# Properties

 Label 1682.2.b.a.1681.2 Level $1682$ Weight $2$ Character 1682.1681 Analytic conductor $13.431$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1682 = 2 \cdot 29^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1682.b (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$13.4308376200$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(i)$$ Defining polynomial: $$x^{2} + 1$$ x^2 + 1 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 58) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## Embedding invariants

 Embedding label 1681.2 Root $$-1.00000i$$ of defining polynomial Character $$\chi$$ $$=$$ 1682.1681 Dual form 1682.2.b.a.1681.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q+1.00000i q^{2} -1.00000i q^{3} -1.00000 q^{4} -1.00000 q^{5} +1.00000 q^{6} -2.00000 q^{7} -1.00000i q^{8} +2.00000 q^{9} +O(q^{10})$$ $$q+1.00000i q^{2} -1.00000i q^{3} -1.00000 q^{4} -1.00000 q^{5} +1.00000 q^{6} -2.00000 q^{7} -1.00000i q^{8} +2.00000 q^{9} -1.00000i q^{10} -3.00000i q^{11} +1.00000i q^{12} +1.00000 q^{13} -2.00000i q^{14} +1.00000i q^{15} +1.00000 q^{16} +8.00000i q^{17} +2.00000i q^{18} +1.00000 q^{20} +2.00000i q^{21} +3.00000 q^{22} +4.00000 q^{23} -1.00000 q^{24} -4.00000 q^{25} +1.00000i q^{26} -5.00000i q^{27} +2.00000 q^{28} -1.00000 q^{30} -3.00000i q^{31} +1.00000i q^{32} -3.00000 q^{33} -8.00000 q^{34} +2.00000 q^{35} -2.00000 q^{36} -8.00000i q^{37} -1.00000i q^{39} +1.00000i q^{40} -2.00000i q^{41} -2.00000 q^{42} -11.0000i q^{43} +3.00000i q^{44} -2.00000 q^{45} +4.00000i q^{46} -13.0000i q^{47} -1.00000i q^{48} -3.00000 q^{49} -4.00000i q^{50} +8.00000 q^{51} -1.00000 q^{52} -11.0000 q^{53} +5.00000 q^{54} +3.00000i q^{55} +2.00000i q^{56} -1.00000i q^{60} -8.00000i q^{61} +3.00000 q^{62} -4.00000 q^{63} -1.00000 q^{64} -1.00000 q^{65} -3.00000i q^{66} +12.0000 q^{67} -8.00000i q^{68} -4.00000i q^{69} +2.00000i q^{70} -2.00000 q^{71} -2.00000i q^{72} -4.00000i q^{73} +8.00000 q^{74} +4.00000i q^{75} +6.00000i q^{77} +1.00000 q^{78} +15.0000i q^{79} -1.00000 q^{80} +1.00000 q^{81} +2.00000 q^{82} +4.00000 q^{83} -2.00000i q^{84} -8.00000i q^{85} +11.0000 q^{86} -3.00000 q^{88} -10.0000i q^{89} -2.00000i q^{90} -2.00000 q^{91} -4.00000 q^{92} -3.00000 q^{93} +13.0000 q^{94} +1.00000 q^{96} +2.00000i q^{97} -3.00000i q^{98} -6.00000i q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{4} - 2 q^{5} + 2 q^{6} - 4 q^{7} + 4 q^{9}+O(q^{10})$$ 2 * q - 2 * q^4 - 2 * q^5 + 2 * q^6 - 4 * q^7 + 4 * q^9 $$2 q - 2 q^{4} - 2 q^{5} + 2 q^{6} - 4 q^{7} + 4 q^{9} + 2 q^{13} + 2 q^{16} + 2 q^{20} + 6 q^{22} + 8 q^{23} - 2 q^{24} - 8 q^{25} + 4 q^{28} - 2 q^{30} - 6 q^{33} - 16 q^{34} + 4 q^{35} - 4 q^{36} - 4 q^{42} - 4 q^{45} - 6 q^{49} + 16 q^{51} - 2 q^{52} - 22 q^{53} + 10 q^{54} + 6 q^{62} - 8 q^{63} - 2 q^{64} - 2 q^{65} + 24 q^{67} - 4 q^{71} + 16 q^{74} + 2 q^{78} - 2 q^{80} + 2 q^{81} + 4 q^{82} + 8 q^{83} + 22 q^{86} - 6 q^{88} - 4 q^{91} - 8 q^{92} - 6 q^{93} + 26 q^{94} + 2 q^{96}+O(q^{100})$$ 2 * q - 2 * q^4 - 2 * q^5 + 2 * q^6 - 4 * q^7 + 4 * q^9 + 2 * q^13 + 2 * q^16 + 2 * q^20 + 6 * q^22 + 8 * q^23 - 2 * q^24 - 8 * q^25 + 4 * q^28 - 2 * q^30 - 6 * q^33 - 16 * q^34 + 4 * q^35 - 4 * q^36 - 4 * q^42 - 4 * q^45 - 6 * q^49 + 16 * q^51 - 2 * q^52 - 22 * q^53 + 10 * q^54 + 6 * q^62 - 8 * q^63 - 2 * q^64 - 2 * q^65 + 24 * q^67 - 4 * q^71 + 16 * q^74 + 2 * q^78 - 2 * q^80 + 2 * q^81 + 4 * q^82 + 8 * q^83 + 22 * q^86 - 6 * q^88 - 4 * q^91 - 8 * q^92 - 6 * q^93 + 26 * q^94 + 2 * q^96

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1682\mathbb{Z}\right)^\times$$.

 $$n$$ $$843$$ $$\chi(n)$$ $$-1$$

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 1.00000i 0.707107i
$$3$$ − 1.00000i − 0.577350i −0.957427 0.288675i $$-0.906785\pi$$
0.957427 0.288675i $$-0.0932147\pi$$
$$4$$ −1.00000 −0.500000
$$5$$ −1.00000 −0.447214 −0.223607 0.974679i $$-0.571783\pi$$
−0.223607 + 0.974679i $$0.571783\pi$$
$$6$$ 1.00000 0.408248
$$7$$ −2.00000 −0.755929 −0.377964 0.925820i $$-0.623376\pi$$
−0.377964 + 0.925820i $$0.623376\pi$$
$$8$$ − 1.00000i − 0.353553i
$$9$$ 2.00000 0.666667
$$10$$ − 1.00000i − 0.316228i
$$11$$ − 3.00000i − 0.904534i −0.891883 0.452267i $$-0.850615\pi$$
0.891883 0.452267i $$-0.149385\pi$$
$$12$$ 1.00000i 0.288675i
$$13$$ 1.00000 0.277350 0.138675 0.990338i $$-0.455716\pi$$
0.138675 + 0.990338i $$0.455716\pi$$
$$14$$ − 2.00000i − 0.534522i
$$15$$ 1.00000i 0.258199i
$$16$$ 1.00000 0.250000
$$17$$ 8.00000i 1.94029i 0.242536 + 0.970143i $$0.422021\pi$$
−0.242536 + 0.970143i $$0.577979\pi$$
$$18$$ 2.00000i 0.471405i
$$19$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$20$$ 1.00000 0.223607
$$21$$ 2.00000i 0.436436i
$$22$$ 3.00000 0.639602
$$23$$ 4.00000 0.834058 0.417029 0.908893i $$-0.363071\pi$$
0.417029 + 0.908893i $$0.363071\pi$$
$$24$$ −1.00000 −0.204124
$$25$$ −4.00000 −0.800000
$$26$$ 1.00000i 0.196116i
$$27$$ − 5.00000i − 0.962250i
$$28$$ 2.00000 0.377964
$$29$$ 0 0
$$30$$ −1.00000 −0.182574
$$31$$ − 3.00000i − 0.538816i −0.963026 0.269408i $$-0.913172\pi$$
0.963026 0.269408i $$-0.0868280\pi$$
$$32$$ 1.00000i 0.176777i
$$33$$ −3.00000 −0.522233
$$34$$ −8.00000 −1.37199
$$35$$ 2.00000 0.338062
$$36$$ −2.00000 −0.333333
$$37$$ − 8.00000i − 1.31519i −0.753371 0.657596i $$-0.771573\pi$$
0.753371 0.657596i $$-0.228427\pi$$
$$38$$ 0 0
$$39$$ − 1.00000i − 0.160128i
$$40$$ 1.00000i 0.158114i
$$41$$ − 2.00000i − 0.312348i −0.987730 0.156174i $$-0.950084\pi$$
0.987730 0.156174i $$-0.0499160\pi$$
$$42$$ −2.00000 −0.308607
$$43$$ − 11.0000i − 1.67748i −0.544529 0.838742i $$-0.683292\pi$$
0.544529 0.838742i $$-0.316708\pi$$
$$44$$ 3.00000i 0.452267i
$$45$$ −2.00000 −0.298142
$$46$$ 4.00000i 0.589768i
$$47$$ − 13.0000i − 1.89624i −0.317905 0.948122i $$-0.602979\pi$$
0.317905 0.948122i $$-0.397021\pi$$
$$48$$ − 1.00000i − 0.144338i
$$49$$ −3.00000 −0.428571
$$50$$ − 4.00000i − 0.565685i
$$51$$ 8.00000 1.12022
$$52$$ −1.00000 −0.138675
$$53$$ −11.0000 −1.51097 −0.755483 0.655168i $$-0.772598\pi$$
−0.755483 + 0.655168i $$0.772598\pi$$
$$54$$ 5.00000 0.680414
$$55$$ 3.00000i 0.404520i
$$56$$ 2.00000i 0.267261i
$$57$$ 0 0
$$58$$ 0 0
$$59$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$60$$ − 1.00000i − 0.129099i
$$61$$ − 8.00000i − 1.02430i −0.858898 0.512148i $$-0.828850\pi$$
0.858898 0.512148i $$-0.171150\pi$$
$$62$$ 3.00000 0.381000
$$63$$ −4.00000 −0.503953
$$64$$ −1.00000 −0.125000
$$65$$ −1.00000 −0.124035
$$66$$ − 3.00000i − 0.369274i
$$67$$ 12.0000 1.46603 0.733017 0.680211i $$-0.238112\pi$$
0.733017 + 0.680211i $$0.238112\pi$$
$$68$$ − 8.00000i − 0.970143i
$$69$$ − 4.00000i − 0.481543i
$$70$$ 2.00000i 0.239046i
$$71$$ −2.00000 −0.237356 −0.118678 0.992933i $$-0.537866\pi$$
−0.118678 + 0.992933i $$0.537866\pi$$
$$72$$ − 2.00000i − 0.235702i
$$73$$ − 4.00000i − 0.468165i −0.972217 0.234082i $$-0.924791\pi$$
0.972217 0.234082i $$-0.0752085\pi$$
$$74$$ 8.00000 0.929981
$$75$$ 4.00000i 0.461880i
$$76$$ 0 0
$$77$$ 6.00000i 0.683763i
$$78$$ 1.00000 0.113228
$$79$$ 15.0000i 1.68763i 0.536633 + 0.843816i $$0.319696\pi$$
−0.536633 + 0.843816i $$0.680304\pi$$
$$80$$ −1.00000 −0.111803
$$81$$ 1.00000 0.111111
$$82$$ 2.00000 0.220863
$$83$$ 4.00000 0.439057 0.219529 0.975606i $$-0.429548\pi$$
0.219529 + 0.975606i $$0.429548\pi$$
$$84$$ − 2.00000i − 0.218218i
$$85$$ − 8.00000i − 0.867722i
$$86$$ 11.0000 1.18616
$$87$$ 0 0
$$88$$ −3.00000 −0.319801
$$89$$ − 10.0000i − 1.06000i −0.847998 0.529999i $$-0.822192\pi$$
0.847998 0.529999i $$-0.177808\pi$$
$$90$$ − 2.00000i − 0.210819i
$$91$$ −2.00000 −0.209657
$$92$$ −4.00000 −0.417029
$$93$$ −3.00000 −0.311086
$$94$$ 13.0000 1.34085
$$95$$ 0 0
$$96$$ 1.00000 0.102062
$$97$$ 2.00000i 0.203069i 0.994832 + 0.101535i $$0.0323753\pi$$
−0.994832 + 0.101535i $$0.967625\pi$$
$$98$$ − 3.00000i − 0.303046i
$$99$$ − 6.00000i − 0.603023i
$$100$$ 4.00000 0.400000
$$101$$ − 8.00000i − 0.796030i −0.917379 0.398015i $$-0.869699\pi$$
0.917379 0.398015i $$-0.130301\pi$$
$$102$$ 8.00000i 0.792118i
$$103$$ 14.0000 1.37946 0.689730 0.724066i $$-0.257729\pi$$
0.689730 + 0.724066i $$0.257729\pi$$
$$104$$ − 1.00000i − 0.0980581i
$$105$$ − 2.00000i − 0.195180i
$$106$$ − 11.0000i − 1.06841i
$$107$$ −2.00000 −0.193347 −0.0966736 0.995316i $$-0.530820\pi$$
−0.0966736 + 0.995316i $$0.530820\pi$$
$$108$$ 5.00000i 0.481125i
$$109$$ −5.00000 −0.478913 −0.239457 0.970907i $$-0.576969\pi$$
−0.239457 + 0.970907i $$0.576969\pi$$
$$110$$ −3.00000 −0.286039
$$111$$ −8.00000 −0.759326
$$112$$ −2.00000 −0.188982
$$113$$ 6.00000i 0.564433i 0.959351 + 0.282216i $$0.0910696\pi$$
−0.959351 + 0.282216i $$0.908930\pi$$
$$114$$ 0 0
$$115$$ −4.00000 −0.373002
$$116$$ 0 0
$$117$$ 2.00000 0.184900
$$118$$ 0 0
$$119$$ − 16.0000i − 1.46672i
$$120$$ 1.00000 0.0912871
$$121$$ 2.00000 0.181818
$$122$$ 8.00000 0.724286
$$123$$ −2.00000 −0.180334
$$124$$ 3.00000i 0.269408i
$$125$$ 9.00000 0.804984
$$126$$ − 4.00000i − 0.356348i
$$127$$ 8.00000i 0.709885i 0.934888 + 0.354943i $$0.115500\pi$$
−0.934888 + 0.354943i $$0.884500\pi$$
$$128$$ − 1.00000i − 0.0883883i
$$129$$ −11.0000 −0.968496
$$130$$ − 1.00000i − 0.0877058i
$$131$$ − 12.0000i − 1.04844i −0.851581 0.524222i $$-0.824356\pi$$
0.851581 0.524222i $$-0.175644\pi$$
$$132$$ 3.00000 0.261116
$$133$$ 0 0
$$134$$ 12.0000i 1.03664i
$$135$$ 5.00000i 0.430331i
$$136$$ 8.00000 0.685994
$$137$$ − 12.0000i − 1.02523i −0.858619 0.512615i $$-0.828677\pi$$
0.858619 0.512615i $$-0.171323\pi$$
$$138$$ 4.00000 0.340503
$$139$$ −20.0000 −1.69638 −0.848189 0.529694i $$-0.822307\pi$$
−0.848189 + 0.529694i $$0.822307\pi$$
$$140$$ −2.00000 −0.169031
$$141$$ −13.0000 −1.09480
$$142$$ − 2.00000i − 0.167836i
$$143$$ − 3.00000i − 0.250873i
$$144$$ 2.00000 0.166667
$$145$$ 0 0
$$146$$ 4.00000 0.331042
$$147$$ 3.00000i 0.247436i
$$148$$ 8.00000i 0.657596i
$$149$$ −15.0000 −1.22885 −0.614424 0.788976i $$-0.710612\pi$$
−0.614424 + 0.788976i $$0.710612\pi$$
$$150$$ −4.00000 −0.326599
$$151$$ −2.00000 −0.162758 −0.0813788 0.996683i $$-0.525932\pi$$
−0.0813788 + 0.996683i $$0.525932\pi$$
$$152$$ 0 0
$$153$$ 16.0000i 1.29352i
$$154$$ −6.00000 −0.483494
$$155$$ 3.00000i 0.240966i
$$156$$ 1.00000i 0.0800641i
$$157$$ − 18.0000i − 1.43656i −0.695756 0.718278i $$-0.744931\pi$$
0.695756 0.718278i $$-0.255069\pi$$
$$158$$ −15.0000 −1.19334
$$159$$ 11.0000i 0.872357i
$$160$$ − 1.00000i − 0.0790569i
$$161$$ −8.00000 −0.630488
$$162$$ 1.00000i 0.0785674i
$$163$$ − 9.00000i − 0.704934i −0.935824 0.352467i $$-0.885343\pi$$
0.935824 0.352467i $$-0.114657\pi$$
$$164$$ 2.00000i 0.156174i
$$165$$ 3.00000 0.233550
$$166$$ 4.00000i 0.310460i
$$167$$ 2.00000 0.154765 0.0773823 0.997001i $$-0.475344\pi$$
0.0773823 + 0.997001i $$0.475344\pi$$
$$168$$ 2.00000 0.154303
$$169$$ −12.0000 −0.923077
$$170$$ 8.00000 0.613572
$$171$$ 0 0
$$172$$ 11.0000i 0.838742i
$$173$$ 6.00000 0.456172 0.228086 0.973641i $$-0.426753\pi$$
0.228086 + 0.973641i $$0.426753\pi$$
$$174$$ 0 0
$$175$$ 8.00000 0.604743
$$176$$ − 3.00000i − 0.226134i
$$177$$ 0 0
$$178$$ 10.0000 0.749532
$$179$$ 10.0000 0.747435 0.373718 0.927543i $$-0.378083\pi$$
0.373718 + 0.927543i $$0.378083\pi$$
$$180$$ 2.00000 0.149071
$$181$$ 7.00000 0.520306 0.260153 0.965567i $$-0.416227\pi$$
0.260153 + 0.965567i $$0.416227\pi$$
$$182$$ − 2.00000i − 0.148250i
$$183$$ −8.00000 −0.591377
$$184$$ − 4.00000i − 0.294884i
$$185$$ 8.00000i 0.588172i
$$186$$ − 3.00000i − 0.219971i
$$187$$ 24.0000 1.75505
$$188$$ 13.0000i 0.948122i
$$189$$ 10.0000i 0.727393i
$$190$$ 0 0
$$191$$ − 8.00000i − 0.578860i −0.957199 0.289430i $$-0.906534\pi$$
0.957199 0.289430i $$-0.0934657\pi$$
$$192$$ 1.00000i 0.0721688i
$$193$$ 14.0000i 1.00774i 0.863779 + 0.503871i $$0.168091\pi$$
−0.863779 + 0.503871i $$0.831909\pi$$
$$194$$ −2.00000 −0.143592
$$195$$ 1.00000i 0.0716115i
$$196$$ 3.00000 0.214286
$$197$$ 18.0000 1.28245 0.641223 0.767354i $$-0.278427\pi$$
0.641223 + 0.767354i $$0.278427\pi$$
$$198$$ 6.00000 0.426401
$$199$$ −10.0000 −0.708881 −0.354441 0.935079i $$-0.615329\pi$$
−0.354441 + 0.935079i $$0.615329\pi$$
$$200$$ 4.00000i 0.282843i
$$201$$ − 12.0000i − 0.846415i
$$202$$ 8.00000 0.562878
$$203$$ 0 0
$$204$$ −8.00000 −0.560112
$$205$$ 2.00000i 0.139686i
$$206$$ 14.0000i 0.975426i
$$207$$ 8.00000 0.556038
$$208$$ 1.00000 0.0693375
$$209$$ 0 0
$$210$$ 2.00000 0.138013
$$211$$ 3.00000i 0.206529i 0.994654 + 0.103264i $$0.0329287\pi$$
−0.994654 + 0.103264i $$0.967071\pi$$
$$212$$ 11.0000 0.755483
$$213$$ 2.00000i 0.137038i
$$214$$ − 2.00000i − 0.136717i
$$215$$ 11.0000i 0.750194i
$$216$$ −5.00000 −0.340207
$$217$$ 6.00000i 0.407307i
$$218$$ − 5.00000i − 0.338643i
$$219$$ −4.00000 −0.270295
$$220$$ − 3.00000i − 0.202260i
$$221$$ 8.00000i 0.538138i
$$222$$ − 8.00000i − 0.536925i
$$223$$ −26.0000 −1.74109 −0.870544 0.492090i $$-0.836233\pi$$
−0.870544 + 0.492090i $$0.836233\pi$$
$$224$$ − 2.00000i − 0.133631i
$$225$$ −8.00000 −0.533333
$$226$$ −6.00000 −0.399114
$$227$$ 18.0000 1.19470 0.597351 0.801980i $$-0.296220\pi$$
0.597351 + 0.801980i $$0.296220\pi$$
$$228$$ 0 0
$$229$$ − 10.0000i − 0.660819i −0.943838 0.330409i $$-0.892813\pi$$
0.943838 0.330409i $$-0.107187\pi$$
$$230$$ − 4.00000i − 0.263752i
$$231$$ 6.00000 0.394771
$$232$$ 0 0
$$233$$ −1.00000 −0.0655122 −0.0327561 0.999463i $$-0.510428\pi$$
−0.0327561 + 0.999463i $$0.510428\pi$$
$$234$$ 2.00000i 0.130744i
$$235$$ 13.0000i 0.848026i
$$236$$ 0 0
$$237$$ 15.0000 0.974355
$$238$$ 16.0000 1.03713
$$239$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$240$$ 1.00000i 0.0645497i
$$241$$ −17.0000 −1.09507 −0.547533 0.836784i $$-0.684433\pi$$
−0.547533 + 0.836784i $$0.684433\pi$$
$$242$$ 2.00000i 0.128565i
$$243$$ − 16.0000i − 1.02640i
$$244$$ 8.00000i 0.512148i
$$245$$ 3.00000 0.191663
$$246$$ − 2.00000i − 0.127515i
$$247$$ 0 0
$$248$$ −3.00000 −0.190500
$$249$$ − 4.00000i − 0.253490i
$$250$$ 9.00000i 0.569210i
$$251$$ 27.0000i 1.70422i 0.523359 + 0.852112i $$0.324679\pi$$
−0.523359 + 0.852112i $$0.675321\pi$$
$$252$$ 4.00000 0.251976
$$253$$ − 12.0000i − 0.754434i
$$254$$ −8.00000 −0.501965
$$255$$ −8.00000 −0.500979
$$256$$ 1.00000 0.0625000
$$257$$ 13.0000 0.810918 0.405459 0.914113i $$-0.367112\pi$$
0.405459 + 0.914113i $$0.367112\pi$$
$$258$$ − 11.0000i − 0.684830i
$$259$$ 16.0000i 0.994192i
$$260$$ 1.00000 0.0620174
$$261$$ 0 0
$$262$$ 12.0000 0.741362
$$263$$ 9.00000i 0.554964i 0.960731 + 0.277482i $$0.0894999\pi$$
−0.960731 + 0.277482i $$0.910500\pi$$
$$264$$ 3.00000i 0.184637i
$$265$$ 11.0000 0.675725
$$266$$ 0 0
$$267$$ −10.0000 −0.611990
$$268$$ −12.0000 −0.733017
$$269$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$270$$ −5.00000 −0.304290
$$271$$ 13.0000i 0.789694i 0.918747 + 0.394847i $$0.129202\pi$$
−0.918747 + 0.394847i $$0.870798\pi$$
$$272$$ 8.00000i 0.485071i
$$273$$ 2.00000i 0.121046i
$$274$$ 12.0000 0.724947
$$275$$ 12.0000i 0.723627i
$$276$$ 4.00000i 0.240772i
$$277$$ −2.00000 −0.120168 −0.0600842 0.998193i $$-0.519137\pi$$
−0.0600842 + 0.998193i $$0.519137\pi$$
$$278$$ − 20.0000i − 1.19952i
$$279$$ − 6.00000i − 0.359211i
$$280$$ − 2.00000i − 0.119523i
$$281$$ 27.0000 1.61068 0.805342 0.592810i $$-0.201981\pi$$
0.805342 + 0.592810i $$0.201981\pi$$
$$282$$ − 13.0000i − 0.774139i
$$283$$ −4.00000 −0.237775 −0.118888 0.992908i $$-0.537933\pi$$
−0.118888 + 0.992908i $$0.537933\pi$$
$$284$$ 2.00000 0.118678
$$285$$ 0 0
$$286$$ 3.00000 0.177394
$$287$$ 4.00000i 0.236113i
$$288$$ 2.00000i 0.117851i
$$289$$ −47.0000 −2.76471
$$290$$ 0 0
$$291$$ 2.00000 0.117242
$$292$$ 4.00000i 0.234082i
$$293$$ 14.0000i 0.817889i 0.912559 + 0.408944i $$0.134103\pi$$
−0.912559 + 0.408944i $$0.865897\pi$$
$$294$$ −3.00000 −0.174964
$$295$$ 0 0
$$296$$ −8.00000 −0.464991
$$297$$ −15.0000 −0.870388
$$298$$ − 15.0000i − 0.868927i
$$299$$ 4.00000 0.231326
$$300$$ − 4.00000i − 0.230940i
$$301$$ 22.0000i 1.26806i
$$302$$ − 2.00000i − 0.115087i
$$303$$ −8.00000 −0.459588
$$304$$ 0 0
$$305$$ 8.00000i 0.458079i
$$306$$ −16.0000 −0.914659
$$307$$ − 7.00000i − 0.399511i −0.979846 0.199756i $$-0.935985\pi$$
0.979846 0.199756i $$-0.0640148\pi$$
$$308$$ − 6.00000i − 0.341882i
$$309$$ − 14.0000i − 0.796432i
$$310$$ −3.00000 −0.170389
$$311$$ − 8.00000i − 0.453638i −0.973937 0.226819i $$-0.927167\pi$$
0.973937 0.226819i $$-0.0728326\pi$$
$$312$$ −1.00000 −0.0566139
$$313$$ 9.00000 0.508710 0.254355 0.967111i $$-0.418137\pi$$
0.254355 + 0.967111i $$0.418137\pi$$
$$314$$ 18.0000 1.01580
$$315$$ 4.00000 0.225374
$$316$$ − 15.0000i − 0.843816i
$$317$$ 12.0000i 0.673987i 0.941507 + 0.336994i $$0.109410\pi$$
−0.941507 + 0.336994i $$0.890590\pi$$
$$318$$ −11.0000 −0.616849
$$319$$ 0 0
$$320$$ 1.00000 0.0559017
$$321$$ 2.00000i 0.111629i
$$322$$ − 8.00000i − 0.445823i
$$323$$ 0 0
$$324$$ −1.00000 −0.0555556
$$325$$ −4.00000 −0.221880
$$326$$ 9.00000 0.498464
$$327$$ 5.00000i 0.276501i
$$328$$ −2.00000 −0.110432
$$329$$ 26.0000i 1.43343i
$$330$$ 3.00000i 0.165145i
$$331$$ 23.0000i 1.26419i 0.774889 + 0.632097i $$0.217806\pi$$
−0.774889 + 0.632097i $$0.782194\pi$$
$$332$$ −4.00000 −0.219529
$$333$$ − 16.0000i − 0.876795i
$$334$$ 2.00000i 0.109435i
$$335$$ −12.0000 −0.655630
$$336$$ 2.00000i 0.109109i
$$337$$ 32.0000i 1.74315i 0.490261 + 0.871576i $$0.336901\pi$$
−0.490261 + 0.871576i $$0.663099\pi$$
$$338$$ − 12.0000i − 0.652714i
$$339$$ 6.00000 0.325875
$$340$$ 8.00000i 0.433861i
$$341$$ −9.00000 −0.487377
$$342$$ 0 0
$$343$$ 20.0000 1.07990
$$344$$ −11.0000 −0.593080
$$345$$ 4.00000i 0.215353i
$$346$$ 6.00000i 0.322562i
$$347$$ 2.00000 0.107366 0.0536828 0.998558i $$-0.482904\pi$$
0.0536828 + 0.998558i $$0.482904\pi$$
$$348$$ 0 0
$$349$$ −15.0000 −0.802932 −0.401466 0.915874i $$-0.631499\pi$$
−0.401466 + 0.915874i $$0.631499\pi$$
$$350$$ 8.00000i 0.427618i
$$351$$ − 5.00000i − 0.266880i
$$352$$ 3.00000 0.159901
$$353$$ 26.0000 1.38384 0.691920 0.721974i $$-0.256765\pi$$
0.691920 + 0.721974i $$0.256765\pi$$
$$354$$ 0 0
$$355$$ 2.00000 0.106149
$$356$$ 10.0000i 0.529999i
$$357$$ −16.0000 −0.846810
$$358$$ 10.0000i 0.528516i
$$359$$ − 25.0000i − 1.31945i −0.751507 0.659725i $$-0.770673\pi$$
0.751507 0.659725i $$-0.229327\pi$$
$$360$$ 2.00000i 0.105409i
$$361$$ 19.0000 1.00000
$$362$$ 7.00000i 0.367912i
$$363$$ − 2.00000i − 0.104973i
$$364$$ 2.00000 0.104828
$$365$$ 4.00000i 0.209370i
$$366$$ − 8.00000i − 0.418167i
$$367$$ − 32.0000i − 1.67039i −0.549957 0.835193i $$-0.685356\pi$$
0.549957 0.835193i $$-0.314644\pi$$
$$368$$ 4.00000 0.208514
$$369$$ − 4.00000i − 0.208232i
$$370$$ −8.00000 −0.415900
$$371$$ 22.0000 1.14218
$$372$$ 3.00000 0.155543
$$373$$ −21.0000 −1.08734 −0.543669 0.839299i $$-0.682965\pi$$
−0.543669 + 0.839299i $$0.682965\pi$$
$$374$$ 24.0000i 1.24101i
$$375$$ − 9.00000i − 0.464758i
$$376$$ −13.0000 −0.670424
$$377$$ 0 0
$$378$$ −10.0000 −0.514344
$$379$$ 20.0000i 1.02733i 0.857991 + 0.513665i $$0.171713\pi$$
−0.857991 + 0.513665i $$0.828287\pi$$
$$380$$ 0 0
$$381$$ 8.00000 0.409852
$$382$$ 8.00000 0.409316
$$383$$ −14.0000 −0.715367 −0.357683 0.933843i $$-0.616433\pi$$
−0.357683 + 0.933843i $$0.616433\pi$$
$$384$$ −1.00000 −0.0510310
$$385$$ − 6.00000i − 0.305788i
$$386$$ −14.0000 −0.712581
$$387$$ − 22.0000i − 1.11832i
$$388$$ − 2.00000i − 0.101535i
$$389$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$390$$ −1.00000 −0.0506370
$$391$$ 32.0000i 1.61831i
$$392$$ 3.00000i 0.151523i
$$393$$ −12.0000 −0.605320
$$394$$ 18.0000i 0.906827i
$$395$$ − 15.0000i − 0.754732i
$$396$$ 6.00000i 0.301511i
$$397$$ −17.0000 −0.853206 −0.426603 0.904439i $$-0.640290\pi$$
−0.426603 + 0.904439i $$0.640290\pi$$
$$398$$ − 10.0000i − 0.501255i
$$399$$ 0 0
$$400$$ −4.00000 −0.200000
$$401$$ 27.0000 1.34832 0.674158 0.738587i $$-0.264507\pi$$
0.674158 + 0.738587i $$0.264507\pi$$
$$402$$ 12.0000 0.598506
$$403$$ − 3.00000i − 0.149441i
$$404$$ 8.00000i 0.398015i
$$405$$ −1.00000 −0.0496904
$$406$$ 0 0
$$407$$ −24.0000 −1.18964
$$408$$ − 8.00000i − 0.396059i
$$409$$ 30.0000i 1.48340i 0.670729 + 0.741702i $$0.265981\pi$$
−0.670729 + 0.741702i $$0.734019\pi$$
$$410$$ −2.00000 −0.0987730
$$411$$ −12.0000 −0.591916
$$412$$ −14.0000 −0.689730
$$413$$ 0 0
$$414$$ 8.00000i 0.393179i
$$415$$ −4.00000 −0.196352
$$416$$ 1.00000i 0.0490290i
$$417$$ 20.0000i 0.979404i
$$418$$ 0 0
$$419$$ 10.0000 0.488532 0.244266 0.969708i $$-0.421453\pi$$
0.244266 + 0.969708i $$0.421453\pi$$
$$420$$ 2.00000i 0.0975900i
$$421$$ − 32.0000i − 1.55958i −0.626038 0.779792i $$-0.715325\pi$$
0.626038 0.779792i $$-0.284675\pi$$
$$422$$ −3.00000 −0.146038
$$423$$ − 26.0000i − 1.26416i
$$424$$ 11.0000i 0.534207i
$$425$$ − 32.0000i − 1.55223i
$$426$$ −2.00000 −0.0969003
$$427$$ 16.0000i 0.774294i
$$428$$ 2.00000 0.0966736
$$429$$ −3.00000 −0.144841
$$430$$ −11.0000 −0.530467
$$431$$ 32.0000 1.54139 0.770693 0.637207i $$-0.219910\pi$$
0.770693 + 0.637207i $$0.219910\pi$$
$$432$$ − 5.00000i − 0.240563i
$$433$$ 16.0000i 0.768911i 0.923144 + 0.384455i $$0.125611\pi$$
−0.923144 + 0.384455i $$0.874389\pi$$
$$434$$ −6.00000 −0.288009
$$435$$ 0 0
$$436$$ 5.00000 0.239457
$$437$$ 0 0
$$438$$ − 4.00000i − 0.191127i
$$439$$ −20.0000 −0.954548 −0.477274 0.878755i $$-0.658375\pi$$
−0.477274 + 0.878755i $$0.658375\pi$$
$$440$$ 3.00000 0.143019
$$441$$ −6.00000 −0.285714
$$442$$ −8.00000 −0.380521
$$443$$ − 4.00000i − 0.190046i −0.995475 0.0950229i $$-0.969708\pi$$
0.995475 0.0950229i $$-0.0302924\pi$$
$$444$$ 8.00000 0.379663
$$445$$ 10.0000i 0.474045i
$$446$$ − 26.0000i − 1.23114i
$$447$$ 15.0000i 0.709476i
$$448$$ 2.00000 0.0944911
$$449$$ − 10.0000i − 0.471929i −0.971762 0.235965i $$-0.924175\pi$$
0.971762 0.235965i $$-0.0758249\pi$$
$$450$$ − 8.00000i − 0.377124i
$$451$$ −6.00000 −0.282529
$$452$$ − 6.00000i − 0.282216i
$$453$$ 2.00000i 0.0939682i
$$454$$ 18.0000i 0.844782i
$$455$$ 2.00000 0.0937614
$$456$$ 0 0
$$457$$ 2.00000 0.0935561 0.0467780 0.998905i $$-0.485105\pi$$
0.0467780 + 0.998905i $$0.485105\pi$$
$$458$$ 10.0000 0.467269
$$459$$ 40.0000 1.86704
$$460$$ 4.00000 0.186501
$$461$$ − 2.00000i − 0.0931493i −0.998915 0.0465746i $$-0.985169\pi$$
0.998915 0.0465746i $$-0.0148305\pi$$
$$462$$ 6.00000i 0.279145i
$$463$$ −4.00000 −0.185896 −0.0929479 0.995671i $$-0.529629\pi$$
−0.0929479 + 0.995671i $$0.529629\pi$$
$$464$$ 0 0
$$465$$ 3.00000 0.139122
$$466$$ − 1.00000i − 0.0463241i
$$467$$ − 27.0000i − 1.24941i −0.780860 0.624705i $$-0.785219\pi$$
0.780860 0.624705i $$-0.214781\pi$$
$$468$$ −2.00000 −0.0924500
$$469$$ −24.0000 −1.10822
$$470$$ −13.0000 −0.599645
$$471$$ −18.0000 −0.829396
$$472$$ 0 0
$$473$$ −33.0000 −1.51734
$$474$$ 15.0000i 0.688973i
$$475$$ 0 0
$$476$$ 16.0000i 0.733359i
$$477$$ −22.0000 −1.00731
$$478$$ 0 0
$$479$$ 5.00000i 0.228456i 0.993455 + 0.114228i $$0.0364394\pi$$
−0.993455 + 0.114228i $$0.963561\pi$$
$$480$$ −1.00000 −0.0456435
$$481$$ − 8.00000i − 0.364769i
$$482$$ − 17.0000i − 0.774329i
$$483$$ 8.00000i 0.364013i
$$484$$ −2.00000 −0.0909091
$$485$$ − 2.00000i − 0.0908153i
$$486$$ 16.0000 0.725775
$$487$$ −22.0000 −0.996915 −0.498458 0.866914i $$-0.666100\pi$$
−0.498458 + 0.866914i $$0.666100\pi$$
$$488$$ −8.00000 −0.362143
$$489$$ −9.00000 −0.406994
$$490$$ 3.00000i 0.135526i
$$491$$ 33.0000i 1.48927i 0.667472 + 0.744635i $$0.267376\pi$$
−0.667472 + 0.744635i $$0.732624\pi$$
$$492$$ 2.00000 0.0901670
$$493$$ 0 0
$$494$$ 0 0
$$495$$ 6.00000i 0.269680i
$$496$$ − 3.00000i − 0.134704i
$$497$$ 4.00000 0.179425
$$498$$ 4.00000 0.179244
$$499$$ 20.0000 0.895323 0.447661 0.894203i $$-0.352257\pi$$
0.447661 + 0.894203i $$0.352257\pi$$
$$500$$ −9.00000 −0.402492
$$501$$ − 2.00000i − 0.0893534i
$$502$$ −27.0000 −1.20507
$$503$$ − 19.0000i − 0.847168i −0.905857 0.423584i $$-0.860772\pi$$
0.905857 0.423584i $$-0.139228\pi$$
$$504$$ 4.00000i 0.178174i
$$505$$ 8.00000i 0.355995i
$$506$$ 12.0000 0.533465
$$507$$ 12.0000i 0.532939i
$$508$$ − 8.00000i − 0.354943i
$$509$$ −15.0000 −0.664863 −0.332432 0.943127i $$-0.607869\pi$$
−0.332432 + 0.943127i $$0.607869\pi$$
$$510$$ − 8.00000i − 0.354246i
$$511$$ 8.00000i 0.353899i
$$512$$ 1.00000i 0.0441942i
$$513$$ 0 0
$$514$$ 13.0000i 0.573405i
$$515$$ −14.0000 −0.616914
$$516$$ 11.0000 0.484248
$$517$$ −39.0000 −1.71522
$$518$$ −16.0000 −0.703000
$$519$$ − 6.00000i − 0.263371i
$$520$$ 1.00000i 0.0438529i
$$521$$ 13.0000 0.569540 0.284770 0.958596i $$-0.408083\pi$$
0.284770 + 0.958596i $$0.408083\pi$$
$$522$$ 0 0
$$523$$ 24.0000 1.04945 0.524723 0.851273i $$-0.324169\pi$$
0.524723 + 0.851273i $$0.324169\pi$$
$$524$$ 12.0000i 0.524222i
$$525$$ − 8.00000i − 0.349149i
$$526$$ −9.00000 −0.392419
$$527$$ 24.0000 1.04546
$$528$$ −3.00000 −0.130558
$$529$$ −7.00000 −0.304348
$$530$$ 11.0000i 0.477809i
$$531$$ 0 0
$$532$$ 0 0
$$533$$ − 2.00000i − 0.0866296i
$$534$$ − 10.0000i − 0.432742i
$$535$$ 2.00000 0.0864675
$$536$$ − 12.0000i − 0.518321i
$$537$$ − 10.0000i − 0.431532i
$$538$$ 0 0
$$539$$ 9.00000i 0.387657i
$$540$$ − 5.00000i − 0.215166i
$$541$$ − 8.00000i − 0.343947i −0.985102 0.171973i $$-0.944986\pi$$
0.985102 0.171973i $$-0.0550143\pi$$
$$542$$ −13.0000 −0.558398
$$543$$ − 7.00000i − 0.300399i
$$544$$ −8.00000 −0.342997
$$545$$ 5.00000 0.214176
$$546$$ −2.00000 −0.0855921
$$547$$ 38.0000 1.62476 0.812381 0.583127i $$-0.198171\pi$$
0.812381 + 0.583127i $$0.198171\pi$$
$$548$$ 12.0000i 0.512615i
$$549$$ − 16.0000i − 0.682863i
$$550$$ −12.0000 −0.511682
$$551$$ 0 0
$$552$$ −4.00000 −0.170251
$$553$$ − 30.0000i − 1.27573i
$$554$$ − 2.00000i − 0.0849719i
$$555$$ 8.00000 0.339581
$$556$$ 20.0000 0.848189
$$557$$ 2.00000 0.0847427 0.0423714 0.999102i $$-0.486509\pi$$
0.0423714 + 0.999102i $$0.486509\pi$$
$$558$$ 6.00000 0.254000
$$559$$ − 11.0000i − 0.465250i
$$560$$ 2.00000 0.0845154
$$561$$ − 24.0000i − 1.01328i
$$562$$ 27.0000i 1.13893i
$$563$$ 11.0000i 0.463595i 0.972764 + 0.231797i $$0.0744606\pi$$
−0.972764 + 0.231797i $$0.925539\pi$$
$$564$$ 13.0000 0.547399
$$565$$ − 6.00000i − 0.252422i
$$566$$ − 4.00000i − 0.168133i
$$567$$ −2.00000 −0.0839921
$$568$$ 2.00000i 0.0839181i
$$569$$ 30.0000i 1.25767i 0.777541 + 0.628833i $$0.216467\pi$$
−0.777541 + 0.628833i $$0.783533\pi$$
$$570$$ 0 0
$$571$$ −28.0000 −1.17176 −0.585882 0.810397i $$-0.699252\pi$$
−0.585882 + 0.810397i $$0.699252\pi$$
$$572$$ 3.00000i 0.125436i
$$573$$ −8.00000 −0.334205
$$574$$ −4.00000 −0.166957
$$575$$ −16.0000 −0.667246
$$576$$ −2.00000 −0.0833333
$$577$$ − 8.00000i − 0.333044i −0.986038 0.166522i $$-0.946746\pi$$
0.986038 0.166522i $$-0.0532537\pi$$
$$578$$ − 47.0000i − 1.95494i
$$579$$ 14.0000 0.581820
$$580$$ 0 0
$$581$$ −8.00000 −0.331896
$$582$$ 2.00000i 0.0829027i
$$583$$ 33.0000i 1.36672i
$$584$$ −4.00000 −0.165521
$$585$$ −2.00000 −0.0826898
$$586$$ −14.0000 −0.578335
$$587$$ 28.0000 1.15568 0.577842 0.816149i $$-0.303895\pi$$
0.577842 + 0.816149i $$0.303895\pi$$
$$588$$ − 3.00000i − 0.123718i
$$589$$ 0 0
$$590$$ 0 0
$$591$$ − 18.0000i − 0.740421i
$$592$$ − 8.00000i − 0.328798i
$$593$$ −39.0000 −1.60154 −0.800769 0.598973i $$-0.795576\pi$$
−0.800769 + 0.598973i $$0.795576\pi$$
$$594$$ − 15.0000i − 0.615457i
$$595$$ 16.0000i 0.655936i
$$596$$ 15.0000 0.614424
$$597$$ 10.0000i 0.409273i
$$598$$ 4.00000i 0.163572i
$$599$$ − 5.00000i − 0.204294i −0.994769 0.102147i $$-0.967429\pi$$
0.994769 0.102147i $$-0.0325713\pi$$
$$600$$ 4.00000 0.163299
$$601$$ 2.00000i 0.0815817i 0.999168 + 0.0407909i $$0.0129877\pi$$
−0.999168 + 0.0407909i $$0.987012\pi$$
$$602$$ −22.0000 −0.896653
$$603$$ 24.0000 0.977356
$$604$$ 2.00000 0.0813788
$$605$$ −2.00000 −0.0813116
$$606$$ − 8.00000i − 0.324978i
$$607$$ − 3.00000i − 0.121766i −0.998145 0.0608831i $$-0.980608\pi$$
0.998145 0.0608831i $$-0.0193917\pi$$
$$608$$ 0 0
$$609$$ 0 0
$$610$$ −8.00000 −0.323911
$$611$$ − 13.0000i − 0.525924i
$$612$$ − 16.0000i − 0.646762i
$$613$$ 31.0000 1.25208 0.626039 0.779792i $$-0.284675\pi$$
0.626039 + 0.779792i $$0.284675\pi$$
$$614$$ 7.00000 0.282497
$$615$$ 2.00000 0.0806478
$$616$$ 6.00000 0.241747
$$617$$ 12.0000i 0.483102i 0.970388 + 0.241551i $$0.0776561\pi$$
−0.970388 + 0.241551i $$0.922344\pi$$
$$618$$ 14.0000 0.563163
$$619$$ 35.0000i 1.40677i 0.710810 + 0.703384i $$0.248329\pi$$
−0.710810 + 0.703384i $$0.751671\pi$$
$$620$$ − 3.00000i − 0.120483i
$$621$$ − 20.0000i − 0.802572i
$$622$$ 8.00000 0.320771
$$623$$ 20.0000i 0.801283i
$$624$$ − 1.00000i − 0.0400320i
$$625$$ 11.0000 0.440000
$$626$$ 9.00000i 0.359712i
$$627$$ 0 0
$$628$$ 18.0000i 0.718278i
$$629$$ 64.0000 2.55185
$$630$$ 4.00000i 0.159364i
$$631$$ 38.0000 1.51276 0.756378 0.654135i $$-0.226967\pi$$
0.756378 + 0.654135i $$0.226967\pi$$
$$632$$ 15.0000 0.596668
$$633$$ 3.00000 0.119239
$$634$$ −12.0000 −0.476581
$$635$$ − 8.00000i − 0.317470i
$$636$$ − 11.0000i − 0.436178i
$$637$$ −3.00000 −0.118864
$$638$$ 0 0
$$639$$ −4.00000 −0.158238
$$640$$ 1.00000i 0.0395285i
$$641$$ − 8.00000i − 0.315981i −0.987441 0.157991i $$-0.949498\pi$$
0.987441 0.157991i $$-0.0505015\pi$$
$$642$$ −2.00000 −0.0789337
$$643$$ −34.0000 −1.34083 −0.670415 0.741987i $$-0.733884\pi$$
−0.670415 + 0.741987i $$0.733884\pi$$
$$644$$ 8.00000 0.315244
$$645$$ 11.0000 0.433125
$$646$$ 0 0
$$647$$ 32.0000 1.25805 0.629025 0.777385i $$-0.283454\pi$$
0.629025 + 0.777385i $$0.283454\pi$$
$$648$$ − 1.00000i − 0.0392837i
$$649$$ 0 0
$$650$$ − 4.00000i − 0.156893i
$$651$$ 6.00000 0.235159
$$652$$ 9.00000i 0.352467i
$$653$$ 26.0000i 1.01746i 0.860927 + 0.508729i $$0.169885\pi$$
−0.860927 + 0.508729i $$0.830115\pi$$
$$654$$ −5.00000 −0.195515
$$655$$ 12.0000i 0.468879i
$$656$$ − 2.00000i − 0.0780869i
$$657$$ − 8.00000i − 0.312110i
$$658$$ −26.0000 −1.01359
$$659$$ 15.0000i 0.584317i 0.956370 + 0.292159i $$0.0943735\pi$$
−0.956370 + 0.292159i $$0.905627\pi$$
$$660$$ −3.00000 −0.116775
$$661$$ −18.0000 −0.700119 −0.350059 0.936727i $$-0.613839\pi$$
−0.350059 + 0.936727i $$0.613839\pi$$
$$662$$ −23.0000 −0.893920
$$663$$ 8.00000 0.310694
$$664$$ − 4.00000i − 0.155230i
$$665$$ 0 0
$$666$$ 16.0000 0.619987
$$667$$ 0 0
$$668$$ −2.00000 −0.0773823
$$669$$ 26.0000i 1.00522i
$$670$$ − 12.0000i − 0.463600i
$$671$$ −24.0000 −0.926510
$$672$$ −2.00000 −0.0771517
$$673$$ −9.00000 −0.346925 −0.173462 0.984841i $$-0.555495\pi$$
−0.173462 + 0.984841i $$0.555495\pi$$
$$674$$ −32.0000 −1.23259
$$675$$ 20.0000i 0.769800i
$$676$$ 12.0000 0.461538
$$677$$ − 38.0000i − 1.46046i −0.683202 0.730229i $$-0.739413\pi$$
0.683202 0.730229i $$-0.260587\pi$$
$$678$$ 6.00000i 0.230429i
$$679$$ − 4.00000i − 0.153506i
$$680$$ −8.00000 −0.306786
$$681$$ − 18.0000i − 0.689761i
$$682$$ − 9.00000i − 0.344628i
$$683$$ 24.0000 0.918334 0.459167 0.888350i $$-0.348148\pi$$
0.459167 + 0.888350i $$0.348148\pi$$
$$684$$ 0 0
$$685$$ 12.0000i 0.458496i
$$686$$ 20.0000i 0.763604i
$$687$$ −10.0000 −0.381524
$$688$$ − 11.0000i − 0.419371i
$$689$$ −11.0000 −0.419067
$$690$$ −4.00000 −0.152277
$$691$$ −28.0000 −1.06517 −0.532585 0.846376i $$-0.678779\pi$$
−0.532585 + 0.846376i $$0.678779\pi$$
$$692$$ −6.00000 −0.228086
$$693$$ 12.0000i 0.455842i
$$694$$ 2.00000i 0.0759190i
$$695$$ 20.0000 0.758643
$$696$$ 0 0
$$697$$ 16.0000 0.606043
$$698$$ − 15.0000i − 0.567758i
$$699$$ 1.00000i 0.0378235i
$$700$$ −8.00000 −0.302372
$$701$$ −27.0000 −1.01978 −0.509888 0.860241i $$-0.670313\pi$$
−0.509888 + 0.860241i $$0.670313\pi$$
$$702$$ 5.00000 0.188713
$$703$$ 0 0
$$704$$ 3.00000i 0.113067i
$$705$$ 13.0000 0.489608
$$706$$ 26.0000i 0.978523i
$$707$$ 16.0000i 0.601742i
$$708$$ 0 0
$$709$$ −15.0000 −0.563337 −0.281668 0.959512i $$-0.590888\pi$$
−0.281668 + 0.959512i $$0.590888\pi$$
$$710$$ 2.00000i 0.0750587i
$$711$$ 30.0000i 1.12509i
$$712$$ −10.0000 −0.374766
$$713$$ − 12.0000i − 0.449404i
$$714$$ − 16.0000i − 0.598785i
$$715$$ 3.00000i 0.112194i
$$716$$ −10.0000 −0.373718
$$717$$ 0 0
$$718$$ 25.0000 0.932992
$$719$$ −50.0000 −1.86469 −0.932343 0.361576i $$-0.882239\pi$$
−0.932343 + 0.361576i $$0.882239\pi$$
$$720$$ −2.00000 −0.0745356
$$721$$ −28.0000 −1.04277
$$722$$ 19.0000i 0.707107i
$$723$$ 17.0000i 0.632237i
$$724$$ −7.00000 −0.260153
$$725$$ 0 0
$$726$$ 2.00000 0.0742270
$$727$$ 8.00000i 0.296704i 0.988935 + 0.148352i $$0.0473968\pi$$
−0.988935 + 0.148352i $$0.952603\pi$$
$$728$$ 2.00000i 0.0741249i
$$729$$ −13.0000 −0.481481
$$730$$ −4.00000 −0.148047
$$731$$ 88.0000 3.25480
$$732$$ 8.00000 0.295689
$$733$$ − 24.0000i − 0.886460i −0.896408 0.443230i $$-0.853832\pi$$
0.896408 0.443230i $$-0.146168\pi$$
$$734$$ 32.0000 1.18114
$$735$$ − 3.00000i − 0.110657i
$$736$$ 4.00000i 0.147442i
$$737$$ − 36.0000i − 1.32608i
$$738$$ 4.00000 0.147242
$$739$$ − 5.00000i − 0.183928i −0.995762 0.0919640i $$-0.970686\pi$$
0.995762 0.0919640i $$-0.0293145\pi$$
$$740$$ − 8.00000i − 0.294086i
$$741$$ 0 0
$$742$$ 22.0000i 0.807645i
$$743$$ − 44.0000i − 1.61420i −0.590412 0.807102i $$-0.701035\pi$$
0.590412 0.807102i $$-0.298965\pi$$
$$744$$ 3.00000i 0.109985i
$$745$$ 15.0000 0.549557
$$746$$ − 21.0000i − 0.768865i
$$747$$ 8.00000 0.292705
$$748$$ −24.0000 −0.877527
$$749$$ 4.00000 0.146157
$$750$$ 9.00000 0.328634
$$751$$ − 32.0000i − 1.16770i −0.811863 0.583848i $$-0.801546\pi$$
0.811863 0.583848i $$-0.198454\pi$$
$$752$$ − 13.0000i − 0.474061i
$$753$$ 27.0000 0.983935
$$754$$ 0 0
$$755$$ 2.00000 0.0727875
$$756$$ − 10.0000i − 0.363696i
$$757$$ 8.00000i 0.290765i 0.989376 + 0.145382i $$0.0464413\pi$$
−0.989376 + 0.145382i $$0.953559\pi$$
$$758$$ −20.0000 −0.726433
$$759$$ −12.0000 −0.435572
$$760$$ 0 0
$$761$$ 42.0000 1.52250 0.761249 0.648459i $$-0.224586\pi$$
0.761249 + 0.648459i $$0.224586\pi$$
$$762$$ 8.00000i 0.289809i
$$763$$ 10.0000 0.362024
$$764$$ 8.00000i 0.289430i
$$765$$ − 16.0000i − 0.578481i
$$766$$ − 14.0000i − 0.505841i
$$767$$ 0 0
$$768$$ − 1.00000i − 0.0360844i
$$769$$ 20.0000i 0.721218i 0.932717 + 0.360609i $$0.117431\pi$$
−0.932717 + 0.360609i $$0.882569\pi$$
$$770$$ 6.00000 0.216225
$$771$$ − 13.0000i − 0.468184i
$$772$$ − 14.0000i − 0.503871i
$$773$$ 14.0000i 0.503545i 0.967786 + 0.251773i $$0.0810135\pi$$
−0.967786 + 0.251773i $$0.918987\pi$$
$$774$$ 22.0000 0.790774
$$775$$ 12.0000i 0.431053i
$$776$$ 2.00000 0.0717958
$$777$$ 16.0000 0.573997
$$778$$ 0 0
$$779$$ 0 0
$$780$$ − 1.00000i − 0.0358057i
$$781$$ 6.00000i 0.214697i
$$782$$ −32.0000 −1.14432
$$783$$ 0 0
$$784$$ −3.00000 −0.107143
$$785$$ 18.0000i 0.642448i
$$786$$ − 12.0000i − 0.428026i
$$787$$ 22.0000 0.784215 0.392108 0.919919i $$-0.371746\pi$$
0.392108 + 0.919919i $$0.371746\pi$$
$$788$$ −18.0000 −0.641223
$$789$$ 9.00000 0.320408
$$790$$ 15.0000 0.533676
$$791$$ − 12.0000i − 0.426671i
$$792$$ −6.00000 −0.213201
$$793$$ − 8.00000i − 0.284088i
$$794$$ − 17.0000i − 0.603307i
$$795$$ − 11.0000i − 0.390130i
$$796$$ 10.0000 0.354441
$$797$$ − 32.0000i − 1.13350i −0.823890 0.566749i $$-0.808201\pi$$
0.823890 0.566749i $$-0.191799\pi$$
$$798$$ 0 0
$$799$$ 104.000 3.67926
$$800$$ − 4.00000i − 0.141421i
$$801$$ − 20.0000i − 0.706665i
$$802$$ 27.0000i 0.953403i
$$803$$ −12.0000 −0.423471
$$804$$ 12.0000i 0.423207i
$$805$$ 8.00000 0.281963
$$806$$ 3.00000 0.105670
$$807$$ 0 0
$$808$$ −8.00000 −0.281439
$$809$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$810$$ − 1.00000i − 0.0351364i
$$811$$ 18.0000 0.632065 0.316033 0.948748i $$-0.397649\pi$$
0.316033 + 0.948748i $$0.397649\pi$$
$$812$$ 0 0
$$813$$ 13.0000 0.455930
$$814$$ − 24.0000i − 0.841200i
$$815$$ 9.00000i 0.315256i
$$816$$ 8.00000 0.280056
$$817$$ 0 0
$$818$$ −30.0000 −1.04893
$$819$$ −4.00000 −0.139771
$$820$$ − 2.00000i − 0.0698430i
$$821$$ 33.0000 1.15171 0.575854 0.817553i $$-0.304670\pi$$
0.575854 + 0.817553i $$0.304670\pi$$
$$822$$ − 12.0000i − 0.418548i
$$823$$ − 16.0000i − 0.557725i −0.960331 0.278862i $$-0.910043\pi$$
0.960331 0.278862i $$-0.0899574\pi$$
$$824$$ − 14.0000i − 0.487713i
$$825$$ 12.0000 0.417786
$$826$$ 0 0
$$827$$ − 13.0000i − 0.452054i −0.974121 0.226027i $$-0.927426\pi$$
0.974121 0.226027i $$-0.0725738\pi$$
$$828$$ −8.00000 −0.278019
$$829$$ 40.0000i 1.38926i 0.719368 + 0.694629i $$0.244431\pi$$
−0.719368 + 0.694629i $$0.755569\pi$$
$$830$$ − 4.00000i − 0.138842i
$$831$$ 2.00000i 0.0693792i
$$832$$ −1.00000 −0.0346688
$$833$$ − 24.0000i − 0.831551i
$$834$$ −20.0000 −0.692543
$$835$$ −2.00000 −0.0692129
$$836$$ 0 0
$$837$$ −15.0000 −0.518476
$$838$$ 10.0000i 0.345444i
$$839$$ − 45.0000i − 1.55357i −0.629764 0.776786i $$-0.716849\pi$$
0.629764 0.776786i $$-0.283151\pi$$
$$840$$ −2.00000 −0.0690066
$$841$$ 0 0
$$842$$ 32.0000 1.10279
$$843$$ − 27.0000i − 0.929929i
$$844$$ − 3.00000i − 0.103264i
$$845$$ 12.0000 0.412813
$$846$$ 26.0000 0.893898
$$847$$ −4.00000 −0.137442
$$848$$ −11.0000 −0.377742
$$849$$ 4.00000i 0.137280i
$$850$$ 32.0000 1.09759
$$851$$ − 32.0000i − 1.09695i
$$852$$ − 2.00000i − 0.0685189i
$$853$$ − 14.0000i − 0.479351i −0.970853 0.239675i $$-0.922959\pi$$
0.970853 0.239675i $$-0.0770410\pi$$
$$854$$ −16.0000 −0.547509
$$855$$ 0 0
$$856$$ 2.00000i 0.0683586i
$$857$$ −27.0000 −0.922302 −0.461151 0.887322i $$-0.652563\pi$$
−0.461151 + 0.887322i $$0.652563\pi$$
$$858$$ − 3.00000i − 0.102418i
$$859$$ 25.0000i 0.852989i 0.904490 + 0.426494i $$0.140252\pi$$
−0.904490 + 0.426494i $$0.859748\pi$$
$$860$$ − 11.0000i − 0.375097i
$$861$$ 4.00000 0.136320
$$862$$ 32.0000i 1.08992i
$$863$$ 46.0000 1.56586 0.782929 0.622111i $$-0.213725\pi$$
0.782929 + 0.622111i $$0.213725\pi$$
$$864$$ 5.00000 0.170103
$$865$$ −6.00000 −0.204006
$$866$$ −16.0000 −0.543702
$$867$$ 47.0000i 1.59620i
$$868$$ − 6.00000i − 0.203653i
$$869$$ 45.0000 1.52652
$$870$$ 0 0
$$871$$ 12.0000 0.406604
$$872$$ 5.00000i 0.169321i
$$873$$ 4.00000i 0.135379i
$$874$$ 0 0
$$875$$ −18.0000 −0.608511
$$876$$ 4.00000 0.135147
$$877$$ 13.0000 0.438979 0.219489 0.975615i $$-0.429561\pi$$
0.219489 + 0.975615i $$0.429561\pi$$
$$878$$ − 20.0000i − 0.674967i
$$879$$ 14.0000 0.472208
$$880$$ 3.00000i 0.101130i
$$881$$ 42.0000i 1.41502i 0.706705 + 0.707508i $$0.250181\pi$$
−0.706705 + 0.707508i $$0.749819\pi$$
$$882$$ − 6.00000i − 0.202031i
$$883$$ 26.0000 0.874970 0.437485 0.899226i $$-0.355869\pi$$
0.437485 + 0.899226i $$0.355869\pi$$
$$884$$ − 8.00000i − 0.269069i
$$885$$ 0 0
$$886$$ 4.00000 0.134383
$$887$$ 33.0000i 1.10803i 0.832506 + 0.554016i $$0.186905\pi$$
−0.832506 + 0.554016i $$0.813095\pi$$
$$888$$ 8.00000i 0.268462i
$$889$$ − 16.0000i − 0.536623i
$$890$$ −10.0000 −0.335201
$$891$$ − 3.00000i − 0.100504i
$$892$$ 26.0000 0.870544
$$893$$ 0 0
$$894$$ −15.0000 −0.501675
$$895$$ −10.0000 −0.334263
$$896$$ 2.00000i 0.0668153i
$$897$$ − 4.00000i − 0.133556i
$$898$$ 10.0000 0.333704
$$899$$ 0 0
$$900$$ 8.00000 0.266667
$$901$$ − 88.0000i − 2.93171i
$$902$$ − 6.00000i − 0.199778i
$$903$$ 22.0000 0.732114
$$904$$ 6.00000 0.199557
$$905$$ −7.00000 −0.232688
$$906$$ −2.00000 −0.0664455
$$907$$ − 28.0000i − 0.929725i −0.885383 0.464862i $$-0.846104\pi$$
0.885383 0.464862i $$-0.153896\pi$$
$$908$$ −18.0000 −0.597351
$$909$$ − 16.0000i − 0.530687i
$$910$$ 2.00000i 0.0662994i
$$911$$ 13.0000i 0.430709i 0.976536 + 0.215355i $$0.0690907\pi$$
−0.976536 + 0.215355i $$0.930909\pi$$
$$912$$ 0 0
$$913$$ − 12.0000i − 0.397142i
$$914$$ 2.00000i 0.0661541i
$$915$$ 8.00000 0.264472
$$916$$ 10.0000i 0.330409i
$$917$$ 24.0000i 0.792550i
$$918$$ 40.0000i 1.32020i
$$919$$ 30.0000 0.989609 0.494804 0.869004i $$-0.335240\pi$$
0.494804 + 0.869004i $$0.335240\pi$$
$$920$$ 4.00000i 0.131876i
$$921$$ −7.00000 −0.230658
$$922$$ 2.00000 0.0658665
$$923$$ −2.00000 −0.0658308
$$924$$ −6.00000 −0.197386
$$925$$ 32.0000i 1.05215i
$$926$$ − 4.00000i − 0.131448i
$$927$$ 28.0000 0.919641
$$928$$ 0 0
$$929$$ −30.0000 −0.984268 −0.492134 0.870519i $$-0.663783\pi$$
−0.492134 + 0.870519i $$0.663783\pi$$
$$930$$ 3.00000i 0.0983739i
$$931$$ 0 0
$$932$$ 1.00000 0.0327561
$$933$$ −8.00000 −0.261908
$$934$$ 27.0000 0.883467
$$935$$ −24.0000 −0.784884
$$936$$ − 2.00000i − 0.0653720i
$$937$$ 2.00000 0.0653372 0.0326686 0.999466i $$-0.489599\pi$$
0.0326686 + 0.999466i $$0.489599\pi$$
$$938$$ − 24.0000i − 0.783628i
$$939$$ − 9.00000i − 0.293704i
$$940$$ − 13.0000i − 0.424013i
$$941$$ −37.0000 −1.20617 −0.603083 0.797679i $$-0.706061\pi$$
−0.603083 + 0.797679i $$0.706061\pi$$
$$942$$ − 18.0000i − 0.586472i
$$943$$ − 8.00000i − 0.260516i
$$944$$ 0 0
$$945$$ − 10.0000i − 0.325300i
$$946$$ − 33.0000i − 1.07292i
$$947$$ 33.0000i 1.07236i 0.844105 + 0.536178i $$0.180132\pi$$
−0.844105 + 0.536178i $$0.819868\pi$$
$$948$$ −15.0000 −0.487177
$$949$$ − 4.00000i − 0.129845i
$$950$$ 0 0
$$951$$ 12.0000 0.389127
$$952$$ −16.0000 −0.518563
$$953$$ −1.00000 −0.0323932 −0.0161966 0.999869i $$-0.505156\pi$$
−0.0161966 + 0.999869i $$0.505156\pi$$
$$954$$ − 22.0000i − 0.712276i
$$955$$ 8.00000i 0.258874i
$$956$$ 0 0
$$957$$ 0 0
$$958$$ −5.00000 −0.161543
$$959$$ 24.0000i 0.775000i
$$960$$ − 1.00000i − 0.0322749i
$$961$$ 22.0000 0.709677
$$962$$ 8.00000 0.257930
$$963$$ −4.00000 −0.128898
$$964$$ 17.0000 0.547533
$$965$$ − 14.0000i − 0.450676i
$$966$$ −8.00000 −0.257396
$$967$$ − 13.0000i − 0.418052i −0.977910 0.209026i $$-0.932971\pi$$
0.977910 0.209026i $$-0.0670293\pi$$
$$968$$ − 2.00000i − 0.0642824i
$$969$$ 0 0
$$970$$ 2.00000 0.0642161
$$971$$ − 28.0000i − 0.898563i −0.893390 0.449281i $$-0.851680\pi$$
0.893390 0.449281i $$-0.148320\pi$$
$$972$$ 16.0000i 0.513200i
$$973$$ 40.0000 1.28234
$$974$$ − 22.0000i − 0.704925i
$$975$$ 4.00000i 0.128103i
$$976$$ − 8.00000i − 0.256074i
$$977$$ 13.0000 0.415907 0.207953 0.978139i $$-0.433320\pi$$
0.207953 + 0.978139i $$0.433320\pi$$
$$978$$ − 9.00000i − 0.287788i
$$979$$ −30.0000 −0.958804
$$980$$ −3.00000 −0.0958315
$$981$$ −10.0000 −0.319275
$$982$$ −33.0000 −1.05307
$$983$$ − 49.0000i − 1.56286i −0.623995 0.781429i $$-0.714491\pi$$
0.623995 0.781429i $$-0.285509\pi$$
$$984$$ 2.00000i 0.0637577i
$$985$$ −18.0000 −0.573528
$$986$$ 0 0
$$987$$ 26.0000 0.827589
$$988$$ 0 0
$$989$$ − 44.0000i − 1.39912i
$$990$$ −6.00000 −0.190693
$$991$$ −22.0000 −0.698853 −0.349427 0.936964i $$-0.613624\pi$$
−0.349427 + 0.936964i $$0.613624\pi$$
$$992$$ 3.00000 0.0952501
$$993$$ 23.0000 0.729883
$$994$$ 4.00000i 0.126872i
$$995$$ 10.0000 0.317021
$$996$$ 4.00000i 0.126745i
$$997$$ 8.00000i 0.253363i 0.991943 + 0.126681i $$0.0404325\pi$$
−0.991943 + 0.126681i $$0.959567\pi$$
$$998$$ 20.0000i 0.633089i
$$999$$ −40.0000 −1.26554
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 1682.2.b.a.1681.2 2
29.12 odd 4 1682.2.a.d.1.1 1
29.17 odd 4 58.2.a.b.1.1 1
29.28 even 2 inner 1682.2.b.a.1681.1 2
87.17 even 4 522.2.a.b.1.1 1
116.75 even 4 464.2.a.e.1.1 1
145.17 even 4 1450.2.b.b.349.2 2
145.104 odd 4 1450.2.a.c.1.1 1
145.133 even 4 1450.2.b.b.349.1 2
203.104 even 4 2842.2.a.e.1.1 1
232.75 even 4 1856.2.a.f.1.1 1
232.133 odd 4 1856.2.a.k.1.1 1
319.307 even 4 7018.2.a.a.1.1 1
348.191 odd 4 4176.2.a.n.1.1 1
377.220 odd 4 9802.2.a.a.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
58.2.a.b.1.1 1 29.17 odd 4
464.2.a.e.1.1 1 116.75 even 4
522.2.a.b.1.1 1 87.17 even 4
1450.2.a.c.1.1 1 145.104 odd 4
1450.2.b.b.349.1 2 145.133 even 4
1450.2.b.b.349.2 2 145.17 even 4
1682.2.a.d.1.1 1 29.12 odd 4
1682.2.b.a.1681.1 2 29.28 even 2 inner
1682.2.b.a.1681.2 2 1.1 even 1 trivial
1856.2.a.f.1.1 1 232.75 even 4
1856.2.a.k.1.1 1 232.133 odd 4
2842.2.a.e.1.1 1 203.104 even 4
4176.2.a.n.1.1 1 348.191 odd 4
7018.2.a.a.1.1 1 319.307 even 4
9802.2.a.a.1.1 1 377.220 odd 4