# Properties

 Label 1728.4.a.bg.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 $$-0.789523\pi$$
−0.789235 + 0.614091i $$0.789523\pi$$
$$8$$ 0 0
$$9$$ 0 0
$$10$$ 0 0
$$11$$ 1.00000 0.0274101 0.0137051 0.999906i $$-0.495637\pi$$
0.0137051 + 0.999906i $$0.495637\pi$$
$$12$$ 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 $$-0.223471\pi$$
0.763516 + 0.645789i $$0.223471\pi$$
$$20$$ 0 0
$$21$$ 0 0
$$22$$ 0 0
$$23$$ 22.9348 0.207923 0.103961 0.994581i $$-0.466848\pi$$
0.103961 + 0.994581i $$0.466848\pi$$
$$24$$ 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 $$-0.595317\pi$$
−0.294992 + 0.955500i $$0.595317\pi$$
$$32$$ 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 $$-0.616167\pi$$
−0.356903 + 0.934142i $$0.616167\pi$$
$$44$$ 0 0
$$45$$ 0 0
$$46$$ 0 0
$$47$$ 251.870 0.781680 0.390840 0.920459i $$-0.372185\pi$$
0.390840 + 0.920459i $$0.372185\pi$$
$$48$$ 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 $$-0.525879\pi$$
−0.0812120 + 0.996697i $$0.525879\pi$$
$$60$$ 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 $$-0.255548\pi$$
0.694675 + 0.719324i $$0.255548\pi$$
$$68$$ 0 0
$$69$$ 0 0
$$70$$ 0 0
$$71$$ −701.196 −1.17207 −0.586033 0.810287i $$-0.699311\pi$$
−0.586033 + 0.810287i $$0.699311\pi$$
$$72$$ 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 $$-0.470828\pi$$
0.0915181 + 0.995803i $$0.470828\pi$$
$$80$$ 0 0
$$81$$ 0 0
$$82$$ 0 0
$$83$$ 160.478 0.212226 0.106113 0.994354i $$-0.466159\pi$$
0.106113 + 0.994354i $$0.466159\pi$$
$$84$$ 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 $$-0.349486\pi$$
0.455429 + 0.890272i $$0.349486\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ −272.087 −0.245828 −0.122914 0.992417i $$-0.539224\pi$$
−0.122914 + 0.992417i $$0.539224\pi$$
$$108$$ 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 $$-0.525869\pi$$
−0.0811808 + 0.996699i $$0.525869\pi$$
$$128$$ 0 0
$$129$$ 0 0
$$130$$ 0 0
$$131$$ −2795.30 −1.86433 −0.932163 0.362038i $$-0.882081\pi$$
−0.932163 + 0.362038i $$0.882081\pi$$
$$132$$ 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 $$-0.529895\pi$$
−0.0937794 + 0.995593i $$0.529895\pi$$
$$140$$ 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 $$-0.127914\pi$$
0.920337 + 0.391126i $$0.127914\pi$$
$$152$$ 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 $$-0.208078\pi$$
0.793842 + 0.608124i $$0.208078\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ 0 0
$$167$$ −2091.65 −0.969203 −0.484601 0.874735i $$-0.661035\pi$$
−0.484601 + 0.874735i $$0.661035\pi$$
$$168$$ 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 $$-0.515418\pi$$
−0.0484191 + 0.998827i $$0.515418\pi$$
$$180$$ 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 $$-0.205402\pi$$
0.798925 + 0.601430i $$0.205402\pi$$
$$192$$ 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 $$-0.462544\pi$$
0.117401 + 0.993085i $$0.462544\pi$$
$$200$$ 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 $$-0.299301\pi$$
0.589560 + 0.807725i $$0.299301\pi$$
$$212$$ 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 $$-0.400200\pi$$
0.308419 + 0.951251i $$0.400200\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ 0 0
$$227$$ 2153.61 0.629692 0.314846 0.949143i $$-0.398047\pi$$
0.314846 + 0.949143i $$0.398047\pi$$
$$228$$ 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 $$-0.219292\pi$$
0.771929 + 0.635709i $$0.219292\pi$$
$$240$$ 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 $$-0.302577\pi$$
0.581217 + 0.813749i $$0.302577\pi$$
$$252$$ 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 $$-0.673941\pi$$
−0.519660 + 0.854373i $$0.673941\pi$$
$$264$$ 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 $$-0.618913\pi$$
−0.364948 + 0.931028i $$0.618913\pi$$
$$272$$ 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 $$-0.656218\pi$$
−0.471307 + 0.881969i $$0.656218\pi$$
$$284$$ 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 $$-0.770415\pi$$
−0.750972 + 0.660334i $$0.770415\pi$$
$$308$$ 0 0
$$309$$ 0 0
$$310$$ 0 0
$$311$$ −2557.63 −0.466334 −0.233167 0.972437i $$-0.574909\pi$$
−0.233167 + 0.972437i $$0.574909\pi$$
$$312$$ 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 $$-0.509445\pi$$
−0.0296673 + 0.999560i $$0.509445\pi$$
$$332$$ 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 $$-0.290918\pi$$
0.610626 + 0.791919i $$0.290918\pi$$
$$348$$ 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 $$-0.408045\pi$$
0.284884 + 0.958562i $$0.408045\pi$$
$$360$$ 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 $$-0.179253\pi$$
0.845583 + 0.533843i $$0.179253\pi$$
$$368$$ 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 $$-0.455358\pi$$
0.139788 + 0.990181i $$0.455358\pi$$
$$380$$ 0 0
$$381$$ 0 0
$$382$$ 0 0
$$383$$ −11905.2 −1.58833 −0.794164 0.607704i $$-0.792091\pi$$
−0.794164 + 0.607704i $$0.792091\pi$$
$$384$$ 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 $$-0.341102\pi$$
0.478718 + 0.877969i $$0.341102\pi$$
$$420$$ 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 $$-0.715394\pi$$
−0.626207 + 0.779657i $$0.715394\pi$$
$$432$$ 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 $$-0.658671\pi$$
−0.478090 + 0.878311i $$0.658671\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ 0 0
$$443$$ 3023.65 0.324285 0.162142 0.986767i $$-0.448160\pi$$
0.162142 + 0.986767i $$0.448160\pi$$
$$444$$ 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 $$-0.555756\pi$$
−0.174267 + 0.984698i $$0.555756\pi$$
$$464$$ 0 0
$$465$$ 0 0
$$466$$ 0 0
$$467$$ 11825.0 1.17172 0.585860 0.810412i $$-0.300757\pi$$
0.585860 + 0.810412i $$0.300757\pi$$
$$468$$ 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 $$-0.474103\pi$$
0.0812692 + 0.996692i $$0.474103\pi$$
$$480$$ 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 $$-0.616969\pi$$
−0.359256 + 0.933239i $$0.616969\pi$$
$$488$$ 0 0
$$489$$ 0 0
$$490$$ 0 0
$$491$$ 20223.6 1.85882 0.929410 0.369049i $$-0.120317\pi$$
0.929410 + 0.369049i $$0.120317\pi$$
$$492$$ 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 $$-0.710696\pi$$
−0.614633 + 0.788813i $$0.710696\pi$$
$$500$$ 0 0
$$501$$ 0 0
$$502$$ 0 0
$$503$$ −13707.6 −1.21509 −0.607546 0.794285i $$-0.707846\pi$$
−0.607546 + 0.794285i $$0.707846\pi$$
$$504$$ 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 $$-0.409690\pi$$
0.279925 + 0.960022i $$0.409690\pi$$
$$524$$ 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 $$-0.344221\pi$$
0.470092 + 0.882617i $$0.344221\pi$$
$$548$$ 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 $$-0.625118\pi$$
−0.383027 + 0.923737i $$0.625118\pi$$
$$564$$ 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 $$-0.580091\pi$$
−0.248965 + 0.968512i $$0.580091\pi$$
$$572$$ 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 $$-0.815900\pi$$
−0.837356 + 0.546658i $$0.815900\pi$$
$$588$$ 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 $$-0.447636\pi$$
0.163766 + 0.986499i $$0.447636\pi$$
$$600$$ 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 $$-0.333045\pi$$
0.500783 + 0.865573i $$0.333045\pi$$
$$608$$ 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 $$-0.296013\pi$$
0.597873 + 0.801591i $$0.296013\pi$$
$$620$$ 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 $$-0.640666\pi$$
−0.427671 + 0.903935i $$0.640666\pi$$
$$632$$ 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 $$-0.531870\pi$$
−0.0999551 + 0.994992i $$0.531870\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 0 0
$$647$$ 8115.35 0.493118 0.246559 0.969128i $$-0.420700\pi$$
0.246559 + 0.969128i $$0.420700\pi$$
$$648$$ 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 $$-0.633936\pi$$
−0.408467 + 0.912773i $$0.633936\pi$$
$$660$$ 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 $$-0.373427\pi$$
0.387243 + 0.921978i $$0.373427\pi$$
$$684$$ 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 $$-0.598059\pi$$
−0.303212 + 0.952923i $$0.598059\pi$$
$$692$$ 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 $$-0.631899\pi$$
−0.402617 + 0.915368i $$0.631899\pi$$
$$720$$ 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 $$-0.660439\pi$$
−0.482961 + 0.875642i $$0.660439\pi$$
$$728$$ 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 $$-0.227951\pi$$
0.754353 + 0.656469i $$0.227951\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ 0 0
$$743$$ −32098.8 −1.58492 −0.792458 0.609927i $$-0.791199\pi$$
−0.792458 + 0.609927i $$0.791199\pi$$
$$744$$ 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 $$-0.573610\pi$$
−0.229197 + 0.973380i $$0.573610\pi$$
$$752$$ 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 $$-0.855563\pi$$
−0.898804 + 0.438350i $$0.855563\pi$$
$$788$$ 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 $$-0.606002\pi$$
−0.326893 + 0.945061i $$0.606002\pi$$
$$812$$ 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 $$-0.578230\pi$$
−0.243300 + 0.969951i $$0.578230\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ 0 0
$$827$$ 36938.8 1.55319 0.776594 0.630001i $$-0.216945\pi$$
0.776594 + 0.630001i $$0.216945\pi$$
$$828$$ 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 $$-0.492621\pi$$
0.0231797 + 0.999731i $$0.492621\pi$$
$$840$$ 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 $$-0.321420\pi$$
0.532056 + 0.846709i $$0.321420\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ 0 0
$$863$$ −46056.3 −1.81666 −0.908329 0.418256i $$-0.862641\pi$$
−0.908329 + 0.418256i $$0.862641\pi$$
$$864$$ 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 $$-0.581789\pi$$
−0.254131 + 0.967170i $$0.581789\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ 0 0
$$887$$ 28027.2 1.06095 0.530474 0.847701i $$-0.322014\pi$$
0.530474 + 0.847701i $$0.322014\pi$$
$$888$$ 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 $$-0.810196\pi$$
−0.827427 + 0.561573i $$0.810196\pi$$
$$908$$ 0 0
$$909$$ 0 0
$$910$$ 0 0
$$911$$ 29822.0 1.08457 0.542287 0.840194i $$-0.317559\pi$$
0.542287 + 0.840194i $$0.317559\pi$$
$$912$$ 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 $$-0.730916\pi$$
−0.663467 + 0.748206i $$0.730916\pi$$
$$920$$ 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 $$-0.258001\pi$$
0.689111 + 0.724656i $$0.258001\pi$$
$$948$$ 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 $$-0.497093\pi$$
0.00913199 + 0.999958i $$0.497093\pi$$
$$968$$ 0 0
$$969$$ 0 0
$$970$$ 0 0
$$971$$ 31024.8 1.02537 0.512685 0.858577i $$-0.328651\pi$$
0.512685 + 0.858577i $$0.328651\pi$$
$$972$$ 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 $$-0.220588\pi$$
0.769334 + 0.638847i $$0.220588\pi$$
$$984$$ 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 $$-0.743861\pi$$
−0.693339 + 0.720612i $$0.743861\pi$$
$$992$$ 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.bg.1.1 2
3.2 odd 2 1728.4.a.bs.1.2 2
4.3 odd 2 1728.4.a.bh.1.1 2
8.3 odd 2 216.4.a.h.1.2 yes 2
8.5 even 2 432.4.a.s.1.2 2
12.11 even 2 1728.4.a.bt.1.2 2
24.5 odd 2 432.4.a.o.1.1 2
24.11 even 2 216.4.a.e.1.1 2
72.11 even 6 648.4.i.s.433.2 4
72.43 odd 6 648.4.i.m.433.1 4
72.59 even 6 648.4.i.s.217.2 4
72.67 odd 6 648.4.i.m.217.1 4

By twisted newform
Twist Min Dim Char Parity Ord Type
216.4.a.e.1.1 2 24.11 even 2
216.4.a.h.1.2 yes 2 8.3 odd 2
432.4.a.o.1.1 2 24.5 odd 2
432.4.a.s.1.2 2 8.5 even 2
648.4.i.m.217.1 4 72.67 odd 6
648.4.i.m.433.1 4 72.43 odd 6
648.4.i.s.217.2 4 72.59 even 6
648.4.i.s.433.2 4 72.11 even 6
1728.4.a.bg.1.1 2 1.1 even 1 trivial
1728.4.a.bh.1.1 2 4.3 odd 2
1728.4.a.bs.1.2 2 3.2 odd 2
1728.4.a.bt.1.2 2 12.11 even 2