Properties

Label 1960.2.a.v.1.1
Level $1960$
Weight $2$
Character 1960.1
Self dual yes
Analytic conductor $15.651$
Analytic rank $0$
Dimension $3$
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: \(3\)
Coefficient field: 3.3.1944.1
Defining polynomial: \(x^{3} - 9 x - 6\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 280)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-2.58423\) of defining polynomial
Character \(\chi\) \(=\) 1960.1

$q$-expansion

\(f(q)\) \(=\) \(q-2.58423 q^{3} -1.00000 q^{5} +3.67822 q^{9} +O(q^{10})\) \(q-2.58423 q^{3} -1.00000 q^{5} +3.67822 q^{9} +1.67822 q^{11} +4.84667 q^{13} +2.58423 q^{15} -2.00000 q^{17} -6.84667 q^{19} -2.26245 q^{23} +1.00000 q^{25} -1.75268 q^{27} +3.32178 q^{29} -9.16845 q^{31} -4.33690 q^{33} -2.84667 q^{37} -12.5249 q^{39} -9.52489 q^{41} +6.58423 q^{43} -3.67822 q^{45} +12.2031 q^{47} +5.16845 q^{51} +7.49023 q^{53} -1.67822 q^{55} +17.6933 q^{57} -8.00000 q^{59} +6.49023 q^{61} -4.84667 q^{65} -5.75268 q^{67} +5.84667 q^{69} +11.6933 q^{73} -2.58423 q^{75} +5.69334 q^{79} -6.50535 q^{81} +12.5842 q^{83} +2.00000 q^{85} -8.58423 q^{87} +5.84667 q^{89} +23.6933 q^{93} +6.84667 q^{95} +2.00000 q^{97} +6.17287 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3q - 3q^{5} + 9q^{9} + O(q^{10}) \) \( 3q - 3q^{5} + 9q^{9} + 3q^{11} - 3q^{13} - 6q^{17} - 3q^{19} + 3q^{23} + 3q^{25} + 18q^{27} + 12q^{29} - 12q^{31} + 18q^{33} + 9q^{37} - 18q^{39} - 9q^{41} + 12q^{43} - 9q^{45} + 15q^{47} + 9q^{53} - 3q^{55} + 18q^{57} - 24q^{59} + 6q^{61} + 3q^{65} + 6q^{67} - 18q^{79} + 27q^{81} + 30q^{83} + 6q^{85} - 18q^{87} + 36q^{93} + 3q^{95} + 6q^{97} + 63q^{99} + O(q^{100}) \)

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
\(3\) −2.58423 −1.49200 −0.746002 0.665944i \(-0.768029\pi\)
−0.746002 + 0.665944i \(0.768029\pi\)
\(4\) 0 0
\(5\) −1.00000 −0.447214
\(6\) 0 0
\(7\) 0 0
\(8\) 0 0
\(9\) 3.67822 1.22607
\(10\) 0 0
\(11\) 1.67822 0.506003 0.253001 0.967466i \(-0.418582\pi\)
0.253001 + 0.967466i \(0.418582\pi\)
\(12\) 0 0
\(13\) 4.84667 1.34422 0.672112 0.740449i \(-0.265387\pi\)
0.672112 + 0.740449i \(0.265387\pi\)
\(14\) 0 0
\(15\) 2.58423 0.667244
\(16\) 0 0
\(17\) −2.00000 −0.485071 −0.242536 0.970143i \(-0.577979\pi\)
−0.242536 + 0.970143i \(0.577979\pi\)
\(18\) 0 0
\(19\) −6.84667 −1.57073 −0.785367 0.619030i \(-0.787526\pi\)
−0.785367 + 0.619030i \(0.787526\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −2.26245 −0.471753 −0.235876 0.971783i \(-0.575796\pi\)
−0.235876 + 0.971783i \(0.575796\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) −1.75268 −0.337303
\(28\) 0 0
\(29\) 3.32178 0.616839 0.308419 0.951250i \(-0.400200\pi\)
0.308419 + 0.951250i \(0.400200\pi\)
\(30\) 0 0
\(31\) −9.16845 −1.64670 −0.823351 0.567532i \(-0.807898\pi\)
−0.823351 + 0.567532i \(0.807898\pi\)
\(32\) 0 0
\(33\) −4.33690 −0.754958
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −2.84667 −0.467990 −0.233995 0.972238i \(-0.575180\pi\)
−0.233995 + 0.972238i \(0.575180\pi\)
\(38\) 0 0
\(39\) −12.5249 −2.00559
\(40\) 0 0
\(41\) −9.52489 −1.48754 −0.743769 0.668437i \(-0.766964\pi\)
−0.743769 + 0.668437i \(0.766964\pi\)
\(42\) 0 0
\(43\) 6.58423 1.00408 0.502042 0.864843i \(-0.332582\pi\)
0.502042 + 0.864843i \(0.332582\pi\)
\(44\) 0 0
\(45\) −3.67822 −0.548317
\(46\) 0 0
\(47\) 12.2031 1.78001 0.890004 0.455954i \(-0.150702\pi\)
0.890004 + 0.455954i \(0.150702\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 0 0
\(51\) 5.16845 0.723728
\(52\) 0 0
\(53\) 7.49023 1.02886 0.514431 0.857532i \(-0.328003\pi\)
0.514431 + 0.857532i \(0.328003\pi\)
\(54\) 0 0
\(55\) −1.67822 −0.226291
\(56\) 0 0
\(57\) 17.6933 2.34354
\(58\) 0 0
\(59\) −8.00000 −1.04151 −0.520756 0.853706i \(-0.674350\pi\)
−0.520756 + 0.853706i \(0.674350\pi\)
\(60\) 0 0
\(61\) 6.49023 0.830989 0.415494 0.909596i \(-0.363609\pi\)
0.415494 + 0.909596i \(0.363609\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −4.84667 −0.601156
\(66\) 0 0
\(67\) −5.75268 −0.702801 −0.351401 0.936225i \(-0.614294\pi\)
−0.351401 + 0.936225i \(0.614294\pi\)
\(68\) 0 0
\(69\) 5.84667 0.703857
\(70\) 0 0
\(71\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(72\) 0 0
\(73\) 11.6933 1.36860 0.684301 0.729199i \(-0.260107\pi\)
0.684301 + 0.729199i \(0.260107\pi\)
\(74\) 0 0
\(75\) −2.58423 −0.298401
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 5.69334 0.640551 0.320276 0.947324i \(-0.396224\pi\)
0.320276 + 0.947324i \(0.396224\pi\)
\(80\) 0 0
\(81\) −6.50535 −0.722817
\(82\) 0 0
\(83\) 12.5842 1.38130 0.690649 0.723190i \(-0.257325\pi\)
0.690649 + 0.723190i \(0.257325\pi\)
\(84\) 0 0
\(85\) 2.00000 0.216930
\(86\) 0 0
\(87\) −8.58423 −0.920326
\(88\) 0 0
\(89\) 5.84667 0.619746 0.309873 0.950778i \(-0.399713\pi\)
0.309873 + 0.950778i \(0.399713\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 23.6933 2.45689
\(94\) 0 0
\(95\) 6.84667 0.702454
\(96\) 0 0
\(97\) 2.00000 0.203069 0.101535 0.994832i \(-0.467625\pi\)
0.101535 + 0.994832i \(0.467625\pi\)
\(98\) 0 0
\(99\) 6.17287 0.620397
\(100\) 0 0
\(101\) 17.0151 1.69307 0.846534 0.532335i \(-0.178685\pi\)
0.846534 + 0.532335i \(0.178685\pi\)
\(102\) 0 0
\(103\) −7.10912 −0.700482 −0.350241 0.936660i \(-0.613900\pi\)
−0.350241 + 0.936660i \(0.613900\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −1.75268 −0.169438 −0.0847188 0.996405i \(-0.526999\pi\)
−0.0847188 + 0.996405i \(0.526999\pi\)
\(108\) 0 0
\(109\) 19.5400 1.87159 0.935797 0.352539i \(-0.114682\pi\)
0.935797 + 0.352539i \(0.114682\pi\)
\(110\) 0 0
\(111\) 7.35644 0.698243
\(112\) 0 0
\(113\) 10.3369 0.972414 0.486207 0.873844i \(-0.338380\pi\)
0.486207 + 0.873844i \(0.338380\pi\)
\(114\) 0 0
\(115\) 2.26245 0.210974
\(116\) 0 0
\(117\) 17.8271 1.64812
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) −8.18357 −0.743961
\(122\) 0 0
\(123\) 24.6145 2.21941
\(124\) 0 0
\(125\) −1.00000 −0.0894427
\(126\) 0 0
\(127\) −19.1836 −1.70227 −0.851133 0.524949i \(-0.824084\pi\)
−0.851133 + 0.524949i \(0.824084\pi\)
\(128\) 0 0
\(129\) −17.0151 −1.49810
\(130\) 0 0
\(131\) 17.8271 1.55756 0.778782 0.627295i \(-0.215838\pi\)
0.778782 + 0.627295i \(0.215838\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) 1.75268 0.150846
\(136\) 0 0
\(137\) 4.00000 0.341743 0.170872 0.985293i \(-0.445342\pi\)
0.170872 + 0.985293i \(0.445342\pi\)
\(138\) 0 0
\(139\) 5.16845 0.438382 0.219191 0.975682i \(-0.429658\pi\)
0.219191 + 0.975682i \(0.429658\pi\)
\(140\) 0 0
\(141\) −31.5356 −2.65578
\(142\) 0 0
\(143\) 8.13379 0.680181
\(144\) 0 0
\(145\) −3.32178 −0.275859
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 0.796886 0.0652834 0.0326417 0.999467i \(-0.489608\pi\)
0.0326417 + 0.999467i \(0.489608\pi\)
\(150\) 0 0
\(151\) −16.5249 −1.34478 −0.672388 0.740199i \(-0.734731\pi\)
−0.672388 + 0.740199i \(0.734731\pi\)
\(152\) 0 0
\(153\) −7.35644 −0.594733
\(154\) 0 0
\(155\) 9.16845 0.736428
\(156\) 0 0
\(157\) −8.20311 −0.654680 −0.327340 0.944907i \(-0.606152\pi\)
−0.327340 + 0.944907i \(0.606152\pi\)
\(158\) 0 0
\(159\) −19.3564 −1.53507
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) 3.69334 0.289285 0.144643 0.989484i \(-0.453797\pi\)
0.144643 + 0.989484i \(0.453797\pi\)
\(164\) 0 0
\(165\) 4.33690 0.337627
\(166\) 0 0
\(167\) 0.262447 0.0203087 0.0101544 0.999948i \(-0.496768\pi\)
0.0101544 + 0.999948i \(0.496768\pi\)
\(168\) 0 0
\(169\) 10.4902 0.806941
\(170\) 0 0
\(171\) −25.1836 −1.92584
\(172\) 0 0
\(173\) 18.8467 1.43289 0.716443 0.697646i \(-0.245769\pi\)
0.716443 + 0.697646i \(0.245769\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 20.6738 1.55394
\(178\) 0 0
\(179\) −12.0151 −0.898052 −0.449026 0.893519i \(-0.648229\pi\)
−0.449026 + 0.893519i \(0.648229\pi\)
\(180\) 0 0
\(181\) 6.03466 0.448553 0.224276 0.974526i \(-0.427998\pi\)
0.224276 + 0.974526i \(0.427998\pi\)
\(182\) 0 0
\(183\) −16.7722 −1.23984
\(184\) 0 0
\(185\) 2.84667 0.209291
\(186\) 0 0
\(187\) −3.35644 −0.245447
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 2.83155 0.204884 0.102442 0.994739i \(-0.467334\pi\)
0.102442 + 0.994739i \(0.467334\pi\)
\(192\) 0 0
\(193\) 12.9805 0.934354 0.467177 0.884164i \(-0.345271\pi\)
0.467177 + 0.884164i \(0.345271\pi\)
\(194\) 0 0
\(195\) 12.5249 0.896926
\(196\) 0 0
\(197\) −4.84667 −0.345311 −0.172656 0.984982i \(-0.555235\pi\)
−0.172656 + 0.984982i \(0.555235\pi\)
\(198\) 0 0
\(199\) 15.3867 1.09073 0.545367 0.838198i \(-0.316390\pi\)
0.545367 + 0.838198i \(0.316390\pi\)
\(200\) 0 0
\(201\) 14.8662 1.04858
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 9.52489 0.665247
\(206\) 0 0
\(207\) −8.32178 −0.578404
\(208\) 0 0
\(209\) −11.4902 −0.794796
\(210\) 0 0
\(211\) 9.18357 0.632223 0.316112 0.948722i \(-0.397623\pi\)
0.316112 + 0.948722i \(0.397623\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) −6.58423 −0.449040
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) −30.2182 −2.04196
\(220\) 0 0
\(221\) −9.69334 −0.652045
\(222\) 0 0
\(223\) 12.9805 0.869236 0.434618 0.900615i \(-0.356883\pi\)
0.434618 + 0.900615i \(0.356883\pi\)
\(224\) 0 0
\(225\) 3.67822 0.245215
\(226\) 0 0
\(227\) 22.0302 1.46220 0.731099 0.682271i \(-0.239008\pi\)
0.731099 + 0.682271i \(0.239008\pi\)
\(228\) 0 0
\(229\) −4.30666 −0.284592 −0.142296 0.989824i \(-0.545448\pi\)
−0.142296 + 0.989824i \(0.545448\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 18.0000 1.17922 0.589610 0.807688i \(-0.299282\pi\)
0.589610 + 0.807688i \(0.299282\pi\)
\(234\) 0 0
\(235\) −12.2031 −0.796043
\(236\) 0 0
\(237\) −14.7129 −0.955705
\(238\) 0 0
\(239\) 10.8618 0.702591 0.351296 0.936265i \(-0.385741\pi\)
0.351296 + 0.936265i \(0.385741\pi\)
\(240\) 0 0
\(241\) −0.203114 −0.0130837 −0.00654187 0.999979i \(-0.502082\pi\)
−0.00654187 + 0.999979i \(0.502082\pi\)
\(242\) 0 0
\(243\) 22.0693 1.41575
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) −33.1836 −2.11142
\(248\) 0 0
\(249\) −32.5205 −2.06090
\(250\) 0 0
\(251\) −5.03466 −0.317785 −0.158893 0.987296i \(-0.550792\pi\)
−0.158893 + 0.987296i \(0.550792\pi\)
\(252\) 0 0
\(253\) −3.79689 −0.238708
\(254\) 0 0
\(255\) −5.16845 −0.323661
\(256\) 0 0
\(257\) −23.3867 −1.45882 −0.729411 0.684076i \(-0.760206\pi\)
−0.729411 + 0.684076i \(0.760206\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) 12.2182 0.756290
\(262\) 0 0
\(263\) 27.1091 1.67162 0.835810 0.549019i \(-0.184999\pi\)
0.835810 + 0.549019i \(0.184999\pi\)
\(264\) 0 0
\(265\) −7.49023 −0.460121
\(266\) 0 0
\(267\) −15.1091 −0.924663
\(268\) 0 0
\(269\) −6.49023 −0.395716 −0.197858 0.980231i \(-0.563399\pi\)
−0.197858 + 0.980231i \(0.563399\pi\)
\(270\) 0 0
\(271\) 23.3867 1.42064 0.710320 0.703879i \(-0.248550\pi\)
0.710320 + 0.703879i \(0.248550\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 1.67822 0.101201
\(276\) 0 0
\(277\) 3.66310 0.220094 0.110047 0.993926i \(-0.464900\pi\)
0.110047 + 0.993926i \(0.464900\pi\)
\(278\) 0 0
\(279\) −33.7236 −2.01898
\(280\) 0 0
\(281\) 13.8965 0.828993 0.414497 0.910051i \(-0.363958\pi\)
0.414497 + 0.910051i \(0.363958\pi\)
\(282\) 0 0
\(283\) 20.3369 1.20890 0.604452 0.796642i \(-0.293392\pi\)
0.604452 + 0.796642i \(0.293392\pi\)
\(284\) 0 0
\(285\) −17.6933 −1.04806
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) −13.0000 −0.764706
\(290\) 0 0
\(291\) −5.16845 −0.302980
\(292\) 0 0
\(293\) −18.8467 −1.10103 −0.550517 0.834824i \(-0.685569\pi\)
−0.550517 + 0.834824i \(0.685569\pi\)
\(294\) 0 0
\(295\) 8.00000 0.465778
\(296\) 0 0
\(297\) −2.94138 −0.170676
\(298\) 0 0
\(299\) −10.9653 −0.634142
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) −43.9709 −2.52606
\(304\) 0 0
\(305\) −6.49023 −0.371629
\(306\) 0 0
\(307\) 2.39623 0.136760 0.0683802 0.997659i \(-0.478217\pi\)
0.0683802 + 0.997659i \(0.478217\pi\)
\(308\) 0 0
\(309\) 18.3716 1.04512
\(310\) 0 0
\(311\) 5.66310 0.321125 0.160562 0.987026i \(-0.448669\pi\)
0.160562 + 0.987026i \(0.448669\pi\)
\(312\) 0 0
\(313\) −29.7236 −1.68008 −0.840038 0.542527i \(-0.817468\pi\)
−0.840038 + 0.542527i \(0.817468\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 19.0498 1.06994 0.534971 0.844870i \(-0.320322\pi\)
0.534971 + 0.844870i \(0.320322\pi\)
\(318\) 0 0
\(319\) 5.57468 0.312122
\(320\) 0 0
\(321\) 4.52931 0.252801
\(322\) 0 0
\(323\) 13.6933 0.761918
\(324\) 0 0
\(325\) 4.84667 0.268845
\(326\) 0 0
\(327\) −50.4958 −2.79242
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) 2.32178 0.127617 0.0638083 0.997962i \(-0.479675\pi\)
0.0638083 + 0.997962i \(0.479675\pi\)
\(332\) 0 0
\(333\) −10.4707 −0.573790
\(334\) 0 0
\(335\) 5.75268 0.314302
\(336\) 0 0
\(337\) −16.4062 −0.893704 −0.446852 0.894608i \(-0.647455\pi\)
−0.446852 + 0.894608i \(0.647455\pi\)
\(338\) 0 0
\(339\) −26.7129 −1.45084
\(340\) 0 0
\(341\) −15.3867 −0.833236
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) −5.84667 −0.314774
\(346\) 0 0
\(347\) 4.58423 0.246094 0.123047 0.992401i \(-0.460733\pi\)
0.123047 + 0.992401i \(0.460733\pi\)
\(348\) 0 0
\(349\) −12.3716 −0.662235 −0.331117 0.943590i \(-0.607426\pi\)
−0.331117 + 0.943590i \(0.607426\pi\)
\(350\) 0 0
\(351\) −8.49465 −0.453411
\(352\) 0 0
\(353\) −19.3867 −1.03185 −0.515925 0.856634i \(-0.672552\pi\)
−0.515925 + 0.856634i \(0.672552\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −9.81201 −0.517858 −0.258929 0.965896i \(-0.583370\pi\)
−0.258929 + 0.965896i \(0.583370\pi\)
\(360\) 0 0
\(361\) 27.8769 1.46721
\(362\) 0 0
\(363\) 21.1482 1.10999
\(364\) 0 0
\(365\) −11.6933 −0.612058
\(366\) 0 0
\(367\) 12.7180 0.663875 0.331937 0.943301i \(-0.392298\pi\)
0.331937 + 0.943301i \(0.392298\pi\)
\(368\) 0 0
\(369\) −35.0347 −1.82383
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) −16.4062 −0.849482 −0.424741 0.905315i \(-0.639635\pi\)
−0.424741 + 0.905315i \(0.639635\pi\)
\(374\) 0 0
\(375\) 2.58423 0.133449
\(376\) 0 0
\(377\) 16.0996 0.829170
\(378\) 0 0
\(379\) 14.4707 0.743309 0.371655 0.928371i \(-0.378791\pi\)
0.371655 + 0.928371i \(0.378791\pi\)
\(380\) 0 0
\(381\) 49.5747 2.53979
\(382\) 0 0
\(383\) 3.54956 0.181374 0.0906871 0.995879i \(-0.471094\pi\)
0.0906871 + 0.995879i \(0.471094\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 24.2182 1.23108
\(388\) 0 0
\(389\) 8.98046 0.455327 0.227664 0.973740i \(-0.426891\pi\)
0.227664 + 0.973740i \(0.426891\pi\)
\(390\) 0 0
\(391\) 4.52489 0.228834
\(392\) 0 0
\(393\) −46.0693 −2.32389
\(394\) 0 0
\(395\) −5.69334 −0.286463
\(396\) 0 0
\(397\) 28.0996 1.41028 0.705139 0.709070i \(-0.250885\pi\)
0.705139 + 0.709070i \(0.250885\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −26.6933 −1.33300 −0.666501 0.745504i \(-0.732209\pi\)
−0.666501 + 0.745504i \(0.732209\pi\)
\(402\) 0 0
\(403\) −44.4365 −2.21354
\(404\) 0 0
\(405\) 6.50535 0.323254
\(406\) 0 0
\(407\) −4.77735 −0.236804
\(408\) 0 0
\(409\) −28.0649 −1.38772 −0.693860 0.720110i \(-0.744091\pi\)
−0.693860 + 0.720110i \(0.744091\pi\)
\(410\) 0 0
\(411\) −10.3369 −0.509882
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) −12.5842 −0.617735
\(416\) 0 0
\(417\) −13.3564 −0.654068
\(418\) 0 0
\(419\) 18.8769 0.922198 0.461099 0.887349i \(-0.347455\pi\)
0.461099 + 0.887349i \(0.347455\pi\)
\(420\) 0 0
\(421\) −28.1836 −1.37358 −0.686792 0.726854i \(-0.740982\pi\)
−0.686792 + 0.726854i \(0.740982\pi\)
\(422\) 0 0
\(423\) 44.8858 2.18242
\(424\) 0 0
\(425\) −2.00000 −0.0970143
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) −21.0195 −1.01483
\(430\) 0 0
\(431\) 2.83155 0.136391 0.0681955 0.997672i \(-0.478276\pi\)
0.0681955 + 0.997672i \(0.478276\pi\)
\(432\) 0 0
\(433\) 2.33690 0.112304 0.0561522 0.998422i \(-0.482117\pi\)
0.0561522 + 0.998422i \(0.482117\pi\)
\(434\) 0 0
\(435\) 8.58423 0.411582
\(436\) 0 0
\(437\) 15.4902 0.740998
\(438\) 0 0
\(439\) 6.06933 0.289673 0.144837 0.989456i \(-0.453734\pi\)
0.144837 + 0.989456i \(0.453734\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 11.6038 0.551312 0.275656 0.961256i \(-0.411105\pi\)
0.275656 + 0.961256i \(0.411105\pi\)
\(444\) 0 0
\(445\) −5.84667 −0.277159
\(446\) 0 0
\(447\) −2.05933 −0.0974031
\(448\) 0 0
\(449\) 4.13821 0.195294 0.0976470 0.995221i \(-0.468868\pi\)
0.0976470 + 0.995221i \(0.468868\pi\)
\(450\) 0 0
\(451\) −15.9849 −0.752698
\(452\) 0 0
\(453\) 42.7040 2.00641
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −12.6436 −0.591441 −0.295720 0.955275i \(-0.595560\pi\)
−0.295720 + 0.955275i \(0.595560\pi\)
\(458\) 0 0
\(459\) 3.50535 0.163616
\(460\) 0 0
\(461\) −20.7129 −0.964695 −0.482348 0.875980i \(-0.660216\pi\)
−0.482348 + 0.875980i \(0.660216\pi\)
\(462\) 0 0
\(463\) −16.3811 −0.761295 −0.380647 0.924720i \(-0.624299\pi\)
−0.380647 + 0.924720i \(0.624299\pi\)
\(464\) 0 0
\(465\) −23.6933 −1.09875
\(466\) 0 0
\(467\) 1.72243 0.0797046 0.0398523 0.999206i \(-0.487311\pi\)
0.0398523 + 0.999206i \(0.487311\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 21.1987 0.976784
\(472\) 0 0
\(473\) 11.0498 0.508070
\(474\) 0 0
\(475\) −6.84667 −0.314147
\(476\) 0 0
\(477\) 27.5507 1.26146
\(478\) 0 0
\(479\) 1.78176 0.0814108 0.0407054 0.999171i \(-0.487039\pi\)
0.0407054 + 0.999171i \(0.487039\pi\)
\(480\) 0 0
\(481\) −13.7969 −0.629084
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) −2.00000 −0.0908153
\(486\) 0 0
\(487\) −11.0195 −0.499343 −0.249672 0.968331i \(-0.580323\pi\)
−0.249672 + 0.968331i \(0.580323\pi\)
\(488\) 0 0
\(489\) −9.54443 −0.431614
\(490\) 0 0
\(491\) −20.9311 −0.944608 −0.472304 0.881436i \(-0.656578\pi\)
−0.472304 + 0.881436i \(0.656578\pi\)
\(492\) 0 0
\(493\) −6.64356 −0.299211
\(494\) 0 0
\(495\) −6.17287 −0.277450
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) −7.50535 −0.335986 −0.167993 0.985788i \(-0.553729\pi\)
−0.167993 + 0.985788i \(0.553729\pi\)
\(500\) 0 0
\(501\) −0.678221 −0.0303007
\(502\) 0 0
\(503\) −20.5842 −0.917805 −0.458903 0.888487i \(-0.651757\pi\)
−0.458903 + 0.888487i \(0.651757\pi\)
\(504\) 0 0
\(505\) −17.0151 −0.757163
\(506\) 0 0
\(507\) −27.1091 −1.20396
\(508\) 0 0
\(509\) −13.6587 −0.605410 −0.302705 0.953084i \(-0.597890\pi\)
−0.302705 + 0.953084i \(0.597890\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) 12.0000 0.529813
\(514\) 0 0
\(515\) 7.10912 0.313265
\(516\) 0 0
\(517\) 20.4795 0.900688
\(518\) 0 0
\(519\) −48.7040 −2.13787
\(520\) 0 0
\(521\) 42.1640 1.84724 0.923620 0.383310i \(-0.125216\pi\)
0.923620 + 0.383310i \(0.125216\pi\)
\(522\) 0 0
\(523\) 30.6436 1.33995 0.669975 0.742384i \(-0.266305\pi\)
0.669975 + 0.742384i \(0.266305\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 18.3369 0.798768
\(528\) 0 0
\(529\) −17.8813 −0.777449
\(530\) 0 0
\(531\) −29.4258 −1.27697
\(532\) 0 0
\(533\) −46.1640 −1.99959
\(534\) 0 0
\(535\) 1.75268 0.0757748
\(536\) 0 0
\(537\) 31.0498 1.33990
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) −33.6889 −1.44840 −0.724200 0.689590i \(-0.757791\pi\)
−0.724200 + 0.689590i \(0.757791\pi\)
\(542\) 0 0
\(543\) −15.5949 −0.669243
\(544\) 0 0
\(545\) −19.5400 −0.837002
\(546\) 0 0
\(547\) −3.03979 −0.129972 −0.0649861 0.997886i \(-0.520700\pi\)
−0.0649861 + 0.997886i \(0.520700\pi\)
\(548\) 0 0
\(549\) 23.8725 1.01885
\(550\) 0 0
\(551\) −22.7431 −0.968890
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) −7.35644 −0.312264
\(556\) 0 0
\(557\) 18.1640 0.769635 0.384817 0.922993i \(-0.374264\pi\)
0.384817 + 0.922993i \(0.374264\pi\)
\(558\) 0 0
\(559\) 31.9116 1.34972
\(560\) 0 0
\(561\) 8.67380 0.366208
\(562\) 0 0
\(563\) 36.9905 1.55896 0.779481 0.626426i \(-0.215483\pi\)
0.779481 + 0.626426i \(0.215483\pi\)
\(564\) 0 0
\(565\) −10.3369 −0.434877
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 32.6093 1.36705 0.683527 0.729925i \(-0.260445\pi\)
0.683527 + 0.729925i \(0.260445\pi\)
\(570\) 0 0
\(571\) 40.4365 1.69221 0.846107 0.533013i \(-0.178940\pi\)
0.846107 + 0.533013i \(0.178940\pi\)
\(572\) 0 0
\(573\) −7.31736 −0.305687
\(574\) 0 0
\(575\) −2.26245 −0.0943505
\(576\) 0 0
\(577\) 38.0302 1.58322 0.791610 0.611027i \(-0.209243\pi\)
0.791610 + 0.611027i \(0.209243\pi\)
\(578\) 0 0
\(579\) −33.5444 −1.39406
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 12.5703 0.520607
\(584\) 0 0
\(585\) −17.8271 −0.737061
\(586\) 0 0
\(587\) −34.0302 −1.40458 −0.702289 0.711892i \(-0.747839\pi\)
−0.702289 + 0.711892i \(0.747839\pi\)
\(588\) 0 0
\(589\) 62.7734 2.58653
\(590\) 0 0
\(591\) 12.5249 0.515205
\(592\) 0 0
\(593\) 33.7236 1.38486 0.692431 0.721484i \(-0.256540\pi\)
0.692431 + 0.721484i \(0.256540\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) −39.7627 −1.62738
\(598\) 0 0
\(599\) 6.30666 0.257683 0.128841 0.991665i \(-0.458874\pi\)
0.128841 + 0.991665i \(0.458874\pi\)
\(600\) 0 0
\(601\) 11.2871 0.460411 0.230206 0.973142i \(-0.426060\pi\)
0.230206 + 0.973142i \(0.426060\pi\)
\(602\) 0 0
\(603\) −21.1596 −0.861686
\(604\) 0 0
\(605\) 8.18357 0.332710
\(606\) 0 0
\(607\) 15.4611 0.627548 0.313774 0.949498i \(-0.398406\pi\)
0.313774 + 0.949498i \(0.398406\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 59.1445 2.39273
\(612\) 0 0
\(613\) −22.1640 −0.895197 −0.447598 0.894235i \(-0.647721\pi\)
−0.447598 + 0.894235i \(0.647721\pi\)
\(614\) 0 0
\(615\) −24.6145 −0.992551
\(616\) 0 0
\(617\) −14.9805 −0.603091 −0.301545 0.953452i \(-0.597502\pi\)
−0.301545 + 0.953452i \(0.597502\pi\)
\(618\) 0 0
\(619\) −23.6391 −0.950137 −0.475069 0.879949i \(-0.657577\pi\)
−0.475069 + 0.879949i \(0.657577\pi\)
\(620\) 0 0
\(621\) 3.96534 0.159123
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) 29.6933 1.18584
\(628\) 0 0
\(629\) 5.69334 0.227008
\(630\) 0 0
\(631\) −13.7818 −0.548643 −0.274322 0.961638i \(-0.588453\pi\)
−0.274322 + 0.961638i \(0.588453\pi\)
\(632\) 0 0
\(633\) −23.7324 −0.943279
\(634\) 0 0
\(635\) 19.1836 0.761277
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 43.8618 1.73244 0.866218 0.499666i \(-0.166544\pi\)
0.866218 + 0.499666i \(0.166544\pi\)
\(642\) 0 0
\(643\) −7.04979 −0.278016 −0.139008 0.990291i \(-0.544391\pi\)
−0.139008 + 0.990291i \(0.544391\pi\)
\(644\) 0 0
\(645\) 17.0151 0.669970
\(646\) 0 0
\(647\) 24.4807 0.962435 0.481217 0.876601i \(-0.340195\pi\)
0.481217 + 0.876601i \(0.340195\pi\)
\(648\) 0 0
\(649\) −13.4258 −0.527008
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 0.816426 0.0319492 0.0159746 0.999872i \(-0.494915\pi\)
0.0159746 + 0.999872i \(0.494915\pi\)
\(654\) 0 0
\(655\) −17.8271 −0.696564
\(656\) 0 0
\(657\) 43.0107 1.67801
\(658\) 0 0
\(659\) 21.2871 0.829228 0.414614 0.909997i \(-0.363917\pi\)
0.414614 + 0.909997i \(0.363917\pi\)
\(660\) 0 0
\(661\) −25.9462 −1.00919 −0.504596 0.863356i \(-0.668359\pi\)
−0.504596 + 0.863356i \(0.668359\pi\)
\(662\) 0 0
\(663\) 25.0498 0.972853
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) −7.51535 −0.290995
\(668\) 0 0
\(669\) −33.5444 −1.29690
\(670\) 0 0
\(671\) 10.8920 0.420483
\(672\) 0 0
\(673\) 22.0693 0.850710 0.425355 0.905027i \(-0.360149\pi\)
0.425355 + 0.905027i \(0.360149\pi\)
\(674\) 0 0
\(675\) −1.75268 −0.0674605
\(676\) 0 0
\(677\) −5.49023 −0.211007 −0.105503 0.994419i \(-0.533645\pi\)
−0.105503 + 0.994419i \(0.533645\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) −56.9311 −2.18161
\(682\) 0 0
\(683\) 19.2278 0.735731 0.367865 0.929879i \(-0.380089\pi\)
0.367865 + 0.929879i \(0.380089\pi\)
\(684\) 0 0
\(685\) −4.00000 −0.152832
\(686\) 0 0
\(687\) 11.1294 0.424612
\(688\) 0 0
\(689\) 36.3027 1.38302
\(690\) 0 0
\(691\) −14.2182 −0.540887 −0.270444 0.962736i \(-0.587170\pi\)
−0.270444 + 0.962736i \(0.587170\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −5.16845 −0.196051
\(696\) 0 0
\(697\) 19.0498 0.721562
\(698\) 0 0
\(699\) −46.5161 −1.75940
\(700\) 0 0
\(701\) 14.1533 0.534564 0.267282 0.963618i \(-0.413875\pi\)
0.267282 + 0.963618i \(0.413875\pi\)
\(702\) 0 0
\(703\) 19.4902 0.735088
\(704\) 0 0
\(705\) 31.5356 1.18770
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) 17.0454 0.640152 0.320076 0.947392i \(-0.396292\pi\)
0.320076 + 0.947392i \(0.396292\pi\)
\(710\) 0 0
\(711\) 20.9414 0.785363
\(712\) 0 0
\(713\) 20.7431 0.776836
\(714\) 0 0
\(715\) −8.13379 −0.304186
\(716\) 0 0
\(717\) −28.0693 −1.04827
\(718\) 0 0
\(719\) 11.5054 0.429077 0.214539 0.976716i \(-0.431175\pi\)
0.214539 + 0.976716i \(0.431175\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) 0.524893 0.0195210
\(724\) 0 0
\(725\) 3.32178 0.123368
\(726\) 0 0
\(727\) −16.4114 −0.608664 −0.304332 0.952566i \(-0.598433\pi\)
−0.304332 + 0.952566i \(0.598433\pi\)
\(728\) 0 0
\(729\) −37.5161 −1.38948
\(730\) 0 0
\(731\) −13.1685 −0.487053
\(732\) 0 0
\(733\) 9.11425 0.336642 0.168321 0.985732i \(-0.446165\pi\)
0.168321 + 0.985732i \(0.446165\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −9.65426 −0.355619
\(738\) 0 0
\(739\) −34.9956 −1.28733 −0.643667 0.765306i \(-0.722588\pi\)
−0.643667 + 0.765306i \(0.722588\pi\)
\(740\) 0 0
\(741\) 85.7538 3.15025
\(742\) 0 0
\(743\) −43.5305 −1.59698 −0.798489 0.602009i \(-0.794367\pi\)
−0.798489 + 0.602009i \(0.794367\pi\)
\(744\) 0 0
\(745\) −0.796886 −0.0291956
\(746\) 0 0
\(747\) 46.2876 1.69357
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) 6.30666 0.230133 0.115067 0.993358i \(-0.463292\pi\)
0.115067 + 0.993358i \(0.463292\pi\)
\(752\) 0 0
\(753\) 13.0107 0.474136
\(754\) 0 0
\(755\) 16.5249 0.601402
\(756\) 0 0
\(757\) 24.3369 0.884540 0.442270 0.896882i \(-0.354173\pi\)
0.442270 + 0.896882i \(0.354173\pi\)
\(758\) 0 0
\(759\) 9.81201 0.356153
\(760\) 0 0
\(761\) 28.2031 1.02236 0.511181 0.859473i \(-0.329208\pi\)
0.511181 + 0.859473i \(0.329208\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) 7.35644 0.265973
\(766\) 0 0
\(767\) −38.7734 −1.40003
\(768\) 0 0
\(769\) 41.8965 1.51082 0.755412 0.655250i \(-0.227437\pi\)
0.755412 + 0.655250i \(0.227437\pi\)
\(770\) 0 0
\(771\) 60.4365 2.17657
\(772\) 0 0
\(773\) −39.8574 −1.43357 −0.716785 0.697294i \(-0.754387\pi\)
−0.716785 + 0.697294i \(0.754387\pi\)
\(774\) 0 0
\(775\) −9.16845 −0.329340
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 65.2138 2.33653
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) −5.82200 −0.208061
\(784\) 0 0
\(785\) 8.20311 0.292782
\(786\) 0 0
\(787\) −35.3274 −1.25928 −0.629642 0.776885i \(-0.716798\pi\)
−0.629642 + 0.776885i \(0.716798\pi\)
\(788\) 0 0
\(789\) −70.0561 −2.49406
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 31.4560 1.11704
\(794\) 0 0
\(795\) 19.3564 0.686502
\(796\) 0 0
\(797\) −29.6933 −1.05179 −0.525896 0.850549i \(-0.676270\pi\)
−0.525896 + 0.850549i \(0.676270\pi\)
\(798\) 0 0
\(799\) −24.4062 −0.863430
\(800\) 0 0
\(801\) 21.5054 0.759854
\(802\) 0 0
\(803\) 19.6240 0.692517
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 16.7722 0.590410
\(808\) 0 0
\(809\) −19.6436 −0.690631 −0.345315 0.938487i \(-0.612228\pi\)
−0.345315 + 0.938487i \(0.612228\pi\)
\(810\) 0 0
\(811\) −0.252452 −0.00886479 −0.00443239 0.999990i \(-0.501411\pi\)
−0.00443239 + 0.999990i \(0.501411\pi\)
\(812\) 0 0
\(813\) −60.4365 −2.11960
\(814\) 0 0
\(815\) −3.69334 −0.129372
\(816\) 0 0
\(817\) −45.0800 −1.57715
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 44.7734 1.56260 0.781301 0.624155i \(-0.214556\pi\)
0.781301 + 0.624155i \(0.214556\pi\)
\(822\) 0 0
\(823\) 3.29711 0.114930 0.0574650 0.998348i \(-0.481698\pi\)
0.0574650 + 0.998348i \(0.481698\pi\)
\(824\) 0 0
\(825\) −4.33690 −0.150992
\(826\) 0 0
\(827\) 9.41577 0.327419 0.163709 0.986509i \(-0.447654\pi\)
0.163709 + 0.986509i \(0.447654\pi\)
\(828\) 0 0
\(829\) 4.71288 0.163685 0.0818426 0.996645i \(-0.473920\pi\)
0.0818426 + 0.996645i \(0.473920\pi\)
\(830\) 0 0
\(831\) −9.46627 −0.328381
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) −0.262447 −0.00908234
\(836\) 0 0
\(837\) 16.0693 0.555437
\(838\) 0 0
\(839\) −22.1880 −0.766015 −0.383007 0.923745i \(-0.625112\pi\)
−0.383007 + 0.923745i \(0.625112\pi\)
\(840\) 0 0
\(841\) −17.9658 −0.619510
\(842\) 0 0
\(843\) −35.9116 −1.23686
\(844\) 0 0
\(845\) −10.4902 −0.360875
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) −52.5551 −1.80369
\(850\) 0 0
\(851\) 6.44044 0.220776
\(852\) 0 0
\(853\) −39.9267 −1.36706 −0.683532 0.729920i \(-0.739557\pi\)
−0.683532 + 0.729920i \(0.739557\pi\)
\(854\) 0 0
\(855\) 25.1836 0.861260
\(856\) 0 0
\(857\) 1.62402 0.0554754 0.0277377 0.999615i \(-0.491170\pi\)
0.0277377 + 0.999615i \(0.491170\pi\)
\(858\) 0 0
\(859\) −8.00000 −0.272956 −0.136478 0.990643i \(-0.543578\pi\)
−0.136478 + 0.990643i \(0.543578\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 18.2234 0.620331 0.310165 0.950683i \(-0.399616\pi\)
0.310165 + 0.950683i \(0.399616\pi\)
\(864\) 0 0
\(865\) −18.8467 −0.640806
\(866\) 0 0
\(867\) 33.5949 1.14094
\(868\) 0 0
\(869\) 9.55469 0.324121
\(870\) 0 0
\(871\) −27.8813 −0.944723
\(872\) 0 0
\(873\) 7.35644 0.248978
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) −50.8769 −1.71799 −0.858996 0.511983i \(-0.828911\pi\)
−0.858996 + 0.511983i \(0.828911\pi\)
\(878\) 0 0
\(879\) 48.7040 1.64275
\(880\) 0 0
\(881\) 19.1187 0.644124 0.322062 0.946719i \(-0.395624\pi\)
0.322062 + 0.946719i \(0.395624\pi\)
\(882\) 0 0
\(883\) 39.3174 1.32313 0.661567 0.749886i \(-0.269892\pi\)
0.661567 + 0.749886i \(0.269892\pi\)
\(884\) 0 0
\(885\) −20.6738 −0.694942
\(886\) 0 0
\(887\) −27.0398 −0.907907 −0.453954 0.891025i \(-0.649987\pi\)
−0.453954 + 0.891025i \(0.649987\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) −10.9174 −0.365747
\(892\) 0 0
\(893\) −83.5507 −2.79592
\(894\) 0 0
\(895\) 12.0151 0.401621
\(896\) 0 0
\(897\) 28.3369 0.946142
\(898\) 0 0
\(899\) −30.4556 −1.01575
\(900\) 0 0
\(901\) −14.9805 −0.499071
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) −6.03466 −0.200599
\(906\) 0 0
\(907\) 32.8025 1.08919 0.544594 0.838700i \(-0.316684\pi\)
0.544594 + 0.838700i \(0.316684\pi\)
\(908\) 0 0
\(909\) 62.5854 2.07583
\(910\) 0 0
\(911\) 6.18799 0.205017 0.102509 0.994732i \(-0.467313\pi\)
0.102509 + 0.994732i \(0.467313\pi\)
\(912\) 0 0
\(913\) 21.1191 0.698941
\(914\) 0 0
\(915\) 16.7722 0.554472
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) −47.7929 −1.57654 −0.788271 0.615328i \(-0.789024\pi\)
−0.788271 + 0.615328i \(0.789024\pi\)
\(920\) 0 0
\(921\) −6.19241 −0.204047
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) −2.84667 −0.0935980
\(926\) 0 0
\(927\) −26.1489 −0.858843
\(928\) 0 0
\(929\) 39.8618 1.30782 0.653912 0.756571i \(-0.273127\pi\)
0.653912 + 0.756571i \(0.273127\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 0 0
\(933\) −14.6347 −0.479119
\(934\) 0 0
\(935\) 3.35644 0.109767
\(936\) 0 0
\(937\) −0.406229 −0.0132709 −0.00663546 0.999978i \(-0.502112\pi\)
−0.00663546 + 0.999978i \(0.502112\pi\)
\(938\) 0 0
\(939\) 76.8125 2.50668
\(940\) 0 0
\(941\) −33.6543 −1.09710 −0.548549 0.836119i \(-0.684820\pi\)
−0.548549 + 0.836119i \(0.684820\pi\)
\(942\) 0 0
\(943\) 21.5496 0.701750
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 5.84110 0.189810 0.0949050 0.995486i \(-0.469745\pi\)
0.0949050 + 0.995486i \(0.469745\pi\)
\(948\) 0 0
\(949\) 56.6738 1.83971
\(950\) 0 0
\(951\) −49.2289 −1.59636
\(952\) 0 0
\(953\) 32.0605 1.03854 0.519271 0.854610i \(-0.326204\pi\)
0.519271 + 0.854610i \(0.326204\pi\)
\(954\) 0 0
\(955\) −2.83155 −0.0916268
\(956\) 0 0
\(957\) −14.4062 −0.465687
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) 53.0605 1.71163
\(962\) 0 0
\(963\) −6.44673 −0.207743
\(964\) 0 0
\(965\) −12.9805 −0.417856
\(966\) 0 0
\(967\) 24.0896 0.774669 0.387334 0.921939i \(-0.373396\pi\)
0.387334 + 0.921939i \(0.373396\pi\)
\(968\) 0 0
\(969\) −35.3867 −1.13678
\(970\) 0 0
\(971\) −18.4707 −0.592753 −0.296376 0.955071i \(-0.595778\pi\)
−0.296376 + 0.955071i \(0.595778\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) −12.5249 −0.401118
\(976\) 0 0
\(977\) 24.3369 0.778606 0.389303 0.921110i \(-0.372716\pi\)
0.389303 + 0.921110i \(0.372716\pi\)
\(978\) 0 0
\(979\) 9.81201 0.313593
\(980\) 0 0
\(981\) 71.8725 2.29471
\(982\) 0 0
\(983\) −8.66868 −0.276488 −0.138244 0.990398i \(-0.544146\pi\)
−0.138244 + 0.990398i \(0.544146\pi\)
\(984\) 0 0
\(985\) 4.84667 0.154428
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) −14.8965 −0.473680
\(990\) 0 0
\(991\) −55.5054 −1.76319 −0.881593 0.472011i \(-0.843528\pi\)
−0.881593 + 0.472011i \(0.843528\pi\)
\(992\) 0 0
\(993\) −6.00000 −0.190404
\(994\) 0 0
\(995\) −15.3867 −0.487791
\(996\) 0 0
\(997\) 35.1191 1.11223 0.556117 0.831104i \(-0.312291\pi\)
0.556117 + 0.831104i \(0.312291\pi\)
\(998\) 0 0
\(999\) 4.98929 0.157854
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.v.1.1 3
4.3 odd 2 3920.2.a.cb.1.3 3
5.4 even 2 9800.2.a.cf.1.3 3
7.2 even 3 1960.2.q.w.361.3 6
7.3 odd 6 280.2.q.e.121.1 yes 6
7.4 even 3 1960.2.q.w.961.3 6
7.5 odd 6 280.2.q.e.81.1 6
7.6 odd 2 1960.2.a.w.1.3 3
21.5 even 6 2520.2.bi.q.361.1 6
21.17 even 6 2520.2.bi.q.1801.1 6
28.3 even 6 560.2.q.l.401.3 6
28.19 even 6 560.2.q.l.81.3 6
28.27 even 2 3920.2.a.cc.1.1 3
35.3 even 12 1400.2.bh.i.849.5 12
35.12 even 12 1400.2.bh.i.249.5 12
35.17 even 12 1400.2.bh.i.849.2 12
35.19 odd 6 1400.2.q.j.1201.3 6
35.24 odd 6 1400.2.q.j.401.3 6
35.33 even 12 1400.2.bh.i.249.2 12
35.34 odd 2 9800.2.a.ce.1.1 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
280.2.q.e.81.1 6 7.5 odd 6
280.2.q.e.121.1 yes 6 7.3 odd 6
560.2.q.l.81.3 6 28.19 even 6
560.2.q.l.401.3 6 28.3 even 6
1400.2.q.j.401.3 6 35.24 odd 6
1400.2.q.j.1201.3 6 35.19 odd 6
1400.2.bh.i.249.2 12 35.33 even 12
1400.2.bh.i.249.5 12 35.12 even 12
1400.2.bh.i.849.2 12 35.17 even 12
1400.2.bh.i.849.5 12 35.3 even 12
1960.2.a.v.1.1 3 1.1 even 1 trivial
1960.2.a.w.1.3 3 7.6 odd 2
1960.2.q.w.361.3 6 7.2 even 3
1960.2.q.w.961.3 6 7.4 even 3
2520.2.bi.q.361.1 6 21.5 even 6
2520.2.bi.q.1801.1 6 21.17 even 6
3920.2.a.cb.1.3 3 4.3 odd 2
3920.2.a.cc.1.1 3 28.27 even 2
9800.2.a.ce.1.1 3 35.34 odd 2
9800.2.a.cf.1.3 3 5.4 even 2