# Properties

 Label 950.3.d.a.949.2 Level $950$ Weight $3$ Character 950.949 Analytic conductor $25.886$ Analytic rank $0$ Dimension $4$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$950 = 2 \cdot 5^{2} \cdot 19$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 950.d (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$25.8856251142$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\zeta_{8})$$ Defining polynomial: $$x^{4} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2\cdot 5^{2}$$ Twist minimal: no (minimal twist has level 38) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## Embedding invariants

 Embedding label 949.2 Root $$-0.707107 - 0.707107i$$ of defining polynomial Character $$\chi$$ $$=$$ 950.949 Dual form 950.3.d.a.949.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q-1.41421 q^{2} +2.82843 q^{3} +2.00000 q^{4} -4.00000 q^{6} +5.00000i q^{7} -2.82843 q^{8} -1.00000 q^{9} +O(q^{10})$$ $$q-1.41421 q^{2} +2.82843 q^{3} +2.00000 q^{4} -4.00000 q^{6} +5.00000i q^{7} -2.82843 q^{8} -1.00000 q^{9} +5.00000 q^{11} +5.65685 q^{12} -16.9706 q^{13} -7.07107i q^{14} +4.00000 q^{16} -25.0000i q^{17} +1.41421 q^{18} -19.0000 q^{19} +14.1421i q^{21} -7.07107 q^{22} +10.0000i q^{23} -8.00000 q^{24} +24.0000 q^{26} -28.2843 q^{27} +10.0000i q^{28} -42.4264i q^{29} -42.4264i q^{31} -5.65685 q^{32} +14.1421 q^{33} +35.3553i q^{34} -2.00000 q^{36} +25.4558 q^{37} +26.8701 q^{38} -48.0000 q^{39} -42.4264i q^{41} -20.0000i q^{42} -5.00000i q^{43} +10.0000 q^{44} -14.1421i q^{46} +5.00000i q^{47} +11.3137 q^{48} +24.0000 q^{49} -70.7107i q^{51} -33.9411 q^{52} -25.4558 q^{53} +40.0000 q^{54} -14.1421i q^{56} -53.7401 q^{57} +60.0000i q^{58} -84.8528i q^{59} +95.0000 q^{61} +60.0000i q^{62} -5.00000i q^{63} +8.00000 q^{64} -20.0000 q^{66} -110.309 q^{67} -50.0000i q^{68} +28.2843i q^{69} +2.82843 q^{72} +25.0000i q^{73} -36.0000 q^{74} -38.0000 q^{76} +25.0000i q^{77} +67.8823 q^{78} -42.4264i q^{79} -71.0000 q^{81} +60.0000i q^{82} +130.000i q^{83} +28.2843i q^{84} +7.07107i q^{86} -120.000i q^{87} -14.1421 q^{88} +127.279i q^{89} -84.8528i q^{91} +20.0000i q^{92} -120.000i q^{93} -7.07107i q^{94} -16.0000 q^{96} -16.9706 q^{97} -33.9411 q^{98} -5.00000 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + 8q^{4} - 16q^{6} - 4q^{9} + O(q^{10})$$ $$4q + 8q^{4} - 16q^{6} - 4q^{9} + 20q^{11} + 16q^{16} - 76q^{19} - 32q^{24} + 96q^{26} - 8q^{36} - 192q^{39} + 40q^{44} + 96q^{49} + 160q^{54} + 380q^{61} + 32q^{64} - 80q^{66} - 144q^{74} - 152q^{76} - 284q^{81} - 64q^{96} - 20q^{99} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/950\mathbb{Z}\right)^\times$$.

 $$n$$ $$77$$ $$401$$ $$\chi(n)$$ $$-1$$ $$-1$$

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ −1.41421 −0.707107
$$3$$ 2.82843 0.942809 0.471405 0.881917i $$-0.343747\pi$$
0.471405 + 0.881917i $$0.343747\pi$$
$$4$$ 2.00000 0.500000
$$5$$ 0 0
$$6$$ −4.00000 −0.666667
$$7$$ 5.00000i 0.714286i 0.934050 + 0.357143i $$0.116249\pi$$
−0.934050 + 0.357143i $$0.883751\pi$$
$$8$$ −2.82843 −0.353553
$$9$$ −1.00000 −0.111111
$$10$$ 0 0
$$11$$ 5.00000 0.454545 0.227273 0.973831i $$-0.427019\pi$$
0.227273 + 0.973831i $$0.427019\pi$$
$$12$$ 5.65685 0.471405
$$13$$ −16.9706 −1.30543 −0.652714 0.757604i $$-0.726370\pi$$
−0.652714 + 0.757604i $$0.726370\pi$$
$$14$$ − 7.07107i − 0.505076i
$$15$$ 0 0
$$16$$ 4.00000 0.250000
$$17$$ − 25.0000i − 1.47059i −0.677748 0.735294i $$-0.737044\pi$$
0.677748 0.735294i $$-0.262956\pi$$
$$18$$ 1.41421 0.0785674
$$19$$ −19.0000 −1.00000
$$20$$ 0 0
$$21$$ 14.1421i 0.673435i
$$22$$ −7.07107 −0.321412
$$23$$ 10.0000i 0.434783i 0.976085 + 0.217391i $$0.0697548\pi$$
−0.976085 + 0.217391i $$0.930245\pi$$
$$24$$ −8.00000 −0.333333
$$25$$ 0 0
$$26$$ 24.0000 0.923077
$$27$$ −28.2843 −1.04757
$$28$$ 10.0000i 0.357143i
$$29$$ − 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 $$-0.760109\pi$$
0.729204 0.684297i $$-0.239891\pi$$
$$32$$ −5.65685 −0.176777
$$33$$ 14.1421 0.428550
$$34$$ 35.3553i 1.03986i
$$35$$ 0 0
$$36$$ −2.00000 −0.0555556
$$37$$ 25.4558 0.687996 0.343998 0.938970i $$-0.388219\pi$$
0.343998 + 0.938970i $$0.388219\pi$$
$$38$$ 26.8701 0.707107
$$39$$ −48.0000 −1.23077
$$40$$ 0 0
$$41$$ − 42.4264i − 1.03479i −0.855747 0.517395i $$-0.826902\pi$$
0.855747 0.517395i $$-0.173098\pi$$
$$42$$ − 20.0000i − 0.476190i
$$43$$ − 5.00000i − 0.116279i −0.998308 0.0581395i $$-0.981483\pi$$
0.998308 0.0581395i $$-0.0185168\pi$$
$$44$$ 10.0000 0.227273
$$45$$ 0 0
$$46$$ − 14.1421i − 0.307438i
$$47$$ 5.00000i 0.106383i 0.998584 + 0.0531915i $$0.0169394\pi$$
−0.998584 + 0.0531915i $$0.983061\pi$$
$$48$$ 11.3137 0.235702
$$49$$ 24.0000 0.489796
$$50$$ 0 0
$$51$$ − 70.7107i − 1.38648i
$$52$$ −33.9411 −0.652714
$$53$$ −25.4558 −0.480299 −0.240149 0.970736i $$-0.577196\pi$$
−0.240149 + 0.970736i $$0.577196\pi$$
$$54$$ 40.0000 0.740741
$$55$$ 0 0
$$56$$ − 14.1421i − 0.252538i
$$57$$ −53.7401 −0.942809
$$58$$ 60.0000i 1.03448i
$$59$$ − 84.8528i − 1.43818i −0.694915 0.719092i $$-0.744558\pi$$
0.694915 0.719092i $$-0.255442\pi$$
$$60$$ 0 0
$$61$$ 95.0000 1.55738 0.778689 0.627411i $$-0.215885\pi$$
0.778689 + 0.627411i $$0.215885\pi$$
$$62$$ 60.0000i 0.967742i
$$63$$ − 5.00000i − 0.0793651i
$$64$$ 8.00000 0.125000
$$65$$ 0 0
$$66$$ −20.0000 −0.303030
$$67$$ −110.309 −1.64640 −0.823199 0.567753i $$-0.807813\pi$$
−0.823199 + 0.567753i $$0.807813\pi$$
$$68$$ − 50.0000i − 0.735294i
$$69$$ 28.2843i 0.409917i
$$70$$ 0 0
$$71$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$72$$ 2.82843 0.0392837
$$73$$ 25.0000i 0.342466i 0.985231 + 0.171233i $$0.0547750\pi$$
−0.985231 + 0.171233i $$0.945225\pi$$
$$74$$ −36.0000 −0.486486
$$75$$ 0 0
$$76$$ −38.0000 −0.500000
$$77$$ 25.0000i 0.324675i
$$78$$ 67.8823 0.870285
$$79$$ − 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
$$82$$ 60.0000i 0.731707i
$$83$$ 130.000i 1.56627i 0.621855 + 0.783133i $$0.286379\pi$$
−0.621855 + 0.783133i $$0.713621\pi$$
$$84$$ 28.2843i 0.336718i
$$85$$ 0 0
$$86$$ 7.07107i 0.0822217i
$$87$$ − 120.000i − 1.37931i
$$88$$ −14.1421 −0.160706
$$89$$ 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
$$92$$ 20.0000i 0.217391i
$$93$$ − 120.000i − 1.29032i
$$94$$ − 7.07107i − 0.0752241i
$$95$$ 0 0
$$96$$ −16.0000 −0.166667
$$97$$ −16.9706 −0.174954 −0.0874771 0.996167i $$-0.527880\pi$$
−0.0874771 + 0.996167i $$0.527880\pi$$
$$98$$ −33.9411 −0.346338
$$99$$ −5.00000 −0.0505051
$$100$$ 0 0
$$101$$ 50.0000 0.495050 0.247525 0.968882i $$-0.420383\pi$$
0.247525 + 0.968882i $$0.420383\pi$$
$$102$$ 100.000i 0.980392i
$$103$$ −16.9706 −0.164763 −0.0823814 0.996601i $$-0.526253\pi$$
−0.0823814 + 0.996601i $$0.526253\pi$$
$$104$$ 48.0000 0.461538
$$105$$ 0 0
$$106$$ 36.0000 0.339623
$$107$$ 101.823 0.951620 0.475810 0.879548i $$-0.342155\pi$$
0.475810 + 0.879548i $$0.342155\pi$$
$$108$$ −56.5685 −0.523783
$$109$$ − 127.279i − 1.16770i −0.811862 0.583850i $$-0.801546\pi$$
0.811862 0.583850i $$-0.198454\pi$$
$$110$$ 0 0
$$111$$ 72.0000 0.648649
$$112$$ 20.0000i 0.178571i
$$113$$ 110.309 0.976183 0.488091 0.872793i $$-0.337693\pi$$
0.488091 + 0.872793i $$0.337693\pi$$
$$114$$ 76.0000 0.666667
$$115$$ 0 0
$$116$$ − 84.8528i − 0.731490i
$$117$$ 16.9706 0.145048
$$118$$ 120.000i 1.01695i
$$119$$ 125.000 1.05042
$$120$$ 0 0
$$121$$ −96.0000 −0.793388
$$122$$ −134.350 −1.10123
$$123$$ − 120.000i − 0.975610i
$$124$$ − 84.8528i − 0.684297i
$$125$$ 0 0
$$126$$ 7.07107i 0.0561196i
$$127$$ −229.103 −1.80396 −0.901979 0.431780i $$-0.857886\pi$$
−0.901979 + 0.431780i $$0.857886\pi$$
$$128$$ −11.3137 −0.0883883
$$129$$ − 14.1421i − 0.109629i
$$130$$ 0 0
$$131$$ −163.000 −1.24427 −0.622137 0.782908i $$-0.713735\pi$$
−0.622137 + 0.782908i $$0.713735\pi$$
$$132$$ 28.2843 0.214275
$$133$$ − 95.0000i − 0.714286i
$$134$$ 156.000 1.16418
$$135$$ 0 0
$$136$$ 70.7107i 0.519931i
$$137$$ 95.0000i 0.693431i 0.937970 + 0.346715i $$0.112703\pi$$
−0.937970 + 0.346715i $$0.887297\pi$$
$$138$$ − 40.0000i − 0.289855i
$$139$$ −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
$$143$$ −84.8528 −0.593376
$$144$$ −4.00000 −0.0277778
$$145$$ 0 0
$$146$$ − 35.3553i − 0.242160i
$$147$$ 67.8823 0.461784
$$148$$ 50.9117 0.343998
$$149$$ −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 $$-0.909344\pi$$
0.959717 0.280970i $$-0.0906560\pi$$
$$152$$ 53.7401 0.353553
$$153$$ 25.0000i 0.163399i
$$154$$ − 35.3553i − 0.229580i
$$155$$ 0 0
$$156$$ −96.0000 −0.615385
$$157$$ − 190.000i − 1.21019i −0.796153 0.605096i $$-0.793135\pi$$
0.796153 0.605096i $$-0.206865\pi$$
$$158$$ 60.0000i 0.379747i
$$159$$ −72.0000 −0.452830
$$160$$ 0 0
$$161$$ −50.0000 −0.310559
$$162$$ 100.409 0.619810
$$163$$ − 110.000i − 0.674847i −0.941353 0.337423i $$-0.890445\pi$$
0.941353 0.337423i $$-0.109555\pi$$
$$164$$ − 84.8528i − 0.517395i
$$165$$ 0 0
$$166$$ − 183.848i − 1.10752i
$$167$$ −59.3970 −0.355670 −0.177835 0.984060i $$-0.556909\pi$$
−0.177835 + 0.984060i $$0.556909\pi$$
$$168$$ − 40.0000i − 0.238095i
$$169$$ 119.000 0.704142
$$170$$ 0 0
$$171$$ 19.0000 0.111111
$$172$$ − 10.0000i − 0.0581395i
$$173$$ −186.676 −1.07905 −0.539527 0.841969i $$-0.681397\pi$$
−0.539527 + 0.841969i $$0.681397\pi$$
$$174$$ 169.706i 0.975320i
$$175$$ 0 0
$$176$$ 20.0000 0.113636
$$177$$ − 240.000i − 1.35593i
$$178$$ − 180.000i − 1.01124i
$$179$$ 127.279i 0.711057i 0.934665 + 0.355529i $$0.115699\pi$$
−0.934665 + 0.355529i $$0.884301\pi$$
$$180$$ 0 0
$$181$$ − 254.558i − 1.40640i −0.710992 0.703200i $$-0.751754\pi$$
0.710992 0.703200i $$-0.248246\pi$$
$$182$$ 120.000i 0.659341i
$$183$$ 268.701 1.46831
$$184$$ − 28.2843i − 0.153719i
$$185$$ 0 0
$$186$$ 169.706i 0.912396i
$$187$$ − 125.000i − 0.668449i
$$188$$ 10.0000i 0.0531915i
$$189$$ − 141.421i − 0.748261i
$$190$$ 0 0
$$191$$ 293.000 1.53403 0.767016 0.641628i $$-0.221741\pi$$
0.767016 + 0.641628i $$0.221741\pi$$
$$192$$ 22.6274 0.117851
$$193$$ 59.3970 0.307756 0.153878 0.988090i $$-0.450824\pi$$
0.153878 + 0.988090i $$0.450824\pi$$
$$194$$ 24.0000 0.123711
$$195$$ 0 0
$$196$$ 48.0000 0.244898
$$197$$ − 70.0000i − 0.355330i −0.984091 0.177665i $$-0.943146\pi$$
0.984091 0.177665i $$-0.0568543\pi$$
$$198$$ 7.07107 0.0357125
$$199$$ −173.000 −0.869347 −0.434673 0.900588i $$-0.643136\pi$$
−0.434673 + 0.900588i $$0.643136\pi$$
$$200$$ 0 0
$$201$$ −312.000 −1.55224
$$202$$ −70.7107 −0.350053
$$203$$ 212.132 1.04499
$$204$$ − 141.421i − 0.693242i
$$205$$ 0 0
$$206$$ 24.0000 0.116505
$$207$$ − 10.0000i − 0.0483092i
$$208$$ −67.8823 −0.326357
$$209$$ −95.0000 −0.454545
$$210$$ 0 0
$$211$$ 84.8528i 0.402146i 0.979576 + 0.201073i $$0.0644429\pi$$
−0.979576 + 0.201073i $$0.935557\pi$$
$$212$$ −50.9117 −0.240149
$$213$$ 0 0
$$214$$ −144.000 −0.672897
$$215$$ 0 0
$$216$$ 80.0000 0.370370
$$217$$ 212.132 0.977567
$$218$$ 180.000i 0.825688i
$$219$$ 70.7107i 0.322880i
$$220$$ 0 0
$$221$$ 424.264i 1.91975i
$$222$$ −101.823 −0.458664
$$223$$ −364.867 −1.63618 −0.818088 0.575094i $$-0.804966\pi$$
−0.818088 + 0.575094i $$0.804966\pi$$
$$224$$ − 28.2843i − 0.126269i
$$225$$ 0 0
$$226$$ −156.000 −0.690265
$$227$$ −67.8823 −0.299041 −0.149520 0.988759i $$-0.547773\pi$$
−0.149520 + 0.988759i $$0.547773\pi$$
$$228$$ −107.480 −0.471405
$$229$$ 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
$$232$$ 120.000i 0.517241i
$$233$$ − 335.000i − 1.43777i −0.695130 0.718884i $$-0.744653\pi$$
0.695130 0.718884i $$-0.255347\pi$$
$$234$$ −24.0000 −0.102564
$$235$$ 0 0
$$236$$ − 169.706i − 0.719092i
$$237$$ − 120.000i − 0.506329i
$$238$$ −176.777 −0.742759
$$239$$ −197.000 −0.824268 −0.412134 0.911123i $$-0.635216\pi$$
−0.412134 + 0.911123i $$0.635216\pi$$
$$240$$ 0 0
$$241$$ 296.985i 1.23230i 0.787628 + 0.616151i $$0.211309\pi$$
−0.787628 + 0.616151i $$0.788691\pi$$
$$242$$ 135.765 0.561010
$$243$$ 53.7401 0.221153
$$244$$ 190.000 0.778689
$$245$$ 0 0
$$246$$ 169.706i 0.689860i
$$247$$ 322.441 1.30543
$$248$$ 120.000i 0.483871i
$$249$$ 367.696i 1.47669i
$$250$$ 0 0
$$251$$ 173.000 0.689243 0.344622 0.938742i $$-0.388007\pi$$
0.344622 + 0.938742i $$0.388007\pi$$
$$252$$ − 10.0000i − 0.0396825i
$$253$$ 50.0000i 0.197628i
$$254$$ 324.000 1.27559
$$255$$ 0 0
$$256$$ 16.0000 0.0625000
$$257$$ 67.8823 0.264133 0.132067 0.991241i $$-0.457839\pi$$
0.132067 + 0.991241i $$0.457839\pi$$
$$258$$ 20.0000i 0.0775194i
$$259$$ 127.279i 0.491426i
$$260$$ 0 0
$$261$$ 42.4264i 0.162553i
$$262$$ 230.517 0.879835
$$263$$ 355.000i 1.34981i 0.737905 + 0.674905i $$0.235815\pi$$
−0.737905 + 0.674905i $$0.764185\pi$$
$$264$$ −40.0000 −0.151515
$$265$$ 0 0
$$266$$ 134.350i 0.505076i
$$267$$ 360.000i 1.34831i
$$268$$ −220.617 −0.823199
$$269$$ 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 $$-0.434946\pi$$
0.202952 + 0.979189i $$0.434946\pi$$
$$272$$ − 100.000i − 0.367647i
$$273$$ − 240.000i − 0.879121i
$$274$$ − 134.350i − 0.490330i
$$275$$ 0 0
$$276$$ 56.5685i 0.204958i
$$277$$ − 265.000i − 0.956679i −0.878175 0.478339i $$-0.841239\pi$$
0.878175 0.478339i $$-0.158761\pi$$
$$278$$ 176.777 0.635887
$$279$$ 42.4264i 0.152066i
$$280$$ 0 0
$$281$$ 424.264i 1.50984i 0.655819 + 0.754918i $$0.272323\pi$$
−0.655819 + 0.754918i $$0.727677\pi$$
$$282$$ − 20.0000i − 0.0709220i
$$283$$ − 125.000i − 0.441696i −0.975308 0.220848i $$-0.929118\pi$$
0.975308 0.220848i $$-0.0708825\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ 120.000 0.419580
$$287$$ 212.132 0.739136
$$288$$ 5.65685 0.0196419
$$289$$ −336.000 −1.16263
$$290$$ 0 0
$$291$$ −48.0000 −0.164948
$$292$$ 50.0000i 0.171233i
$$293$$ 186.676 0.637120 0.318560 0.947903i $$-0.396801\pi$$
0.318560 + 0.947903i $$0.396801\pi$$
$$294$$ −96.0000 −0.326531
$$295$$ 0 0
$$296$$ −72.0000 −0.243243
$$297$$ −141.421 −0.476166
$$298$$ 304.056 1.02032
$$299$$ − 169.706i − 0.567577i
$$300$$ 0 0
$$301$$ 25.0000 0.0830565
$$302$$ 120.000i 0.397351i
$$303$$ 141.421 0.466737
$$304$$ −76.0000 −0.250000
$$305$$ 0 0
$$306$$ − 35.3553i − 0.115540i
$$307$$ −280.014 −0.912099 −0.456049 0.889955i $$-0.650736\pi$$
−0.456049 + 0.889955i $$0.650736\pi$$
$$308$$ 50.0000i 0.162338i
$$309$$ −48.0000 −0.155340
$$310$$ 0 0
$$311$$ −235.000 −0.755627 −0.377814 0.925882i $$-0.623324\pi$$
−0.377814 + 0.925882i $$0.623324\pi$$
$$312$$ 135.765 0.435143
$$313$$ 310.000i 0.990415i 0.868775 + 0.495208i $$0.164908\pi$$
−0.868775 + 0.495208i $$0.835092\pi$$
$$314$$ 268.701i 0.855734i
$$315$$ 0 0
$$316$$ − 84.8528i − 0.268522i
$$317$$ 186.676 0.588884 0.294442 0.955669i $$-0.404866\pi$$
0.294442 + 0.955669i $$0.404866\pi$$
$$318$$ 101.823 0.320199
$$319$$ − 212.132i − 0.664991i
$$320$$ 0 0
$$321$$ 288.000 0.897196
$$322$$ 70.7107 0.219598
$$323$$ 475.000i 1.47059i
$$324$$ −142.000 −0.438272
$$325$$ 0 0
$$326$$ 155.563i 0.477189i
$$327$$ − 360.000i − 1.10092i
$$328$$ 120.000i 0.365854i
$$329$$ −25.0000 −0.0759878
$$330$$ 0 0
$$331$$ 296.985i 0.897235i 0.893724 + 0.448618i $$0.148083\pi$$
−0.893724 + 0.448618i $$0.851917\pi$$
$$332$$ 260.000i 0.783133i
$$333$$ −25.4558 −0.0764440
$$334$$ 84.0000 0.251497
$$335$$ 0 0
$$336$$ 56.5685i 0.168359i
$$337$$ 526.087 1.56109 0.780545 0.625099i $$-0.214942\pi$$
0.780545 + 0.625099i $$0.214942\pi$$
$$338$$ −168.291 −0.497904
$$339$$ 312.000 0.920354
$$340$$ 0 0
$$341$$ − 212.132i − 0.622088i
$$342$$ −26.8701 −0.0785674
$$343$$ 365.000i 1.06414i
$$344$$ 14.1421i 0.0411109i
$$345$$ 0 0
$$346$$ 264.000 0.763006
$$347$$ 125.000i 0.360231i 0.983646 + 0.180115i $$0.0576471\pi$$
−0.983646 + 0.180115i $$0.942353\pi$$
$$348$$ − 240.000i − 0.689655i
$$349$$ −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
$$352$$ −28.2843 −0.0803530
$$353$$ − 410.000i − 1.16147i −0.814092 0.580737i $$-0.802765\pi$$
0.814092 0.580737i $$-0.197235\pi$$
$$354$$ 339.411i 0.958789i
$$355$$ 0 0
$$356$$ 254.558i 0.715052i
$$357$$ 353.553 0.990346
$$358$$ − 180.000i − 0.502793i
$$359$$ 475.000 1.32312 0.661560 0.749892i $$-0.269895\pi$$
0.661560 + 0.749892i $$0.269895\pi$$
$$360$$ 0 0
$$361$$ 361.000 1.00000
$$362$$ 360.000i 0.994475i
$$363$$ −271.529 −0.748014
$$364$$ − 169.706i − 0.466224i
$$365$$ 0 0
$$366$$ −380.000 −1.03825
$$367$$ 230.000i 0.626703i 0.949637 + 0.313351i $$0.101452\pi$$
−0.949637 + 0.313351i $$0.898548\pi$$
$$368$$ 40.0000i 0.108696i
$$369$$ 42.4264i 0.114977i
$$370$$ 0 0
$$371$$ − 127.279i − 0.343071i
$$372$$ − 240.000i − 0.645161i
$$373$$ −67.8823 −0.181990 −0.0909950 0.995851i $$-0.529005\pi$$
−0.0909950 + 0.995851i $$0.529005\pi$$
$$374$$ 176.777i 0.472665i
$$375$$ 0 0
$$376$$ − 14.1421i − 0.0376121i
$$377$$ 720.000i 1.90981i
$$378$$ 200.000i 0.529101i
$$379$$ 254.558i 0.671658i 0.941923 + 0.335829i $$0.109016\pi$$
−0.941923 + 0.335829i $$0.890984\pi$$
$$380$$ 0 0
$$381$$ −648.000 −1.70079
$$382$$ −414.365 −1.08472
$$383$$ −144.250 −0.376631 −0.188316 0.982109i $$-0.560303\pi$$
−0.188316 + 0.982109i $$0.560303\pi$$
$$384$$ −32.0000 −0.0833333
$$385$$ 0 0
$$386$$ −84.0000 −0.217617
$$387$$ 5.00000i 0.0129199i
$$388$$ −33.9411 −0.0874771
$$389$$ 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
$$392$$ −67.8823 −0.173169
$$393$$ −461.034 −1.17311
$$394$$ 98.9949i 0.251256i
$$395$$ 0 0
$$396$$ −10.0000 −0.0252525
$$397$$ 335.000i 0.843829i 0.906636 + 0.421914i $$0.138642\pi$$
−0.906636 + 0.421914i $$0.861358\pi$$
$$398$$ 244.659 0.614721
$$399$$ − 268.701i − 0.673435i
$$400$$ 0 0
$$401$$ 212.132i 0.529008i 0.964385 + 0.264504i $$0.0852082\pi$$
−0.964385 + 0.264504i $$0.914792\pi$$
$$402$$ 441.235 1.09760
$$403$$ 720.000i 1.78660i
$$404$$ 100.000 0.247525
$$405$$ 0 0
$$406$$ −300.000 −0.738916
$$407$$ 127.279 0.312725
$$408$$ 200.000i 0.490196i
$$409$$ − 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
$$412$$ −33.9411 −0.0823814
$$413$$ 424.264 1.02727
$$414$$ 14.1421i 0.0341597i
$$415$$ 0 0
$$416$$ 96.0000 0.230769
$$417$$ −353.553 −0.847850
$$418$$ 134.350 0.321412
$$419$$ −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 $$0.885259\pi$$
$$422$$ − 120.000i − 0.284360i
$$423$$ − 5.00000i − 0.0118203i
$$424$$ 72.0000 0.169811
$$425$$ 0 0
$$426$$ 0 0
$$427$$ 475.000i 1.11241i
$$428$$ 203.647 0.475810
$$429$$ −240.000 −0.559441
$$430$$ 0 0
$$431$$ − 509.117i − 1.18125i −0.806948 0.590623i $$-0.798882\pi$$
0.806948 0.590623i $$-0.201118\pi$$
$$432$$ −113.137 −0.261891
$$433$$ −229.103 −0.529105 −0.264553 0.964371i $$-0.585224\pi$$
−0.264553 + 0.964371i $$0.585224\pi$$
$$434$$ −300.000 −0.691244
$$435$$ 0 0
$$436$$ − 254.558i − 0.583850i
$$437$$ − 190.000i − 0.434783i
$$438$$ − 100.000i − 0.228311i
$$439$$ − 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
$$442$$ − 600.000i − 1.35747i
$$443$$ − 365.000i − 0.823928i −0.911200 0.411964i $$-0.864843\pi$$
0.911200 0.411964i $$-0.135157\pi$$
$$444$$ 144.000 0.324324
$$445$$ 0 0
$$446$$ 516.000 1.15695
$$447$$ −608.112 −1.36043
$$448$$ 40.0000i 0.0892857i
$$449$$ 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
$$452$$ 220.617 0.488091
$$453$$ − 240.000i − 0.529801i
$$454$$ 96.0000 0.211454
$$455$$ 0 0
$$456$$ 152.000 0.333333
$$457$$ − 265.000i − 0.579869i −0.957047 0.289934i $$-0.906367\pi$$
0.957047 0.289934i $$-0.0936335\pi$$
$$458$$ −205.061 −0.447731
$$459$$ 707.107i 1.54054i
$$460$$ 0 0
$$461$$ −553.000 −1.19957 −0.599783 0.800163i $$-0.704746\pi$$
−0.599783 + 0.800163i $$0.704746\pi$$
$$462$$ − 100.000i − 0.216450i
$$463$$ − 485.000i − 1.04752i −0.851867 0.523758i $$-0.824530\pi$$
0.851867 0.523758i $$-0.175470\pi$$
$$464$$ − 169.706i − 0.365745i
$$465$$ 0 0
$$466$$ 473.762i 1.01666i
$$467$$ − 115.000i − 0.246253i −0.992391 0.123126i $$-0.960708\pi$$
0.992391 0.123126i $$-0.0392920\pi$$
$$468$$ 33.9411 0.0725238
$$469$$ − 551.543i − 1.17600i
$$470$$ 0 0
$$471$$ − 537.401i − 1.14098i
$$472$$ 240.000i 0.508475i
$$473$$ − 25.0000i − 0.0528541i
$$474$$ 169.706i 0.358029i
$$475$$ 0 0
$$476$$ 250.000 0.525210
$$477$$ 25.4558 0.0533665
$$478$$ 278.600 0.582845
$$479$$ 490.000 1.02296 0.511482 0.859294i $$-0.329097\pi$$
0.511482 + 0.859294i $$0.329097\pi$$
$$480$$ 0 0
$$481$$ −432.000 −0.898129
$$482$$ − 420.000i − 0.871369i
$$483$$ −141.421 −0.292798
$$484$$ −192.000 −0.396694
$$485$$ 0 0
$$486$$ −76.0000 −0.156379
$$487$$ 610.940 1.25450 0.627249 0.778819i $$-0.284181\pi$$
0.627249 + 0.778819i $$0.284181\pi$$
$$488$$ −268.701 −0.550616
$$489$$ − 311.127i − 0.636252i
$$490$$ 0 0
$$491$$ −82.0000 −0.167006 −0.0835031 0.996508i $$-0.526611\pi$$
−0.0835031 + 0.996508i $$0.526611\pi$$
$$492$$ − 240.000i − 0.487805i
$$493$$ −1060.66 −2.15144
$$494$$ −456.000 −0.923077
$$495$$ 0 0
$$496$$ − 169.706i − 0.342148i
$$497$$ 0 0
$$498$$ − 520.000i − 1.04418i
$$499$$ −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
$$502$$ −244.659 −0.487368
$$503$$ 250.000i 0.497018i 0.968630 + 0.248509i $$0.0799405\pi$$
−0.968630 + 0.248509i $$0.920059\pi$$
$$504$$ 14.1421i 0.0280598i
$$505$$ 0 0
$$506$$ − 70.7107i − 0.139744i
$$507$$ 336.583 0.663871
$$508$$ −458.205 −0.901979
$$509$$ 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
$$512$$ −22.6274 −0.0441942
$$513$$ 537.401 1.04757
$$514$$ −96.0000 −0.186770
$$515$$ 0 0
$$516$$ − 28.2843i − 0.0548145i
$$517$$ 25.0000i 0.0483559i
$$518$$ − 180.000i − 0.347490i
$$519$$ −528.000 −1.01734
$$520$$ 0 0
$$521$$ − 127.279i − 0.244298i −0.992512 0.122149i $$-0.961021\pi$$
0.992512 0.122149i $$-0.0389786\pi$$
$$522$$ − 60.0000i − 0.114943i
$$523$$ 356.382 0.681418 0.340709 0.940169i $$-0.389333\pi$$
0.340709 + 0.940169i $$0.389333\pi$$
$$524$$ −326.000 −0.622137
$$525$$ 0 0
$$526$$ − 502.046i − 0.954460i
$$527$$ −1060.66 −2.01264
$$528$$ 56.5685 0.107137
$$529$$ 429.000 0.810964
$$530$$ 0 0
$$531$$ 84.8528i 0.159798i
$$532$$ − 190.000i − 0.357143i
$$533$$ 720.000i 1.35084i
$$534$$ − 509.117i − 0.953402i
$$535$$ 0 0
$$536$$ 312.000 0.582090
$$537$$ 360.000i 0.670391i
$$538$$ − 540.000i − 1.00372i
$$539$$ 120.000 0.222635
$$540$$ 0 0
$$541$$ −25.0000 −0.0462107 −0.0231054 0.999733i $$-0.507355\pi$$
−0.0231054 + 0.999733i $$0.507355\pi$$
$$542$$ −155.563 −0.287018
$$543$$ − 720.000i − 1.32597i
$$544$$ 141.421i 0.259966i
$$545$$ 0 0
$$546$$ 339.411i 0.621632i
$$547$$ −16.9706 −0.0310248 −0.0155124 0.999880i $$-0.504938\pi$$
−0.0155124 + 0.999880i $$0.504938\pi$$
$$548$$ 190.000i 0.346715i
$$549$$ −95.0000 −0.173042
$$550$$ 0 0
$$551$$ 806.102i 1.46298i
$$552$$ − 80.0000i − 0.144928i
$$553$$ 212.132 0.383602
$$554$$ 374.767i 0.676474i
$$555$$ 0 0
$$556$$ −250.000 −0.449640
$$557$$ − 745.000i − 1.33752i −0.743477 0.668761i $$-0.766825\pi$$
0.743477 0.668761i $$-0.233175\pi$$
$$558$$ − 60.0000i − 0.107527i
$$559$$ 84.8528i 0.151794i
$$560$$ 0 0
$$561$$ − 353.553i − 0.630220i
$$562$$ − 600.000i − 1.06762i
$$563$$ 313.955 0.557647 0.278824 0.960342i $$-0.410056\pi$$
0.278824 + 0.960342i $$0.410056\pi$$
$$564$$ 28.2843i 0.0501494i
$$565$$ 0 0
$$566$$ 176.777i 0.312326i
$$567$$ − 355.000i − 0.626102i
$$568$$ 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 $$-0.113634\pi$$
0.936953 + 0.349456i $$0.113634\pi$$
$$572$$ −169.706 −0.296688
$$573$$ 828.729 1.44630
$$574$$ −300.000 −0.522648
$$575$$ 0 0
$$576$$ −8.00000 −0.0138889
$$577$$ − 25.0000i − 0.0433276i −0.999765 0.0216638i $$-0.993104\pi$$
0.999765 0.0216638i $$-0.00689633\pi$$
$$578$$ 475.176 0.822103
$$579$$ 168.000 0.290155
$$580$$ 0 0
$$581$$ −650.000 −1.11876
$$582$$ 67.8823 0.116636
$$583$$ −127.279 −0.218318
$$584$$ − 70.7107i − 0.121080i
$$585$$ 0 0
$$586$$ −264.000 −0.450512
$$587$$ 725.000i 1.23509i 0.786534 + 0.617547i $$0.211873\pi$$
−0.786534 + 0.617547i $$0.788127\pi$$
$$588$$ 135.765 0.230892
$$589$$ 806.102i 1.36859i
$$590$$ 0 0
$$591$$ − 197.990i − 0.335008i
$$592$$ 101.823 0.171999
$$593$$ − 650.000i − 1.09612i −0.836439 0.548061i $$-0.815366\pi$$
0.836439 0.548061i $$-0.184634\pi$$
$$594$$ 200.000 0.336700
$$595$$ 0 0
$$596$$ −430.000 −0.721477
$$597$$ −489.318 −0.819628
$$598$$ 240.000i 0.401338i
$$599$$ 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 $$-0.750529\pi$$
0.708281 0.705930i $$-0.249471\pi$$
$$602$$ −35.3553 −0.0587298
$$603$$ 110.309 0.182933
$$604$$ − 169.706i − 0.280970i
$$605$$ 0 0
$$606$$ −200.000 −0.330033
$$607$$ 271.529 0.447329 0.223665 0.974666i $$-0.428198\pi$$
0.223665 + 0.974666i $$0.428198\pi$$
$$608$$ 107.480 0.176777
$$609$$ 600.000 0.985222
$$610$$ 0 0
$$611$$ − 84.8528i − 0.138875i
$$612$$ 50.0000i 0.0816993i
$$613$$ − 1055.00i − 1.72104i −0.509413 0.860522i $$-0.670138\pi$$
0.509413 0.860522i $$-0.329862\pi$$
$$614$$ 396.000 0.644951
$$615$$ 0 0
$$616$$ − 70.7107i − 0.114790i
$$617$$ − 505.000i − 0.818476i −0.912428 0.409238i $$-0.865794\pi$$
0.912428 0.409238i $$-0.134206\pi$$
$$618$$ 67.8823 0.109842
$$619$$ 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
$$622$$ 332.340 0.534309
$$623$$ −636.396 −1.02150
$$624$$ −192.000 −0.307692
$$625$$ 0 0
$$626$$ − 438.406i − 0.700329i
$$627$$ −268.701 −0.428550
$$628$$ − 380.000i − 0.605096i
$$629$$ − 636.396i − 1.01176i
$$630$$ 0 0
$$631$$ −475.000 −0.752773 −0.376387 0.926463i $$-0.622834\pi$$
−0.376387 + 0.926463i $$0.622834\pi$$
$$632$$ 120.000i 0.189873i
$$633$$ 240.000i 0.379147i
$$634$$ −264.000 −0.416404
$$635$$ 0 0
$$636$$ −144.000 −0.226415
$$637$$ −407.294 −0.639393
$$638$$ 300.000i 0.470219i
$$639$$ 0 0
$$640$$ 0 0
$$641$$ − 848.528i − 1.32376i −0.749611 0.661878i $$-0.769760\pi$$
0.749611 0.661878i $$-0.230240\pi$$
$$642$$ −407.294 −0.634414
$$643$$ 955.000i 1.48523i 0.669721 + 0.742613i $$0.266414\pi$$
−0.669721 + 0.742613i $$0.733586\pi$$
$$644$$ −100.000 −0.155280
$$645$$ 0 0
$$646$$ − 671.751i − 1.03986i
$$647$$ 965.000i 1.49150i 0.666226 + 0.745750i $$0.267908\pi$$
−0.666226 + 0.745750i $$0.732092\pi$$
$$648$$ 200.818 0.309905
$$649$$ − 424.264i − 0.653720i
$$650$$ 0 0
$$651$$ 600.000 0.921659
$$652$$ − 220.000i − 0.337423i
$$653$$ − 935.000i − 1.43185i −0.698176 0.715926i $$-0.746005\pi$$
0.698176 0.715926i $$-0.253995\pi$$
$$654$$ 509.117i 0.778466i
$$655$$ 0 0
$$656$$ − 169.706i − 0.258698i
$$657$$ − 25.0000i − 0.0380518i
$$658$$ 35.3553 0.0537315
$$659$$ 84.8528i 0.128760i 0.997925 + 0.0643800i $$0.0205070\pi$$
−0.997925 + 0.0643800i $$0.979493\pi$$
$$660$$ 0 0
$$661$$ − 678.823i − 1.02696i −0.858101 0.513481i $$-0.828356\pi$$
0.858101 0.513481i $$-0.171644\pi$$
$$662$$ − 420.000i − 0.634441i
$$663$$ 1200.00i 1.80995i
$$664$$ − 367.696i − 0.553758i
$$665$$ 0 0
$$666$$ 36.0000 0.0540541
$$667$$ 424.264 0.636078
$$668$$ −118.794 −0.177835
$$669$$ −1032.00 −1.54260
$$670$$ 0 0
$$671$$ 475.000 0.707899
$$672$$ − 80.0000i − 0.119048i
$$673$$ −186.676 −0.277379 −0.138690 0.990336i $$-0.544289\pi$$
−0.138690 + 0.990336i $$0.544289\pi$$
$$674$$ −744.000 −1.10386
$$675$$ 0 0
$$676$$ 238.000 0.352071
$$677$$ −907.925 −1.34110 −0.670550 0.741864i $$-0.733942\pi$$
−0.670550 + 0.741864i $$0.733942\pi$$
$$678$$ −441.235 −0.650789
$$679$$ − 84.8528i − 0.124967i
$$680$$ 0 0
$$681$$ −192.000 −0.281938
$$682$$ 300.000i 0.439883i
$$683$$ 1120.06 1.63991 0.819954 0.572429i $$-0.193999\pi$$
0.819954 + 0.572429i $$0.193999\pi$$
$$684$$ 38.0000 0.0555556
$$685$$ 0 0
$$686$$ − 516.188i − 0.752461i
$$687$$ 410.122 0.596975
$$688$$ − 20.0000i − 0.0290698i
$$689$$ 432.000 0.626996
$$690$$ 0 0
$$691$$ −715.000 −1.03473 −0.517366 0.855764i $$-0.673087\pi$$
−0.517366 + 0.855764i $$0.673087\pi$$
$$692$$ −373.352 −0.539527
$$693$$ − 25.0000i − 0.0360750i
$$694$$ − 176.777i − 0.254721i
$$695$$ 0 0
$$696$$ 339.411i 0.487660i
$$697$$ −1060.66 −1.52175
$$698$$ 32.5269 0.0466002
$$699$$ − 947.523i − 1.35554i
$$700$$ 0 0
$$701$$ −430.000 −0.613409 −0.306705 0.951805i $$-0.599226\pi$$
−0.306705 + 0.951805i $$0.599226\pi$$
$$702$$ −678.823 −0.966984
$$703$$ −483.661 −0.687996
$$704$$ 40.0000 0.0568182
$$705$$ 0 0
$$706$$ 579.828i 0.821285i
$$707$$ 250.000i 0.353607i
$$708$$ − 480.000i − 0.677966i
$$709$$ 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
$$712$$ − 360.000i − 0.505618i
$$713$$ 424.264 0.595041
$$714$$ −500.000 −0.700280
$$715$$ 0 0
$$716$$ 254.558i 0.355529i
$$717$$ −557.200 −0.777127
$$718$$ −671.751 −0.935587
$$719$$ 115.000 0.159944 0.0799722 0.996797i $$-0.474517\pi$$
0.0799722 + 0.996797i $$0.474517\pi$$
$$720$$ 0 0
$$721$$ − 84.8528i − 0.117688i
$$722$$ −510.531 −0.707107
$$723$$ 840.000i 1.16183i
$$724$$ − 509.117i − 0.703200i
$$725$$ 0 0
$$726$$ 384.000 0.528926
$$727$$ − 1075.00i − 1.47868i −0.673333 0.739340i $$-0.735138\pi$$
0.673333 0.739340i $$-0.264862\pi$$
$$728$$ 240.000i 0.329670i
$$729$$ 791.000 1.08505
$$730$$ 0 0
$$731$$ −125.000 −0.170999
$$732$$ 537.401 0.734155
$$733$$ − 530.000i − 0.723056i −0.932361 0.361528i $$-0.882255\pi$$
0.932361 0.361528i $$-0.117745\pi$$
$$734$$ − 325.269i − 0.443146i
$$735$$ 0 0
$$736$$ − 56.5685i − 0.0768594i
$$737$$ −551.543 −0.748363
$$738$$ − 60.0000i − 0.0813008i
$$739$$ 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
$$742$$ 180.000i 0.242588i
$$743$$ −958.837 −1.29049 −0.645247 0.763974i $$-0.723245\pi$$
−0.645247 + 0.763974i $$0.723245\pi$$
$$744$$ 339.411i 0.456198i
$$745$$ 0 0
$$746$$ 96.0000 0.128686
$$747$$ − 130.000i − 0.174029i
$$748$$ − 250.000i − 0.334225i
$$749$$ 509.117i 0.679729i
$$750$$ 0 0
$$751$$ − 169.706i − 0.225973i −0.993597 0.112986i $$-0.963958\pi$$
0.993597 0.112986i $$-0.0360417\pi$$
$$752$$ 20.0000i 0.0265957i
$$753$$ 489.318 0.649825
$$754$$ − 1018.23i − 1.35044i
$$755$$ 0 0
$$756$$ − 282.843i − 0.374131i
$$757$$ 1055.00i 1.39366i 0.717237 + 0.696830i $$0.245407\pi$$
−0.717237 + 0.696830i $$0.754593\pi$$
$$758$$ − 360.000i − 0.474934i
$$759$$ 141.421i 0.186326i
$$760$$ 0 0
$$761$$ 215.000 0.282523 0.141261 0.989972i $$-0.454884\pi$$
0.141261 + 0.989972i $$0.454884\pi$$
$$762$$ 916.410 1.20264
$$763$$ 636.396 0.834071
$$764$$ 586.000 0.767016
$$765$$ 0 0
$$766$$ 204.000 0.266319
$$767$$ 1440.00i 1.87744i
$$768$$ 45.2548 0.0589256
$$769$$ 145.000 0.188557 0.0942783 0.995546i $$-0.469946\pi$$
0.0942783 + 0.995546i $$0.469946\pi$$
$$770$$ 0 0
$$771$$ 192.000 0.249027
$$772$$ 118.794 0.153878
$$773$$ 407.294 0.526900 0.263450 0.964673i $$-0.415140\pi$$
0.263450 + 0.964673i $$0.415140\pi$$
$$774$$ − 7.07107i − 0.00913575i
$$775$$ 0 0
$$776$$ 48.0000 0.0618557
$$777$$ 360.000i 0.463320i
$$778$$ −782.060 −1.00522
$$779$$ 806.102i 1.03479i
$$780$$ 0 0
$$781$$ 0 0
$$782$$ −353.553 −0.452114
$$783$$ 1200.00i 1.53257i
$$784$$ 96.0000 0.122449
$$785$$ 0 0
$$786$$ 652.000 0.829517
$$787$$ −186.676 −0.237200 −0.118600 0.992942i $$-0.537841\pi$$
−0.118600 + 0.992942i $$0.537841\pi$$
$$788$$ − 140.000i − 0.177665i
$$789$$ 1004.09i 1.27261i
$$790$$ 0 0
$$791$$ 551.543i 0.697273i
$$792$$ 14.1421 0.0178562
$$793$$ −1612.20 −2.03304
$$794$$ − 473.762i − 0.596677i
$$795$$ 0 0
$$796$$ −346.000 −0.434673
$$797$$ 704.278 0.883662 0.441831 0.897098i $$-0.354329\pi$$
0.441831 + 0.897098i $$0.354329\pi$$
$$798$$ 380.000i 0.476190i
$$799$$ 125.000 0.156446
$$800$$ 0 0
$$801$$ − 127.279i − 0.158900i
$$802$$ − 300.000i − 0.374065i
$$803$$ 125.000i 0.155666i
$$804$$ −624.000 −0.776119
$$805$$ 0 0
$$806$$ − 1018.23i − 1.26332i
$$807$$ 1080.00i 1.33829i
$$808$$ −141.421 −0.175026
$$809$$ 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 $$0.101630\pi$$
−0.949462 + 0.313882i $$0.898370\pi$$
$$812$$ 424.264 0.522493
$$813$$ 311.127 0.382690
$$814$$ −180.000 −0.221130
$$815$$ 0 0
$$816$$ − 282.843i − 0.346621i
$$817$$ 95.0000i 0.116279i
$$818$$ 1020.00i 1.24694i
$$819$$ 84.8528i 0.103605i
$$820$$ 0 0
$$821$$ 167.000 0.203410 0.101705 0.994815i $$-0.467570\pi$$
0.101705 + 0.994815i $$0.467570\pi$$
$$822$$ − 380.000i − 0.462287i
$$823$$ 1315.00i 1.59781i 0.601455 + 0.798906i $$0.294588\pi$$
−0.601455 + 0.798906i $$0.705412\pi$$
$$824$$ 48.0000 0.0582524
$$825$$ 0 0
$$826$$ −600.000 −0.726392
$$827$$ 534.573 0.646400 0.323200 0.946331i $$-0.395241\pi$$
0.323200 + 0.946331i $$0.395241\pi$$
$$828$$ − 20.0000i − 0.0241546i
$$829$$ − 763.675i − 0.921201i −0.887608 0.460600i $$-0.847634\pi$$
0.887608 0.460600i $$-0.152366\pi$$
$$830$$ 0 0
$$831$$ − 749.533i − 0.901965i
$$832$$ −135.765 −0.163178
$$833$$ − 600.000i − 0.720288i
$$834$$ 500.000 0.599520
$$835$$ 0 0
$$836$$ −190.000 −0.227273
$$837$$ 1200.00i 1.43369i
$$838$$ 87.6812 0.104632
$$839$$ 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
$$842$$ − 420.000i − 0.498812i
$$843$$ 1200.00i 1.42349i
$$844$$ 169.706i 0.201073i
$$845$$ 0 0
$$846$$ 7.07107i 0.00835824i
$$847$$ − 480.000i − 0.566706i
$$848$$ −101.823 −0.120075
$$849$$ − 353.553i − 0.416435i
$$850$$ 0 0
$$851$$ 254.558i 0.299129i
$$852$$ 0 0
$$853$$ − 770.000i − 0.902696i −0.892348 0.451348i $$-0.850943\pi$$
0.892348 0.451348i $$-0.149057\pi$$
$$854$$ − 671.751i − 0.786594i
$$855$$ 0 0
$$856$$ −288.000 −0.336449
$$857$$ 1255.82 1.46537 0.732685 0.680568i $$-0.238267\pi$$
0.732685 + 0.680568i $$0.238267\pi$$
$$858$$ 339.411 0.395584
$$859$$ −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
$$862$$ 720.000i 0.835267i
$$863$$ 992.778 1.15038 0.575190 0.818020i $$-0.304928\pi$$
0.575190 + 0.818020i $$0.304928\pi$$
$$864$$ 160.000 0.185185
$$865$$ 0 0
$$866$$ 324.000 0.374134
$$867$$ −950.352 −1.09614
$$868$$ 424.264 0.488783
$$869$$ − 212.132i − 0.244111i
$$870$$ 0 0
$$871$$ 1872.00 2.14925
$$872$$ 360.000i 0.412844i
$$873$$ 16.9706 0.0194394
$$874$$ 268.701i 0.307438i
$$875$$ 0 0
$$876$$ 141.421i 0.161440i
$$877$$ −186.676 −0.212858 −0.106429 0.994320i $$-0.533942\pi$$
−0.106429 + 0.994320i $$0.533942\pi$$
$$878$$ 1140.00i 1.29841i
$$879$$ 528.000 0.600683
$$880$$ 0 0
$$881$$ −25.0000 −0.0283768 −0.0141884 0.999899i $$-0.504516\pi$$
−0.0141884 + 0.999899i $$0.504516\pi$$
$$882$$ 33.9411 0.0384820
$$883$$ − 965.000i − 1.09287i −0.837503 0.546433i $$-0.815985\pi$$
0.837503 0.546433i $$-0.184015\pi$$
$$884$$ 848.528i 0.959873i
$$885$$ 0 0
$$886$$ 516.188i 0.582605i
$$887$$ 780.646 0.880097 0.440048 0.897974i $$-0.354961\pi$$
0.440048 + 0.897974i $$0.354961\pi$$
$$888$$ −203.647 −0.229332
$$889$$ − 1145.51i − 1.28854i
$$890$$ 0 0
$$891$$ −355.000 −0.398429
$$892$$ −729.734 −0.818088
$$893$$ − 95.0000i − 0.106383i
$$894$$ 860.000 0.961969
$$895$$ 0 0
$$896$$ − 56.5685i − 0.0631345i
$$897$$ − 480.000i − 0.535117i
$$898$$ − 1080.00i − 1.20267i
$$899$$ −1800.00 −2.00222
$$900$$ 0 0
$$901$$ 636.396i 0.706322i
$$902$$ 300.000i 0.332594i
$$903$$ 70.7107 0.0783064
$$904$$ −312.000 −0.345133
$$905$$ 0 0
$$906$$ 339.411i 0.374626i
$$907$$ 313.955 0.346147 0.173074 0.984909i $$-0.444630\pi$$
0.173074 + 0.984909i $$0.444630\pi$$
$$908$$ −135.765 −0.149520
$$909$$ −50.0000 −0.0550055
$$910$$ 0 0
$$911$$ 933.381i 1.02457i 0.858816 + 0.512284i $$0.171200\pi$$
−0.858816 + 0.512284i $$0.828800\pi$$
$$912$$ −214.960 −0.235702
$$913$$ 650.000i 0.711939i
$$914$$ 374.767i 0.410029i
$$915$$ 0 0
$$916$$ 290.000 0.316594
$$917$$ − 815.000i − 0.888768i
$$918$$ − 1000.00i − 1.08932i
$$919$$ 538.000 0.585419 0.292709 0.956201i $$-0.405443\pi$$
0.292709 + 0.956201i $$0.405443\pi$$
$$920$$ 0 0
$$921$$ −792.000 −0.859935
$$922$$ 782.060 0.848221
$$923$$ 0 0
$$924$$ 141.421i 0.153053i
$$925$$ 0 0
$$926$$ 685.894i 0.740706i
$$927$$ 16.9706 0.0183070
$$928$$ 240.000i 0.258621i
$$929$$ 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
$$932$$ − 670.000i − 0.718884i
$$933$$ −664.680 −0.712412
$$934$$ 162.635i 0.174127i
$$935$$ 0 0
$$936$$ −48.0000 −0.0512821
$$937$$ 335.000i 0.357524i 0.983892 + 0.178762i $$0.0572092\pi$$
−0.983892 + 0.178762i $$0.942791\pi$$
$$938$$ 780.000i 0.831557i
$$939$$ 876.812i 0.933773i
$$940$$ 0 0
$$941$$ − 424.264i − 0.450865i −0.974259 0.225433i $$-0.927620\pi$$
0.974259 0.225433i $$-0.0723795\pi$$
$$942$$ 760.000i 0.806794i
$$943$$ 424.264 0.449909
$$944$$ − 339.411i − 0.359546i
$$945$$ 0 0
$$946$$ 35.3553i 0.0373735i
$$947$$ − 1210.00i − 1.27772i −0.769323 0.638860i $$-0.779406\pi$$
0.769323 0.638860i $$-0.220594\pi$$
$$948$$ − 240.000i − 0.253165i
$$949$$ − 424.264i − 0.447064i
$$950$$ 0 0
$$951$$ 528.000 0.555205
$$952$$ −353.553 −0.371380
$$953$$ 992.778 1.04174 0.520870 0.853636i $$-0.325608\pi$$
0.520870 + 0.853636i $$0.325608\pi$$
$$954$$ −36.0000 −0.0377358
$$955$$ 0 0
$$956$$ −394.000 −0.412134
$$957$$ − 600.000i − 0.626959i
$$958$$ −692.965 −0.723345
$$959$$ −475.000 −0.495308
$$960$$ 0 0
$$961$$ −839.000 −0.873049
$$962$$ 610.940 0.635073
$$963$$ −101.823 −0.105736
$$964$$ 593.970i 0.616151i
$$965$$ 0 0
$$966$$ 200.000 0.207039
$$967$$ 350.000i 0.361944i 0.983488 + 0.180972i $$0.0579244\pi$$
−0.983488 + 0.180972i $$0.942076\pi$$
$$968$$ 271.529 0.280505
$$969$$ 1343.50i 1.38648i
$$970$$ 0 0
$$971$$ 254.558i 0.262161i 0.991372 + 0.131081i $$0.0418447\pi$$
−0.991372 + 0.131081i $$0.958155\pi$$
$$972$$ 107.480 0.110576
$$973$$ − 625.000i − 0.642343i
$$974$$ −864.000 −0.887064
$$975$$ 0 0
$$976$$ 380.000 0.389344
$$977$$ 398.808 0.408197 0.204098 0.978950i $$-0.434574\pi$$
0.204098 + 0.978950i $$0.434574\pi$$
$$978$$ 440.000i 0.449898i
$$979$$ 636.396i 0.650047i
$$980$$ 0 0
$$981$$ 127.279i 0.129744i
$$982$$ 115.966 0.118091
$$983$$ −695.793 −0.707826 −0.353913 0.935278i $$-0.615149\pi$$
−0.353913 + 0.935278i $$0.615149\pi$$
$$984$$ 339.411i 0.344930i
$$985$$ 0 0
$$986$$ 1500.00 1.52130
$$987$$ −70.7107 −0.0716420
$$988$$ 644.881 0.652714
$$989$$ 50.0000 0.0505561
$$990$$ 0 0
$$991$$ 381.838i 0.385305i 0.981267 + 0.192653i $$0.0617091\pi$$
−0.981267 + 0.192653i $$0.938291\pi$$
$$992$$ 240.000i 0.241935i
$$993$$ 840.000i 0.845921i
$$994$$ 0 0
$$995$$ 0 0
$$996$$ 735.391i 0.738344i
$$997$$ − 265.000i − 0.265797i −0.991130 0.132899i $$-0.957572\pi$$
0.991130 0.132899i $$-0.0424285\pi$$
$$998$$ 685.894 0.687268
$$999$$ −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 950.3.d.a.949.2 4
5.2 odd 4 950.3.c.a.151.1 2
5.3 odd 4 38.3.b.a.37.2 yes 2
5.4 even 2 inner 950.3.d.a.949.3 4
15.8 even 4 342.3.d.a.37.1 2
19.18 odd 2 inner 950.3.d.a.949.4 4
20.3 even 4 304.3.e.c.113.1 2
40.3 even 4 1216.3.e.i.1025.2 2
40.13 odd 4 1216.3.e.j.1025.1 2
60.23 odd 4 2736.3.o.h.721.1 2
95.18 even 4 38.3.b.a.37.1 2
95.37 even 4 950.3.c.a.151.2 2
95.94 odd 2 inner 950.3.d.a.949.1 4
285.113 odd 4 342.3.d.a.37.2 2
380.303 odd 4 304.3.e.c.113.2 2
760.493 even 4 1216.3.e.j.1025.2 2
760.683 odd 4 1216.3.e.i.1025.1 2
1140.683 even 4 2736.3.o.h.721.2 2

By twisted newform
Twist Min Dim Char Parity Ord Type
38.3.b.a.37.1 2 95.18 even 4
38.3.b.a.37.2 yes 2 5.3 odd 4
304.3.e.c.113.1 2 20.3 even 4
304.3.e.c.113.2 2 380.303 odd 4
342.3.d.a.37.1 2 15.8 even 4
342.3.d.a.37.2 2 285.113 odd 4
950.3.c.a.151.1 2 5.2 odd 4
950.3.c.a.151.2 2 95.37 even 4
950.3.d.a.949.1 4 95.94 odd 2 inner
950.3.d.a.949.2 4 1.1 even 1 trivial
950.3.d.a.949.3 4 5.4 even 2 inner
950.3.d.a.949.4 4 19.18 odd 2 inner
1216.3.e.i.1025.1 2 760.683 odd 4
1216.3.e.i.1025.2 2 40.3 even 4
1216.3.e.j.1025.1 2 40.13 odd 4
1216.3.e.j.1025.2 2 760.493 even 4
2736.3.o.h.721.1 2 60.23 odd 4
2736.3.o.h.721.2 2 1140.683 even 4