# Properties

 Label 950.3.d.a.949.4 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 949.4 Root $$0.707107 + 0.707107i$$ of defining polynomial Character $$\chi$$ $$=$$ 950.949 Dual form 950.3.d.a.949.3

## $q$-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 $$-0.656253\pi$$
−0.471405 + 0.881917i $$0.656253\pi$$
$$4$$ 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 $$-0.273630\pi$$
0.652714 + 0.757604i $$0.273630\pi$$
$$14$$ 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 $$0.261175\pi$$
−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 $$0.760109\pi$$
$$32$$ 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 $$-0.611781\pi$$
−0.343998 + 0.938970i $$0.611781\pi$$
$$38$$ −26.8701 −0.707107
$$39$$ −48.0000 −1.23077
$$40$$ 0 0
$$41$$ 42.4264i 1.03479i 0.855747 + 0.517395i $$0.173098\pi$$
−0.855747 + 0.517395i $$0.826902\pi$$
$$42$$ − 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 $$-0.422804\pi$$
0.240149 + 0.970736i $$0.422804\pi$$
$$54$$ 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 $$0.255442\pi$$
−0.694915 + 0.719092i $$0.744558\pi$$
$$60$$ 0 0
$$61$$ 95.0000 1.55738 0.778689 0.627411i $$-0.215885\pi$$
0.778689 + 0.627411i $$0.215885\pi$$
$$62$$ 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 $$-0.192187\pi$$
0.823199 + 0.567753i $$0.192187\pi$$
$$68$$ − 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 $$0.0865351\pi$$
−0.963274 + 0.268522i $$0.913465\pi$$
$$80$$ 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 $$-0.746403\pi$$
0.699071 0.715052i $$-0.253597\pi$$
$$90$$ 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 $$-0.472120\pi$$
0.0874771 + 0.996167i $$0.472120\pi$$
$$98$$ 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 $$-0.473747\pi$$
0.0823814 + 0.996601i $$0.473747\pi$$
$$104$$ 48.0000 0.461538
$$105$$ 0 0
$$106$$ 36.0000 0.339623
$$107$$ −101.823 −0.951620 −0.475810 0.879548i $$-0.657845\pi$$
−0.475810 + 0.879548i $$0.657845\pi$$
$$108$$ 56.5685 0.523783
$$109$$ 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
$$112$$ 20.0000i 0.178571i
$$113$$ −110.309 −0.976183 −0.488091 0.872793i $$-0.662307\pi$$
−0.488091 + 0.872793i $$0.662307\pi$$
$$114$$ 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 $$-0.142114\pi$$
0.901979 + 0.431780i $$0.142114\pi$$
$$128$$ 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 $$0.0906560\pi$$
−0.959717 + 0.280970i $$0.909344\pi$$
$$152$$ −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 $$-0.443091\pi$$
0.177835 + 0.984060i $$0.443091\pi$$
$$168$$ − 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 $$-0.318603\pi$$
0.539527 + 0.841969i $$0.318603\pi$$
$$174$$ − 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 $$-0.884301\pi$$
0.934665 0.355529i $$-0.115699\pi$$
$$180$$ 0 0
$$181$$ 254.558i 1.40640i 0.710992 + 0.703200i $$0.248246\pi$$
−0.710992 + 0.703200i $$0.751754\pi$$
$$182$$ 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 $$-0.549176\pi$$
−0.153878 + 0.988090i $$0.549176\pi$$
$$194$$ 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 $$-0.935557\pi$$
0.979576 0.201073i $$-0.0644429\pi$$
$$212$$ 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 $$-0.195034\pi$$
0.818088 + 0.575094i $$0.195034\pi$$
$$224$$ 28.2843i 0.126269i
$$225$$ 0 0
$$226$$ −156.000 −0.690265
$$227$$ 67.8823 0.299041 0.149520 0.988759i $$-0.452227\pi$$
0.149520 + 0.988759i $$0.452227\pi$$
$$228$$ 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 $$-0.788691\pi$$
0.787628 0.616151i $$-0.211309\pi$$
$$242$$ −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 $$-0.542161\pi$$
−0.132067 + 0.991241i $$0.542161\pi$$
$$258$$ 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 $$-0.748814\pi$$
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 $$0.434946\pi$$
$$272$$ − 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 $$-0.727677\pi$$
0.655819 0.754918i $$-0.272323\pi$$
$$282$$ − 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 $$-0.603199\pi$$
−0.318560 + 0.947903i $$0.603199\pi$$
$$294$$ −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 $$-0.349264\pi$$
0.456049 + 0.889955i $$0.349264\pi$$
$$308$$ 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 $$-0.595134\pi$$
−0.294442 + 0.955669i $$0.595134\pi$$
$$318$$ −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 $$-0.851917\pi$$
0.893724 0.448618i $$-0.148083\pi$$
$$332$$ 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 $$-0.785058\pi$$
−0.780545 + 0.625099i $$0.785058\pi$$
$$338$$ 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 $$-0.470995\pi$$
0.0909950 + 0.995851i $$0.470995\pi$$
$$374$$ − 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 $$-0.890984\pi$$
0.941923 0.335829i $$-0.109016\pi$$
$$380$$ 0 0
$$381$$ −648.000 −1.70079
$$382$$ 414.365 1.08472
$$383$$ 144.250 0.376631 0.188316 0.982109i $$-0.439697\pi$$
0.188316 + 0.982109i $$0.439697\pi$$
$$384$$ −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 $$-0.914792\pi$$
0.964385 0.264504i $$-0.0852082\pi$$
$$402$$ −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 $$0.343616\pi$$
−0.471769 + 0.881722i $$0.656384\pi$$
$$410$$ 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 $$-0.885259\pi$$
0.935731 0.352714i $$-0.114741\pi$$
$$422$$ − 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 $$0.201118\pi$$
−0.806948 + 0.590623i $$0.798882\pi$$
$$432$$ 113.137 0.261891
$$433$$ 229.103 0.529105 0.264553 0.964371i $$-0.414776\pi$$
0.264553 + 0.964371i $$0.414776\pi$$
$$434$$ −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 $$0.370286\pi$$
−0.396323 + 0.918111i $$0.629714\pi$$
$$440$$ 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 $$-0.676349\pi$$
0.526108 0.850418i $$-0.323651\pi$$
$$450$$ 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 $$-0.715819\pi$$
−0.627249 + 0.778819i $$0.715819\pi$$
$$488$$ 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 $$-0.946687\pi$$
0.986007 0.166705i $$-0.0533127\pi$$
$$510$$ 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 $$0.0389786\pi$$
−0.992512 + 0.122149i $$0.961021\pi$$
$$522$$ − 60.0000i − 0.114943i
$$523$$ −356.382 −0.681418 −0.340709 0.940169i $$-0.610667\pi$$
−0.340709 + 0.940169i $$0.610667\pi$$
$$524$$ −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 $$-0.495062\pi$$
0.0155124 + 0.999880i $$0.495062\pi$$
$$548$$ 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 $$-0.589944\pi$$
−0.278824 + 0.960342i $$0.589944\pi$$
$$564$$ − 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 $$0.121608\pi$$
−0.927905 + 0.372816i $$0.878392\pi$$
$$570$$ 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 $$-0.920259\pi$$
0.968785 0.247901i $$-0.0797406\pi$$
$$600$$ 0 0
$$601$$ 848.528i 1.41186i 0.708281 + 0.705930i $$0.249471\pi$$
−0.708281 + 0.705930i $$0.750529\pi$$
$$602$$ 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 $$-0.571802\pi$$
−0.223665 + 0.974666i $$0.571802\pi$$
$$608$$ −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 $$0.230240\pi$$
−0.749611 + 0.661878i $$0.769760\pi$$
$$642$$ 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 $$-0.979493\pi$$
0.997925 0.0643800i $$-0.0205070\pi$$
$$660$$ 0 0
$$661$$ 678.823i 1.02696i 0.858101 + 0.513481i $$0.171644\pi$$
−0.858101 + 0.513481i $$0.828356\pi$$
$$662$$ − 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 $$-0.455711\pi$$
0.138690 + 0.990336i $$0.455711\pi$$
$$674$$ −744.000 −1.10386
$$675$$ 0 0
$$676$$ 238.000 0.352071
$$677$$ 907.925 1.34110 0.670550 0.741864i $$-0.266058\pi$$
0.670550 + 0.741864i $$0.266058\pi$$
$$678$$ 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 $$-0.806001\pi$$
−0.819954 + 0.572429i $$0.806001\pi$$
$$684$$ 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 $$-0.276755\pi$$
0.645247 + 0.763974i $$0.276755\pi$$
$$744$$ − 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 $$0.0360417\pi$$
−0.993597 + 0.112986i $$0.963958\pi$$
$$752$$ 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 $$-0.584860\pi$$
−0.263450 + 0.964673i $$0.584860\pi$$
$$774$$ 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 $$-0.462159\pi$$
0.118600 + 0.992942i $$0.462159\pi$$
$$788$$ − 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 $$-0.645671\pi$$
−0.441831 + 0.897098i $$0.645671\pi$$
$$798$$ 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 $$-0.898370\pi$$
0.949462 0.313882i $$-0.101630\pi$$
$$812$$ −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 $$-0.604759\pi$$
−0.323200 + 0.946331i $$0.604759\pi$$
$$828$$ − 20.0000i − 0.0241546i
$$829$$ 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
$$832$$ 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 $$-0.935168\pi$$
0.979330 0.202271i $$-0.0648323\pi$$
$$840$$ 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 $$-0.761733\pi$$
−0.732685 + 0.680568i $$0.761733\pi$$
$$858$$ −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 $$-0.695072\pi$$
−0.575190 + 0.818020i $$0.695072\pi$$
$$864$$ 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 $$-0.466058\pi$$
0.106429 + 0.994320i $$0.466058\pi$$
$$878$$ 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 $$-0.645039\pi$$
−0.440048 + 0.897974i $$0.645039\pi$$
$$888$$ 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 $$-0.555370\pi$$
−0.173074 + 0.984909i $$0.555370\pi$$
$$908$$ 135.765 0.149520
$$909$$ −50.0000 −0.0550055
$$910$$ 0 0
$$911$$ − 933.381i − 1.02457i −0.858816 0.512284i $$-0.828800\pi$$
0.858816 0.512284i $$-0.171200\pi$$
$$912$$ 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 $$0.0723795\pi$$
−0.974259 + 0.225433i $$0.927620\pi$$
$$942$$ 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 $$-0.674392\pi$$
−0.520870 + 0.853636i $$0.674392\pi$$
$$954$$ −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 $$-0.958155\pi$$
0.991372 0.131081i $$-0.0418447\pi$$
$$972$$ −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 $$-0.565426\pi$$
−0.204098 + 0.978950i $$0.565426\pi$$
$$978$$ 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 $$-0.384851\pi$$
0.353913 + 0.935278i $$0.384851\pi$$
$$984$$ − 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 $$-0.938291\pi$$
0.981267 0.192653i $$-0.0617091\pi$$
$$992$$ 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.949.4 4
5.2 odd 4 950.3.c.a.151.2 2
5.3 odd 4 38.3.b.a.37.1 2
5.4 even 2 inner 950.3.d.a.949.1 4
15.8 even 4 342.3.d.a.37.2 2
19.18 odd 2 inner 950.3.d.a.949.2 4
20.3 even 4 304.3.e.c.113.2 2
40.3 even 4 1216.3.e.i.1025.1 2
40.13 odd 4 1216.3.e.j.1025.2 2
60.23 odd 4 2736.3.o.h.721.2 2
95.18 even 4 38.3.b.a.37.2 yes 2
95.37 even 4 950.3.c.a.151.1 2
95.94 odd 2 inner 950.3.d.a.949.3 4
285.113 odd 4 342.3.d.a.37.1 2
380.303 odd 4 304.3.e.c.113.1 2
760.493 even 4 1216.3.e.j.1025.1 2
760.683 odd 4 1216.3.e.i.1025.2 2
1140.683 even 4 2736.3.o.h.721.1 2

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