$ 0 0
$$400$$ 0 0
$$401$$ −4829.15 −0.601388 −0.300694 0.953721i $$-0.597218\pi$$
−0.300694 + 0.953721i $$0.597218\pi$$
$$402$$ 0 0
$$403$$ 5390.43 0.666294
$$404$$ 0 0
$$405$$ 0 0
$$406$$ 0 0
$$407$$ 105.065 0.0127958
$$408$$ 0 0
$$409$$ 11484.9 1.38849 0.694245 0.719739i $$-0.255738\pi$$
0.694245 + 0.719739i $$0.255738\pi$$
$$410$$ 0 0
$$411$$ 0 0
$$412$$ 0 0
$$413$$ 2151.85 0.256381
$$414$$ 0 0
$$415$$ 3407.54 0.403059
$$416$$ 0 0
$$417$$ 0 0
$$418$$ 0 0
$$419$$ −8211.65 −0.957435 −0.478718 0.877969i $$-0.658898\pi$$
−0.478718 + 0.877969i $$0.658898\pi$$
$$420$$ 0 0
$$421$$ −9788.13 −1.13312 −0.566561 0.824020i $$-0.691726\pi$$
−0.566561 + 0.824020i $$0.691726\pi$$
$$422$$ 0 0
$$423$$ 0 0
$$424$$ 0 0
$$425$$ −31588.1 −3.60529
$$426$$ 0 0
$$427$$ 17758.8 2.01267
$$428$$ 0 0
$$429$$ 0 0
$$430$$ 0 0
$$431$$ 11206.3 1.25241 0.626207 0.779657i $$-0.284606\pi$$
0.626207 + 0.779657i $$0.284606\pi$$
$$432$$ 0 0
$$433$$ 719.306 0.0798329 0.0399165 0.999203i $$-0.487291\pi$$
0.0399165 + 0.999203i $$0.487291\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 0 0
$$437$$ 2900.50 0.317505
$$438$$ 0 0
$$439$$ 8795.00 0.956179 0.478090 0.878311i $$-0.341329\pi$$
0.478090 + 0.878311i $$0.341329\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ 0 0
$$443$$ −3023.65 −0.324285 −0.162142 0.986767i $$-0.551840\pi$$
−0.162142 + 0.986767i $$0.551840\pi$$
$$444$$ 0 0
$$445$$ −9134.87 −0.973111
$$446$$ 0 0
$$447$$ 0 0
$$448$$ 0 0
$$449$$ 3137.67 0.329790 0.164895 0.986311i $$-0.447272\pi$$
0.164895 + 0.986311i $$0.447272\pi$$
$$450$$ 0 0
$$451$$ −16.3369 −0.00170571
$$452$$ 0 0
$$453$$ 0 0
$$454$$ 0 0
$$455$$ −32858.7 −3.38558
$$456$$ 0 0
$$457$$ −9015.74 −0.922841 −0.461421 0.887182i $$-0.652660\pi$$
−0.461421 + 0.887182i $$0.652660\pi$$
$$458$$ 0 0
$$459$$ 0 0
$$460$$ 0 0
$$461$$ 12262.6 1.23888 0.619442 0.785042i $$-0.287359\pi$$
0.619442 + 0.785042i $$0.287359\pi$$
$$462$$ 0 0
$$463$$ 3472.30 0.348534 0.174267 0.984698i $$-0.444244\pi$$
0.174267 + 0.984698i $$0.444244\pi$$
$$464$$ 0 0
$$465$$ 0 0
$$466$$ 0 0
$$467$$ −11825.0 −1.17172 −0.585860 0.810412i $$-0.699243\pi$$
−0.585860 + 0.810412i $$0.699243\pi$$
$$468$$ 0 0
$$469$$ −22274.5 −2.19305
$$470$$ 0 0
$$471$$ 0 0
$$472$$ 0 0
$$473$$ −201.272 −0.0195655
$$474$$ 0 0
$$475$$ −41211.9 −3.98090
$$476$$ 0 0
$$477$$ 0 0
$$478$$ 0 0
$$479$$ −1703.96 −0.162538 −0.0812692 0.996692i $$-0.525897\pi$$
−0.0812692 + 0.996692i $$0.525897\pi$$
$$480$$ 0 0
$$481$$ −5561.60 −0.527208
$$482$$ 0 0
$$483$$ 0 0
$$484$$ 0 0
$$485$$ 661.015 0.0618869
$$486$$ 0 0
$$487$$ 7721.95 0.718512 0.359256 0.933239i $$-0.383031\pi$$
0.359256 + 0.933239i $$0.383031\pi$$
$$488$$ 0 0
$$489$$ 0 0
$$490$$ 0 0
$$491$$ −20223.6 −1.85882 −0.929410 0.369049i $$-0.879683\pi$$
−0.929410 + 0.369049i $$0.879683\pi$$
$$492$$ 0 0
$$493$$ −12944.0 −1.18249
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 0 0
$$497$$ 20498.5 1.85007
$$498$$ 0 0
$$499$$ 13702.4 1.22927 0.614633 0.788813i $$-0.289304\pi$$
0.614633 + 0.788813i $$0.289304\pi$$
$$500$$ 0 0
$$501$$ 0 0
$$502$$ 0 0
$$503$$ 13707.6 1.21509 0.607546 0.794285i $$-0.292154\pi$$
0.607546 + 0.794285i $$0.292154\pi$$
$$504$$ 0 0
$$505$$ −20874.9 −1.83945
$$506$$ 0 0
$$507$$ 0 0
$$508$$ 0 0
$$509$$ −12339.2 −1.07451 −0.537254 0.843421i $$-0.680538\pi$$
−0.537254 + 0.843421i $$0.680538\pi$$
$$510$$ 0 0
$$511$$ −8390.07 −0.726330
$$512$$ 0 0
$$513$$ 0 0
$$514$$ 0 0
$$515$$ 20217.7 1.72990
$$516$$ 0 0
$$517$$ 251.870 0.0214259
$$518$$ 0 0
$$519$$ 0 0
$$520$$ 0 0
$$521$$ 1416.70 0.119130 0.0595649 0.998224i $$-0.481029\pi$$
0.0595649 + 0.998224i $$0.481029\pi$$
$$522$$ 0 0
$$523$$ −6696.15 −0.559851 −0.279925 0.960022i $$-0.590310\pi$$
−0.279925 + 0.960022i $$0.590310\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ −9871.02 −0.815917
$$528$$ 0 0
$$529$$ −11641.0 −0.956768
$$530$$ 0 0
$$531$$ 0 0
$$532$$ 0 0
$$533$$ 864.789 0.0702780
$$534$$ 0 0
$$535$$ −5777.40 −0.466876
$$536$$ 0 0
$$537$$ 0 0
$$538$$ 0 0
$$539$$ −511.609 −0.0408841
$$540$$ 0 0
$$541$$ −7105.69 −0.564691 −0.282345 0.959313i $$-0.591112\pi$$
−0.282345 + 0.959313i $$0.591112\pi$$
$$542$$ 0 0
$$543$$ 0 0
$$544$$ 0 0
$$545$$ 28787.8 2.26263
$$546$$ 0 0
$$547$$ −12028.0 −0.940185 −0.470092 0.882617i $$-0.655779\pi$$
−0.470092 + 0.882617i $$0.655779\pi$$
$$548$$ 0 0
$$549$$ 0 0
$$550$$ 0 0
$$551$$ −16887.5 −1.30569
$$552$$ 0 0
$$553$$ −3757.17 −0.288917
$$554$$ 0 0
$$555$$ 0 0
$$556$$ 0 0
$$557$$ −14529.5 −1.10527 −0.552634 0.833424i $$-0.686377\pi$$
−0.552634 + 0.833424i $$0.686377\pi$$
$$558$$ 0 0
$$559$$ 10654.3 0.806131
$$560$$ 0 0
$$561$$ 0 0
$$562$$ 0 0
$$563$$ 10233.4 0.766053 0.383027 0.923737i $$-0.374882\pi$$
0.383027 + 0.923737i $$0.374882\pi$$
$$564$$ 0 0
$$565$$ 41158.5 3.06469
$$566$$ 0 0
$$567$$ 0 0
$$568$$ 0 0
$$569$$ 319.460 0.0235368 0.0117684 0.999931i $$-0.496254\pi$$
0.0117684 + 0.999931i $$0.496254\pi$$
$$570$$ 0 0
$$571$$ 6793.96 0.497931 0.248965 0.968512i $$-0.419909\pi$$
0.248965 + 0.968512i $$0.419909\pi$$
$$572$$ 0 0
$$573$$ 0 0
$$574$$ 0 0
$$575$$ −7473.74 −0.542046
$$576$$ 0 0
$$577$$ −11145.6 −0.804156 −0.402078 0.915605i $$-0.631712\pi$$
−0.402078 + 0.915605i $$0.631712\pi$$
$$578$$ 0 0
$$579$$ 0 0
$$580$$ 0 0
$$581$$ −4691.36 −0.334992
$$582$$ 0 0
$$583$$ −148.038 −0.0105165
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 0 0
$$587$$ 23817.6 1.67471 0.837356 0.546658i $$-0.184100\pi$$
0.837356 + 0.546658i $$0.184100\pi$$
$$588$$ 0 0
$$589$$ −12878.4 −0.900924
$$590$$ 0 0
$$591$$ 0 0
$$592$$ 0 0
$$593$$ −16416.2 −1.13682 −0.568408 0.822747i $$-0.692441\pi$$
−0.568408 + 0.822747i $$0.692441\pi$$
$$594$$ 0 0
$$595$$ 60171.2 4.14585
$$596$$ 0 0
$$597$$ 0 0
$$598$$ 0 0
$$599$$ −4801.70 −0.327533 −0.163766 0.986499i $$-0.552364\pi$$
−0.163766 + 0.986499i $$0.552364\pi$$
$$600$$ 0 0
$$601$$ 4524.21 0.307066 0.153533 0.988144i $$-0.450935\pi$$
0.153533 + 0.988144i $$0.450935\pi$$
$$602$$ 0 0
$$603$$ 0 0
$$604$$ 0 0
$$605$$ 28240.8 1.89777
$$606$$ 0 0
$$607$$ −14978.3 −1.00157 −0.500783 0.865573i $$-0.666955\pi$$
−0.500783 + 0.865573i $$0.666955\pi$$
$$608$$ 0 0
$$609$$ 0 0
$$610$$ 0 0
$$611$$ −13332.6 −0.882784
$$612$$ 0 0
$$613$$ −11651.6 −0.767709 −0.383854 0.923394i $$-0.625404\pi$$
−0.383854 + 0.923394i $$0.625404\pi$$
$$614$$ 0 0
$$615$$ 0 0
$$616$$ 0 0
$$617$$ −341.120 −0.0222576 −0.0111288 0.999938i $$-0.503542\pi$$
−0.0111288 + 0.999938i $$0.503542\pi$$
$$618$$ 0 0
$$619$$ −18415.1 −1.19575 −0.597873 0.801591i $$-0.703987\pi$$
−0.597873 + 0.801591i $$0.703987\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ 0 0
$$623$$ 12576.5 0.808776
$$624$$ 0 0
$$625$$ 49832.2 3.18926
$$626$$ 0 0
$$627$$ 0 0
$$628$$ 0 0
$$629$$ 10184.5 0.645599
$$630$$ 0 0
$$631$$ 13557.6 0.855341 0.427671 0.903935i $$-0.359334\pi$$
0.427671 + 0.903935i $$0.359334\pi$$
$$632$$ 0 0
$$633$$ 0 0
$$634$$ 0 0
$$635$$ −4934.17 −0.308357
$$636$$ 0 0
$$637$$ 27081.9 1.68449
$$638$$ 0 0
$$639$$ 0 0
$$640$$ 0 0
$$641$$ 9682.55 0.596627 0.298313 0.954468i $$-0.403576\pi$$
0.298313 + 0.954468i $$0.403576\pi$$
$$642$$ 0 0
$$643$$ 3259.50 0.199910 0.0999551 0.994992i $$-0.468130\pi$$
0.0999551 + 0.994992i $$0.468130\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 0 0
$$647$$ −8115.35 −0.493118 −0.246559 0.969128i $$-0.579300\pi$$
−0.246559 + 0.969128i $$0.579300\pi$$
$$648$$ 0 0
$$649$$ −73.6085 −0.00445206
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 0 0
$$653$$ 3427.74 0.205418 0.102709 0.994711i $$-0.467249\pi$$
0.102709 + 0.994711i $$0.467249\pi$$
$$654$$ 0 0
$$655$$ −59354.6 −3.54073
$$656$$ 0 0
$$657$$ 0 0
$$658$$ 0 0
$$659$$ 13820.2 0.816934 0.408467 0.912773i $$-0.366064\pi$$
0.408467 + 0.912773i $$0.366064\pi$$
$$660$$ 0 0
$$661$$ 22719.7 1.33690 0.668451 0.743756i $$-0.266957\pi$$
0.668451 + 0.743756i $$0.266957\pi$$
$$662$$ 0 0
$$663$$ 0 0
$$664$$ 0 0
$$665$$ 78503.2 4.57778
$$666$$ 0 0
$$667$$ −3062.54 −0.177784
$$668$$ 0 0
$$669$$ 0 0
$$670$$ 0 0
$$671$$ −607.478 −0.0349500
$$672$$ 0 0
$$673$$ −5133.17 −0.294011 −0.147005 0.989136i $$-0.546963\pi$$
−0.147005 + 0.989136i $$0.546963\pi$$
$$674$$ 0 0
$$675$$ 0 0
$$676$$ 0 0
$$677$$ −9295.88 −0.527725 −0.263862 0.964560i $$-0.584996\pi$$
−0.263862 + 0.964560i $$0.584996\pi$$
$$678$$ 0 0
$$679$$ −910.059 −0.0514357
$$680$$ 0 0
$$681$$ 0 0
$$682$$ 0 0
$$683$$ −13824.3 −0.774486 −0.387243 0.921978i $$-0.626573\pi$$
−0.387243 + 0.921978i $$0.626573\pi$$
$$684$$ 0 0
$$685$$ 26787.0 1.49413
$$686$$ 0 0
$$687$$ 0 0
$$688$$ 0 0
$$689$$ 7836.35 0.433296
$$690$$ 0 0
$$691$$ 11015.2 0.606424 0.303212 0.952923i $$-0.401941\pi$$
0.303212 + 0.952923i $$0.401941\pi$$
$$692$$ 0 0
$$693$$ 0 0
$$694$$ 0 0
$$695$$ −6526.57 −0.356212
$$696$$ 0 0
$$697$$ −1583.61 −0.0860596
$$698$$ 0 0
$$699$$ 0 0
$$700$$ 0 0
$$701$$ 460.523 0.0248127 0.0124064 0.999923i $$-0.496051\pi$$
0.0124064 + 0.999923i $$0.496051\pi$$
$$702$$ 0 0
$$703$$ 13287.3 0.712861
$$704$$ 0 0
$$705$$ 0 0
$$706$$ 0 0
$$707$$ 28739.7 1.52881
$$708$$ 0 0
$$709$$ −26445.0 −1.40080 −0.700398 0.713752i $$-0.746994\pi$$
−0.700398 + 0.713752i $$0.746994\pi$$
$$710$$ 0 0
$$711$$ 0 0
$$712$$ 0 0
$$713$$ −2335.48 −0.122671
$$714$$ 0 0
$$715$$ 1124.00 0.0587905
$$716$$ 0 0
$$717$$ 0 0
$$718$$ 0 0
$$719$$ 15524.4 0.805235 0.402617 0.915368i $$-0.368101\pi$$
0.402617 + 0.915368i $$0.368101\pi$$
$$720$$ 0 0
$$721$$ −27834.9 −1.43776
$$722$$ 0 0
$$723$$ 0 0
$$724$$ 0 0
$$725$$ 43514.2 2.22907
$$726$$ 0 0
$$727$$ 18934.0 0.965922 0.482961 0.875642i $$-0.339561\pi$$
0.482961 + 0.875642i $$0.339561\pi$$
$$728$$ 0 0
$$729$$ 0 0
$$730$$ 0 0
$$731$$ −19510.2 −0.987156
$$732$$ 0 0
$$733$$ 16779.9 0.845540 0.422770 0.906237i $$-0.361058\pi$$
0.422770 + 0.906237i $$0.361058\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ 761.945 0.0380823
$$738$$ 0 0
$$739$$ −30309.0 −1.50871 −0.754353 0.656469i $$-0.772049\pi$$
−0.754353 + 0.656469i $$0.772049\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ 0 0
$$743$$ 32098.8 1.58492 0.792458 0.609927i $$-0.208801\pi$$
0.792458 + 0.609927i $$0.208801\pi$$
$$744$$ 0 0
$$745$$ 33926.2 1.66840
$$746$$ 0 0
$$747$$ 0 0
$$748$$ 0 0
$$749$$ 7954.09 0.388032
$$750$$ 0 0
$$751$$ 9434.07 0.458394 0.229197 0.973380i $$-0.426390\pi$$
0.229197 + 0.973380i $$0.426390\pi$$
$$752$$ 0 0
$$753$$ 0 0
$$754$$ 0 0
$$755$$ 72521.7 3.49581
$$756$$ 0 0
$$757$$ −1280.65 −0.0614876 −0.0307438 0.999527i $$-0.509788\pi$$
−0.0307438 + 0.999527i $$0.509788\pi$$
$$758$$ 0 0
$$759$$ 0 0
$$760$$ 0 0
$$761$$ 19624.7 0.934818 0.467409 0.884041i $$-0.345188\pi$$
0.467409 + 0.884041i $$0.345188\pi$$
$$762$$ 0 0
$$763$$ −39633.9 −1.88053
$$764$$ 0 0
$$765$$ 0 0
$$766$$ 0 0
$$767$$ 3896.45 0.183432
$$768$$ 0 0
$$769$$ 17644.9 0.827426 0.413713 0.910407i $$-0.364232\pi$$
0.413713 + 0.910407i $$0.364232\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ 0 0
$$773$$ 22541.7 1.04886 0.524430 0.851454i $$-0.324278\pi$$
0.524430 + 0.851454i $$0.324278\pi$$
$$774$$ 0 0
$$775$$ 33183.8 1.53806
$$776$$ 0 0
$$777$$ 0 0
$$778$$ 0 0
$$779$$ −2066.08 −0.0950258
$$780$$ 0 0
$$781$$ −701.196 −0.0321264
$$782$$ 0 0
$$783$$ 0 0
$$784$$ 0 0
$$785$$ 10124.8 0.460342
$$786$$ 0 0
$$787$$ 39687.8 1.79761 0.898804 0.438350i $$-0.144437\pi$$
0.898804 + 0.438350i $$0.144437\pi$$
$$788$$ 0 0
$$789$$ 0 0
$$790$$ 0 0
$$791$$ −56665.4 −2.54714
$$792$$ 0 0
$$793$$ 32156.7 1.44000
$$794$$ 0 0
$$795$$ 0 0
$$796$$ 0 0
$$797$$ 1991.97 0.0885309 0.0442654 0.999020i $$-0.485905\pi$$
0.0442654 + 0.999020i $$0.485905\pi$$
$$798$$ 0 0
$$799$$ 24414.9 1.08102
$$800$$ 0 0
$$801$$ 0 0
$$802$$ 0 0
$$803$$ 287.000 0.0126127
$$804$$ 0 0
$$805$$ 14236.5 0.623317
$$806$$ 0 0
$$807$$ 0 0
$$808$$ 0 0
$$809$$ −21679.4 −0.942160 −0.471080 0.882090i $$-0.656136\pi$$
−0.471080 + 0.882090i $$0.656136\pi$$
$$810$$ 0 0
$$811$$ 15099.7 0.653786 0.326893 0.945061i $$-0.393998\pi$$
0.326893 + 0.945061i $$0.393998\pi$$
$$812$$ 0 0
$$813$$ 0 0
$$814$$ 0 0
$$815$$ 70157.0 3.01533
$$816$$ 0 0
$$817$$ −25454.3 −1.09000
$$818$$ 0 0
$$819$$ 0 0
$$820$$ 0 0
$$821$$ −9902.77 −0.420961 −0.210480 0.977598i $$-0.567503\pi$$
−0.210480 + 0.977598i $$0.567503\pi$$
$$822$$ 0 0
$$823$$ 11488.7 0.486600 0.243300 0.969951i $$-0.421770\pi$$
0.243300 + 0.969951i $$0.421770\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ 0 0
$$827$$ −36938.8 −1.55319 −0.776594 0.630001i $$-0.783055\pi$$
−0.776594 + 0.630001i $$0.783055\pi$$
$$828$$ 0 0
$$829$$ −3205.00 −0.134275 −0.0671376 0.997744i $$-0.521387\pi$$
−0.0671376 + 0.997744i $$0.521387\pi$$
$$830$$ 0 0
$$831$$ 0 0
$$832$$ 0 0
$$833$$ −49592.6 −2.06277
$$834$$ 0 0
$$835$$ −44413.5 −1.84071
$$836$$ 0 0
$$837$$ 0 0
$$838$$ 0 0
$$839$$ −1126.63 −0.0463594 −0.0231797 0.999731i $$-0.507379\pi$$
−0.0231797 + 0.999731i $$0.507379\pi$$
$$840$$ 0 0
$$841$$ −6558.04 −0.268893
$$842$$ 0 0
$$843$$ 0 0
$$844$$ 0 0
$$845$$ −12848.2 −0.523069
$$846$$ 0 0
$$847$$ −38880.8 −1.57728
$$848$$ 0 0
$$849$$ 0 0
$$850$$ 0 0
$$851$$ 2409.65 0.0970641
$$852$$ 0 0
$$853$$ −6254.66 −0.251061 −0.125531 0.992090i $$-0.540063\pi$$
−0.125531 + 0.992090i $$0.540063\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ 0 0
$$857$$ 19069.4 0.760090 0.380045 0.924968i $$-0.375908\pi$$
0.380045 + 0.924968i $$0.375908\pi$$
$$858$$ 0 0
$$859$$ −26790.3 −1.06411 −0.532056 0.846709i $$-0.678580\pi$$
−0.532056 + 0.846709i $$0.678580\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ 0 0
$$863$$ 46056.3 1.81666 0.908329 0.418256i $$-0.137359\pi$$
0.908329 + 0.418256i $$0.137359\pi$$
$$864$$ 0 0
$$865$$ 62099.2 2.44097
$$866$$ 0 0
$$867$$ 0 0
$$868$$ 0 0
$$869$$ 128.522 0.00501704
$$870$$ 0 0
$$871$$ −40333.4 −1.56905
$$872$$ 0 0
$$873$$ 0 0
$$874$$ 0 0
$$875$$ −124688. −4.81738
$$876$$ 0 0
$$877$$ −36490.2 −1.40500 −0.702500 0.711684i $$-0.747933\pi$$
−0.702500 + 0.711684i $$0.747933\pi$$
$$878$$ 0 0
$$879$$ 0 0
$$880$$ 0 0
$$881$$ −38136.3 −1.45839 −0.729197 0.684303i $$-0.760106\pi$$
−0.729197 + 0.684303i $$0.760106\pi$$
$$882$$ 0 0
$$883$$ 13336.1 0.508262 0.254131 0.967170i $$-0.418211\pi$$
0.254131 + 0.967170i $$0.418211\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ 0 0
$$887$$ −28027.2 −1.06095 −0.530474 0.847701i $$-0.677986\pi$$
−0.530474 + 0.847701i $$0.677986\pi$$
$$888$$ 0 0
$$889$$ 6793.17 0.256283
$$890$$ 0 0
$$891$$ 0 0
$$892$$ 0 0
$$893$$ 31853.3 1.19365
$$894$$ 0 0
$$895$$ −4924.38 −0.183915
$$896$$ 0 0
$$897$$ 0 0
$$898$$ 0 0
$$899$$ 13597.8 0.504464
$$900$$ 0 0
$$901$$ −14350.0 −0.530598
$$902$$ 0 0
$$903$$ 0 0
$$904$$ 0 0
$$905$$ 48662.1 1.78738
$$906$$ 0 0
$$907$$ 45203.3 1.65485 0.827427 0.561573i $$-0.189804\pi$$
0.827427 + 0.561573i $$0.189804\pi$$
$$908$$ 0 0
$$909$$ 0 0
$$910$$ 0 0
$$911$$ −29822.0 −1.08457 −0.542287 0.840194i $$-0.682441\pi$$
−0.542287 + 0.840194i $$0.682441\pi$$
$$912$$ 0 0
$$913$$ 160.478 0.00581714
$$914$$ 0 0
$$915$$ 0 0
$$916$$ 0 0
$$917$$ 81717.0 2.94279
$$918$$ 0 0
$$919$$ 36967.7 1.32693 0.663467 0.748206i $$-0.269084\pi$$
0.663467 + 0.748206i $$0.269084\pi$$
$$920$$ 0 0
$$921$$ 0 0
$$922$$ 0 0
$$923$$ 37117.6 1.32366
$$924$$ 0 0
$$925$$ −34237.6 −1.21700
$$926$$ 0 0
$$927$$ 0 0
$$928$$ 0 0
$$929$$ 50409.4 1.78028 0.890139 0.455690i $$-0.150607\pi$$
0.890139 + 0.455690i $$0.150607\pi$$
$$930$$ 0 0
$$931$$ −64701.8 −2.27767
$$932$$ 0 0
$$933$$ 0 0
$$934$$ 0 0
$$935$$ −2058.28 −0.0719925
$$936$$ 0 0
$$937$$ −30156.5 −1.05141 −0.525705 0.850667i $$-0.676198\pi$$
−0.525705 + 0.850667i $$0.676198\pi$$
$$938$$ 0 0
$$939$$ 0 0
$$940$$ 0 0
$$941$$ 42221.1 1.46267 0.731333 0.682020i $$-0.238898\pi$$
0.731333 + 0.682020i $$0.238898\pi$$
$$942$$ 0 0
$$943$$ −374.682 −0.0129388
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ −40164.6 −1.37822 −0.689111 0.724656i $$-0.741999\pi$$
−0.689111 + 0.724656i $$0.741999\pi$$
$$948$$ 0 0
$$949$$ −15192.3 −0.519665
$$950$$ 0 0
$$951$$ 0 0
$$952$$ 0 0
$$953$$ 7132.10 0.242425 0.121213 0.992627i $$-0.461322\pi$$
0.121213 + 0.992627i $$0.461322\pi$$
$$954$$ 0 0
$$955$$ 89559.5 3.03464
$$956$$ 0 0
$$957$$ 0 0
$$958$$ 0 0
$$959$$ −36879.3 −1.24181
$$960$$ 0 0
$$961$$ −19421.3 −0.651919
$$962$$ 0 0
$$963$$ 0 0
$$964$$ 0 0
$$965$$ 25233.0 0.841740
$$966$$ 0 0
$$967$$ −549.206 −0.0182640 −0.00913199 0.999958i $$-0.502907\pi$$
−0.00913199 + 0.999958i $$0.502907\pi$$
$$968$$ 0 0
$$969$$ 0 0
$$970$$ 0 0
$$971$$ −31024.8 −1.02537 −0.512685 0.858577i $$-0.671349\pi$$
−0.512685 + 0.858577i $$0.671349\pi$$
$$972$$ 0 0
$$973$$ 8985.52 0.296056
$$974$$ 0 0
$$975$$ 0 0
$$976$$ 0 0
$$977$$ 7827.03 0.256304 0.128152 0.991755i $$-0.459096\pi$$
0.128152 + 0.991755i $$0.459096\pi$$
$$978$$ 0 0
$$979$$ −430.206 −0.0140444
$$980$$ 0 0
$$981$$ 0 0
$$982$$ 0 0
$$983$$ −47421.5 −1.53867 −0.769334 0.638847i $$-0.779412\pi$$
−0.769334 + 0.638847i $$0.779412\pi$$
$$984$$ 0 0
$$985$$ −66466.8 −2.15006
$$986$$ 0 0
$$987$$ 0 0
$$988$$ 0 0
$$989$$ −4616.11 −0.148416
$$990$$ 0 0
$$991$$ 43260.0 1.38668 0.693339 0.720612i $$-0.256139\pi$$
0.693339 + 0.720612i $$0.256139\pi$$
$$992$$ 0 0
$$993$$ 0 0
$$994$$ 0 0
$$995$$ 13996.1 0.445936
$$996$$ 0 0
$$997$$ 40209.1 1.27727 0.638634 0.769511i $$-0.279500\pi$$
0.638634 + 0.769511i $$0.279500\pi$$
$$998$$ 0 0
$$999$$ 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 1728.4.a.bh.1.1 2
3.2 odd 2 1728.4.a.bt.1.2 2
4.3 odd 2 1728.4.a.bg.1.1 2
8.3 odd 2 432.4.a.s.1.2 2
8.5 even 2 216.4.a.h.1.2 yes 2
12.11 even 2 1728.4.a.bs.1.2 2
24.5 odd 2 216.4.a.e.1.1 2
24.11 even 2 432.4.a.o.1.1 2
72.5 odd 6 648.4.i.s.217.2 4
72.13 even 6 648.4.i.m.217.1 4
72.29 odd 6 648.4.i.s.433.2 4
72.61 even 6 648.4.i.m.433.1 4

By twisted newform
Twist Min Dim Char Parity Ord Type
216.4.a.e.1.1 2 24.5 odd 2
216.4.a.h.1.2 yes 2 8.5 even 2
432.4.a.o.1.1 2 24.11 even 2
432.4.a.s.1.2 2 8.3 odd 2
648.4.i.m.217.1 4 72.13 even 6
648.4.i.m.433.1 4 72.61 even 6
648.4.i.s.217.2 4 72.5 odd 6
648.4.i.s.433.2 4 72.29 odd 6
1728.4.a.bg.1.1 2 4.3 odd 2
1728.4.a.bh.1.1 2 1.1 even 1 trivial
1728.4.a.bs.1.2 2 12.11 even 2
1728.4.a.bt.1.2 2 3.2 odd 2