Properties

Label 1575.2.d.k.1324.2
Level $1575$
Weight $2$
Character 1575.1324
Analytic conductor $12.576$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 1575 = 3^{2} \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1575.d (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(12.5764383184\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(i, \sqrt{5})\)
Defining polynomial: \(x^{4} + 3 x^{2} + 1\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 175)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 1324.2
Root \(-0.618034i\) of defining polynomial
Character \(\chi\) \(=\) 1575.1324
Dual form 1575.2.d.k.1324.3

$q$-expansion

\(f(q)\) \(=\) \(q-0.618034i q^{2} +1.61803 q^{4} -1.00000i q^{7} -2.23607i q^{8} +O(q^{10})\) \(q-0.618034i q^{2} +1.61803 q^{4} -1.00000i q^{7} -2.23607i q^{8} +0.236068 q^{11} +1.23607i q^{13} -0.618034 q^{14} +1.85410 q^{16} +2.47214i q^{17} +4.47214 q^{19} -0.145898i q^{22} -6.23607i q^{23} +0.763932 q^{26} -1.61803i q^{28} +5.00000 q^{29} +3.70820 q^{31} -5.61803i q^{32} +1.52786 q^{34} -3.00000i q^{37} -2.76393i q^{38} -4.76393 q^{41} +1.76393i q^{43} +0.381966 q^{44} -3.85410 q^{46} -2.00000i q^{47} -1.00000 q^{49} +2.00000i q^{52} -8.47214i q^{53} -2.23607 q^{56} -3.09017i q^{58} +11.7082 q^{59} -9.70820 q^{61} -2.29180i q^{62} +0.236068 q^{64} +4.23607i q^{67} +4.00000i q^{68} -8.70820 q^{71} -8.76393i q^{73} -1.85410 q^{74} +7.23607 q^{76} -0.236068i q^{77} +11.1803 q^{79} +2.94427i q^{82} +7.70820i q^{83} +1.09017 q^{86} -0.527864i q^{88} +17.2361 q^{89} +1.23607 q^{91} -10.0902i q^{92} -1.23607 q^{94} -5.23607i q^{97} +0.618034i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 2q^{4} + O(q^{10}) \) \( 4q + 2q^{4} - 8q^{11} + 2q^{14} - 6q^{16} + 12q^{26} + 20q^{29} - 12q^{31} + 24q^{34} - 28q^{41} + 6q^{44} - 2q^{46} - 4q^{49} + 20q^{59} - 12q^{61} - 8q^{64} - 8q^{71} + 6q^{74} + 20q^{76} - 18q^{86} + 60q^{89} - 4q^{91} + 4q^{94} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1575\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(451\) \(1226\)
\(\chi(n)\) \(-1\) \(1\) \(1\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) − 0.618034i − 0.437016i −0.975835 0.218508i \(-0.929881\pi\)
0.975835 0.218508i \(-0.0701190\pi\)
\(3\) 0 0
\(4\) 1.61803 0.809017
\(5\) 0 0
\(6\) 0 0
\(7\) − 1.00000i − 0.377964i
\(8\) − 2.23607i − 0.790569i
\(9\) 0 0
\(10\) 0 0
\(11\) 0.236068 0.0711772 0.0355886 0.999367i \(-0.488669\pi\)
0.0355886 + 0.999367i \(0.488669\pi\)
\(12\) 0 0
\(13\) 1.23607i 0.342824i 0.985199 + 0.171412i \(0.0548329\pi\)
−0.985199 + 0.171412i \(0.945167\pi\)
\(14\) −0.618034 −0.165177
\(15\) 0 0
\(16\) 1.85410 0.463525
\(17\) 2.47214i 0.599581i 0.954005 + 0.299791i \(0.0969168\pi\)
−0.954005 + 0.299791i \(0.903083\pi\)
\(18\) 0 0
\(19\) 4.47214 1.02598 0.512989 0.858395i \(-0.328538\pi\)
0.512989 + 0.858395i \(0.328538\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) − 0.145898i − 0.0311056i
\(23\) − 6.23607i − 1.30031i −0.759802 0.650155i \(-0.774704\pi\)
0.759802 0.650155i \(-0.225296\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0.763932 0.149819
\(27\) 0 0
\(28\) − 1.61803i − 0.305780i
\(29\) 5.00000 0.928477 0.464238 0.885710i \(-0.346328\pi\)
0.464238 + 0.885710i \(0.346328\pi\)
\(30\) 0 0
\(31\) 3.70820 0.666013 0.333007 0.942925i \(-0.391937\pi\)
0.333007 + 0.942925i \(0.391937\pi\)
\(32\) − 5.61803i − 0.993137i
\(33\) 0 0
\(34\) 1.52786 0.262027
\(35\) 0 0
\(36\) 0 0
\(37\) − 3.00000i − 0.493197i −0.969118 0.246598i \(-0.920687\pi\)
0.969118 0.246598i \(-0.0793129\pi\)
\(38\) − 2.76393i − 0.448369i
\(39\) 0 0
\(40\) 0 0
\(41\) −4.76393 −0.744001 −0.372001 0.928232i \(-0.621328\pi\)
−0.372001 + 0.928232i \(0.621328\pi\)
\(42\) 0 0
\(43\) 1.76393i 0.268997i 0.990914 + 0.134499i \(0.0429424\pi\)
−0.990914 + 0.134499i \(0.957058\pi\)
\(44\) 0.381966 0.0575835
\(45\) 0 0
\(46\) −3.85410 −0.568256
\(47\) − 2.00000i − 0.291730i −0.989305 0.145865i \(-0.953403\pi\)
0.989305 0.145865i \(-0.0465965\pi\)
\(48\) 0 0
\(49\) −1.00000 −0.142857
\(50\) 0 0
\(51\) 0 0
\(52\) 2.00000i 0.277350i
\(53\) − 8.47214i − 1.16374i −0.813283 0.581869i \(-0.802322\pi\)
0.813283 0.581869i \(-0.197678\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) −2.23607 −0.298807
\(57\) 0 0
\(58\) − 3.09017i − 0.405759i
\(59\) 11.7082 1.52428 0.762139 0.647413i \(-0.224149\pi\)
0.762139 + 0.647413i \(0.224149\pi\)
\(60\) 0 0
\(61\) −9.70820 −1.24301 −0.621504 0.783411i \(-0.713478\pi\)
−0.621504 + 0.783411i \(0.713478\pi\)
\(62\) − 2.29180i − 0.291058i
\(63\) 0 0
\(64\) 0.236068 0.0295085
\(65\) 0 0
\(66\) 0 0
\(67\) 4.23607i 0.517518i 0.965942 + 0.258759i \(0.0833136\pi\)
−0.965942 + 0.258759i \(0.916686\pi\)
\(68\) 4.00000i 0.485071i
\(69\) 0 0
\(70\) 0 0
\(71\) −8.70820 −1.03347 −0.516737 0.856144i \(-0.672853\pi\)
−0.516737 + 0.856144i \(0.672853\pi\)
\(72\) 0 0
\(73\) − 8.76393i − 1.02574i −0.858466 0.512870i \(-0.828582\pi\)
0.858466 0.512870i \(-0.171418\pi\)
\(74\) −1.85410 −0.215535
\(75\) 0 0
\(76\) 7.23607 0.830034
\(77\) − 0.236068i − 0.0269024i
\(78\) 0 0
\(79\) 11.1803 1.25789 0.628943 0.777451i \(-0.283488\pi\)
0.628943 + 0.777451i \(0.283488\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 2.94427i 0.325140i
\(83\) 7.70820i 0.846085i 0.906110 + 0.423043i \(0.139038\pi\)
−0.906110 + 0.423043i \(0.860962\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 1.09017 0.117556
\(87\) 0 0
\(88\) − 0.527864i − 0.0562705i
\(89\) 17.2361 1.82702 0.913510 0.406817i \(-0.133361\pi\)
0.913510 + 0.406817i \(0.133361\pi\)
\(90\) 0 0
\(91\) 1.23607 0.129575
\(92\) − 10.0902i − 1.05197i
\(93\) 0 0
\(94\) −1.23607 −0.127491
\(95\) 0 0
\(96\) 0 0
\(97\) − 5.23607i − 0.531642i −0.964022 0.265821i \(-0.914357\pi\)
0.964022 0.265821i \(-0.0856430\pi\)
\(98\) 0.618034i 0.0624309i
\(99\) 0 0
\(100\) 0 0
\(101\) −4.76393 −0.474029 −0.237014 0.971506i \(-0.576169\pi\)
−0.237014 + 0.971506i \(0.576169\pi\)
\(102\) 0 0
\(103\) 8.47214i 0.834784i 0.908726 + 0.417392i \(0.137056\pi\)
−0.908726 + 0.417392i \(0.862944\pi\)
\(104\) 2.76393 0.271026
\(105\) 0 0
\(106\) −5.23607 −0.508572
\(107\) 8.00000i 0.773389i 0.922208 + 0.386695i \(0.126383\pi\)
−0.922208 + 0.386695i \(0.873617\pi\)
\(108\) 0 0
\(109\) −8.41641 −0.806146 −0.403073 0.915168i \(-0.632058\pi\)
−0.403073 + 0.915168i \(0.632058\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) − 1.85410i − 0.175196i
\(113\) 14.4164i 1.35618i 0.734978 + 0.678091i \(0.237192\pi\)
−0.734978 + 0.678091i \(0.762808\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 8.09017 0.751153
\(117\) 0 0
\(118\) − 7.23607i − 0.666134i
\(119\) 2.47214 0.226620
\(120\) 0 0
\(121\) −10.9443 −0.994934
\(122\) 6.00000i 0.543214i
\(123\) 0 0
\(124\) 6.00000 0.538816
\(125\) 0 0
\(126\) 0 0
\(127\) − 13.6525i − 1.21146i −0.795670 0.605731i \(-0.792881\pi\)
0.795670 0.605731i \(-0.207119\pi\)
\(128\) − 11.3820i − 1.00603i
\(129\) 0 0
\(130\) 0 0
\(131\) 16.9443 1.48043 0.740214 0.672371i \(-0.234724\pi\)
0.740214 + 0.672371i \(0.234724\pi\)
\(132\) 0 0
\(133\) − 4.47214i − 0.387783i
\(134\) 2.61803 0.226164
\(135\) 0 0
\(136\) 5.52786 0.474010
\(137\) − 10.9443i − 0.935032i −0.883985 0.467516i \(-0.845149\pi\)
0.883985 0.467516i \(-0.154851\pi\)
\(138\) 0 0
\(139\) −10.6525 −0.903531 −0.451766 0.892137i \(-0.649206\pi\)
−0.451766 + 0.892137i \(0.649206\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 5.38197i 0.451645i
\(143\) 0.291796i 0.0244012i
\(144\) 0 0
\(145\) 0 0
\(146\) −5.41641 −0.448265
\(147\) 0 0
\(148\) − 4.85410i − 0.399005i
\(149\) 3.94427 0.323127 0.161564 0.986862i \(-0.448346\pi\)
0.161564 + 0.986862i \(0.448346\pi\)
\(150\) 0 0
\(151\) −20.2361 −1.64679 −0.823394 0.567470i \(-0.807922\pi\)
−0.823394 + 0.567470i \(0.807922\pi\)
\(152\) − 10.0000i − 0.811107i
\(153\) 0 0
\(154\) −0.145898 −0.0117568
\(155\) 0 0
\(156\) 0 0
\(157\) − 0.763932i − 0.0609684i −0.999535 0.0304842i \(-0.990295\pi\)
0.999535 0.0304842i \(-0.00970493\pi\)
\(158\) − 6.90983i − 0.549717i
\(159\) 0 0
\(160\) 0 0
\(161\) −6.23607 −0.491471
\(162\) 0 0
\(163\) − 1.52786i − 0.119672i −0.998208 0.0598358i \(-0.980942\pi\)
0.998208 0.0598358i \(-0.0190577\pi\)
\(164\) −7.70820 −0.601910
\(165\) 0 0
\(166\) 4.76393 0.369753
\(167\) 5.23607i 0.405179i 0.979264 + 0.202590i \(0.0649357\pi\)
−0.979264 + 0.202590i \(0.935064\pi\)
\(168\) 0 0
\(169\) 11.4721 0.882472
\(170\) 0 0
\(171\) 0 0
\(172\) 2.85410i 0.217623i
\(173\) 11.5279i 0.876447i 0.898866 + 0.438224i \(0.144392\pi\)
−0.898866 + 0.438224i \(0.855608\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0.437694 0.0329924
\(177\) 0 0
\(178\) − 10.6525i − 0.798437i
\(179\) −23.4164 −1.75022 −0.875112 0.483920i \(-0.839213\pi\)
−0.875112 + 0.483920i \(0.839213\pi\)
\(180\) 0 0
\(181\) 8.18034 0.608040 0.304020 0.952666i \(-0.401671\pi\)
0.304020 + 0.952666i \(0.401671\pi\)
\(182\) − 0.763932i − 0.0566264i
\(183\) 0 0
\(184\) −13.9443 −1.02799
\(185\) 0 0
\(186\) 0 0
\(187\) 0.583592i 0.0426765i
\(188\) − 3.23607i − 0.236015i
\(189\) 0 0
\(190\) 0 0
\(191\) −6.47214 −0.468307 −0.234154 0.972200i \(-0.575232\pi\)
−0.234154 + 0.972200i \(0.575232\pi\)
\(192\) 0 0
\(193\) 12.4164i 0.893753i 0.894596 + 0.446876i \(0.147464\pi\)
−0.894596 + 0.446876i \(0.852536\pi\)
\(194\) −3.23607 −0.232336
\(195\) 0 0
\(196\) −1.61803 −0.115574
\(197\) − 1.47214i − 0.104885i −0.998624 0.0524427i \(-0.983299\pi\)
0.998624 0.0524427i \(-0.0167007\pi\)
\(198\) 0 0
\(199\) −7.23607 −0.512951 −0.256476 0.966551i \(-0.582561\pi\)
−0.256476 + 0.966551i \(0.582561\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 2.94427i 0.207158i
\(203\) − 5.00000i − 0.350931i
\(204\) 0 0
\(205\) 0 0
\(206\) 5.23607 0.364814
\(207\) 0 0
\(208\) 2.29180i 0.158907i
\(209\) 1.05573 0.0730262
\(210\) 0 0
\(211\) 12.0000 0.826114 0.413057 0.910705i \(-0.364461\pi\)
0.413057 + 0.910705i \(0.364461\pi\)
\(212\) − 13.7082i − 0.941483i
\(213\) 0 0
\(214\) 4.94427 0.337983
\(215\) 0 0
\(216\) 0 0
\(217\) − 3.70820i − 0.251729i
\(218\) 5.20163i 0.352299i
\(219\) 0 0
\(220\) 0 0
\(221\) −3.05573 −0.205551
\(222\) 0 0
\(223\) 20.1803i 1.35138i 0.737188 + 0.675688i \(0.236153\pi\)
−0.737188 + 0.675688i \(0.763847\pi\)
\(224\) −5.61803 −0.375371
\(225\) 0 0
\(226\) 8.90983 0.592673
\(227\) 21.4164i 1.42146i 0.703466 + 0.710728i \(0.251635\pi\)
−0.703466 + 0.710728i \(0.748365\pi\)
\(228\) 0 0
\(229\) 4.47214 0.295527 0.147764 0.989023i \(-0.452793\pi\)
0.147764 + 0.989023i \(0.452793\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) − 11.1803i − 0.734025i
\(233\) − 7.94427i − 0.520447i −0.965548 0.260223i \(-0.916204\pi\)
0.965548 0.260223i \(-0.0837962\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 18.9443 1.23317
\(237\) 0 0
\(238\) − 1.52786i − 0.0990367i
\(239\) −5.52786 −0.357568 −0.178784 0.983888i \(-0.557216\pi\)
−0.178784 + 0.983888i \(0.557216\pi\)
\(240\) 0 0
\(241\) −3.52786 −0.227250 −0.113625 0.993524i \(-0.536246\pi\)
−0.113625 + 0.993524i \(0.536246\pi\)
\(242\) 6.76393i 0.434802i
\(243\) 0 0
\(244\) −15.7082 −1.00561
\(245\) 0 0
\(246\) 0 0
\(247\) 5.52786i 0.351730i
\(248\) − 8.29180i − 0.526530i
\(249\) 0 0
\(250\) 0 0
\(251\) −6.47214 −0.408518 −0.204259 0.978917i \(-0.565478\pi\)
−0.204259 + 0.978917i \(0.565478\pi\)
\(252\) 0 0
\(253\) − 1.47214i − 0.0925524i
\(254\) −8.43769 −0.529428
\(255\) 0 0
\(256\) −6.56231 −0.410144
\(257\) − 12.6525i − 0.789240i −0.918844 0.394620i \(-0.870876\pi\)
0.918844 0.394620i \(-0.129124\pi\)
\(258\) 0 0
\(259\) −3.00000 −0.186411
\(260\) 0 0
\(261\) 0 0
\(262\) − 10.4721i − 0.646971i
\(263\) − 16.2361i − 1.00116i −0.865691 0.500579i \(-0.833120\pi\)
0.865691 0.500579i \(-0.166880\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) −2.76393 −0.169468
\(267\) 0 0
\(268\) 6.85410i 0.418681i
\(269\) −11.7082 −0.713862 −0.356931 0.934131i \(-0.616177\pi\)
−0.356931 + 0.934131i \(0.616177\pi\)
\(270\) 0 0
\(271\) 23.7082 1.44017 0.720085 0.693885i \(-0.244103\pi\)
0.720085 + 0.693885i \(0.244103\pi\)
\(272\) 4.58359i 0.277921i
\(273\) 0 0
\(274\) −6.76393 −0.408624
\(275\) 0 0
\(276\) 0 0
\(277\) 19.8885i 1.19499i 0.801874 + 0.597493i \(0.203837\pi\)
−0.801874 + 0.597493i \(0.796163\pi\)
\(278\) 6.58359i 0.394858i
\(279\) 0 0
\(280\) 0 0
\(281\) 15.3607 0.916341 0.458171 0.888864i \(-0.348505\pi\)
0.458171 + 0.888864i \(0.348505\pi\)
\(282\) 0 0
\(283\) 17.4164i 1.03530i 0.855593 + 0.517649i \(0.173193\pi\)
−0.855593 + 0.517649i \(0.826807\pi\)
\(284\) −14.0902 −0.836098
\(285\) 0 0
\(286\) 0.180340 0.0106637
\(287\) 4.76393i 0.281206i
\(288\) 0 0
\(289\) 10.8885 0.640503
\(290\) 0 0
\(291\) 0 0
\(292\) − 14.1803i − 0.829842i
\(293\) 31.1246i 1.81832i 0.416448 + 0.909160i \(0.363275\pi\)
−0.416448 + 0.909160i \(0.636725\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) −6.70820 −0.389906
\(297\) 0 0
\(298\) − 2.43769i − 0.141212i
\(299\) 7.70820 0.445777
\(300\) 0 0
\(301\) 1.76393 0.101671
\(302\) 12.5066i 0.719673i
\(303\) 0 0
\(304\) 8.29180 0.475567
\(305\) 0 0
\(306\) 0 0
\(307\) − 4.58359i − 0.261599i −0.991409 0.130800i \(-0.958246\pi\)
0.991409 0.130800i \(-0.0417545\pi\)
\(308\) − 0.381966i − 0.0217645i
\(309\) 0 0
\(310\) 0 0
\(311\) −24.3607 −1.38137 −0.690684 0.723157i \(-0.742690\pi\)
−0.690684 + 0.723157i \(0.742690\pi\)
\(312\) 0 0
\(313\) 19.5279i 1.10378i 0.833917 + 0.551890i \(0.186093\pi\)
−0.833917 + 0.551890i \(0.813907\pi\)
\(314\) −0.472136 −0.0266442
\(315\) 0 0
\(316\) 18.0902 1.01765
\(317\) 25.3607i 1.42440i 0.701978 + 0.712199i \(0.252301\pi\)
−0.701978 + 0.712199i \(0.747699\pi\)
\(318\) 0 0
\(319\) 1.18034 0.0660863
\(320\) 0 0
\(321\) 0 0
\(322\) 3.85410i 0.214781i
\(323\) 11.0557i 0.615157i
\(324\) 0 0
\(325\) 0 0
\(326\) −0.944272 −0.0522984
\(327\) 0 0
\(328\) 10.6525i 0.588185i
\(329\) −2.00000 −0.110264
\(330\) 0 0
\(331\) −24.7082 −1.35809 −0.679043 0.734099i \(-0.737605\pi\)
−0.679043 + 0.734099i \(0.737605\pi\)
\(332\) 12.4721i 0.684497i
\(333\) 0 0
\(334\) 3.23607 0.177070
\(335\) 0 0
\(336\) 0 0
\(337\) 16.4721i 0.897294i 0.893709 + 0.448647i \(0.148094\pi\)
−0.893709 + 0.448647i \(0.851906\pi\)
\(338\) − 7.09017i − 0.385654i
\(339\) 0 0
\(340\) 0 0
\(341\) 0.875388 0.0474049
\(342\) 0 0
\(343\) 1.00000i 0.0539949i
\(344\) 3.94427 0.212661
\(345\) 0 0
\(346\) 7.12461 0.383022
\(347\) 20.2361i 1.08633i 0.839626 + 0.543165i \(0.182774\pi\)
−0.839626 + 0.543165i \(0.817226\pi\)
\(348\) 0 0
\(349\) −4.47214 −0.239388 −0.119694 0.992811i \(-0.538191\pi\)
−0.119694 + 0.992811i \(0.538191\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) − 1.32624i − 0.0706887i
\(353\) 2.18034i 0.116048i 0.998315 + 0.0580239i \(0.0184800\pi\)
−0.998315 + 0.0580239i \(0.981520\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 27.8885 1.47809
\(357\) 0 0
\(358\) 14.4721i 0.764876i
\(359\) −30.1246 −1.58992 −0.794958 0.606664i \(-0.792507\pi\)
−0.794958 + 0.606664i \(0.792507\pi\)
\(360\) 0 0
\(361\) 1.00000 0.0526316
\(362\) − 5.05573i − 0.265723i
\(363\) 0 0
\(364\) 2.00000 0.104828
\(365\) 0 0
\(366\) 0 0
\(367\) 37.1246i 1.93789i 0.247278 + 0.968944i \(0.420464\pi\)
−0.247278 + 0.968944i \(0.579536\pi\)
\(368\) − 11.5623i − 0.602727i
\(369\) 0 0
\(370\) 0 0
\(371\) −8.47214 −0.439851
\(372\) 0 0
\(373\) − 37.8328i − 1.95891i −0.201665 0.979454i \(-0.564635\pi\)
0.201665 0.979454i \(-0.435365\pi\)
\(374\) 0.360680 0.0186503
\(375\) 0 0
\(376\) −4.47214 −0.230633
\(377\) 6.18034i 0.318304i
\(378\) 0 0
\(379\) 11.1803 0.574295 0.287148 0.957886i \(-0.407293\pi\)
0.287148 + 0.957886i \(0.407293\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 4.00000i 0.204658i
\(383\) 33.2361i 1.69828i 0.528165 + 0.849142i \(0.322880\pi\)
−0.528165 + 0.849142i \(0.677120\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 7.67376 0.390584
\(387\) 0 0
\(388\) − 8.47214i − 0.430108i
\(389\) 2.88854 0.146455 0.0732275 0.997315i \(-0.476670\pi\)
0.0732275 + 0.997315i \(0.476670\pi\)
\(390\) 0 0
\(391\) 15.4164 0.779641
\(392\) 2.23607i 0.112938i
\(393\) 0 0
\(394\) −0.909830 −0.0458366
\(395\) 0 0
\(396\) 0 0
\(397\) − 9.05573i − 0.454494i −0.973837 0.227247i \(-0.927028\pi\)
0.973837 0.227247i \(-0.0729725\pi\)
\(398\) 4.47214i 0.224168i
\(399\) 0 0
\(400\) 0 0
\(401\) −2.52786 −0.126236 −0.0631178 0.998006i \(-0.520104\pi\)
−0.0631178 + 0.998006i \(0.520104\pi\)
\(402\) 0 0
\(403\) 4.58359i 0.228325i
\(404\) −7.70820 −0.383497
\(405\) 0 0
\(406\) −3.09017 −0.153363
\(407\) − 0.708204i − 0.0351044i
\(408\) 0 0
\(409\) −24.4721 −1.21007 −0.605035 0.796199i \(-0.706841\pi\)
−0.605035 + 0.796199i \(0.706841\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 13.7082i 0.675355i
\(413\) − 11.7082i − 0.576123i
\(414\) 0 0
\(415\) 0 0
\(416\) 6.94427 0.340471
\(417\) 0 0
\(418\) − 0.652476i − 0.0319136i
\(419\) −26.1803 −1.27899 −0.639497 0.768794i \(-0.720857\pi\)
−0.639497 + 0.768794i \(0.720857\pi\)
\(420\) 0 0
\(421\) −13.0000 −0.633581 −0.316791 0.948495i \(-0.602605\pi\)
−0.316791 + 0.948495i \(0.602605\pi\)
\(422\) − 7.41641i − 0.361025i
\(423\) 0 0
\(424\) −18.9443 −0.920015
\(425\) 0 0
\(426\) 0 0
\(427\) 9.70820i 0.469813i
\(428\) 12.9443i 0.625685i
\(429\) 0 0
\(430\) 0 0
\(431\) −17.5279 −0.844288 −0.422144 0.906529i \(-0.638722\pi\)
−0.422144 + 0.906529i \(0.638722\pi\)
\(432\) 0 0
\(433\) − 28.3607i − 1.36293i −0.731852 0.681464i \(-0.761344\pi\)
0.731852 0.681464i \(-0.238656\pi\)
\(434\) −2.29180 −0.110010
\(435\) 0 0
\(436\) −13.6180 −0.652186
\(437\) − 27.8885i − 1.33409i
\(438\) 0 0
\(439\) 8.29180 0.395746 0.197873 0.980228i \(-0.436597\pi\)
0.197873 + 0.980228i \(0.436597\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 1.88854i 0.0898289i
\(443\) 19.4164i 0.922501i 0.887270 + 0.461251i \(0.152599\pi\)
−0.887270 + 0.461251i \(0.847401\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 12.4721 0.590573
\(447\) 0 0
\(448\) − 0.236068i − 0.0111532i
\(449\) 20.5279 0.968770 0.484385 0.874855i \(-0.339043\pi\)
0.484385 + 0.874855i \(0.339043\pi\)
\(450\) 0 0
\(451\) −1.12461 −0.0529559
\(452\) 23.3262i 1.09717i
\(453\) 0 0
\(454\) 13.2361 0.621199
\(455\) 0 0
\(456\) 0 0
\(457\) 12.5279i 0.586029i 0.956108 + 0.293014i \(0.0946584\pi\)
−0.956108 + 0.293014i \(0.905342\pi\)
\(458\) − 2.76393i − 0.129150i
\(459\) 0 0
\(460\) 0 0
\(461\) 14.1803 0.660444 0.330222 0.943903i \(-0.392876\pi\)
0.330222 + 0.943903i \(0.392876\pi\)
\(462\) 0 0
\(463\) − 13.8885i − 0.645455i −0.946492 0.322728i \(-0.895400\pi\)
0.946492 0.322728i \(-0.104600\pi\)
\(464\) 9.27051 0.430373
\(465\) 0 0
\(466\) −4.90983 −0.227443
\(467\) 6.94427i 0.321343i 0.987008 + 0.160671i \(0.0513659\pi\)
−0.987008 + 0.160671i \(0.948634\pi\)
\(468\) 0 0
\(469\) 4.23607 0.195603
\(470\) 0 0
\(471\) 0 0
\(472\) − 26.1803i − 1.20505i
\(473\) 0.416408i 0.0191465i
\(474\) 0 0
\(475\) 0 0
\(476\) 4.00000 0.183340
\(477\) 0 0
\(478\) 3.41641i 0.156263i
\(479\) −26.1803 −1.19621 −0.598105 0.801418i \(-0.704079\pi\)
−0.598105 + 0.801418i \(0.704079\pi\)
\(480\) 0 0
\(481\) 3.70820 0.169080
\(482\) 2.18034i 0.0993118i
\(483\) 0 0
\(484\) −17.7082 −0.804918
\(485\) 0 0
\(486\) 0 0
\(487\) − 5.76393i − 0.261189i −0.991436 0.130594i \(-0.958311\pi\)
0.991436 0.130594i \(-0.0416885\pi\)
\(488\) 21.7082i 0.982684i
\(489\) 0 0
\(490\) 0 0
\(491\) 5.76393 0.260123 0.130061 0.991506i \(-0.458483\pi\)
0.130061 + 0.991506i \(0.458483\pi\)
\(492\) 0 0
\(493\) 12.3607i 0.556697i
\(494\) 3.41641 0.153711
\(495\) 0 0
\(496\) 6.87539 0.308714
\(497\) 8.70820i 0.390616i
\(498\) 0 0
\(499\) −11.0557 −0.494922 −0.247461 0.968898i \(-0.579596\pi\)
−0.247461 + 0.968898i \(0.579596\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 4.00000i 0.178529i
\(503\) 8.11146i 0.361672i 0.983513 + 0.180836i \(0.0578803\pi\)
−0.983513 + 0.180836i \(0.942120\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) −0.909830 −0.0404469
\(507\) 0 0
\(508\) − 22.0902i − 0.980093i
\(509\) 40.6525 1.80189 0.900945 0.433934i \(-0.142875\pi\)
0.900945 + 0.433934i \(0.142875\pi\)
\(510\) 0 0
\(511\) −8.76393 −0.387694
\(512\) − 18.7082i − 0.826794i
\(513\) 0 0
\(514\) −7.81966 −0.344910
\(515\) 0 0
\(516\) 0 0
\(517\) − 0.472136i − 0.0207645i
\(518\) 1.85410i 0.0814646i
\(519\) 0 0
\(520\) 0 0
\(521\) −22.0000 −0.963837 −0.481919 0.876216i \(-0.660060\pi\)
−0.481919 + 0.876216i \(0.660060\pi\)
\(522\) 0 0
\(523\) 16.3607i 0.715403i 0.933836 + 0.357701i \(0.116439\pi\)
−0.933836 + 0.357701i \(0.883561\pi\)
\(524\) 27.4164 1.19769
\(525\) 0 0
\(526\) −10.0344 −0.437522
\(527\) 9.16718i 0.399329i
\(528\) 0 0
\(529\) −15.8885 −0.690806
\(530\) 0 0
\(531\) 0 0
\(532\) − 7.23607i − 0.313723i
\(533\) − 5.88854i − 0.255061i
\(534\) 0 0
\(535\) 0 0
\(536\) 9.47214 0.409134
\(537\) 0 0
\(538\) 7.23607i 0.311969i
\(539\) −0.236068 −0.0101682
\(540\) 0 0
\(541\) 15.9443 0.685498 0.342749 0.939427i \(-0.388642\pi\)
0.342749 + 0.939427i \(0.388642\pi\)
\(542\) − 14.6525i − 0.629378i
\(543\) 0 0
\(544\) 13.8885 0.595466
\(545\) 0 0
\(546\) 0 0
\(547\) 9.76393i 0.417476i 0.977972 + 0.208738i \(0.0669355\pi\)
−0.977972 + 0.208738i \(0.933064\pi\)
\(548\) − 17.7082i − 0.756457i
\(549\) 0 0
\(550\) 0 0
\(551\) 22.3607 0.952597
\(552\) 0 0
\(553\) − 11.1803i − 0.475436i
\(554\) 12.2918 0.522228
\(555\) 0 0
\(556\) −17.2361 −0.730972
\(557\) − 9.11146i − 0.386065i −0.981192 0.193032i \(-0.938168\pi\)
0.981192 0.193032i \(-0.0618322\pi\)
\(558\) 0 0
\(559\) −2.18034 −0.0922186
\(560\) 0 0
\(561\) 0 0
\(562\) − 9.49342i − 0.400456i
\(563\) − 17.4164i − 0.734014i −0.930218 0.367007i \(-0.880382\pi\)
0.930218 0.367007i \(-0.119618\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 10.7639 0.452442
\(567\) 0 0
\(568\) 19.4721i 0.817033i
\(569\) −3.94427 −0.165352 −0.0826762 0.996576i \(-0.526347\pi\)
−0.0826762 + 0.996576i \(0.526347\pi\)
\(570\) 0 0
\(571\) 36.5967 1.53153 0.765763 0.643123i \(-0.222362\pi\)
0.765763 + 0.643123i \(0.222362\pi\)
\(572\) 0.472136i 0.0197410i
\(573\) 0 0
\(574\) 2.94427 0.122892
\(575\) 0 0
\(576\) 0 0
\(577\) 2.00000i 0.0832611i 0.999133 + 0.0416305i \(0.0132552\pi\)
−0.999133 + 0.0416305i \(0.986745\pi\)
\(578\) − 6.72949i − 0.279910i
\(579\) 0 0
\(580\) 0 0
\(581\) 7.70820 0.319790
\(582\) 0 0
\(583\) − 2.00000i − 0.0828315i
\(584\) −19.5967 −0.810919
\(585\) 0 0
\(586\) 19.2361 0.794635
\(587\) − 24.7639i − 1.02212i −0.859546 0.511058i \(-0.829254\pi\)
0.859546 0.511058i \(-0.170746\pi\)
\(588\) 0 0
\(589\) 16.5836 0.683315
\(590\) 0 0
\(591\) 0 0
\(592\) − 5.56231i − 0.228609i
\(593\) 37.3050i 1.53193i 0.642882 + 0.765965i \(0.277739\pi\)
−0.642882 + 0.765965i \(0.722261\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 6.38197 0.261416
\(597\) 0 0
\(598\) − 4.76393i − 0.194812i
\(599\) −11.1803 −0.456816 −0.228408 0.973565i \(-0.573352\pi\)
−0.228408 + 0.973565i \(0.573352\pi\)
\(600\) 0 0
\(601\) −36.9443 −1.50699 −0.753494 0.657455i \(-0.771633\pi\)
−0.753494 + 0.657455i \(0.771633\pi\)
\(602\) − 1.09017i − 0.0444320i
\(603\) 0 0
\(604\) −32.7426 −1.33228
\(605\) 0 0
\(606\) 0 0
\(607\) 7.12461i 0.289179i 0.989492 + 0.144590i \(0.0461862\pi\)
−0.989492 + 0.144590i \(0.953814\pi\)
\(608\) − 25.1246i − 1.01894i
\(609\) 0 0
\(610\) 0 0
\(611\) 2.47214 0.100012
\(612\) 0 0
\(613\) − 44.4164i − 1.79396i −0.442069 0.896981i \(-0.645755\pi\)
0.442069 0.896981i \(-0.354245\pi\)
\(614\) −2.83282 −0.114323
\(615\) 0 0
\(616\) −0.527864 −0.0212682
\(617\) − 5.94427i − 0.239307i −0.992816 0.119654i \(-0.961822\pi\)
0.992816 0.119654i \(-0.0381784\pi\)
\(618\) 0 0
\(619\) 11.7082 0.470592 0.235296 0.971924i \(-0.424394\pi\)
0.235296 + 0.971924i \(0.424394\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 15.0557i 0.603680i
\(623\) − 17.2361i − 0.690548i
\(624\) 0 0
\(625\) 0 0
\(626\) 12.0689 0.482370
\(627\) 0 0
\(628\) − 1.23607i − 0.0493245i
\(629\) 7.41641 0.295712
\(630\) 0 0
\(631\) 27.6525 1.10083 0.550414 0.834892i \(-0.314470\pi\)
0.550414 + 0.834892i \(0.314470\pi\)
\(632\) − 25.0000i − 0.994447i
\(633\) 0 0
\(634\) 15.6738 0.622485
\(635\) 0 0
\(636\) 0 0
\(637\) − 1.23607i − 0.0489748i
\(638\) − 0.729490i − 0.0288808i
\(639\) 0 0
\(640\) 0 0
\(641\) −43.8328 −1.73129 −0.865646 0.500656i \(-0.833092\pi\)
−0.865646 + 0.500656i \(0.833092\pi\)
\(642\) 0 0
\(643\) 18.4721i 0.728470i 0.931307 + 0.364235i \(0.118669\pi\)
−0.931307 + 0.364235i \(0.881331\pi\)
\(644\) −10.0902 −0.397608
\(645\) 0 0
\(646\) 6.83282 0.268834
\(647\) − 19.8885i − 0.781899i −0.920412 0.390950i \(-0.872147\pi\)
0.920412 0.390950i \(-0.127853\pi\)
\(648\) 0 0
\(649\) 2.76393 0.108494
\(650\) 0 0
\(651\) 0 0
\(652\) − 2.47214i − 0.0968163i
\(653\) − 25.0557i − 0.980506i −0.871580 0.490253i \(-0.836904\pi\)
0.871580 0.490253i \(-0.163096\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) −8.83282 −0.344864
\(657\) 0 0
\(658\) 1.23607i 0.0481869i
\(659\) 17.8885 0.696839 0.348419 0.937339i \(-0.386719\pi\)
0.348419 + 0.937339i \(0.386719\pi\)
\(660\) 0 0
\(661\) −42.7214 −1.66167 −0.830834 0.556520i \(-0.812136\pi\)
−0.830834 + 0.556520i \(0.812136\pi\)
\(662\) 15.2705i 0.593505i
\(663\) 0 0
\(664\) 17.2361 0.668889
\(665\) 0 0
\(666\) 0 0
\(667\) − 31.1803i − 1.20731i
\(668\) 8.47214i 0.327797i
\(669\) 0 0
\(670\) 0 0
\(671\) −2.29180 −0.0884738
\(672\) 0 0
\(673\) 19.5279i 0.752744i 0.926469 + 0.376372i \(0.122829\pi\)
−0.926469 + 0.376372i \(0.877171\pi\)
\(674\) 10.1803 0.392132
\(675\) 0 0
\(676\) 18.5623 0.713935
\(677\) − 14.3607i − 0.551926i −0.961168 0.275963i \(-0.911003\pi\)
0.961168 0.275963i \(-0.0889967\pi\)
\(678\) 0 0
\(679\) −5.23607 −0.200942
\(680\) 0 0
\(681\) 0 0
\(682\) − 0.541020i − 0.0207167i
\(683\) − 14.1246i − 0.540463i −0.962795 0.270232i \(-0.912900\pi\)
0.962795 0.270232i \(-0.0871003\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0.618034 0.0235966
\(687\) 0 0
\(688\) 3.27051i 0.124687i
\(689\) 10.4721 0.398957
\(690\) 0 0
\(691\) −4.18034 −0.159028 −0.0795138 0.996834i \(-0.525337\pi\)
−0.0795138 + 0.996834i \(0.525337\pi\)
\(692\) 18.6525i 0.709061i
\(693\) 0 0
\(694\) 12.5066 0.474743
\(695\) 0 0
\(696\) 0 0
\(697\) − 11.7771i − 0.446089i
\(698\) 2.76393i 0.104616i
\(699\) 0 0
\(700\) 0 0
\(701\) 29.0557 1.09742 0.548710 0.836013i \(-0.315119\pi\)
0.548710 + 0.836013i \(0.315119\pi\)
\(702\) 0 0
\(703\) − 13.4164i − 0.506009i
\(704\) 0.0557281 0.00210033
\(705\) 0 0
\(706\) 1.34752 0.0507147
\(707\) 4.76393i 0.179166i
\(708\) 0 0
\(709\) 12.1115 0.454855 0.227428 0.973795i \(-0.426968\pi\)
0.227428 + 0.973795i \(0.426968\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) − 38.5410i − 1.44439i
\(713\) − 23.1246i − 0.866024i
\(714\) 0 0
\(715\) 0 0
\(716\) −37.8885 −1.41596
\(717\) 0 0
\(718\) 18.6180i 0.694819i
\(719\) 16.1803 0.603425 0.301712 0.953399i \(-0.402442\pi\)
0.301712 + 0.953399i \(0.402442\pi\)
\(720\) 0 0
\(721\) 8.47214 0.315519
\(722\) − 0.618034i − 0.0230008i
\(723\) 0 0
\(724\) 13.2361 0.491915
\(725\) 0 0
\(726\) 0 0
\(727\) 3.05573i 0.113331i 0.998393 + 0.0566653i \(0.0180468\pi\)
−0.998393 + 0.0566653i \(0.981953\pi\)
\(728\) − 2.76393i − 0.102438i
\(729\) 0 0
\(730\) 0 0
\(731\) −4.36068 −0.161286
\(732\) 0 0
\(733\) 4.00000i 0.147743i 0.997268 + 0.0738717i \(0.0235355\pi\)
−0.997268 + 0.0738717i \(0.976464\pi\)
\(734\) 22.9443 0.846889
\(735\) 0 0
\(736\) −35.0344 −1.29139
\(737\) 1.00000i 0.0368355i
\(738\) 0 0
\(739\) −25.6525 −0.943642 −0.471821 0.881694i \(-0.656403\pi\)
−0.471821 + 0.881694i \(0.656403\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 5.23607i 0.192222i
\(743\) 10.4721i 0.384185i 0.981377 + 0.192093i \(0.0615274\pi\)
−0.981377 + 0.192093i \(0.938473\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) −23.3820 −0.856075
\(747\) 0 0
\(748\) 0.944272i 0.0345260i
\(749\) 8.00000 0.292314
\(750\) 0 0
\(751\) 3.05573 0.111505 0.0557526 0.998445i \(-0.482244\pi\)
0.0557526 + 0.998445i \(0.482244\pi\)
\(752\) − 3.70820i − 0.135224i
\(753\) 0 0
\(754\) 3.81966 0.139104
\(755\) 0 0
\(756\) 0 0
\(757\) − 19.5836i − 0.711778i −0.934528 0.355889i \(-0.884178\pi\)
0.934528 0.355889i \(-0.115822\pi\)
\(758\) − 6.90983i − 0.250976i
\(759\) 0 0
\(760\) 0 0
\(761\) −27.7771 −1.00692 −0.503459 0.864019i \(-0.667940\pi\)
−0.503459 + 0.864019i \(0.667940\pi\)
\(762\) 0 0
\(763\) 8.41641i 0.304694i
\(764\) −10.4721 −0.378869
\(765\) 0 0
\(766\) 20.5410 0.742177
\(767\) 14.4721i 0.522559i
\(768\) 0 0
\(769\) 43.0132 1.55109 0.775547 0.631290i \(-0.217474\pi\)
0.775547 + 0.631290i \(0.217474\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 20.0902i 0.723061i
\(773\) − 50.1803i − 1.80486i −0.430835 0.902431i \(-0.641781\pi\)
0.430835 0.902431i \(-0.358219\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) −11.7082 −0.420300
\(777\) 0 0
\(778\) − 1.78522i − 0.0640032i
\(779\) −21.3050 −0.763329
\(780\) 0 0
\(781\) −2.05573 −0.0735597
\(782\) − 9.52786i − 0.340716i
\(783\) 0 0
\(784\) −1.85410 −0.0662179
\(785\) 0 0
\(786\) 0 0
\(787\) − 40.7639i − 1.45308i −0.687126 0.726539i \(-0.741128\pi\)
0.687126 0.726539i \(-0.258872\pi\)
\(788\) − 2.38197i − 0.0848540i
\(789\) 0 0
\(790\) 0 0
\(791\) 14.4164 0.512588
\(792\) 0 0
\(793\) − 12.0000i − 0.426132i
\(794\) −5.59675 −0.198621
\(795\) 0 0
\(796\) −11.7082 −0.414986
\(797\) − 35.4164i − 1.25451i −0.778813 0.627257i \(-0.784178\pi\)
0.778813 0.627257i \(-0.215822\pi\)
\(798\) 0 0
\(799\) 4.94427 0.174916
\(800\) 0 0
\(801\) 0 0
\(802\) 1.56231i 0.0551669i
\(803\) − 2.06888i − 0.0730093i
\(804\) 0 0
\(805\) 0 0
\(806\) 2.83282 0.0997817
\(807\) 0 0
\(808\) 10.6525i 0.374753i
\(809\) 29.4721 1.03619 0.518093 0.855325i \(-0.326642\pi\)
0.518093 + 0.855325i \(0.326642\pi\)
\(810\) 0 0
\(811\) −42.7214 −1.50015 −0.750075 0.661353i \(-0.769983\pi\)
−0.750075 + 0.661353i \(0.769983\pi\)
\(812\) − 8.09017i − 0.283909i
\(813\) 0 0
\(814\) −0.437694 −0.0153412
\(815\) 0 0
\(816\) 0 0
\(817\) 7.88854i 0.275985i
\(818\) 15.1246i 0.528820i
\(819\) 0 0
\(820\) 0 0
\(821\) −28.8328 −1.00627 −0.503136 0.864207i \(-0.667821\pi\)
−0.503136 + 0.864207i \(0.667821\pi\)
\(822\) 0 0
\(823\) − 31.6525i − 1.10334i −0.834064 0.551668i \(-0.813992\pi\)
0.834064 0.551668i \(-0.186008\pi\)
\(824\) 18.9443 0.659955
\(825\) 0 0
\(826\) −7.23607 −0.251775
\(827\) 41.5410i 1.44452i 0.691620 + 0.722261i \(0.256897\pi\)
−0.691620 + 0.722261i \(0.743103\pi\)
\(828\) 0 0
\(829\) 7.63932 0.265325 0.132662 0.991161i \(-0.457647\pi\)
0.132662 + 0.991161i \(0.457647\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0.291796i 0.0101162i
\(833\) − 2.47214i − 0.0856544i
\(834\) 0 0
\(835\) 0 0
\(836\) 1.70820 0.0590795
\(837\) 0 0
\(838\) 16.1803i 0.558941i
\(839\) −30.6525 −1.05824 −0.529120 0.848547i \(-0.677478\pi\)
−0.529120 + 0.848547i \(0.677478\pi\)
\(840\) 0 0
\(841\) −4.00000 −0.137931
\(842\) 8.03444i 0.276885i
\(843\) 0 0
\(844\) 19.4164 0.668340
\(845\) 0 0
\(846\) 0 0
\(847\) 10.9443i 0.376050i
\(848\) − 15.7082i − 0.539422i
\(849\) 0 0
\(850\) 0 0
\(851\) −18.7082 −0.641309
\(852\) 0 0
\(853\) 27.4164i 0.938720i 0.883007 + 0.469360i \(0.155515\pi\)
−0.883007 + 0.469360i \(0.844485\pi\)
\(854\) 6.00000 0.205316
\(855\) 0 0
\(856\) 17.8885 0.611418
\(857\) − 15.8197i − 0.540389i −0.962806 0.270195i \(-0.912912\pi\)
0.962806 0.270195i \(-0.0870881\pi\)
\(858\) 0 0
\(859\) −22.3607 −0.762937 −0.381468 0.924382i \(-0.624581\pi\)
−0.381468 + 0.924382i \(0.624581\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 10.8328i 0.368967i
\(863\) − 18.3475i − 0.624557i −0.949991 0.312278i \(-0.898908\pi\)
0.949991 0.312278i \(-0.101092\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) −17.5279 −0.595621
\(867\) 0 0
\(868\) − 6.00000i − 0.203653i
\(869\) 2.63932 0.0895328
\(870\) 0 0
\(871\) −5.23607 −0.177417
\(872\) 18.8197i 0.637314i
\(873\) 0 0
\(874\) −17.2361 −0.583019
\(875\) 0 0
\(876\) 0 0
\(877\) − 30.3607i − 1.02521i −0.858625 0.512604i \(-0.828681\pi\)
0.858625 0.512604i \(-0.171319\pi\)
\(878\) − 5.12461i − 0.172947i
\(879\) 0 0
\(880\) 0 0
\(881\) −5.81966 −0.196069 −0.0980347 0.995183i \(-0.531256\pi\)
−0.0980347 + 0.995183i \(0.531256\pi\)
\(882\) 0 0
\(883\) − 1.40325i − 0.0472232i −0.999721 0.0236116i \(-0.992483\pi\)
0.999721 0.0236116i \(-0.00751650\pi\)
\(884\) −4.94427 −0.166294
\(885\) 0 0
\(886\) 12.0000 0.403148
\(887\) − 21.3475i − 0.716780i −0.933572 0.358390i \(-0.883326\pi\)
0.933572 0.358390i \(-0.116674\pi\)
\(888\) 0 0
\(889\) −13.6525 −0.457889
\(890\) 0 0
\(891\) 0 0
\(892\) 32.6525i 1.09329i
\(893\) − 8.94427i − 0.299309i
\(894\) 0 0
\(895\) 0 0
\(896\) −11.3820 −0.380245
\(897\) 0 0
\(898\) − 12.6869i − 0.423368i
\(899\) 18.5410 0.618378
\(900\) 0 0
\(901\) 20.9443 0.697755
\(902\) 0.695048i 0.0231426i
\(903\) 0 0
\(904\) 32.2361 1.07216
\(905\) 0 0
\(906\) 0 0
\(907\) − 34.8328i − 1.15660i −0.815823 0.578302i \(-0.803715\pi\)
0.815823 0.578302i \(-0.196285\pi\)
\(908\) 34.6525i 1.14998i
\(909\) 0 0
\(910\) 0 0
\(911\) −0.819660 −0.0271566 −0.0135783 0.999908i \(-0.504322\pi\)
−0.0135783 + 0.999908i \(0.504322\pi\)
\(912\) 0 0
\(913\) 1.81966i 0.0602220i
\(914\) 7.74265 0.256104
\(915\) 0 0
\(916\) 7.23607 0.239086
\(917\) − 16.9443i − 0.559549i
\(918\) 0 0
\(919\) 27.7639 0.915848 0.457924 0.888991i \(-0.348593\pi\)
0.457924 + 0.888991i \(0.348593\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) − 8.76393i − 0.288625i
\(923\) − 10.7639i − 0.354299i
\(924\) 0 0
\(925\) 0 0
\(926\) −8.58359 −0.282074
\(927\) 0 0
\(928\) − 28.0902i − 0.922105i
\(929\) 38.2918 1.25631 0.628157 0.778087i \(-0.283810\pi\)
0.628157 + 0.778087i \(0.283810\pi\)
\(930\) 0 0
\(931\) −4.47214 −0.146568
\(932\) − 12.8541i − 0.421050i
\(933\) 0 0
\(934\) 4.29180 0.140432
\(935\) 0 0
\(936\) 0 0
\(937\) − 35.2361i − 1.15111i −0.817762 0.575556i \(-0.804786\pi\)
0.817762 0.575556i \(-0.195214\pi\)
\(938\) − 2.61803i − 0.0854818i
\(939\) 0 0
\(940\) 0 0
\(941\) 5.23607 0.170691 0.0853455 0.996351i \(-0.472801\pi\)
0.0853455 + 0.996351i \(0.472801\pi\)
\(942\) 0 0
\(943\) 29.7082i 0.967432i
\(944\) 21.7082 0.706542
\(945\) 0 0
\(946\) 0.257354 0.00836731
\(947\) 34.8328i 1.13191i 0.824435 + 0.565957i \(0.191493\pi\)
−0.824435 + 0.565957i \(0.808507\pi\)
\(948\) 0 0
\(949\) 10.8328 0.351648
\(950\) 0 0
\(951\) 0 0
\(952\) − 5.52786i − 0.179159i
\(953\) − 3.47214i − 0.112474i −0.998417 0.0562368i \(-0.982090\pi\)
0.998417 0.0562368i \(-0.0179102\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) −8.94427 −0.289278
\(957\) 0 0
\(958\) 16.1803i 0.522763i
\(959\) −10.9443 −0.353409
\(960\) 0 0
\(961\) −17.2492 −0.556427
\(962\) − 2.29180i − 0.0738905i
\(963\) 0 0
\(964\) −5.70820 −0.183849
\(965\) 0 0
\(966\) 0 0
\(967\) 14.1115i 0.453794i 0.973919 + 0.226897i \(0.0728580\pi\)
−0.973919 + 0.226897i \(0.927142\pi\)
\(968\) 24.4721i 0.786564i
\(969\) 0 0
\(970\) 0 0
\(971\) 18.0000 0.577647 0.288824 0.957382i \(-0.406736\pi\)
0.288824 + 0.957382i \(0.406736\pi\)
\(972\) 0 0
\(973\) 10.6525i 0.341503i
\(974\) −3.56231 −0.114144
\(975\) 0 0
\(976\) −18.0000 −0.576166
\(977\) − 11.4721i − 0.367026i −0.983017 0.183513i \(-0.941253\pi\)
0.983017 0.183513i \(-0.0587470\pi\)
\(978\) 0 0
\(979\) 4.06888 0.130042
\(980\) 0 0
\(981\) 0 0
\(982\) − 3.56231i − 0.113678i
\(983\) 34.5410i 1.10169i 0.834608 + 0.550844i \(0.185694\pi\)
−0.834608 + 0.550844i \(0.814306\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 7.63932 0.243286
\(987\) 0 0
\(988\) 8.94427i 0.284555i
\(989\) 11.0000 0.349780
\(990\) 0 0
\(991\) 13.1803 0.418687 0.209344 0.977842i \(-0.432867\pi\)
0.209344 + 0.977842i \(0.432867\pi\)
\(992\) − 20.8328i − 0.661443i
\(993\) 0 0
\(994\) 5.38197 0.170706
\(995\) 0 0
\(996\) 0 0
\(997\) 45.4164i 1.43835i 0.694828 + 0.719176i \(0.255481\pi\)
−0.694828 + 0.719176i \(0.744519\pi\)
\(998\) 6.83282i 0.216289i
\(999\) 0 0
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 1575.2.d.k.1324.2 4
3.2 odd 2 175.2.b.c.99.3 4
5.2 odd 4 1575.2.a.n.1.2 2
5.3 odd 4 1575.2.a.s.1.1 2
5.4 even 2 inner 1575.2.d.k.1324.3 4
12.11 even 2 2800.2.g.s.449.4 4
15.2 even 4 175.2.a.e.1.1 yes 2
15.8 even 4 175.2.a.d.1.2 2
15.14 odd 2 175.2.b.c.99.2 4
21.20 even 2 1225.2.b.k.99.3 4
60.23 odd 4 2800.2.a.bh.1.1 2
60.47 odd 4 2800.2.a.bp.1.2 2
60.59 even 2 2800.2.g.s.449.1 4
105.62 odd 4 1225.2.a.u.1.1 2
105.83 odd 4 1225.2.a.n.1.2 2
105.104 even 2 1225.2.b.k.99.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
175.2.a.d.1.2 2 15.8 even 4
175.2.a.e.1.1 yes 2 15.2 even 4
175.2.b.c.99.2 4 15.14 odd 2
175.2.b.c.99.3 4 3.2 odd 2
1225.2.a.n.1.2 2 105.83 odd 4
1225.2.a.u.1.1 2 105.62 odd 4
1225.2.b.k.99.2 4 105.104 even 2
1225.2.b.k.99.3 4 21.20 even 2
1575.2.a.n.1.2 2 5.2 odd 4
1575.2.a.s.1.1 2 5.3 odd 4
1575.2.d.k.1324.2 4 1.1 even 1 trivial
1575.2.d.k.1324.3 4 5.4 even 2 inner
2800.2.a.bh.1.1 2 60.23 odd 4
2800.2.a.bp.1.2 2 60.47 odd 4
2800.2.g.s.449.1 4 60.59 even 2
2800.2.g.s.449.4 4 12.11 even 2