# Properties

 Label 38.3.b.a.37.1 Level $38$ Weight $3$ Character 38.37 Analytic conductor $1.035$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$38 = 2 \cdot 19$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 38.b (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$1.03542500457$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-2})$$ Defining polynomial: $$x^{2} + 2$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## Embedding invariants

 Embedding label 37.1 Root $$-1.41421i$$ of defining polynomial Character $$\chi$$ $$=$$ 38.37 Dual form 38.3.b.a.37.2

## $q$-expansion

 $$f(q)$$ $$=$$ $$q-1.41421i q^{2} -2.82843i q^{3} -2.00000 q^{4} -1.00000 q^{5} -4.00000 q^{6} +5.00000 q^{7} +2.82843i q^{8} +1.00000 q^{9} +O(q^{10})$$ $$q-1.41421i q^{2} -2.82843i q^{3} -2.00000 q^{4} -1.00000 q^{5} -4.00000 q^{6} +5.00000 q^{7} +2.82843i q^{8} +1.00000 q^{9} +1.41421i q^{10} +5.00000 q^{11} +5.65685i q^{12} +16.9706i q^{13} -7.07107i q^{14} +2.82843i q^{15} +4.00000 q^{16} -25.0000 q^{17} -1.41421i q^{18} +19.0000 q^{19} +2.00000 q^{20} -14.1421i q^{21} -7.07107i q^{22} -10.0000 q^{23} +8.00000 q^{24} -24.0000 q^{25} +24.0000 q^{26} -28.2843i q^{27} -10.0000 q^{28} -42.4264i q^{29} +4.00000 q^{30} +42.4264i q^{31} -5.65685i q^{32} -14.1421i q^{33} +35.3553i q^{34} -5.00000 q^{35} -2.00000 q^{36} +25.4558i q^{37} -26.8701i q^{38} +48.0000 q^{39} -2.82843i q^{40} +42.4264i q^{41} -20.0000 q^{42} +5.00000 q^{43} -10.0000 q^{44} -1.00000 q^{45} +14.1421i q^{46} +5.00000 q^{47} -11.3137i q^{48} -24.0000 q^{49} +33.9411i q^{50} +70.7107i q^{51} -33.9411i q^{52} +25.4558i q^{53} -40.0000 q^{54} -5.00000 q^{55} +14.1421i q^{56} -53.7401i q^{57} -60.0000 q^{58} -84.8528i q^{59} -5.65685i q^{60} +95.0000 q^{61} +60.0000 q^{62} +5.00000 q^{63} -8.00000 q^{64} -16.9706i q^{65} -20.0000 q^{66} -110.309i q^{67} +50.0000 q^{68} +28.2843i q^{69} +7.07107i q^{70} +2.82843i q^{72} -25.0000 q^{73} +36.0000 q^{74} +67.8823i q^{75} -38.0000 q^{76} +25.0000 q^{77} -67.8823i q^{78} -42.4264i q^{79} -4.00000 q^{80} -71.0000 q^{81} +60.0000 q^{82} -130.000 q^{83} +28.2843i q^{84} +25.0000 q^{85} -7.07107i q^{86} -120.000 q^{87} +14.1421i q^{88} +127.279i q^{89} +1.41421i q^{90} +84.8528i q^{91} +20.0000 q^{92} +120.000 q^{93} -7.07107i q^{94} -19.0000 q^{95} -16.0000 q^{96} -16.9706i q^{97} +33.9411i q^{98} +5.00000 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 4q^{4} - 2q^{5} - 8q^{6} + 10q^{7} + 2q^{9} + O(q^{10})$$ $$2q - 4q^{4} - 2q^{5} - 8q^{6} + 10q^{7} + 2q^{9} + 10q^{11} + 8q^{16} - 50q^{17} + 38q^{19} + 4q^{20} - 20q^{23} + 16q^{24} - 48q^{25} + 48q^{26} - 20q^{28} + 8q^{30} - 10q^{35} - 4q^{36} + 96q^{39} - 40q^{42} + 10q^{43} - 20q^{44} - 2q^{45} + 10q^{47} - 48q^{49} - 80q^{54} - 10q^{55} - 120q^{58} + 190q^{61} + 120q^{62} + 10q^{63} - 16q^{64} - 40q^{66} + 100q^{68} - 50q^{73} + 72q^{74} - 76q^{76} + 50q^{77} - 8q^{80} - 142q^{81} + 120q^{82} - 260q^{83} + 50q^{85} - 240q^{87} + 40q^{92} + 240q^{93} - 38q^{95} - 32q^{96} + 10q^{99} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/38\mathbb{Z}\right)^\times$$.

 $$n$$ $$21$$ $$\chi(n)$$ $$-1$$

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ − 1.41421i − 0.707107i
$$3$$ − 2.82843i − 0.942809i −0.881917 0.471405i $$-0.843747\pi$$
0.881917 0.471405i $$-0.156253\pi$$
$$4$$ −2.00000 −0.500000
$$5$$ −1.00000 −0.200000 −0.100000 0.994987i $$-0.531884\pi$$
−0.100000 + 0.994987i $$0.531884\pi$$
$$6$$ −4.00000 −0.666667
$$7$$ 5.00000 0.714286 0.357143 0.934050i $$-0.383751\pi$$
0.357143 + 0.934050i $$0.383751\pi$$
$$8$$ 2.82843i 0.353553i
$$9$$ 1.00000 0.111111
$$10$$ 1.41421i 0.141421i
$$11$$ 5.00000 0.454545 0.227273 0.973831i $$-0.427019\pi$$
0.227273 + 0.973831i $$0.427019\pi$$
$$12$$ 5.65685i 0.471405i
$$13$$ 16.9706i 1.30543i 0.757604 + 0.652714i $$0.226370\pi$$
−0.757604 + 0.652714i $$0.773630\pi$$
$$14$$ − 7.07107i − 0.505076i
$$15$$ 2.82843i 0.188562i
$$16$$ 4.00000 0.250000
$$17$$ −25.0000 −1.47059 −0.735294 0.677748i $$-0.762956\pi$$
−0.735294 + 0.677748i $$0.762956\pi$$
$$18$$ − 1.41421i − 0.0785674i
$$19$$ 19.0000 1.00000
$$20$$ 2.00000 0.100000
$$21$$ − 14.1421i − 0.673435i
$$22$$ − 7.07107i − 0.321412i
$$23$$ −10.0000 −0.434783 −0.217391 0.976085i $$-0.569755\pi$$
−0.217391 + 0.976085i $$0.569755\pi$$
$$24$$ 8.00000 0.333333
$$25$$ −24.0000 −0.960000
$$26$$ 24.0000 0.923077
$$27$$ − 28.2843i − 1.04757i
$$28$$ −10.0000 −0.357143
$$29$$ − 42.4264i − 1.46298i −0.681852 0.731490i $$-0.738825\pi$$
0.681852 0.731490i $$-0.261175\pi$$
$$30$$ 4.00000 0.133333
$$31$$ 42.4264i 1.36859i 0.729204 + 0.684297i $$0.239891\pi$$
−0.729204 + 0.684297i $$0.760109\pi$$
$$32$$ − 5.65685i − 0.176777i
$$33$$ − 14.1421i − 0.428550i
$$34$$ 35.3553i 1.03986i
$$35$$ −5.00000 −0.142857
$$36$$ −2.00000 −0.0555556
$$37$$ 25.4558i 0.687996i 0.938970 + 0.343998i $$0.111781\pi$$
−0.938970 + 0.343998i $$0.888219\pi$$
$$38$$ − 26.8701i − 0.707107i
$$39$$ 48.0000 1.23077
$$40$$ − 2.82843i − 0.0707107i
$$41$$ 42.4264i 1.03479i 0.855747 + 0.517395i $$0.173098\pi$$
−0.855747 + 0.517395i $$0.826902\pi$$
$$42$$ −20.0000 −0.476190
$$43$$ 5.00000 0.116279 0.0581395 0.998308i $$-0.481483\pi$$
0.0581395 + 0.998308i $$0.481483\pi$$
$$44$$ −10.0000 −0.227273
$$45$$ −1.00000 −0.0222222
$$46$$ 14.1421i 0.307438i
$$47$$ 5.00000 0.106383 0.0531915 0.998584i $$-0.483061\pi$$
0.0531915 + 0.998584i $$0.483061\pi$$
$$48$$ − 11.3137i − 0.235702i
$$49$$ −24.0000 −0.489796
$$50$$ 33.9411i 0.678823i
$$51$$ 70.7107i 1.38648i
$$52$$ − 33.9411i − 0.652714i
$$53$$ 25.4558i 0.480299i 0.970736 + 0.240149i $$0.0771965\pi$$
−0.970736 + 0.240149i $$0.922804\pi$$
$$54$$ −40.0000 −0.740741
$$55$$ −5.00000 −0.0909091
$$56$$ 14.1421i 0.252538i
$$57$$ − 53.7401i − 0.942809i
$$58$$ −60.0000 −1.03448
$$59$$ − 84.8528i − 1.43818i −0.694915 0.719092i $$-0.744558\pi$$
0.694915 0.719092i $$-0.255442\pi$$
$$60$$ − 5.65685i − 0.0942809i
$$61$$ 95.0000 1.55738 0.778689 0.627411i $$-0.215885\pi$$
0.778689 + 0.627411i $$0.215885\pi$$
$$62$$ 60.0000 0.967742
$$63$$ 5.00000 0.0793651
$$64$$ −8.00000 −0.125000
$$65$$ − 16.9706i − 0.261086i
$$66$$ −20.0000 −0.303030
$$67$$ − 110.309i − 1.64640i −0.567753 0.823199i $$-0.692187\pi$$
0.567753 0.823199i $$-0.307813\pi$$
$$68$$ 50.0000 0.735294
$$69$$ 28.2843i 0.409917i
$$70$$ 7.07107i 0.101015i
$$71$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$72$$ 2.82843i 0.0392837i
$$73$$ −25.0000 −0.342466 −0.171233 0.985231i $$-0.554775\pi$$
−0.171233 + 0.985231i $$0.554775\pi$$
$$74$$ 36.0000 0.486486
$$75$$ 67.8823i 0.905097i
$$76$$ −38.0000 −0.500000
$$77$$ 25.0000 0.324675
$$78$$ − 67.8823i − 0.870285i
$$79$$ − 42.4264i − 0.537043i −0.963274 0.268522i $$-0.913465\pi$$
0.963274 0.268522i $$-0.0865351\pi$$
$$80$$ −4.00000 −0.0500000
$$81$$ −71.0000 −0.876543
$$82$$ 60.0000 0.731707
$$83$$ −130.000 −1.56627 −0.783133 0.621855i $$-0.786379\pi$$
−0.783133 + 0.621855i $$0.786379\pi$$
$$84$$ 28.2843i 0.336718i
$$85$$ 25.0000 0.294118
$$86$$ − 7.07107i − 0.0822217i
$$87$$ −120.000 −1.37931
$$88$$ 14.1421i 0.160706i
$$89$$ 127.279i 1.43010i 0.699071 + 0.715052i $$0.253597\pi$$
−0.699071 + 0.715052i $$0.746403\pi$$
$$90$$ 1.41421i 0.0157135i
$$91$$ 84.8528i 0.932449i
$$92$$ 20.0000 0.217391
$$93$$ 120.000 1.29032
$$94$$ − 7.07107i − 0.0752241i
$$95$$ −19.0000 −0.200000
$$96$$ −16.0000 −0.166667
$$97$$ − 16.9706i − 0.174954i −0.996167 0.0874771i $$-0.972120\pi$$
0.996167 0.0874771i $$-0.0278805\pi$$
$$98$$ 33.9411i 0.346338i
$$99$$ 5.00000 0.0505051
$$100$$ 48.0000 0.480000
$$101$$ 50.0000 0.495050 0.247525 0.968882i $$-0.420383\pi$$
0.247525 + 0.968882i $$0.420383\pi$$
$$102$$ 100.000 0.980392
$$103$$ 16.9706i 0.164763i 0.996601 + 0.0823814i $$0.0262526\pi$$
−0.996601 + 0.0823814i $$0.973747\pi$$
$$104$$ −48.0000 −0.461538
$$105$$ 14.1421i 0.134687i
$$106$$ 36.0000 0.339623
$$107$$ 101.823i 0.951620i 0.879548 + 0.475810i $$0.157845\pi$$
−0.879548 + 0.475810i $$0.842155\pi$$
$$108$$ 56.5685i 0.523783i
$$109$$ − 127.279i − 1.16770i −0.811862 0.583850i $$-0.801546\pi$$
0.811862 0.583850i $$-0.198454\pi$$
$$110$$ 7.07107i 0.0642824i
$$111$$ 72.0000 0.648649
$$112$$ 20.0000 0.178571
$$113$$ − 110.309i − 0.976183i −0.872793 0.488091i $$-0.837693\pi$$
0.872793 0.488091i $$-0.162307\pi$$
$$114$$ −76.0000 −0.666667
$$115$$ 10.0000 0.0869565
$$116$$ 84.8528i 0.731490i
$$117$$ 16.9706i 0.145048i
$$118$$ −120.000 −1.01695
$$119$$ −125.000 −1.05042
$$120$$ −8.00000 −0.0666667
$$121$$ −96.0000 −0.793388
$$122$$ − 134.350i − 1.10123i
$$123$$ 120.000 0.975610
$$124$$ − 84.8528i − 0.684297i
$$125$$ 49.0000 0.392000
$$126$$ − 7.07107i − 0.0561196i
$$127$$ − 229.103i − 1.80396i −0.431780 0.901979i $$-0.642114\pi$$
0.431780 0.901979i $$-0.357886\pi$$
$$128$$ 11.3137i 0.0883883i
$$129$$ − 14.1421i − 0.109629i
$$130$$ −24.0000 −0.184615
$$131$$ −163.000 −1.24427 −0.622137 0.782908i $$-0.713735\pi$$
−0.622137 + 0.782908i $$0.713735\pi$$
$$132$$ 28.2843i 0.214275i
$$133$$ 95.0000 0.714286
$$134$$ −156.000 −1.16418
$$135$$ 28.2843i 0.209513i
$$136$$ − 70.7107i − 0.519931i
$$137$$ 95.0000 0.693431 0.346715 0.937970i $$-0.387297\pi$$
0.346715 + 0.937970i $$0.387297\pi$$
$$138$$ 40.0000 0.289855
$$139$$ 125.000 0.899281 0.449640 0.893210i $$-0.351552\pi$$
0.449640 + 0.893210i $$0.351552\pi$$
$$140$$ 10.0000 0.0714286
$$141$$ − 14.1421i − 0.100299i
$$142$$ 0 0
$$143$$ 84.8528i 0.593376i
$$144$$ 4.00000 0.0277778
$$145$$ 42.4264i 0.292596i
$$146$$ 35.3553i 0.242160i
$$147$$ 67.8823i 0.461784i
$$148$$ − 50.9117i − 0.343998i
$$149$$ 215.000 1.44295 0.721477 0.692439i $$-0.243464\pi$$
0.721477 + 0.692439i $$0.243464\pi$$
$$150$$ 96.0000 0.640000
$$151$$ 84.8528i 0.561939i 0.959717 + 0.280970i $$0.0906560\pi$$
−0.959717 + 0.280970i $$0.909344\pi$$
$$152$$ 53.7401i 0.353553i
$$153$$ −25.0000 −0.163399
$$154$$ − 35.3553i − 0.229580i
$$155$$ − 42.4264i − 0.273719i
$$156$$ −96.0000 −0.615385
$$157$$ −190.000 −1.21019 −0.605096 0.796153i $$-0.706865\pi$$
−0.605096 + 0.796153i $$0.706865\pi$$
$$158$$ −60.0000 −0.379747
$$159$$ 72.0000 0.452830
$$160$$ 5.65685i 0.0353553i
$$161$$ −50.0000 −0.310559
$$162$$ 100.409i 0.619810i
$$163$$ 110.000 0.674847 0.337423 0.941353i $$-0.390445\pi$$
0.337423 + 0.941353i $$0.390445\pi$$
$$164$$ − 84.8528i − 0.517395i
$$165$$ 14.1421i 0.0857099i
$$166$$ 183.848i 1.10752i
$$167$$ − 59.3970i − 0.355670i −0.984060 0.177835i $$-0.943091\pi$$
0.984060 0.177835i $$-0.0569094\pi$$
$$168$$ 40.0000 0.238095
$$169$$ −119.000 −0.704142
$$170$$ − 35.3553i − 0.207973i
$$171$$ 19.0000 0.111111
$$172$$ −10.0000 −0.0581395
$$173$$ 186.676i 1.07905i 0.841969 + 0.539527i $$0.181397\pi$$
−0.841969 + 0.539527i $$0.818603\pi$$
$$174$$ 169.706i 0.975320i
$$175$$ −120.000 −0.685714
$$176$$ 20.0000 0.113636
$$177$$ −240.000 −1.35593
$$178$$ 180.000 1.01124
$$179$$ 127.279i 0.711057i 0.934665 + 0.355529i $$0.115699\pi$$
−0.934665 + 0.355529i $$0.884301\pi$$
$$180$$ 2.00000 0.0111111
$$181$$ 254.558i 1.40640i 0.710992 + 0.703200i $$0.248246\pi$$
−0.710992 + 0.703200i $$0.751754\pi$$
$$182$$ 120.000 0.659341
$$183$$ − 268.701i − 1.46831i
$$184$$ − 28.2843i − 0.153719i
$$185$$ − 25.4558i − 0.137599i
$$186$$ − 169.706i − 0.912396i
$$187$$ −125.000 −0.668449
$$188$$ −10.0000 −0.0531915
$$189$$ − 141.421i − 0.748261i
$$190$$ 26.8701i 0.141421i
$$191$$ 293.000 1.53403 0.767016 0.641628i $$-0.221741\pi$$
0.767016 + 0.641628i $$0.221741\pi$$
$$192$$ 22.6274i 0.117851i
$$193$$ − 59.3970i − 0.307756i −0.988090 0.153878i $$-0.950824\pi$$
0.988090 0.153878i $$-0.0491763\pi$$
$$194$$ −24.0000 −0.123711
$$195$$ −48.0000 −0.246154
$$196$$ 48.0000 0.244898
$$197$$ −70.0000 −0.355330 −0.177665 0.984091i $$-0.556854\pi$$
−0.177665 + 0.984091i $$0.556854\pi$$
$$198$$ − 7.07107i − 0.0357125i
$$199$$ 173.000 0.869347 0.434673 0.900588i $$-0.356864\pi$$
0.434673 + 0.900588i $$0.356864\pi$$
$$200$$ − 67.8823i − 0.339411i
$$201$$ −312.000 −1.55224
$$202$$ − 70.7107i − 0.350053i
$$203$$ − 212.132i − 1.04499i
$$204$$ − 141.421i − 0.693242i
$$205$$ − 42.4264i − 0.206958i
$$206$$ 24.0000 0.116505
$$207$$ −10.0000 −0.0483092
$$208$$ 67.8823i 0.326357i
$$209$$ 95.0000 0.454545
$$210$$ 20.0000 0.0952381
$$211$$ − 84.8528i − 0.402146i −0.979576 0.201073i $$-0.935557\pi$$
0.979576 0.201073i $$-0.0644429\pi$$
$$212$$ − 50.9117i − 0.240149i
$$213$$ 0 0
$$214$$ 144.000 0.672897
$$215$$ −5.00000 −0.0232558
$$216$$ 80.0000 0.370370
$$217$$ 212.132i 0.977567i
$$218$$ −180.000 −0.825688
$$219$$ 70.7107i 0.322880i
$$220$$ 10.0000 0.0454545
$$221$$ − 424.264i − 1.91975i
$$222$$ − 101.823i − 0.458664i
$$223$$ 364.867i 1.63618i 0.575094 + 0.818088i $$0.304966\pi$$
−0.575094 + 0.818088i $$0.695034\pi$$
$$224$$ − 28.2843i − 0.126269i
$$225$$ −24.0000 −0.106667
$$226$$ −156.000 −0.690265
$$227$$ − 67.8823i − 0.299041i −0.988759 0.149520i $$-0.952227\pi$$
0.988759 0.149520i $$-0.0477730\pi$$
$$228$$ 107.480i 0.471405i
$$229$$ −145.000 −0.633188 −0.316594 0.948561i $$-0.602539\pi$$
−0.316594 + 0.948561i $$0.602539\pi$$
$$230$$ − 14.1421i − 0.0614875i
$$231$$ − 70.7107i − 0.306107i
$$232$$ 120.000 0.517241
$$233$$ 335.000 1.43777 0.718884 0.695130i $$-0.244653\pi$$
0.718884 + 0.695130i $$0.244653\pi$$
$$234$$ 24.0000 0.102564
$$235$$ −5.00000 −0.0212766
$$236$$ 169.706i 0.719092i
$$237$$ −120.000 −0.506329
$$238$$ 176.777i 0.742759i
$$239$$ 197.000 0.824268 0.412134 0.911123i $$-0.364784\pi$$
0.412134 + 0.911123i $$0.364784\pi$$
$$240$$ 11.3137i 0.0471405i
$$241$$ − 296.985i − 1.23230i −0.787628 0.616151i $$-0.788691\pi$$
0.787628 0.616151i $$-0.211309\pi$$
$$242$$ 135.765i 0.561010i
$$243$$ − 53.7401i − 0.221153i
$$244$$ −190.000 −0.778689
$$245$$ 24.0000 0.0979592
$$246$$ − 169.706i − 0.689860i
$$247$$ 322.441i 1.30543i
$$248$$ −120.000 −0.483871
$$249$$ 367.696i 1.47669i
$$250$$ − 69.2965i − 0.277186i
$$251$$ 173.000 0.689243 0.344622 0.938742i $$-0.388007\pi$$
0.344622 + 0.938742i $$0.388007\pi$$
$$252$$ −10.0000 −0.0396825
$$253$$ −50.0000 −0.197628
$$254$$ −324.000 −1.27559
$$255$$ − 70.7107i − 0.277297i
$$256$$ 16.0000 0.0625000
$$257$$ 67.8823i 0.264133i 0.991241 + 0.132067i $$0.0421613\pi$$
−0.991241 + 0.132067i $$0.957839\pi$$
$$258$$ −20.0000 −0.0775194
$$259$$ 127.279i 0.491426i
$$260$$ 33.9411i 0.130543i
$$261$$ − 42.4264i − 0.162553i
$$262$$ 230.517i 0.879835i
$$263$$ −355.000 −1.34981 −0.674905 0.737905i $$-0.735815\pi$$
−0.674905 + 0.737905i $$0.735815\pi$$
$$264$$ 40.0000 0.151515
$$265$$ − 25.4558i − 0.0960598i
$$266$$ − 134.350i − 0.505076i
$$267$$ 360.000 1.34831
$$268$$ 220.617i 0.823199i
$$269$$ 381.838i 1.41947i 0.704468 + 0.709735i $$0.251186\pi$$
−0.704468 + 0.709735i $$0.748814\pi$$
$$270$$ 40.0000 0.148148
$$271$$ 110.000 0.405904 0.202952 0.979189i $$-0.434946\pi$$
0.202952 + 0.979189i $$0.434946\pi$$
$$272$$ −100.000 −0.367647
$$273$$ 240.000 0.879121
$$274$$ − 134.350i − 0.490330i
$$275$$ −120.000 −0.436364
$$276$$ − 56.5685i − 0.204958i
$$277$$ −265.000 −0.956679 −0.478339 0.878175i $$-0.658761\pi$$
−0.478339 + 0.878175i $$0.658761\pi$$
$$278$$ − 176.777i − 0.635887i
$$279$$ 42.4264i 0.152066i
$$280$$ − 14.1421i − 0.0505076i
$$281$$ − 424.264i − 1.50984i −0.655819 0.754918i $$-0.727677\pi$$
0.655819 0.754918i $$-0.272323\pi$$
$$282$$ −20.0000 −0.0709220
$$283$$ 125.000 0.441696 0.220848 0.975308i $$-0.429118\pi$$
0.220848 + 0.975308i $$0.429118\pi$$
$$284$$ 0 0
$$285$$ 53.7401i 0.188562i
$$286$$ 120.000 0.419580
$$287$$ 212.132i 0.739136i
$$288$$ − 5.65685i − 0.0196419i
$$289$$ 336.000 1.16263
$$290$$ 60.0000 0.206897
$$291$$ −48.0000 −0.164948
$$292$$ 50.0000 0.171233
$$293$$ − 186.676i − 0.637120i −0.947903 0.318560i $$-0.896801\pi$$
0.947903 0.318560i $$-0.103199\pi$$
$$294$$ 96.0000 0.326531
$$295$$ 84.8528i 0.287637i
$$296$$ −72.0000 −0.243243
$$297$$ − 141.421i − 0.476166i
$$298$$ − 304.056i − 1.02032i
$$299$$ − 169.706i − 0.567577i
$$300$$ − 135.765i − 0.452548i
$$301$$ 25.0000 0.0830565
$$302$$ 120.000 0.397351
$$303$$ − 141.421i − 0.466737i
$$304$$ 76.0000 0.250000
$$305$$ −95.0000 −0.311475
$$306$$ 35.3553i 0.115540i
$$307$$ − 280.014i − 0.912099i −0.889955 0.456049i $$-0.849264\pi$$
0.889955 0.456049i $$-0.150736\pi$$
$$308$$ −50.0000 −0.162338
$$309$$ 48.0000 0.155340
$$310$$ −60.0000 −0.193548
$$311$$ −235.000 −0.755627 −0.377814 0.925882i $$-0.623324\pi$$
−0.377814 + 0.925882i $$0.623324\pi$$
$$312$$ 135.765i 0.435143i
$$313$$ −310.000 −0.990415 −0.495208 0.868775i $$-0.664908\pi$$
−0.495208 + 0.868775i $$0.664908\pi$$
$$314$$ 268.701i 0.855734i
$$315$$ −5.00000 −0.0158730
$$316$$ 84.8528i 0.268522i
$$317$$ 186.676i 0.588884i 0.955669 + 0.294442i $$0.0951338\pi$$
−0.955669 + 0.294442i $$0.904866\pi$$
$$318$$ − 101.823i − 0.320199i
$$319$$ − 212.132i − 0.664991i
$$320$$ 8.00000 0.0250000
$$321$$ 288.000 0.897196
$$322$$ 70.7107i 0.219598i
$$323$$ −475.000 −1.47059
$$324$$ 142.000 0.438272
$$325$$ − 407.294i − 1.25321i
$$326$$ − 155.563i − 0.477189i
$$327$$ −360.000 −1.10092
$$328$$ −120.000 −0.365854
$$329$$ 25.0000 0.0759878
$$330$$ 20.0000 0.0606061
$$331$$ − 296.985i − 0.897235i −0.893724 0.448618i $$-0.851917\pi$$
0.893724 0.448618i $$-0.148083\pi$$
$$332$$ 260.000 0.783133
$$333$$ 25.4558i 0.0764440i
$$334$$ −84.0000 −0.251497
$$335$$ 110.309i 0.329280i
$$336$$ − 56.5685i − 0.168359i
$$337$$ 526.087i 1.56109i 0.625099 + 0.780545i $$0.285058\pi$$
−0.625099 + 0.780545i $$0.714942\pi$$
$$338$$ 168.291i 0.497904i
$$339$$ −312.000 −0.920354
$$340$$ −50.0000 −0.147059
$$341$$ 212.132i 0.622088i
$$342$$ − 26.8701i − 0.0785674i
$$343$$ −365.000 −1.06414
$$344$$ 14.1421i 0.0411109i
$$345$$ − 28.2843i − 0.0819834i
$$346$$ 264.000 0.763006
$$347$$ 125.000 0.360231 0.180115 0.983646i $$-0.442353\pi$$
0.180115 + 0.983646i $$0.442353\pi$$
$$348$$ 240.000 0.689655
$$349$$ 23.0000 0.0659026 0.0329513 0.999457i $$-0.489509\pi$$
0.0329513 + 0.999457i $$0.489509\pi$$
$$350$$ 169.706i 0.484873i
$$351$$ 480.000 1.36752
$$352$$ − 28.2843i − 0.0803530i
$$353$$ 410.000 1.16147 0.580737 0.814092i $$-0.302765\pi$$
0.580737 + 0.814092i $$0.302765\pi$$
$$354$$ 339.411i 0.958789i
$$355$$ 0 0
$$356$$ − 254.558i − 0.715052i
$$357$$ 353.553i 0.990346i
$$358$$ 180.000 0.502793
$$359$$ −475.000 −1.32312 −0.661560 0.749892i $$-0.730105\pi$$
−0.661560 + 0.749892i $$0.730105\pi$$
$$360$$ − 2.82843i − 0.00785674i
$$361$$ 361.000 1.00000
$$362$$ 360.000 0.994475
$$363$$ 271.529i 0.748014i
$$364$$ − 169.706i − 0.466224i
$$365$$ 25.0000 0.0684932
$$366$$ −380.000 −1.03825
$$367$$ 230.000 0.626703 0.313351 0.949637i $$-0.398548\pi$$
0.313351 + 0.949637i $$0.398548\pi$$
$$368$$ −40.0000 −0.108696
$$369$$ 42.4264i 0.114977i
$$370$$ −36.0000 −0.0972973
$$371$$ 127.279i 0.343071i
$$372$$ −240.000 −0.645161
$$373$$ 67.8823i 0.181990i 0.995851 + 0.0909950i $$0.0290047\pi$$
−0.995851 + 0.0909950i $$0.970995\pi$$
$$374$$ 176.777i 0.472665i
$$375$$ − 138.593i − 0.369581i
$$376$$ 14.1421i 0.0376121i
$$377$$ 720.000 1.90981
$$378$$ −200.000 −0.529101
$$379$$ 254.558i 0.671658i 0.941923 + 0.335829i $$0.109016\pi$$
−0.941923 + 0.335829i $$0.890984\pi$$
$$380$$ 38.0000 0.100000
$$381$$ −648.000 −1.70079
$$382$$ − 414.365i − 1.08472i
$$383$$ 144.250i 0.376631i 0.982109 + 0.188316i $$0.0603028\pi$$
−0.982109 + 0.188316i $$0.939697\pi$$
$$384$$ 32.0000 0.0833333
$$385$$ −25.0000 −0.0649351
$$386$$ −84.0000 −0.217617
$$387$$ 5.00000 0.0129199
$$388$$ 33.9411i 0.0874771i
$$389$$ −553.000 −1.42159 −0.710797 0.703397i $$-0.751666\pi$$
−0.710797 + 0.703397i $$0.751666\pi$$
$$390$$ 67.8823i 0.174057i
$$391$$ 250.000 0.639386
$$392$$ − 67.8823i − 0.173169i
$$393$$ 461.034i 1.17311i
$$394$$ 98.9949i 0.251256i
$$395$$ 42.4264i 0.107409i
$$396$$ −10.0000 −0.0252525
$$397$$ 335.000 0.843829 0.421914 0.906636i $$-0.361358\pi$$
0.421914 + 0.906636i $$0.361358\pi$$
$$398$$ − 244.659i − 0.614721i
$$399$$ − 268.701i − 0.673435i
$$400$$ −96.0000 −0.240000
$$401$$ − 212.132i − 0.529008i −0.964385 0.264504i $$-0.914792\pi$$
0.964385 0.264504i $$-0.0852082\pi$$
$$402$$ 441.235i 1.09760i
$$403$$ −720.000 −1.78660
$$404$$ −100.000 −0.247525
$$405$$ 71.0000 0.175309
$$406$$ −300.000 −0.738916
$$407$$ 127.279i 0.312725i
$$408$$ −200.000 −0.490196
$$409$$ − 721.249i − 1.76344i −0.471769 0.881722i $$-0.656384\pi$$
0.471769 0.881722i $$-0.343616\pi$$
$$410$$ −60.0000 −0.146341
$$411$$ − 268.701i − 0.653773i
$$412$$ − 33.9411i − 0.0823814i
$$413$$ − 424.264i − 1.02727i
$$414$$ 14.1421i 0.0341597i
$$415$$ 130.000 0.313253
$$416$$ 96.0000 0.230769
$$417$$ − 353.553i − 0.847850i
$$418$$ − 134.350i − 0.321412i
$$419$$ 62.0000 0.147971 0.0739857 0.997259i $$-0.476428\pi$$
0.0739857 + 0.997259i $$0.476428\pi$$
$$420$$ − 28.2843i − 0.0673435i
$$421$$ − 296.985i − 0.705427i −0.935731 0.352714i $$-0.885259\pi$$
0.935731 0.352714i $$-0.114741\pi$$
$$422$$ −120.000 −0.284360
$$423$$ 5.00000 0.0118203
$$424$$ −72.0000 −0.169811
$$425$$ 600.000 1.41176
$$426$$ 0 0
$$427$$ 475.000 1.11241
$$428$$ − 203.647i − 0.475810i
$$429$$ 240.000 0.559441
$$430$$ 7.07107i 0.0164443i
$$431$$ 509.117i 1.18125i 0.806948 + 0.590623i $$0.201118\pi$$
−0.806948 + 0.590623i $$0.798882\pi$$
$$432$$ − 113.137i − 0.261891i
$$433$$ 229.103i 0.529105i 0.964371 + 0.264553i $$0.0852243\pi$$
−0.964371 + 0.264553i $$0.914776\pi$$
$$434$$ 300.000 0.691244
$$435$$ 120.000 0.275862
$$436$$ 254.558i 0.583850i
$$437$$ −190.000 −0.434783
$$438$$ 100.000 0.228311
$$439$$ − 806.102i − 1.83622i −0.396323 0.918111i $$-0.629714\pi$$
0.396323 0.918111i $$-0.370286\pi$$
$$440$$ − 14.1421i − 0.0321412i
$$441$$ −24.0000 −0.0544218
$$442$$ −600.000 −1.35747
$$443$$ 365.000 0.823928 0.411964 0.911200i $$-0.364843\pi$$
0.411964 + 0.911200i $$0.364843\pi$$
$$444$$ −144.000 −0.324324
$$445$$ − 127.279i − 0.286021i
$$446$$ 516.000 1.15695
$$447$$ − 608.112i − 1.36043i
$$448$$ −40.0000 −0.0892857
$$449$$ 763.675i 1.70084i 0.526108 + 0.850418i $$0.323651\pi$$
−0.526108 + 0.850418i $$0.676349\pi$$
$$450$$ 33.9411i 0.0754247i
$$451$$ 212.132i 0.470359i
$$452$$ 220.617i 0.488091i
$$453$$ 240.000 0.529801
$$454$$ −96.0000 −0.211454
$$455$$ − 84.8528i − 0.186490i
$$456$$ 152.000 0.333333
$$457$$ −265.000 −0.579869 −0.289934 0.957047i $$-0.593633\pi$$
−0.289934 + 0.957047i $$0.593633\pi$$
$$458$$ 205.061i 0.447731i
$$459$$ 707.107i 1.54054i
$$460$$ −20.0000 −0.0434783
$$461$$ −553.000 −1.19957 −0.599783 0.800163i $$-0.704746\pi$$
−0.599783 + 0.800163i $$0.704746\pi$$
$$462$$ −100.000 −0.216450
$$463$$ 485.000 1.04752 0.523758 0.851867i $$-0.324530\pi$$
0.523758 + 0.851867i $$0.324530\pi$$
$$464$$ − 169.706i − 0.365745i
$$465$$ −120.000 −0.258065
$$466$$ − 473.762i − 1.01666i
$$467$$ −115.000 −0.246253 −0.123126 0.992391i $$-0.539292\pi$$
−0.123126 + 0.992391i $$0.539292\pi$$
$$468$$ − 33.9411i − 0.0725238i
$$469$$ − 551.543i − 1.17600i
$$470$$ 7.07107i 0.0150448i
$$471$$ 537.401i 1.14098i
$$472$$ 240.000 0.508475
$$473$$ 25.0000 0.0528541
$$474$$ 169.706i 0.358029i
$$475$$ −456.000 −0.960000
$$476$$ 250.000 0.525210
$$477$$ 25.4558i 0.0533665i
$$478$$ − 278.600i − 0.582845i
$$479$$ −490.000 −1.02296 −0.511482 0.859294i $$-0.670903\pi$$
−0.511482 + 0.859294i $$0.670903\pi$$
$$480$$ 16.0000 0.0333333
$$481$$ −432.000 −0.898129
$$482$$ −420.000 −0.871369
$$483$$ 141.421i 0.292798i
$$484$$ 192.000 0.396694
$$485$$ 16.9706i 0.0349909i
$$486$$ −76.0000 −0.156379
$$487$$ 610.940i 1.25450i 0.778819 + 0.627249i $$0.215819\pi$$
−0.778819 + 0.627249i $$0.784181\pi$$
$$488$$ 268.701i 0.550616i
$$489$$ − 311.127i − 0.636252i
$$490$$ − 33.9411i − 0.0692676i
$$491$$ −82.0000 −0.167006 −0.0835031 0.996508i $$-0.526611\pi$$
−0.0835031 + 0.996508i $$0.526611\pi$$
$$492$$ −240.000 −0.487805
$$493$$ 1060.66i 2.15144i
$$494$$ 456.000 0.923077
$$495$$ −5.00000 −0.0101010
$$496$$ 169.706i 0.342148i
$$497$$ 0 0
$$498$$ 520.000 1.04418
$$499$$ 485.000 0.971944 0.485972 0.873974i $$-0.338466\pi$$
0.485972 + 0.873974i $$0.338466\pi$$
$$500$$ −98.0000 −0.196000
$$501$$ −168.000 −0.335329
$$502$$ − 244.659i − 0.487368i
$$503$$ −250.000 −0.497018 −0.248509 0.968630i $$-0.579941\pi$$
−0.248509 + 0.968630i $$0.579941\pi$$
$$504$$ 14.1421i 0.0280598i
$$505$$ −50.0000 −0.0990099
$$506$$ 70.7107i 0.139744i
$$507$$ 336.583i 0.663871i
$$508$$ 458.205i 0.901979i
$$509$$ 169.706i 0.333410i 0.986007 + 0.166705i $$0.0533127\pi$$
−0.986007 + 0.166705i $$0.946687\pi$$
$$510$$ −100.000 −0.196078
$$511$$ −125.000 −0.244618
$$512$$ − 22.6274i − 0.0441942i
$$513$$ − 537.401i − 1.04757i
$$514$$ 96.0000 0.186770
$$515$$ − 16.9706i − 0.0329525i
$$516$$ 28.2843i 0.0548145i
$$517$$ 25.0000 0.0483559
$$518$$ 180.000 0.347490
$$519$$ 528.000 1.01734
$$520$$ 48.0000 0.0923077
$$521$$ 127.279i 0.244298i 0.992512 + 0.122149i $$0.0389786\pi$$
−0.992512 + 0.122149i $$0.961021\pi$$
$$522$$ −60.0000 −0.114943
$$523$$ − 356.382i − 0.681418i −0.940169 0.340709i $$-0.889333\pi$$
0.940169 0.340709i $$-0.110667\pi$$
$$524$$ 326.000 0.622137
$$525$$ 339.411i 0.646498i
$$526$$ 502.046i 0.954460i
$$527$$ − 1060.66i − 2.01264i
$$528$$ − 56.5685i − 0.107137i
$$529$$ −429.000 −0.810964
$$530$$ −36.0000 −0.0679245
$$531$$ − 84.8528i − 0.159798i
$$532$$ −190.000 −0.357143
$$533$$ −720.000 −1.35084
$$534$$ − 509.117i − 0.953402i
$$535$$ − 101.823i − 0.190324i
$$536$$ 312.000 0.582090
$$537$$ 360.000 0.670391
$$538$$ 540.000 1.00372
$$539$$ −120.000 −0.222635
$$540$$ − 56.5685i − 0.104757i
$$541$$ −25.0000 −0.0462107 −0.0231054 0.999733i $$-0.507355\pi$$
−0.0231054 + 0.999733i $$0.507355\pi$$
$$542$$ − 155.563i − 0.287018i
$$543$$ 720.000 1.32597
$$544$$ 141.421i 0.259966i
$$545$$ 127.279i 0.233540i
$$546$$ − 339.411i − 0.621632i
$$547$$ − 16.9706i − 0.0310248i −0.999880 0.0155124i $$-0.995062\pi$$
0.999880 0.0155124i $$-0.00493795\pi$$
$$548$$ −190.000 −0.346715
$$549$$ 95.0000 0.173042
$$550$$ 169.706i 0.308556i
$$551$$ − 806.102i − 1.46298i
$$552$$ −80.0000 −0.144928
$$553$$ − 212.132i − 0.383602i
$$554$$ 374.767i 0.676474i
$$555$$ −72.0000 −0.129730
$$556$$ −250.000 −0.449640
$$557$$ −745.000 −1.33752 −0.668761 0.743477i $$-0.733175\pi$$
−0.668761 + 0.743477i $$0.733175\pi$$
$$558$$ 60.0000 0.107527
$$559$$ 84.8528i 0.151794i
$$560$$ −20.0000 −0.0357143
$$561$$ 353.553i 0.630220i
$$562$$ −600.000 −1.06762
$$563$$ − 313.955i − 0.557647i −0.960342 0.278824i $$-0.910056\pi$$
0.960342 0.278824i $$-0.0899445\pi$$
$$564$$ 28.2843i 0.0501494i
$$565$$ 110.309i 0.195237i
$$566$$ − 176.777i − 0.312326i
$$567$$ −355.000 −0.626102
$$568$$ 0 0
$$569$$ − 424.264i − 0.745631i −0.927905 0.372816i $$-0.878392\pi$$
0.927905 0.372816i $$-0.121608\pi$$
$$570$$ 76.0000 0.133333
$$571$$ 1070.00 1.87391 0.936953 0.349456i $$-0.113634\pi$$
0.936953 + 0.349456i $$0.113634\pi$$
$$572$$ − 169.706i − 0.296688i
$$573$$ − 828.729i − 1.44630i
$$574$$ 300.000 0.522648
$$575$$ 240.000 0.417391
$$576$$ −8.00000 −0.0138889
$$577$$ −25.0000 −0.0433276 −0.0216638 0.999765i $$-0.506896\pi$$
−0.0216638 + 0.999765i $$0.506896\pi$$
$$578$$ − 475.176i − 0.822103i
$$579$$ −168.000 −0.290155
$$580$$ − 84.8528i − 0.146298i
$$581$$ −650.000 −1.11876
$$582$$ 67.8823i 0.116636i
$$583$$ 127.279i 0.218318i
$$584$$ − 70.7107i − 0.121080i
$$585$$ − 16.9706i − 0.0290095i
$$586$$ −264.000 −0.450512
$$587$$ 725.000 1.23509 0.617547 0.786534i $$-0.288127\pi$$
0.617547 + 0.786534i $$0.288127\pi$$
$$588$$ − 135.765i − 0.230892i
$$589$$ 806.102i 1.36859i
$$590$$ 120.000 0.203390
$$591$$ 197.990i 0.335008i
$$592$$ 101.823i 0.171999i
$$593$$ 650.000 1.09612 0.548061 0.836439i $$-0.315366\pi$$
0.548061 + 0.836439i $$0.315366\pi$$
$$594$$ −200.000 −0.336700
$$595$$ 125.000 0.210084
$$596$$ −430.000 −0.721477
$$597$$ − 489.318i − 0.819628i
$$598$$ −240.000 −0.401338
$$599$$ 296.985i 0.495801i 0.968785 + 0.247901i $$0.0797406\pi$$
−0.968785 + 0.247901i $$0.920259\pi$$
$$600$$ −192.000 −0.320000
$$601$$ 848.528i 1.41186i 0.708281 + 0.705930i $$0.249471\pi$$
−0.708281 + 0.705930i $$0.750529\pi$$
$$602$$ − 35.3553i − 0.0587298i
$$603$$ − 110.309i − 0.182933i
$$604$$ − 169.706i − 0.280970i
$$605$$ 96.0000 0.158678
$$606$$ −200.000 −0.330033
$$607$$ 271.529i 0.447329i 0.974666 + 0.223665i $$0.0718021\pi$$
−0.974666 + 0.223665i $$0.928198\pi$$
$$608$$ − 107.480i − 0.176777i
$$609$$ −600.000 −0.985222
$$610$$ 134.350i 0.220246i
$$611$$ 84.8528i 0.138875i
$$612$$ 50.0000 0.0816993
$$613$$ 1055.00 1.72104 0.860522 0.509413i $$-0.170138\pi$$
0.860522 + 0.509413i $$0.170138\pi$$
$$614$$ −396.000 −0.644951
$$615$$ −120.000 −0.195122
$$616$$ 70.7107i 0.114790i
$$617$$ −505.000 −0.818476 −0.409238 0.912428i $$-0.634206\pi$$
−0.409238 + 0.912428i $$0.634206\pi$$
$$618$$ − 67.8823i − 0.109842i
$$619$$ −130.000 −0.210016 −0.105008 0.994471i $$-0.533487\pi$$
−0.105008 + 0.994471i $$0.533487\pi$$
$$620$$ 84.8528i 0.136859i
$$621$$ 282.843i 0.455463i
$$622$$ 332.340i 0.534309i
$$623$$ 636.396i 1.02150i
$$624$$ 192.000 0.307692
$$625$$ 551.000 0.881600
$$626$$ 438.406i 0.700329i
$$627$$ − 268.701i − 0.428550i
$$628$$ 380.000 0.605096
$$629$$ − 636.396i − 1.01176i
$$630$$ 7.07107i 0.0112239i
$$631$$ −475.000 −0.752773 −0.376387 0.926463i $$-0.622834\pi$$
−0.376387 + 0.926463i $$0.622834\pi$$
$$632$$ 120.000 0.189873
$$633$$ −240.000 −0.379147
$$634$$ 264.000 0.416404
$$635$$ 229.103i 0.360791i
$$636$$ −144.000 −0.226415
$$637$$ − 407.294i − 0.639393i
$$638$$ −300.000 −0.470219
$$639$$ 0 0
$$640$$ − 11.3137i − 0.0176777i
$$641$$ 848.528i 1.32376i 0.749611 + 0.661878i $$0.230240\pi$$
−0.749611 + 0.661878i $$0.769760\pi$$
$$642$$ − 407.294i − 0.634414i
$$643$$ −955.000 −1.48523 −0.742613 0.669721i $$-0.766414\pi$$
−0.742613 + 0.669721i $$0.766414\pi$$
$$644$$ 100.000 0.155280
$$645$$ 14.1421i 0.0219258i
$$646$$ 671.751i 1.03986i
$$647$$ 965.000 1.49150 0.745750 0.666226i $$-0.232092\pi$$
0.745750 + 0.666226i $$0.232092\pi$$
$$648$$ − 200.818i − 0.309905i
$$649$$ − 424.264i − 0.653720i
$$650$$ −576.000 −0.886154
$$651$$ 600.000 0.921659
$$652$$ −220.000 −0.337423
$$653$$ 935.000 1.43185 0.715926 0.698176i $$-0.246005\pi$$
0.715926 + 0.698176i $$0.246005\pi$$
$$654$$ 509.117i 0.778466i
$$655$$ 163.000 0.248855
$$656$$ 169.706i 0.258698i
$$657$$ −25.0000 −0.0380518
$$658$$ − 35.3553i − 0.0537315i
$$659$$ 84.8528i 0.128760i 0.997925 + 0.0643800i $$0.0205070\pi$$
−0.997925 + 0.0643800i $$0.979493\pi$$
$$660$$ − 28.2843i − 0.0428550i
$$661$$ 678.823i 1.02696i 0.858101 + 0.513481i $$0.171644\pi$$
−0.858101 + 0.513481i $$0.828356\pi$$
$$662$$ −420.000 −0.634441
$$663$$ −1200.00 −1.80995
$$664$$ − 367.696i − 0.553758i
$$665$$ −95.0000 −0.142857
$$666$$ 36.0000 0.0540541
$$667$$ 424.264i 0.636078i
$$668$$ 118.794i 0.177835i
$$669$$ 1032.00 1.54260
$$670$$ 156.000 0.232836
$$671$$ 475.000 0.707899
$$672$$ −80.0000 −0.119048
$$673$$ 186.676i 0.277379i 0.990336 + 0.138690i $$0.0442890\pi$$
−0.990336 + 0.138690i $$0.955711\pi$$
$$674$$ 744.000 1.10386
$$675$$ 678.823i 1.00566i
$$676$$ 238.000 0.352071
$$677$$ − 907.925i − 1.34110i −0.741864 0.670550i $$-0.766058\pi$$
0.741864 0.670550i $$-0.233942\pi$$
$$678$$ 441.235i 0.650789i
$$679$$ − 84.8528i − 0.124967i
$$680$$ 70.7107i 0.103986i
$$681$$ −192.000 −0.281938
$$682$$ 300.000 0.439883
$$683$$ − 1120.06i − 1.63991i −0.572429 0.819954i $$-0.693999\pi$$
0.572429 0.819954i $$-0.306001\pi$$
$$684$$ −38.0000 −0.0555556
$$685$$ −95.0000 −0.138686
$$686$$ 516.188i 0.752461i
$$687$$ 410.122i 0.596975i
$$688$$ 20.0000 0.0290698
$$689$$ −432.000 −0.626996
$$690$$ −40.0000 −0.0579710
$$691$$ −715.000 −1.03473 −0.517366 0.855764i $$-0.673087\pi$$
−0.517366 + 0.855764i $$0.673087\pi$$
$$692$$ − 373.352i − 0.539527i
$$693$$ 25.0000 0.0360750
$$694$$ − 176.777i − 0.254721i
$$695$$ −125.000 −0.179856
$$696$$ − 339.411i − 0.487660i
$$697$$ − 1060.66i − 1.52175i
$$698$$ − 32.5269i − 0.0466002i
$$699$$ − 947.523i − 1.35554i
$$700$$ 240.000 0.342857
$$701$$ −430.000 −0.613409 −0.306705 0.951805i $$-0.599226\pi$$
−0.306705 + 0.951805i $$0.599226\pi$$
$$702$$ − 678.823i − 0.966984i
$$703$$ 483.661i 0.687996i
$$704$$ −40.0000 −0.0568182
$$705$$ 14.1421i 0.0200598i
$$706$$ − 579.828i − 0.821285i
$$707$$ 250.000 0.353607
$$708$$ 480.000 0.677966
$$709$$ −382.000 −0.538787 −0.269394 0.963030i $$-0.586823\pi$$
−0.269394 + 0.963030i $$0.586823\pi$$
$$710$$ 0 0
$$711$$ − 42.4264i − 0.0596715i
$$712$$ −360.000 −0.505618
$$713$$ − 424.264i − 0.595041i
$$714$$ 500.000 0.700280
$$715$$ − 84.8528i − 0.118675i
$$716$$ − 254.558i − 0.355529i
$$717$$ − 557.200i − 0.777127i
$$718$$ 671.751i 0.935587i
$$719$$ −115.000 −0.159944 −0.0799722 0.996797i $$-0.525483\pi$$
−0.0799722 + 0.996797i $$0.525483\pi$$
$$720$$ −4.00000 −0.00555556
$$721$$ 84.8528i 0.117688i
$$722$$ − 510.531i − 0.707107i
$$723$$ −840.000 −1.16183
$$724$$ − 509.117i − 0.703200i
$$725$$ 1018.23i 1.40446i
$$726$$ 384.000 0.528926
$$727$$ −1075.00 −1.47868 −0.739340 0.673333i $$-0.764862\pi$$
−0.739340 + 0.673333i $$0.764862\pi$$
$$728$$ −240.000 −0.329670
$$729$$ −791.000 −1.08505
$$730$$ − 35.3553i − 0.0484320i
$$731$$ −125.000 −0.170999
$$732$$ 537.401i 0.734155i
$$733$$ 530.000 0.723056 0.361528 0.932361i $$-0.382255\pi$$
0.361528 + 0.932361i $$0.382255\pi$$
$$734$$ − 325.269i − 0.443146i
$$735$$ − 67.8823i − 0.0923568i
$$736$$ 56.5685i 0.0768594i
$$737$$ − 551.543i − 0.748363i
$$738$$ 60.0000 0.0813008
$$739$$ −547.000 −0.740189 −0.370095 0.928994i $$-0.620675\pi$$
−0.370095 + 0.928994i $$0.620675\pi$$
$$740$$ 50.9117i 0.0687996i
$$741$$ 912.000 1.23077
$$742$$ 180.000 0.242588
$$743$$ 958.837i 1.29049i 0.763974 + 0.645247i $$0.223245\pi$$
−0.763974 + 0.645247i $$0.776755\pi$$
$$744$$ 339.411i 0.456198i
$$745$$ −215.000 −0.288591
$$746$$ 96.0000 0.128686
$$747$$ −130.000 −0.174029
$$748$$ 250.000 0.334225
$$749$$ 509.117i 0.679729i
$$750$$ −196.000 −0.261333
$$751$$ 169.706i 0.225973i 0.993597 + 0.112986i $$0.0360417\pi$$
−0.993597 + 0.112986i $$0.963958\pi$$
$$752$$ 20.0000 0.0265957
$$753$$ − 489.318i − 0.649825i
$$754$$ − 1018.23i − 1.35044i
$$755$$ − 84.8528i − 0.112388i
$$756$$ 282.843i 0.374131i
$$757$$ 1055.00 1.39366 0.696830 0.717237i $$-0.254593\pi$$
0.696830 + 0.717237i $$0.254593\pi$$
$$758$$ 360.000 0.474934
$$759$$ 141.421i 0.186326i
$$760$$ − 53.7401i − 0.0707107i
$$761$$ 215.000 0.282523 0.141261 0.989972i $$-0.454884\pi$$
0.141261 + 0.989972i $$0.454884\pi$$
$$762$$ 916.410i 1.20264i
$$763$$ − 636.396i − 0.834071i
$$764$$ −586.000 −0.767016
$$765$$ 25.0000 0.0326797
$$766$$ 204.000 0.266319
$$767$$ 1440.00 1.87744
$$768$$ − 45.2548i − 0.0589256i
$$769$$ −145.000 −0.188557 −0.0942783 0.995546i $$-0.530054\pi$$
−0.0942783 + 0.995546i $$0.530054\pi$$
$$770$$ 35.3553i 0.0459160i
$$771$$ 192.000 0.249027
$$772$$ 118.794i 0.153878i
$$773$$ − 407.294i − 0.526900i −0.964673 0.263450i $$-0.915140\pi$$
0.964673 0.263450i $$-0.0848604\pi$$
$$774$$ − 7.07107i − 0.00913575i
$$775$$ − 1018.23i − 1.31385i
$$776$$ 48.0000 0.0618557
$$777$$ 360.000 0.463320
$$778$$ 782.060i 1.00522i
$$779$$ 806.102i 1.03479i
$$780$$ 96.0000 0.123077
$$781$$ 0 0
$$782$$ − 353.553i − 0.452114i
$$783$$ −1200.00 −1.53257
$$784$$ −96.0000 −0.122449
$$785$$ 190.000 0.242038
$$786$$ 652.000 0.829517
$$787$$ − 186.676i − 0.237200i −0.992942 0.118600i $$-0.962159\pi$$
0.992942 0.118600i $$-0.0378406\pi$$
$$788$$ 140.000 0.177665
$$789$$ 1004.09i 1.27261i
$$790$$ 60.0000 0.0759494
$$791$$ − 551.543i − 0.697273i
$$792$$ 14.1421i 0.0178562i
$$793$$ 1612.20i 2.03304i
$$794$$ − 473.762i − 0.596677i
$$795$$ −72.0000 −0.0905660
$$796$$ −346.000 −0.434673
$$797$$ 704.278i 0.883662i 0.897098 + 0.441831i $$0.145671\pi$$
−0.897098 + 0.441831i $$0.854329\pi$$
$$798$$ −380.000 −0.476190
$$799$$ −125.000 −0.156446
$$800$$ 135.765i 0.169706i
$$801$$ 127.279i 0.158900i
$$802$$ −300.000 −0.374065
$$803$$ −125.000 −0.155666
$$804$$ 624.000 0.776119
$$805$$ 50.0000 0.0621118
$$806$$ 1018.23i 1.26332i
$$807$$ 1080.00 1.33829
$$808$$ 141.421i 0.175026i
$$809$$ −457.000 −0.564895 −0.282447 0.959283i $$-0.591146\pi$$
−0.282447 + 0.959283i $$0.591146\pi$$
$$810$$ − 100.409i − 0.123962i
$$811$$ − 509.117i − 0.627764i −0.949462 0.313882i $$-0.898370\pi$$
0.949462 0.313882i $$-0.101630\pi$$
$$812$$ 424.264i 0.522493i
$$813$$ − 311.127i − 0.382690i
$$814$$ 180.000 0.221130
$$815$$ −110.000 −0.134969
$$816$$ 282.843i 0.346621i
$$817$$ 95.0000 0.116279
$$818$$ −1020.00 −1.24694
$$819$$ 84.8528i 0.103605i
$$820$$ 84.8528i 0.103479i
$$821$$ 167.000 0.203410 0.101705 0.994815i $$-0.467570\pi$$
0.101705 + 0.994815i $$0.467570\pi$$
$$822$$ −380.000 −0.462287
$$823$$ −1315.00 −1.59781 −0.798906 0.601455i $$-0.794588\pi$$
−0.798906 + 0.601455i $$0.794588\pi$$
$$824$$ −48.0000 −0.0582524
$$825$$ 339.411i 0.411408i
$$826$$ −600.000 −0.726392
$$827$$ 534.573i 0.646400i 0.946331 + 0.323200i $$0.104759\pi$$
−0.946331 + 0.323200i $$0.895241\pi$$
$$828$$ 20.0000 0.0241546
$$829$$ − 763.675i − 0.921201i −0.887608 0.460600i $$-0.847634\pi$$
0.887608 0.460600i $$-0.152366\pi$$
$$830$$ − 183.848i − 0.221503i
$$831$$ 749.533i 0.901965i
$$832$$ − 135.765i − 0.163178i
$$833$$ 600.000 0.720288
$$834$$ −500.000 −0.599520
$$835$$ 59.3970i 0.0711341i
$$836$$ −190.000 −0.227273
$$837$$ 1200.00 1.43369
$$838$$ − 87.6812i − 0.104632i
$$839$$ 339.411i 0.404543i 0.979330 + 0.202271i $$0.0648323\pi$$
−0.979330 + 0.202271i $$0.935168\pi$$
$$840$$ −40.0000 −0.0476190
$$841$$ −959.000 −1.14031
$$842$$ −420.000 −0.498812
$$843$$ −1200.00 −1.42349
$$844$$ 169.706i 0.201073i
$$845$$ 119.000 0.140828
$$846$$ − 7.07107i − 0.00835824i
$$847$$ −480.000 −0.566706
$$848$$ 101.823i 0.120075i
$$849$$ − 353.553i − 0.416435i
$$850$$ − 848.528i − 0.998268i
$$851$$ − 254.558i − 0.299129i
$$852$$ 0 0
$$853$$ 770.000 0.902696 0.451348 0.892348i $$-0.350943\pi$$
0.451348 + 0.892348i $$0.350943\pi$$
$$854$$ − 671.751i − 0.786594i
$$855$$ −19.0000 −0.0222222
$$856$$ −288.000 −0.336449
$$857$$ 1255.82i 1.46537i 0.680568 + 0.732685i $$0.261733\pi$$
−0.680568 + 0.732685i $$0.738267\pi$$
$$858$$ − 339.411i − 0.395584i
$$859$$ 557.000 0.648428 0.324214 0.945984i $$-0.394900\pi$$
0.324214 + 0.945984i $$0.394900\pi$$
$$860$$ 10.0000 0.0116279
$$861$$ 600.000 0.696864
$$862$$ 720.000 0.835267
$$863$$ − 992.778i − 1.15038i −0.818020 0.575190i $$-0.804928\pi$$
0.818020 0.575190i $$-0.195072\pi$$
$$864$$ −160.000 −0.185185
$$865$$ − 186.676i − 0.215811i
$$866$$ 324.000 0.374134
$$867$$ − 950.352i − 1.09614i
$$868$$ − 424.264i − 0.488783i
$$869$$ − 212.132i − 0.244111i
$$870$$ − 169.706i − 0.195064i
$$871$$ 1872.00 2.14925
$$872$$ 360.000 0.412844
$$873$$ − 16.9706i − 0.0194394i
$$874$$ 268.701i 0.307438i
$$875$$ 245.000 0.280000
$$876$$ − 141.421i − 0.161440i
$$877$$ − 186.676i − 0.212858i −0.994320 0.106429i $$-0.966058\pi$$
0.994320 0.106429i $$-0.0339416\pi$$
$$878$$ −1140.00 −1.29841
$$879$$ −528.000 −0.600683
$$880$$ −20.0000 −0.0227273
$$881$$ −25.0000 −0.0283768 −0.0141884 0.999899i $$-0.504516\pi$$
−0.0141884 + 0.999899i $$0.504516\pi$$
$$882$$ 33.9411i 0.0384820i
$$883$$ 965.000 1.09287 0.546433 0.837503i $$-0.315985\pi$$
0.546433 + 0.837503i $$0.315985\pi$$
$$884$$ 848.528i 0.959873i
$$885$$ 240.000 0.271186
$$886$$ − 516.188i − 0.582605i
$$887$$ 780.646i 0.880097i 0.897974 + 0.440048i $$0.145039\pi$$
−0.897974 + 0.440048i $$0.854961\pi$$
$$888$$ 203.647i 0.229332i
$$889$$ − 1145.51i − 1.28854i
$$890$$ −180.000 −0.202247
$$891$$ −355.000 −0.398429
$$892$$ − 729.734i − 0.818088i
$$893$$ 95.0000 0.106383
$$894$$ −860.000 −0.961969
$$895$$ − 127.279i − 0.142211i
$$896$$ 56.5685i 0.0631345i
$$897$$ −480.000 −0.535117
$$898$$ 1080.00 1.20267
$$899$$ 1800.00 2.00222
$$900$$ 48.0000 0.0533333
$$901$$ − 636.396i − 0.706322i
$$902$$ 300.000 0.332594
$$903$$ − 70.7107i − 0.0783064i
$$904$$ 312.000 0.345133
$$905$$ − 254.558i − 0.281280i
$$906$$ − 339.411i − 0.374626i
$$907$$ 313.955i 0.346147i 0.984909 + 0.173074i $$0.0553698\pi$$
−0.984909 + 0.173074i $$0.944630\pi$$
$$908$$ 135.765i 0.149520i
$$909$$ 50.0000 0.0550055
$$910$$ −120.000 −0.131868
$$911$$ − 933.381i − 1.02457i −0.858816 0.512284i $$-0.828800\pi$$
0.858816 0.512284i $$-0.171200\pi$$
$$912$$ − 214.960i − 0.235702i
$$913$$ −650.000 −0.711939
$$914$$ 374.767i 0.410029i
$$915$$ 268.701i 0.293662i
$$916$$ 290.000 0.316594
$$917$$ −815.000 −0.888768
$$918$$ 1000.00 1.08932
$$919$$ −538.000 −0.585419 −0.292709 0.956201i $$-0.594557\pi$$
−0.292709 + 0.956201i $$0.594557\pi$$
$$920$$ 28.2843i 0.0307438i
$$921$$ −792.000 −0.859935
$$922$$ 782.060i 0.848221i
$$923$$ 0 0
$$924$$ 141.421i 0.153053i
$$925$$ − 610.940i − 0.660476i
$$926$$ − 685.894i − 0.740706i
$$927$$ 16.9706i 0.0183070i
$$928$$ −240.000 −0.258621
$$929$$ −742.000 −0.798708 −0.399354 0.916797i $$-0.630766\pi$$
−0.399354 + 0.916797i $$0.630766\pi$$
$$930$$ 169.706i 0.182479i
$$931$$ −456.000 −0.489796
$$932$$ −670.000 −0.718884
$$933$$ 664.680i 0.712412i
$$934$$ 162.635i 0.174127i
$$935$$ 125.000 0.133690
$$936$$ −48.0000 −0.0512821
$$937$$ 335.000 0.357524 0.178762 0.983892i $$-0.442791\pi$$
0.178762 + 0.983892i $$0.442791\pi$$
$$938$$ −780.000 −0.831557
$$939$$ 876.812i 0.933773i
$$940$$ 10.0000 0.0106383
$$941$$ 424.264i 0.450865i 0.974259 + 0.225433i $$0.0723795\pi$$
−0.974259 + 0.225433i $$0.927620\pi$$
$$942$$ 760.000 0.806794
$$943$$ − 424.264i − 0.449909i
$$944$$ − 339.411i − 0.359546i
$$945$$ 141.421i 0.149652i
$$946$$ − 35.3553i − 0.0373735i
$$947$$ −1210.00 −1.27772 −0.638860 0.769323i $$-0.720594\pi$$
−0.638860 + 0.769323i $$0.720594\pi$$
$$948$$ 240.000 0.253165
$$949$$ − 424.264i − 0.447064i
$$950$$ 644.881i 0.678823i
$$951$$ 528.000 0.555205
$$952$$ − 353.553i − 0.371380i
$$953$$ − 992.778i − 1.04174i −0.853636 0.520870i $$-0.825608\pi$$
0.853636 0.520870i $$-0.174392\pi$$
$$954$$ 36.0000 0.0377358
$$955$$ −293.000 −0.306806
$$956$$ −394.000 −0.412134
$$957$$ −600.000 −0.626959
$$958$$ 692.965i 0.723345i
$$959$$ 475.000 0.495308
$$960$$ − 22.6274i − 0.0235702i
$$961$$ −839.000 −0.873049
$$962$$ 610.940i 0.635073i
$$963$$ 101.823i 0.105736i
$$964$$ 593.970i 0.616151i
$$965$$ 59.3970i 0.0615513i
$$966$$ 200.000 0.207039
$$967$$ 350.000 0.361944 0.180972 0.983488i $$-0.442076\pi$$
0.180972 + 0.983488i $$0.442076\pi$$
$$968$$ − 271.529i − 0.280505i
$$969$$ 1343.50i 1.38648i
$$970$$ 24.0000 0.0247423
$$971$$ − 254.558i − 0.262161i −0.991372 0.131081i $$-0.958155\pi$$
0.991372 0.131081i $$-0.0418447\pi$$
$$972$$ 107.480i 0.110576i
$$973$$ 625.000 0.642343
$$974$$ 864.000 0.887064
$$975$$ −1152.00 −1.18154
$$976$$ 380.000 0.389344
$$977$$ 398.808i 0.408197i 0.978950 + 0.204098i $$0.0654262\pi$$
−0.978950 + 0.204098i $$0.934574\pi$$
$$978$$ −440.000 −0.449898
$$979$$ 636.396i 0.650047i
$$980$$ −48.0000 −0.0489796
$$981$$ − 127.279i − 0.129744i
$$982$$ 115.966i 0.118091i
$$983$$ 695.793i 0.707826i 0.935278 + 0.353913i $$0.115149\pi$$
−0.935278 + 0.353913i $$0.884851\pi$$
$$984$$ 339.411i 0.344930i
$$985$$ 70.0000 0.0710660
$$986$$ 1500.00 1.52130
$$987$$ − 70.7107i − 0.0716420i
$$988$$ − 644.881i − 0.652714i
$$989$$ −50.0000 −0.0505561
$$990$$ 7.07107i 0.00714249i
$$991$$ − 381.838i − 0.385305i −0.981267 0.192653i $$-0.938291\pi$$
0.981267 0.192653i $$-0.0617091\pi$$
$$992$$ 240.000 0.241935
$$993$$ −840.000 −0.845921
$$994$$ 0 0
$$995$$ −173.000 −0.173869
$$996$$ − 735.391i − 0.738344i
$$997$$ −265.000 −0.265797 −0.132899 0.991130i $$-0.542428\pi$$
−0.132899 + 0.991130i $$0.542428\pi$$
$$998$$ − 685.894i − 0.687268i
$$999$$ 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 38.3.b.a.37.1 2
3.2 odd 2 342.3.d.a.37.2 2
4.3 odd 2 304.3.e.c.113.2 2
5.2 odd 4 950.3.d.a.949.4 4
5.3 odd 4 950.3.d.a.949.1 4
5.4 even 2 950.3.c.a.151.2 2
8.3 odd 2 1216.3.e.i.1025.1 2
8.5 even 2 1216.3.e.j.1025.2 2
12.11 even 2 2736.3.o.h.721.2 2
19.18 odd 2 inner 38.3.b.a.37.2 yes 2
57.56 even 2 342.3.d.a.37.1 2
76.75 even 2 304.3.e.c.113.1 2
95.18 even 4 950.3.d.a.949.3 4
95.37 even 4 950.3.d.a.949.2 4
95.94 odd 2 950.3.c.a.151.1 2
152.37 odd 2 1216.3.e.j.1025.1 2
152.75 even 2 1216.3.e.i.1025.2 2
228.227 odd 2 2736.3.o.h.721.1 2

By twisted newform
Twist Min Dim Char Parity Ord Type
38.3.b.a.37.1 2 1.1 even 1 trivial
38.3.b.a.37.2 yes 2 19.18 odd 2 inner
304.3.e.c.113.1 2 76.75 even 2
304.3.e.c.113.2 2 4.3 odd 2
342.3.d.a.37.1 2 57.56 even 2
342.3.d.a.37.2 2 3.2 odd 2
950.3.c.a.151.1 2 95.94 odd 2
950.3.c.a.151.2 2 5.4 even 2
950.3.d.a.949.1 4 5.3 odd 4
950.3.d.a.949.2 4 95.37 even 4
950.3.d.a.949.3 4 95.18 even 4
950.3.d.a.949.4 4 5.2 odd 4
1216.3.e.i.1025.1 2 8.3 odd 2
1216.3.e.i.1025.2 2 152.75 even 2
1216.3.e.j.1025.1 2 152.37 odd 2
1216.3.e.j.1025.2 2 8.5 even 2
2736.3.o.h.721.1 2 228.227 odd 2
2736.3.o.h.721.2 2 12.11 even 2