Properties

Label 1014.2.b.e.337.4
Level $1014$
Weight $2$
Character 1014.337
Analytic conductor $8.097$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 1014 = 2 \cdot 3 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1014.b (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(8.09683076496\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{12})\)
Defining polynomial: \(x^{4} - x^{2} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 78)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 337.4
Root \(0.866025 + 0.500000i\) of defining polynomial
Character \(\chi\) \(=\) 1014.337
Dual form 1014.2.b.e.337.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000i q^{2} +1.00000 q^{3} -1.00000 q^{4} +3.73205i q^{5} +1.00000i q^{6} +2.73205i q^{7} -1.00000i q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000i q^{2} +1.00000 q^{3} -1.00000 q^{4} +3.73205i q^{5} +1.00000i q^{6} +2.73205i q^{7} -1.00000i q^{8} +1.00000 q^{9} -3.73205 q^{10} -1.26795i q^{11} -1.00000 q^{12} -2.73205 q^{14} +3.73205i q^{15} +1.00000 q^{16} +5.73205 q^{17} +1.00000i q^{18} +4.73205i q^{19} -3.73205i q^{20} +2.73205i q^{21} +1.26795 q^{22} -4.19615 q^{23} -1.00000i q^{24} -8.92820 q^{25} +1.00000 q^{27} -2.73205i q^{28} -4.46410 q^{29} -3.73205 q^{30} +1.46410i q^{31} +1.00000i q^{32} -1.26795i q^{33} +5.73205i q^{34} -10.1962 q^{35} -1.00000 q^{36} -3.53590i q^{37} -4.73205 q^{38} +3.73205 q^{40} -9.39230i q^{41} -2.73205 q^{42} +9.66025 q^{43} +1.26795i q^{44} +3.73205i q^{45} -4.19615i q^{46} -2.19615i q^{47} +1.00000 q^{48} -0.464102 q^{49} -8.92820i q^{50} +5.73205 q^{51} -6.46410 q^{53} +1.00000i q^{54} +4.73205 q^{55} +2.73205 q^{56} +4.73205i q^{57} -4.46410i q^{58} -8.00000i q^{59} -3.73205i q^{60} -9.19615 q^{61} -1.46410 q^{62} +2.73205i q^{63} -1.00000 q^{64} +1.26795 q^{66} +13.1244i q^{67} -5.73205 q^{68} -4.19615 q^{69} -10.1962i q^{70} +4.73205i q^{71} -1.00000i q^{72} +6.26795i q^{73} +3.53590 q^{74} -8.92820 q^{75} -4.73205i q^{76} +3.46410 q^{77} -2.53590 q^{79} +3.73205i q^{80} +1.00000 q^{81} +9.39230 q^{82} -0.196152i q^{83} -2.73205i q^{84} +21.3923i q^{85} +9.66025i q^{86} -4.46410 q^{87} -1.26795 q^{88} +9.46410i q^{89} -3.73205 q^{90} +4.19615 q^{92} +1.46410i q^{93} +2.19615 q^{94} -17.6603 q^{95} +1.00000i q^{96} -6.00000i q^{97} -0.464102i q^{98} -1.26795i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 4 q^{3} - 4 q^{4} + 4 q^{9} + O(q^{10}) \) \( 4 q + 4 q^{3} - 4 q^{4} + 4 q^{9} - 8 q^{10} - 4 q^{12} - 4 q^{14} + 4 q^{16} + 16 q^{17} + 12 q^{22} + 4 q^{23} - 8 q^{25} + 4 q^{27} - 4 q^{29} - 8 q^{30} - 20 q^{35} - 4 q^{36} - 12 q^{38} + 8 q^{40} - 4 q^{42} + 4 q^{43} + 4 q^{48} + 12 q^{49} + 16 q^{51} - 12 q^{53} + 12 q^{55} + 4 q^{56} - 16 q^{61} + 8 q^{62} - 4 q^{64} + 12 q^{66} - 16 q^{68} + 4 q^{69} + 28 q^{74} - 8 q^{75} - 24 q^{79} + 4 q^{81} - 4 q^{82} - 4 q^{87} - 12 q^{88} - 8 q^{90} - 4 q^{92} - 12 q^{94} - 36 q^{95} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1014\mathbb{Z}\right)^\times\).

\(n\) \(677\) \(847\)
\(\chi(n)\) \(1\) \(-1\)

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 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000i 0.707107i
\(3\) 1.00000 0.577350
\(4\) −1.00000 −0.500000
\(5\) 3.73205i 1.66902i 0.550990 + 0.834512i \(0.314250\pi\)
−0.550990 + 0.834512i \(0.685750\pi\)
\(6\) 1.00000i 0.408248i
\(7\) 2.73205i 1.03262i 0.856402 + 0.516309i \(0.172694\pi\)
−0.856402 + 0.516309i \(0.827306\pi\)
\(8\) − 1.00000i − 0.353553i
\(9\) 1.00000 0.333333
\(10\) −3.73205 −1.18018
\(11\) − 1.26795i − 0.382301i −0.981561 0.191151i \(-0.938778\pi\)
0.981561 0.191151i \(-0.0612219\pi\)
\(12\) −1.00000 −0.288675
\(13\) 0 0
\(14\) −2.73205 −0.730171
\(15\) 3.73205i 0.963611i
\(16\) 1.00000 0.250000
\(17\) 5.73205 1.39023 0.695113 0.718900i \(-0.255354\pi\)
0.695113 + 0.718900i \(0.255354\pi\)
\(18\) 1.00000i 0.235702i
\(19\) 4.73205i 1.08561i 0.839860 + 0.542803i \(0.182637\pi\)
−0.839860 + 0.542803i \(0.817363\pi\)
\(20\) − 3.73205i − 0.834512i
\(21\) 2.73205i 0.596182i
\(22\) 1.26795 0.270328
\(23\) −4.19615 −0.874958 −0.437479 0.899229i \(-0.644129\pi\)
−0.437479 + 0.899229i \(0.644129\pi\)
\(24\) − 1.00000i − 0.204124i
\(25\) −8.92820 −1.78564
\(26\) 0 0
\(27\) 1.00000 0.192450
\(28\) − 2.73205i − 0.516309i
\(29\) −4.46410 −0.828963 −0.414481 0.910058i \(-0.636037\pi\)
−0.414481 + 0.910058i \(0.636037\pi\)
\(30\) −3.73205 −0.681376
\(31\) 1.46410i 0.262960i 0.991319 + 0.131480i \(0.0419730\pi\)
−0.991319 + 0.131480i \(0.958027\pi\)
\(32\) 1.00000i 0.176777i
\(33\) − 1.26795i − 0.220722i
\(34\) 5.73205i 0.983039i
\(35\) −10.1962 −1.72346
\(36\) −1.00000 −0.166667
\(37\) − 3.53590i − 0.581298i −0.956830 0.290649i \(-0.906129\pi\)
0.956830 0.290649i \(-0.0938712\pi\)
\(38\) −4.73205 −0.767640
\(39\) 0 0
\(40\) 3.73205 0.590089
\(41\) − 9.39230i − 1.46683i −0.679780 0.733416i \(-0.737925\pi\)
0.679780 0.733416i \(-0.262075\pi\)
\(42\) −2.73205 −0.421565
\(43\) 9.66025 1.47317 0.736587 0.676342i \(-0.236436\pi\)
0.736587 + 0.676342i \(0.236436\pi\)
\(44\) 1.26795i 0.191151i
\(45\) 3.73205i 0.556341i
\(46\) − 4.19615i − 0.618689i
\(47\) − 2.19615i − 0.320342i −0.987089 0.160171i \(-0.948795\pi\)
0.987089 0.160171i \(-0.0512045\pi\)
\(48\) 1.00000 0.144338
\(49\) −0.464102 −0.0663002
\(50\) − 8.92820i − 1.26264i
\(51\) 5.73205 0.802648
\(52\) 0 0
\(53\) −6.46410 −0.887913 −0.443956 0.896048i \(-0.646425\pi\)
−0.443956 + 0.896048i \(0.646425\pi\)
\(54\) 1.00000i 0.136083i
\(55\) 4.73205 0.638070
\(56\) 2.73205 0.365086
\(57\) 4.73205i 0.626775i
\(58\) − 4.46410i − 0.586165i
\(59\) − 8.00000i − 1.04151i −0.853706 0.520756i \(-0.825650\pi\)
0.853706 0.520756i \(-0.174350\pi\)
\(60\) − 3.73205i − 0.481806i
\(61\) −9.19615 −1.17745 −0.588723 0.808335i \(-0.700369\pi\)
−0.588723 + 0.808335i \(0.700369\pi\)
\(62\) −1.46410 −0.185941
\(63\) 2.73205i 0.344206i
\(64\) −1.00000 −0.125000
\(65\) 0 0
\(66\) 1.26795 0.156074
\(67\) 13.1244i 1.60340i 0.597730 + 0.801698i \(0.296070\pi\)
−0.597730 + 0.801698i \(0.703930\pi\)
\(68\) −5.73205 −0.695113
\(69\) −4.19615 −0.505157
\(70\) − 10.1962i − 1.21867i
\(71\) 4.73205i 0.561591i 0.959768 + 0.280796i \(0.0905983\pi\)
−0.959768 + 0.280796i \(0.909402\pi\)
\(72\) − 1.00000i − 0.117851i
\(73\) 6.26795i 0.733608i 0.930298 + 0.366804i \(0.119548\pi\)
−0.930298 + 0.366804i \(0.880452\pi\)
\(74\) 3.53590 0.411040
\(75\) −8.92820 −1.03094
\(76\) − 4.73205i − 0.542803i
\(77\) 3.46410 0.394771
\(78\) 0 0
\(79\) −2.53590 −0.285311 −0.142655 0.989772i \(-0.545564\pi\)
−0.142655 + 0.989772i \(0.545564\pi\)
\(80\) 3.73205i 0.417256i
\(81\) 1.00000 0.111111
\(82\) 9.39230 1.03721
\(83\) − 0.196152i − 0.0215305i −0.999942 0.0107653i \(-0.996573\pi\)
0.999942 0.0107653i \(-0.00342676\pi\)
\(84\) − 2.73205i − 0.298091i
\(85\) 21.3923i 2.32032i
\(86\) 9.66025i 1.04169i
\(87\) −4.46410 −0.478602
\(88\) −1.26795 −0.135164
\(89\) 9.46410i 1.00319i 0.865102 + 0.501596i \(0.167254\pi\)
−0.865102 + 0.501596i \(0.832746\pi\)
\(90\) −3.73205 −0.393393
\(91\) 0 0
\(92\) 4.19615 0.437479
\(93\) 1.46410i 0.151820i
\(94\) 2.19615 0.226516
\(95\) −17.6603 −1.81190
\(96\) 1.00000i 0.102062i
\(97\) − 6.00000i − 0.609208i −0.952479 0.304604i \(-0.901476\pi\)
0.952479 0.304604i \(-0.0985241\pi\)
\(98\) − 0.464102i − 0.0468813i
\(99\) − 1.26795i − 0.127434i
\(100\) 8.92820 0.892820
\(101\) −1.92820 −0.191863 −0.0959317 0.995388i \(-0.530583\pi\)
−0.0959317 + 0.995388i \(0.530583\pi\)
\(102\) 5.73205i 0.567558i
\(103\) 15.2679 1.50440 0.752198 0.658937i \(-0.228994\pi\)
0.752198 + 0.658937i \(0.228994\pi\)
\(104\) 0 0
\(105\) −10.1962 −0.995043
\(106\) − 6.46410i − 0.627849i
\(107\) 10.1962 0.985699 0.492850 0.870114i \(-0.335955\pi\)
0.492850 + 0.870114i \(0.335955\pi\)
\(108\) −1.00000 −0.0962250
\(109\) 1.46410i 0.140236i 0.997539 + 0.0701178i \(0.0223375\pi\)
−0.997539 + 0.0701178i \(0.977662\pi\)
\(110\) 4.73205i 0.451183i
\(111\) − 3.53590i − 0.335613i
\(112\) 2.73205i 0.258155i
\(113\) −1.33975 −0.126033 −0.0630163 0.998012i \(-0.520072\pi\)
−0.0630163 + 0.998012i \(0.520072\pi\)
\(114\) −4.73205 −0.443197
\(115\) − 15.6603i − 1.46033i
\(116\) 4.46410 0.414481
\(117\) 0 0
\(118\) 8.00000 0.736460
\(119\) 15.6603i 1.43557i
\(120\) 3.73205 0.340688
\(121\) 9.39230 0.853846
\(122\) − 9.19615i − 0.832581i
\(123\) − 9.39230i − 0.846876i
\(124\) − 1.46410i − 0.131480i
\(125\) − 14.6603i − 1.31125i
\(126\) −2.73205 −0.243390
\(127\) 9.85641 0.874615 0.437307 0.899312i \(-0.355932\pi\)
0.437307 + 0.899312i \(0.355932\pi\)
\(128\) − 1.00000i − 0.0883883i
\(129\) 9.66025 0.850538
\(130\) 0 0
\(131\) 6.53590 0.571044 0.285522 0.958372i \(-0.407833\pi\)
0.285522 + 0.958372i \(0.407833\pi\)
\(132\) 1.26795i 0.110361i
\(133\) −12.9282 −1.12102
\(134\) −13.1244 −1.13377
\(135\) 3.73205i 0.321204i
\(136\) − 5.73205i − 0.491519i
\(137\) − 11.9282i − 1.01910i −0.860442 0.509548i \(-0.829813\pi\)
0.860442 0.509548i \(-0.170187\pi\)
\(138\) − 4.19615i − 0.357200i
\(139\) 17.8564 1.51456 0.757280 0.653090i \(-0.226528\pi\)
0.757280 + 0.653090i \(0.226528\pi\)
\(140\) 10.1962 0.861732
\(141\) − 2.19615i − 0.184949i
\(142\) −4.73205 −0.397105
\(143\) 0 0
\(144\) 1.00000 0.0833333
\(145\) − 16.6603i − 1.38356i
\(146\) −6.26795 −0.518739
\(147\) −0.464102 −0.0382785
\(148\) 3.53590i 0.290649i
\(149\) 13.1962i 1.08107i 0.841321 + 0.540535i \(0.181778\pi\)
−0.841321 + 0.540535i \(0.818222\pi\)
\(150\) − 8.92820i − 0.728985i
\(151\) − 6.73205i − 0.547847i −0.961752 0.273923i \(-0.911679\pi\)
0.961752 0.273923i \(-0.0883214\pi\)
\(152\) 4.73205 0.383820
\(153\) 5.73205 0.463409
\(154\) 3.46410i 0.279145i
\(155\) −5.46410 −0.438887
\(156\) 0 0
\(157\) 7.58846 0.605625 0.302812 0.953050i \(-0.402074\pi\)
0.302812 + 0.953050i \(0.402074\pi\)
\(158\) − 2.53590i − 0.201745i
\(159\) −6.46410 −0.512637
\(160\) −3.73205 −0.295045
\(161\) − 11.4641i − 0.903498i
\(162\) 1.00000i 0.0785674i
\(163\) 13.4641i 1.05459i 0.849682 + 0.527295i \(0.176794\pi\)
−0.849682 + 0.527295i \(0.823206\pi\)
\(164\) 9.39230i 0.733416i
\(165\) 4.73205 0.368390
\(166\) 0.196152 0.0152244
\(167\) 9.46410i 0.732354i 0.930545 + 0.366177i \(0.119334\pi\)
−0.930545 + 0.366177i \(0.880666\pi\)
\(168\) 2.73205 0.210782
\(169\) 0 0
\(170\) −21.3923 −1.64071
\(171\) 4.73205i 0.361869i
\(172\) −9.66025 −0.736587
\(173\) −4.39230 −0.333941 −0.166970 0.985962i \(-0.553398\pi\)
−0.166970 + 0.985962i \(0.553398\pi\)
\(174\) − 4.46410i − 0.338423i
\(175\) − 24.3923i − 1.84388i
\(176\) − 1.26795i − 0.0955753i
\(177\) − 8.00000i − 0.601317i
\(178\) −9.46410 −0.709364
\(179\) 16.0526 1.19982 0.599912 0.800066i \(-0.295202\pi\)
0.599912 + 0.800066i \(0.295202\pi\)
\(180\) − 3.73205i − 0.278171i
\(181\) −19.1962 −1.42684 −0.713419 0.700737i \(-0.752855\pi\)
−0.713419 + 0.700737i \(0.752855\pi\)
\(182\) 0 0
\(183\) −9.19615 −0.679799
\(184\) 4.19615i 0.309344i
\(185\) 13.1962 0.970200
\(186\) −1.46410 −0.107353
\(187\) − 7.26795i − 0.531485i
\(188\) 2.19615i 0.160171i
\(189\) 2.73205i 0.198727i
\(190\) − 17.6603i − 1.28121i
\(191\) −6.92820 −0.501307 −0.250654 0.968077i \(-0.580646\pi\)
−0.250654 + 0.968077i \(0.580646\pi\)
\(192\) −1.00000 −0.0721688
\(193\) 11.7321i 0.844491i 0.906481 + 0.422246i \(0.138758\pi\)
−0.906481 + 0.422246i \(0.861242\pi\)
\(194\) 6.00000 0.430775
\(195\) 0 0
\(196\) 0.464102 0.0331501
\(197\) − 17.8564i − 1.27222i −0.771600 0.636108i \(-0.780543\pi\)
0.771600 0.636108i \(-0.219457\pi\)
\(198\) 1.26795 0.0901092
\(199\) 14.1962 1.00634 0.503169 0.864188i \(-0.332167\pi\)
0.503169 + 0.864188i \(0.332167\pi\)
\(200\) 8.92820i 0.631319i
\(201\) 13.1244i 0.925721i
\(202\) − 1.92820i − 0.135668i
\(203\) − 12.1962i − 0.856002i
\(204\) −5.73205 −0.401324
\(205\) 35.0526 2.44818
\(206\) 15.2679i 1.06377i
\(207\) −4.19615 −0.291653
\(208\) 0 0
\(209\) 6.00000 0.415029
\(210\) − 10.1962i − 0.703601i
\(211\) 16.3923 1.12849 0.564246 0.825606i \(-0.309167\pi\)
0.564246 + 0.825606i \(0.309167\pi\)
\(212\) 6.46410 0.443956
\(213\) 4.73205i 0.324235i
\(214\) 10.1962i 0.696995i
\(215\) 36.0526i 2.45876i
\(216\) − 1.00000i − 0.0680414i
\(217\) −4.00000 −0.271538
\(218\) −1.46410 −0.0991615
\(219\) 6.26795i 0.423549i
\(220\) −4.73205 −0.319035
\(221\) 0 0
\(222\) 3.53590 0.237314
\(223\) − 26.9282i − 1.80325i −0.432523 0.901623i \(-0.642377\pi\)
0.432523 0.901623i \(-0.357623\pi\)
\(224\) −2.73205 −0.182543
\(225\) −8.92820 −0.595214
\(226\) − 1.33975i − 0.0891186i
\(227\) − 12.1962i − 0.809487i −0.914430 0.404744i \(-0.867361\pi\)
0.914430 0.404744i \(-0.132639\pi\)
\(228\) − 4.73205i − 0.313388i
\(229\) 11.8564i 0.783493i 0.920073 + 0.391747i \(0.128129\pi\)
−0.920073 + 0.391747i \(0.871871\pi\)
\(230\) 15.6603 1.03261
\(231\) 3.46410 0.227921
\(232\) 4.46410i 0.293083i
\(233\) 7.85641 0.514690 0.257345 0.966320i \(-0.417152\pi\)
0.257345 + 0.966320i \(0.417152\pi\)
\(234\) 0 0
\(235\) 8.19615 0.534658
\(236\) 8.00000i 0.520756i
\(237\) −2.53590 −0.164724
\(238\) −15.6603 −1.01510
\(239\) 7.66025i 0.495501i 0.968824 + 0.247750i \(0.0796913\pi\)
−0.968824 + 0.247750i \(0.920309\pi\)
\(240\) 3.73205i 0.240903i
\(241\) − 13.5885i − 0.875309i −0.899143 0.437655i \(-0.855809\pi\)
0.899143 0.437655i \(-0.144191\pi\)
\(242\) 9.39230i 0.603760i
\(243\) 1.00000 0.0641500
\(244\) 9.19615 0.588723
\(245\) − 1.73205i − 0.110657i
\(246\) 9.39230 0.598831
\(247\) 0 0
\(248\) 1.46410 0.0929705
\(249\) − 0.196152i − 0.0124307i
\(250\) 14.6603 0.927196
\(251\) −13.4641 −0.849847 −0.424923 0.905229i \(-0.639699\pi\)
−0.424923 + 0.905229i \(0.639699\pi\)
\(252\) − 2.73205i − 0.172103i
\(253\) 5.32051i 0.334497i
\(254\) 9.85641i 0.618446i
\(255\) 21.3923i 1.33964i
\(256\) 1.00000 0.0625000
\(257\) −9.33975 −0.582597 −0.291299 0.956632i \(-0.594087\pi\)
−0.291299 + 0.956632i \(0.594087\pi\)
\(258\) 9.66025i 0.601421i
\(259\) 9.66025 0.600259
\(260\) 0 0
\(261\) −4.46410 −0.276321
\(262\) 6.53590i 0.403789i
\(263\) −10.0526 −0.619867 −0.309934 0.950758i \(-0.600307\pi\)
−0.309934 + 0.950758i \(0.600307\pi\)
\(264\) −1.26795 −0.0780369
\(265\) − 24.1244i − 1.48195i
\(266\) − 12.9282i − 0.792679i
\(267\) 9.46410i 0.579194i
\(268\) − 13.1244i − 0.801698i
\(269\) 5.46410 0.333152 0.166576 0.986029i \(-0.446729\pi\)
0.166576 + 0.986029i \(0.446729\pi\)
\(270\) −3.73205 −0.227125
\(271\) 21.8564i 1.32768i 0.747874 + 0.663841i \(0.231075\pi\)
−0.747874 + 0.663841i \(0.768925\pi\)
\(272\) 5.73205 0.347557
\(273\) 0 0
\(274\) 11.9282 0.720609
\(275\) 11.3205i 0.682652i
\(276\) 4.19615 0.252579
\(277\) 5.73205 0.344406 0.172203 0.985062i \(-0.444912\pi\)
0.172203 + 0.985062i \(0.444912\pi\)
\(278\) 17.8564i 1.07096i
\(279\) 1.46410i 0.0876535i
\(280\) 10.1962i 0.609337i
\(281\) − 12.3205i − 0.734980i −0.930027 0.367490i \(-0.880217\pi\)
0.930027 0.367490i \(-0.119783\pi\)
\(282\) 2.19615 0.130779
\(283\) −25.6603 −1.52534 −0.762672 0.646786i \(-0.776113\pi\)
−0.762672 + 0.646786i \(0.776113\pi\)
\(284\) − 4.73205i − 0.280796i
\(285\) −17.6603 −1.04610
\(286\) 0 0
\(287\) 25.6603 1.51468
\(288\) 1.00000i 0.0589256i
\(289\) 15.8564 0.932730
\(290\) 16.6603 0.978324
\(291\) − 6.00000i − 0.351726i
\(292\) − 6.26795i − 0.366804i
\(293\) − 30.5167i − 1.78280i −0.453215 0.891401i \(-0.649723\pi\)
0.453215 0.891401i \(-0.350277\pi\)
\(294\) − 0.464102i − 0.0270670i
\(295\) 29.8564 1.73831
\(296\) −3.53590 −0.205520
\(297\) − 1.26795i − 0.0735739i
\(298\) −13.1962 −0.764433
\(299\) 0 0
\(300\) 8.92820 0.515470
\(301\) 26.3923i 1.52123i
\(302\) 6.73205 0.387386
\(303\) −1.92820 −0.110772
\(304\) 4.73205i 0.271402i
\(305\) − 34.3205i − 1.96519i
\(306\) 5.73205i 0.327680i
\(307\) 22.5885i 1.28919i 0.764524 + 0.644596i \(0.222974\pi\)
−0.764524 + 0.644596i \(0.777026\pi\)
\(308\) −3.46410 −0.197386
\(309\) 15.2679 0.868563
\(310\) − 5.46410i − 0.310340i
\(311\) 1.66025 0.0941444 0.0470722 0.998891i \(-0.485011\pi\)
0.0470722 + 0.998891i \(0.485011\pi\)
\(312\) 0 0
\(313\) 6.53590 0.369431 0.184715 0.982792i \(-0.440864\pi\)
0.184715 + 0.982792i \(0.440864\pi\)
\(314\) 7.58846i 0.428241i
\(315\) −10.1962 −0.574488
\(316\) 2.53590 0.142655
\(317\) 20.6603i 1.16040i 0.814476 + 0.580198i \(0.197025\pi\)
−0.814476 + 0.580198i \(0.802975\pi\)
\(318\) − 6.46410i − 0.362489i
\(319\) 5.66025i 0.316913i
\(320\) − 3.73205i − 0.208628i
\(321\) 10.1962 0.569094
\(322\) 11.4641 0.638869
\(323\) 27.1244i 1.50924i
\(324\) −1.00000 −0.0555556
\(325\) 0 0
\(326\) −13.4641 −0.745708
\(327\) 1.46410i 0.0809650i
\(328\) −9.39230 −0.518603
\(329\) 6.00000 0.330791
\(330\) 4.73205i 0.260491i
\(331\) 20.0000i 1.09930i 0.835395 + 0.549650i \(0.185239\pi\)
−0.835395 + 0.549650i \(0.814761\pi\)
\(332\) 0.196152i 0.0107653i
\(333\) − 3.53590i − 0.193766i
\(334\) −9.46410 −0.517853
\(335\) −48.9808 −2.67610
\(336\) 2.73205i 0.149046i
\(337\) 20.8564 1.13612 0.568060 0.822987i \(-0.307694\pi\)
0.568060 + 0.822987i \(0.307694\pi\)
\(338\) 0 0
\(339\) −1.33975 −0.0727650
\(340\) − 21.3923i − 1.16016i
\(341\) 1.85641 0.100530
\(342\) −4.73205 −0.255880
\(343\) 17.8564i 0.964155i
\(344\) − 9.66025i − 0.520846i
\(345\) − 15.6603i − 0.843120i
\(346\) − 4.39230i − 0.236132i
\(347\) 33.1244 1.77821 0.889104 0.457705i \(-0.151328\pi\)
0.889104 + 0.457705i \(0.151328\pi\)
\(348\) 4.46410 0.239301
\(349\) − 15.3205i − 0.820088i −0.912066 0.410044i \(-0.865513\pi\)
0.912066 0.410044i \(-0.134487\pi\)
\(350\) 24.3923 1.30382
\(351\) 0 0
\(352\) 1.26795 0.0675819
\(353\) − 21.7846i − 1.15948i −0.814802 0.579739i \(-0.803155\pi\)
0.814802 0.579739i \(-0.196845\pi\)
\(354\) 8.00000 0.425195
\(355\) −17.6603 −0.937309
\(356\) − 9.46410i − 0.501596i
\(357\) 15.6603i 0.828829i
\(358\) 16.0526i 0.848404i
\(359\) 1.12436i 0.0593412i 0.999560 + 0.0296706i \(0.00944584\pi\)
−0.999560 + 0.0296706i \(0.990554\pi\)
\(360\) 3.73205 0.196696
\(361\) −3.39230 −0.178542
\(362\) − 19.1962i − 1.00893i
\(363\) 9.39230 0.492968
\(364\) 0 0
\(365\) −23.3923 −1.22441
\(366\) − 9.19615i − 0.480691i
\(367\) −11.2679 −0.588182 −0.294091 0.955777i \(-0.595017\pi\)
−0.294091 + 0.955777i \(0.595017\pi\)
\(368\) −4.19615 −0.218740
\(369\) − 9.39230i − 0.488944i
\(370\) 13.1962i 0.686035i
\(371\) − 17.6603i − 0.916875i
\(372\) − 1.46410i − 0.0759101i
\(373\) −13.7321 −0.711019 −0.355509 0.934673i \(-0.615693\pi\)
−0.355509 + 0.934673i \(0.615693\pi\)
\(374\) 7.26795 0.375817
\(375\) − 14.6603i − 0.757052i
\(376\) −2.19615 −0.113258
\(377\) 0 0
\(378\) −2.73205 −0.140522
\(379\) − 5.46410i − 0.280672i −0.990104 0.140336i \(-0.955182\pi\)
0.990104 0.140336i \(-0.0448183\pi\)
\(380\) 17.6603 0.905952
\(381\) 9.85641 0.504959
\(382\) − 6.92820i − 0.354478i
\(383\) − 1.46410i − 0.0748121i −0.999300 0.0374060i \(-0.988091\pi\)
0.999300 0.0374060i \(-0.0119095\pi\)
\(384\) − 1.00000i − 0.0510310i
\(385\) 12.9282i 0.658882i
\(386\) −11.7321 −0.597146
\(387\) 9.66025 0.491058
\(388\) 6.00000i 0.304604i
\(389\) −11.7846 −0.597503 −0.298752 0.954331i \(-0.596570\pi\)
−0.298752 + 0.954331i \(0.596570\pi\)
\(390\) 0 0
\(391\) −24.0526 −1.21639
\(392\) 0.464102i 0.0234407i
\(393\) 6.53590 0.329692
\(394\) 17.8564 0.899593
\(395\) − 9.46410i − 0.476191i
\(396\) 1.26795i 0.0637168i
\(397\) − 20.3923i − 1.02346i −0.859146 0.511730i \(-0.829005\pi\)
0.859146 0.511730i \(-0.170995\pi\)
\(398\) 14.1962i 0.711589i
\(399\) −12.9282 −0.647220
\(400\) −8.92820 −0.446410
\(401\) − 8.07180i − 0.403086i −0.979480 0.201543i \(-0.935404\pi\)
0.979480 0.201543i \(-0.0645956\pi\)
\(402\) −13.1244 −0.654583
\(403\) 0 0
\(404\) 1.92820 0.0959317
\(405\) 3.73205i 0.185447i
\(406\) 12.1962 0.605285
\(407\) −4.48334 −0.222231
\(408\) − 5.73205i − 0.283779i
\(409\) 17.7321i 0.876793i 0.898782 + 0.438397i \(0.144454\pi\)
−0.898782 + 0.438397i \(0.855546\pi\)
\(410\) 35.0526i 1.73112i
\(411\) − 11.9282i − 0.588375i
\(412\) −15.2679 −0.752198
\(413\) 21.8564 1.07548
\(414\) − 4.19615i − 0.206230i
\(415\) 0.732051 0.0359350
\(416\) 0 0
\(417\) 17.8564 0.874432
\(418\) 6.00000i 0.293470i
\(419\) −17.4641 −0.853177 −0.426589 0.904446i \(-0.640285\pi\)
−0.426589 + 0.904446i \(0.640285\pi\)
\(420\) 10.1962 0.497521
\(421\) − 22.7128i − 1.10695i −0.832864 0.553477i \(-0.813301\pi\)
0.832864 0.553477i \(-0.186699\pi\)
\(422\) 16.3923i 0.797965i
\(423\) − 2.19615i − 0.106781i
\(424\) 6.46410i 0.313925i
\(425\) −51.1769 −2.48244
\(426\) −4.73205 −0.229269
\(427\) − 25.1244i − 1.21585i
\(428\) −10.1962 −0.492850
\(429\) 0 0
\(430\) −36.0526 −1.73861
\(431\) 13.1244i 0.632178i 0.948730 + 0.316089i \(0.102370\pi\)
−0.948730 + 0.316089i \(0.897630\pi\)
\(432\) 1.00000 0.0481125
\(433\) −12.8564 −0.617839 −0.308920 0.951088i \(-0.599967\pi\)
−0.308920 + 0.951088i \(0.599967\pi\)
\(434\) − 4.00000i − 0.192006i
\(435\) − 16.6603i − 0.798798i
\(436\) − 1.46410i − 0.0701178i
\(437\) − 19.8564i − 0.949861i
\(438\) −6.26795 −0.299494
\(439\) 0.339746 0.0162152 0.00810760 0.999967i \(-0.497419\pi\)
0.00810760 + 0.999967i \(0.497419\pi\)
\(440\) − 4.73205i − 0.225592i
\(441\) −0.464102 −0.0221001
\(442\) 0 0
\(443\) 15.6077 0.741544 0.370772 0.928724i \(-0.379093\pi\)
0.370772 + 0.928724i \(0.379093\pi\)
\(444\) 3.53590i 0.167806i
\(445\) −35.3205 −1.67435
\(446\) 26.9282 1.27509
\(447\) 13.1962i 0.624157i
\(448\) − 2.73205i − 0.129077i
\(449\) − 11.3205i − 0.534248i −0.963662 0.267124i \(-0.913927\pi\)
0.963662 0.267124i \(-0.0860733\pi\)
\(450\) − 8.92820i − 0.420880i
\(451\) −11.9090 −0.560771
\(452\) 1.33975 0.0630163
\(453\) − 6.73205i − 0.316299i
\(454\) 12.1962 0.572394
\(455\) 0 0
\(456\) 4.73205 0.221599
\(457\) 1.33975i 0.0626707i 0.999509 + 0.0313353i \(0.00997598\pi\)
−0.999509 + 0.0313353i \(0.990024\pi\)
\(458\) −11.8564 −0.554013
\(459\) 5.73205 0.267549
\(460\) 15.6603i 0.730163i
\(461\) 22.2679i 1.03712i 0.855041 + 0.518561i \(0.173532\pi\)
−0.855041 + 0.518561i \(0.826468\pi\)
\(462\) 3.46410i 0.161165i
\(463\) − 10.0526i − 0.467182i −0.972335 0.233591i \(-0.924952\pi\)
0.972335 0.233591i \(-0.0750477\pi\)
\(464\) −4.46410 −0.207241
\(465\) −5.46410 −0.253392
\(466\) 7.85641i 0.363941i
\(467\) −18.5885 −0.860171 −0.430086 0.902788i \(-0.641517\pi\)
−0.430086 + 0.902788i \(0.641517\pi\)
\(468\) 0 0
\(469\) −35.8564 −1.65570
\(470\) 8.19615i 0.378060i
\(471\) 7.58846 0.349658
\(472\) −8.00000 −0.368230
\(473\) − 12.2487i − 0.563196i
\(474\) − 2.53590i − 0.116478i
\(475\) − 42.2487i − 1.93850i
\(476\) − 15.6603i − 0.717787i
\(477\) −6.46410 −0.295971
\(478\) −7.66025 −0.350372
\(479\) − 33.4641i − 1.52901i −0.644616 0.764507i \(-0.722983\pi\)
0.644616 0.764507i \(-0.277017\pi\)
\(480\) −3.73205 −0.170344
\(481\) 0 0
\(482\) 13.5885 0.618937
\(483\) − 11.4641i − 0.521635i
\(484\) −9.39230 −0.426923
\(485\) 22.3923 1.01678
\(486\) 1.00000i 0.0453609i
\(487\) − 3.12436i − 0.141578i −0.997491 0.0707890i \(-0.977448\pi\)
0.997491 0.0707890i \(-0.0225517\pi\)
\(488\) 9.19615i 0.416290i
\(489\) 13.4641i 0.608868i
\(490\) 1.73205 0.0782461
\(491\) −8.73205 −0.394072 −0.197036 0.980396i \(-0.563132\pi\)
−0.197036 + 0.980396i \(0.563132\pi\)
\(492\) 9.39230i 0.423438i
\(493\) −25.5885 −1.15245
\(494\) 0 0
\(495\) 4.73205 0.212690
\(496\) 1.46410i 0.0657401i
\(497\) −12.9282 −0.579909
\(498\) 0.196152 0.00878980
\(499\) 32.0000i 1.43252i 0.697835 + 0.716258i \(0.254147\pi\)
−0.697835 + 0.716258i \(0.745853\pi\)
\(500\) 14.6603i 0.655626i
\(501\) 9.46410i 0.422825i
\(502\) − 13.4641i − 0.600932i
\(503\) 40.9808 1.82724 0.913621 0.406567i \(-0.133274\pi\)
0.913621 + 0.406567i \(0.133274\pi\)
\(504\) 2.73205 0.121695
\(505\) − 7.19615i − 0.320225i
\(506\) −5.32051 −0.236525
\(507\) 0 0
\(508\) −9.85641 −0.437307
\(509\) 13.7321i 0.608662i 0.952566 + 0.304331i \(0.0984330\pi\)
−0.952566 + 0.304331i \(0.901567\pi\)
\(510\) −21.3923 −0.947267
\(511\) −17.1244 −0.757537
\(512\) 1.00000i 0.0441942i
\(513\) 4.73205i 0.208925i
\(514\) − 9.33975i − 0.411959i
\(515\) 56.9808i 2.51087i
\(516\) −9.66025 −0.425269
\(517\) −2.78461 −0.122467
\(518\) 9.66025i 0.424447i
\(519\) −4.39230 −0.192801
\(520\) 0 0
\(521\) 41.4449 1.81573 0.907866 0.419260i \(-0.137710\pi\)
0.907866 + 0.419260i \(0.137710\pi\)
\(522\) − 4.46410i − 0.195388i
\(523\) −22.4449 −0.981445 −0.490723 0.871316i \(-0.663267\pi\)
−0.490723 + 0.871316i \(0.663267\pi\)
\(524\) −6.53590 −0.285522
\(525\) − 24.3923i − 1.06457i
\(526\) − 10.0526i − 0.438312i
\(527\) 8.39230i 0.365575i
\(528\) − 1.26795i − 0.0551804i
\(529\) −5.39230 −0.234448
\(530\) 24.1244 1.04790
\(531\) − 8.00000i − 0.347170i
\(532\) 12.9282 0.560509
\(533\) 0 0
\(534\) −9.46410 −0.409552
\(535\) 38.0526i 1.64516i
\(536\) 13.1244 0.566886
\(537\) 16.0526 0.692719
\(538\) 5.46410i 0.235574i
\(539\) 0.588457i 0.0253466i
\(540\) − 3.73205i − 0.160602i
\(541\) − 5.67949i − 0.244180i −0.992519 0.122090i \(-0.961040\pi\)
0.992519 0.122090i \(-0.0389597\pi\)
\(542\) −21.8564 −0.938813
\(543\) −19.1962 −0.823786
\(544\) 5.73205i 0.245760i
\(545\) −5.46410 −0.234056
\(546\) 0 0
\(547\) −4.19615 −0.179415 −0.0897073 0.995968i \(-0.528593\pi\)
−0.0897073 + 0.995968i \(0.528593\pi\)
\(548\) 11.9282i 0.509548i
\(549\) −9.19615 −0.392482
\(550\) −11.3205 −0.482708
\(551\) − 21.1244i − 0.899928i
\(552\) 4.19615i 0.178600i
\(553\) − 6.92820i − 0.294617i
\(554\) 5.73205i 0.243532i
\(555\) 13.1962 0.560145
\(556\) −17.8564 −0.757280
\(557\) − 42.3731i − 1.79540i −0.440603 0.897702i \(-0.645235\pi\)
0.440603 0.897702i \(-0.354765\pi\)
\(558\) −1.46410 −0.0619804
\(559\) 0 0
\(560\) −10.1962 −0.430866
\(561\) − 7.26795i − 0.306853i
\(562\) 12.3205 0.519709
\(563\) −34.9282 −1.47205 −0.736024 0.676955i \(-0.763299\pi\)
−0.736024 + 0.676955i \(0.763299\pi\)
\(564\) 2.19615i 0.0924747i
\(565\) − 5.00000i − 0.210352i
\(566\) − 25.6603i − 1.07858i
\(567\) 2.73205i 0.114735i
\(568\) 4.73205 0.198552
\(569\) −30.6410 −1.28454 −0.642269 0.766479i \(-0.722007\pi\)
−0.642269 + 0.766479i \(0.722007\pi\)
\(570\) − 17.6603i − 0.739707i
\(571\) −14.0526 −0.588081 −0.294041 0.955793i \(-0.595000\pi\)
−0.294041 + 0.955793i \(0.595000\pi\)
\(572\) 0 0
\(573\) −6.92820 −0.289430
\(574\) 25.6603i 1.07104i
\(575\) 37.4641 1.56236
\(576\) −1.00000 −0.0416667
\(577\) 3.73205i 0.155367i 0.996978 + 0.0776837i \(0.0247524\pi\)
−0.996978 + 0.0776837i \(0.975248\pi\)
\(578\) 15.8564i 0.659540i
\(579\) 11.7321i 0.487567i
\(580\) 16.6603i 0.691779i
\(581\) 0.535898 0.0222328
\(582\) 6.00000 0.248708
\(583\) 8.19615i 0.339450i
\(584\) 6.26795 0.259370
\(585\) 0 0
\(586\) 30.5167 1.26063
\(587\) 16.0000i 0.660391i 0.943913 + 0.330195i \(0.107115\pi\)
−0.943913 + 0.330195i \(0.892885\pi\)
\(588\) 0.464102 0.0191392
\(589\) −6.92820 −0.285472
\(590\) 29.8564i 1.22917i
\(591\) − 17.8564i − 0.734514i
\(592\) − 3.53590i − 0.145325i
\(593\) 9.14359i 0.375482i 0.982219 + 0.187741i \(0.0601166\pi\)
−0.982219 + 0.187741i \(0.939883\pi\)
\(594\) 1.26795 0.0520246
\(595\) −58.4449 −2.39601
\(596\) − 13.1962i − 0.540535i
\(597\) 14.1962 0.581010
\(598\) 0 0
\(599\) −2.53590 −0.103614 −0.0518070 0.998657i \(-0.516498\pi\)
−0.0518070 + 0.998657i \(0.516498\pi\)
\(600\) 8.92820i 0.364492i
\(601\) 7.92820 0.323398 0.161699 0.986840i \(-0.448303\pi\)
0.161699 + 0.986840i \(0.448303\pi\)
\(602\) −26.3923 −1.07567
\(603\) 13.1244i 0.534465i
\(604\) 6.73205i 0.273923i
\(605\) 35.0526i 1.42509i
\(606\) − 1.92820i − 0.0783279i
\(607\) −40.7846 −1.65540 −0.827698 0.561174i \(-0.810350\pi\)
−0.827698 + 0.561174i \(0.810350\pi\)
\(608\) −4.73205 −0.191910
\(609\) − 12.1962i − 0.494213i
\(610\) 34.3205 1.38960
\(611\) 0 0
\(612\) −5.73205 −0.231704
\(613\) − 9.39230i − 0.379352i −0.981847 0.189676i \(-0.939256\pi\)
0.981847 0.189676i \(-0.0607437\pi\)
\(614\) −22.5885 −0.911596
\(615\) 35.0526 1.41346
\(616\) − 3.46410i − 0.139573i
\(617\) − 13.2487i − 0.533373i −0.963783 0.266687i \(-0.914071\pi\)
0.963783 0.266687i \(-0.0859288\pi\)
\(618\) 15.2679i 0.614167i
\(619\) − 17.4641i − 0.701942i −0.936386 0.350971i \(-0.885852\pi\)
0.936386 0.350971i \(-0.114148\pi\)
\(620\) 5.46410 0.219444
\(621\) −4.19615 −0.168386
\(622\) 1.66025i 0.0665701i
\(623\) −25.8564 −1.03592
\(624\) 0 0
\(625\) 10.0718 0.402872
\(626\) 6.53590i 0.261227i
\(627\) 6.00000 0.239617
\(628\) −7.58846 −0.302812
\(629\) − 20.2679i − 0.808136i
\(630\) − 10.1962i − 0.406224i
\(631\) 7.71281i 0.307042i 0.988145 + 0.153521i \(0.0490613\pi\)
−0.988145 + 0.153521i \(0.950939\pi\)
\(632\) 2.53590i 0.100873i
\(633\) 16.3923 0.651536
\(634\) −20.6603 −0.820524
\(635\) 36.7846i 1.45975i
\(636\) 6.46410 0.256318
\(637\) 0 0
\(638\) −5.66025 −0.224092
\(639\) 4.73205i 0.187197i
\(640\) 3.73205 0.147522
\(641\) 25.9808 1.02618 0.513089 0.858335i \(-0.328501\pi\)
0.513089 + 0.858335i \(0.328501\pi\)
\(642\) 10.1962i 0.402410i
\(643\) − 13.8564i − 0.546443i −0.961951 0.273222i \(-0.911911\pi\)
0.961951 0.273222i \(-0.0880892\pi\)
\(644\) 11.4641i 0.451749i
\(645\) 36.0526i 1.41957i
\(646\) −27.1244 −1.06719
\(647\) 22.2487 0.874687 0.437344 0.899295i \(-0.355919\pi\)
0.437344 + 0.899295i \(0.355919\pi\)
\(648\) − 1.00000i − 0.0392837i
\(649\) −10.1436 −0.398171
\(650\) 0 0
\(651\) −4.00000 −0.156772
\(652\) − 13.4641i − 0.527295i
\(653\) −17.4641 −0.683423 −0.341712 0.939805i \(-0.611007\pi\)
−0.341712 + 0.939805i \(0.611007\pi\)
\(654\) −1.46410 −0.0572509
\(655\) 24.3923i 0.953086i
\(656\) − 9.39230i − 0.366708i
\(657\) 6.26795i 0.244536i
\(658\) 6.00000i 0.233904i
\(659\) −10.2487 −0.399233 −0.199617 0.979874i \(-0.563970\pi\)
−0.199617 + 0.979874i \(0.563970\pi\)
\(660\) −4.73205 −0.184195
\(661\) − 11.3923i − 0.443109i −0.975148 0.221555i \(-0.928887\pi\)
0.975148 0.221555i \(-0.0711131\pi\)
\(662\) −20.0000 −0.777322
\(663\) 0 0
\(664\) −0.196152 −0.00761219
\(665\) − 48.2487i − 1.87100i
\(666\) 3.53590 0.137013
\(667\) 18.7321 0.725308
\(668\) − 9.46410i − 0.366177i
\(669\) − 26.9282i − 1.04110i
\(670\) − 48.9808i − 1.89229i
\(671\) 11.6603i 0.450139i
\(672\) −2.73205 −0.105391
\(673\) −27.9282 −1.07655 −0.538277 0.842768i \(-0.680924\pi\)
−0.538277 + 0.842768i \(0.680924\pi\)
\(674\) 20.8564i 0.803359i
\(675\) −8.92820 −0.343647
\(676\) 0 0
\(677\) −45.4641 −1.74733 −0.873664 0.486530i \(-0.838262\pi\)
−0.873664 + 0.486530i \(0.838262\pi\)
\(678\) − 1.33975i − 0.0514526i
\(679\) 16.3923 0.629079
\(680\) 21.3923 0.820357
\(681\) − 12.1962i − 0.467358i
\(682\) 1.85641i 0.0710855i
\(683\) 10.1436i 0.388134i 0.980988 + 0.194067i \(0.0621679\pi\)
−0.980988 + 0.194067i \(0.937832\pi\)
\(684\) − 4.73205i − 0.180934i
\(685\) 44.5167 1.70089
\(686\) −17.8564 −0.681761
\(687\) 11.8564i 0.452350i
\(688\) 9.66025 0.368294
\(689\) 0 0
\(690\) 15.6603 0.596176
\(691\) 43.6603i 1.66091i 0.557082 + 0.830457i \(0.311921\pi\)
−0.557082 + 0.830457i \(0.688079\pi\)
\(692\) 4.39230 0.166970
\(693\) 3.46410 0.131590
\(694\) 33.1244i 1.25738i
\(695\) 66.6410i 2.52784i
\(696\) 4.46410i 0.169211i
\(697\) − 53.8372i − 2.03923i
\(698\) 15.3205 0.579890
\(699\) 7.85641 0.297157
\(700\) 24.3923i 0.921942i
\(701\) 3.32051 0.125414 0.0627069 0.998032i \(-0.480027\pi\)
0.0627069 + 0.998032i \(0.480027\pi\)
\(702\) 0 0
\(703\) 16.7321 0.631061
\(704\) 1.26795i 0.0477876i
\(705\) 8.19615 0.308685
\(706\) 21.7846 0.819875
\(707\) − 5.26795i − 0.198122i
\(708\) 8.00000i 0.300658i
\(709\) − 13.1436i − 0.493618i −0.969064 0.246809i \(-0.920618\pi\)
0.969064 0.246809i \(-0.0793820\pi\)
\(710\) − 17.6603i − 0.662778i
\(711\) −2.53590 −0.0951036
\(712\) 9.46410 0.354682
\(713\) − 6.14359i − 0.230079i
\(714\) −15.6603 −0.586070
\(715\) 0 0
\(716\) −16.0526 −0.599912
\(717\) 7.66025i 0.286077i
\(718\) −1.12436 −0.0419606
\(719\) 29.4641 1.09883 0.549413 0.835551i \(-0.314851\pi\)
0.549413 + 0.835551i \(0.314851\pi\)
\(720\) 3.73205i 0.139085i
\(721\) 41.7128i 1.55347i
\(722\) − 3.39230i − 0.126249i
\(723\) − 13.5885i − 0.505360i
\(724\) 19.1962 0.713419
\(725\) 39.8564 1.48023
\(726\) 9.39230i 0.348581i
\(727\) 30.9808 1.14901 0.574506 0.818500i \(-0.305194\pi\)
0.574506 + 0.818500i \(0.305194\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) − 23.3923i − 0.865788i
\(731\) 55.3731 2.04805
\(732\) 9.19615 0.339900
\(733\) 19.0000i 0.701781i 0.936416 + 0.350891i \(0.114121\pi\)
−0.936416 + 0.350891i \(0.885879\pi\)
\(734\) − 11.2679i − 0.415908i
\(735\) − 1.73205i − 0.0638877i
\(736\) − 4.19615i − 0.154672i
\(737\) 16.6410 0.612980
\(738\) 9.39230 0.345736
\(739\) − 2.92820i − 0.107716i −0.998549 0.0538578i \(-0.982848\pi\)
0.998549 0.0538578i \(-0.0171518\pi\)
\(740\) −13.1962 −0.485100
\(741\) 0 0
\(742\) 17.6603 0.648328
\(743\) − 48.3923i − 1.77534i −0.460479 0.887671i \(-0.652322\pi\)
0.460479 0.887671i \(-0.347678\pi\)
\(744\) 1.46410 0.0536766
\(745\) −49.2487 −1.80433
\(746\) − 13.7321i − 0.502766i
\(747\) − 0.196152i − 0.00717684i
\(748\) 7.26795i 0.265743i
\(749\) 27.8564i 1.01785i
\(750\) 14.6603 0.535317
\(751\) 49.9090 1.82120 0.910602 0.413284i \(-0.135618\pi\)
0.910602 + 0.413284i \(0.135618\pi\)
\(752\) − 2.19615i − 0.0800854i
\(753\) −13.4641 −0.490659
\(754\) 0 0
\(755\) 25.1244 0.914369
\(756\) − 2.73205i − 0.0993637i
\(757\) 20.9282 0.760648 0.380324 0.924853i \(-0.375812\pi\)
0.380324 + 0.924853i \(0.375812\pi\)
\(758\) 5.46410 0.198465
\(759\) 5.32051i 0.193122i
\(760\) 17.6603i 0.640605i
\(761\) − 11.3205i − 0.410368i −0.978723 0.205184i \(-0.934221\pi\)
0.978723 0.205184i \(-0.0657793\pi\)
\(762\) 9.85641i 0.357060i
\(763\) −4.00000 −0.144810
\(764\) 6.92820 0.250654
\(765\) 21.3923i 0.773440i
\(766\) 1.46410 0.0529001
\(767\) 0 0
\(768\) 1.00000 0.0360844
\(769\) − 43.8564i − 1.58150i −0.612138 0.790751i \(-0.709690\pi\)
0.612138 0.790751i \(-0.290310\pi\)
\(770\) −12.9282 −0.465900
\(771\) −9.33975 −0.336363
\(772\) − 11.7321i − 0.422246i
\(773\) − 48.9282i − 1.75983i −0.475136 0.879913i \(-0.657601\pi\)
0.475136 0.879913i \(-0.342399\pi\)
\(774\) 9.66025i 0.347231i
\(775\) − 13.0718i − 0.469553i
\(776\) −6.00000 −0.215387
\(777\) 9.66025 0.346560
\(778\) − 11.7846i − 0.422499i
\(779\) 44.4449 1.59240
\(780\) 0 0
\(781\) 6.00000 0.214697
\(782\) − 24.0526i − 0.860118i
\(783\) −4.46410 −0.159534
\(784\) −0.464102 −0.0165751
\(785\) 28.3205i 1.01080i
\(786\) 6.53590i 0.233128i
\(787\) − 4.67949i − 0.166806i −0.996516 0.0834029i \(-0.973421\pi\)
0.996516 0.0834029i \(-0.0265789\pi\)
\(788\) 17.8564i 0.636108i
\(789\) −10.0526 −0.357881
\(790\) 9.46410 0.336718
\(791\) − 3.66025i − 0.130144i
\(792\) −1.26795 −0.0450546
\(793\) 0 0
\(794\) 20.3923 0.723696
\(795\) − 24.1244i − 0.855603i
\(796\) −14.1962 −0.503169
\(797\) 34.0000 1.20434 0.602171 0.798367i \(-0.294303\pi\)
0.602171 + 0.798367i \(0.294303\pi\)
\(798\) − 12.9282i − 0.457653i
\(799\) − 12.5885i − 0.445348i
\(800\) − 8.92820i − 0.315660i
\(801\) 9.46410i 0.334398i
\(802\) 8.07180 0.285025
\(803\) 7.94744 0.280459
\(804\) − 13.1244i − 0.462860i
\(805\) 42.7846 1.50796
\(806\) 0 0
\(807\) 5.46410 0.192345
\(808\) 1.92820i 0.0678340i
\(809\) −53.5885 −1.88407 −0.942035 0.335515i \(-0.891090\pi\)
−0.942035 + 0.335515i \(0.891090\pi\)
\(810\) −3.73205 −0.131131
\(811\) 17.1769i 0.603163i 0.953440 + 0.301582i \(0.0975145\pi\)
−0.953440 + 0.301582i \(0.902485\pi\)
\(812\) 12.1962i 0.428001i
\(813\) 21.8564i 0.766538i
\(814\) − 4.48334i − 0.157141i
\(815\) −50.2487 −1.76014
\(816\) 5.73205 0.200662
\(817\) 45.7128i 1.59929i
\(818\) −17.7321 −0.619987
\(819\) 0 0
\(820\) −35.0526 −1.22409
\(821\) − 0.928203i − 0.0323945i −0.999869 0.0161973i \(-0.994844\pi\)
0.999869 0.0161973i \(-0.00515597\pi\)
\(822\) 11.9282 0.416044
\(823\) −41.5692 −1.44901 −0.724506 0.689269i \(-0.757932\pi\)
−0.724506 + 0.689269i \(0.757932\pi\)
\(824\) − 15.2679i − 0.531884i
\(825\) 11.3205i 0.394130i
\(826\) 21.8564i 0.760482i
\(827\) − 26.5359i − 0.922744i −0.887207 0.461372i \(-0.847357\pi\)
0.887207 0.461372i \(-0.152643\pi\)
\(828\) 4.19615 0.145826
\(829\) 12.1244 0.421096 0.210548 0.977583i \(-0.432475\pi\)
0.210548 + 0.977583i \(0.432475\pi\)
\(830\) 0.732051i 0.0254099i
\(831\) 5.73205 0.198843
\(832\) 0 0
\(833\) −2.66025 −0.0921723
\(834\) 17.8564i 0.618317i
\(835\) −35.3205 −1.22232
\(836\) −6.00000 −0.207514
\(837\) 1.46410i 0.0506068i
\(838\) − 17.4641i − 0.603287i
\(839\) − 41.8564i − 1.44504i −0.691348 0.722522i \(-0.742983\pi\)
0.691348 0.722522i \(-0.257017\pi\)
\(840\) 10.1962i 0.351801i
\(841\) −9.07180 −0.312821
\(842\) 22.7128 0.782735
\(843\) − 12.3205i − 0.424341i
\(844\) −16.3923 −0.564246
\(845\) 0 0
\(846\) 2.19615 0.0755053
\(847\) 25.6603i 0.881697i
\(848\) −6.46410 −0.221978
\(849\) −25.6603 −0.880658
\(850\) − 51.1769i − 1.75535i
\(851\) 14.8372i 0.508612i
\(852\) − 4.73205i − 0.162117i
\(853\) 54.1769i 1.85498i 0.373845 + 0.927491i \(0.378039\pi\)
−0.373845 + 0.927491i \(0.621961\pi\)
\(854\) 25.1244 0.859738
\(855\) −17.6603 −0.603968
\(856\) − 10.1962i − 0.348497i
\(857\) −39.4449 −1.34741 −0.673705 0.739000i \(-0.735298\pi\)
−0.673705 + 0.739000i \(0.735298\pi\)
\(858\) 0 0
\(859\) −47.1244 −1.60786 −0.803931 0.594722i \(-0.797262\pi\)
−0.803931 + 0.594722i \(0.797262\pi\)
\(860\) − 36.0526i − 1.22938i
\(861\) 25.6603 0.874499
\(862\) −13.1244 −0.447017
\(863\) − 17.1244i − 0.582920i −0.956583 0.291460i \(-0.905859\pi\)
0.956583 0.291460i \(-0.0941410\pi\)
\(864\) 1.00000i 0.0340207i
\(865\) − 16.3923i − 0.557355i
\(866\) − 12.8564i − 0.436878i
\(867\) 15.8564 0.538512
\(868\) 4.00000 0.135769
\(869\) 3.21539i 0.109075i
\(870\) 16.6603 0.564836
\(871\) 0 0
\(872\) 1.46410 0.0495807
\(873\) − 6.00000i − 0.203069i
\(874\) 19.8564 0.671653
\(875\) 40.0526 1.35402
\(876\) − 6.26795i − 0.211774i
\(877\) 23.9282i 0.807998i 0.914759 + 0.403999i \(0.132380\pi\)
−0.914759 + 0.403999i \(0.867620\pi\)
\(878\) 0.339746i 0.0114659i
\(879\) − 30.5167i − 1.02930i
\(880\) 4.73205 0.159517
\(881\) 27.8372 0.937858 0.468929 0.883236i \(-0.344640\pi\)
0.468929 + 0.883236i \(0.344640\pi\)
\(882\) − 0.464102i − 0.0156271i
\(883\) −42.9282 −1.44465 −0.722325 0.691554i \(-0.756926\pi\)
−0.722325 + 0.691554i \(0.756926\pi\)
\(884\) 0 0
\(885\) 29.8564 1.00361
\(886\) 15.6077i 0.524351i
\(887\) 37.8564 1.27109 0.635547 0.772062i \(-0.280775\pi\)
0.635547 + 0.772062i \(0.280775\pi\)
\(888\) −3.53590 −0.118657
\(889\) 26.9282i 0.903143i
\(890\) − 35.3205i − 1.18395i
\(891\) − 1.26795i − 0.0424779i
\(892\) 26.9282i 0.901623i
\(893\) 10.3923 0.347765
\(894\) −13.1962 −0.441345
\(895\) 59.9090i 2.00254i
\(896\) 2.73205 0.0912714
\(897\) 0 0
\(898\) 11.3205 0.377770
\(899\) − 6.53590i − 0.217984i
\(900\) 8.92820 0.297607
\(901\) −37.0526 −1.23440
\(902\) − 11.9090i − 0.396525i
\(903\) 26.3923i 0.878281i
\(904\) 1.33975i 0.0445593i
\(905\) − 71.6410i − 2.38143i
\(906\) 6.73205 0.223657
\(907\) −36.3923 −1.20839 −0.604193 0.796838i \(-0.706505\pi\)
−0.604193 + 0.796838i \(0.706505\pi\)
\(908\) 12.1962i 0.404744i
\(909\) −1.92820 −0.0639545
\(910\) 0 0
\(911\) −2.53590 −0.0840181 −0.0420090 0.999117i \(-0.513376\pi\)
−0.0420090 + 0.999117i \(0.513376\pi\)
\(912\) 4.73205i 0.156694i
\(913\) −0.248711 −0.00823114
\(914\) −1.33975 −0.0443149
\(915\) − 34.3205i − 1.13460i
\(916\) − 11.8564i − 0.391747i
\(917\) 17.8564i 0.589670i
\(918\) 5.73205i 0.189186i
\(919\) 45.9615 1.51613 0.758065 0.652179i \(-0.226145\pi\)
0.758065 + 0.652179i \(0.226145\pi\)
\(920\) −15.6603 −0.516303
\(921\) 22.5885i 0.744315i
\(922\) −22.2679 −0.733356
\(923\) 0 0
\(924\) −3.46410 −0.113961
\(925\) 31.5692i 1.03799i
\(926\) 10.0526 0.330348
\(927\) 15.2679 0.501465
\(928\) − 4.46410i − 0.146541i
\(929\) − 39.2487i − 1.28771i −0.765148 0.643854i \(-0.777334\pi\)
0.765148 0.643854i \(-0.222666\pi\)
\(930\) − 5.46410i − 0.179175i
\(931\) − 2.19615i − 0.0719760i
\(932\) −7.85641 −0.257345
\(933\) 1.66025 0.0543543
\(934\) − 18.5885i − 0.608233i
\(935\) 27.1244 0.887061
\(936\) 0 0
\(937\) −5.24871 −0.171468 −0.0857340 0.996318i \(-0.527324\pi\)
−0.0857340 + 0.996318i \(0.527324\pi\)
\(938\) − 35.8564i − 1.17075i
\(939\) 6.53590 0.213291
\(940\) −8.19615 −0.267329
\(941\) 12.6410i 0.412085i 0.978543 + 0.206043i \(0.0660586\pi\)
−0.978543 + 0.206043i \(0.933941\pi\)
\(942\) 7.58846i 0.247245i
\(943\) 39.4115i 1.28342i
\(944\) − 8.00000i − 0.260378i
\(945\) −10.1962 −0.331681
\(946\) 12.2487 0.398240
\(947\) − 21.0718i − 0.684741i −0.939565 0.342371i \(-0.888770\pi\)
0.939565 0.342371i \(-0.111230\pi\)
\(948\) 2.53590 0.0823622
\(949\) 0 0
\(950\) 42.2487 1.37073
\(951\) 20.6603i 0.669955i
\(952\) 15.6603 0.507552
\(953\) −41.5692 −1.34656 −0.673280 0.739388i \(-0.735115\pi\)
−0.673280 + 0.739388i \(0.735115\pi\)
\(954\) − 6.46410i − 0.209283i
\(955\) − 25.8564i − 0.836694i
\(956\) − 7.66025i − 0.247750i
\(957\) 5.66025i 0.182970i
\(958\) 33.4641 1.08118
\(959\) 32.5885 1.05234
\(960\) − 3.73205i − 0.120451i
\(961\) 28.8564 0.930852
\(962\) 0 0
\(963\) 10.1962 0.328566
\(964\) 13.5885i 0.437655i
\(965\) −43.7846 −1.40948
\(966\) 11.4641 0.368851
\(967\) − 43.1244i − 1.38679i −0.720560 0.693393i \(-0.756115\pi\)
0.720560 0.693393i \(-0.243885\pi\)
\(968\) − 9.39230i − 0.301880i
\(969\) 27.1244i 0.871360i
\(970\) 22.3923i 0.718974i
\(971\) 30.2487 0.970727 0.485364 0.874312i \(-0.338687\pi\)
0.485364 + 0.874312i \(0.338687\pi\)
\(972\) −1.00000 −0.0320750
\(973\) 48.7846i 1.56396i
\(974\) 3.12436 0.100111
\(975\) 0 0
\(976\) −9.19615 −0.294362
\(977\) 45.9282i 1.46937i 0.678407 + 0.734687i \(0.262671\pi\)
−0.678407 + 0.734687i \(0.737329\pi\)
\(978\) −13.4641 −0.430534
\(979\) 12.0000 0.383522
\(980\) 1.73205i 0.0553283i
\(981\) 1.46410i 0.0467452i
\(982\) − 8.73205i − 0.278651i
\(983\) − 20.7846i − 0.662926i −0.943468 0.331463i \(-0.892458\pi\)
0.943468 0.331463i \(-0.107542\pi\)
\(984\) −9.39230 −0.299416
\(985\) 66.6410 2.12336
\(986\) − 25.5885i − 0.814902i
\(987\) 6.00000 0.190982
\(988\) 0 0
\(989\) −40.5359 −1.28897
\(990\) 4.73205i 0.150394i
\(991\) 22.5885 0.717546 0.358773 0.933425i \(-0.383195\pi\)
0.358773 + 0.933425i \(0.383195\pi\)
\(992\) −1.46410 −0.0464853
\(993\) 20.0000i 0.634681i
\(994\) − 12.9282i − 0.410058i
\(995\) 52.9808i 1.67960i
\(996\) 0.196152i 0.00621533i
\(997\) 21.3397 0.675837 0.337918 0.941175i \(-0.390277\pi\)
0.337918 + 0.941175i \(0.390277\pi\)
\(998\) −32.0000 −1.01294
\(999\) − 3.53590i − 0.111871i
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 1014.2.b.e.337.4 4
3.2 odd 2 3042.2.b.i.1351.1 4
13.2 odd 12 1014.2.e.g.529.2 4
13.3 even 3 78.2.i.a.43.1 4
13.4 even 6 78.2.i.a.49.1 yes 4
13.5 odd 4 1014.2.a.k.1.2 2
13.6 odd 12 1014.2.e.g.991.2 4
13.7 odd 12 1014.2.e.i.991.1 4
13.8 odd 4 1014.2.a.i.1.1 2
13.9 even 3 1014.2.i.a.361.2 4
13.10 even 6 1014.2.i.a.823.2 4
13.11 odd 12 1014.2.e.i.529.1 4
13.12 even 2 inner 1014.2.b.e.337.1 4
39.5 even 4 3042.2.a.p.1.1 2
39.8 even 4 3042.2.a.y.1.2 2
39.17 odd 6 234.2.l.c.127.2 4
39.29 odd 6 234.2.l.c.199.2 4
39.38 odd 2 3042.2.b.i.1351.4 4
52.3 odd 6 624.2.bv.e.433.2 4
52.31 even 4 8112.2.a.bp.1.2 2
52.43 odd 6 624.2.bv.e.49.1 4
52.47 even 4 8112.2.a.bj.1.1 2
65.3 odd 12 1950.2.y.b.199.1 4
65.4 even 6 1950.2.bc.d.751.2 4
65.17 odd 12 1950.2.y.b.49.1 4
65.29 even 6 1950.2.bc.d.901.2 4
65.42 odd 12 1950.2.y.g.199.2 4
65.43 odd 12 1950.2.y.g.49.2 4
156.95 even 6 1872.2.by.h.1297.2 4
156.107 even 6 1872.2.by.h.433.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
78.2.i.a.43.1 4 13.3 even 3
78.2.i.a.49.1 yes 4 13.4 even 6
234.2.l.c.127.2 4 39.17 odd 6
234.2.l.c.199.2 4 39.29 odd 6
624.2.bv.e.49.1 4 52.43 odd 6
624.2.bv.e.433.2 4 52.3 odd 6
1014.2.a.i.1.1 2 13.8 odd 4
1014.2.a.k.1.2 2 13.5 odd 4
1014.2.b.e.337.1 4 13.12 even 2 inner
1014.2.b.e.337.4 4 1.1 even 1 trivial
1014.2.e.g.529.2 4 13.2 odd 12
1014.2.e.g.991.2 4 13.6 odd 12
1014.2.e.i.529.1 4 13.11 odd 12
1014.2.e.i.991.1 4 13.7 odd 12
1014.2.i.a.361.2 4 13.9 even 3
1014.2.i.a.823.2 4 13.10 even 6
1872.2.by.h.433.1 4 156.107 even 6
1872.2.by.h.1297.2 4 156.95 even 6
1950.2.y.b.49.1 4 65.17 odd 12
1950.2.y.b.199.1 4 65.3 odd 12
1950.2.y.g.49.2 4 65.43 odd 12
1950.2.y.g.199.2 4 65.42 odd 12
1950.2.bc.d.751.2 4 65.4 even 6
1950.2.bc.d.901.2 4 65.29 even 6
3042.2.a.p.1.1 2 39.5 even 4
3042.2.a.y.1.2 2 39.8 even 4
3042.2.b.i.1351.1 4 3.2 odd 2
3042.2.b.i.1351.4 4 39.38 odd 2
8112.2.a.bj.1.1 2 52.47 even 4
8112.2.a.bp.1.2 2 52.31 even 4