# Properties

 Label 2736.3.o.h.721.1 Level $2736$ Weight $3$ Character 2736.721 Analytic conductor $74.551$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$2736 = 2^{4} \cdot 3^{2} \cdot 19$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 2736.o (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$74.5506003290$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-2})$$ Defining polynomial: $$x^{2} + 2$$ Coefficient ring: $$\Z[a_1, \ldots, a_{29}]$$ Coefficient ring index: $$2\cdot 3$$ Twist minimal: no (minimal twist has level 38) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## Embedding invariants

 Embedding label 721.1 Root $$1.41421i$$ of defining polynomial Character $$\chi$$ $$=$$ 2736.721 Dual form 2736.3.o.h.721.2

## $q$-expansion

 $$f(q)$$ $$=$$ $$q+1.00000 q^{5} -5.00000 q^{7} +O(q^{10})$$ $$q+1.00000 q^{5} -5.00000 q^{7} +5.00000 q^{11} -16.9706i q^{13} +25.0000 q^{17} -19.0000 q^{19} -10.0000 q^{23} -24.0000 q^{25} -42.4264i q^{29} +42.4264i q^{31} -5.00000 q^{35} -25.4558i q^{37} +42.4264i q^{41} -5.00000 q^{43} +5.00000 q^{47} -24.0000 q^{49} +25.4558i q^{53} +5.00000 q^{55} +84.8528i q^{59} +95.0000 q^{61} -16.9706i q^{65} -110.309i q^{67} -25.0000 q^{73} -25.0000 q^{77} -42.4264i q^{79} -130.000 q^{83} +25.0000 q^{85} +127.279i q^{89} +84.8528i q^{91} -19.0000 q^{95} +16.9706i q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + 2q^{5} - 10q^{7} + O(q^{10})$$ $$2q + 2q^{5} - 10q^{7} + 10q^{11} + 50q^{17} - 38q^{19} - 20q^{23} - 48q^{25} - 10q^{35} - 10q^{43} + 10q^{47} - 48q^{49} + 10q^{55} + 190q^{61} - 50q^{73} - 50q^{77} - 260q^{83} + 50q^{85} - 38q^{95} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/2736\mathbb{Z}\right)^\times$$.

 $$n$$ $$1009$$ $$1217$$ $$1711$$ $$2053$$ $$\chi(n)$$ $$-1$$ $$1$$ $$1$$ $$1$$

## 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
$$5$$ 1.00000 0.200000 0.100000 0.994987i $$-0.468116\pi$$
0.100000 + 0.994987i $$0.468116\pi$$
$$6$$ 0 0
$$7$$ −5.00000 −0.714286 −0.357143 0.934050i $$-0.616249\pi$$
−0.357143 + 0.934050i $$0.616249\pi$$
$$8$$ 0 0
$$9$$ 0 0
$$10$$ 0 0
$$11$$ 5.00000 0.454545 0.227273 0.973831i $$-0.427019\pi$$
0.227273 + 0.973831i $$0.427019\pi$$
$$12$$ 0 0
$$13$$ − 16.9706i − 1.30543i −0.757604 0.652714i $$-0.773630\pi$$
0.757604 0.652714i $$-0.226370\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 0 0
$$17$$ 25.0000 1.47059 0.735294 0.677748i $$-0.237044\pi$$
0.735294 + 0.677748i $$0.237044\pi$$
$$18$$ 0 0
$$19$$ −19.0000 −1.00000
$$20$$ 0 0
$$21$$ 0 0
$$22$$ 0 0
$$23$$ −10.0000 −0.434783 −0.217391 0.976085i $$-0.569755\pi$$
−0.217391 + 0.976085i $$0.569755\pi$$
$$24$$ 0 0
$$25$$ −24.0000 −0.960000
$$26$$ 0 0
$$27$$ 0 0
$$28$$ 0 0
$$29$$ − 42.4264i − 1.46298i −0.681852 0.731490i $$-0.738825\pi$$
0.681852 0.731490i $$-0.261175\pi$$
$$30$$ 0 0
$$31$$ 42.4264i 1.36859i 0.729204 + 0.684297i $$0.239891\pi$$
−0.729204 + 0.684297i $$0.760109\pi$$
$$32$$ 0 0
$$33$$ 0 0
$$34$$ 0 0
$$35$$ −5.00000 −0.142857
$$36$$ 0 0
$$37$$ − 25.4558i − 0.687996i −0.938970 0.343998i $$-0.888219\pi$$
0.938970 0.343998i $$-0.111781\pi$$
$$38$$ 0 0
$$39$$ 0 0
$$40$$ 0 0
$$41$$ 42.4264i 1.03479i 0.855747 + 0.517395i $$0.173098\pi$$
−0.855747 + 0.517395i $$0.826902\pi$$
$$42$$ 0 0
$$43$$ −5.00000 −0.116279 −0.0581395 0.998308i $$-0.518517\pi$$
−0.0581395 + 0.998308i $$0.518517\pi$$
$$44$$ 0 0
$$45$$ 0 0
$$46$$ 0 0
$$47$$ 5.00000 0.106383 0.0531915 0.998584i $$-0.483061\pi$$
0.0531915 + 0.998584i $$0.483061\pi$$
$$48$$ 0 0
$$49$$ −24.0000 −0.489796
$$50$$ 0 0
$$51$$ 0 0
$$52$$ 0 0
$$53$$ 25.4558i 0.480299i 0.970736 + 0.240149i $$0.0771965\pi$$
−0.970736 + 0.240149i $$0.922804\pi$$
$$54$$ 0 0
$$55$$ 5.00000 0.0909091
$$56$$ 0 0
$$57$$ 0 0
$$58$$ 0 0
$$59$$ 84.8528i 1.43818i 0.694915 + 0.719092i $$0.255442\pi$$
−0.694915 + 0.719092i $$0.744558\pi$$
$$60$$ 0 0
$$61$$ 95.0000 1.55738 0.778689 0.627411i $$-0.215885\pi$$
0.778689 + 0.627411i $$0.215885\pi$$
$$62$$ 0 0
$$63$$ 0 0
$$64$$ 0 0
$$65$$ − 16.9706i − 0.261086i
$$66$$ 0 0
$$67$$ − 110.309i − 1.64640i −0.567753 0.823199i $$-0.692187\pi$$
0.567753 0.823199i $$-0.307813\pi$$
$$68$$ 0 0
$$69$$ 0 0
$$70$$ 0 0
$$71$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$72$$ 0 0
$$73$$ −25.0000 −0.342466 −0.171233 0.985231i $$-0.554775\pi$$
−0.171233 + 0.985231i $$0.554775\pi$$
$$74$$ 0 0
$$75$$ 0 0
$$76$$ 0 0
$$77$$ −25.0000 −0.324675
$$78$$ 0 0
$$79$$ − 42.4264i − 0.537043i −0.963274 0.268522i $$-0.913465\pi$$
0.963274 0.268522i $$-0.0865351\pi$$
$$80$$ 0 0
$$81$$ 0 0
$$82$$ 0 0
$$83$$ −130.000 −1.56627 −0.783133 0.621855i $$-0.786379\pi$$
−0.783133 + 0.621855i $$0.786379\pi$$
$$84$$ 0 0
$$85$$ 25.0000 0.294118
$$86$$ 0 0
$$87$$ 0 0
$$88$$ 0 0
$$89$$ 127.279i 1.43010i 0.699071 + 0.715052i $$0.253597\pi$$
−0.699071 + 0.715052i $$0.746403\pi$$
$$90$$ 0 0
$$91$$ 84.8528i 0.932449i
$$92$$ 0 0
$$93$$ 0 0
$$94$$ 0 0
$$95$$ −19.0000 −0.200000
$$96$$ 0 0
$$97$$ 16.9706i 0.174954i 0.996167 + 0.0874771i $$0.0278805\pi$$
−0.996167 + 0.0874771i $$0.972120\pi$$
$$98$$ 0 0
$$99$$ 0 0
$$100$$ 0 0
$$101$$ −50.0000 −0.495050 −0.247525 0.968882i $$-0.579617\pi$$
−0.247525 + 0.968882i $$0.579617\pi$$
$$102$$ 0 0
$$103$$ 16.9706i 0.164763i 0.996601 + 0.0823814i $$0.0262526\pi$$
−0.996601 + 0.0823814i $$0.973747\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ − 101.823i − 0.951620i −0.879548 0.475810i $$-0.842155\pi$$
0.879548 0.475810i $$-0.157845\pi$$
$$108$$ 0 0
$$109$$ 127.279i 1.16770i 0.811862 + 0.583850i $$0.198454\pi$$
−0.811862 + 0.583850i $$0.801546\pi$$
$$110$$ 0 0
$$111$$ 0 0
$$112$$ 0 0
$$113$$ − 110.309i − 0.976183i −0.872793 0.488091i $$-0.837693\pi$$
0.872793 0.488091i $$-0.162307\pi$$
$$114$$ 0 0
$$115$$ −10.0000 −0.0869565
$$116$$ 0 0
$$117$$ 0 0
$$118$$ 0 0
$$119$$ −125.000 −1.05042
$$120$$ 0 0
$$121$$ −96.0000 −0.793388
$$122$$ 0 0
$$123$$ 0 0
$$124$$ 0 0
$$125$$ −49.0000 −0.392000
$$126$$ 0 0
$$127$$ − 229.103i − 1.80396i −0.431780 0.901979i $$-0.642114\pi$$
0.431780 0.901979i $$-0.357886\pi$$
$$128$$ 0 0
$$129$$ 0 0
$$130$$ 0 0
$$131$$ −163.000 −1.24427 −0.622137 0.782908i $$-0.713735\pi$$
−0.622137 + 0.782908i $$0.713735\pi$$
$$132$$ 0 0
$$133$$ 95.0000 0.714286
$$134$$ 0 0
$$135$$ 0 0
$$136$$ 0 0
$$137$$ −95.0000 −0.693431 −0.346715 0.937970i $$-0.612703\pi$$
−0.346715 + 0.937970i $$0.612703\pi$$
$$138$$ 0 0
$$139$$ −125.000 −0.899281 −0.449640 0.893210i $$-0.648448\pi$$
−0.449640 + 0.893210i $$0.648448\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ 0 0
$$143$$ − 84.8528i − 0.593376i
$$144$$ 0 0
$$145$$ − 42.4264i − 0.292596i
$$146$$ 0 0
$$147$$ 0 0
$$148$$ 0 0
$$149$$ −215.000 −1.44295 −0.721477 0.692439i $$-0.756536\pi$$
−0.721477 + 0.692439i $$0.756536\pi$$
$$150$$ 0 0
$$151$$ 84.8528i 0.561939i 0.959717 + 0.280970i $$0.0906560\pi$$
−0.959717 + 0.280970i $$0.909344\pi$$
$$152$$ 0 0
$$153$$ 0 0
$$154$$ 0 0
$$155$$ 42.4264i 0.273719i
$$156$$ 0 0
$$157$$ −190.000 −1.21019 −0.605096 0.796153i $$-0.706865\pi$$
−0.605096 + 0.796153i $$0.706865\pi$$
$$158$$ 0 0
$$159$$ 0 0
$$160$$ 0 0
$$161$$ 50.0000 0.310559
$$162$$ 0 0
$$163$$ −110.000 −0.674847 −0.337423 0.941353i $$-0.609555\pi$$
−0.337423 + 0.941353i $$0.609555\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ 0 0
$$167$$ 59.3970i 0.355670i 0.984060 + 0.177835i $$0.0569094\pi$$
−0.984060 + 0.177835i $$0.943091\pi$$
$$168$$ 0 0
$$169$$ −119.000 −0.704142
$$170$$ 0 0
$$171$$ 0 0
$$172$$ 0 0
$$173$$ 186.676i 1.07905i 0.841969 + 0.539527i $$0.181397\pi$$
−0.841969 + 0.539527i $$0.818603\pi$$
$$174$$ 0 0
$$175$$ 120.000 0.685714
$$176$$ 0 0
$$177$$ 0 0
$$178$$ 0 0
$$179$$ − 127.279i − 0.711057i −0.934665 0.355529i $$-0.884301\pi$$
0.934665 0.355529i $$-0.115699\pi$$
$$180$$ 0 0
$$181$$ − 254.558i − 1.40640i −0.710992 0.703200i $$-0.751754\pi$$
0.710992 0.703200i $$-0.248246\pi$$
$$182$$ 0 0
$$183$$ 0 0
$$184$$ 0 0
$$185$$ − 25.4558i − 0.137599i
$$186$$ 0 0
$$187$$ 125.000 0.668449
$$188$$ 0 0
$$189$$ 0 0
$$190$$ 0 0
$$191$$ 293.000 1.53403 0.767016 0.641628i $$-0.221741\pi$$
0.767016 + 0.641628i $$0.221741\pi$$
$$192$$ 0 0
$$193$$ 59.3970i 0.307756i 0.988090 + 0.153878i $$0.0491763\pi$$
−0.988090 + 0.153878i $$0.950824\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ 0 0
$$197$$ 70.0000 0.355330 0.177665 0.984091i $$-0.443146\pi$$
0.177665 + 0.984091i $$0.443146\pi$$
$$198$$ 0 0
$$199$$ −173.000 −0.869347 −0.434673 0.900588i $$-0.643136\pi$$
−0.434673 + 0.900588i $$0.643136\pi$$
$$200$$ 0 0
$$201$$ 0 0
$$202$$ 0 0
$$203$$ 212.132i 1.04499i
$$204$$ 0 0
$$205$$ 42.4264i 0.206958i
$$206$$ 0 0
$$207$$ 0 0
$$208$$ 0 0
$$209$$ −95.0000 −0.454545
$$210$$ 0 0
$$211$$ − 84.8528i − 0.402146i −0.979576 0.201073i $$-0.935557\pi$$
0.979576 0.201073i $$-0.0644429\pi$$
$$212$$ 0 0
$$213$$ 0 0
$$214$$ 0 0
$$215$$ −5.00000 −0.0232558
$$216$$ 0 0
$$217$$ − 212.132i − 0.977567i
$$218$$ 0 0
$$219$$ 0 0
$$220$$ 0 0
$$221$$ − 424.264i − 1.91975i
$$222$$ 0 0
$$223$$ 364.867i 1.63618i 0.575094 + 0.818088i $$0.304966\pi$$
−0.575094 + 0.818088i $$0.695034\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ 0 0
$$227$$ 67.8823i 0.299041i 0.988759 + 0.149520i $$0.0477730\pi$$
−0.988759 + 0.149520i $$0.952227\pi$$
$$228$$ 0 0
$$229$$ −145.000 −0.633188 −0.316594 0.948561i $$-0.602539\pi$$
−0.316594 + 0.948561i $$0.602539\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 0 0
$$233$$ −335.000 −1.43777 −0.718884 0.695130i $$-0.755347\pi$$
−0.718884 + 0.695130i $$0.755347\pi$$
$$234$$ 0 0
$$235$$ 5.00000 0.0212766
$$236$$ 0 0
$$237$$ 0 0
$$238$$ 0 0
$$239$$ 197.000 0.824268 0.412134 0.911123i $$-0.364784\pi$$
0.412134 + 0.911123i $$0.364784\pi$$
$$240$$ 0 0
$$241$$ 296.985i 1.23230i 0.787628 + 0.616151i $$0.211309\pi$$
−0.787628 + 0.616151i $$0.788691\pi$$
$$242$$ 0 0
$$243$$ 0 0
$$244$$ 0 0
$$245$$ −24.0000 −0.0979592
$$246$$ 0 0
$$247$$ 322.441i 1.30543i
$$248$$ 0 0
$$249$$ 0 0
$$250$$ 0 0
$$251$$ 173.000 0.689243 0.344622 0.938742i $$-0.388007\pi$$
0.344622 + 0.938742i $$0.388007\pi$$
$$252$$ 0 0
$$253$$ −50.0000 −0.197628
$$254$$ 0 0
$$255$$ 0 0
$$256$$ 0 0
$$257$$ 67.8823i 0.264133i 0.991241 + 0.132067i $$0.0421613\pi$$
−0.991241 + 0.132067i $$0.957839\pi$$
$$258$$ 0 0
$$259$$ 127.279i 0.491426i
$$260$$ 0 0
$$261$$ 0 0
$$262$$ 0 0
$$263$$ −355.000 −1.34981 −0.674905 0.737905i $$-0.735815\pi$$
−0.674905 + 0.737905i $$0.735815\pi$$
$$264$$ 0 0
$$265$$ 25.4558i 0.0960598i
$$266$$ 0 0
$$267$$ 0 0
$$268$$ 0 0
$$269$$ 381.838i 1.41947i 0.704468 + 0.709735i $$0.251186\pi$$
−0.704468 + 0.709735i $$0.748814\pi$$
$$270$$ 0 0
$$271$$ −110.000 −0.405904 −0.202952 0.979189i $$-0.565054\pi$$
−0.202952 + 0.979189i $$0.565054\pi$$
$$272$$ 0 0
$$273$$ 0 0
$$274$$ 0 0
$$275$$ −120.000 −0.436364
$$276$$ 0 0
$$277$$ −265.000 −0.956679 −0.478339 0.878175i $$-0.658761\pi$$
−0.478339 + 0.878175i $$0.658761\pi$$
$$278$$ 0 0
$$279$$ 0 0
$$280$$ 0 0
$$281$$ − 424.264i − 1.50984i −0.655819 0.754918i $$-0.727677\pi$$
0.655819 0.754918i $$-0.272323\pi$$
$$282$$ 0 0
$$283$$ −125.000 −0.441696 −0.220848 0.975308i $$-0.570882\pi$$
−0.220848 + 0.975308i $$0.570882\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ 0 0
$$287$$ − 212.132i − 0.739136i
$$288$$ 0 0
$$289$$ 336.000 1.16263
$$290$$ 0 0
$$291$$ 0 0
$$292$$ 0 0
$$293$$ − 186.676i − 0.637120i −0.947903 0.318560i $$-0.896801\pi$$
0.947903 0.318560i $$-0.103199\pi$$
$$294$$ 0 0
$$295$$ 84.8528i 0.287637i
$$296$$ 0 0
$$297$$ 0 0
$$298$$ 0 0
$$299$$ 169.706i 0.567577i
$$300$$ 0 0
$$301$$ 25.0000 0.0830565
$$302$$ 0 0
$$303$$ 0 0
$$304$$ 0 0
$$305$$ 95.0000 0.311475
$$306$$ 0 0
$$307$$ − 280.014i − 0.912099i −0.889955 0.456049i $$-0.849264\pi$$
0.889955 0.456049i $$-0.150736\pi$$
$$308$$ 0 0
$$309$$ 0 0
$$310$$ 0 0
$$311$$ −235.000 −0.755627 −0.377814 0.925882i $$-0.623324\pi$$
−0.377814 + 0.925882i $$0.623324\pi$$
$$312$$ 0 0
$$313$$ −310.000 −0.990415 −0.495208 0.868775i $$-0.664908\pi$$
−0.495208 + 0.868775i $$0.664908\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ 186.676i 0.588884i 0.955669 + 0.294442i $$0.0951338\pi$$
−0.955669 + 0.294442i $$0.904866\pi$$
$$318$$ 0 0
$$319$$ − 212.132i − 0.664991i
$$320$$ 0 0
$$321$$ 0 0
$$322$$ 0 0
$$323$$ −475.000 −1.47059
$$324$$ 0 0
$$325$$ 407.294i 1.25321i
$$326$$ 0 0
$$327$$ 0 0
$$328$$ 0 0
$$329$$ −25.0000 −0.0759878
$$330$$ 0 0
$$331$$ − 296.985i − 0.897235i −0.893724 0.448618i $$-0.851917\pi$$
0.893724 0.448618i $$-0.148083\pi$$
$$332$$ 0 0
$$333$$ 0 0
$$334$$ 0 0
$$335$$ − 110.309i − 0.329280i
$$336$$ 0 0
$$337$$ − 526.087i − 1.56109i −0.625099 0.780545i $$-0.714942\pi$$
0.625099 0.780545i $$-0.285058\pi$$
$$338$$ 0 0
$$339$$ 0 0
$$340$$ 0 0
$$341$$ 212.132i 0.622088i
$$342$$ 0 0
$$343$$ 365.000 1.06414
$$344$$ 0 0
$$345$$ 0 0
$$346$$ 0 0
$$347$$ 125.000 0.360231 0.180115 0.983646i $$-0.442353\pi$$
0.180115 + 0.983646i $$0.442353\pi$$
$$348$$ 0 0
$$349$$ 23.0000 0.0659026 0.0329513 0.999457i $$-0.489509\pi$$
0.0329513 + 0.999457i $$0.489509\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ 0 0
$$353$$ −410.000 −1.16147 −0.580737 0.814092i $$-0.697235\pi$$
−0.580737 + 0.814092i $$0.697235\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ 0 0
$$357$$ 0 0
$$358$$ 0 0
$$359$$ −475.000 −1.32312 −0.661560 0.749892i $$-0.730105\pi$$
−0.661560 + 0.749892i $$0.730105\pi$$
$$360$$ 0 0
$$361$$ 361.000 1.00000
$$362$$ 0 0
$$363$$ 0 0
$$364$$ 0 0
$$365$$ −25.0000 −0.0684932
$$366$$ 0 0
$$367$$ −230.000 −0.626703 −0.313351 0.949637i $$-0.601452\pi$$
−0.313351 + 0.949637i $$0.601452\pi$$
$$368$$ 0 0
$$369$$ 0 0
$$370$$ 0 0
$$371$$ − 127.279i − 0.343071i
$$372$$ 0 0
$$373$$ − 67.8823i − 0.181990i −0.995851 0.0909950i $$-0.970995\pi$$
0.995851 0.0909950i $$-0.0290047\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ 0 0
$$377$$ −720.000 −1.90981
$$378$$ 0 0
$$379$$ 254.558i 0.671658i 0.941923 + 0.335829i $$0.109016\pi$$
−0.941923 + 0.335829i $$0.890984\pi$$
$$380$$ 0 0
$$381$$ 0 0
$$382$$ 0 0
$$383$$ − 144.250i − 0.376631i −0.982109 0.188316i $$-0.939697\pi$$
0.982109 0.188316i $$-0.0603028\pi$$
$$384$$ 0 0
$$385$$ −25.0000 −0.0649351
$$386$$ 0 0
$$387$$ 0 0
$$388$$ 0 0
$$389$$ 553.000 1.42159 0.710797 0.703397i $$-0.248334\pi$$
0.710797 + 0.703397i $$0.248334\pi$$
$$390$$ 0 0
$$391$$ −250.000 −0.639386
$$392$$ 0 0
$$393$$ 0 0
$$394$$ 0 0
$$395$$ − 42.4264i − 0.107409i
$$396$$ 0 0
$$397$$ 335.000 0.843829 0.421914 0.906636i $$-0.361358\pi$$
0.421914 + 0.906636i $$0.361358\pi$$
$$398$$ 0 0
$$399$$ 0 0
$$400$$ 0 0
$$401$$ − 212.132i − 0.529008i −0.964385 0.264504i $$-0.914792\pi$$
0.964385 0.264504i $$-0.0852082\pi$$
$$402$$ 0 0
$$403$$ 720.000 1.78660
$$404$$ 0 0
$$405$$ 0 0
$$406$$ 0 0
$$407$$ − 127.279i − 0.312725i
$$408$$ 0 0
$$409$$ 721.249i 1.76344i 0.471769 + 0.881722i $$0.343616\pi$$
−0.471769 + 0.881722i $$0.656384\pi$$
$$410$$ 0 0
$$411$$ 0 0
$$412$$ 0 0
$$413$$ − 424.264i − 1.02727i
$$414$$ 0 0
$$415$$ −130.000 −0.313253
$$416$$ 0 0
$$417$$ 0 0
$$418$$ 0 0
$$419$$ 62.0000 0.147971 0.0739857 0.997259i $$-0.476428\pi$$
0.0739857 + 0.997259i $$0.476428\pi$$
$$420$$ 0 0
$$421$$ 296.985i 0.705427i 0.935731 + 0.352714i $$0.114741\pi$$
−0.935731 + 0.352714i $$0.885259\pi$$
$$422$$ 0 0
$$423$$ 0 0
$$424$$ 0 0
$$425$$ −600.000 −1.41176
$$426$$ 0 0
$$427$$ −475.000 −1.11241
$$428$$ 0 0
$$429$$ 0 0
$$430$$ 0 0
$$431$$ − 509.117i − 1.18125i −0.806948 0.590623i $$-0.798882\pi$$
0.806948 0.590623i $$-0.201118\pi$$
$$432$$ 0 0
$$433$$ − 229.103i − 0.529105i −0.964371 0.264553i $$-0.914776\pi$$
0.964371 0.264553i $$-0.0852243\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 0 0
$$437$$ 190.000 0.434783
$$438$$ 0 0
$$439$$ − 806.102i − 1.83622i −0.396323 0.918111i $$-0.629714\pi$$
0.396323 0.918111i $$-0.370286\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ 0 0
$$443$$ 365.000 0.823928 0.411964 0.911200i $$-0.364843\pi$$
0.411964 + 0.911200i $$0.364843\pi$$
$$444$$ 0 0
$$445$$ 127.279i 0.286021i
$$446$$ 0 0
$$447$$ 0 0
$$448$$ 0 0
$$449$$ 763.675i 1.70084i 0.526108 + 0.850418i $$0.323651\pi$$
−0.526108 + 0.850418i $$0.676349\pi$$
$$450$$ 0 0
$$451$$ 212.132i 0.470359i
$$452$$ 0 0
$$453$$ 0 0
$$454$$ 0 0
$$455$$ 84.8528i 0.186490i
$$456$$ 0 0
$$457$$ −265.000 −0.579869 −0.289934 0.957047i $$-0.593633\pi$$
−0.289934 + 0.957047i $$0.593633\pi$$
$$458$$ 0 0
$$459$$ 0 0
$$460$$ 0 0
$$461$$ 553.000 1.19957 0.599783 0.800163i $$-0.295254\pi$$
0.599783 + 0.800163i $$0.295254\pi$$
$$462$$ 0 0
$$463$$ −485.000 −1.04752 −0.523758 0.851867i $$-0.675470\pi$$
−0.523758 + 0.851867i $$0.675470\pi$$
$$464$$ 0 0
$$465$$ 0 0
$$466$$ 0 0
$$467$$ −115.000 −0.246253 −0.123126 0.992391i $$-0.539292\pi$$
−0.123126 + 0.992391i $$0.539292\pi$$
$$468$$ 0 0
$$469$$ 551.543i 1.17600i
$$470$$ 0 0
$$471$$ 0 0
$$472$$ 0 0
$$473$$ −25.0000 −0.0528541
$$474$$ 0 0
$$475$$ 456.000 0.960000
$$476$$ 0 0
$$477$$ 0 0
$$478$$ 0 0
$$479$$ −490.000 −1.02296 −0.511482 0.859294i $$-0.670903\pi$$
−0.511482 + 0.859294i $$0.670903\pi$$
$$480$$ 0 0
$$481$$ −432.000 −0.898129
$$482$$ 0 0
$$483$$ 0 0
$$484$$ 0 0
$$485$$ 16.9706i 0.0349909i
$$486$$ 0 0
$$487$$ 610.940i 1.25450i 0.778819 + 0.627249i $$0.215819\pi$$
−0.778819 + 0.627249i $$0.784181\pi$$
$$488$$ 0 0
$$489$$ 0 0
$$490$$ 0 0
$$491$$ −82.0000 −0.167006 −0.0835031 0.996508i $$-0.526611\pi$$
−0.0835031 + 0.996508i $$0.526611\pi$$
$$492$$ 0 0
$$493$$ − 1060.66i − 2.15144i
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 0 0
$$497$$ 0 0
$$498$$ 0 0
$$499$$ −485.000 −0.971944 −0.485972 0.873974i $$-0.661534\pi$$
−0.485972 + 0.873974i $$0.661534\pi$$
$$500$$ 0 0
$$501$$ 0 0
$$502$$ 0 0
$$503$$ −250.000 −0.497018 −0.248509 0.968630i $$-0.579941\pi$$
−0.248509 + 0.968630i $$0.579941\pi$$
$$504$$ 0 0
$$505$$ −50.0000 −0.0990099
$$506$$ 0 0
$$507$$ 0 0
$$508$$ 0 0
$$509$$ 169.706i 0.333410i 0.986007 + 0.166705i $$0.0533127\pi$$
−0.986007 + 0.166705i $$0.946687\pi$$
$$510$$ 0 0
$$511$$ 125.000 0.244618
$$512$$ 0 0
$$513$$ 0 0
$$514$$ 0 0
$$515$$ 16.9706i 0.0329525i
$$516$$ 0 0
$$517$$ 25.0000 0.0483559
$$518$$ 0 0
$$519$$ 0 0
$$520$$ 0 0
$$521$$ 127.279i 0.244298i 0.992512 + 0.122149i $$0.0389786\pi$$
−0.992512 + 0.122149i $$0.961021\pi$$
$$522$$ 0 0
$$523$$ − 356.382i − 0.681418i −0.940169 0.340709i $$-0.889333\pi$$
0.940169 0.340709i $$-0.110667\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ 1060.66i 2.01264i
$$528$$ 0 0
$$529$$ −429.000 −0.810964
$$530$$ 0 0
$$531$$ 0 0
$$532$$ 0 0
$$533$$ 720.000 1.35084
$$534$$ 0 0
$$535$$ − 101.823i − 0.190324i
$$536$$ 0 0
$$537$$ 0 0
$$538$$ 0 0
$$539$$ −120.000 −0.222635
$$540$$ 0 0
$$541$$ −25.0000 −0.0462107 −0.0231054 0.999733i $$-0.507355\pi$$
−0.0231054 + 0.999733i $$0.507355\pi$$
$$542$$ 0 0
$$543$$ 0 0
$$544$$ 0 0
$$545$$ 127.279i 0.233540i
$$546$$ 0 0
$$547$$ − 16.9706i − 0.0310248i −0.999880 0.0155124i $$-0.995062\pi$$
0.999880 0.0155124i $$-0.00493795\pi$$
$$548$$ 0 0
$$549$$ 0 0
$$550$$ 0 0
$$551$$ 806.102i 1.46298i
$$552$$ 0 0
$$553$$ 212.132i 0.383602i
$$554$$ 0 0
$$555$$ 0 0
$$556$$ 0 0
$$557$$ 745.000 1.33752 0.668761 0.743477i $$-0.266825\pi$$
0.668761 + 0.743477i $$0.266825\pi$$
$$558$$ 0 0
$$559$$ 84.8528i 0.151794i
$$560$$ 0 0
$$561$$ 0 0
$$562$$ 0 0
$$563$$ 313.955i 0.557647i 0.960342 + 0.278824i $$0.0899445\pi$$
−0.960342 + 0.278824i $$0.910056\pi$$
$$564$$ 0 0
$$565$$ − 110.309i − 0.195237i
$$566$$ 0 0
$$567$$ 0 0
$$568$$ 0 0
$$569$$ − 424.264i − 0.745631i −0.927905 0.372816i $$-0.878392\pi$$
0.927905 0.372816i $$-0.121608\pi$$
$$570$$ 0 0
$$571$$ −1070.00 −1.87391 −0.936953 0.349456i $$-0.886366\pi$$
−0.936953 + 0.349456i $$0.886366\pi$$
$$572$$ 0 0
$$573$$ 0 0
$$574$$ 0 0
$$575$$ 240.000 0.417391
$$576$$ 0 0
$$577$$ −25.0000 −0.0433276 −0.0216638 0.999765i $$-0.506896\pi$$
−0.0216638 + 0.999765i $$0.506896\pi$$
$$578$$ 0 0
$$579$$ 0 0
$$580$$ 0 0
$$581$$ 650.000 1.11876
$$582$$ 0 0
$$583$$ 127.279i 0.218318i
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 0 0
$$587$$ 725.000 1.23509 0.617547 0.786534i $$-0.288127\pi$$
0.617547 + 0.786534i $$0.288127\pi$$
$$588$$ 0 0
$$589$$ − 806.102i − 1.36859i
$$590$$ 0 0
$$591$$ 0 0
$$592$$ 0 0
$$593$$ −650.000 −1.09612 −0.548061 0.836439i $$-0.684634\pi$$
−0.548061 + 0.836439i $$0.684634\pi$$
$$594$$ 0 0
$$595$$ −125.000 −0.210084
$$596$$ 0 0
$$597$$ 0 0
$$598$$ 0 0
$$599$$ − 296.985i − 0.495801i −0.968785 0.247901i $$-0.920259\pi$$
0.968785 0.247901i $$-0.0797406\pi$$
$$600$$ 0 0
$$601$$ − 848.528i − 1.41186i −0.708281 0.705930i $$-0.750529\pi$$
0.708281 0.705930i $$-0.249471\pi$$
$$602$$ 0 0
$$603$$ 0 0
$$604$$ 0 0
$$605$$ −96.0000 −0.158678
$$606$$ 0 0
$$607$$ 271.529i 0.447329i 0.974666 + 0.223665i $$0.0718021\pi$$
−0.974666 + 0.223665i $$0.928198\pi$$
$$608$$ 0 0
$$609$$ 0 0
$$610$$ 0 0
$$611$$ − 84.8528i − 0.138875i
$$612$$ 0 0
$$613$$ 1055.00 1.72104 0.860522 0.509413i $$-0.170138\pi$$
0.860522 + 0.509413i $$0.170138\pi$$
$$614$$ 0 0
$$615$$ 0 0
$$616$$ 0 0
$$617$$ 505.000 0.818476 0.409238 0.912428i $$-0.365794\pi$$
0.409238 + 0.912428i $$0.365794\pi$$
$$618$$ 0 0
$$619$$ 130.000 0.210016 0.105008 0.994471i $$-0.466513\pi$$
0.105008 + 0.994471i $$0.466513\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ 0 0
$$623$$ − 636.396i − 1.02150i
$$624$$ 0 0
$$625$$ 551.000 0.881600
$$626$$ 0 0
$$627$$ 0 0
$$628$$ 0 0
$$629$$ − 636.396i − 1.01176i
$$630$$ 0 0
$$631$$ 475.000 0.752773 0.376387 0.926463i $$-0.377166\pi$$
0.376387 + 0.926463i $$0.377166\pi$$
$$632$$ 0 0
$$633$$ 0 0
$$634$$ 0 0
$$635$$ − 229.103i − 0.360791i
$$636$$ 0 0
$$637$$ 407.294i 0.639393i
$$638$$ 0 0
$$639$$ 0 0
$$640$$ 0 0
$$641$$ 848.528i 1.32376i 0.749611 + 0.661878i $$0.230240\pi$$
−0.749611 + 0.661878i $$0.769760\pi$$
$$642$$ 0 0
$$643$$ 955.000 1.48523 0.742613 0.669721i $$-0.233586\pi$$
0.742613 + 0.669721i $$0.233586\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 0 0
$$647$$ 965.000 1.49150 0.745750 0.666226i $$-0.232092\pi$$
0.745750 + 0.666226i $$0.232092\pi$$
$$648$$ 0 0
$$649$$ 424.264i 0.653720i
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 0 0
$$653$$ −935.000 −1.43185 −0.715926 0.698176i $$-0.753995\pi$$
−0.715926 + 0.698176i $$0.753995\pi$$
$$654$$ 0 0
$$655$$ −163.000 −0.248855
$$656$$ 0 0
$$657$$ 0 0
$$658$$ 0 0
$$659$$ − 84.8528i − 0.128760i −0.997925 0.0643800i $$-0.979493\pi$$
0.997925 0.0643800i $$-0.0205070\pi$$
$$660$$ 0 0
$$661$$ − 678.823i − 1.02696i −0.858101 0.513481i $$-0.828356\pi$$
0.858101 0.513481i $$-0.171644\pi$$
$$662$$ 0 0
$$663$$ 0 0
$$664$$ 0 0
$$665$$ 95.0000 0.142857
$$666$$ 0 0
$$667$$ 424.264i 0.636078i
$$668$$ 0 0
$$669$$ 0 0
$$670$$ 0 0
$$671$$ 475.000 0.707899
$$672$$ 0 0
$$673$$ − 186.676i − 0.277379i −0.990336 0.138690i $$-0.955711\pi$$
0.990336 0.138690i $$-0.0442890\pi$$
$$674$$ 0 0
$$675$$ 0 0
$$676$$ 0 0
$$677$$ − 907.925i − 1.34110i −0.741864 0.670550i $$-0.766058\pi$$
0.741864 0.670550i $$-0.233942\pi$$
$$678$$ 0 0
$$679$$ − 84.8528i − 0.124967i
$$680$$ 0 0
$$681$$ 0 0
$$682$$ 0 0
$$683$$ 1120.06i 1.63991i 0.572429 + 0.819954i $$0.306001\pi$$
−0.572429 + 0.819954i $$0.693999\pi$$
$$684$$ 0 0
$$685$$ −95.0000 −0.138686
$$686$$ 0 0
$$687$$ 0 0
$$688$$ 0 0
$$689$$ 432.000 0.626996
$$690$$ 0 0
$$691$$ 715.000 1.03473 0.517366 0.855764i $$-0.326913\pi$$
0.517366 + 0.855764i $$0.326913\pi$$
$$692$$ 0 0
$$693$$ 0 0
$$694$$ 0 0
$$695$$ −125.000 −0.179856
$$696$$ 0 0
$$697$$ 1060.66i 1.52175i
$$698$$ 0 0
$$699$$ 0 0
$$700$$ 0 0
$$701$$ 430.000 0.613409 0.306705 0.951805i $$-0.400774\pi$$
0.306705 + 0.951805i $$0.400774\pi$$
$$702$$ 0 0
$$703$$ 483.661i 0.687996i
$$704$$ 0 0
$$705$$ 0 0
$$706$$ 0 0
$$707$$ 250.000 0.353607
$$708$$ 0 0
$$709$$ −382.000 −0.538787 −0.269394 0.963030i $$-0.586823\pi$$
−0.269394 + 0.963030i $$0.586823\pi$$
$$710$$ 0 0
$$711$$ 0 0
$$712$$ 0 0
$$713$$ − 424.264i − 0.595041i
$$714$$ 0 0
$$715$$ − 84.8528i − 0.118675i
$$716$$ 0 0
$$717$$ 0 0
$$718$$ 0 0
$$719$$ −115.000 −0.159944 −0.0799722 0.996797i $$-0.525483\pi$$
−0.0799722 + 0.996797i $$0.525483\pi$$
$$720$$ 0 0
$$721$$ − 84.8528i − 0.117688i
$$722$$ 0 0
$$723$$ 0 0
$$724$$ 0 0
$$725$$ 1018.23i 1.40446i
$$726$$ 0 0
$$727$$ 1075.00 1.47868 0.739340 0.673333i $$-0.235138\pi$$
0.739340 + 0.673333i $$0.235138\pi$$
$$728$$ 0 0
$$729$$ 0 0
$$730$$ 0 0
$$731$$ −125.000 −0.170999
$$732$$ 0 0
$$733$$ 530.000 0.723056 0.361528 0.932361i $$-0.382255\pi$$
0.361528 + 0.932361i $$0.382255\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ − 551.543i − 0.748363i
$$738$$ 0 0
$$739$$ 547.000 0.740189 0.370095 0.928994i $$-0.379325\pi$$
0.370095 + 0.928994i $$0.379325\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ 0 0
$$743$$ − 958.837i − 1.29049i −0.763974 0.645247i $$-0.776755\pi$$
0.763974 0.645247i $$-0.223245\pi$$
$$744$$ 0 0
$$745$$ −215.000 −0.288591
$$746$$ 0 0
$$747$$ 0 0
$$748$$ 0 0
$$749$$ 509.117i 0.679729i
$$750$$ 0 0
$$751$$ 169.706i 0.225973i 0.993597 + 0.112986i $$0.0360417\pi$$
−0.993597 + 0.112986i $$0.963958\pi$$
$$752$$ 0 0
$$753$$ 0 0
$$754$$ 0 0
$$755$$ 84.8528i 0.112388i
$$756$$ 0 0
$$757$$ 1055.00 1.39366 0.696830 0.717237i $$-0.254593\pi$$
0.696830 + 0.717237i $$0.254593\pi$$
$$758$$ 0 0
$$759$$ 0 0
$$760$$ 0 0
$$761$$ −215.000 −0.282523 −0.141261 0.989972i $$-0.545116\pi$$
−0.141261 + 0.989972i $$0.545116\pi$$
$$762$$ 0 0
$$763$$ − 636.396i − 0.834071i
$$764$$ 0 0
$$765$$ 0 0
$$766$$ 0 0
$$767$$ 1440.00 1.87744
$$768$$ 0 0
$$769$$ −145.000 −0.188557 −0.0942783 0.995546i $$-0.530054\pi$$
−0.0942783 + 0.995546i $$0.530054\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ 0 0
$$773$$ − 407.294i − 0.526900i −0.964673 0.263450i $$-0.915140\pi$$
0.964673 0.263450i $$-0.0848604\pi$$
$$774$$ 0 0
$$775$$ − 1018.23i − 1.31385i
$$776$$ 0 0
$$777$$ 0 0
$$778$$ 0 0
$$779$$ − 806.102i − 1.03479i
$$780$$ 0 0
$$781$$ 0 0
$$782$$ 0 0
$$783$$ 0 0
$$784$$ 0 0
$$785$$ −190.000 −0.242038
$$786$$ 0 0
$$787$$ − 186.676i − 0.237200i −0.992942 0.118600i $$-0.962159\pi$$
0.992942 0.118600i $$-0.0378406\pi$$
$$788$$ 0 0
$$789$$ 0 0
$$790$$ 0 0
$$791$$ 551.543i 0.697273i
$$792$$ 0 0
$$793$$ − 1612.20i − 2.03304i
$$794$$ 0 0
$$795$$ 0 0
$$796$$ 0 0
$$797$$ 704.278i 0.883662i 0.897098 + 0.441831i $$0.145671\pi$$
−0.897098 + 0.441831i $$0.854329\pi$$
$$798$$ 0 0
$$799$$ 125.000 0.156446
$$800$$ 0 0
$$801$$ 0 0
$$802$$ 0 0
$$803$$ −125.000 −0.155666
$$804$$ 0 0
$$805$$ 50.0000 0.0621118
$$806$$ 0 0
$$807$$ 0 0
$$808$$ 0 0
$$809$$ 457.000 0.564895 0.282447 0.959283i $$-0.408854\pi$$
0.282447 + 0.959283i $$0.408854\pi$$
$$810$$ 0 0
$$811$$ − 509.117i − 0.627764i −0.949462 0.313882i $$-0.898370\pi$$
0.949462 0.313882i $$-0.101630\pi$$
$$812$$ 0 0
$$813$$ 0 0
$$814$$ 0 0
$$815$$ −110.000 −0.134969
$$816$$ 0 0
$$817$$ 95.0000 0.116279
$$818$$ 0 0
$$819$$ 0 0
$$820$$ 0 0
$$821$$ −167.000 −0.203410 −0.101705 0.994815i $$-0.532430\pi$$
−0.101705 + 0.994815i $$0.532430\pi$$
$$822$$ 0 0
$$823$$ 1315.00 1.59781 0.798906 0.601455i $$-0.205412\pi$$
0.798906 + 0.601455i $$0.205412\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ 0 0
$$827$$ − 534.573i − 0.646400i −0.946331 0.323200i $$-0.895241\pi$$
0.946331 0.323200i $$-0.104759\pi$$
$$828$$ 0 0
$$829$$ 763.675i 0.921201i 0.887608 + 0.460600i $$0.152366\pi$$
−0.887608 + 0.460600i $$0.847634\pi$$
$$830$$ 0 0
$$831$$ 0 0
$$832$$ 0 0
$$833$$ −600.000 −0.720288
$$834$$ 0 0
$$835$$ 59.3970i 0.0711341i
$$836$$ 0 0
$$837$$ 0 0
$$838$$ 0 0
$$839$$ − 339.411i − 0.404543i −0.979330 0.202271i $$-0.935168\pi$$
0.979330 0.202271i $$-0.0648323\pi$$
$$840$$ 0 0
$$841$$ −959.000 −1.14031
$$842$$ 0 0
$$843$$ 0 0
$$844$$ 0 0
$$845$$ −119.000 −0.140828
$$846$$ 0 0
$$847$$ 480.000 0.566706
$$848$$ 0 0
$$849$$ 0 0
$$850$$ 0 0
$$851$$ 254.558i 0.299129i
$$852$$ 0 0
$$853$$ 770.000 0.902696 0.451348 0.892348i $$-0.350943\pi$$
0.451348 + 0.892348i $$0.350943\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ 0 0
$$857$$ 1255.82i 1.46537i 0.680568 + 0.732685i $$0.261733\pi$$
−0.680568 + 0.732685i $$0.738267\pi$$
$$858$$ 0 0
$$859$$ −557.000 −0.648428 −0.324214 0.945984i $$-0.605100\pi$$
−0.324214 + 0.945984i $$0.605100\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ 0 0
$$863$$ 992.778i 1.15038i 0.818020 + 0.575190i $$0.195072\pi$$
−0.818020 + 0.575190i $$0.804928\pi$$
$$864$$ 0 0
$$865$$ 186.676i 0.215811i
$$866$$ 0 0
$$867$$ 0 0
$$868$$ 0 0
$$869$$ − 212.132i − 0.244111i
$$870$$ 0 0
$$871$$ −1872.00 −2.14925
$$872$$ 0 0
$$873$$ 0 0
$$874$$ 0 0
$$875$$ 245.000 0.280000
$$876$$ 0 0
$$877$$ 186.676i 0.212858i 0.994320 + 0.106429i $$0.0339416\pi$$
−0.994320 + 0.106429i $$0.966058\pi$$
$$878$$ 0 0
$$879$$ 0 0
$$880$$ 0 0
$$881$$ 25.0000 0.0283768 0.0141884 0.999899i $$-0.495484\pi$$
0.0141884 + 0.999899i $$0.495484\pi$$
$$882$$ 0 0
$$883$$ −965.000 −1.09287 −0.546433 0.837503i $$-0.684015\pi$$
−0.546433 + 0.837503i $$0.684015\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ 0 0
$$887$$ − 780.646i − 0.880097i −0.897974 0.440048i $$-0.854961\pi$$
0.897974 0.440048i $$-0.145039\pi$$
$$888$$ 0 0
$$889$$ 1145.51i 1.28854i
$$890$$ 0 0
$$891$$ 0 0
$$892$$ 0 0
$$893$$ −95.0000 −0.106383
$$894$$ 0 0
$$895$$ − 127.279i − 0.142211i
$$896$$ 0 0
$$897$$ 0 0
$$898$$ 0 0
$$899$$ 1800.00 2.00222
$$900$$ 0 0
$$901$$ 636.396i 0.706322i
$$902$$ 0 0
$$903$$ 0 0
$$904$$ 0 0
$$905$$ − 254.558i − 0.281280i
$$906$$ 0 0
$$907$$ 313.955i 0.346147i 0.984909 + 0.173074i $$0.0553698\pi$$
−0.984909 + 0.173074i $$0.944630\pi$$
$$908$$ 0 0
$$909$$ 0 0
$$910$$ 0 0
$$911$$ 933.381i 1.02457i 0.858816 + 0.512284i $$0.171200\pi$$
−0.858816 + 0.512284i $$0.828800\pi$$
$$912$$ 0 0
$$913$$ −650.000 −0.711939
$$914$$ 0 0
$$915$$ 0 0
$$916$$ 0 0
$$917$$ 815.000 0.888768
$$918$$ 0 0
$$919$$ 538.000 0.585419 0.292709 0.956201i $$-0.405443\pi$$
0.292709 + 0.956201i $$0.405443\pi$$
$$920$$ 0 0
$$921$$ 0 0
$$922$$ 0 0
$$923$$ 0 0
$$924$$ 0 0
$$925$$ 610.940i 0.660476i
$$926$$ 0 0
$$927$$ 0 0
$$928$$ 0 0
$$929$$ 742.000 0.798708 0.399354 0.916797i $$-0.369234\pi$$
0.399354 + 0.916797i $$0.369234\pi$$
$$930$$ 0 0
$$931$$ 456.000 0.489796
$$932$$ 0 0
$$933$$ 0 0
$$934$$ 0 0
$$935$$ 125.000 0.133690
$$936$$ 0 0
$$937$$ 335.000 0.357524 0.178762 0.983892i $$-0.442791\pi$$
0.178762 + 0.983892i $$0.442791\pi$$
$$938$$ 0 0
$$939$$ 0 0
$$940$$ 0 0
$$941$$ 424.264i 0.450865i 0.974259 + 0.225433i $$0.0723795\pi$$
−0.974259 + 0.225433i $$0.927620\pi$$
$$942$$ 0 0
$$943$$ − 424.264i − 0.449909i
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ −1210.00 −1.27772 −0.638860 0.769323i $$-0.720594\pi$$
−0.638860 + 0.769323i $$0.720594\pi$$
$$948$$ 0 0
$$949$$ 424.264i 0.447064i
$$950$$ 0 0
$$951$$ 0 0
$$952$$ 0 0
$$953$$ − 992.778i − 1.04174i −0.853636 0.520870i $$-0.825608\pi$$
0.853636 0.520870i $$-0.174392\pi$$
$$954$$ 0 0
$$955$$ 293.000 0.306806
$$956$$ 0 0
$$957$$ 0 0
$$958$$ 0 0
$$959$$ 475.000 0.495308
$$960$$ 0 0
$$961$$ −839.000 −0.873049
$$962$$ 0 0
$$963$$ 0 0
$$964$$ 0 0
$$965$$ 59.3970i 0.0615513i
$$966$$ 0 0
$$967$$ −350.000 −0.361944 −0.180972 0.983488i $$-0.557924\pi$$
−0.180972 + 0.983488i $$0.557924\pi$$
$$968$$ 0 0
$$969$$ 0 0
$$970$$ 0 0
$$971$$ 254.558i 0.262161i 0.991372 + 0.131081i $$0.0418447\pi$$
−0.991372 + 0.131081i $$0.958155\pi$$
$$972$$ 0 0
$$973$$ 625.000 0.642343
$$974$$ 0 0
$$975$$ 0 0
$$976$$ 0 0
$$977$$ 398.808i 0.408197i 0.978950 + 0.204098i $$0.0654262\pi$$
−0.978950 + 0.204098i $$0.934574\pi$$
$$978$$ 0 0
$$979$$ 636.396i 0.650047i
$$980$$ 0 0
$$981$$ 0 0
$$982$$ 0 0
$$983$$ − 695.793i − 0.707826i −0.935278 0.353913i $$-0.884851\pi$$
0.935278 0.353913i $$-0.115149\pi$$
$$984$$ 0 0
$$985$$ 70.0000 0.0710660
$$986$$ 0 0
$$987$$ 0 0
$$988$$ 0 0
$$989$$ 50.0000 0.0505561
$$990$$ 0 0
$$991$$ − 381.838i − 0.385305i −0.981267 0.192653i $$-0.938291\pi$$
0.981267 0.192653i $$-0.0617091\pi$$
$$992$$ 0 0
$$993$$ 0 0
$$994$$ 0 0
$$995$$ −173.000 −0.173869
$$996$$ 0 0
$$997$$ −265.000 −0.265797 −0.132899 0.991130i $$-0.542428\pi$$
−0.132899 + 0.991130i $$0.542428\pi$$
$$998$$ 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 2736.3.o.h.721.1 2
3.2 odd 2 304.3.e.c.113.1 2
4.3 odd 2 342.3.d.a.37.1 2
12.11 even 2 38.3.b.a.37.2 yes 2
19.18 odd 2 inner 2736.3.o.h.721.2 2
24.5 odd 2 1216.3.e.i.1025.2 2
24.11 even 2 1216.3.e.j.1025.1 2
57.56 even 2 304.3.e.c.113.2 2
60.23 odd 4 950.3.d.a.949.3 4
60.47 odd 4 950.3.d.a.949.2 4
60.59 even 2 950.3.c.a.151.1 2
76.75 even 2 342.3.d.a.37.2 2
228.227 odd 2 38.3.b.a.37.1 2
456.227 odd 2 1216.3.e.j.1025.2 2
456.341 even 2 1216.3.e.i.1025.1 2
1140.227 even 4 950.3.d.a.949.4 4
1140.683 even 4 950.3.d.a.949.1 4
1140.1139 odd 2 950.3.c.a.151.2 2

By twisted newform
Twist Min Dim Char Parity Ord Type
38.3.b.a.37.1 2 228.227 odd 2
38.3.b.a.37.2 yes 2 12.11 even 2
304.3.e.c.113.1 2 3.2 odd 2
304.3.e.c.113.2 2 57.56 even 2
342.3.d.a.37.1 2 4.3 odd 2
342.3.d.a.37.2 2 76.75 even 2
950.3.c.a.151.1 2 60.59 even 2
950.3.c.a.151.2 2 1140.1139 odd 2
950.3.d.a.949.1 4 1140.683 even 4
950.3.d.a.949.2 4 60.47 odd 4
950.3.d.a.949.3 4 60.23 odd 4
950.3.d.a.949.4 4 1140.227 even 4
1216.3.e.i.1025.1 2 456.341 even 2
1216.3.e.i.1025.2 2 24.5 odd 2
1216.3.e.j.1025.1 2 24.11 even 2
1216.3.e.j.1025.2 2 456.227 odd 2
2736.3.o.h.721.1 2 1.1 even 1 trivial
2736.3.o.h.721.2 2 19.18 odd 2 inner