Properties

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

$q$-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 \(0.0701190\pi\)
−0.975835 + 0.218508i \(0.929881\pi\)
\(3\) 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 \(-0.945167\pi\)
0.985199 0.171412i \(-0.0548329\pi\)
\(14\) −0.618034 −0.165177
\(15\) 0 0
\(16\) 1.85410 0.463525
\(17\) − 2.47214i − 0.599581i −0.954005 0.299791i \(-0.903083\pi\)
0.954005 0.299791i \(-0.0969168\pi\)
\(18\) 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 \(0.225296\pi\)
−0.759802 + 0.650155i \(0.774704\pi\)
\(24\) 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 \(0.0793129\pi\)
−0.969118 + 0.246598i \(0.920687\pi\)
\(38\) 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 \(-0.957058\pi\)
0.990914 0.134499i \(-0.0429424\pi\)
\(44\) 0.381966 0.0575835
\(45\) 0 0
\(46\) −3.85410 −0.568256
\(47\) 2.00000i 0.291730i 0.989305 + 0.145865i \(0.0465965\pi\)
−0.989305 + 0.145865i \(0.953403\pi\)
\(48\) 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 \(0.197678\pi\)
−0.813283 + 0.581869i \(0.802322\pi\)
\(54\) 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 \(-0.916686\pi\)
0.965942 0.258759i \(-0.0833136\pi\)
\(68\) − 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 \(0.171418\pi\)
−0.858466 + 0.512870i \(0.828582\pi\)
\(74\) −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 \(-0.860962\pi\)
0.906110 0.423043i \(-0.139038\pi\)
\(84\) 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 \(0.0856430\pi\)
−0.964022 + 0.265821i \(0.914357\pi\)
\(98\) − 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 \(-0.862944\pi\)
0.908726 0.417392i \(-0.137056\pi\)
\(104\) 2.76393 0.271026
\(105\) 0 0
\(106\) −5.23607 −0.508572
\(107\) − 8.00000i − 0.773389i −0.922208 0.386695i \(-0.873617\pi\)
0.922208 0.386695i \(-0.126383\pi\)
\(108\) 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 \(-0.762808\pi\)
0.734978 0.678091i \(-0.237192\pi\)
\(114\) 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 \(0.207119\pi\)
−0.795670 + 0.605731i \(0.792881\pi\)
\(128\) 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 \(0.154851\pi\)
−0.883985 + 0.467516i \(0.845149\pi\)
\(138\) 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 \(0.00970493\pi\)
−0.999535 + 0.0304842i \(0.990295\pi\)
\(158\) 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 \(0.0190577\pi\)
−0.998208 + 0.0598358i \(0.980942\pi\)
\(164\) −7.70820 −0.601910
\(165\) 0 0
\(166\) 4.76393 0.369753
\(167\) − 5.23607i − 0.405179i −0.979264 0.202590i \(-0.935064\pi\)
0.979264 0.202590i \(-0.0649357\pi\)
\(168\) 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 \(-0.855608\pi\)
0.898866 0.438224i \(-0.144392\pi\)
\(174\) 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 \(-0.852536\pi\)
0.894596 0.446876i \(-0.147464\pi\)
\(194\) −3.23607 −0.232336
\(195\) 0 0
\(196\) −1.61803 −0.115574
\(197\) 1.47214i 0.104885i 0.998624 + 0.0524427i \(0.0167007\pi\)
−0.998624 + 0.0524427i \(0.983299\pi\)
\(198\) 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 \(-0.763847\pi\)
0.737188 0.675688i \(-0.236153\pi\)
\(224\) −5.61803 −0.375371
\(225\) 0 0
\(226\) 8.90983 0.592673
\(227\) − 21.4164i − 1.42146i −0.703466 0.710728i \(-0.748365\pi\)
0.703466 0.710728i \(-0.251635\pi\)
\(228\) 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 \(0.0837962\pi\)
−0.965548 + 0.260223i \(0.916204\pi\)
\(234\) 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 \(0.129124\pi\)
−0.918844 + 0.394620i \(0.870876\pi\)
\(258\) 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 \(0.166880\pi\)
−0.865691 + 0.500579i \(0.833120\pi\)
\(264\) 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 \(-0.796163\pi\)
0.801874 0.597493i \(-0.203837\pi\)
\(278\) − 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 \(-0.826807\pi\)
0.855593 0.517649i \(-0.173193\pi\)
\(284\) −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 \(-0.636725\pi\)
0.416448 0.909160i \(-0.363275\pi\)
\(294\) 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 \(0.0417545\pi\)
−0.991409 + 0.130800i \(0.958246\pi\)
\(308\) 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 \(-0.813907\pi\)
0.833917 0.551890i \(-0.186093\pi\)
\(314\) −0.472136 −0.0266442
\(315\) 0 0
\(316\) 18.0902 1.01765
\(317\) − 25.3607i − 1.42440i −0.701978 0.712199i \(-0.747699\pi\)
0.701978 0.712199i \(-0.252301\pi\)
\(318\) 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 \(-0.851906\pi\)
0.893709 0.448647i \(-0.148094\pi\)
\(338\) 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 \(-0.817226\pi\)
0.839626 0.543165i \(-0.182774\pi\)
\(348\) 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 \(-0.981520\pi\)
0.998315 0.0580239i \(-0.0184800\pi\)
\(354\) 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 \(-0.579536\pi\)
0.247278 0.968944i \(-0.420464\pi\)
\(368\) 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 \(0.435365\pi\)
−0.201665 + 0.979454i \(0.564635\pi\)
\(374\) 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 \(-0.677120\pi\)
0.528165 0.849142i \(-0.322880\pi\)
\(384\) 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 \(0.0729725\pi\)
−0.973837 + 0.227247i \(0.927028\pi\)
\(398\) − 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 \(0.238656\pi\)
−0.731852 + 0.681464i \(0.761344\pi\)
\(434\) −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 \(-0.847401\pi\)
0.887270 0.461251i \(-0.152599\pi\)
\(444\) 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 \(-0.905342\pi\)
0.956108 0.293014i \(-0.0946584\pi\)
\(458\) 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 \(0.104600\pi\)
−0.946492 + 0.322728i \(0.895400\pi\)
\(464\) 9.27051 0.430373
\(465\) 0 0
\(466\) −4.90983 −0.227443
\(467\) − 6.94427i − 0.321343i −0.987008 0.160671i \(-0.948634\pi\)
0.987008 0.160671i \(-0.0513659\pi\)
\(468\) 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 \(0.0416885\pi\)
−0.991436 + 0.130594i \(0.958311\pi\)
\(488\) − 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 \(-0.942120\pi\)
0.983513 0.180836i \(-0.0578803\pi\)
\(504\) 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 \(-0.883561\pi\)
0.933836 0.357701i \(-0.116439\pi\)
\(524\) 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 \(-0.933064\pi\)
0.977972 0.208738i \(-0.0669355\pi\)
\(548\) 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 \(0.0618322\pi\)
−0.981192 + 0.193032i \(0.938168\pi\)
\(558\) 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 \(0.119618\pi\)
−0.930218 + 0.367007i \(0.880382\pi\)
\(564\) 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 \(-0.986745\pi\)
0.999133 0.0416305i \(-0.0132552\pi\)
\(578\) 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 \(0.170746\pi\)
−0.859546 + 0.511058i \(0.829254\pi\)
\(588\) 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 \(-0.722261\pi\)
0.642882 0.765965i \(-0.277739\pi\)
\(594\) 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 \(-0.953814\pi\)
0.989492 0.144590i \(-0.0461862\pi\)
\(608\) 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 \(0.354245\pi\)
−0.442069 + 0.896981i \(0.645755\pi\)
\(614\) −2.83282 −0.114323
\(615\) 0 0
\(616\) −0.527864 −0.0212682
\(617\) 5.94427i 0.239307i 0.992816 + 0.119654i \(0.0381784\pi\)
−0.992816 + 0.119654i \(0.961822\pi\)
\(618\) 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 \(-0.881331\pi\)
0.931307 0.364235i \(-0.118669\pi\)
\(644\) −10.0902 −0.397608
\(645\) 0 0
\(646\) 6.83282 0.268834
\(647\) 19.8885i 0.781899i 0.920412 + 0.390950i \(0.127853\pi\)
−0.920412 + 0.390950i \(0.872147\pi\)
\(648\) 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 \(0.163096\pi\)
−0.871580 + 0.490253i \(0.836904\pi\)
\(654\) 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 \(-0.877171\pi\)
0.926469 0.376372i \(-0.122829\pi\)
\(674\) 10.1803 0.392132
\(675\) 0 0
\(676\) 18.5623 0.713935
\(677\) 14.3607i 0.551926i 0.961168 + 0.275963i \(0.0889967\pi\)
−0.961168 + 0.275963i \(0.911003\pi\)
\(678\) 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 \(0.0871003\pi\)
−0.962795 + 0.270232i \(0.912900\pi\)
\(684\) 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 \(-0.981953\pi\)
0.998393 0.0566653i \(-0.0180468\pi\)
\(728\) 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 \(-0.976464\pi\)
0.997268 0.0738717i \(-0.0235355\pi\)
\(734\) 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 \(-0.938473\pi\)
0.981377 0.192093i \(-0.0615274\pi\)
\(744\) 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 \(0.115822\pi\)
−0.934528 + 0.355889i \(0.884178\pi\)
\(758\) 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 \(0.358219\pi\)
−0.430835 + 0.902431i \(0.641781\pi\)
\(774\) 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 \(0.258872\pi\)
−0.687126 + 0.726539i \(0.741128\pi\)
\(788\) 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 \(0.215822\pi\)
−0.778813 + 0.627257i \(0.784178\pi\)
\(798\) 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 \(0.186008\pi\)
−0.834064 + 0.551668i \(0.813992\pi\)
\(824\) 18.9443 0.659955
\(825\) 0 0
\(826\) −7.23607 −0.251775
\(827\) − 41.5410i − 1.44452i −0.691620 0.722261i \(-0.743103\pi\)
0.691620 0.722261i \(-0.256897\pi\)
\(828\) 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 \(-0.844485\pi\)
0.883007 0.469360i \(-0.155515\pi\)
\(854\) 6.00000 0.205316
\(855\) 0 0
\(856\) 17.8885 0.611418
\(857\) 15.8197i 0.540389i 0.962806 + 0.270195i \(0.0870881\pi\)
−0.962806 + 0.270195i \(0.912912\pi\)
\(858\) 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 \(0.101092\pi\)
−0.949991 + 0.312278i \(0.898908\pi\)
\(864\) 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 \(0.171319\pi\)
−0.858625 + 0.512604i \(0.828681\pi\)
\(878\) 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 \(0.00751650\pi\)
−0.999721 + 0.0236116i \(0.992483\pi\)
\(884\) −4.94427 −0.166294
\(885\) 0 0
\(886\) 12.0000 0.403148
\(887\) 21.3475i 0.716780i 0.933572 + 0.358390i \(0.116674\pi\)
−0.933572 + 0.358390i \(0.883326\pi\)
\(888\) 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 \(0.196285\pi\)
−0.815823 + 0.578302i \(0.803715\pi\)
\(908\) − 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 \(0.195214\pi\)
−0.817762 + 0.575556i \(0.804786\pi\)
\(938\) 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 \(-0.808507\pi\)
0.824435 0.565957i \(-0.191493\pi\)
\(948\) 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 \(0.0179102\pi\)
−0.998417 + 0.0562368i \(0.982090\pi\)
\(954\) 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 \(-0.927142\pi\)
0.973919 0.226897i \(-0.0728580\pi\)
\(968\) − 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 \(0.0587470\pi\)
−0.983017 + 0.183513i \(0.941253\pi\)
\(978\) 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 \(-0.814306\pi\)
0.834608 0.550844i \(-0.185694\pi\)
\(984\) 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 \(-0.744519\pi\)
0.694828 0.719176i \(-0.255481\pi\)
\(998\) − 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.1324.3 4
3.2 odd 2 175.2.b.c.99.2 4
5.2 odd 4 1575.2.a.s.1.1 2
5.3 odd 4 1575.2.a.n.1.2 2
5.4 even 2 inner 1575.2.d.k.1324.2 4
12.11 even 2 2800.2.g.s.449.1 4
15.2 even 4 175.2.a.d.1.2 2
15.8 even 4 175.2.a.e.1.1 yes 2
15.14 odd 2 175.2.b.c.99.3 4
21.20 even 2 1225.2.b.k.99.2 4
60.23 odd 4 2800.2.a.bp.1.2 2
60.47 odd 4 2800.2.a.bh.1.1 2
60.59 even 2 2800.2.g.s.449.4 4
105.62 odd 4 1225.2.a.n.1.2 2
105.83 odd 4 1225.2.a.u.1.1 2
105.104 even 2 1225.2.b.k.99.3 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
175.2.a.d.1.2 2 15.2 even 4
175.2.a.e.1.1 yes 2 15.8 even 4
175.2.b.c.99.2 4 3.2 odd 2
175.2.b.c.99.3 4 15.14 odd 2
1225.2.a.n.1.2 2 105.62 odd 4
1225.2.a.u.1.1 2 105.83 odd 4
1225.2.b.k.99.2 4 21.20 even 2
1225.2.b.k.99.3 4 105.104 even 2
1575.2.a.n.1.2 2 5.3 odd 4
1575.2.a.s.1.1 2 5.2 odd 4
1575.2.d.k.1324.2 4 5.4 even 2 inner
1575.2.d.k.1324.3 4 1.1 even 1 trivial
2800.2.a.bh.1.1 2 60.47 odd 4
2800.2.a.bp.1.2 2 60.23 odd 4
2800.2.g.s.449.1 4 12.11 even 2
2800.2.g.s.449.4 4 60.59 even 2