Properties

Label 1960.2.a.y.1.1
Level $1960$
Weight $2$
Character 1960.1
Self dual yes
Analytic conductor $15.651$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 1960 = 2^{3} \cdot 5 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1960.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(15.6506787962\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: 4.4.16448.2
Defining polynomial: \(x^{4} - 2 x^{3} - 7 x^{2} + 8 x + 14\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-1.87996\) of defining polynomial
Character \(\chi\) \(=\) 1960.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.87996 q^{3} +1.00000 q^{5} +0.534253 q^{9} +O(q^{10})\) \(q-1.87996 q^{3} +1.00000 q^{5} +0.534253 q^{9} -3.29417 q^{11} +4.19292 q^{13} -1.87996 q^{15} -1.43737 q^{17} +1.24445 q^{19} -0.272828 q^{23} +1.00000 q^{25} +4.63551 q^{27} -2.36268 q^{29} -3.72717 q^{31} +6.19292 q^{33} +0.169761 q^{37} -7.88252 q^{39} +11.6630 q^{41} -10.1458 q^{43} +0.534253 q^{45} +3.12441 q^{47} +2.70220 q^{51} +9.24701 q^{53} -3.29417 q^{55} -2.33952 q^{57} +9.07107 q^{59} +7.27102 q^{61} +4.19292 q^{65} -13.1439 q^{67} +0.512907 q^{69} +6.87474 q^{71} -15.2496 q^{73} -1.87996 q^{75} +14.9510 q^{79} -10.3173 q^{81} +0.167199 q^{83} -1.43737 q^{85} +4.44175 q^{87} +3.09869 q^{89} +7.00694 q^{93} +1.24445 q^{95} +6.60894 q^{97} -1.75992 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 2q^{3} + 4q^{5} + 6q^{9} + O(q^{10}) \) \( 4q + 2q^{3} + 4q^{5} + 6q^{9} + 2q^{11} + 10q^{13} + 2q^{15} + 6q^{17} - 4q^{23} + 4q^{25} + 14q^{27} - 2q^{29} - 12q^{31} + 18q^{33} + 14q^{39} + 12q^{41} - 8q^{43} + 6q^{45} - 2q^{47} + 2q^{51} - 4q^{53} + 2q^{55} - 8q^{57} + 8q^{59} + 20q^{61} + 10q^{65} - 8q^{67} + 24q^{69} + 4q^{71} + 16q^{73} + 2q^{75} + 22q^{79} - 20q^{81} + 36q^{83} + 6q^{85} - 18q^{87} + 40q^{89} - 32q^{93} + 26q^{97} + 12q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) −1.87996 −1.08540 −0.542698 0.839928i \(-0.682597\pi\)
−0.542698 + 0.839928i \(0.682597\pi\)
\(4\) 0 0
\(5\) 1.00000 0.447214
\(6\) 0 0
\(7\) 0 0
\(8\) 0 0
\(9\) 0.534253 0.178084
\(10\) 0 0
\(11\) −3.29417 −0.993231 −0.496615 0.867971i \(-0.665424\pi\)
−0.496615 + 0.867971i \(0.665424\pi\)
\(12\) 0 0
\(13\) 4.19292 1.16291 0.581453 0.813580i \(-0.302484\pi\)
0.581453 + 0.813580i \(0.302484\pi\)
\(14\) 0 0
\(15\) −1.87996 −0.485404
\(16\) 0 0
\(17\) −1.43737 −0.348614 −0.174307 0.984691i \(-0.555768\pi\)
−0.174307 + 0.984691i \(0.555768\pi\)
\(18\) 0 0
\(19\) 1.24445 0.285497 0.142748 0.989759i \(-0.454406\pi\)
0.142748 + 0.989759i \(0.454406\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −0.272828 −0.0568887 −0.0284443 0.999595i \(-0.509055\pi\)
−0.0284443 + 0.999595i \(0.509055\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) 4.63551 0.892104
\(28\) 0 0
\(29\) −2.36268 −0.438739 −0.219369 0.975642i \(-0.570400\pi\)
−0.219369 + 0.975642i \(0.570400\pi\)
\(30\) 0 0
\(31\) −3.72717 −0.669420 −0.334710 0.942321i \(-0.608638\pi\)
−0.334710 + 0.942321i \(0.608638\pi\)
\(32\) 0 0
\(33\) 6.19292 1.07805
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 0.169761 0.0279085 0.0139543 0.999903i \(-0.495558\pi\)
0.0139543 + 0.999903i \(0.495558\pi\)
\(38\) 0 0
\(39\) −7.88252 −1.26221
\(40\) 0 0
\(41\) 11.6630 1.82146 0.910730 0.413001i \(-0.135520\pi\)
0.910730 + 0.413001i \(0.135520\pi\)
\(42\) 0 0
\(43\) −10.1458 −1.54721 −0.773607 0.633666i \(-0.781549\pi\)
−0.773607 + 0.633666i \(0.781549\pi\)
\(44\) 0 0
\(45\) 0.534253 0.0796417
\(46\) 0 0
\(47\) 3.12441 0.455743 0.227871 0.973691i \(-0.426823\pi\)
0.227871 + 0.973691i \(0.426823\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 0 0
\(51\) 2.70220 0.378384
\(52\) 0 0
\(53\) 9.24701 1.27018 0.635088 0.772440i \(-0.280964\pi\)
0.635088 + 0.772440i \(0.280964\pi\)
\(54\) 0 0
\(55\) −3.29417 −0.444186
\(56\) 0 0
\(57\) −2.33952 −0.309877
\(58\) 0 0
\(59\) 9.07107 1.18095 0.590476 0.807055i \(-0.298940\pi\)
0.590476 + 0.807055i \(0.298940\pi\)
\(60\) 0 0
\(61\) 7.27102 0.930958 0.465479 0.885059i \(-0.345882\pi\)
0.465479 + 0.885059i \(0.345882\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 4.19292 0.520068
\(66\) 0 0
\(67\) −13.1439 −1.60579 −0.802894 0.596121i \(-0.796708\pi\)
−0.802894 + 0.596121i \(0.796708\pi\)
\(68\) 0 0
\(69\) 0.512907 0.0617467
\(70\) 0 0
\(71\) 6.87474 0.815882 0.407941 0.913008i \(-0.366247\pi\)
0.407941 + 0.913008i \(0.366247\pi\)
\(72\) 0 0
\(73\) −15.2496 −1.78483 −0.892414 0.451218i \(-0.850990\pi\)
−0.892414 + 0.451218i \(0.850990\pi\)
\(74\) 0 0
\(75\) −1.87996 −0.217079
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 14.9510 1.68212 0.841061 0.540940i \(-0.181931\pi\)
0.841061 + 0.540940i \(0.181931\pi\)
\(80\) 0 0
\(81\) −10.3173 −1.14637
\(82\) 0 0
\(83\) 0.167199 0.0183524 0.00917622 0.999958i \(-0.497079\pi\)
0.00917622 + 0.999958i \(0.497079\pi\)
\(84\) 0 0
\(85\) −1.43737 −0.155905
\(86\) 0 0
\(87\) 4.44175 0.476205
\(88\) 0 0
\(89\) 3.09869 0.328461 0.164230 0.986422i \(-0.447486\pi\)
0.164230 + 0.986422i \(0.447486\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 7.00694 0.726585
\(94\) 0 0
\(95\) 1.24445 0.127678
\(96\) 0 0
\(97\) 6.60894 0.671037 0.335518 0.942034i \(-0.391089\pi\)
0.335518 + 0.942034i \(0.391089\pi\)
\(98\) 0 0
\(99\) −1.75992 −0.176879
\(100\) 0 0
\(101\) 18.8372 1.87437 0.937185 0.348834i \(-0.113422\pi\)
0.937185 + 0.348834i \(0.113422\pi\)
\(102\) 0 0
\(103\) −1.46756 −0.144603 −0.0723014 0.997383i \(-0.523034\pi\)
−0.0723014 + 0.997383i \(0.523034\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 13.5625 1.31114 0.655570 0.755135i \(-0.272428\pi\)
0.655570 + 0.755135i \(0.272428\pi\)
\(108\) 0 0
\(109\) 14.8140 1.41893 0.709463 0.704743i \(-0.248938\pi\)
0.709463 + 0.704743i \(0.248938\pi\)
\(110\) 0 0
\(111\) −0.319144 −0.0302918
\(112\) 0 0
\(113\) −13.1903 −1.24084 −0.620418 0.784271i \(-0.713037\pi\)
−0.620418 + 0.784271i \(0.713037\pi\)
\(114\) 0 0
\(115\) −0.272828 −0.0254414
\(116\) 0 0
\(117\) 2.24008 0.207095
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) −0.148415 −0.0134923
\(122\) 0 0
\(123\) −21.9261 −1.97701
\(124\) 0 0
\(125\) 1.00000 0.0894427
\(126\) 0 0
\(127\) 6.86118 0.608831 0.304416 0.952539i \(-0.401539\pi\)
0.304416 + 0.952539i \(0.401539\pi\)
\(128\) 0 0
\(129\) 19.0736 1.67934
\(130\) 0 0
\(131\) −0.345708 −0.0302047 −0.0151023 0.999886i \(-0.504807\pi\)
−0.0151023 + 0.999886i \(0.504807\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) 4.63551 0.398961
\(136\) 0 0
\(137\) 11.3137 0.966595 0.483298 0.875456i \(-0.339439\pi\)
0.483298 + 0.875456i \(0.339439\pi\)
\(138\) 0 0
\(139\) 2.68885 0.228066 0.114033 0.993477i \(-0.463623\pi\)
0.114033 + 0.993477i \(0.463623\pi\)
\(140\) 0 0
\(141\) −5.87377 −0.494661
\(142\) 0 0
\(143\) −13.8122 −1.15503
\(144\) 0 0
\(145\) −2.36268 −0.196210
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 13.1340 1.07598 0.537990 0.842951i \(-0.319184\pi\)
0.537990 + 0.842951i \(0.319184\pi\)
\(150\) 0 0
\(151\) 3.01140 0.245065 0.122532 0.992465i \(-0.460898\pi\)
0.122532 + 0.992465i \(0.460898\pi\)
\(152\) 0 0
\(153\) −0.767920 −0.0620826
\(154\) 0 0
\(155\) −3.72717 −0.299374
\(156\) 0 0
\(157\) 19.4631 1.55332 0.776662 0.629918i \(-0.216911\pi\)
0.776662 + 0.629918i \(0.216911\pi\)
\(158\) 0 0
\(159\) −17.3840 −1.37864
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) 13.4922 1.05679 0.528396 0.848998i \(-0.322794\pi\)
0.528396 + 0.848998i \(0.322794\pi\)
\(164\) 0 0
\(165\) 6.19292 0.482118
\(166\) 0 0
\(167\) 12.3011 0.951889 0.475944 0.879475i \(-0.342106\pi\)
0.475944 + 0.879475i \(0.342106\pi\)
\(168\) 0 0
\(169\) 4.58057 0.352351
\(170\) 0 0
\(171\) 0.664852 0.0508425
\(172\) 0 0
\(173\) 2.48975 0.189292 0.0946461 0.995511i \(-0.469828\pi\)
0.0946461 + 0.995511i \(0.469828\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) −17.0533 −1.28180
\(178\) 0 0
\(179\) 15.1340 1.13117 0.565584 0.824690i \(-0.308651\pi\)
0.565584 + 0.824690i \(0.308651\pi\)
\(180\) 0 0
\(181\) 11.0637 0.822357 0.411179 0.911555i \(-0.365117\pi\)
0.411179 + 0.911555i \(0.365117\pi\)
\(182\) 0 0
\(183\) −13.6692 −1.01046
\(184\) 0 0
\(185\) 0.169761 0.0124811
\(186\) 0 0
\(187\) 4.73495 0.346254
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −22.4517 −1.62455 −0.812274 0.583277i \(-0.801770\pi\)
−0.812274 + 0.583277i \(0.801770\pi\)
\(192\) 0 0
\(193\) 3.11482 0.224210 0.112105 0.993696i \(-0.464241\pi\)
0.112105 + 0.993696i \(0.464241\pi\)
\(194\) 0 0
\(195\) −7.88252 −0.564479
\(196\) 0 0
\(197\) 1.38765 0.0988659 0.0494330 0.998777i \(-0.484259\pi\)
0.0494330 + 0.998777i \(0.484259\pi\)
\(198\) 0 0
\(199\) −0.413463 −0.0293096 −0.0146548 0.999893i \(-0.504665\pi\)
−0.0146548 + 0.999893i \(0.504665\pi\)
\(200\) 0 0
\(201\) 24.7101 1.74292
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 11.6630 0.814582
\(206\) 0 0
\(207\) −0.145759 −0.0101310
\(208\) 0 0
\(209\) −4.09944 −0.283564
\(210\) 0 0
\(211\) 24.6203 1.69493 0.847464 0.530853i \(-0.178128\pi\)
0.847464 + 0.530853i \(0.178128\pi\)
\(212\) 0 0
\(213\) −12.9242 −0.885555
\(214\) 0 0
\(215\) −10.1458 −0.691935
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) 28.6686 1.93724
\(220\) 0 0
\(221\) −6.02678 −0.405405
\(222\) 0 0
\(223\) −17.9065 −1.19911 −0.599555 0.800334i \(-0.704656\pi\)
−0.599555 + 0.800334i \(0.704656\pi\)
\(224\) 0 0
\(225\) 0.534253 0.0356168
\(226\) 0 0
\(227\) −12.5486 −0.832878 −0.416439 0.909164i \(-0.636722\pi\)
−0.416439 + 0.909164i \(0.636722\pi\)
\(228\) 0 0
\(229\) −15.8354 −1.04643 −0.523215 0.852201i \(-0.675268\pi\)
−0.523215 + 0.852201i \(0.675268\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −23.3792 −1.53162 −0.765811 0.643065i \(-0.777662\pi\)
−0.765811 + 0.643065i \(0.777662\pi\)
\(234\) 0 0
\(235\) 3.12441 0.203814
\(236\) 0 0
\(237\) −28.1073 −1.82577
\(238\) 0 0
\(239\) 20.9135 1.35278 0.676390 0.736544i \(-0.263544\pi\)
0.676390 + 0.736544i \(0.263544\pi\)
\(240\) 0 0
\(241\) −3.35746 −0.216273 −0.108137 0.994136i \(-0.534488\pi\)
−0.108137 + 0.994136i \(0.534488\pi\)
\(242\) 0 0
\(243\) 5.48966 0.352162
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 5.21789 0.332006
\(248\) 0 0
\(249\) −0.314327 −0.0199197
\(250\) 0 0
\(251\) 16.5717 1.04600 0.522999 0.852333i \(-0.324813\pi\)
0.522999 + 0.852333i \(0.324813\pi\)
\(252\) 0 0
\(253\) 0.898744 0.0565036
\(254\) 0 0
\(255\) 2.70220 0.169218
\(256\) 0 0
\(257\) 13.1414 0.819737 0.409869 0.912145i \(-0.365575\pi\)
0.409869 + 0.912145i \(0.365575\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) −1.26227 −0.0781324
\(262\) 0 0
\(263\) −3.75480 −0.231531 −0.115765 0.993277i \(-0.536932\pi\)
−0.115765 + 0.993277i \(0.536932\pi\)
\(264\) 0 0
\(265\) 9.24701 0.568040
\(266\) 0 0
\(267\) −5.82542 −0.356510
\(268\) 0 0
\(269\) 16.4952 1.00573 0.502866 0.864365i \(-0.332279\pi\)
0.502866 + 0.864365i \(0.332279\pi\)
\(270\) 0 0
\(271\) −22.5420 −1.36933 −0.684665 0.728857i \(-0.740052\pi\)
−0.684665 + 0.728857i \(0.740052\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −3.29417 −0.198646
\(276\) 0 0
\(277\) −14.9982 −0.901154 −0.450577 0.892738i \(-0.648782\pi\)
−0.450577 + 0.892738i \(0.648782\pi\)
\(278\) 0 0
\(279\) −1.99125 −0.119213
\(280\) 0 0
\(281\) −28.0283 −1.67203 −0.836014 0.548709i \(-0.815120\pi\)
−0.836014 + 0.548709i \(0.815120\pi\)
\(282\) 0 0
\(283\) −26.1715 −1.55573 −0.777866 0.628430i \(-0.783698\pi\)
−0.777866 + 0.628430i \(0.783698\pi\)
\(284\) 0 0
\(285\) −2.33952 −0.138581
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) −14.9340 −0.878468
\(290\) 0 0
\(291\) −12.4246 −0.728340
\(292\) 0 0
\(293\) 15.6061 0.911716 0.455858 0.890052i \(-0.349332\pi\)
0.455858 + 0.890052i \(0.349332\pi\)
\(294\) 0 0
\(295\) 9.07107 0.528138
\(296\) 0 0
\(297\) −15.2702 −0.886065
\(298\) 0 0
\(299\) −1.14395 −0.0661562
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) −35.4132 −2.03443
\(304\) 0 0
\(305\) 7.27102 0.416337
\(306\) 0 0
\(307\) 21.3055 1.21597 0.607984 0.793949i \(-0.291978\pi\)
0.607984 + 0.793949i \(0.291978\pi\)
\(308\) 0 0
\(309\) 2.75895 0.156951
\(310\) 0 0
\(311\) −14.8799 −0.843760 −0.421880 0.906652i \(-0.638630\pi\)
−0.421880 + 0.906652i \(0.638630\pi\)
\(312\) 0 0
\(313\) 2.16273 0.122245 0.0611224 0.998130i \(-0.480532\pi\)
0.0611224 + 0.998130i \(0.480532\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −21.4595 −1.20528 −0.602642 0.798012i \(-0.705885\pi\)
−0.602642 + 0.798012i \(0.705885\pi\)
\(318\) 0 0
\(319\) 7.78308 0.435769
\(320\) 0 0
\(321\) −25.4970 −1.42311
\(322\) 0 0
\(323\) −1.78874 −0.0995282
\(324\) 0 0
\(325\) 4.19292 0.232581
\(326\) 0 0
\(327\) −27.8498 −1.54010
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) −11.6692 −0.641399 −0.320699 0.947181i \(-0.603918\pi\)
−0.320699 + 0.947181i \(0.603918\pi\)
\(332\) 0 0
\(333\) 0.0906953 0.00497007
\(334\) 0 0
\(335\) −13.1439 −0.718131
\(336\) 0 0
\(337\) 17.1403 0.933693 0.466846 0.884338i \(-0.345390\pi\)
0.466846 + 0.884338i \(0.345390\pi\)
\(338\) 0 0
\(339\) 24.7972 1.34680
\(340\) 0 0
\(341\) 12.2780 0.664888
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) 0.512907 0.0276140
\(346\) 0 0
\(347\) 5.20070 0.279188 0.139594 0.990209i \(-0.455420\pi\)
0.139594 + 0.990209i \(0.455420\pi\)
\(348\) 0 0
\(349\) −33.8645 −1.81272 −0.906362 0.422501i \(-0.861152\pi\)
−0.906362 + 0.422501i \(0.861152\pi\)
\(350\) 0 0
\(351\) 19.4363 1.03743
\(352\) 0 0
\(353\) −23.2451 −1.23721 −0.618606 0.785701i \(-0.712302\pi\)
−0.618606 + 0.785701i \(0.712302\pi\)
\(354\) 0 0
\(355\) 6.87474 0.364873
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −20.2843 −1.07056 −0.535281 0.844674i \(-0.679794\pi\)
−0.535281 + 0.844674i \(0.679794\pi\)
\(360\) 0 0
\(361\) −17.4513 −0.918491
\(362\) 0 0
\(363\) 0.279014 0.0146445
\(364\) 0 0
\(365\) −15.2496 −0.798199
\(366\) 0 0
\(367\) −13.2738 −0.692887 −0.346443 0.938071i \(-0.612611\pi\)
−0.346443 + 0.938071i \(0.612611\pi\)
\(368\) 0 0
\(369\) 6.23101 0.324373
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) −35.7083 −1.84891 −0.924453 0.381297i \(-0.875478\pi\)
−0.924453 + 0.381297i \(0.875478\pi\)
\(374\) 0 0
\(375\) −1.87996 −0.0970808
\(376\) 0 0
\(377\) −9.90652 −0.510212
\(378\) 0 0
\(379\) −12.6878 −0.651728 −0.325864 0.945417i \(-0.605655\pi\)
−0.325864 + 0.945417i \(0.605655\pi\)
\(380\) 0 0
\(381\) −12.8987 −0.660823
\(382\) 0 0
\(383\) 36.5707 1.86867 0.934337 0.356391i \(-0.115993\pi\)
0.934337 + 0.356391i \(0.115993\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) −5.42040 −0.275534
\(388\) 0 0
\(389\) 30.1085 1.52656 0.763282 0.646066i \(-0.223587\pi\)
0.763282 + 0.646066i \(0.223587\pi\)
\(390\) 0 0
\(391\) 0.392156 0.0198322
\(392\) 0 0
\(393\) 0.649918 0.0327840
\(394\) 0 0
\(395\) 14.9510 0.752268
\(396\) 0 0
\(397\) −7.46756 −0.374786 −0.187393 0.982285i \(-0.560004\pi\)
−0.187393 + 0.982285i \(0.560004\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −11.7331 −0.585925 −0.292963 0.956124i \(-0.594641\pi\)
−0.292963 + 0.956124i \(0.594641\pi\)
\(402\) 0 0
\(403\) −15.6277 −0.778473
\(404\) 0 0
\(405\) −10.3173 −0.512672
\(406\) 0 0
\(407\) −0.559222 −0.0277196
\(408\) 0 0
\(409\) 16.8601 0.833679 0.416840 0.908980i \(-0.363138\pi\)
0.416840 + 0.908980i \(0.363138\pi\)
\(410\) 0 0
\(411\) −21.2693 −1.04914
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) 0.167199 0.00820746
\(416\) 0 0
\(417\) −5.05494 −0.247541
\(418\) 0 0
\(419\) −10.3884 −0.507507 −0.253753 0.967269i \(-0.581665\pi\)
−0.253753 + 0.967269i \(0.581665\pi\)
\(420\) 0 0
\(421\) 13.7758 0.671393 0.335696 0.941970i \(-0.391028\pi\)
0.335696 + 0.941970i \(0.391028\pi\)
\(422\) 0 0
\(423\) 1.66923 0.0811606
\(424\) 0 0
\(425\) −1.43737 −0.0697228
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) 25.9664 1.25367
\(430\) 0 0
\(431\) 30.4568 1.46705 0.733527 0.679661i \(-0.237873\pi\)
0.733527 + 0.679661i \(0.237873\pi\)
\(432\) 0 0
\(433\) 25.9182 1.24555 0.622774 0.782402i \(-0.286006\pi\)
0.622774 + 0.782402i \(0.286006\pi\)
\(434\) 0 0
\(435\) 4.44175 0.212965
\(436\) 0 0
\(437\) −0.339522 −0.0162415
\(438\) 0 0
\(439\) −26.9414 −1.28584 −0.642922 0.765931i \(-0.722278\pi\)
−0.642922 + 0.765931i \(0.722278\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −16.3858 −0.778515 −0.389257 0.921129i \(-0.627268\pi\)
−0.389257 + 0.921129i \(0.627268\pi\)
\(444\) 0 0
\(445\) 3.09869 0.146892
\(446\) 0 0
\(447\) −24.6914 −1.16786
\(448\) 0 0
\(449\) 14.2152 0.670858 0.335429 0.942065i \(-0.391119\pi\)
0.335429 + 0.942065i \(0.391119\pi\)
\(450\) 0 0
\(451\) −38.4201 −1.80913
\(452\) 0 0
\(453\) −5.66132 −0.265992
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −14.5093 −0.678716 −0.339358 0.940657i \(-0.610210\pi\)
−0.339358 + 0.940657i \(0.610210\pi\)
\(458\) 0 0
\(459\) −6.66295 −0.311000
\(460\) 0 0
\(461\) −17.8933 −0.833374 −0.416687 0.909050i \(-0.636809\pi\)
−0.416687 + 0.909050i \(0.636809\pi\)
\(462\) 0 0
\(463\) 15.2657 0.709457 0.354729 0.934969i \(-0.384573\pi\)
0.354729 + 0.934969i \(0.384573\pi\)
\(464\) 0 0
\(465\) 7.00694 0.324939
\(466\) 0 0
\(467\) 23.5492 1.08973 0.544863 0.838525i \(-0.316582\pi\)
0.544863 + 0.838525i \(0.316582\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) −36.5898 −1.68597
\(472\) 0 0
\(473\) 33.4219 1.53674
\(474\) 0 0
\(475\) 1.24445 0.0570994
\(476\) 0 0
\(477\) 4.94024 0.226198
\(478\) 0 0
\(479\) 4.65867 0.212860 0.106430 0.994320i \(-0.466058\pi\)
0.106430 + 0.994320i \(0.466058\pi\)
\(480\) 0 0
\(481\) 0.711794 0.0324550
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 6.60894 0.300097
\(486\) 0 0
\(487\) 11.5210 0.522068 0.261034 0.965330i \(-0.415937\pi\)
0.261034 + 0.965330i \(0.415937\pi\)
\(488\) 0 0
\(489\) −25.3648 −1.14704
\(490\) 0 0
\(491\) −14.2771 −0.644317 −0.322158 0.946686i \(-0.604408\pi\)
−0.322158 + 0.946686i \(0.604408\pi\)
\(492\) 0 0
\(493\) 3.39605 0.152950
\(494\) 0 0
\(495\) −1.75992 −0.0791026
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) 36.9790 1.65541 0.827703 0.561167i \(-0.189647\pi\)
0.827703 + 0.561167i \(0.189647\pi\)
\(500\) 0 0
\(501\) −23.1256 −1.03318
\(502\) 0 0
\(503\) −8.08985 −0.360709 −0.180354 0.983602i \(-0.557724\pi\)
−0.180354 + 0.983602i \(0.557724\pi\)
\(504\) 0 0
\(505\) 18.8372 0.838243
\(506\) 0 0
\(507\) −8.61129 −0.382441
\(508\) 0 0
\(509\) −28.0806 −1.24465 −0.622325 0.782759i \(-0.713812\pi\)
−0.622325 + 0.782759i \(0.713812\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) 5.76867 0.254693
\(514\) 0 0
\(515\) −1.46756 −0.0646684
\(516\) 0 0
\(517\) −10.2924 −0.452658
\(518\) 0 0
\(519\) −4.68063 −0.205457
\(520\) 0 0
\(521\) −12.5810 −0.551182 −0.275591 0.961275i \(-0.588874\pi\)
−0.275591 + 0.961275i \(0.588874\pi\)
\(522\) 0 0
\(523\) −1.48453 −0.0649140 −0.0324570 0.999473i \(-0.510333\pi\)
−0.0324570 + 0.999473i \(0.510333\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 5.35733 0.233369
\(528\) 0 0
\(529\) −22.9256 −0.996764
\(530\) 0 0
\(531\) 4.84624 0.210309
\(532\) 0 0
\(533\) 48.9022 2.11819
\(534\) 0 0
\(535\) 13.5625 0.586360
\(536\) 0 0
\(537\) −28.4513 −1.22777
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) 18.9135 0.813153 0.406577 0.913617i \(-0.366722\pi\)
0.406577 + 0.913617i \(0.366722\pi\)
\(542\) 0 0
\(543\) −20.7993 −0.892583
\(544\) 0 0
\(545\) 14.8140 0.634563
\(546\) 0 0
\(547\) −17.4403 −0.745693 −0.372846 0.927893i \(-0.621618\pi\)
−0.372846 + 0.927893i \(0.621618\pi\)
\(548\) 0 0
\(549\) 3.88456 0.165789
\(550\) 0 0
\(551\) −2.94024 −0.125259
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) −0.319144 −0.0135469
\(556\) 0 0
\(557\) 19.2106 0.813981 0.406990 0.913432i \(-0.366578\pi\)
0.406990 + 0.913432i \(0.366578\pi\)
\(558\) 0 0
\(559\) −42.5403 −1.79926
\(560\) 0 0
\(561\) −8.90152 −0.375823
\(562\) 0 0
\(563\) 33.7952 1.42430 0.712150 0.702028i \(-0.247722\pi\)
0.712150 + 0.702028i \(0.247722\pi\)
\(564\) 0 0
\(565\) −13.1903 −0.554919
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 4.63635 0.194366 0.0971830 0.995267i \(-0.469017\pi\)
0.0971830 + 0.995267i \(0.469017\pi\)
\(570\) 0 0
\(571\) 43.8681 1.83582 0.917912 0.396785i \(-0.129874\pi\)
0.917912 + 0.396785i \(0.129874\pi\)
\(572\) 0 0
\(573\) 42.2083 1.76328
\(574\) 0 0
\(575\) −0.272828 −0.0113777
\(576\) 0 0
\(577\) 9.27755 0.386230 0.193115 0.981176i \(-0.438141\pi\)
0.193115 + 0.981176i \(0.438141\pi\)
\(578\) 0 0
\(579\) −5.85574 −0.243356
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) −30.4613 −1.26158
\(584\) 0 0
\(585\) 2.24008 0.0926158
\(586\) 0 0
\(587\) 28.9971 1.19684 0.598420 0.801183i \(-0.295796\pi\)
0.598420 + 0.801183i \(0.295796\pi\)
\(588\) 0 0
\(589\) −4.63829 −0.191117
\(590\) 0 0
\(591\) −2.60873 −0.107309
\(592\) 0 0
\(593\) −25.5228 −1.04809 −0.524047 0.851689i \(-0.675578\pi\)
−0.524047 + 0.851689i \(0.675578\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 0.777294 0.0318125
\(598\) 0 0
\(599\) 31.0364 1.26811 0.634057 0.773287i \(-0.281389\pi\)
0.634057 + 0.773287i \(0.281389\pi\)
\(600\) 0 0
\(601\) 3.56479 0.145411 0.0727054 0.997353i \(-0.476837\pi\)
0.0727054 + 0.997353i \(0.476837\pi\)
\(602\) 0 0
\(603\) −7.02219 −0.285966
\(604\) 0 0
\(605\) −0.148415 −0.00603393
\(606\) 0 0
\(607\) −6.72620 −0.273008 −0.136504 0.990640i \(-0.543587\pi\)
−0.136504 + 0.990640i \(0.543587\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 13.1004 0.529986
\(612\) 0 0
\(613\) −19.7422 −0.797381 −0.398691 0.917085i \(-0.630535\pi\)
−0.398691 + 0.917085i \(0.630535\pi\)
\(614\) 0 0
\(615\) −21.9261 −0.884144
\(616\) 0 0
\(617\) −16.3958 −0.660069 −0.330035 0.943969i \(-0.607060\pi\)
−0.330035 + 0.943969i \(0.607060\pi\)
\(618\) 0 0
\(619\) 35.5510 1.42892 0.714458 0.699678i \(-0.246673\pi\)
0.714458 + 0.699678i \(0.246673\pi\)
\(620\) 0 0
\(621\) −1.26470 −0.0507506
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) 7.70679 0.307780
\(628\) 0 0
\(629\) −0.244010 −0.00972930
\(630\) 0 0
\(631\) −9.58569 −0.381600 −0.190800 0.981629i \(-0.561108\pi\)
−0.190800 + 0.981629i \(0.561108\pi\)
\(632\) 0 0
\(633\) −46.2851 −1.83967
\(634\) 0 0
\(635\) 6.86118 0.272278
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) 3.67285 0.145296
\(640\) 0 0
\(641\) 19.5145 0.770778 0.385389 0.922754i \(-0.374067\pi\)
0.385389 + 0.922754i \(0.374067\pi\)
\(642\) 0 0
\(643\) −18.9895 −0.748872 −0.374436 0.927253i \(-0.622164\pi\)
−0.374436 + 0.927253i \(0.622164\pi\)
\(644\) 0 0
\(645\) 19.0736 0.751023
\(646\) 0 0
\(647\) 41.2570 1.62198 0.810989 0.585061i \(-0.198930\pi\)
0.810989 + 0.585061i \(0.198930\pi\)
\(648\) 0 0
\(649\) −29.8817 −1.17296
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −19.7495 −0.772857 −0.386429 0.922319i \(-0.626291\pi\)
−0.386429 + 0.922319i \(0.626291\pi\)
\(654\) 0 0
\(655\) −0.345708 −0.0135079
\(656\) 0 0
\(657\) −8.14713 −0.317850
\(658\) 0 0
\(659\) 10.1830 0.396672 0.198336 0.980134i \(-0.436446\pi\)
0.198336 + 0.980134i \(0.436446\pi\)
\(660\) 0 0
\(661\) −2.79086 −0.108552 −0.0542759 0.998526i \(-0.517285\pi\)
−0.0542759 + 0.998526i \(0.517285\pi\)
\(662\) 0 0
\(663\) 11.3301 0.440025
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 0.644606 0.0249593
\(668\) 0 0
\(669\) 33.6636 1.30151
\(670\) 0 0
\(671\) −23.9520 −0.924657
\(672\) 0 0
\(673\) 18.2879 0.704947 0.352473 0.935822i \(-0.385341\pi\)
0.352473 + 0.935822i \(0.385341\pi\)
\(674\) 0 0
\(675\) 4.63551 0.178421
\(676\) 0 0
\(677\) 24.0523 0.924404 0.462202 0.886775i \(-0.347059\pi\)
0.462202 + 0.886775i \(0.347059\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 23.5908 0.904002
\(682\) 0 0
\(683\) −38.2200 −1.46245 −0.731224 0.682137i \(-0.761051\pi\)
−0.731224 + 0.682137i \(0.761051\pi\)
\(684\) 0 0
\(685\) 11.3137 0.432275
\(686\) 0 0
\(687\) 29.7699 1.13579
\(688\) 0 0
\(689\) 38.7720 1.47709
\(690\) 0 0
\(691\) 26.4623 1.00667 0.503337 0.864090i \(-0.332105\pi\)
0.503337 + 0.864090i \(0.332105\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 2.68885 0.101994
\(696\) 0 0
\(697\) −16.7641 −0.634986
\(698\) 0 0
\(699\) 43.9520 1.66242
\(700\) 0 0
\(701\) −34.5136 −1.30356 −0.651780 0.758408i \(-0.725977\pi\)
−0.651780 + 0.758408i \(0.725977\pi\)
\(702\) 0 0
\(703\) 0.211260 0.00796780
\(704\) 0 0
\(705\) −5.87377 −0.221219
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) 6.40900 0.240695 0.120347 0.992732i \(-0.461599\pi\)
0.120347 + 0.992732i \(0.461599\pi\)
\(710\) 0 0
\(711\) 7.98763 0.299559
\(712\) 0 0
\(713\) 1.01688 0.0380824
\(714\) 0 0
\(715\) −13.8122 −0.516547
\(716\) 0 0
\(717\) −39.3165 −1.46830
\(718\) 0 0
\(719\) 38.6070 1.43980 0.719900 0.694078i \(-0.244188\pi\)
0.719900 + 0.694078i \(0.244188\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) 6.31190 0.234742
\(724\) 0 0
\(725\) −2.36268 −0.0877477
\(726\) 0 0
\(727\) −22.5280 −0.835516 −0.417758 0.908558i \(-0.637184\pi\)
−0.417758 + 0.908558i \(0.637184\pi\)
\(728\) 0 0
\(729\) 20.6317 0.764136
\(730\) 0 0
\(731\) 14.5832 0.539380
\(732\) 0 0
\(733\) −7.50150 −0.277074 −0.138537 0.990357i \(-0.544240\pi\)
−0.138537 + 0.990357i \(0.544240\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 43.2985 1.59492
\(738\) 0 0
\(739\) 35.0864 1.29067 0.645336 0.763899i \(-0.276717\pi\)
0.645336 + 0.763899i \(0.276717\pi\)
\(740\) 0 0
\(741\) −9.80943 −0.360358
\(742\) 0 0
\(743\) −2.42902 −0.0891122 −0.0445561 0.999007i \(-0.514187\pi\)
−0.0445561 + 0.999007i \(0.514187\pi\)
\(744\) 0 0
\(745\) 13.1340 0.481193
\(746\) 0 0
\(747\) 0.0893263 0.00326828
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) 20.9598 0.764833 0.382417 0.923990i \(-0.375092\pi\)
0.382417 + 0.923990i \(0.375092\pi\)
\(752\) 0 0
\(753\) −31.1542 −1.13532
\(754\) 0 0
\(755\) 3.01140 0.109596
\(756\) 0 0
\(757\) −35.9748 −1.30753 −0.653763 0.756699i \(-0.726811\pi\)
−0.653763 + 0.756699i \(0.726811\pi\)
\(758\) 0 0
\(759\) −1.68960 −0.0613288
\(760\) 0 0
\(761\) −2.45752 −0.0890852 −0.0445426 0.999007i \(-0.514183\pi\)
−0.0445426 + 0.999007i \(0.514183\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) −0.767920 −0.0277642
\(766\) 0 0
\(767\) 38.0343 1.37334
\(768\) 0 0
\(769\) 21.6994 0.782501 0.391250 0.920284i \(-0.372043\pi\)
0.391250 + 0.920284i \(0.372043\pi\)
\(770\) 0 0
\(771\) −24.7053 −0.889739
\(772\) 0 0
\(773\) 15.6980 0.564618 0.282309 0.959324i \(-0.408900\pi\)
0.282309 + 0.959324i \(0.408900\pi\)
\(774\) 0 0
\(775\) −3.72717 −0.133884
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 14.5141 0.520022
\(780\) 0 0
\(781\) −22.6466 −0.810359
\(782\) 0 0
\(783\) −10.9522 −0.391400
\(784\) 0 0
\(785\) 19.4631 0.694668
\(786\) 0 0
\(787\) −24.5074 −0.873594 −0.436797 0.899560i \(-0.643887\pi\)
−0.436797 + 0.899560i \(0.643887\pi\)
\(788\) 0 0
\(789\) 7.05887 0.251302
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 30.4868 1.08262
\(794\) 0 0
\(795\) −17.3840 −0.616548
\(796\) 0 0
\(797\) 5.73557 0.203164 0.101582 0.994827i \(-0.467610\pi\)
0.101582 + 0.994827i \(0.467610\pi\)
\(798\) 0 0
\(799\) −4.49094 −0.158878
\(800\) 0 0
\(801\) 1.65549 0.0584937
\(802\) 0 0
\(803\) 50.2348 1.77275
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) −31.0104 −1.09162
\(808\) 0 0
\(809\) 3.48431 0.122502 0.0612510 0.998122i \(-0.480491\pi\)
0.0612510 + 0.998122i \(0.480491\pi\)
\(810\) 0 0
\(811\) 25.9953 0.912819 0.456409 0.889770i \(-0.349135\pi\)
0.456409 + 0.889770i \(0.349135\pi\)
\(812\) 0 0
\(813\) 42.3781 1.48627
\(814\) 0 0
\(815\) 13.4922 0.472612
\(816\) 0 0
\(817\) −12.6259 −0.441725
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −50.4988 −1.76242 −0.881210 0.472725i \(-0.843271\pi\)
−0.881210 + 0.472725i \(0.843271\pi\)
\(822\) 0 0
\(823\) −44.8699 −1.56407 −0.782034 0.623236i \(-0.785818\pi\)
−0.782034 + 0.623236i \(0.785818\pi\)
\(824\) 0 0
\(825\) 6.19292 0.215610
\(826\) 0 0
\(827\) −15.7323 −0.547066 −0.273533 0.961863i \(-0.588192\pi\)
−0.273533 + 0.961863i \(0.588192\pi\)
\(828\) 0 0
\(829\) −51.7726 −1.79814 −0.899068 0.437808i \(-0.855755\pi\)
−0.899068 + 0.437808i \(0.855755\pi\)
\(830\) 0 0
\(831\) 28.1960 0.978109
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) 12.3011 0.425697
\(836\) 0 0
\(837\) −17.2773 −0.597192
\(838\) 0 0
\(839\) −2.86118 −0.0987788 −0.0493894 0.998780i \(-0.515728\pi\)
−0.0493894 + 0.998780i \(0.515728\pi\)
\(840\) 0 0
\(841\) −23.4177 −0.807508
\(842\) 0 0
\(843\) 52.6921 1.81481
\(844\) 0 0
\(845\) 4.58057 0.157576
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) 49.2014 1.68859
\(850\) 0 0
\(851\) −0.0463156 −0.00158768
\(852\) 0 0
\(853\) 45.8645 1.57037 0.785185 0.619261i \(-0.212568\pi\)
0.785185 + 0.619261i \(0.212568\pi\)
\(854\) 0 0
\(855\) 0.664852 0.0227375
\(856\) 0 0
\(857\) −23.1501 −0.790793 −0.395397 0.918510i \(-0.629393\pi\)
−0.395397 + 0.918510i \(0.629393\pi\)
\(858\) 0 0
\(859\) −8.46384 −0.288783 −0.144391 0.989521i \(-0.546122\pi\)
−0.144391 + 0.989521i \(0.546122\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) −35.0923 −1.19456 −0.597278 0.802034i \(-0.703751\pi\)
−0.597278 + 0.802034i \(0.703751\pi\)
\(864\) 0 0
\(865\) 2.48975 0.0846540
\(866\) 0 0
\(867\) 28.0753 0.953486
\(868\) 0 0
\(869\) −49.2513 −1.67074
\(870\) 0 0
\(871\) −55.1115 −1.86738
\(872\) 0 0
\(873\) 3.53085 0.119501
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 19.3372 0.652971 0.326486 0.945202i \(-0.394136\pi\)
0.326486 + 0.945202i \(0.394136\pi\)
\(878\) 0 0
\(879\) −29.3388 −0.989573
\(880\) 0 0
\(881\) −33.0573 −1.11373 −0.556865 0.830603i \(-0.687996\pi\)
−0.556865 + 0.830603i \(0.687996\pi\)
\(882\) 0 0
\(883\) −28.6238 −0.963267 −0.481634 0.876373i \(-0.659956\pi\)
−0.481634 + 0.876373i \(0.659956\pi\)
\(884\) 0 0
\(885\) −17.0533 −0.573239
\(886\) 0 0
\(887\) 19.1289 0.642285 0.321142 0.947031i \(-0.395933\pi\)
0.321142 + 0.947031i \(0.395933\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) 33.9871 1.13861
\(892\) 0 0
\(893\) 3.88818 0.130113
\(894\) 0 0
\(895\) 15.1340 0.505874
\(896\) 0 0
\(897\) 2.15058 0.0718057
\(898\) 0 0
\(899\) 8.80611 0.293700
\(900\) 0 0
\(901\) −13.2914 −0.442801
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 11.0637 0.367769
\(906\) 0 0
\(907\) −50.9781 −1.69270 −0.846350 0.532627i \(-0.821205\pi\)
−0.846350 + 0.532627i \(0.821205\pi\)
\(908\) 0 0
\(909\) 10.0638 0.333796
\(910\) 0 0
\(911\) −5.52585 −0.183080 −0.0915399 0.995801i \(-0.529179\pi\)
−0.0915399 + 0.995801i \(0.529179\pi\)
\(912\) 0 0
\(913\) −0.550782 −0.0182282
\(914\) 0 0
\(915\) −13.6692 −0.451891
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) −40.6326 −1.34035 −0.670173 0.742205i \(-0.733780\pi\)
−0.670173 + 0.742205i \(0.733780\pi\)
\(920\) 0 0
\(921\) −40.0535 −1.31981
\(922\) 0 0
\(923\) 28.8252 0.948794
\(924\) 0 0
\(925\) 0.169761 0.00558171
\(926\) 0 0
\(927\) −0.784047 −0.0257515
\(928\) 0 0
\(929\) −15.5879 −0.511423 −0.255711 0.966753i \(-0.582310\pi\)
−0.255711 + 0.966753i \(0.582310\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 0 0
\(933\) 27.9736 0.915814
\(934\) 0 0
\(935\) 4.73495 0.154849
\(936\) 0 0
\(937\) −20.4794 −0.669034 −0.334517 0.942390i \(-0.608573\pi\)
−0.334517 + 0.942390i \(0.608573\pi\)
\(938\) 0 0
\(939\) −4.06585 −0.132684
\(940\) 0 0
\(941\) 16.7872 0.547248 0.273624 0.961837i \(-0.411778\pi\)
0.273624 + 0.961837i \(0.411778\pi\)
\(942\) 0 0
\(943\) −3.18201 −0.103620
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −3.43928 −0.111761 −0.0558807 0.998437i \(-0.517797\pi\)
−0.0558807 + 0.998437i \(0.517797\pi\)
\(948\) 0 0
\(949\) −63.9402 −2.07559
\(950\) 0 0
\(951\) 40.3430 1.30821
\(952\) 0 0
\(953\) −30.8610 −0.999686 −0.499843 0.866116i \(-0.666609\pi\)
−0.499843 + 0.866116i \(0.666609\pi\)
\(954\) 0 0
\(955\) −22.4517 −0.726520
\(956\) 0 0
\(957\) −14.6319 −0.472982
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) −17.1082 −0.551877
\(962\) 0 0
\(963\) 7.24582 0.233493
\(964\) 0 0
\(965\) 3.11482 0.100270
\(966\) 0 0
\(967\) −34.3195 −1.10364 −0.551820 0.833964i \(-0.686066\pi\)
−0.551820 + 0.833964i \(0.686066\pi\)
\(968\) 0 0
\(969\) 3.36276 0.108027
\(970\) 0 0
\(971\) 31.6203 1.01475 0.507373 0.861727i \(-0.330617\pi\)
0.507373 + 0.861727i \(0.330617\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) −7.88252 −0.252443
\(976\) 0 0
\(977\) −18.9454 −0.606116 −0.303058 0.952972i \(-0.598008\pi\)
−0.303058 + 0.952972i \(0.598008\pi\)
\(978\) 0 0
\(979\) −10.2076 −0.326237
\(980\) 0 0
\(981\) 7.91443 0.252688
\(982\) 0 0
\(983\) 15.2187 0.485402 0.242701 0.970101i \(-0.421967\pi\)
0.242701 + 0.970101i \(0.421967\pi\)
\(984\) 0 0
\(985\) 1.38765 0.0442142
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 2.76805 0.0880189
\(990\) 0 0
\(991\) 31.3069 0.994496 0.497248 0.867608i \(-0.334344\pi\)
0.497248 + 0.867608i \(0.334344\pi\)
\(992\) 0 0
\(993\) 21.9377 0.696172
\(994\) 0 0
\(995\) −0.413463 −0.0131077
\(996\) 0 0
\(997\) 40.1054 1.27015 0.635076 0.772450i \(-0.280969\pi\)
0.635076 + 0.772450i \(0.280969\pi\)
\(998\) 0 0
\(999\) 0.786929 0.0248973
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 1960.2.a.y.1.1 yes 4
4.3 odd 2 3920.2.a.cd.1.4 4
5.4 even 2 9800.2.a.cl.1.4 4
7.2 even 3 1960.2.q.x.361.4 8
7.3 odd 6 1960.2.q.y.961.1 8
7.4 even 3 1960.2.q.x.961.4 8
7.5 odd 6 1960.2.q.y.361.1 8
7.6 odd 2 1960.2.a.x.1.4 4
28.27 even 2 3920.2.a.ce.1.1 4
35.34 odd 2 9800.2.a.cs.1.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1960.2.a.x.1.4 4 7.6 odd 2
1960.2.a.y.1.1 yes 4 1.1 even 1 trivial
1960.2.q.x.361.4 8 7.2 even 3
1960.2.q.x.961.4 8 7.4 even 3
1960.2.q.y.361.1 8 7.5 odd 6
1960.2.q.y.961.1 8 7.3 odd 6
3920.2.a.cd.1.4 4 4.3 odd 2
3920.2.a.ce.1.1 4 28.27 even 2
9800.2.a.cl.1.4 4 5.4 even 2
9800.2.a.cs.1.1 4 35.34 odd 2