Properties

Label 5070.2.a.bm.1.3
Level $5070$
Weight $2$
Character 5070.1
Self dual yes
Analytic conductor $40.484$
Analytic rank $0$
Dimension $3$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 5070 = 2 \cdot 3 \cdot 5 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5070.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(40.4841538248\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: \(\Q(\zeta_{14})^+\)
Defining polynomial: \(x^{3} - x^{2} - 2 x + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(-1.24698\) of defining polynomial
Character \(\chi\) \(=\) 5070.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} +1.00000 q^{6} +4.24698 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} +1.00000 q^{6} +4.24698 q^{7} -1.00000 q^{8} +1.00000 q^{9} -1.00000 q^{10} -0.801938 q^{11} -1.00000 q^{12} -4.24698 q^{14} -1.00000 q^{15} +1.00000 q^{16} +7.43296 q^{17} -1.00000 q^{18} +3.74094 q^{19} +1.00000 q^{20} -4.24698 q^{21} +0.801938 q^{22} -2.54288 q^{23} +1.00000 q^{24} +1.00000 q^{25} -1.00000 q^{27} +4.24698 q^{28} +5.63102 q^{29} +1.00000 q^{30} -1.69202 q^{31} -1.00000 q^{32} +0.801938 q^{33} -7.43296 q^{34} +4.24698 q^{35} +1.00000 q^{36} -9.67456 q^{37} -3.74094 q^{38} -1.00000 q^{40} +8.85086 q^{41} +4.24698 q^{42} -6.29590 q^{43} -0.801938 q^{44} +1.00000 q^{45} +2.54288 q^{46} +7.53750 q^{47} -1.00000 q^{48} +11.0368 q^{49} -1.00000 q^{50} -7.43296 q^{51} +13.5646 q^{53} +1.00000 q^{54} -0.801938 q^{55} -4.24698 q^{56} -3.74094 q^{57} -5.63102 q^{58} +9.58211 q^{59} -1.00000 q^{60} -10.7899 q^{61} +1.69202 q^{62} +4.24698 q^{63} +1.00000 q^{64} -0.801938 q^{66} +7.82908 q^{67} +7.43296 q^{68} +2.54288 q^{69} -4.24698 q^{70} -2.04892 q^{71} -1.00000 q^{72} +2.17092 q^{73} +9.67456 q^{74} -1.00000 q^{75} +3.74094 q^{76} -3.40581 q^{77} -16.2349 q^{79} +1.00000 q^{80} +1.00000 q^{81} -8.85086 q^{82} -3.61356 q^{83} -4.24698 q^{84} +7.43296 q^{85} +6.29590 q^{86} -5.63102 q^{87} +0.801938 q^{88} -8.24459 q^{89} -1.00000 q^{90} -2.54288 q^{92} +1.69202 q^{93} -7.53750 q^{94} +3.74094 q^{95} +1.00000 q^{96} +6.07069 q^{97} -11.0368 q^{98} -0.801938 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3q - 3q^{2} - 3q^{3} + 3q^{4} + 3q^{5} + 3q^{6} + 8q^{7} - 3q^{8} + 3q^{9} + O(q^{10}) \) \( 3q - 3q^{2} - 3q^{3} + 3q^{4} + 3q^{5} + 3q^{6} + 8q^{7} - 3q^{8} + 3q^{9} - 3q^{10} + 2q^{11} - 3q^{12} - 8q^{14} - 3q^{15} + 3q^{16} + 3q^{17} - 3q^{18} - 3q^{19} + 3q^{20} - 8q^{21} - 2q^{22} + 11q^{23} + 3q^{24} + 3q^{25} - 3q^{27} + 8q^{28} + 2q^{29} + 3q^{30} - 3q^{32} - 2q^{33} - 3q^{34} + 8q^{35} + 3q^{36} - 8q^{37} + 3q^{38} - 3q^{40} + 13q^{41} + 8q^{42} - 5q^{43} + 2q^{44} + 3q^{45} - 11q^{46} + 7q^{47} - 3q^{48} + 5q^{49} - 3q^{50} - 3q^{51} + 19q^{53} + 3q^{54} + 2q^{55} - 8q^{56} + 3q^{57} - 2q^{58} + 23q^{59} - 3q^{60} - 9q^{61} + 8q^{63} + 3q^{64} + 2q^{66} + 13q^{67} + 3q^{68} - 11q^{69} - 8q^{70} + 3q^{71} - 3q^{72} + 17q^{73} + 8q^{74} - 3q^{75} - 3q^{76} + 3q^{77} - 25q^{79} + 3q^{80} + 3q^{81} - 13q^{82} + 20q^{83} - 8q^{84} + 3q^{85} + 5q^{86} - 2q^{87} - 2q^{88} + 21q^{89} - 3q^{90} + 11q^{92} - 7q^{94} - 3q^{95} + 3q^{96} + 6q^{97} - 5q^{98} + 2q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.00000 −0.707107
\(3\) −1.00000 −0.577350
\(4\) 1.00000 0.500000
\(5\) 1.00000 0.447214
\(6\) 1.00000 0.408248
\(7\) 4.24698 1.60521 0.802604 0.596513i \(-0.203447\pi\)
0.802604 + 0.596513i \(0.203447\pi\)
\(8\) −1.00000 −0.353553
\(9\) 1.00000 0.333333
\(10\) −1.00000 −0.316228
\(11\) −0.801938 −0.241793 −0.120897 0.992665i \(-0.538577\pi\)
−0.120897 + 0.992665i \(0.538577\pi\)
\(12\) −1.00000 −0.288675
\(13\) 0 0
\(14\) −4.24698 −1.13505
\(15\) −1.00000 −0.258199
\(16\) 1.00000 0.250000
\(17\) 7.43296 1.80276 0.901379 0.433031i \(-0.142556\pi\)
0.901379 + 0.433031i \(0.142556\pi\)
\(18\) −1.00000 −0.235702
\(19\) 3.74094 0.858230 0.429115 0.903250i \(-0.358825\pi\)
0.429115 + 0.903250i \(0.358825\pi\)
\(20\) 1.00000 0.223607
\(21\) −4.24698 −0.926767
\(22\) 0.801938 0.170974
\(23\) −2.54288 −0.530226 −0.265113 0.964217i \(-0.585409\pi\)
−0.265113 + 0.964217i \(0.585409\pi\)
\(24\) 1.00000 0.204124
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) −1.00000 −0.192450
\(28\) 4.24698 0.802604
\(29\) 5.63102 1.04565 0.522827 0.852439i \(-0.324877\pi\)
0.522827 + 0.852439i \(0.324877\pi\)
\(30\) 1.00000 0.182574
\(31\) −1.69202 −0.303896 −0.151948 0.988388i \(-0.548555\pi\)
−0.151948 + 0.988388i \(0.548555\pi\)
\(32\) −1.00000 −0.176777
\(33\) 0.801938 0.139599
\(34\) −7.43296 −1.27474
\(35\) 4.24698 0.717871
\(36\) 1.00000 0.166667
\(37\) −9.67456 −1.59049 −0.795244 0.606289i \(-0.792657\pi\)
−0.795244 + 0.606289i \(0.792657\pi\)
\(38\) −3.74094 −0.606860
\(39\) 0 0
\(40\) −1.00000 −0.158114
\(41\) 8.85086 1.38227 0.691136 0.722725i \(-0.257111\pi\)
0.691136 + 0.722725i \(0.257111\pi\)
\(42\) 4.24698 0.655323
\(43\) −6.29590 −0.960115 −0.480058 0.877237i \(-0.659384\pi\)
−0.480058 + 0.877237i \(0.659384\pi\)
\(44\) −0.801938 −0.120897
\(45\) 1.00000 0.149071
\(46\) 2.54288 0.374927
\(47\) 7.53750 1.09946 0.549729 0.835343i \(-0.314731\pi\)
0.549729 + 0.835343i \(0.314731\pi\)
\(48\) −1.00000 −0.144338
\(49\) 11.0368 1.57669
\(50\) −1.00000 −0.141421
\(51\) −7.43296 −1.04082
\(52\) 0 0
\(53\) 13.5646 1.86325 0.931624 0.363424i \(-0.118392\pi\)
0.931624 + 0.363424i \(0.118392\pi\)
\(54\) 1.00000 0.136083
\(55\) −0.801938 −0.108133
\(56\) −4.24698 −0.567527
\(57\) −3.74094 −0.495499
\(58\) −5.63102 −0.739389
\(59\) 9.58211 1.24748 0.623742 0.781630i \(-0.285612\pi\)
0.623742 + 0.781630i \(0.285612\pi\)
\(60\) −1.00000 −0.129099
\(61\) −10.7899 −1.38150 −0.690750 0.723094i \(-0.742719\pi\)
−0.690750 + 0.723094i \(0.742719\pi\)
\(62\) 1.69202 0.214887
\(63\) 4.24698 0.535069
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) −0.801938 −0.0987117
\(67\) 7.82908 0.956475 0.478237 0.878231i \(-0.341276\pi\)
0.478237 + 0.878231i \(0.341276\pi\)
\(68\) 7.43296 0.901379
\(69\) 2.54288 0.306126
\(70\) −4.24698 −0.507611
\(71\) −2.04892 −0.243162 −0.121581 0.992582i \(-0.538796\pi\)
−0.121581 + 0.992582i \(0.538796\pi\)
\(72\) −1.00000 −0.117851
\(73\) 2.17092 0.254086 0.127043 0.991897i \(-0.459451\pi\)
0.127043 + 0.991897i \(0.459451\pi\)
\(74\) 9.67456 1.12465
\(75\) −1.00000 −0.115470
\(76\) 3.74094 0.429115
\(77\) −3.40581 −0.388128
\(78\) 0 0
\(79\) −16.2349 −1.82657 −0.913284 0.407323i \(-0.866462\pi\)
−0.913284 + 0.407323i \(0.866462\pi\)
\(80\) 1.00000 0.111803
\(81\) 1.00000 0.111111
\(82\) −8.85086 −0.977413
\(83\) −3.61356 −0.396640 −0.198320 0.980137i \(-0.563549\pi\)
−0.198320 + 0.980137i \(0.563549\pi\)
\(84\) −4.24698 −0.463383
\(85\) 7.43296 0.806218
\(86\) 6.29590 0.678904
\(87\) −5.63102 −0.603709
\(88\) 0.801938 0.0854868
\(89\) −8.24459 −0.873924 −0.436962 0.899480i \(-0.643946\pi\)
−0.436962 + 0.899480i \(0.643946\pi\)
\(90\) −1.00000 −0.105409
\(91\) 0 0
\(92\) −2.54288 −0.265113
\(93\) 1.69202 0.175454
\(94\) −7.53750 −0.777434
\(95\) 3.74094 0.383812
\(96\) 1.00000 0.102062
\(97\) 6.07069 0.616385 0.308192 0.951324i \(-0.400276\pi\)
0.308192 + 0.951324i \(0.400276\pi\)
\(98\) −11.0368 −1.11489
\(99\) −0.801938 −0.0805978
\(100\) 1.00000 0.100000
\(101\) −5.38404 −0.535732 −0.267866 0.963456i \(-0.586318\pi\)
−0.267866 + 0.963456i \(0.586318\pi\)
\(102\) 7.43296 0.735973
\(103\) −8.92154 −0.879066 −0.439533 0.898227i \(-0.644856\pi\)
−0.439533 + 0.898227i \(0.644856\pi\)
\(104\) 0 0
\(105\) −4.24698 −0.414463
\(106\) −13.5646 −1.31751
\(107\) 13.9976 1.35320 0.676600 0.736351i \(-0.263453\pi\)
0.676600 + 0.736351i \(0.263453\pi\)
\(108\) −1.00000 −0.0962250
\(109\) 13.8605 1.32760 0.663800 0.747910i \(-0.268943\pi\)
0.663800 + 0.747910i \(0.268943\pi\)
\(110\) 0.801938 0.0764618
\(111\) 9.67456 0.918269
\(112\) 4.24698 0.401302
\(113\) 2.19806 0.206776 0.103388 0.994641i \(-0.467032\pi\)
0.103388 + 0.994641i \(0.467032\pi\)
\(114\) 3.74094 0.350371
\(115\) −2.54288 −0.237124
\(116\) 5.63102 0.522827
\(117\) 0 0
\(118\) −9.58211 −0.882104
\(119\) 31.5676 2.89380
\(120\) 1.00000 0.0912871
\(121\) −10.3569 −0.941536
\(122\) 10.7899 0.976868
\(123\) −8.85086 −0.798055
\(124\) −1.69202 −0.151948
\(125\) 1.00000 0.0894427
\(126\) −4.24698 −0.378351
\(127\) −4.19136 −0.371923 −0.185961 0.982557i \(-0.559540\pi\)
−0.185961 + 0.982557i \(0.559540\pi\)
\(128\) −1.00000 −0.0883883
\(129\) 6.29590 0.554323
\(130\) 0 0
\(131\) −10.8140 −0.944825 −0.472413 0.881378i \(-0.656617\pi\)
−0.472413 + 0.881378i \(0.656617\pi\)
\(132\) 0.801938 0.0697997
\(133\) 15.8877 1.37764
\(134\) −7.82908 −0.676330
\(135\) −1.00000 −0.0860663
\(136\) −7.43296 −0.637371
\(137\) 12.8509 1.09792 0.548961 0.835848i \(-0.315023\pi\)
0.548961 + 0.835848i \(0.315023\pi\)
\(138\) −2.54288 −0.216464
\(139\) 16.6233 1.40997 0.704983 0.709224i \(-0.250955\pi\)
0.704983 + 0.709224i \(0.250955\pi\)
\(140\) 4.24698 0.358935
\(141\) −7.53750 −0.634772
\(142\) 2.04892 0.171941
\(143\) 0 0
\(144\) 1.00000 0.0833333
\(145\) 5.63102 0.467631
\(146\) −2.17092 −0.179666
\(147\) −11.0368 −0.910303
\(148\) −9.67456 −0.795244
\(149\) −1.74094 −0.142623 −0.0713116 0.997454i \(-0.522718\pi\)
−0.0713116 + 0.997454i \(0.522718\pi\)
\(150\) 1.00000 0.0816497
\(151\) −1.75840 −0.143096 −0.0715482 0.997437i \(-0.522794\pi\)
−0.0715482 + 0.997437i \(0.522794\pi\)
\(152\) −3.74094 −0.303430
\(153\) 7.43296 0.600919
\(154\) 3.40581 0.274448
\(155\) −1.69202 −0.135906
\(156\) 0 0
\(157\) −15.9825 −1.27555 −0.637773 0.770225i \(-0.720144\pi\)
−0.637773 + 0.770225i \(0.720144\pi\)
\(158\) 16.2349 1.29158
\(159\) −13.5646 −1.07575
\(160\) −1.00000 −0.0790569
\(161\) −10.7995 −0.851123
\(162\) −1.00000 −0.0785674
\(163\) −5.18598 −0.406197 −0.203099 0.979158i \(-0.565101\pi\)
−0.203099 + 0.979158i \(0.565101\pi\)
\(164\) 8.85086 0.691136
\(165\) 0.801938 0.0624308
\(166\) 3.61356 0.280467
\(167\) −5.29590 −0.409809 −0.204904 0.978782i \(-0.565688\pi\)
−0.204904 + 0.978782i \(0.565688\pi\)
\(168\) 4.24698 0.327662
\(169\) 0 0
\(170\) −7.43296 −0.570082
\(171\) 3.74094 0.286077
\(172\) −6.29590 −0.480058
\(173\) −19.6558 −1.49440 −0.747201 0.664599i \(-0.768603\pi\)
−0.747201 + 0.664599i \(0.768603\pi\)
\(174\) 5.63102 0.426887
\(175\) 4.24698 0.321041
\(176\) −0.801938 −0.0604483
\(177\) −9.58211 −0.720235
\(178\) 8.24459 0.617958
\(179\) −17.9095 −1.33862 −0.669308 0.742985i \(-0.733409\pi\)
−0.669308 + 0.742985i \(0.733409\pi\)
\(180\) 1.00000 0.0745356
\(181\) −8.74094 −0.649709 −0.324854 0.945764i \(-0.605315\pi\)
−0.324854 + 0.945764i \(0.605315\pi\)
\(182\) 0 0
\(183\) 10.7899 0.797609
\(184\) 2.54288 0.187463
\(185\) −9.67456 −0.711288
\(186\) −1.69202 −0.124065
\(187\) −5.96077 −0.435895
\(188\) 7.53750 0.549729
\(189\) −4.24698 −0.308922
\(190\) −3.74094 −0.271396
\(191\) −3.76510 −0.272433 −0.136217 0.990679i \(-0.543494\pi\)
−0.136217 + 0.990679i \(0.543494\pi\)
\(192\) −1.00000 −0.0721688
\(193\) 21.9584 1.58060 0.790299 0.612721i \(-0.209925\pi\)
0.790299 + 0.612721i \(0.209925\pi\)
\(194\) −6.07069 −0.435850
\(195\) 0 0
\(196\) 11.0368 0.788345
\(197\) −18.2446 −1.29987 −0.649936 0.759989i \(-0.725205\pi\)
−0.649936 + 0.759989i \(0.725205\pi\)
\(198\) 0.801938 0.0569912
\(199\) −1.77479 −0.125812 −0.0629058 0.998019i \(-0.520037\pi\)
−0.0629058 + 0.998019i \(0.520037\pi\)
\(200\) −1.00000 −0.0707107
\(201\) −7.82908 −0.552221
\(202\) 5.38404 0.378820
\(203\) 23.9148 1.67849
\(204\) −7.43296 −0.520411
\(205\) 8.85086 0.618171
\(206\) 8.92154 0.621593
\(207\) −2.54288 −0.176742
\(208\) 0 0
\(209\) −3.00000 −0.207514
\(210\) 4.24698 0.293069
\(211\) 0.0978347 0.00673522 0.00336761 0.999994i \(-0.498928\pi\)
0.00336761 + 0.999994i \(0.498928\pi\)
\(212\) 13.5646 0.931624
\(213\) 2.04892 0.140390
\(214\) −13.9976 −0.956857
\(215\) −6.29590 −0.429377
\(216\) 1.00000 0.0680414
\(217\) −7.18598 −0.487816
\(218\) −13.8605 −0.938754
\(219\) −2.17092 −0.146697
\(220\) −0.801938 −0.0540666
\(221\) 0 0
\(222\) −9.67456 −0.649314
\(223\) −19.5080 −1.30635 −0.653175 0.757207i \(-0.726563\pi\)
−0.653175 + 0.757207i \(0.726563\pi\)
\(224\) −4.24698 −0.283763
\(225\) 1.00000 0.0666667
\(226\) −2.19806 −0.146213
\(227\) 3.23059 0.214422 0.107211 0.994236i \(-0.465808\pi\)
0.107211 + 0.994236i \(0.465808\pi\)
\(228\) −3.74094 −0.247750
\(229\) −4.58748 −0.303149 −0.151575 0.988446i \(-0.548434\pi\)
−0.151575 + 0.988446i \(0.548434\pi\)
\(230\) 2.54288 0.167672
\(231\) 3.40581 0.224086
\(232\) −5.63102 −0.369695
\(233\) −6.18359 −0.405100 −0.202550 0.979272i \(-0.564923\pi\)
−0.202550 + 0.979272i \(0.564923\pi\)
\(234\) 0 0
\(235\) 7.53750 0.491692
\(236\) 9.58211 0.623742
\(237\) 16.2349 1.05457
\(238\) −31.5676 −2.04623
\(239\) 8.42998 0.545290 0.272645 0.962115i \(-0.412102\pi\)
0.272645 + 0.962115i \(0.412102\pi\)
\(240\) −1.00000 −0.0645497
\(241\) 20.8485 1.34297 0.671484 0.741019i \(-0.265657\pi\)
0.671484 + 0.741019i \(0.265657\pi\)
\(242\) 10.3569 0.665766
\(243\) −1.00000 −0.0641500
\(244\) −10.7899 −0.690750
\(245\) 11.0368 0.705118
\(246\) 8.85086 0.564310
\(247\) 0 0
\(248\) 1.69202 0.107443
\(249\) 3.61356 0.229000
\(250\) −1.00000 −0.0632456
\(251\) −6.39075 −0.403380 −0.201690 0.979449i \(-0.564643\pi\)
−0.201690 + 0.979449i \(0.564643\pi\)
\(252\) 4.24698 0.267535
\(253\) 2.03923 0.128205
\(254\) 4.19136 0.262989
\(255\) −7.43296 −0.465470
\(256\) 1.00000 0.0625000
\(257\) 9.78687 0.610488 0.305244 0.952274i \(-0.401262\pi\)
0.305244 + 0.952274i \(0.401262\pi\)
\(258\) −6.29590 −0.391965
\(259\) −41.0877 −2.55306
\(260\) 0 0
\(261\) 5.63102 0.348552
\(262\) 10.8140 0.668092
\(263\) 14.0465 0.866146 0.433073 0.901359i \(-0.357429\pi\)
0.433073 + 0.901359i \(0.357429\pi\)
\(264\) −0.801938 −0.0493559
\(265\) 13.5646 0.833270
\(266\) −15.8877 −0.974137
\(267\) 8.24459 0.504561
\(268\) 7.82908 0.478237
\(269\) −17.2597 −1.05234 −0.526170 0.850380i \(-0.676372\pi\)
−0.526170 + 0.850380i \(0.676372\pi\)
\(270\) 1.00000 0.0608581
\(271\) −17.1782 −1.04350 −0.521751 0.853098i \(-0.674721\pi\)
−0.521751 + 0.853098i \(0.674721\pi\)
\(272\) 7.43296 0.450689
\(273\) 0 0
\(274\) −12.8509 −0.776349
\(275\) −0.801938 −0.0483587
\(276\) 2.54288 0.153063
\(277\) −21.1444 −1.27044 −0.635221 0.772331i \(-0.719091\pi\)
−0.635221 + 0.772331i \(0.719091\pi\)
\(278\) −16.6233 −0.996996
\(279\) −1.69202 −0.101299
\(280\) −4.24698 −0.253806
\(281\) 4.09246 0.244136 0.122068 0.992522i \(-0.461047\pi\)
0.122068 + 0.992522i \(0.461047\pi\)
\(282\) 7.53750 0.448852
\(283\) 12.1468 0.722049 0.361025 0.932556i \(-0.382427\pi\)
0.361025 + 0.932556i \(0.382427\pi\)
\(284\) −2.04892 −0.121581
\(285\) −3.74094 −0.221594
\(286\) 0 0
\(287\) 37.5894 2.21883
\(288\) −1.00000 −0.0589256
\(289\) 38.2489 2.24994
\(290\) −5.63102 −0.330665
\(291\) −6.07069 −0.355870
\(292\) 2.17092 0.127043
\(293\) 20.6789 1.20807 0.604036 0.796957i \(-0.293558\pi\)
0.604036 + 0.796957i \(0.293558\pi\)
\(294\) 11.0368 0.643681
\(295\) 9.58211 0.557892
\(296\) 9.67456 0.562323
\(297\) 0.801938 0.0465331
\(298\) 1.74094 0.100850
\(299\) 0 0
\(300\) −1.00000 −0.0577350
\(301\) −26.7385 −1.54118
\(302\) 1.75840 0.101184
\(303\) 5.38404 0.309305
\(304\) 3.74094 0.214558
\(305\) −10.7899 −0.617825
\(306\) −7.43296 −0.424914
\(307\) 11.3123 0.645627 0.322813 0.946463i \(-0.395371\pi\)
0.322813 + 0.946463i \(0.395371\pi\)
\(308\) −3.40581 −0.194064
\(309\) 8.92154 0.507529
\(310\) 1.69202 0.0961004
\(311\) −26.2282 −1.48726 −0.743632 0.668589i \(-0.766899\pi\)
−0.743632 + 0.668589i \(0.766899\pi\)
\(312\) 0 0
\(313\) 29.5023 1.66757 0.833785 0.552089i \(-0.186169\pi\)
0.833785 + 0.552089i \(0.186169\pi\)
\(314\) 15.9825 0.901947
\(315\) 4.24698 0.239290
\(316\) −16.2349 −0.913284
\(317\) −7.06398 −0.396753 −0.198376 0.980126i \(-0.563567\pi\)
−0.198376 + 0.980126i \(0.563567\pi\)
\(318\) 13.5646 0.760668
\(319\) −4.51573 −0.252832
\(320\) 1.00000 0.0559017
\(321\) −13.9976 −0.781270
\(322\) 10.7995 0.601835
\(323\) 27.8062 1.54718
\(324\) 1.00000 0.0555556
\(325\) 0 0
\(326\) 5.18598 0.287225
\(327\) −13.8605 −0.766490
\(328\) −8.85086 −0.488707
\(329\) 32.0116 1.76486
\(330\) −0.801938 −0.0441452
\(331\) 10.3134 0.566873 0.283437 0.958991i \(-0.408525\pi\)
0.283437 + 0.958991i \(0.408525\pi\)
\(332\) −3.61356 −0.198320
\(333\) −9.67456 −0.530163
\(334\) 5.29590 0.289779
\(335\) 7.82908 0.427749
\(336\) −4.24698 −0.231692
\(337\) 30.3163 1.65144 0.825718 0.564083i \(-0.190770\pi\)
0.825718 + 0.564083i \(0.190770\pi\)
\(338\) 0 0
\(339\) −2.19806 −0.119382
\(340\) 7.43296 0.403109
\(341\) 1.35690 0.0734800
\(342\) −3.74094 −0.202287
\(343\) 17.1444 0.925708
\(344\) 6.29590 0.339452
\(345\) 2.54288 0.136904
\(346\) 19.6558 1.05670
\(347\) 28.8592 1.54924 0.774622 0.632425i \(-0.217940\pi\)
0.774622 + 0.632425i \(0.217940\pi\)
\(348\) −5.63102 −0.301854
\(349\) −27.4566 −1.46972 −0.734860 0.678218i \(-0.762752\pi\)
−0.734860 + 0.678218i \(0.762752\pi\)
\(350\) −4.24698 −0.227011
\(351\) 0 0
\(352\) 0.801938 0.0427434
\(353\) 0.885772 0.0471449 0.0235724 0.999722i \(-0.492496\pi\)
0.0235724 + 0.999722i \(0.492496\pi\)
\(354\) 9.58211 0.509283
\(355\) −2.04892 −0.108745
\(356\) −8.24459 −0.436962
\(357\) −31.5676 −1.67074
\(358\) 17.9095 0.946544
\(359\) 7.07069 0.373177 0.186588 0.982438i \(-0.440257\pi\)
0.186588 + 0.982438i \(0.440257\pi\)
\(360\) −1.00000 −0.0527046
\(361\) −5.00538 −0.263441
\(362\) 8.74094 0.459414
\(363\) 10.3569 0.543596
\(364\) 0 0
\(365\) 2.17092 0.113631
\(366\) −10.7899 −0.563995
\(367\) 27.3588 1.42812 0.714059 0.700085i \(-0.246855\pi\)
0.714059 + 0.700085i \(0.246855\pi\)
\(368\) −2.54288 −0.132557
\(369\) 8.85086 0.460757
\(370\) 9.67456 0.502957
\(371\) 57.6088 2.99090
\(372\) 1.69202 0.0877272
\(373\) 18.3370 0.949456 0.474728 0.880133i \(-0.342546\pi\)
0.474728 + 0.880133i \(0.342546\pi\)
\(374\) 5.96077 0.308224
\(375\) −1.00000 −0.0516398
\(376\) −7.53750 −0.388717
\(377\) 0 0
\(378\) 4.24698 0.218441
\(379\) −14.3690 −0.738085 −0.369042 0.929413i \(-0.620314\pi\)
−0.369042 + 0.929413i \(0.620314\pi\)
\(380\) 3.74094 0.191906
\(381\) 4.19136 0.214730
\(382\) 3.76510 0.192639
\(383\) 34.6920 1.77268 0.886340 0.463035i \(-0.153240\pi\)
0.886340 + 0.463035i \(0.153240\pi\)
\(384\) 1.00000 0.0510310
\(385\) −3.40581 −0.173576
\(386\) −21.9584 −1.11765
\(387\) −6.29590 −0.320038
\(388\) 6.07069 0.308192
\(389\) −32.6872 −1.65731 −0.828654 0.559761i \(-0.810893\pi\)
−0.828654 + 0.559761i \(0.810893\pi\)
\(390\) 0 0
\(391\) −18.9011 −0.955870
\(392\) −11.0368 −0.557444
\(393\) 10.8140 0.545495
\(394\) 18.2446 0.919149
\(395\) −16.2349 −0.816866
\(396\) −0.801938 −0.0402989
\(397\) 12.3080 0.617720 0.308860 0.951108i \(-0.400053\pi\)
0.308860 + 0.951108i \(0.400053\pi\)
\(398\) 1.77479 0.0889622
\(399\) −15.8877 −0.795379
\(400\) 1.00000 0.0500000
\(401\) 19.2524 0.961417 0.480708 0.876881i \(-0.340380\pi\)
0.480708 + 0.876881i \(0.340380\pi\)
\(402\) 7.82908 0.390479
\(403\) 0 0
\(404\) −5.38404 −0.267866
\(405\) 1.00000 0.0496904
\(406\) −23.9148 −1.18687
\(407\) 7.75840 0.384569
\(408\) 7.43296 0.367986
\(409\) −22.8672 −1.13071 −0.565356 0.824847i \(-0.691261\pi\)
−0.565356 + 0.824847i \(0.691261\pi\)
\(410\) −8.85086 −0.437113
\(411\) −12.8509 −0.633886
\(412\) −8.92154 −0.439533
\(413\) 40.6950 2.00247
\(414\) 2.54288 0.124976
\(415\) −3.61356 −0.177383
\(416\) 0 0
\(417\) −16.6233 −0.814044
\(418\) 3.00000 0.146735
\(419\) −17.1806 −0.839327 −0.419664 0.907680i \(-0.637852\pi\)
−0.419664 + 0.907680i \(0.637852\pi\)
\(420\) −4.24698 −0.207231
\(421\) 36.9922 1.80289 0.901445 0.432893i \(-0.142507\pi\)
0.901445 + 0.432893i \(0.142507\pi\)
\(422\) −0.0978347 −0.00476252
\(423\) 7.53750 0.366486
\(424\) −13.5646 −0.658757
\(425\) 7.43296 0.360552
\(426\) −2.04892 −0.0992704
\(427\) −45.8243 −2.21759
\(428\) 13.9976 0.676600
\(429\) 0 0
\(430\) 6.29590 0.303615
\(431\) −19.0834 −0.919213 −0.459607 0.888123i \(-0.652010\pi\)
−0.459607 + 0.888123i \(0.652010\pi\)
\(432\) −1.00000 −0.0481125
\(433\) 11.5084 0.553060 0.276530 0.961005i \(-0.410815\pi\)
0.276530 + 0.961005i \(0.410815\pi\)
\(434\) 7.18598 0.344938
\(435\) −5.63102 −0.269987
\(436\) 13.8605 0.663800
\(437\) −9.51275 −0.455056
\(438\) 2.17092 0.103730
\(439\) 8.87502 0.423582 0.211791 0.977315i \(-0.432070\pi\)
0.211791 + 0.977315i \(0.432070\pi\)
\(440\) 0.801938 0.0382309
\(441\) 11.0368 0.525564
\(442\) 0 0
\(443\) 24.4838 1.16326 0.581630 0.813453i \(-0.302415\pi\)
0.581630 + 0.813453i \(0.302415\pi\)
\(444\) 9.67456 0.459134
\(445\) −8.24459 −0.390831
\(446\) 19.5080 0.923729
\(447\) 1.74094 0.0823436
\(448\) 4.24698 0.200651
\(449\) 10.2567 0.484042 0.242021 0.970271i \(-0.422190\pi\)
0.242021 + 0.970271i \(0.422190\pi\)
\(450\) −1.00000 −0.0471405
\(451\) −7.09783 −0.334224
\(452\) 2.19806 0.103388
\(453\) 1.75840 0.0826167
\(454\) −3.23059 −0.151619
\(455\) 0 0
\(456\) 3.74094 0.175186
\(457\) 38.6359 1.80731 0.903656 0.428259i \(-0.140873\pi\)
0.903656 + 0.428259i \(0.140873\pi\)
\(458\) 4.58748 0.214359
\(459\) −7.43296 −0.346941
\(460\) −2.54288 −0.118562
\(461\) 24.8073 1.15539 0.577696 0.816252i \(-0.303952\pi\)
0.577696 + 0.816252i \(0.303952\pi\)
\(462\) −3.40581 −0.158453
\(463\) −18.0713 −0.839844 −0.419922 0.907560i \(-0.637942\pi\)
−0.419922 + 0.907560i \(0.637942\pi\)
\(464\) 5.63102 0.261414
\(465\) 1.69202 0.0784656
\(466\) 6.18359 0.286449
\(467\) −16.8834 −0.781270 −0.390635 0.920546i \(-0.627744\pi\)
−0.390635 + 0.920546i \(0.627744\pi\)
\(468\) 0 0
\(469\) 33.2500 1.53534
\(470\) −7.53750 −0.347679
\(471\) 15.9825 0.736437
\(472\) −9.58211 −0.441052
\(473\) 5.04892 0.232149
\(474\) −16.2349 −0.745694
\(475\) 3.74094 0.171646
\(476\) 31.5676 1.44690
\(477\) 13.5646 0.621083
\(478\) −8.42998 −0.385578
\(479\) 13.9879 0.639124 0.319562 0.947565i \(-0.396464\pi\)
0.319562 + 0.947565i \(0.396464\pi\)
\(480\) 1.00000 0.0456435
\(481\) 0 0
\(482\) −20.8485 −0.949621
\(483\) 10.7995 0.491396
\(484\) −10.3569 −0.470768
\(485\) 6.07069 0.275656
\(486\) 1.00000 0.0453609
\(487\) 23.2760 1.05474 0.527369 0.849636i \(-0.323178\pi\)
0.527369 + 0.849636i \(0.323178\pi\)
\(488\) 10.7899 0.488434
\(489\) 5.18598 0.234518
\(490\) −11.0368 −0.498593
\(491\) −32.5918 −1.47085 −0.735424 0.677608i \(-0.763017\pi\)
−0.735424 + 0.677608i \(0.763017\pi\)
\(492\) −8.85086 −0.399027
\(493\) 41.8552 1.88506
\(494\) 0 0
\(495\) −0.801938 −0.0360444
\(496\) −1.69202 −0.0759740
\(497\) −8.70171 −0.390325
\(498\) −3.61356 −0.161928
\(499\) −1.48666 −0.0665522 −0.0332761 0.999446i \(-0.510594\pi\)
−0.0332761 + 0.999446i \(0.510594\pi\)
\(500\) 1.00000 0.0447214
\(501\) 5.29590 0.236603
\(502\) 6.39075 0.285233
\(503\) 38.3629 1.71052 0.855258 0.518203i \(-0.173399\pi\)
0.855258 + 0.518203i \(0.173399\pi\)
\(504\) −4.24698 −0.189176
\(505\) −5.38404 −0.239587
\(506\) −2.03923 −0.0906548
\(507\) 0 0
\(508\) −4.19136 −0.185961
\(509\) 0.770479 0.0341509 0.0170754 0.999854i \(-0.494564\pi\)
0.0170754 + 0.999854i \(0.494564\pi\)
\(510\) 7.43296 0.329137
\(511\) 9.21983 0.407861
\(512\) −1.00000 −0.0441942
\(513\) −3.74094 −0.165166
\(514\) −9.78687 −0.431680
\(515\) −8.92154 −0.393130
\(516\) 6.29590 0.277161
\(517\) −6.04461 −0.265841
\(518\) 41.0877 1.80529
\(519\) 19.6558 0.862793
\(520\) 0 0
\(521\) −13.7657 −0.603086 −0.301543 0.953453i \(-0.597502\pi\)
−0.301543 + 0.953453i \(0.597502\pi\)
\(522\) −5.63102 −0.246463
\(523\) −17.5991 −0.769554 −0.384777 0.923009i \(-0.625722\pi\)
−0.384777 + 0.923009i \(0.625722\pi\)
\(524\) −10.8140 −0.472413
\(525\) −4.24698 −0.185353
\(526\) −14.0465 −0.612458
\(527\) −12.5767 −0.547851
\(528\) 0.801938 0.0348999
\(529\) −16.5338 −0.718860
\(530\) −13.5646 −0.589211
\(531\) 9.58211 0.415828
\(532\) 15.8877 0.688819
\(533\) 0 0
\(534\) −8.24459 −0.356778
\(535\) 13.9976 0.605169
\(536\) −7.82908 −0.338165
\(537\) 17.9095 0.772850
\(538\) 17.2597 0.744116
\(539\) −8.85086 −0.381233
\(540\) −1.00000 −0.0430331
\(541\) −20.9259 −0.899673 −0.449836 0.893111i \(-0.648518\pi\)
−0.449836 + 0.893111i \(0.648518\pi\)
\(542\) 17.1782 0.737867
\(543\) 8.74094 0.375110
\(544\) −7.43296 −0.318686
\(545\) 13.8605 0.593720
\(546\) 0 0
\(547\) 42.5599 1.81973 0.909864 0.414906i \(-0.136186\pi\)
0.909864 + 0.414906i \(0.136186\pi\)
\(548\) 12.8509 0.548961
\(549\) −10.7899 −0.460500
\(550\) 0.801938 0.0341947
\(551\) 21.0653 0.897412
\(552\) −2.54288 −0.108232
\(553\) −68.9493 −2.93202
\(554\) 21.1444 0.898338
\(555\) 9.67456 0.410662
\(556\) 16.6233 0.704983
\(557\) 33.1637 1.40519 0.702596 0.711589i \(-0.252024\pi\)
0.702596 + 0.711589i \(0.252024\pi\)
\(558\) 1.69202 0.0716290
\(559\) 0 0
\(560\) 4.24698 0.179468
\(561\) 5.96077 0.251664
\(562\) −4.09246 −0.172630
\(563\) −7.27173 −0.306467 −0.153234 0.988190i \(-0.548969\pi\)
−0.153234 + 0.988190i \(0.548969\pi\)
\(564\) −7.53750 −0.317386
\(565\) 2.19806 0.0924732
\(566\) −12.1468 −0.510566
\(567\) 4.24698 0.178356
\(568\) 2.04892 0.0859707
\(569\) 38.9269 1.63190 0.815951 0.578122i \(-0.196214\pi\)
0.815951 + 0.578122i \(0.196214\pi\)
\(570\) 3.74094 0.156691
\(571\) 38.6045 1.61555 0.807773 0.589493i \(-0.200673\pi\)
0.807773 + 0.589493i \(0.200673\pi\)
\(572\) 0 0
\(573\) 3.76510 0.157289
\(574\) −37.5894 −1.56895
\(575\) −2.54288 −0.106045
\(576\) 1.00000 0.0416667
\(577\) 27.8646 1.16002 0.580009 0.814610i \(-0.303049\pi\)
0.580009 + 0.814610i \(0.303049\pi\)
\(578\) −38.2489 −1.59094
\(579\) −21.9584 −0.912559
\(580\) 5.63102 0.233815
\(581\) −15.3467 −0.636690
\(582\) 6.07069 0.251638
\(583\) −10.8780 −0.450521
\(584\) −2.17092 −0.0898331
\(585\) 0 0
\(586\) −20.6789 −0.854236
\(587\) −26.1648 −1.07994 −0.539968 0.841685i \(-0.681564\pi\)
−0.539968 + 0.841685i \(0.681564\pi\)
\(588\) −11.0368 −0.455151
\(589\) −6.32975 −0.260813
\(590\) −9.58211 −0.394489
\(591\) 18.2446 0.750482
\(592\) −9.67456 −0.397622
\(593\) −8.61117 −0.353618 −0.176809 0.984245i \(-0.556578\pi\)
−0.176809 + 0.984245i \(0.556578\pi\)
\(594\) −0.801938 −0.0329039
\(595\) 31.5676 1.29415
\(596\) −1.74094 −0.0713116
\(597\) 1.77479 0.0726373
\(598\) 0 0
\(599\) 11.1709 0.456431 0.228216 0.973611i \(-0.426711\pi\)
0.228216 + 0.973611i \(0.426711\pi\)
\(600\) 1.00000 0.0408248
\(601\) 8.91856 0.363796 0.181898 0.983317i \(-0.441776\pi\)
0.181898 + 0.983317i \(0.441776\pi\)
\(602\) 26.7385 1.08978
\(603\) 7.82908 0.318825
\(604\) −1.75840 −0.0715482
\(605\) −10.3569 −0.421068
\(606\) −5.38404 −0.218712
\(607\) −28.0277 −1.13761 −0.568805 0.822472i \(-0.692594\pi\)
−0.568805 + 0.822472i \(0.692594\pi\)
\(608\) −3.74094 −0.151715
\(609\) −23.9148 −0.969078
\(610\) 10.7899 0.436869
\(611\) 0 0
\(612\) 7.43296 0.300460
\(613\) 1.10885 0.0447861 0.0223930 0.999749i \(-0.492871\pi\)
0.0223930 + 0.999749i \(0.492871\pi\)
\(614\) −11.3123 −0.456527
\(615\) −8.85086 −0.356901
\(616\) 3.40581 0.137224
\(617\) −16.2905 −0.655832 −0.327916 0.944707i \(-0.606346\pi\)
−0.327916 + 0.944707i \(0.606346\pi\)
\(618\) −8.92154 −0.358877
\(619\) −23.2252 −0.933500 −0.466750 0.884389i \(-0.654575\pi\)
−0.466750 + 0.884389i \(0.654575\pi\)
\(620\) −1.69202 −0.0679532
\(621\) 2.54288 0.102042
\(622\) 26.2282 1.05165
\(623\) −35.0146 −1.40283
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) −29.5023 −1.17915
\(627\) 3.00000 0.119808
\(628\) −15.9825 −0.637773
\(629\) −71.9106 −2.86727
\(630\) −4.24698 −0.169204
\(631\) 0.357961 0.0142502 0.00712510 0.999975i \(-0.497732\pi\)
0.00712510 + 0.999975i \(0.497732\pi\)
\(632\) 16.2349 0.645790
\(633\) −0.0978347 −0.00388858
\(634\) 7.06398 0.280547
\(635\) −4.19136 −0.166329
\(636\) −13.5646 −0.537873
\(637\) 0 0
\(638\) 4.51573 0.178779
\(639\) −2.04892 −0.0810539
\(640\) −1.00000 −0.0395285
\(641\) 33.2814 1.31454 0.657269 0.753656i \(-0.271712\pi\)
0.657269 + 0.753656i \(0.271712\pi\)
\(642\) 13.9976 0.552441
\(643\) −48.4172 −1.90939 −0.954693 0.297592i \(-0.903817\pi\)
−0.954693 + 0.297592i \(0.903817\pi\)
\(644\) −10.7995 −0.425562
\(645\) 6.29590 0.247901
\(646\) −27.8062 −1.09402
\(647\) −47.4814 −1.86669 −0.933343 0.358985i \(-0.883123\pi\)
−0.933343 + 0.358985i \(0.883123\pi\)
\(648\) −1.00000 −0.0392837
\(649\) −7.68425 −0.301633
\(650\) 0 0
\(651\) 7.18598 0.281641
\(652\) −5.18598 −0.203099
\(653\) −25.4336 −0.995292 −0.497646 0.867380i \(-0.665802\pi\)
−0.497646 + 0.867380i \(0.665802\pi\)
\(654\) 13.8605 0.541990
\(655\) −10.8140 −0.422539
\(656\) 8.85086 0.345568
\(657\) 2.17092 0.0846955
\(658\) −32.0116 −1.24794
\(659\) −2.55496 −0.0995270 −0.0497635 0.998761i \(-0.515847\pi\)
−0.0497635 + 0.998761i \(0.515847\pi\)
\(660\) 0.801938 0.0312154
\(661\) 26.3394 1.02449 0.512243 0.858841i \(-0.328815\pi\)
0.512243 + 0.858841i \(0.328815\pi\)
\(662\) −10.3134 −0.400840
\(663\) 0 0
\(664\) 3.61356 0.140233
\(665\) 15.8877 0.616098
\(666\) 9.67456 0.374882
\(667\) −14.3190 −0.554434
\(668\) −5.29590 −0.204904
\(669\) 19.5080 0.754221
\(670\) −7.82908 −0.302464
\(671\) 8.65279 0.334037
\(672\) 4.24698 0.163831
\(673\) −23.7767 −0.916525 −0.458262 0.888817i \(-0.651528\pi\)
−0.458262 + 0.888817i \(0.651528\pi\)
\(674\) −30.3163 −1.16774
\(675\) −1.00000 −0.0384900
\(676\) 0 0
\(677\) −40.3250 −1.54981 −0.774907 0.632075i \(-0.782203\pi\)
−0.774907 + 0.632075i \(0.782203\pi\)
\(678\) 2.19806 0.0844161
\(679\) 25.7821 0.989426
\(680\) −7.43296 −0.285041
\(681\) −3.23059 −0.123796
\(682\) −1.35690 −0.0519582
\(683\) −4.51573 −0.172790 −0.0863948 0.996261i \(-0.527535\pi\)
−0.0863948 + 0.996261i \(0.527535\pi\)
\(684\) 3.74094 0.143038
\(685\) 12.8509 0.491006
\(686\) −17.1444 −0.654575
\(687\) 4.58748 0.175023
\(688\) −6.29590 −0.240029
\(689\) 0 0
\(690\) −2.54288 −0.0968057
\(691\) 5.44398 0.207099 0.103549 0.994624i \(-0.466980\pi\)
0.103549 + 0.994624i \(0.466980\pi\)
\(692\) −19.6558 −0.747201
\(693\) −3.40581 −0.129376
\(694\) −28.8592 −1.09548
\(695\) 16.6233 0.630556
\(696\) 5.63102 0.213443
\(697\) 65.7881 2.49190
\(698\) 27.4566 1.03925
\(699\) 6.18359 0.233885
\(700\) 4.24698 0.160521
\(701\) −19.8670 −0.750366 −0.375183 0.926951i \(-0.622420\pi\)
−0.375183 + 0.926951i \(0.622420\pi\)
\(702\) 0 0
\(703\) −36.1919 −1.36501
\(704\) −0.801938 −0.0302242
\(705\) −7.53750 −0.283879
\(706\) −0.885772 −0.0333365
\(707\) −22.8659 −0.859961
\(708\) −9.58211 −0.360118
\(709\) 5.64204 0.211891 0.105946 0.994372i \(-0.466213\pi\)
0.105946 + 0.994372i \(0.466213\pi\)
\(710\) 2.04892 0.0768945
\(711\) −16.2349 −0.608856
\(712\) 8.24459 0.308979
\(713\) 4.30260 0.161134
\(714\) 31.5676 1.18139
\(715\) 0 0
\(716\) −17.9095 −0.669308
\(717\) −8.42998 −0.314823
\(718\) −7.07069 −0.263876
\(719\) 9.49289 0.354025 0.177013 0.984209i \(-0.443357\pi\)
0.177013 + 0.984209i \(0.443357\pi\)
\(720\) 1.00000 0.0372678
\(721\) −37.8896 −1.41108
\(722\) 5.00538 0.186281
\(723\) −20.8485 −0.775363
\(724\) −8.74094 −0.324854
\(725\) 5.63102 0.209131
\(726\) −10.3569 −0.384380
\(727\) −18.2338 −0.676255 −0.338128 0.941100i \(-0.609794\pi\)
−0.338128 + 0.941100i \(0.609794\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) −2.17092 −0.0803492
\(731\) −46.7972 −1.73086
\(732\) 10.7899 0.398805
\(733\) 36.4172 1.34510 0.672549 0.740052i \(-0.265199\pi\)
0.672549 + 0.740052i \(0.265199\pi\)
\(734\) −27.3588 −1.00983
\(735\) −11.0368 −0.407100
\(736\) 2.54288 0.0937317
\(737\) −6.27844 −0.231269
\(738\) −8.85086 −0.325804
\(739\) −48.2312 −1.77421 −0.887106 0.461565i \(-0.847288\pi\)
−0.887106 + 0.461565i \(0.847288\pi\)
\(740\) −9.67456 −0.355644
\(741\) 0 0
\(742\) −57.6088 −2.11488
\(743\) 41.6859 1.52931 0.764654 0.644441i \(-0.222910\pi\)
0.764654 + 0.644441i \(0.222910\pi\)
\(744\) −1.69202 −0.0620325
\(745\) −1.74094 −0.0637831
\(746\) −18.3370 −0.671367
\(747\) −3.61356 −0.132213
\(748\) −5.96077 −0.217947
\(749\) 59.4476 2.17217
\(750\) 1.00000 0.0365148
\(751\) −0.486663 −0.0177586 −0.00887930 0.999961i \(-0.502826\pi\)
−0.00887930 + 0.999961i \(0.502826\pi\)
\(752\) 7.53750 0.274864
\(753\) 6.39075 0.232892
\(754\) 0 0
\(755\) −1.75840 −0.0639946
\(756\) −4.24698 −0.154461
\(757\) −46.2784 −1.68202 −0.841009 0.541021i \(-0.818038\pi\)
−0.841009 + 0.541021i \(0.818038\pi\)
\(758\) 14.3690 0.521905
\(759\) −2.03923 −0.0740193
\(760\) −3.74094 −0.135698
\(761\) 9.76915 0.354131 0.177066 0.984199i \(-0.443339\pi\)
0.177066 + 0.984199i \(0.443339\pi\)
\(762\) −4.19136 −0.151837
\(763\) 58.8654 2.13107
\(764\) −3.76510 −0.136217
\(765\) 7.43296 0.268739
\(766\) −34.6920 −1.25347
\(767\) 0 0
\(768\) −1.00000 −0.0360844
\(769\) −18.5888 −0.670330 −0.335165 0.942160i \(-0.608792\pi\)
−0.335165 + 0.942160i \(0.608792\pi\)
\(770\) 3.40581 0.122737
\(771\) −9.78687 −0.352466
\(772\) 21.9584 0.790299
\(773\) 23.4480 0.843367 0.421683 0.906743i \(-0.361439\pi\)
0.421683 + 0.906743i \(0.361439\pi\)
\(774\) 6.29590 0.226301
\(775\) −1.69202 −0.0607792
\(776\) −6.07069 −0.217925
\(777\) 41.0877 1.47401
\(778\) 32.6872 1.17189
\(779\) 33.1105 1.18631
\(780\) 0 0
\(781\) 1.64310 0.0587949
\(782\) 18.9011 0.675902
\(783\) −5.63102 −0.201236
\(784\) 11.0368 0.394173
\(785\) −15.9825 −0.570441
\(786\) −10.8140 −0.385723
\(787\) 28.0030 0.998199 0.499099 0.866545i \(-0.333664\pi\)
0.499099 + 0.866545i \(0.333664\pi\)
\(788\) −18.2446 −0.649936
\(789\) −14.0465 −0.500070
\(790\) 16.2349 0.577612
\(791\) 9.33513 0.331919
\(792\) 0.801938 0.0284956
\(793\) 0 0
\(794\) −12.3080 −0.436794
\(795\) −13.5646 −0.481088
\(796\) −1.77479 −0.0629058
\(797\) 42.9452 1.52120 0.760599 0.649222i \(-0.224905\pi\)
0.760599 + 0.649222i \(0.224905\pi\)
\(798\) 15.8877 0.562418
\(799\) 56.0259 1.98206
\(800\) −1.00000 −0.0353553
\(801\) −8.24459 −0.291308
\(802\) −19.2524 −0.679824
\(803\) −1.74094 −0.0614364
\(804\) −7.82908 −0.276111
\(805\) −10.7995 −0.380634
\(806\) 0 0
\(807\) 17.2597 0.607569
\(808\) 5.38404 0.189410
\(809\) 36.8622 1.29601 0.648003 0.761638i \(-0.275604\pi\)
0.648003 + 0.761638i \(0.275604\pi\)
\(810\) −1.00000 −0.0351364
\(811\) 27.9952 0.983045 0.491522 0.870865i \(-0.336441\pi\)
0.491522 + 0.870865i \(0.336441\pi\)
\(812\) 23.9148 0.839246
\(813\) 17.1782 0.602466
\(814\) −7.75840 −0.271932
\(815\) −5.18598 −0.181657
\(816\) −7.43296 −0.260206
\(817\) −23.5526 −0.824000
\(818\) 22.8672 0.799535
\(819\) 0 0
\(820\) 8.85086 0.309085
\(821\) 18.5593 0.647723 0.323861 0.946105i \(-0.395019\pi\)
0.323861 + 0.946105i \(0.395019\pi\)
\(822\) 12.8509 0.448225
\(823\) −43.2234 −1.50667 −0.753337 0.657635i \(-0.771557\pi\)
−0.753337 + 0.657635i \(0.771557\pi\)
\(824\) 8.92154 0.310797
\(825\) 0.801938 0.0279199
\(826\) −40.6950 −1.41596
\(827\) −16.7162 −0.581279 −0.290639 0.956833i \(-0.593868\pi\)
−0.290639 + 0.956833i \(0.593868\pi\)
\(828\) −2.54288 −0.0883711
\(829\) −10.6735 −0.370706 −0.185353 0.982672i \(-0.559343\pi\)
−0.185353 + 0.982672i \(0.559343\pi\)
\(830\) 3.61356 0.125429
\(831\) 21.1444 0.733490
\(832\) 0 0
\(833\) 82.0364 2.84239
\(834\) 16.6233 0.575616
\(835\) −5.29590 −0.183272
\(836\) −3.00000 −0.103757
\(837\) 1.69202 0.0584848
\(838\) 17.1806 0.593494
\(839\) −44.2465 −1.52756 −0.763780 0.645477i \(-0.776659\pi\)
−0.763780 + 0.645477i \(0.776659\pi\)
\(840\) 4.24698 0.146535
\(841\) 2.70841 0.0933936
\(842\) −36.9922 −1.27484
\(843\) −4.09246 −0.140952
\(844\) 0.0978347 0.00336761
\(845\) 0 0
\(846\) −7.53750 −0.259145
\(847\) −43.9855 −1.51136
\(848\) 13.5646 0.465812
\(849\) −12.1468 −0.416875
\(850\) −7.43296 −0.254948
\(851\) 24.6012 0.843319
\(852\) 2.04892 0.0701948
\(853\) −42.5437 −1.45667 −0.728335 0.685221i \(-0.759705\pi\)
−0.728335 + 0.685221i \(0.759705\pi\)
\(854\) 45.8243 1.56808
\(855\) 3.74094 0.127937
\(856\) −13.9976 −0.478428
\(857\) 8.95061 0.305747 0.152873 0.988246i \(-0.451147\pi\)
0.152873 + 0.988246i \(0.451147\pi\)
\(858\) 0 0
\(859\) −24.8810 −0.848928 −0.424464 0.905445i \(-0.639538\pi\)
−0.424464 + 0.905445i \(0.639538\pi\)
\(860\) −6.29590 −0.214688
\(861\) −37.5894 −1.28104
\(862\) 19.0834 0.649982
\(863\) 53.7531 1.82978 0.914889 0.403705i \(-0.132278\pi\)
0.914889 + 0.403705i \(0.132278\pi\)
\(864\) 1.00000 0.0340207
\(865\) −19.6558 −0.668316
\(866\) −11.5084 −0.391072
\(867\) −38.2489 −1.29900
\(868\) −7.18598 −0.243908
\(869\) 13.0194 0.441652
\(870\) 5.63102 0.190910
\(871\) 0 0
\(872\) −13.8605 −0.469377
\(873\) 6.07069 0.205462
\(874\) 9.51275 0.321773
\(875\) 4.24698 0.143574
\(876\) −2.17092 −0.0733484
\(877\) −36.1782 −1.22165 −0.610826 0.791765i \(-0.709162\pi\)
−0.610826 + 0.791765i \(0.709162\pi\)
\(878\) −8.87502 −0.299517
\(879\) −20.6789 −0.697481
\(880\) −0.801938 −0.0270333
\(881\) −24.3075 −0.818941 −0.409470 0.912323i \(-0.634286\pi\)
−0.409470 + 0.912323i \(0.634286\pi\)
\(882\) −11.0368 −0.371630
\(883\) 33.9764 1.14340 0.571699 0.820463i \(-0.306285\pi\)
0.571699 + 0.820463i \(0.306285\pi\)
\(884\) 0 0
\(885\) −9.58211 −0.322099
\(886\) −24.4838 −0.822549
\(887\) −4.97929 −0.167188 −0.0835942 0.996500i \(-0.526640\pi\)
−0.0835942 + 0.996500i \(0.526640\pi\)
\(888\) −9.67456 −0.324657
\(889\) −17.8006 −0.597013
\(890\) 8.24459 0.276359
\(891\) −0.801938 −0.0268659
\(892\) −19.5080 −0.653175
\(893\) 28.1973 0.943588
\(894\) −1.74094 −0.0582257
\(895\) −17.9095 −0.598647
\(896\) −4.24698 −0.141882
\(897\) 0 0
\(898\) −10.2567 −0.342269
\(899\) −9.52781 −0.317770
\(900\) 1.00000 0.0333333
\(901\) 100.825 3.35898
\(902\) 7.09783 0.236332
\(903\) 26.7385 0.889803
\(904\) −2.19806 −0.0731065
\(905\) −8.74094 −0.290559
\(906\) −1.75840 −0.0584188
\(907\) −24.1919 −0.803280 −0.401640 0.915798i \(-0.631560\pi\)
−0.401640 + 0.915798i \(0.631560\pi\)
\(908\) 3.23059 0.107211
\(909\) −5.38404 −0.178577
\(910\) 0 0
\(911\) −25.3284 −0.839168 −0.419584 0.907717i \(-0.637824\pi\)
−0.419584 + 0.907717i \(0.637824\pi\)
\(912\) −3.74094 −0.123875
\(913\) 2.89785 0.0959050
\(914\) −38.6359 −1.27796
\(915\) 10.7899 0.356702
\(916\) −4.58748 −0.151575
\(917\) −45.9269 −1.51664
\(918\) 7.43296 0.245324
\(919\) 24.4316 0.805925 0.402963 0.915216i \(-0.367980\pi\)
0.402963 + 0.915216i \(0.367980\pi\)
\(920\) 2.54288 0.0838362
\(921\) −11.3123 −0.372753
\(922\) −24.8073 −0.816985
\(923\) 0 0
\(924\) 3.40581 0.112043
\(925\) −9.67456 −0.318098
\(926\) 18.0713 0.593859
\(927\) −8.92154 −0.293022
\(928\) −5.63102 −0.184847
\(929\) 1.78448 0.0585469 0.0292734 0.999571i \(-0.490681\pi\)
0.0292734 + 0.999571i \(0.490681\pi\)
\(930\) −1.69202 −0.0554836
\(931\) 41.2881 1.35316
\(932\) −6.18359 −0.202550
\(933\) 26.2282 0.858672
\(934\) 16.8834 0.552441
\(935\) −5.96077 −0.194938
\(936\) 0 0
\(937\) 46.9463 1.53367 0.766834 0.641845i \(-0.221831\pi\)
0.766834 + 0.641845i \(0.221831\pi\)
\(938\) −33.2500 −1.08565
\(939\) −29.5023 −0.962772
\(940\) 7.53750 0.245846
\(941\) 27.4047 0.893369 0.446685 0.894691i \(-0.352605\pi\)
0.446685 + 0.894691i \(0.352605\pi\)
\(942\) −15.9825 −0.520739
\(943\) −22.5066 −0.732917
\(944\) 9.58211 0.311871
\(945\) −4.24698 −0.138154
\(946\) −5.04892 −0.164154
\(947\) −34.6055 −1.12453 −0.562264 0.826958i \(-0.690070\pi\)
−0.562264 + 0.826958i \(0.690070\pi\)
\(948\) 16.2349 0.527285
\(949\) 0 0
\(950\) −3.74094 −0.121372
\(951\) 7.06398 0.229065
\(952\) −31.5676 −1.02311
\(953\) 42.3435 1.37164 0.685820 0.727771i \(-0.259444\pi\)
0.685820 + 0.727771i \(0.259444\pi\)
\(954\) −13.5646 −0.439172
\(955\) −3.76510 −0.121836
\(956\) 8.42998 0.272645
\(957\) 4.51573 0.145973
\(958\) −13.9879 −0.451929
\(959\) 54.5773 1.76239
\(960\) −1.00000 −0.0322749
\(961\) −28.1371 −0.907647
\(962\) 0 0
\(963\) 13.9976 0.451067
\(964\) 20.8485 0.671484
\(965\) 21.9584 0.706865
\(966\) −10.7995 −0.347470
\(967\) −31.6437 −1.01759 −0.508796 0.860887i \(-0.669909\pi\)
−0.508796 + 0.860887i \(0.669909\pi\)
\(968\) 10.3569 0.332883
\(969\) −27.8062 −0.893265
\(970\) −6.07069 −0.194918
\(971\) 8.41849 0.270162 0.135081 0.990835i \(-0.456871\pi\)
0.135081 + 0.990835i \(0.456871\pi\)
\(972\) −1.00000 −0.0320750
\(973\) 70.5986 2.26329
\(974\) −23.2760 −0.745813
\(975\) 0 0
\(976\) −10.7899 −0.345375
\(977\) −22.0814 −0.706448 −0.353224 0.935539i \(-0.614915\pi\)
−0.353224 + 0.935539i \(0.614915\pi\)
\(978\) −5.18598 −0.165829
\(979\) 6.61165 0.211309
\(980\) 11.0368 0.352559
\(981\) 13.8605 0.442533
\(982\) 32.5918 1.04005
\(983\) −32.7217 −1.04366 −0.521830 0.853050i \(-0.674750\pi\)
−0.521830 + 0.853050i \(0.674750\pi\)
\(984\) 8.85086 0.282155
\(985\) −18.2446 −0.581321
\(986\) −41.8552 −1.33294
\(987\) −32.0116 −1.01894
\(988\) 0 0
\(989\) 16.0097 0.509078
\(990\) 0.801938 0.0254873
\(991\) −52.9329 −1.68147 −0.840734 0.541448i \(-0.817876\pi\)
−0.840734 + 0.541448i \(0.817876\pi\)
\(992\) 1.69202 0.0537217
\(993\) −10.3134 −0.327284
\(994\) 8.70171 0.276001
\(995\) −1.77479 −0.0562646
\(996\) 3.61356 0.114500
\(997\) −10.8823 −0.344646 −0.172323 0.985040i \(-0.555127\pi\)
−0.172323 + 0.985040i \(0.555127\pi\)
\(998\) 1.48666 0.0470595
\(999\) 9.67456 0.306090
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 5070.2.a.bm.1.3 3
13.5 odd 4 5070.2.b.u.1351.6 6
13.8 odd 4 5070.2.b.u.1351.1 6
13.12 even 2 5070.2.a.br.1.1 yes 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
5070.2.a.bm.1.3 3 1.1 even 1 trivial
5070.2.a.br.1.1 yes 3 13.12 even 2
5070.2.b.u.1351.1 6 13.8 odd 4
5070.2.b.u.1351.6 6 13.5 odd 4