Properties

Label 4025.2.a.a.1.1
Level $4025$
Weight $2$
Character 4025.1
Self dual yes
Analytic conductor $32.140$
Analytic rank $2$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 4025 = 5^{2} \cdot 7 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4025.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(32.1397868136\)
Analytic rank: \(2\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 805)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 4025.1

$q$-expansion

\(f(q)\) \(=\) \(q-2.00000 q^{2} -3.00000 q^{3} +2.00000 q^{4} +6.00000 q^{6} +1.00000 q^{7} +6.00000 q^{9} +O(q^{10})\) \(q-2.00000 q^{2} -3.00000 q^{3} +2.00000 q^{4} +6.00000 q^{6} +1.00000 q^{7} +6.00000 q^{9} -1.00000 q^{11} -6.00000 q^{12} -7.00000 q^{13} -2.00000 q^{14} -4.00000 q^{16} -3.00000 q^{17} -12.0000 q^{18} -8.00000 q^{19} -3.00000 q^{21} +2.00000 q^{22} -1.00000 q^{23} +14.0000 q^{26} -9.00000 q^{27} +2.00000 q^{28} -5.00000 q^{29} -2.00000 q^{31} +8.00000 q^{32} +3.00000 q^{33} +6.00000 q^{34} +12.0000 q^{36} +4.00000 q^{37} +16.0000 q^{38} +21.0000 q^{39} -8.00000 q^{41} +6.00000 q^{42} -6.00000 q^{43} -2.00000 q^{44} +2.00000 q^{46} -3.00000 q^{47} +12.0000 q^{48} +1.00000 q^{49} +9.00000 q^{51} -14.0000 q^{52} -2.00000 q^{53} +18.0000 q^{54} +24.0000 q^{57} +10.0000 q^{58} +2.00000 q^{59} -14.0000 q^{61} +4.00000 q^{62} +6.00000 q^{63} -8.00000 q^{64} -6.00000 q^{66} +4.00000 q^{67} -6.00000 q^{68} +3.00000 q^{69} +8.00000 q^{71} +6.00000 q^{73} -8.00000 q^{74} -16.0000 q^{76} -1.00000 q^{77} -42.0000 q^{78} -3.00000 q^{79} +9.00000 q^{81} +16.0000 q^{82} -12.0000 q^{83} -6.00000 q^{84} +12.0000 q^{86} +15.0000 q^{87} -2.00000 q^{89} -7.00000 q^{91} -2.00000 q^{92} +6.00000 q^{93} +6.00000 q^{94} -24.0000 q^{96} -7.00000 q^{97} -2.00000 q^{98} -6.00000 q^{99} +O(q^{100})\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.00000 −1.41421 −0.707107 0.707107i \(-0.750000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(3\) −3.00000 −1.73205 −0.866025 0.500000i \(-0.833333\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(4\) 2.00000 1.00000
\(5\) 0 0
\(6\) 6.00000 2.44949
\(7\) 1.00000 0.377964
\(8\) 0 0
\(9\) 6.00000 2.00000
\(10\) 0 0
\(11\) −1.00000 −0.301511 −0.150756 0.988571i \(-0.548171\pi\)
−0.150756 + 0.988571i \(0.548171\pi\)
\(12\) −6.00000 −1.73205
\(13\) −7.00000 −1.94145 −0.970725 0.240192i \(-0.922790\pi\)
−0.970725 + 0.240192i \(0.922790\pi\)
\(14\) −2.00000 −0.534522
\(15\) 0 0
\(16\) −4.00000 −1.00000
\(17\) −3.00000 −0.727607 −0.363803 0.931476i \(-0.618522\pi\)
−0.363803 + 0.931476i \(0.618522\pi\)
\(18\) −12.0000 −2.82843
\(19\) −8.00000 −1.83533 −0.917663 0.397360i \(-0.869927\pi\)
−0.917663 + 0.397360i \(0.869927\pi\)
\(20\) 0 0
\(21\) −3.00000 −0.654654
\(22\) 2.00000 0.426401
\(23\) −1.00000 −0.208514
\(24\) 0 0
\(25\) 0 0
\(26\) 14.0000 2.74563
\(27\) −9.00000 −1.73205
\(28\) 2.00000 0.377964
\(29\) −5.00000 −0.928477 −0.464238 0.885710i \(-0.653672\pi\)
−0.464238 + 0.885710i \(0.653672\pi\)
\(30\) 0 0
\(31\) −2.00000 −0.359211 −0.179605 0.983739i \(-0.557482\pi\)
−0.179605 + 0.983739i \(0.557482\pi\)
\(32\) 8.00000 1.41421
\(33\) 3.00000 0.522233
\(34\) 6.00000 1.02899
\(35\) 0 0
\(36\) 12.0000 2.00000
\(37\) 4.00000 0.657596 0.328798 0.944400i \(-0.393356\pi\)
0.328798 + 0.944400i \(0.393356\pi\)
\(38\) 16.0000 2.59554
\(39\) 21.0000 3.36269
\(40\) 0 0
\(41\) −8.00000 −1.24939 −0.624695 0.780869i \(-0.714777\pi\)
−0.624695 + 0.780869i \(0.714777\pi\)
\(42\) 6.00000 0.925820
\(43\) −6.00000 −0.914991 −0.457496 0.889212i \(-0.651253\pi\)
−0.457496 + 0.889212i \(0.651253\pi\)
\(44\) −2.00000 −0.301511
\(45\) 0 0
\(46\) 2.00000 0.294884
\(47\) −3.00000 −0.437595 −0.218797 0.975770i \(-0.570213\pi\)
−0.218797 + 0.975770i \(0.570213\pi\)
\(48\) 12.0000 1.73205
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) 9.00000 1.26025
\(52\) −14.0000 −1.94145
\(53\) −2.00000 −0.274721 −0.137361 0.990521i \(-0.543862\pi\)
−0.137361 + 0.990521i \(0.543862\pi\)
\(54\) 18.0000 2.44949
\(55\) 0 0
\(56\) 0 0
\(57\) 24.0000 3.17888
\(58\) 10.0000 1.31306
\(59\) 2.00000 0.260378 0.130189 0.991489i \(-0.458442\pi\)
0.130189 + 0.991489i \(0.458442\pi\)
\(60\) 0 0
\(61\) −14.0000 −1.79252 −0.896258 0.443533i \(-0.853725\pi\)
−0.896258 + 0.443533i \(0.853725\pi\)
\(62\) 4.00000 0.508001
\(63\) 6.00000 0.755929
\(64\) −8.00000 −1.00000
\(65\) 0 0
\(66\) −6.00000 −0.738549
\(67\) 4.00000 0.488678 0.244339 0.969690i \(-0.421429\pi\)
0.244339 + 0.969690i \(0.421429\pi\)
\(68\) −6.00000 −0.727607
\(69\) 3.00000 0.361158
\(70\) 0 0
\(71\) 8.00000 0.949425 0.474713 0.880141i \(-0.342552\pi\)
0.474713 + 0.880141i \(0.342552\pi\)
\(72\) 0 0
\(73\) 6.00000 0.702247 0.351123 0.936329i \(-0.385800\pi\)
0.351123 + 0.936329i \(0.385800\pi\)
\(74\) −8.00000 −0.929981
\(75\) 0 0
\(76\) −16.0000 −1.83533
\(77\) −1.00000 −0.113961
\(78\) −42.0000 −4.75556
\(79\) −3.00000 −0.337526 −0.168763 0.985657i \(-0.553977\pi\)
−0.168763 + 0.985657i \(0.553977\pi\)
\(80\) 0 0
\(81\) 9.00000 1.00000
\(82\) 16.0000 1.76690
\(83\) −12.0000 −1.31717 −0.658586 0.752506i \(-0.728845\pi\)
−0.658586 + 0.752506i \(0.728845\pi\)
\(84\) −6.00000 −0.654654
\(85\) 0 0
\(86\) 12.0000 1.29399
\(87\) 15.0000 1.60817
\(88\) 0 0
\(89\) −2.00000 −0.212000 −0.106000 0.994366i \(-0.533804\pi\)
−0.106000 + 0.994366i \(0.533804\pi\)
\(90\) 0 0
\(91\) −7.00000 −0.733799
\(92\) −2.00000 −0.208514
\(93\) 6.00000 0.622171
\(94\) 6.00000 0.618853
\(95\) 0 0
\(96\) −24.0000 −2.44949
\(97\) −7.00000 −0.710742 −0.355371 0.934725i \(-0.615646\pi\)
−0.355371 + 0.934725i \(0.615646\pi\)
\(98\) −2.00000 −0.202031
\(99\) −6.00000 −0.603023
\(100\) 0 0
\(101\) −14.0000 −1.39305 −0.696526 0.717532i \(-0.745272\pi\)
−0.696526 + 0.717532i \(0.745272\pi\)
\(102\) −18.0000 −1.78227
\(103\) −1.00000 −0.0985329 −0.0492665 0.998786i \(-0.515688\pi\)
−0.0492665 + 0.998786i \(0.515688\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 4.00000 0.388514
\(107\) −18.0000 −1.74013 −0.870063 0.492941i \(-0.835922\pi\)
−0.870063 + 0.492941i \(0.835922\pi\)
\(108\) −18.0000 −1.73205
\(109\) 11.0000 1.05361 0.526804 0.849987i \(-0.323390\pi\)
0.526804 + 0.849987i \(0.323390\pi\)
\(110\) 0 0
\(111\) −12.0000 −1.13899
\(112\) −4.00000 −0.377964
\(113\) −18.0000 −1.69330 −0.846649 0.532152i \(-0.821383\pi\)
−0.846649 + 0.532152i \(0.821383\pi\)
\(114\) −48.0000 −4.49561
\(115\) 0 0
\(116\) −10.0000 −0.928477
\(117\) −42.0000 −3.88290
\(118\) −4.00000 −0.368230
\(119\) −3.00000 −0.275010
\(120\) 0 0
\(121\) −10.0000 −0.909091
\(122\) 28.0000 2.53500
\(123\) 24.0000 2.16401
\(124\) −4.00000 −0.359211
\(125\) 0 0
\(126\) −12.0000 −1.06904
\(127\) 4.00000 0.354943 0.177471 0.984126i \(-0.443208\pi\)
0.177471 + 0.984126i \(0.443208\pi\)
\(128\) 0 0
\(129\) 18.0000 1.58481
\(130\) 0 0
\(131\) 12.0000 1.04844 0.524222 0.851581i \(-0.324356\pi\)
0.524222 + 0.851581i \(0.324356\pi\)
\(132\) 6.00000 0.522233
\(133\) −8.00000 −0.693688
\(134\) −8.00000 −0.691095
\(135\) 0 0
\(136\) 0 0
\(137\) 6.00000 0.512615 0.256307 0.966595i \(-0.417494\pi\)
0.256307 + 0.966595i \(0.417494\pi\)
\(138\) −6.00000 −0.510754
\(139\) 8.00000 0.678551 0.339276 0.940687i \(-0.389818\pi\)
0.339276 + 0.940687i \(0.389818\pi\)
\(140\) 0 0
\(141\) 9.00000 0.757937
\(142\) −16.0000 −1.34269
\(143\) 7.00000 0.585369
\(144\) −24.0000 −2.00000
\(145\) 0 0
\(146\) −12.0000 −0.993127
\(147\) −3.00000 −0.247436
\(148\) 8.00000 0.657596
\(149\) −2.00000 −0.163846 −0.0819232 0.996639i \(-0.526106\pi\)
−0.0819232 + 0.996639i \(0.526106\pi\)
\(150\) 0 0
\(151\) −9.00000 −0.732410 −0.366205 0.930534i \(-0.619343\pi\)
−0.366205 + 0.930534i \(0.619343\pi\)
\(152\) 0 0
\(153\) −18.0000 −1.45521
\(154\) 2.00000 0.161165
\(155\) 0 0
\(156\) 42.0000 3.36269
\(157\) −2.00000 −0.159617 −0.0798087 0.996810i \(-0.525431\pi\)
−0.0798087 + 0.996810i \(0.525431\pi\)
\(158\) 6.00000 0.477334
\(159\) 6.00000 0.475831
\(160\) 0 0
\(161\) −1.00000 −0.0788110
\(162\) −18.0000 −1.41421
\(163\) −24.0000 −1.87983 −0.939913 0.341415i \(-0.889094\pi\)
−0.939913 + 0.341415i \(0.889094\pi\)
\(164\) −16.0000 −1.24939
\(165\) 0 0
\(166\) 24.0000 1.86276
\(167\) −3.00000 −0.232147 −0.116073 0.993241i \(-0.537031\pi\)
−0.116073 + 0.993241i \(0.537031\pi\)
\(168\) 0 0
\(169\) 36.0000 2.76923
\(170\) 0 0
\(171\) −48.0000 −3.67065
\(172\) −12.0000 −0.914991
\(173\) 15.0000 1.14043 0.570214 0.821496i \(-0.306860\pi\)
0.570214 + 0.821496i \(0.306860\pi\)
\(174\) −30.0000 −2.27429
\(175\) 0 0
\(176\) 4.00000 0.301511
\(177\) −6.00000 −0.450988
\(178\) 4.00000 0.299813
\(179\) 24.0000 1.79384 0.896922 0.442189i \(-0.145798\pi\)
0.896922 + 0.442189i \(0.145798\pi\)
\(180\) 0 0
\(181\) −14.0000 −1.04061 −0.520306 0.853980i \(-0.674182\pi\)
−0.520306 + 0.853980i \(0.674182\pi\)
\(182\) 14.0000 1.03775
\(183\) 42.0000 3.10473
\(184\) 0 0
\(185\) 0 0
\(186\) −12.0000 −0.879883
\(187\) 3.00000 0.219382
\(188\) −6.00000 −0.437595
\(189\) −9.00000 −0.654654
\(190\) 0 0
\(191\) −9.00000 −0.651217 −0.325609 0.945505i \(-0.605569\pi\)
−0.325609 + 0.945505i \(0.605569\pi\)
\(192\) 24.0000 1.73205
\(193\) 2.00000 0.143963 0.0719816 0.997406i \(-0.477068\pi\)
0.0719816 + 0.997406i \(0.477068\pi\)
\(194\) 14.0000 1.00514
\(195\) 0 0
\(196\) 2.00000 0.142857
\(197\) −28.0000 −1.99492 −0.997459 0.0712470i \(-0.977302\pi\)
−0.997459 + 0.0712470i \(0.977302\pi\)
\(198\) 12.0000 0.852803
\(199\) 26.0000 1.84309 0.921546 0.388270i \(-0.126927\pi\)
0.921546 + 0.388270i \(0.126927\pi\)
\(200\) 0 0
\(201\) −12.0000 −0.846415
\(202\) 28.0000 1.97007
\(203\) −5.00000 −0.350931
\(204\) 18.0000 1.26025
\(205\) 0 0
\(206\) 2.00000 0.139347
\(207\) −6.00000 −0.417029
\(208\) 28.0000 1.94145
\(209\) 8.00000 0.553372
\(210\) 0 0
\(211\) 15.0000 1.03264 0.516321 0.856395i \(-0.327301\pi\)
0.516321 + 0.856395i \(0.327301\pi\)
\(212\) −4.00000 −0.274721
\(213\) −24.0000 −1.64445
\(214\) 36.0000 2.46091
\(215\) 0 0
\(216\) 0 0
\(217\) −2.00000 −0.135769
\(218\) −22.0000 −1.49003
\(219\) −18.0000 −1.21633
\(220\) 0 0
\(221\) 21.0000 1.41261
\(222\) 24.0000 1.61077
\(223\) 13.0000 0.870544 0.435272 0.900299i \(-0.356652\pi\)
0.435272 + 0.900299i \(0.356652\pi\)
\(224\) 8.00000 0.534522
\(225\) 0 0
\(226\) 36.0000 2.39468
\(227\) −13.0000 −0.862840 −0.431420 0.902151i \(-0.641987\pi\)
−0.431420 + 0.902151i \(0.641987\pi\)
\(228\) 48.0000 3.17888
\(229\) −16.0000 −1.05731 −0.528655 0.848837i \(-0.677303\pi\)
−0.528655 + 0.848837i \(0.677303\pi\)
\(230\) 0 0
\(231\) 3.00000 0.197386
\(232\) 0 0
\(233\) −10.0000 −0.655122 −0.327561 0.944830i \(-0.606227\pi\)
−0.327561 + 0.944830i \(0.606227\pi\)
\(234\) 84.0000 5.49125
\(235\) 0 0
\(236\) 4.00000 0.260378
\(237\) 9.00000 0.584613
\(238\) 6.00000 0.388922
\(239\) −17.0000 −1.09964 −0.549819 0.835284i \(-0.685303\pi\)
−0.549819 + 0.835284i \(0.685303\pi\)
\(240\) 0 0
\(241\) −6.00000 −0.386494 −0.193247 0.981150i \(-0.561902\pi\)
−0.193247 + 0.981150i \(0.561902\pi\)
\(242\) 20.0000 1.28565
\(243\) 0 0
\(244\) −28.0000 −1.79252
\(245\) 0 0
\(246\) −48.0000 −3.06037
\(247\) 56.0000 3.56319
\(248\) 0 0
\(249\) 36.0000 2.28141
\(250\) 0 0
\(251\) −20.0000 −1.26239 −0.631194 0.775625i \(-0.717435\pi\)
−0.631194 + 0.775625i \(0.717435\pi\)
\(252\) 12.0000 0.755929
\(253\) 1.00000 0.0628695
\(254\) −8.00000 −0.501965
\(255\) 0 0
\(256\) 16.0000 1.00000
\(257\) 14.0000 0.873296 0.436648 0.899632i \(-0.356166\pi\)
0.436648 + 0.899632i \(0.356166\pi\)
\(258\) −36.0000 −2.24126
\(259\) 4.00000 0.248548
\(260\) 0 0
\(261\) −30.0000 −1.85695
\(262\) −24.0000 −1.48272
\(263\) −28.0000 −1.72655 −0.863277 0.504730i \(-0.831592\pi\)
−0.863277 + 0.504730i \(0.831592\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 16.0000 0.981023
\(267\) 6.00000 0.367194
\(268\) 8.00000 0.488678
\(269\) −30.0000 −1.82913 −0.914566 0.404436i \(-0.867468\pi\)
−0.914566 + 0.404436i \(0.867468\pi\)
\(270\) 0 0
\(271\) −2.00000 −0.121491 −0.0607457 0.998153i \(-0.519348\pi\)
−0.0607457 + 0.998153i \(0.519348\pi\)
\(272\) 12.0000 0.727607
\(273\) 21.0000 1.27098
\(274\) −12.0000 −0.724947
\(275\) 0 0
\(276\) 6.00000 0.361158
\(277\) 24.0000 1.44202 0.721010 0.692925i \(-0.243678\pi\)
0.721010 + 0.692925i \(0.243678\pi\)
\(278\) −16.0000 −0.959616
\(279\) −12.0000 −0.718421
\(280\) 0 0
\(281\) −15.0000 −0.894825 −0.447412 0.894328i \(-0.647654\pi\)
−0.447412 + 0.894328i \(0.647654\pi\)
\(282\) −18.0000 −1.07188
\(283\) 1.00000 0.0594438 0.0297219 0.999558i \(-0.490538\pi\)
0.0297219 + 0.999558i \(0.490538\pi\)
\(284\) 16.0000 0.949425
\(285\) 0 0
\(286\) −14.0000 −0.827837
\(287\) −8.00000 −0.472225
\(288\) 48.0000 2.82843
\(289\) −8.00000 −0.470588
\(290\) 0 0
\(291\) 21.0000 1.23104
\(292\) 12.0000 0.702247
\(293\) 9.00000 0.525786 0.262893 0.964825i \(-0.415323\pi\)
0.262893 + 0.964825i \(0.415323\pi\)
\(294\) 6.00000 0.349927
\(295\) 0 0
\(296\) 0 0
\(297\) 9.00000 0.522233
\(298\) 4.00000 0.231714
\(299\) 7.00000 0.404820
\(300\) 0 0
\(301\) −6.00000 −0.345834
\(302\) 18.0000 1.03578
\(303\) 42.0000 2.41284
\(304\) 32.0000 1.83533
\(305\) 0 0
\(306\) 36.0000 2.05798
\(307\) 11.0000 0.627803 0.313902 0.949456i \(-0.398364\pi\)
0.313902 + 0.949456i \(0.398364\pi\)
\(308\) −2.00000 −0.113961
\(309\) 3.00000 0.170664
\(310\) 0 0
\(311\) 4.00000 0.226819 0.113410 0.993548i \(-0.463823\pi\)
0.113410 + 0.993548i \(0.463823\pi\)
\(312\) 0 0
\(313\) −9.00000 −0.508710 −0.254355 0.967111i \(-0.581863\pi\)
−0.254355 + 0.967111i \(0.581863\pi\)
\(314\) 4.00000 0.225733
\(315\) 0 0
\(316\) −6.00000 −0.337526
\(317\) 12.0000 0.673987 0.336994 0.941507i \(-0.390590\pi\)
0.336994 + 0.941507i \(0.390590\pi\)
\(318\) −12.0000 −0.672927
\(319\) 5.00000 0.279946
\(320\) 0 0
\(321\) 54.0000 3.01399
\(322\) 2.00000 0.111456
\(323\) 24.0000 1.33540
\(324\) 18.0000 1.00000
\(325\) 0 0
\(326\) 48.0000 2.65847
\(327\) −33.0000 −1.82490
\(328\) 0 0
\(329\) −3.00000 −0.165395
\(330\) 0 0
\(331\) 8.00000 0.439720 0.219860 0.975531i \(-0.429440\pi\)
0.219860 + 0.975531i \(0.429440\pi\)
\(332\) −24.0000 −1.31717
\(333\) 24.0000 1.31519
\(334\) 6.00000 0.328305
\(335\) 0 0
\(336\) 12.0000 0.654654
\(337\) 14.0000 0.762629 0.381314 0.924445i \(-0.375472\pi\)
0.381314 + 0.924445i \(0.375472\pi\)
\(338\) −72.0000 −3.91628
\(339\) 54.0000 2.93288
\(340\) 0 0
\(341\) 2.00000 0.108306
\(342\) 96.0000 5.19109
\(343\) 1.00000 0.0539949
\(344\) 0 0
\(345\) 0 0
\(346\) −30.0000 −1.61281
\(347\) −12.0000 −0.644194 −0.322097 0.946707i \(-0.604388\pi\)
−0.322097 + 0.946707i \(0.604388\pi\)
\(348\) 30.0000 1.60817
\(349\) −28.0000 −1.49881 −0.749403 0.662114i \(-0.769659\pi\)
−0.749403 + 0.662114i \(0.769659\pi\)
\(350\) 0 0
\(351\) 63.0000 3.36269
\(352\) −8.00000 −0.426401
\(353\) −25.0000 −1.33062 −0.665308 0.746569i \(-0.731700\pi\)
−0.665308 + 0.746569i \(0.731700\pi\)
\(354\) 12.0000 0.637793
\(355\) 0 0
\(356\) −4.00000 −0.212000
\(357\) 9.00000 0.476331
\(358\) −48.0000 −2.53688
\(359\) −24.0000 −1.26667 −0.633336 0.773877i \(-0.718315\pi\)
−0.633336 + 0.773877i \(0.718315\pi\)
\(360\) 0 0
\(361\) 45.0000 2.36842
\(362\) 28.0000 1.47165
\(363\) 30.0000 1.57459
\(364\) −14.0000 −0.733799
\(365\) 0 0
\(366\) −84.0000 −4.39075
\(367\) 7.00000 0.365397 0.182699 0.983169i \(-0.441517\pi\)
0.182699 + 0.983169i \(0.441517\pi\)
\(368\) 4.00000 0.208514
\(369\) −48.0000 −2.49878
\(370\) 0 0
\(371\) −2.00000 −0.103835
\(372\) 12.0000 0.622171
\(373\) −26.0000 −1.34623 −0.673114 0.739538i \(-0.735044\pi\)
−0.673114 + 0.739538i \(0.735044\pi\)
\(374\) −6.00000 −0.310253
\(375\) 0 0
\(376\) 0 0
\(377\) 35.0000 1.80259
\(378\) 18.0000 0.925820
\(379\) −12.0000 −0.616399 −0.308199 0.951322i \(-0.599726\pi\)
−0.308199 + 0.951322i \(0.599726\pi\)
\(380\) 0 0
\(381\) −12.0000 −0.614779
\(382\) 18.0000 0.920960
\(383\) 16.0000 0.817562 0.408781 0.912633i \(-0.365954\pi\)
0.408781 + 0.912633i \(0.365954\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) −4.00000 −0.203595
\(387\) −36.0000 −1.82998
\(388\) −14.0000 −0.710742
\(389\) 19.0000 0.963338 0.481669 0.876353i \(-0.340031\pi\)
0.481669 + 0.876353i \(0.340031\pi\)
\(390\) 0 0
\(391\) 3.00000 0.151717
\(392\) 0 0
\(393\) −36.0000 −1.81596
\(394\) 56.0000 2.82124
\(395\) 0 0
\(396\) −12.0000 −0.603023
\(397\) −25.0000 −1.25471 −0.627357 0.778732i \(-0.715863\pi\)
−0.627357 + 0.778732i \(0.715863\pi\)
\(398\) −52.0000 −2.60652
\(399\) 24.0000 1.20150
\(400\) 0 0
\(401\) −25.0000 −1.24844 −0.624220 0.781248i \(-0.714583\pi\)
−0.624220 + 0.781248i \(0.714583\pi\)
\(402\) 24.0000 1.19701
\(403\) 14.0000 0.697390
\(404\) −28.0000 −1.39305
\(405\) 0 0
\(406\) 10.0000 0.496292
\(407\) −4.00000 −0.198273
\(408\) 0 0
\(409\) −14.0000 −0.692255 −0.346128 0.938187i \(-0.612504\pi\)
−0.346128 + 0.938187i \(0.612504\pi\)
\(410\) 0 0
\(411\) −18.0000 −0.887875
\(412\) −2.00000 −0.0985329
\(413\) 2.00000 0.0984136
\(414\) 12.0000 0.589768
\(415\) 0 0
\(416\) −56.0000 −2.74563
\(417\) −24.0000 −1.17529
\(418\) −16.0000 −0.782586
\(419\) 30.0000 1.46560 0.732798 0.680446i \(-0.238214\pi\)
0.732798 + 0.680446i \(0.238214\pi\)
\(420\) 0 0
\(421\) 7.00000 0.341159 0.170580 0.985344i \(-0.445436\pi\)
0.170580 + 0.985344i \(0.445436\pi\)
\(422\) −30.0000 −1.46038
\(423\) −18.0000 −0.875190
\(424\) 0 0
\(425\) 0 0
\(426\) 48.0000 2.32561
\(427\) −14.0000 −0.677507
\(428\) −36.0000 −1.74013
\(429\) −21.0000 −1.01389
\(430\) 0 0
\(431\) 19.0000 0.915198 0.457599 0.889159i \(-0.348710\pi\)
0.457599 + 0.889159i \(0.348710\pi\)
\(432\) 36.0000 1.73205
\(433\) −2.00000 −0.0961139 −0.0480569 0.998845i \(-0.515303\pi\)
−0.0480569 + 0.998845i \(0.515303\pi\)
\(434\) 4.00000 0.192006
\(435\) 0 0
\(436\) 22.0000 1.05361
\(437\) 8.00000 0.382692
\(438\) 36.0000 1.72015
\(439\) 26.0000 1.24091 0.620456 0.784241i \(-0.286947\pi\)
0.620456 + 0.784241i \(0.286947\pi\)
\(440\) 0 0
\(441\) 6.00000 0.285714
\(442\) −42.0000 −1.99774
\(443\) −10.0000 −0.475114 −0.237557 0.971374i \(-0.576347\pi\)
−0.237557 + 0.971374i \(0.576347\pi\)
\(444\) −24.0000 −1.13899
\(445\) 0 0
\(446\) −26.0000 −1.23114
\(447\) 6.00000 0.283790
\(448\) −8.00000 −0.377964
\(449\) −9.00000 −0.424736 −0.212368 0.977190i \(-0.568118\pi\)
−0.212368 + 0.977190i \(0.568118\pi\)
\(450\) 0 0
\(451\) 8.00000 0.376705
\(452\) −36.0000 −1.69330
\(453\) 27.0000 1.26857
\(454\) 26.0000 1.22024
\(455\) 0 0
\(456\) 0 0
\(457\) 8.00000 0.374224 0.187112 0.982339i \(-0.440087\pi\)
0.187112 + 0.982339i \(0.440087\pi\)
\(458\) 32.0000 1.49526
\(459\) 27.0000 1.26025
\(460\) 0 0
\(461\) 16.0000 0.745194 0.372597 0.927993i \(-0.378467\pi\)
0.372597 + 0.927993i \(0.378467\pi\)
\(462\) −6.00000 −0.279145
\(463\) 34.0000 1.58011 0.790057 0.613033i \(-0.210051\pi\)
0.790057 + 0.613033i \(0.210051\pi\)
\(464\) 20.0000 0.928477
\(465\) 0 0
\(466\) 20.0000 0.926482
\(467\) 21.0000 0.971764 0.485882 0.874024i \(-0.338498\pi\)
0.485882 + 0.874024i \(0.338498\pi\)
\(468\) −84.0000 −3.88290
\(469\) 4.00000 0.184703
\(470\) 0 0
\(471\) 6.00000 0.276465
\(472\) 0 0
\(473\) 6.00000 0.275880
\(474\) −18.0000 −0.826767
\(475\) 0 0
\(476\) −6.00000 −0.275010
\(477\) −12.0000 −0.549442
\(478\) 34.0000 1.55512
\(479\) 6.00000 0.274147 0.137073 0.990561i \(-0.456230\pi\)
0.137073 + 0.990561i \(0.456230\pi\)
\(480\) 0 0
\(481\) −28.0000 −1.27669
\(482\) 12.0000 0.546585
\(483\) 3.00000 0.136505
\(484\) −20.0000 −0.909091
\(485\) 0 0
\(486\) 0 0
\(487\) −2.00000 −0.0906287 −0.0453143 0.998973i \(-0.514429\pi\)
−0.0453143 + 0.998973i \(0.514429\pi\)
\(488\) 0 0
\(489\) 72.0000 3.25595
\(490\) 0 0
\(491\) −9.00000 −0.406164 −0.203082 0.979162i \(-0.565096\pi\)
−0.203082 + 0.979162i \(0.565096\pi\)
\(492\) 48.0000 2.16401
\(493\) 15.0000 0.675566
\(494\) −112.000 −5.03912
\(495\) 0 0
\(496\) 8.00000 0.359211
\(497\) 8.00000 0.358849
\(498\) −72.0000 −3.22640
\(499\) 25.0000 1.11915 0.559577 0.828778i \(-0.310964\pi\)
0.559577 + 0.828778i \(0.310964\pi\)
\(500\) 0 0
\(501\) 9.00000 0.402090
\(502\) 40.0000 1.78529
\(503\) −39.0000 −1.73892 −0.869462 0.494000i \(-0.835534\pi\)
−0.869462 + 0.494000i \(0.835534\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) −2.00000 −0.0889108
\(507\) −108.000 −4.79645
\(508\) 8.00000 0.354943
\(509\) −12.0000 −0.531891 −0.265945 0.963988i \(-0.585684\pi\)
−0.265945 + 0.963988i \(0.585684\pi\)
\(510\) 0 0
\(511\) 6.00000 0.265424
\(512\) −32.0000 −1.41421
\(513\) 72.0000 3.17888
\(514\) −28.0000 −1.23503
\(515\) 0 0
\(516\) 36.0000 1.58481
\(517\) 3.00000 0.131940
\(518\) −8.00000 −0.351500
\(519\) −45.0000 −1.97528
\(520\) 0 0
\(521\) 30.0000 1.31432 0.657162 0.753749i \(-0.271757\pi\)
0.657162 + 0.753749i \(0.271757\pi\)
\(522\) 60.0000 2.62613
\(523\) 16.0000 0.699631 0.349816 0.936819i \(-0.386244\pi\)
0.349816 + 0.936819i \(0.386244\pi\)
\(524\) 24.0000 1.04844
\(525\) 0 0
\(526\) 56.0000 2.44172
\(527\) 6.00000 0.261364
\(528\) −12.0000 −0.522233
\(529\) 1.00000 0.0434783
\(530\) 0 0
\(531\) 12.0000 0.520756
\(532\) −16.0000 −0.693688
\(533\) 56.0000 2.42563
\(534\) −12.0000 −0.519291
\(535\) 0 0
\(536\) 0 0
\(537\) −72.0000 −3.10703
\(538\) 60.0000 2.58678
\(539\) −1.00000 −0.0430730
\(540\) 0 0
\(541\) −17.0000 −0.730887 −0.365444 0.930834i \(-0.619083\pi\)
−0.365444 + 0.930834i \(0.619083\pi\)
\(542\) 4.00000 0.171815
\(543\) 42.0000 1.80239
\(544\) −24.0000 −1.02899
\(545\) 0 0
\(546\) −42.0000 −1.79743
\(547\) 30.0000 1.28271 0.641354 0.767245i \(-0.278373\pi\)
0.641354 + 0.767245i \(0.278373\pi\)
\(548\) 12.0000 0.512615
\(549\) −84.0000 −3.58503
\(550\) 0 0
\(551\) 40.0000 1.70406
\(552\) 0 0
\(553\) −3.00000 −0.127573
\(554\) −48.0000 −2.03932
\(555\) 0 0
\(556\) 16.0000 0.678551
\(557\) −2.00000 −0.0847427 −0.0423714 0.999102i \(-0.513491\pi\)
−0.0423714 + 0.999102i \(0.513491\pi\)
\(558\) 24.0000 1.01600
\(559\) 42.0000 1.77641
\(560\) 0 0
\(561\) −9.00000 −0.379980
\(562\) 30.0000 1.26547
\(563\) −4.00000 −0.168580 −0.0842900 0.996441i \(-0.526862\pi\)
−0.0842900 + 0.996441i \(0.526862\pi\)
\(564\) 18.0000 0.757937
\(565\) 0 0
\(566\) −2.00000 −0.0840663
\(567\) 9.00000 0.377964
\(568\) 0 0
\(569\) −18.0000 −0.754599 −0.377300 0.926091i \(-0.623147\pi\)
−0.377300 + 0.926091i \(0.623147\pi\)
\(570\) 0 0
\(571\) −16.0000 −0.669579 −0.334790 0.942293i \(-0.608665\pi\)
−0.334790 + 0.942293i \(0.608665\pi\)
\(572\) 14.0000 0.585369
\(573\) 27.0000 1.12794
\(574\) 16.0000 0.667827
\(575\) 0 0
\(576\) −48.0000 −2.00000
\(577\) 37.0000 1.54033 0.770165 0.637845i \(-0.220174\pi\)
0.770165 + 0.637845i \(0.220174\pi\)
\(578\) 16.0000 0.665512
\(579\) −6.00000 −0.249351
\(580\) 0 0
\(581\) −12.0000 −0.497844
\(582\) −42.0000 −1.74096
\(583\) 2.00000 0.0828315
\(584\) 0 0
\(585\) 0 0
\(586\) −18.0000 −0.743573
\(587\) −44.0000 −1.81607 −0.908037 0.418890i \(-0.862419\pi\)
−0.908037 + 0.418890i \(0.862419\pi\)
\(588\) −6.00000 −0.247436
\(589\) 16.0000 0.659269
\(590\) 0 0
\(591\) 84.0000 3.45530
\(592\) −16.0000 −0.657596
\(593\) 25.0000 1.02663 0.513313 0.858201i \(-0.328418\pi\)
0.513313 + 0.858201i \(0.328418\pi\)
\(594\) −18.0000 −0.738549
\(595\) 0 0
\(596\) −4.00000 −0.163846
\(597\) −78.0000 −3.19233
\(598\) −14.0000 −0.572503
\(599\) 21.0000 0.858037 0.429018 0.903296i \(-0.358860\pi\)
0.429018 + 0.903296i \(0.358860\pi\)
\(600\) 0 0
\(601\) 24.0000 0.978980 0.489490 0.872009i \(-0.337183\pi\)
0.489490 + 0.872009i \(0.337183\pi\)
\(602\) 12.0000 0.489083
\(603\) 24.0000 0.977356
\(604\) −18.0000 −0.732410
\(605\) 0 0
\(606\) −84.0000 −3.41227
\(607\) 43.0000 1.74532 0.872658 0.488332i \(-0.162394\pi\)
0.872658 + 0.488332i \(0.162394\pi\)
\(608\) −64.0000 −2.59554
\(609\) 15.0000 0.607831
\(610\) 0 0
\(611\) 21.0000 0.849569
\(612\) −36.0000 −1.45521
\(613\) −16.0000 −0.646234 −0.323117 0.946359i \(-0.604731\pi\)
−0.323117 + 0.946359i \(0.604731\pi\)
\(614\) −22.0000 −0.887848
\(615\) 0 0
\(616\) 0 0
\(617\) −4.00000 −0.161034 −0.0805170 0.996753i \(-0.525657\pi\)
−0.0805170 + 0.996753i \(0.525657\pi\)
\(618\) −6.00000 −0.241355
\(619\) −18.0000 −0.723481 −0.361741 0.932279i \(-0.617817\pi\)
−0.361741 + 0.932279i \(0.617817\pi\)
\(620\) 0 0
\(621\) 9.00000 0.361158
\(622\) −8.00000 −0.320771
\(623\) −2.00000 −0.0801283
\(624\) −84.0000 −3.36269
\(625\) 0 0
\(626\) 18.0000 0.719425
\(627\) −24.0000 −0.958468
\(628\) −4.00000 −0.159617
\(629\) −12.0000 −0.478471
\(630\) 0 0
\(631\) −17.0000 −0.676759 −0.338380 0.941010i \(-0.609879\pi\)
−0.338380 + 0.941010i \(0.609879\pi\)
\(632\) 0 0
\(633\) −45.0000 −1.78859
\(634\) −24.0000 −0.953162
\(635\) 0 0
\(636\) 12.0000 0.475831
\(637\) −7.00000 −0.277350
\(638\) −10.0000 −0.395904
\(639\) 48.0000 1.89885
\(640\) 0 0
\(641\) 26.0000 1.02694 0.513469 0.858108i \(-0.328360\pi\)
0.513469 + 0.858108i \(0.328360\pi\)
\(642\) −108.000 −4.26242
\(643\) 23.0000 0.907031 0.453516 0.891248i \(-0.350170\pi\)
0.453516 + 0.891248i \(0.350170\pi\)
\(644\) −2.00000 −0.0788110
\(645\) 0 0
\(646\) −48.0000 −1.88853
\(647\) 8.00000 0.314512 0.157256 0.987558i \(-0.449735\pi\)
0.157256 + 0.987558i \(0.449735\pi\)
\(648\) 0 0
\(649\) −2.00000 −0.0785069
\(650\) 0 0
\(651\) 6.00000 0.235159
\(652\) −48.0000 −1.87983
\(653\) −36.0000 −1.40879 −0.704394 0.709809i \(-0.748781\pi\)
−0.704394 + 0.709809i \(0.748781\pi\)
\(654\) 66.0000 2.58080
\(655\) 0 0
\(656\) 32.0000 1.24939
\(657\) 36.0000 1.40449
\(658\) 6.00000 0.233904
\(659\) −33.0000 −1.28550 −0.642749 0.766077i \(-0.722206\pi\)
−0.642749 + 0.766077i \(0.722206\pi\)
\(660\) 0 0
\(661\) 40.0000 1.55582 0.777910 0.628376i \(-0.216280\pi\)
0.777910 + 0.628376i \(0.216280\pi\)
\(662\) −16.0000 −0.621858
\(663\) −63.0000 −2.44672
\(664\) 0 0
\(665\) 0 0
\(666\) −48.0000 −1.85996
\(667\) 5.00000 0.193601
\(668\) −6.00000 −0.232147
\(669\) −39.0000 −1.50783
\(670\) 0 0
\(671\) 14.0000 0.540464
\(672\) −24.0000 −0.925820
\(673\) −28.0000 −1.07932 −0.539660 0.841883i \(-0.681447\pi\)
−0.539660 + 0.841883i \(0.681447\pi\)
\(674\) −28.0000 −1.07852
\(675\) 0 0
\(676\) 72.0000 2.76923
\(677\) −33.0000 −1.26829 −0.634147 0.773213i \(-0.718648\pi\)
−0.634147 + 0.773213i \(0.718648\pi\)
\(678\) −108.000 −4.14772
\(679\) −7.00000 −0.268635
\(680\) 0 0
\(681\) 39.0000 1.49448
\(682\) −4.00000 −0.153168
\(683\) 28.0000 1.07139 0.535695 0.844411i \(-0.320050\pi\)
0.535695 + 0.844411i \(0.320050\pi\)
\(684\) −96.0000 −3.67065
\(685\) 0 0
\(686\) −2.00000 −0.0763604
\(687\) 48.0000 1.83131
\(688\) 24.0000 0.914991
\(689\) 14.0000 0.533358
\(690\) 0 0
\(691\) 10.0000 0.380418 0.190209 0.981744i \(-0.439083\pi\)
0.190209 + 0.981744i \(0.439083\pi\)
\(692\) 30.0000 1.14043
\(693\) −6.00000 −0.227921
\(694\) 24.0000 0.911028
\(695\) 0 0
\(696\) 0 0
\(697\) 24.0000 0.909065
\(698\) 56.0000 2.11963
\(699\) 30.0000 1.13470
\(700\) 0 0
\(701\) 9.00000 0.339925 0.169963 0.985451i \(-0.445635\pi\)
0.169963 + 0.985451i \(0.445635\pi\)
\(702\) −126.000 −4.75556
\(703\) −32.0000 −1.20690
\(704\) 8.00000 0.301511
\(705\) 0 0
\(706\) 50.0000 1.88177
\(707\) −14.0000 −0.526524
\(708\) −12.0000 −0.450988
\(709\) −39.0000 −1.46468 −0.732338 0.680941i \(-0.761571\pi\)
−0.732338 + 0.680941i \(0.761571\pi\)
\(710\) 0 0
\(711\) −18.0000 −0.675053
\(712\) 0 0
\(713\) 2.00000 0.0749006
\(714\) −18.0000 −0.673633
\(715\) 0 0
\(716\) 48.0000 1.79384
\(717\) 51.0000 1.90463
\(718\) 48.0000 1.79134
\(719\) −22.0000 −0.820462 −0.410231 0.911982i \(-0.634552\pi\)
−0.410231 + 0.911982i \(0.634552\pi\)
\(720\) 0 0
\(721\) −1.00000 −0.0372419
\(722\) −90.0000 −3.34945
\(723\) 18.0000 0.669427
\(724\) −28.0000 −1.04061
\(725\) 0 0
\(726\) −60.0000 −2.22681
\(727\) 32.0000 1.18681 0.593407 0.804902i \(-0.297782\pi\)
0.593407 + 0.804902i \(0.297782\pi\)
\(728\) 0 0
\(729\) −27.0000 −1.00000
\(730\) 0 0
\(731\) 18.0000 0.665754
\(732\) 84.0000 3.10473
\(733\) 43.0000 1.58824 0.794121 0.607760i \(-0.207932\pi\)
0.794121 + 0.607760i \(0.207932\pi\)
\(734\) −14.0000 −0.516749
\(735\) 0 0
\(736\) −8.00000 −0.294884
\(737\) −4.00000 −0.147342
\(738\) 96.0000 3.53381
\(739\) −19.0000 −0.698926 −0.349463 0.936950i \(-0.613636\pi\)
−0.349463 + 0.936950i \(0.613636\pi\)
\(740\) 0 0
\(741\) −168.000 −6.17163
\(742\) 4.00000 0.146845
\(743\) −10.0000 −0.366864 −0.183432 0.983032i \(-0.558721\pi\)
−0.183432 + 0.983032i \(0.558721\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 52.0000 1.90386
\(747\) −72.0000 −2.63434
\(748\) 6.00000 0.219382
\(749\) −18.0000 −0.657706
\(750\) 0 0
\(751\) −31.0000 −1.13121 −0.565603 0.824678i \(-0.691357\pi\)
−0.565603 + 0.824678i \(0.691357\pi\)
\(752\) 12.0000 0.437595
\(753\) 60.0000 2.18652
\(754\) −70.0000 −2.54925
\(755\) 0 0
\(756\) −18.0000 −0.654654
\(757\) 8.00000 0.290765 0.145382 0.989376i \(-0.453559\pi\)
0.145382 + 0.989376i \(0.453559\pi\)
\(758\) 24.0000 0.871719
\(759\) −3.00000 −0.108893
\(760\) 0 0
\(761\) 12.0000 0.435000 0.217500 0.976060i \(-0.430210\pi\)
0.217500 + 0.976060i \(0.430210\pi\)
\(762\) 24.0000 0.869428
\(763\) 11.0000 0.398227
\(764\) −18.0000 −0.651217
\(765\) 0 0
\(766\) −32.0000 −1.15621
\(767\) −14.0000 −0.505511
\(768\) −48.0000 −1.73205
\(769\) −40.0000 −1.44244 −0.721218 0.692708i \(-0.756418\pi\)
−0.721218 + 0.692708i \(0.756418\pi\)
\(770\) 0 0
\(771\) −42.0000 −1.51259
\(772\) 4.00000 0.143963
\(773\) 31.0000 1.11499 0.557496 0.830179i \(-0.311762\pi\)
0.557496 + 0.830179i \(0.311762\pi\)
\(774\) 72.0000 2.58799
\(775\) 0 0
\(776\) 0 0
\(777\) −12.0000 −0.430498
\(778\) −38.0000 −1.36237
\(779\) 64.0000 2.29304
\(780\) 0 0
\(781\) −8.00000 −0.286263
\(782\) −6.00000 −0.214560
\(783\) 45.0000 1.60817
\(784\) −4.00000 −0.142857
\(785\) 0 0
\(786\) 72.0000 2.56815
\(787\) −53.0000 −1.88925 −0.944623 0.328158i \(-0.893572\pi\)
−0.944623 + 0.328158i \(0.893572\pi\)
\(788\) −56.0000 −1.99492
\(789\) 84.0000 2.99048
\(790\) 0 0
\(791\) −18.0000 −0.640006
\(792\) 0 0
\(793\) 98.0000 3.48008
\(794\) 50.0000 1.77443
\(795\) 0 0
\(796\) 52.0000 1.84309
\(797\) −35.0000 −1.23976 −0.619882 0.784695i \(-0.712819\pi\)
−0.619882 + 0.784695i \(0.712819\pi\)
\(798\) −48.0000 −1.69918
\(799\) 9.00000 0.318397
\(800\) 0 0
\(801\) −12.0000 −0.423999
\(802\) 50.0000 1.76556
\(803\) −6.00000 −0.211735
\(804\) −24.0000 −0.846415
\(805\) 0 0
\(806\) −28.0000 −0.986258
\(807\) 90.0000 3.16815
\(808\) 0 0
\(809\) −15.0000 −0.527372 −0.263686 0.964609i \(-0.584938\pi\)
−0.263686 + 0.964609i \(0.584938\pi\)
\(810\) 0 0
\(811\) 6.00000 0.210688 0.105344 0.994436i \(-0.466406\pi\)
0.105344 + 0.994436i \(0.466406\pi\)
\(812\) −10.0000 −0.350931
\(813\) 6.00000 0.210429
\(814\) 8.00000 0.280400
\(815\) 0 0
\(816\) −36.0000 −1.26025
\(817\) 48.0000 1.67931
\(818\) 28.0000 0.978997
\(819\) −42.0000 −1.46760
\(820\) 0 0
\(821\) −11.0000 −0.383903 −0.191951 0.981404i \(-0.561482\pi\)
−0.191951 + 0.981404i \(0.561482\pi\)
\(822\) 36.0000 1.25564
\(823\) −26.0000 −0.906303 −0.453152 0.891434i \(-0.649700\pi\)
−0.453152 + 0.891434i \(0.649700\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) −4.00000 −0.139178
\(827\) 20.0000 0.695468 0.347734 0.937593i \(-0.386951\pi\)
0.347734 + 0.937593i \(0.386951\pi\)
\(828\) −12.0000 −0.417029
\(829\) −24.0000 −0.833554 −0.416777 0.909009i \(-0.636840\pi\)
−0.416777 + 0.909009i \(0.636840\pi\)
\(830\) 0 0
\(831\) −72.0000 −2.49765
\(832\) 56.0000 1.94145
\(833\) −3.00000 −0.103944
\(834\) 48.0000 1.66210
\(835\) 0 0
\(836\) 16.0000 0.553372
\(837\) 18.0000 0.622171
\(838\) −60.0000 −2.07267
\(839\) −12.0000 −0.414286 −0.207143 0.978311i \(-0.566417\pi\)
−0.207143 + 0.978311i \(0.566417\pi\)
\(840\) 0 0
\(841\) −4.00000 −0.137931
\(842\) −14.0000 −0.482472
\(843\) 45.0000 1.54988
\(844\) 30.0000 1.03264
\(845\) 0 0
\(846\) 36.0000 1.23771
\(847\) −10.0000 −0.343604
\(848\) 8.00000 0.274721
\(849\) −3.00000 −0.102960
\(850\) 0 0
\(851\) −4.00000 −0.137118
\(852\) −48.0000 −1.64445
\(853\) 2.00000 0.0684787 0.0342393 0.999414i \(-0.489099\pi\)
0.0342393 + 0.999414i \(0.489099\pi\)
\(854\) 28.0000 0.958140
\(855\) 0 0
\(856\) 0 0
\(857\) 50.0000 1.70797 0.853984 0.520300i \(-0.174180\pi\)
0.853984 + 0.520300i \(0.174180\pi\)
\(858\) 42.0000 1.43386
\(859\) 40.0000 1.36478 0.682391 0.730987i \(-0.260940\pi\)
0.682391 + 0.730987i \(0.260940\pi\)
\(860\) 0 0
\(861\) 24.0000 0.817918
\(862\) −38.0000 −1.29429
\(863\) −30.0000 −1.02121 −0.510606 0.859815i \(-0.670579\pi\)
−0.510606 + 0.859815i \(0.670579\pi\)
\(864\) −72.0000 −2.44949
\(865\) 0 0
\(866\) 4.00000 0.135926
\(867\) 24.0000 0.815083
\(868\) −4.00000 −0.135769
\(869\) 3.00000 0.101768
\(870\) 0 0
\(871\) −28.0000 −0.948744
\(872\) 0 0
\(873\) −42.0000 −1.42148
\(874\) −16.0000 −0.541208
\(875\) 0 0
\(876\) −36.0000 −1.21633
\(877\) 32.0000 1.08056 0.540282 0.841484i \(-0.318318\pi\)
0.540282 + 0.841484i \(0.318318\pi\)
\(878\) −52.0000 −1.75491
\(879\) −27.0000 −0.910687
\(880\) 0 0
\(881\) 22.0000 0.741199 0.370599 0.928793i \(-0.379152\pi\)
0.370599 + 0.928793i \(0.379152\pi\)
\(882\) −12.0000 −0.404061
\(883\) 32.0000 1.07689 0.538443 0.842662i \(-0.319013\pi\)
0.538443 + 0.842662i \(0.319013\pi\)
\(884\) 42.0000 1.41261
\(885\) 0 0
\(886\) 20.0000 0.671913
\(887\) 8.00000 0.268614 0.134307 0.990940i \(-0.457119\pi\)
0.134307 + 0.990940i \(0.457119\pi\)
\(888\) 0 0
\(889\) 4.00000 0.134156
\(890\) 0 0
\(891\) −9.00000 −0.301511
\(892\) 26.0000 0.870544
\(893\) 24.0000 0.803129
\(894\) −12.0000 −0.401340
\(895\) 0 0
\(896\) 0 0
\(897\) −21.0000 −0.701170
\(898\) 18.0000 0.600668
\(899\) 10.0000 0.333519
\(900\) 0 0
\(901\) 6.00000 0.199889
\(902\) −16.0000 −0.532742
\(903\) 18.0000 0.599002
\(904\) 0 0
\(905\) 0 0
\(906\) −54.0000 −1.79403
\(907\) 28.0000 0.929725 0.464862 0.885383i \(-0.346104\pi\)
0.464862 + 0.885383i \(0.346104\pi\)
\(908\) −26.0000 −0.862840
\(909\) −84.0000 −2.78610
\(910\) 0 0
\(911\) 8.00000 0.265052 0.132526 0.991180i \(-0.457691\pi\)
0.132526 + 0.991180i \(0.457691\pi\)
\(912\) −96.0000 −3.17888
\(913\) 12.0000 0.397142
\(914\) −16.0000 −0.529233
\(915\) 0 0
\(916\) −32.0000 −1.05731
\(917\) 12.0000 0.396275
\(918\) −54.0000 −1.78227
\(919\) −7.00000 −0.230909 −0.115454 0.993313i \(-0.536832\pi\)
−0.115454 + 0.993313i \(0.536832\pi\)
\(920\) 0 0
\(921\) −33.0000 −1.08739
\(922\) −32.0000 −1.05386
\(923\) −56.0000 −1.84326
\(924\) 6.00000 0.197386
\(925\) 0 0
\(926\) −68.0000 −2.23462
\(927\) −6.00000 −0.197066
\(928\) −40.0000 −1.31306
\(929\) −6.00000 −0.196854 −0.0984268 0.995144i \(-0.531381\pi\)
−0.0984268 + 0.995144i \(0.531381\pi\)
\(930\) 0 0
\(931\) −8.00000 −0.262189
\(932\) −20.0000 −0.655122
\(933\) −12.0000 −0.392862
\(934\) −42.0000 −1.37428
\(935\) 0 0
\(936\) 0 0
\(937\) −59.0000 −1.92745 −0.963723 0.266904i \(-0.913999\pi\)
−0.963723 + 0.266904i \(0.913999\pi\)
\(938\) −8.00000 −0.261209
\(939\) 27.0000 0.881112
\(940\) 0 0
\(941\) −12.0000 −0.391189 −0.195594 0.980685i \(-0.562664\pi\)
−0.195594 + 0.980685i \(0.562664\pi\)
\(942\) −12.0000 −0.390981
\(943\) 8.00000 0.260516
\(944\) −8.00000 −0.260378
\(945\) 0 0
\(946\) −12.0000 −0.390154
\(947\) 28.0000 0.909878 0.454939 0.890523i \(-0.349661\pi\)
0.454939 + 0.890523i \(0.349661\pi\)
\(948\) 18.0000 0.584613
\(949\) −42.0000 −1.36338
\(950\) 0 0
\(951\) −36.0000 −1.16738
\(952\) 0 0
\(953\) −26.0000 −0.842223 −0.421111 0.907009i \(-0.638360\pi\)
−0.421111 + 0.907009i \(0.638360\pi\)
\(954\) 24.0000 0.777029
\(955\) 0 0
\(956\) −34.0000 −1.09964
\(957\) −15.0000 −0.484881
\(958\) −12.0000 −0.387702
\(959\) 6.00000 0.193750
\(960\) 0 0
\(961\) −27.0000 −0.870968
\(962\) 56.0000 1.80551
\(963\) −108.000 −3.48025
\(964\) −12.0000 −0.386494
\(965\) 0 0
\(966\) −6.00000 −0.193047
\(967\) 16.0000 0.514525 0.257263 0.966342i \(-0.417179\pi\)
0.257263 + 0.966342i \(0.417179\pi\)
\(968\) 0 0
\(969\) −72.0000 −2.31297
\(970\) 0 0
\(971\) 36.0000 1.15529 0.577647 0.816286i \(-0.303971\pi\)
0.577647 + 0.816286i \(0.303971\pi\)
\(972\) 0 0
\(973\) 8.00000 0.256468
\(974\) 4.00000 0.128168
\(975\) 0 0
\(976\) 56.0000 1.79252
\(977\) −24.0000 −0.767828 −0.383914 0.923369i \(-0.625424\pi\)
−0.383914 + 0.923369i \(0.625424\pi\)
\(978\) −144.000 −4.60461
\(979\) 2.00000 0.0639203
\(980\) 0 0
\(981\) 66.0000 2.10722
\(982\) 18.0000 0.574403
\(983\) 21.0000 0.669796 0.334898 0.942254i \(-0.391298\pi\)
0.334898 + 0.942254i \(0.391298\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) −30.0000 −0.955395
\(987\) 9.00000 0.286473
\(988\) 112.000 3.56319
\(989\) 6.00000 0.190789
\(990\) 0 0
\(991\) −40.0000 −1.27064 −0.635321 0.772248i \(-0.719132\pi\)
−0.635321 + 0.772248i \(0.719132\pi\)
\(992\) −16.0000 −0.508001
\(993\) −24.0000 −0.761617
\(994\) −16.0000 −0.507489
\(995\) 0 0
\(996\) 72.0000 2.28141
\(997\) 7.00000 0.221692 0.110846 0.993838i \(-0.464644\pi\)
0.110846 + 0.993838i \(0.464644\pi\)
\(998\) −50.0000 −1.58272
\(999\) −36.0000 −1.13899
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 4025.2.a.a.1.1 1
5.4 even 2 805.2.a.d.1.1 1
15.14 odd 2 7245.2.a.c.1.1 1
35.34 odd 2 5635.2.a.g.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
805.2.a.d.1.1 1 5.4 even 2
4025.2.a.a.1.1 1 1.1 even 1 trivial
5635.2.a.g.1.1 1 35.34 odd 2
7245.2.a.c.1.1 1 15.14 odd 2