# Properties

 Label 1216.3.e.j.1025.2 Level $1216$ Weight $3$ Character 1216.1025 Analytic conductor $33.134$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1216 = 2^{6} \cdot 19$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 1216.e (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$33.1336001462$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-2})$$ Defining polynomial: $$x^{2} + 2$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 38) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## Embedding invariants

 Embedding label 1025.2 Root $$1.41421i$$ of defining polynomial Character $$\chi$$ $$=$$ 1216.1025 Dual form 1216.3.e.j.1025.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q+2.82843i q^{3} +1.00000 q^{5} +5.00000 q^{7} +1.00000 q^{9} +O(q^{10})$$ $$q+2.82843i q^{3} +1.00000 q^{5} +5.00000 q^{7} +1.00000 q^{9} -5.00000 q^{11} -16.9706i q^{13} +2.82843i q^{15} -25.0000 q^{17} -19.0000 q^{19} +14.1421i q^{21} -10.0000 q^{23} -24.0000 q^{25} +28.2843i q^{27} +42.4264i q^{29} +42.4264i q^{31} -14.1421i q^{33} +5.00000 q^{35} -25.4558i q^{37} +48.0000 q^{39} +42.4264i q^{41} -5.00000 q^{43} +1.00000 q^{45} +5.00000 q^{47} -24.0000 q^{49} -70.7107i q^{51} -25.4558i q^{53} -5.00000 q^{55} -53.7401i q^{57} +84.8528i q^{59} -95.0000 q^{61} +5.00000 q^{63} -16.9706i q^{65} +110.309i q^{67} -28.2843i q^{69} -25.0000 q^{73} -67.8823i q^{75} -25.0000 q^{77} -42.4264i q^{79} -71.0000 q^{81} +130.000 q^{83} -25.0000 q^{85} -120.000 q^{87} +127.279i q^{89} -84.8528i q^{91} -120.000 q^{93} -19.0000 q^{95} -16.9706i q^{97} -5.00000 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + 2q^{5} + 10q^{7} + 2q^{9} + O(q^{10})$$ $$2q + 2q^{5} + 10q^{7} + 2q^{9} - 10q^{11} - 50q^{17} - 38q^{19} - 20q^{23} - 48q^{25} + 10q^{35} + 96q^{39} - 10q^{43} + 2q^{45} + 10q^{47} - 48q^{49} - 10q^{55} - 190q^{61} + 10q^{63} - 50q^{73} - 50q^{77} - 142q^{81} + 260q^{83} - 50q^{85} - 240q^{87} - 240q^{93} - 38q^{95} - 10q^{99} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1216\mathbb{Z}\right)^\times$$.

 $$n$$ $$191$$ $$705$$ $$837$$ $$\chi(n)$$ $$1$$ $$-1$$ $$1$$

## 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
$$3$$ 2.82843i 0.942809i 0.881917 + 0.471405i $$0.156253\pi$$
−0.881917 + 0.471405i $$0.843747\pi$$
$$4$$ 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 $$-0.383751\pi$$
0.357143 + 0.934050i $$0.383751\pi$$
$$8$$ 0 0
$$9$$ 1.00000 0.111111
$$10$$ 0 0
$$11$$ −5.00000 −0.454545 −0.227273 0.973831i $$-0.572981\pi$$
−0.227273 + 0.973831i $$0.572981\pi$$
$$12$$ 0 0
$$13$$ − 16.9706i − 1.30543i −0.757604 0.652714i $$-0.773630\pi$$
0.757604 0.652714i $$-0.226370\pi$$
$$14$$ 0 0
$$15$$ 2.82843i 0.188562i
$$16$$ 0 0
$$17$$ −25.0000 −1.47059 −0.735294 0.677748i $$-0.762956\pi$$
−0.735294 + 0.677748i $$0.762956\pi$$
$$18$$ 0 0
$$19$$ −19.0000 −1.00000
$$20$$ 0 0
$$21$$ 14.1421i 0.673435i
$$22$$ 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
$$27$$ 28.2843i 1.04757i
$$28$$ 0 0
$$29$$ 42.4264i 1.46298i 0.681852 + 0.731490i $$0.261175\pi$$
−0.681852 + 0.731490i $$0.738825\pi$$
$$30$$ 0 0
$$31$$ 42.4264i 1.36859i 0.729204 + 0.684297i $$0.239891\pi$$
−0.729204 + 0.684297i $$0.760109\pi$$
$$32$$ 0 0
$$33$$ − 14.1421i − 0.428550i
$$34$$ 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
$$39$$ 48.0000 1.23077
$$40$$ 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
$$45$$ 1.00000 0.0222222
$$46$$ 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
$$51$$ − 70.7107i − 1.38648i
$$52$$ 0 0
$$53$$ − 25.4558i − 0.480299i −0.970736 0.240149i $$-0.922804\pi$$
0.970736 0.240149i $$-0.0771965\pi$$
$$54$$ 0 0
$$55$$ −5.00000 −0.0909091
$$56$$ 0 0
$$57$$ − 53.7401i − 0.942809i
$$58$$ 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 $$-0.784115\pi$$
−0.778689 + 0.627411i $$0.784115\pi$$
$$62$$ 0 0
$$63$$ 5.00000 0.0793651
$$64$$ 0 0
$$65$$ − 16.9706i − 0.261086i
$$66$$ 0 0
$$67$$ 110.309i 1.64640i 0.567753 + 0.823199i $$0.307813\pi$$
−0.567753 + 0.823199i $$0.692187\pi$$
$$68$$ 0 0
$$69$$ − 28.2843i − 0.409917i
$$70$$ 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
$$75$$ − 67.8823i − 0.905097i
$$76$$ 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
$$81$$ −71.0000 −0.876543
$$82$$ 0 0
$$83$$ 130.000 1.56627 0.783133 0.621855i $$-0.213621\pi$$
0.783133 + 0.621855i $$0.213621\pi$$
$$84$$ 0 0
$$85$$ −25.0000 −0.294118
$$86$$ 0 0
$$87$$ −120.000 −1.37931
$$88$$ 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
$$93$$ −120.000 −1.29032
$$94$$ 0 0
$$95$$ −19.0000 −0.200000
$$96$$ 0 0
$$97$$ − 16.9706i − 0.174954i −0.996167 0.0874771i $$-0.972120\pi$$
0.996167 0.0874771i $$-0.0278805\pi$$
$$98$$ 0 0
$$99$$ −5.00000 −0.0505051
$$100$$ 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
$$105$$ 14.1421i 0.134687i
$$106$$ 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
$$111$$ 72.0000 0.648649
$$112$$ 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
$$117$$ − 16.9706i − 0.145048i
$$118$$ 0 0
$$119$$ −125.000 −1.05042
$$120$$ 0 0
$$121$$ −96.0000 −0.793388
$$122$$ 0 0
$$123$$ −120.000 −0.975610
$$124$$ 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
$$129$$ − 14.1421i − 0.109629i
$$130$$ 0 0
$$131$$ 163.000 1.24427 0.622137 0.782908i $$-0.286265\pi$$
0.622137 + 0.782908i $$0.286265\pi$$
$$132$$ 0 0
$$133$$ −95.0000 −0.714286
$$134$$ 0 0
$$135$$ 28.2843i 0.209513i
$$136$$ 0 0
$$137$$ 95.0000 0.693431 0.346715 0.937970i $$-0.387297\pi$$
0.346715 + 0.937970i $$0.387297\pi$$
$$138$$ 0 0
$$139$$ −125.000 −0.899281 −0.449640 0.893210i $$-0.648448\pi$$
−0.449640 + 0.893210i $$0.648448\pi$$
$$140$$ 0 0
$$141$$ 14.1421i 0.100299i
$$142$$ 0 0
$$143$$ 84.8528i 0.593376i
$$144$$ 0 0
$$145$$ 42.4264i 0.292596i
$$146$$ 0 0
$$147$$ − 67.8823i − 0.461784i
$$148$$ 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
$$153$$ −25.0000 −0.163399
$$154$$ 0 0
$$155$$ 42.4264i 0.273719i
$$156$$ 0 0
$$157$$ 190.000 1.21019 0.605096 0.796153i $$-0.293135\pi$$
0.605096 + 0.796153i $$0.293135\pi$$
$$158$$ 0 0
$$159$$ 72.0000 0.452830
$$160$$ 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
$$165$$ − 14.1421i − 0.0857099i
$$166$$ 0 0
$$167$$ − 59.3970i − 0.355670i −0.984060 0.177835i $$-0.943091\pi$$
0.984060 0.177835i $$-0.0569094\pi$$
$$168$$ 0 0
$$169$$ −119.000 −0.704142
$$170$$ 0 0
$$171$$ −19.0000 −0.111111
$$172$$ 0 0
$$173$$ − 186.676i − 1.07905i −0.841969 0.539527i $$-0.818603\pi$$
0.841969 0.539527i $$-0.181397\pi$$
$$174$$ 0 0
$$175$$ −120.000 −0.685714
$$176$$ 0 0
$$177$$ −240.000 −1.35593
$$178$$ 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
$$183$$ − 268.701i − 1.46831i
$$184$$ 0 0
$$185$$ − 25.4558i − 0.137599i
$$186$$ 0 0
$$187$$ 125.000 0.668449
$$188$$ 0 0
$$189$$ 141.421i 0.748261i
$$190$$ 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 $$-0.950824\pi$$
0.988090 0.153878i $$-0.0491763\pi$$
$$194$$ 0 0
$$195$$ 48.0000 0.246154
$$196$$ 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 $$-0.356864\pi$$
0.434673 + 0.900588i $$0.356864\pi$$
$$200$$ 0 0
$$201$$ −312.000 −1.55224
$$202$$ 0 0
$$203$$ 212.132i 1.04499i
$$204$$ 0 0
$$205$$ 42.4264i 0.206958i
$$206$$ 0 0
$$207$$ −10.0000 −0.0483092
$$208$$ 0 0
$$209$$ 95.0000 0.454545
$$210$$ 0 0
$$211$$ 84.8528i 0.402146i 0.979576 + 0.201073i $$0.0644429\pi$$
−0.979576 + 0.201073i $$0.935557\pi$$
$$212$$ 0 0
$$213$$ 0 0
$$214$$ 0 0
$$215$$ −5.00000 −0.0232558
$$216$$ 0 0
$$217$$ 212.132i 0.977567i
$$218$$ 0 0
$$219$$ − 70.7107i − 0.322880i
$$220$$ 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
$$225$$ −24.0000 −0.106667
$$226$$ 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 $$-0.397461\pi$$
0.316594 + 0.948561i $$0.397461\pi$$
$$230$$ 0 0
$$231$$ − 70.7107i − 0.306107i
$$232$$ 0 0
$$233$$ 335.000 1.43777 0.718884 0.695130i $$-0.244653\pi$$
0.718884 + 0.695130i $$0.244653\pi$$
$$234$$ 0 0
$$235$$ 5.00000 0.0212766
$$236$$ 0 0
$$237$$ 120.000 0.506329
$$238$$ 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 $$-0.788691\pi$$
0.787628 0.616151i $$-0.211309\pi$$
$$242$$ 0 0
$$243$$ 53.7401i 0.221153i
$$244$$ 0 0
$$245$$ −24.0000 −0.0979592
$$246$$ 0 0
$$247$$ 322.441i 1.30543i
$$248$$ 0 0
$$249$$ 367.696i 1.47669i
$$250$$ 0 0
$$251$$ −173.000 −0.689243 −0.344622 0.938742i $$-0.611993\pi$$
−0.344622 + 0.938742i $$0.611993\pi$$
$$252$$ 0 0
$$253$$ 50.0000 0.197628
$$254$$ 0 0
$$255$$ − 70.7107i − 0.277297i
$$256$$ 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
$$261$$ 42.4264i 0.162553i
$$262$$ 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
$$267$$ −360.000 −1.34831
$$268$$ 0 0
$$269$$ − 381.838i − 1.41947i −0.704468 0.709735i $$-0.748814\pi$$
0.704468 0.709735i $$-0.251186\pi$$
$$270$$ 0 0
$$271$$ 110.000 0.405904 0.202952 0.979189i $$-0.434946\pi$$
0.202952 + 0.979189i $$0.434946\pi$$
$$272$$ 0 0
$$273$$ 240.000 0.879121
$$274$$ 0 0
$$275$$ 120.000 0.436364
$$276$$ 0 0
$$277$$ 265.000 0.956679 0.478339 0.878175i $$-0.341239\pi$$
0.478339 + 0.878175i $$0.341239\pi$$
$$278$$ 0 0
$$279$$ 42.4264i 0.152066i
$$280$$ 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
$$285$$ − 53.7401i − 0.188562i
$$286$$ 0 0
$$287$$ 212.132i 0.739136i
$$288$$ 0 0
$$289$$ 336.000 1.16263
$$290$$ 0 0
$$291$$ 48.0000 0.164948
$$292$$ 0 0
$$293$$ 186.676i 0.637120i 0.947903 + 0.318560i $$0.103199\pi$$
−0.947903 + 0.318560i $$0.896801\pi$$
$$294$$ 0 0
$$295$$ 84.8528i 0.287637i
$$296$$ 0 0
$$297$$ − 141.421i − 0.476166i
$$298$$ 0 0
$$299$$ 169.706i 0.567577i
$$300$$ 0 0
$$301$$ −25.0000 −0.0830565
$$302$$ 0 0
$$303$$ − 141.421i − 0.466737i
$$304$$ 0 0
$$305$$ −95.0000 −0.311475
$$306$$ 0 0
$$307$$ 280.014i 0.912099i 0.889955 + 0.456049i $$0.150736\pi$$
−0.889955 + 0.456049i $$0.849264\pi$$
$$308$$ 0 0
$$309$$ −48.0000 −0.155340
$$310$$ 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
$$315$$ 5.00000 0.0158730
$$316$$ 0 0
$$317$$ − 186.676i − 0.588884i −0.955669 0.294442i $$-0.904866\pi$$
0.955669 0.294442i $$-0.0951338\pi$$
$$318$$ 0 0
$$319$$ − 212.132i − 0.664991i
$$320$$ 0 0
$$321$$ 288.000 0.897196
$$322$$ 0 0
$$323$$ 475.000 1.47059
$$324$$ 0 0
$$325$$ 407.294i 1.25321i
$$326$$ 0 0
$$327$$ −360.000 −1.10092
$$328$$ 0 0
$$329$$ 25.0000 0.0759878
$$330$$ 0 0
$$331$$ 296.985i 0.897235i 0.893724 + 0.448618i $$0.148083\pi$$
−0.893724 + 0.448618i $$0.851917\pi$$
$$332$$ 0 0
$$333$$ − 25.4558i − 0.0764440i
$$334$$ 0 0
$$335$$ 110.309i 0.329280i
$$336$$ 0 0
$$337$$ 526.087i 1.56109i 0.625099 + 0.780545i $$0.285058\pi$$
−0.625099 + 0.780545i $$0.714942\pi$$
$$338$$ 0 0
$$339$$ 312.000 0.920354
$$340$$ 0 0
$$341$$ − 212.132i − 0.622088i
$$342$$ 0 0
$$343$$ −365.000 −1.06414
$$344$$ 0 0
$$345$$ − 28.2843i − 0.0819834i
$$346$$ 0 0
$$347$$ −125.000 −0.360231 −0.180115 0.983646i $$-0.557647\pi$$
−0.180115 + 0.983646i $$0.557647\pi$$
$$348$$ 0 0
$$349$$ −23.0000 −0.0659026 −0.0329513 0.999457i $$-0.510491\pi$$
−0.0329513 + 0.999457i $$0.510491\pi$$
$$350$$ 0 0
$$351$$ 480.000 1.36752
$$352$$ 0 0
$$353$$ 410.000 1.16147 0.580737 0.814092i $$-0.302765\pi$$
0.580737 + 0.814092i $$0.302765\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ 0 0
$$357$$ − 353.553i − 0.990346i
$$358$$ 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
$$363$$ − 271.529i − 0.748014i
$$364$$ 0 0
$$365$$ −25.0000 −0.0684932
$$366$$ 0 0
$$367$$ 230.000 0.626703 0.313351 0.949637i $$-0.398548\pi$$
0.313351 + 0.949637i $$0.398548\pi$$
$$368$$ 0 0
$$369$$ 42.4264i 0.114977i
$$370$$ 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
$$375$$ − 138.593i − 0.369581i
$$376$$ 0 0
$$377$$ 720.000 1.90981
$$378$$ 0 0
$$379$$ − 254.558i − 0.671658i −0.941923 0.335829i $$-0.890984\pi$$
0.941923 0.335829i $$-0.109016\pi$$
$$380$$ 0 0
$$381$$ 648.000 1.70079
$$382$$ 0 0
$$383$$ 144.250i 0.376631i 0.982109 + 0.188316i $$0.0603028\pi$$
−0.982109 + 0.188316i $$0.939697\pi$$
$$384$$ 0 0
$$385$$ −25.0000 −0.0649351
$$386$$ 0 0
$$387$$ −5.00000 −0.0129199
$$388$$ 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
$$393$$ 461.034i 1.17311i
$$394$$ 0 0
$$395$$ − 42.4264i − 0.107409i
$$396$$ 0 0
$$397$$ −335.000 −0.843829 −0.421914 0.906636i $$-0.638642\pi$$
−0.421914 + 0.906636i $$0.638642\pi$$
$$398$$ 0 0
$$399$$ − 268.701i − 0.673435i
$$400$$ 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
$$405$$ −71.0000 −0.175309
$$406$$ 0 0
$$407$$ 127.279i 0.312725i
$$408$$ 0 0
$$409$$ − 721.249i − 1.76344i −0.471769 0.881722i $$-0.656384\pi$$
0.471769 0.881722i $$-0.343616\pi$$
$$410$$ 0 0
$$411$$ 268.701i 0.653773i
$$412$$ 0 0
$$413$$ 424.264i 1.02727i
$$414$$ 0 0
$$415$$ 130.000 0.313253
$$416$$ 0 0
$$417$$ − 353.553i − 0.847850i
$$418$$ 0 0
$$419$$ −62.0000 −0.147971 −0.0739857 0.997259i $$-0.523572\pi$$
−0.0739857 + 0.997259i $$0.523572\pi$$
$$420$$ 0 0
$$421$$ 296.985i 0.705427i 0.935731 + 0.352714i $$0.114741\pi$$
−0.935731 + 0.352714i $$0.885259\pi$$
$$422$$ 0 0
$$423$$ 5.00000 0.0118203
$$424$$ 0 0
$$425$$ 600.000 1.41176
$$426$$ 0 0
$$427$$ −475.000 −1.11241
$$428$$ 0 0
$$429$$ −240.000 −0.559441
$$430$$ 0 0
$$431$$ 509.117i 1.18125i 0.806948 + 0.590623i $$0.201118\pi$$
−0.806948 + 0.590623i $$0.798882\pi$$
$$432$$ 0 0
$$433$$ 229.103i 0.529105i 0.964371 + 0.264553i $$0.0852243\pi$$
−0.964371 + 0.264553i $$0.914776\pi$$
$$434$$ 0 0
$$435$$ −120.000 −0.275862
$$436$$ 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
$$441$$ −24.0000 −0.0544218
$$442$$ 0 0
$$443$$ −365.000 −0.823928 −0.411964 0.911200i $$-0.635157\pi$$
−0.411964 + 0.911200i $$0.635157\pi$$
$$444$$ 0 0
$$445$$ 127.279i 0.286021i
$$446$$ 0 0
$$447$$ − 608.112i − 1.36043i
$$448$$ 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
$$453$$ −240.000 −0.529801
$$454$$ 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
$$459$$ − 707.107i − 1.54054i
$$460$$ 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 $$-0.324530\pi$$
0.523758 + 0.851867i $$0.324530\pi$$
$$464$$ 0 0
$$465$$ −120.000 −0.258065
$$466$$ 0 0
$$467$$ 115.000 0.246253 0.123126 0.992391i $$-0.460708\pi$$
0.123126 + 0.992391i $$0.460708\pi$$
$$468$$ 0 0
$$469$$ 551.543i 1.17600i
$$470$$ 0 0
$$471$$ 537.401i 1.14098i
$$472$$ 0 0
$$473$$ 25.0000 0.0528541
$$474$$ 0 0
$$475$$ 456.000 0.960000
$$476$$ 0 0
$$477$$ − 25.4558i − 0.0533665i
$$478$$ 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
$$483$$ − 141.421i − 0.292798i
$$484$$ 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
$$489$$ − 311.127i − 0.636252i
$$490$$ 0 0
$$491$$ 82.0000 0.167006 0.0835031 0.996508i $$-0.473389\pi$$
0.0835031 + 0.996508i $$0.473389\pi$$
$$492$$ 0 0
$$493$$ − 1060.66i − 2.15144i
$$494$$ 0 0
$$495$$ −5.00000 −0.0101010
$$496$$ 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
$$501$$ 168.000 0.335329
$$502$$ 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
$$507$$ − 336.583i − 0.663871i
$$508$$ 0 0
$$509$$ − 169.706i − 0.333410i −0.986007 0.166705i $$-0.946687\pi$$
0.986007 0.166705i $$-0.0533127\pi$$
$$510$$ 0 0
$$511$$ −125.000 −0.244618
$$512$$ 0 0
$$513$$ − 537.401i − 1.04757i
$$514$$ 0 0
$$515$$ 16.9706i 0.0329525i
$$516$$ 0 0
$$517$$ −25.0000 −0.0483559
$$518$$ 0 0
$$519$$ 528.000 1.01734
$$520$$ 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 $$0.110667\pi$$
−0.940169 + 0.340709i $$0.889333\pi$$
$$524$$ 0 0
$$525$$ − 339.411i − 0.646498i
$$526$$ 0 0
$$527$$ − 1060.66i − 2.01264i
$$528$$ 0 0
$$529$$ −429.000 −0.810964
$$530$$ 0 0
$$531$$ 84.8528i 0.159798i
$$532$$ 0 0
$$533$$ 720.000 1.35084
$$534$$ 0 0
$$535$$ − 101.823i − 0.190324i
$$536$$ 0 0
$$537$$ 360.000 0.670391
$$538$$ 0 0
$$539$$ 120.000 0.222635
$$540$$ 0 0
$$541$$ 25.0000 0.0462107 0.0231054 0.999733i $$-0.492645\pi$$
0.0231054 + 0.999733i $$0.492645\pi$$
$$542$$ 0 0
$$543$$ 720.000 1.32597
$$544$$ 0 0
$$545$$ 127.279i 0.233540i
$$546$$ 0 0
$$547$$ 16.9706i 0.0310248i 0.999880 + 0.0155124i $$0.00493795\pi$$
−0.999880 + 0.0155124i $$0.995062\pi$$
$$548$$ 0 0
$$549$$ −95.0000 −0.173042
$$550$$ 0 0
$$551$$ − 806.102i − 1.46298i
$$552$$ 0 0
$$553$$ − 212.132i − 0.383602i
$$554$$ 0 0
$$555$$ 72.0000 0.129730
$$556$$ 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
$$561$$ 353.553i 0.630220i
$$562$$ 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
$$567$$ −355.000 −0.626102
$$568$$ 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
$$573$$ 828.729i 1.44630i
$$574$$ 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
$$579$$ 168.000 0.290155
$$580$$ 0 0
$$581$$ 650.000 1.11876
$$582$$ 0 0
$$583$$ 127.279i 0.218318i
$$584$$ 0 0
$$585$$ − 16.9706i − 0.0290095i
$$586$$ 0 0
$$587$$ −725.000 −1.23509 −0.617547 0.786534i $$-0.711873\pi$$
−0.617547 + 0.786534i $$0.711873\pi$$
$$588$$ 0 0
$$589$$ − 806.102i − 1.36859i
$$590$$ 0 0
$$591$$ 197.990i 0.335008i
$$592$$ 0 0
$$593$$ 650.000 1.09612 0.548061 0.836439i $$-0.315366\pi$$
0.548061 + 0.836439i $$0.315366\pi$$
$$594$$ 0 0
$$595$$ −125.000 −0.210084
$$596$$ 0 0
$$597$$ 489.318i 0.819628i
$$598$$ 0 0
$$599$$ 296.985i 0.495801i 0.968785 + 0.247901i $$0.0797406\pi$$
−0.968785 + 0.247901i $$0.920259\pi$$
$$600$$ 0 0
$$601$$ 848.528i 1.41186i 0.708281 + 0.705930i $$0.249471\pi$$
−0.708281 + 0.705930i $$0.750529\pi$$
$$602$$ 0 0
$$603$$ 110.309i 0.182933i
$$604$$ 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
$$609$$ −600.000 −0.985222
$$610$$ 0 0
$$611$$ − 84.8528i − 0.138875i
$$612$$ 0 0
$$613$$ −1055.00 −1.72104 −0.860522 0.509413i $$-0.829862\pi$$
−0.860522 + 0.509413i $$0.829862\pi$$
$$614$$ 0 0
$$615$$ −120.000 −0.195122
$$616$$ 0 0
$$617$$ −505.000 −0.818476 −0.409238 0.912428i $$-0.634206\pi$$
−0.409238 + 0.912428i $$0.634206\pi$$
$$618$$ 0 0
$$619$$ 130.000 0.210016 0.105008 0.994471i $$-0.466513\pi$$
0.105008 + 0.994471i $$0.466513\pi$$
$$620$$ 0 0
$$621$$ − 282.843i − 0.455463i
$$622$$ 0 0
$$623$$ 636.396i 1.02150i
$$624$$ 0 0
$$625$$ 551.000 0.881600
$$626$$ 0 0
$$627$$ 268.701i 0.428550i
$$628$$ 0 0
$$629$$ 636.396i 1.01176i
$$630$$ 0 0
$$631$$ −475.000 −0.752773 −0.376387 0.926463i $$-0.622834\pi$$
−0.376387 + 0.926463i $$0.622834\pi$$
$$632$$ 0 0
$$633$$ −240.000 −0.379147
$$634$$ 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
$$645$$ − 14.1421i − 0.0219258i
$$646$$ 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
$$651$$ −600.000 −0.921659
$$652$$ 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
$$657$$ −25.0000 −0.0380518
$$658$$ 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
$$663$$ −1200.00 −1.80995
$$664$$ 0 0
$$665$$ −95.0000 −0.142857
$$666$$ 0 0
$$667$$ − 424.264i − 0.636078i
$$668$$ 0 0
$$669$$ −1032.00 −1.54260
$$670$$ 0 0
$$671$$ 475.000 0.707899
$$672$$ 0 0
$$673$$ 186.676i 0.277379i 0.990336 + 0.138690i $$0.0442890\pi$$
−0.990336 + 0.138690i $$0.955711\pi$$
$$674$$ 0 0
$$675$$ − 678.823i − 1.00566i
$$676$$ 0 0
$$677$$ 907.925i 1.34110i 0.741864 + 0.670550i $$0.233942\pi$$
−0.741864 + 0.670550i $$0.766058\pi$$
$$678$$ 0 0
$$679$$ − 84.8528i − 0.124967i
$$680$$ 0 0
$$681$$ −192.000 −0.281938
$$682$$ 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
$$687$$ 410.122i 0.596975i
$$688$$ 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
$$693$$ −25.0000 −0.0360750
$$694$$ 0 0
$$695$$ −125.000 −0.179856
$$696$$ 0 0
$$697$$ − 1060.66i − 1.52175i
$$698$$ 0 0
$$699$$ 947.523i 1.35554i
$$700$$ 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
$$705$$ 14.1421i 0.0200598i
$$706$$ 0 0
$$707$$ −250.000 −0.353607
$$708$$ 0 0
$$709$$ 382.000 0.538787 0.269394 0.963030i $$-0.413177\pi$$
0.269394 + 0.963030i $$0.413177\pi$$
$$710$$ 0 0
$$711$$ − 42.4264i − 0.0596715i
$$712$$ 0 0
$$713$$ − 424.264i − 0.595041i
$$714$$ 0 0
$$715$$ 84.8528i 0.118675i
$$716$$ 0 0
$$717$$ 557.200i 0.777127i
$$718$$ 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
$$723$$ 840.000 1.16183
$$724$$ 0 0
$$725$$ − 1018.23i − 1.40446i
$$726$$ 0 0
$$727$$ −1075.00 −1.47868 −0.739340 0.673333i $$-0.764862\pi$$
−0.739340 + 0.673333i $$0.764862\pi$$
$$728$$ 0 0
$$729$$ −791.000 −1.08505
$$730$$ 0 0
$$731$$ 125.000 0.170999
$$732$$ 0 0
$$733$$ −530.000 −0.723056 −0.361528 0.932361i $$-0.617745\pi$$
−0.361528 + 0.932361i $$0.617745\pi$$
$$734$$ 0 0
$$735$$ − 67.8823i − 0.0923568i
$$736$$ 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
$$741$$ −912.000 −1.23077
$$742$$ 0 0
$$743$$ 958.837i 1.29049i 0.763974 + 0.645247i $$0.223245\pi$$
−0.763974 + 0.645247i $$0.776755\pi$$
$$744$$ 0 0
$$745$$ −215.000 −0.288591
$$746$$ 0 0
$$747$$ 130.000 0.174029
$$748$$ 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
$$753$$ − 489.318i − 0.649825i
$$754$$ 0 0
$$755$$ 84.8528i 0.112388i
$$756$$ 0 0
$$757$$ −1055.00 −1.39366 −0.696830 0.717237i $$-0.745407\pi$$
−0.696830 + 0.717237i $$0.745407\pi$$
$$758$$ 0 0
$$759$$ 141.421i 0.186326i
$$760$$ 0 0
$$761$$ 215.000 0.282523 0.141261 0.989972i $$-0.454884\pi$$
0.141261 + 0.989972i $$0.454884\pi$$
$$762$$ 0 0
$$763$$ 636.396i 0.834071i
$$764$$ 0 0
$$765$$ −25.0000 −0.0326797
$$766$$ 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
$$771$$ −192.000 −0.249027
$$772$$ 0 0
$$773$$ 407.294i 0.526900i 0.964673 + 0.263450i $$0.0848604\pi$$
−0.964673 + 0.263450i $$0.915140\pi$$
$$774$$ 0 0
$$775$$ − 1018.23i − 1.31385i
$$776$$ 0 0
$$777$$ 360.000 0.463320
$$778$$ 0 0
$$779$$ − 806.102i − 1.03479i
$$780$$ 0 0
$$781$$ 0 0
$$782$$ 0 0
$$783$$ −1200.00 −1.53257
$$784$$ 0 0
$$785$$ 190.000 0.242038
$$786$$ 0 0
$$787$$ 186.676i 0.237200i 0.992942 + 0.118600i $$0.0378406\pi$$
−0.992942 + 0.118600i $$0.962159\pi$$
$$788$$ 0 0
$$789$$ − 1004.09i − 1.27261i
$$790$$ 0 0
$$791$$ − 551.543i − 0.697273i
$$792$$ 0 0
$$793$$ 1612.20i 2.03304i
$$794$$ 0 0
$$795$$ 72.0000 0.0905660
$$796$$ 0 0
$$797$$ − 704.278i − 0.883662i −0.897098 0.441831i $$-0.854329\pi$$
0.897098 0.441831i $$-0.145671\pi$$
$$798$$ 0 0
$$799$$ −125.000 −0.156446
$$800$$ 0 0
$$801$$ 127.279i 0.158900i
$$802$$ 0 0
$$803$$ 125.000 0.155666
$$804$$ 0 0
$$805$$ −50.0000 −0.0621118
$$806$$ 0 0
$$807$$ 1080.00 1.33829
$$808$$ 0 0
$$809$$ −457.000 −0.564895 −0.282447 0.959283i $$-0.591146\pi$$
−0.282447 + 0.959283i $$0.591146\pi$$
$$810$$ 0 0
$$811$$ 509.117i 0.627764i 0.949462 + 0.313882i $$0.101630\pi$$
−0.949462 + 0.313882i $$0.898370\pi$$
$$812$$ 0 0
$$813$$ 311.127i 0.382690i
$$814$$ 0 0
$$815$$ −110.000 −0.134969
$$816$$ 0 0
$$817$$ 95.0000 0.116279
$$818$$ 0 0
$$819$$ − 84.8528i − 0.103605i
$$820$$ 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 $$-0.794588\pi$$
−0.798906 + 0.601455i $$0.794588\pi$$
$$824$$ 0 0
$$825$$ 339.411i 0.411408i
$$826$$ 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
$$831$$ 749.533i 0.901965i
$$832$$ 0 0
$$833$$ 600.000 0.720288
$$834$$ 0 0
$$835$$ − 59.3970i − 0.0711341i
$$836$$ 0 0
$$837$$ −1200.00 −1.43369
$$838$$ 0 0
$$839$$ 339.411i 0.404543i 0.979330 + 0.202271i $$0.0648323\pi$$
−0.979330 + 0.202271i $$0.935168\pi$$
$$840$$ 0 0
$$841$$ −959.000 −1.14031
$$842$$ 0 0
$$843$$ 1200.00 1.42349
$$844$$ 0 0
$$845$$ −119.000 −0.140828
$$846$$ 0 0
$$847$$ −480.000 −0.566706
$$848$$ 0 0
$$849$$ − 353.553i − 0.416435i
$$850$$ 0 0
$$851$$ 254.558i 0.299129i
$$852$$ 0 0
$$853$$ −770.000 −0.902696 −0.451348 0.892348i $$-0.649057\pi$$
−0.451348 + 0.892348i $$0.649057\pi$$
$$854$$ 0 0
$$855$$ −19.0000 −0.0222222
$$856$$ 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
$$861$$ −600.000 −0.696864
$$862$$ 0 0
$$863$$ − 992.778i − 1.15038i −0.818020 0.575190i $$-0.804928\pi$$
0.818020 0.575190i $$-0.195072\pi$$
$$864$$ 0 0
$$865$$ − 186.676i − 0.215811i
$$866$$ 0 0
$$867$$ 950.352i 1.09614i
$$868$$ 0 0
$$869$$ 212.132i 0.244111i
$$870$$ 0 0
$$871$$ 1872.00 2.14925
$$872$$ 0 0
$$873$$ − 16.9706i − 0.0194394i
$$874$$ 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
$$879$$ −528.000 −0.600683
$$880$$ 0 0
$$881$$ −25.0000 −0.0283768 −0.0141884 0.999899i $$-0.504516\pi$$
−0.0141884 + 0.999899i $$0.504516\pi$$
$$882$$ 0 0
$$883$$ −965.000 −1.09287 −0.546433 0.837503i $$-0.684015\pi$$
−0.546433 + 0.837503i $$0.684015\pi$$
$$884$$ 0 0
$$885$$ −240.000 −0.271186
$$886$$ 0 0
$$887$$ 780.646i 0.880097i 0.897974 + 0.440048i $$0.145039\pi$$
−0.897974 + 0.440048i $$0.854961\pi$$
$$888$$ 0 0
$$889$$ − 1145.51i − 1.28854i
$$890$$ 0 0
$$891$$ 355.000 0.398429
$$892$$ 0 0
$$893$$ −95.0000 −0.106383
$$894$$ 0 0
$$895$$ − 127.279i − 0.142211i
$$896$$ 0 0
$$897$$ −480.000 −0.535117
$$898$$ 0 0
$$899$$ −1800.00 −2.00222
$$900$$ 0 0
$$901$$ 636.396i 0.706322i
$$902$$ 0 0
$$903$$ − 70.7107i − 0.0783064i
$$904$$ 0 0
$$905$$ − 254.558i − 0.281280i
$$906$$ 0 0
$$907$$ − 313.955i − 0.346147i −0.984909 0.173074i $$-0.944630\pi$$
0.984909 0.173074i $$-0.0553698\pi$$
$$908$$ 0 0
$$909$$ −50.0000 −0.0550055
$$910$$ 0 0
$$911$$ − 933.381i − 1.02457i −0.858816 0.512284i $$-0.828800\pi$$
0.858816 0.512284i $$-0.171200\pi$$
$$912$$ 0 0
$$913$$ −650.000 −0.711939
$$914$$ 0 0
$$915$$ − 268.701i − 0.293662i
$$916$$ 0 0
$$917$$ 815.000 0.888768
$$918$$ 0 0
$$919$$ −538.000 −0.585419 −0.292709 0.956201i $$-0.594557\pi$$
−0.292709 + 0.956201i $$0.594557\pi$$
$$920$$ 0 0
$$921$$ −792.000 −0.859935
$$922$$ 0 0
$$923$$ 0 0
$$924$$ 0 0
$$925$$ 610.940i 0.660476i
$$926$$ 0 0
$$927$$ 16.9706i 0.0183070i
$$928$$ 0 0
$$929$$ −742.000 −0.798708 −0.399354 0.916797i $$-0.630766\pi$$
−0.399354 + 0.916797i $$0.630766\pi$$
$$930$$ 0 0
$$931$$ 456.000 0.489796
$$932$$ 0 0
$$933$$ − 664.680i − 0.712412i
$$934$$ 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
$$939$$ − 876.812i − 0.933773i
$$940$$ 0 0
$$941$$ − 424.264i − 0.450865i −0.974259 0.225433i $$-0.927620\pi$$
0.974259 0.225433i $$-0.0723795\pi$$
$$942$$ 0 0
$$943$$ − 424.264i − 0.449909i
$$944$$ 0 0
$$945$$ 141.421i 0.149652i
$$946$$ 0 0
$$947$$ 1210.00 1.27772 0.638860 0.769323i $$-0.279406\pi$$
0.638860 + 0.769323i $$0.279406\pi$$
$$948$$ 0 0
$$949$$ 424.264i 0.447064i
$$950$$ 0 0
$$951$$ 528.000 0.555205
$$952$$ 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
$$957$$ 600.000 0.626959
$$958$$ 0 0
$$959$$ 475.000 0.495308
$$960$$ 0 0
$$961$$ −839.000 −0.873049
$$962$$ 0 0
$$963$$ − 101.823i − 0.105736i
$$964$$ 0 0
$$965$$ − 59.3970i − 0.0615513i
$$966$$ 0 0
$$967$$ 350.000 0.361944 0.180972 0.983488i $$-0.442076\pi$$
0.180972 + 0.983488i $$0.442076\pi$$
$$968$$ 0 0
$$969$$ 1343.50i 1.38648i
$$970$$ 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
$$975$$ −1152.00 −1.18154
$$976$$ 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
$$981$$ 127.279i 0.129744i
$$982$$ 0 0
$$983$$ 695.793i 0.707826i 0.935278 + 0.353913i $$0.115149\pi$$
−0.935278 + 0.353913i $$0.884851\pi$$
$$984$$ 0 0
$$985$$ 70.0000 0.0710660
$$986$$ 0 0
$$987$$ 70.7107i 0.0716420i
$$988$$ 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
$$993$$ −840.000 −0.845921
$$994$$ 0 0
$$995$$ 173.000 0.173869
$$996$$ 0 0
$$997$$ 265.000 0.265797 0.132899 0.991130i $$-0.457572\pi$$
0.132899 + 0.991130i $$0.457572\pi$$
$$998$$ 0 0
$$999$$ 720.000 0.720721
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 1216.3.e.j.1025.2 2
4.3 odd 2 1216.3.e.i.1025.1 2
8.3 odd 2 304.3.e.c.113.2 2
8.5 even 2 38.3.b.a.37.1 2
19.18 odd 2 inner 1216.3.e.j.1025.1 2
24.5 odd 2 342.3.d.a.37.2 2
24.11 even 2 2736.3.o.h.721.2 2
40.13 odd 4 950.3.d.a.949.1 4
40.29 even 2 950.3.c.a.151.2 2
40.37 odd 4 950.3.d.a.949.4 4
76.75 even 2 1216.3.e.i.1025.2 2
152.37 odd 2 38.3.b.a.37.2 yes 2
152.75 even 2 304.3.e.c.113.1 2
456.227 odd 2 2736.3.o.h.721.1 2
456.341 even 2 342.3.d.a.37.1 2
760.37 even 4 950.3.d.a.949.2 4
760.189 odd 2 950.3.c.a.151.1 2
760.493 even 4 950.3.d.a.949.3 4

By twisted newform
Twist Min Dim Char Parity Ord Type
38.3.b.a.37.1 2 8.5 even 2
38.3.b.a.37.2 yes 2 152.37 odd 2
304.3.e.c.113.1 2 152.75 even 2
304.3.e.c.113.2 2 8.3 odd 2
342.3.d.a.37.1 2 456.341 even 2
342.3.d.a.37.2 2 24.5 odd 2
950.3.c.a.151.1 2 760.189 odd 2
950.3.c.a.151.2 2 40.29 even 2
950.3.d.a.949.1 4 40.13 odd 4
950.3.d.a.949.2 4 760.37 even 4
950.3.d.a.949.3 4 760.493 even 4
950.3.d.a.949.4 4 40.37 odd 4
1216.3.e.i.1025.1 2 4.3 odd 2
1216.3.e.i.1025.2 2 76.75 even 2
1216.3.e.j.1025.1 2 19.18 odd 2 inner
1216.3.e.j.1025.2 2 1.1 even 1 trivial
2736.3.o.h.721.1 2 456.227 odd 2
2736.3.o.h.721.2 2 24.11 even 2