Properties

Label 1575.2.a.s.1.1
Level $1575$
Weight $2$
Character 1575.1
Self dual yes
Analytic conductor $12.576$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 1575 = 3^{2} \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1575.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(12.5764383184\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{10})^+\)
Defining polynomial: \(x^{2} - x - 1\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 175)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-0.618034\) of defining polynomial
Character \(\chi\) \(=\) 1575.1

$q$-expansion

\(f(q)\) \(=\) \(q-0.618034 q^{2} -1.61803 q^{4} -1.00000 q^{7} +2.23607 q^{8} +O(q^{10})\) \(q-0.618034 q^{2} -1.61803 q^{4} -1.00000 q^{7} +2.23607 q^{8} +0.236068 q^{11} -1.23607 q^{13} +0.618034 q^{14} +1.85410 q^{16} +2.47214 q^{17} -4.47214 q^{19} -0.145898 q^{22} +6.23607 q^{23} +0.763932 q^{26} +1.61803 q^{28} -5.00000 q^{29} +3.70820 q^{31} -5.61803 q^{32} -1.52786 q^{34} -3.00000 q^{37} +2.76393 q^{38} -4.76393 q^{41} -1.76393 q^{43} -0.381966 q^{44} -3.85410 q^{46} -2.00000 q^{47} +1.00000 q^{49} +2.00000 q^{52} +8.47214 q^{53} -2.23607 q^{56} +3.09017 q^{58} -11.7082 q^{59} -9.70820 q^{61} -2.29180 q^{62} -0.236068 q^{64} +4.23607 q^{67} -4.00000 q^{68} -8.70820 q^{71} +8.76393 q^{73} +1.85410 q^{74} +7.23607 q^{76} -0.236068 q^{77} -11.1803 q^{79} +2.94427 q^{82} -7.70820 q^{83} +1.09017 q^{86} +0.527864 q^{88} -17.2361 q^{89} +1.23607 q^{91} -10.0902 q^{92} +1.23607 q^{94} -5.23607 q^{97} -0.618034 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + q^{2} - q^{4} - 2q^{7} + O(q^{10}) \) \( 2q + q^{2} - q^{4} - 2q^{7} - 4q^{11} + 2q^{13} - q^{14} - 3q^{16} - 4q^{17} - 7q^{22} + 8q^{23} + 6q^{26} + q^{28} - 10q^{29} - 6q^{31} - 9q^{32} - 12q^{34} - 6q^{37} + 10q^{38} - 14q^{41} - 8q^{43} - 3q^{44} - q^{46} - 4q^{47} + 2q^{49} + 4q^{52} + 8q^{53} - 5q^{58} - 10q^{59} - 6q^{61} - 18q^{62} + 4q^{64} + 4q^{67} - 8q^{68} - 4q^{71} + 22q^{73} - 3q^{74} + 10q^{76} + 4q^{77} - 12q^{82} - 2q^{83} - 9q^{86} + 10q^{88} - 30q^{89} - 2q^{91} - 9q^{92} - 2q^{94} - 6q^{97} + q^{98} + O(q^{100}) \)

Coefficient data

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

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.618034 −0.437016 −0.218508 0.975835i \(-0.570119\pi\)
−0.218508 + 0.975835i \(0.570119\pi\)
\(3\) 0 0
\(4\) −1.61803 −0.809017
\(5\) 0 0
\(6\) 0 0
\(7\) −1.00000 −0.377964
\(8\) 2.23607 0.790569
\(9\) 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
\(13\) −1.23607 −0.342824 −0.171412 0.985199i \(-0.554833\pi\)
−0.171412 + 0.985199i \(0.554833\pi\)
\(14\) 0.618034 0.165177
\(15\) 0 0
\(16\) 1.85410 0.463525
\(17\) 2.47214 0.599581 0.299791 0.954005i \(-0.403083\pi\)
0.299791 + 0.954005i \(0.403083\pi\)
\(18\) 0 0
\(19\) −4.47214 −1.02598 −0.512989 0.858395i \(-0.671462\pi\)
−0.512989 + 0.858395i \(0.671462\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) −0.145898 −0.0311056
\(23\) 6.23607 1.30031 0.650155 0.759802i \(-0.274704\pi\)
0.650155 + 0.759802i \(0.274704\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0.763932 0.149819
\(27\) 0 0
\(28\) 1.61803 0.305780
\(29\) −5.00000 −0.928477 −0.464238 0.885710i \(-0.653672\pi\)
−0.464238 + 0.885710i \(0.653672\pi\)
\(30\) 0 0
\(31\) 3.70820 0.666013 0.333007 0.942925i \(-0.391937\pi\)
0.333007 + 0.942925i \(0.391937\pi\)
\(32\) −5.61803 −0.993137
\(33\) 0 0
\(34\) −1.52786 −0.262027
\(35\) 0 0
\(36\) 0 0
\(37\) −3.00000 −0.493197 −0.246598 0.969118i \(-0.579313\pi\)
−0.246598 + 0.969118i \(0.579313\pi\)
\(38\) 2.76393 0.448369
\(39\) 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
\(43\) −1.76393 −0.268997 −0.134499 0.990914i \(-0.542942\pi\)
−0.134499 + 0.990914i \(0.542942\pi\)
\(44\) −0.381966 −0.0575835
\(45\) 0 0
\(46\) −3.85410 −0.568256
\(47\) −2.00000 −0.291730 −0.145865 0.989305i \(-0.546597\pi\)
−0.145865 + 0.989305i \(0.546597\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) 0 0
\(52\) 2.00000 0.277350
\(53\) 8.47214 1.16374 0.581869 0.813283i \(-0.302322\pi\)
0.581869 + 0.813283i \(0.302322\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) −2.23607 −0.298807
\(57\) 0 0
\(58\) 3.09017 0.405759
\(59\) −11.7082 −1.52428 −0.762139 0.647413i \(-0.775851\pi\)
−0.762139 + 0.647413i \(0.775851\pi\)
\(60\) 0 0
\(61\) −9.70820 −1.24301 −0.621504 0.783411i \(-0.713478\pi\)
−0.621504 + 0.783411i \(0.713478\pi\)
\(62\) −2.29180 −0.291058
\(63\) 0 0
\(64\) −0.236068 −0.0295085
\(65\) 0 0
\(66\) 0 0
\(67\) 4.23607 0.517518 0.258759 0.965942i \(-0.416686\pi\)
0.258759 + 0.965942i \(0.416686\pi\)
\(68\) −4.00000 −0.485071
\(69\) 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
\(73\) 8.76393 1.02574 0.512870 0.858466i \(-0.328582\pi\)
0.512870 + 0.858466i \(0.328582\pi\)
\(74\) 1.85410 0.215535
\(75\) 0 0
\(76\) 7.23607 0.830034
\(77\) −0.236068 −0.0269024
\(78\) 0 0
\(79\) −11.1803 −1.25789 −0.628943 0.777451i \(-0.716512\pi\)
−0.628943 + 0.777451i \(0.716512\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 2.94427 0.325140
\(83\) −7.70820 −0.846085 −0.423043 0.906110i \(-0.639038\pi\)
−0.423043 + 0.906110i \(0.639038\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 1.09017 0.117556
\(87\) 0 0
\(88\) 0.527864 0.0562705
\(89\) −17.2361 −1.82702 −0.913510 0.406817i \(-0.866639\pi\)
−0.913510 + 0.406817i \(0.866639\pi\)
\(90\) 0 0
\(91\) 1.23607 0.129575
\(92\) −10.0902 −1.05197
\(93\) 0 0
\(94\) 1.23607 0.127491
\(95\) 0 0
\(96\) 0 0
\(97\) −5.23607 −0.531642 −0.265821 0.964022i \(-0.585643\pi\)
−0.265821 + 0.964022i \(0.585643\pi\)
\(98\) −0.618034 −0.0624309
\(99\) 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
\(103\) −8.47214 −0.834784 −0.417392 0.908726i \(-0.637056\pi\)
−0.417392 + 0.908726i \(0.637056\pi\)
\(104\) −2.76393 −0.271026
\(105\) 0 0
\(106\) −5.23607 −0.508572
\(107\) 8.00000 0.773389 0.386695 0.922208i \(-0.373617\pi\)
0.386695 + 0.922208i \(0.373617\pi\)
\(108\) 0 0
\(109\) 8.41641 0.806146 0.403073 0.915168i \(-0.367942\pi\)
0.403073 + 0.915168i \(0.367942\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) −1.85410 −0.175196
\(113\) −14.4164 −1.35618 −0.678091 0.734978i \(-0.737192\pi\)
−0.678091 + 0.734978i \(0.737192\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 8.09017 0.751153
\(117\) 0 0
\(118\) 7.23607 0.666134
\(119\) −2.47214 −0.226620
\(120\) 0 0
\(121\) −10.9443 −0.994934
\(122\) 6.00000 0.543214
\(123\) 0 0
\(124\) −6.00000 −0.538816
\(125\) 0 0
\(126\) 0 0
\(127\) −13.6525 −1.21146 −0.605731 0.795670i \(-0.707119\pi\)
−0.605731 + 0.795670i \(0.707119\pi\)
\(128\) 11.3820 1.00603
\(129\) 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
\(133\) 4.47214 0.387783
\(134\) −2.61803 −0.226164
\(135\) 0 0
\(136\) 5.52786 0.474010
\(137\) −10.9443 −0.935032 −0.467516 0.883985i \(-0.654851\pi\)
−0.467516 + 0.883985i \(0.654851\pi\)
\(138\) 0 0
\(139\) 10.6525 0.903531 0.451766 0.892137i \(-0.350794\pi\)
0.451766 + 0.892137i \(0.350794\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 5.38197 0.451645
\(143\) −0.291796 −0.0244012
\(144\) 0 0
\(145\) 0 0
\(146\) −5.41641 −0.448265
\(147\) 0 0
\(148\) 4.85410 0.399005
\(149\) −3.94427 −0.323127 −0.161564 0.986862i \(-0.551654\pi\)
−0.161564 + 0.986862i \(0.551654\pi\)
\(150\) 0 0
\(151\) −20.2361 −1.64679 −0.823394 0.567470i \(-0.807922\pi\)
−0.823394 + 0.567470i \(0.807922\pi\)
\(152\) −10.0000 −0.811107
\(153\) 0 0
\(154\) 0.145898 0.0117568
\(155\) 0 0
\(156\) 0 0
\(157\) −0.763932 −0.0609684 −0.0304842 0.999535i \(-0.509705\pi\)
−0.0304842 + 0.999535i \(0.509705\pi\)
\(158\) 6.90983 0.549717
\(159\) 0 0
\(160\) 0 0
\(161\) −6.23607 −0.491471
\(162\) 0 0
\(163\) 1.52786 0.119672 0.0598358 0.998208i \(-0.480942\pi\)
0.0598358 + 0.998208i \(0.480942\pi\)
\(164\) 7.70820 0.601910
\(165\) 0 0
\(166\) 4.76393 0.369753
\(167\) 5.23607 0.405179 0.202590 0.979264i \(-0.435064\pi\)
0.202590 + 0.979264i \(0.435064\pi\)
\(168\) 0 0
\(169\) −11.4721 −0.882472
\(170\) 0 0
\(171\) 0 0
\(172\) 2.85410 0.217623
\(173\) −11.5279 −0.876447 −0.438224 0.898866i \(-0.644392\pi\)
−0.438224 + 0.898866i \(0.644392\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0.437694 0.0329924
\(177\) 0 0
\(178\) 10.6525 0.798437
\(179\) 23.4164 1.75022 0.875112 0.483920i \(-0.160787\pi\)
0.875112 + 0.483920i \(0.160787\pi\)
\(180\) 0 0
\(181\) 8.18034 0.608040 0.304020 0.952666i \(-0.401671\pi\)
0.304020 + 0.952666i \(0.401671\pi\)
\(182\) −0.763932 −0.0566264
\(183\) 0 0
\(184\) 13.9443 1.02799
\(185\) 0 0
\(186\) 0 0
\(187\) 0.583592 0.0426765
\(188\) 3.23607 0.236015
\(189\) 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
\(193\) −12.4164 −0.893753 −0.446876 0.894596i \(-0.647464\pi\)
−0.446876 + 0.894596i \(0.647464\pi\)
\(194\) 3.23607 0.232336
\(195\) 0 0
\(196\) −1.61803 −0.115574
\(197\) −1.47214 −0.104885 −0.0524427 0.998624i \(-0.516701\pi\)
−0.0524427 + 0.998624i \(0.516701\pi\)
\(198\) 0 0
\(199\) 7.23607 0.512951 0.256476 0.966551i \(-0.417439\pi\)
0.256476 + 0.966551i \(0.417439\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 2.94427 0.207158
\(203\) 5.00000 0.350931
\(204\) 0 0
\(205\) 0 0
\(206\) 5.23607 0.364814
\(207\) 0 0
\(208\) −2.29180 −0.158907
\(209\) −1.05573 −0.0730262
\(210\) 0 0
\(211\) 12.0000 0.826114 0.413057 0.910705i \(-0.364461\pi\)
0.413057 + 0.910705i \(0.364461\pi\)
\(212\) −13.7082 −0.941483
\(213\) 0 0
\(214\) −4.94427 −0.337983
\(215\) 0 0
\(216\) 0 0
\(217\) −3.70820 −0.251729
\(218\) −5.20163 −0.352299
\(219\) 0 0
\(220\) 0 0
\(221\) −3.05573 −0.205551
\(222\) 0 0
\(223\) −20.1803 −1.35138 −0.675688 0.737188i \(-0.736153\pi\)
−0.675688 + 0.737188i \(0.736153\pi\)
\(224\) 5.61803 0.375371
\(225\) 0 0
\(226\) 8.90983 0.592673
\(227\) 21.4164 1.42146 0.710728 0.703466i \(-0.248365\pi\)
0.710728 + 0.703466i \(0.248365\pi\)
\(228\) 0 0
\(229\) −4.47214 −0.295527 −0.147764 0.989023i \(-0.547207\pi\)
−0.147764 + 0.989023i \(0.547207\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) −11.1803 −0.734025
\(233\) 7.94427 0.520447 0.260223 0.965548i \(-0.416204\pi\)
0.260223 + 0.965548i \(0.416204\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 18.9443 1.23317
\(237\) 0 0
\(238\) 1.52786 0.0990367
\(239\) 5.52786 0.357568 0.178784 0.983888i \(-0.442784\pi\)
0.178784 + 0.983888i \(0.442784\pi\)
\(240\) 0 0
\(241\) −3.52786 −0.227250 −0.113625 0.993524i \(-0.536246\pi\)
−0.113625 + 0.993524i \(0.536246\pi\)
\(242\) 6.76393 0.434802
\(243\) 0 0
\(244\) 15.7082 1.00561
\(245\) 0 0
\(246\) 0 0
\(247\) 5.52786 0.351730
\(248\) 8.29180 0.526530
\(249\) 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
\(253\) 1.47214 0.0925524
\(254\) 8.43769 0.529428
\(255\) 0 0
\(256\) −6.56231 −0.410144
\(257\) −12.6525 −0.789240 −0.394620 0.918844i \(-0.629124\pi\)
−0.394620 + 0.918844i \(0.629124\pi\)
\(258\) 0 0
\(259\) 3.00000 0.186411
\(260\) 0 0
\(261\) 0 0
\(262\) −10.4721 −0.646971
\(263\) 16.2361 1.00116 0.500579 0.865691i \(-0.333120\pi\)
0.500579 + 0.865691i \(0.333120\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) −2.76393 −0.169468
\(267\) 0 0
\(268\) −6.85410 −0.418681
\(269\) 11.7082 0.713862 0.356931 0.934131i \(-0.383823\pi\)
0.356931 + 0.934131i \(0.383823\pi\)
\(270\) 0 0
\(271\) 23.7082 1.44017 0.720085 0.693885i \(-0.244103\pi\)
0.720085 + 0.693885i \(0.244103\pi\)
\(272\) 4.58359 0.277921
\(273\) 0 0
\(274\) 6.76393 0.408624
\(275\) 0 0
\(276\) 0 0
\(277\) 19.8885 1.19499 0.597493 0.801874i \(-0.296163\pi\)
0.597493 + 0.801874i \(0.296163\pi\)
\(278\) −6.58359 −0.394858
\(279\) 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
\(283\) −17.4164 −1.03530 −0.517649 0.855593i \(-0.673193\pi\)
−0.517649 + 0.855593i \(0.673193\pi\)
\(284\) 14.0902 0.836098
\(285\) 0 0
\(286\) 0.180340 0.0106637
\(287\) 4.76393 0.281206
\(288\) 0 0
\(289\) −10.8885 −0.640503
\(290\) 0 0
\(291\) 0 0
\(292\) −14.1803 −0.829842
\(293\) −31.1246 −1.81832 −0.909160 0.416448i \(-0.863275\pi\)
−0.909160 + 0.416448i \(0.863275\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) −6.70820 −0.389906
\(297\) 0 0
\(298\) 2.43769 0.141212
\(299\) −7.70820 −0.445777
\(300\) 0 0
\(301\) 1.76393 0.101671
\(302\) 12.5066 0.719673
\(303\) 0 0
\(304\) −8.29180 −0.475567
\(305\) 0 0
\(306\) 0 0
\(307\) −4.58359 −0.261599 −0.130800 0.991409i \(-0.541754\pi\)
−0.130800 + 0.991409i \(0.541754\pi\)
\(308\) 0.381966 0.0217645
\(309\) 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
\(313\) −19.5279 −1.10378 −0.551890 0.833917i \(-0.686093\pi\)
−0.551890 + 0.833917i \(0.686093\pi\)
\(314\) 0.472136 0.0266442
\(315\) 0 0
\(316\) 18.0902 1.01765
\(317\) 25.3607 1.42440 0.712199 0.701978i \(-0.247699\pi\)
0.712199 + 0.701978i \(0.247699\pi\)
\(318\) 0 0
\(319\) −1.18034 −0.0660863
\(320\) 0 0
\(321\) 0 0
\(322\) 3.85410 0.214781
\(323\) −11.0557 −0.615157
\(324\) 0 0
\(325\) 0 0
\(326\) −0.944272 −0.0522984
\(327\) 0 0
\(328\) −10.6525 −0.588185
\(329\) 2.00000 0.110264
\(330\) 0 0
\(331\) −24.7082 −1.35809 −0.679043 0.734099i \(-0.737605\pi\)
−0.679043 + 0.734099i \(0.737605\pi\)
\(332\) 12.4721 0.684497
\(333\) 0 0
\(334\) −3.23607 −0.177070
\(335\) 0 0
\(336\) 0 0
\(337\) 16.4721 0.897294 0.448647 0.893709i \(-0.351906\pi\)
0.448647 + 0.893709i \(0.351906\pi\)
\(338\) 7.09017 0.385654
\(339\) 0 0
\(340\) 0 0
\(341\) 0.875388 0.0474049
\(342\) 0 0
\(343\) −1.00000 −0.0539949
\(344\) −3.94427 −0.212661
\(345\) 0 0
\(346\) 7.12461 0.383022
\(347\) 20.2361 1.08633 0.543165 0.839626i \(-0.317226\pi\)
0.543165 + 0.839626i \(0.317226\pi\)
\(348\) 0 0
\(349\) 4.47214 0.239388 0.119694 0.992811i \(-0.461809\pi\)
0.119694 + 0.992811i \(0.461809\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) −1.32624 −0.0706887
\(353\) −2.18034 −0.116048 −0.0580239 0.998315i \(-0.518480\pi\)
−0.0580239 + 0.998315i \(0.518480\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 27.8885 1.47809
\(357\) 0 0
\(358\) −14.4721 −0.764876
\(359\) 30.1246 1.58992 0.794958 0.606664i \(-0.207493\pi\)
0.794958 + 0.606664i \(0.207493\pi\)
\(360\) 0 0
\(361\) 1.00000 0.0526316
\(362\) −5.05573 −0.265723
\(363\) 0 0
\(364\) −2.00000 −0.104828
\(365\) 0 0
\(366\) 0 0
\(367\) 37.1246 1.93789 0.968944 0.247278i \(-0.0795362\pi\)
0.968944 + 0.247278i \(0.0795362\pi\)
\(368\) 11.5623 0.602727
\(369\) 0 0
\(370\) 0 0
\(371\) −8.47214 −0.439851
\(372\) 0 0
\(373\) 37.8328 1.95891 0.979454 0.201665i \(-0.0646354\pi\)
0.979454 + 0.201665i \(0.0646354\pi\)
\(374\) −0.360680 −0.0186503
\(375\) 0 0
\(376\) −4.47214 −0.230633
\(377\) 6.18034 0.318304
\(378\) 0 0
\(379\) −11.1803 −0.574295 −0.287148 0.957886i \(-0.592707\pi\)
−0.287148 + 0.957886i \(0.592707\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 4.00000 0.204658
\(383\) −33.2361 −1.69828 −0.849142 0.528165i \(-0.822880\pi\)
−0.849142 + 0.528165i \(0.822880\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 7.67376 0.390584
\(387\) 0 0
\(388\) 8.47214 0.430108
\(389\) −2.88854 −0.146455 −0.0732275 0.997315i \(-0.523330\pi\)
−0.0732275 + 0.997315i \(0.523330\pi\)
\(390\) 0 0
\(391\) 15.4164 0.779641
\(392\) 2.23607 0.112938
\(393\) 0 0
\(394\) 0.909830 0.0458366
\(395\) 0 0
\(396\) 0 0
\(397\) −9.05573 −0.454494 −0.227247 0.973837i \(-0.572972\pi\)
−0.227247 + 0.973837i \(0.572972\pi\)
\(398\) −4.47214 −0.224168
\(399\) 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
\(403\) −4.58359 −0.228325
\(404\) 7.70820 0.383497
\(405\) 0 0
\(406\) −3.09017 −0.153363
\(407\) −0.708204 −0.0351044
\(408\) 0 0
\(409\) 24.4721 1.21007 0.605035 0.796199i \(-0.293159\pi\)
0.605035 + 0.796199i \(0.293159\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 13.7082 0.675355
\(413\) 11.7082 0.576123
\(414\) 0 0
\(415\) 0 0
\(416\) 6.94427 0.340471
\(417\) 0 0
\(418\) 0.652476 0.0319136
\(419\) 26.1803 1.27899 0.639497 0.768794i \(-0.279143\pi\)
0.639497 + 0.768794i \(0.279143\pi\)
\(420\) 0 0
\(421\) −13.0000 −0.633581 −0.316791 0.948495i \(-0.602605\pi\)
−0.316791 + 0.948495i \(0.602605\pi\)
\(422\) −7.41641 −0.361025
\(423\) 0 0
\(424\) 18.9443 0.920015
\(425\) 0 0
\(426\) 0 0
\(427\) 9.70820 0.469813
\(428\) −12.9443 −0.625685
\(429\) 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
\(433\) 28.3607 1.36293 0.681464 0.731852i \(-0.261344\pi\)
0.681464 + 0.731852i \(0.261344\pi\)
\(434\) 2.29180 0.110010
\(435\) 0 0
\(436\) −13.6180 −0.652186
\(437\) −27.8885 −1.33409
\(438\) 0 0
\(439\) −8.29180 −0.395746 −0.197873 0.980228i \(-0.563403\pi\)
−0.197873 + 0.980228i \(0.563403\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 1.88854 0.0898289
\(443\) −19.4164 −0.922501 −0.461251 0.887270i \(-0.652599\pi\)
−0.461251 + 0.887270i \(0.652599\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 12.4721 0.590573
\(447\) 0 0
\(448\) 0.236068 0.0111532
\(449\) −20.5279 −0.968770 −0.484385 0.874855i \(-0.660957\pi\)
−0.484385 + 0.874855i \(0.660957\pi\)
\(450\) 0 0
\(451\) −1.12461 −0.0529559
\(452\) 23.3262 1.09717
\(453\) 0 0
\(454\) −13.2361 −0.621199
\(455\) 0 0
\(456\) 0 0
\(457\) 12.5279 0.586029 0.293014 0.956108i \(-0.405342\pi\)
0.293014 + 0.956108i \(0.405342\pi\)
\(458\) 2.76393 0.129150
\(459\) 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
\(463\) 13.8885 0.645455 0.322728 0.946492i \(-0.395400\pi\)
0.322728 + 0.946492i \(0.395400\pi\)
\(464\) −9.27051 −0.430373
\(465\) 0 0
\(466\) −4.90983 −0.227443
\(467\) 6.94427 0.321343 0.160671 0.987008i \(-0.448634\pi\)
0.160671 + 0.987008i \(0.448634\pi\)
\(468\) 0 0
\(469\) −4.23607 −0.195603
\(470\) 0 0
\(471\) 0 0
\(472\) −26.1803 −1.20505
\(473\) −0.416408 −0.0191465
\(474\) 0 0
\(475\) 0 0
\(476\) 4.00000 0.183340
\(477\) 0 0
\(478\) −3.41641 −0.156263
\(479\) 26.1803 1.19621 0.598105 0.801418i \(-0.295921\pi\)
0.598105 + 0.801418i \(0.295921\pi\)
\(480\) 0 0
\(481\) 3.70820 0.169080
\(482\) 2.18034 0.0993118
\(483\) 0 0
\(484\) 17.7082 0.804918
\(485\) 0 0
\(486\) 0 0
\(487\) −5.76393 −0.261189 −0.130594 0.991436i \(-0.541689\pi\)
−0.130594 + 0.991436i \(0.541689\pi\)
\(488\) −21.7082 −0.982684
\(489\) 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
\(493\) −12.3607 −0.556697
\(494\) −3.41641 −0.153711
\(495\) 0 0
\(496\) 6.87539 0.308714
\(497\) 8.70820 0.390616
\(498\) 0 0
\(499\) 11.0557 0.494922 0.247461 0.968898i \(-0.420404\pi\)
0.247461 + 0.968898i \(0.420404\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 4.00000 0.178529
\(503\) −8.11146 −0.361672 −0.180836 0.983513i \(-0.557880\pi\)
−0.180836 + 0.983513i \(0.557880\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) −0.909830 −0.0404469
\(507\) 0 0
\(508\) 22.0902 0.980093
\(509\) −40.6525 −1.80189 −0.900945 0.433934i \(-0.857125\pi\)
−0.900945 + 0.433934i \(0.857125\pi\)
\(510\) 0 0
\(511\) −8.76393 −0.387694
\(512\) −18.7082 −0.826794
\(513\) 0 0
\(514\) 7.81966 0.344910
\(515\) 0 0
\(516\) 0 0
\(517\) −0.472136 −0.0207645
\(518\) −1.85410 −0.0814646
\(519\) 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
\(523\) −16.3607 −0.715403 −0.357701 0.933836i \(-0.616439\pi\)
−0.357701 + 0.933836i \(0.616439\pi\)
\(524\) −27.4164 −1.19769
\(525\) 0 0
\(526\) −10.0344 −0.437522
\(527\) 9.16718 0.399329
\(528\) 0 0
\(529\) 15.8885 0.690806
\(530\) 0 0
\(531\) 0 0
\(532\) −7.23607 −0.313723
\(533\) 5.88854 0.255061
\(534\) 0 0
\(535\) 0 0
\(536\) 9.47214 0.409134
\(537\) 0 0
\(538\) −7.23607 −0.311969
\(539\) 0.236068 0.0101682
\(540\) 0 0
\(541\) 15.9443 0.685498 0.342749 0.939427i \(-0.388642\pi\)
0.342749 + 0.939427i \(0.388642\pi\)
\(542\) −14.6525 −0.629378
\(543\) 0 0
\(544\) −13.8885 −0.595466
\(545\) 0 0
\(546\) 0 0
\(547\) 9.76393 0.417476 0.208738 0.977972i \(-0.433064\pi\)
0.208738 + 0.977972i \(0.433064\pi\)
\(548\) 17.7082 0.756457
\(549\) 0 0
\(550\) 0 0
\(551\) 22.3607 0.952597
\(552\) 0 0
\(553\) 11.1803 0.475436
\(554\) −12.2918 −0.522228
\(555\) 0 0
\(556\) −17.2361 −0.730972
\(557\) −9.11146 −0.386065 −0.193032 0.981192i \(-0.561832\pi\)
−0.193032 + 0.981192i \(0.561832\pi\)
\(558\) 0 0
\(559\) 2.18034 0.0922186
\(560\) 0 0
\(561\) 0 0
\(562\) −9.49342 −0.400456
\(563\) 17.4164 0.734014 0.367007 0.930218i \(-0.380382\pi\)
0.367007 + 0.930218i \(0.380382\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 10.7639 0.452442
\(567\) 0 0
\(568\) −19.4721 −0.817033
\(569\) 3.94427 0.165352 0.0826762 0.996576i \(-0.473653\pi\)
0.0826762 + 0.996576i \(0.473653\pi\)
\(570\) 0 0
\(571\) 36.5967 1.53153 0.765763 0.643123i \(-0.222362\pi\)
0.765763 + 0.643123i \(0.222362\pi\)
\(572\) 0.472136 0.0197410
\(573\) 0 0
\(574\) −2.94427 −0.122892
\(575\) 0 0
\(576\) 0 0
\(577\) 2.00000 0.0832611 0.0416305 0.999133i \(-0.486745\pi\)
0.0416305 + 0.999133i \(0.486745\pi\)
\(578\) 6.72949 0.279910
\(579\) 0 0
\(580\) 0 0
\(581\) 7.70820 0.319790
\(582\) 0 0
\(583\) 2.00000 0.0828315
\(584\) 19.5967 0.810919
\(585\) 0 0
\(586\) 19.2361 0.794635
\(587\) −24.7639 −1.02212 −0.511058 0.859546i \(-0.670746\pi\)
−0.511058 + 0.859546i \(0.670746\pi\)
\(588\) 0 0
\(589\) −16.5836 −0.683315
\(590\) 0 0
\(591\) 0 0
\(592\) −5.56231 −0.228609
\(593\) −37.3050 −1.53193 −0.765965 0.642882i \(-0.777739\pi\)
−0.765965 + 0.642882i \(0.777739\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 6.38197 0.261416
\(597\) 0 0
\(598\) 4.76393 0.194812
\(599\) 11.1803 0.456816 0.228408 0.973565i \(-0.426648\pi\)
0.228408 + 0.973565i \(0.426648\pi\)
\(600\) 0 0
\(601\) −36.9443 −1.50699 −0.753494 0.657455i \(-0.771633\pi\)
−0.753494 + 0.657455i \(0.771633\pi\)
\(602\) −1.09017 −0.0444320
\(603\) 0 0
\(604\) 32.7426 1.33228
\(605\) 0 0
\(606\) 0 0
\(607\) 7.12461 0.289179 0.144590 0.989492i \(-0.453814\pi\)
0.144590 + 0.989492i \(0.453814\pi\)
\(608\) 25.1246 1.01894
\(609\) 0 0
\(610\) 0 0
\(611\) 2.47214 0.100012
\(612\) 0 0
\(613\) 44.4164 1.79396 0.896981 0.442069i \(-0.145755\pi\)
0.896981 + 0.442069i \(0.145755\pi\)
\(614\) 2.83282 0.114323
\(615\) 0 0
\(616\) −0.527864 −0.0212682
\(617\) −5.94427 −0.239307 −0.119654 0.992816i \(-0.538178\pi\)
−0.119654 + 0.992816i \(0.538178\pi\)
\(618\) 0 0
\(619\) −11.7082 −0.470592 −0.235296 0.971924i \(-0.575606\pi\)
−0.235296 + 0.971924i \(0.575606\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 15.0557 0.603680
\(623\) 17.2361 0.690548
\(624\) 0 0
\(625\) 0 0
\(626\) 12.0689 0.482370
\(627\) 0 0
\(628\) 1.23607 0.0493245
\(629\) −7.41641 −0.295712
\(630\) 0 0
\(631\) 27.6525 1.10083 0.550414 0.834892i \(-0.314470\pi\)
0.550414 + 0.834892i \(0.314470\pi\)
\(632\) −25.0000 −0.994447
\(633\) 0 0
\(634\) −15.6738 −0.622485
\(635\) 0 0
\(636\) 0 0
\(637\) −1.23607 −0.0489748
\(638\) 0.729490 0.0288808
\(639\) 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
\(643\) −18.4721 −0.728470 −0.364235 0.931307i \(-0.618669\pi\)
−0.364235 + 0.931307i \(0.618669\pi\)
\(644\) 10.0902 0.397608
\(645\) 0 0
\(646\) 6.83282 0.268834
\(647\) −19.8885 −0.781899 −0.390950 0.920412i \(-0.627853\pi\)
−0.390950 + 0.920412i \(0.627853\pi\)
\(648\) 0 0
\(649\) −2.76393 −0.108494
\(650\) 0 0
\(651\) 0 0
\(652\) −2.47214 −0.0968163
\(653\) 25.0557 0.980506 0.490253 0.871580i \(-0.336904\pi\)
0.490253 + 0.871580i \(0.336904\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) −8.83282 −0.344864
\(657\) 0 0
\(658\) −1.23607 −0.0481869
\(659\) −17.8885 −0.696839 −0.348419 0.937339i \(-0.613281\pi\)
−0.348419 + 0.937339i \(0.613281\pi\)
\(660\) 0 0
\(661\) −42.7214 −1.66167 −0.830834 0.556520i \(-0.812136\pi\)
−0.830834 + 0.556520i \(0.812136\pi\)
\(662\) 15.2705 0.593505
\(663\) 0 0
\(664\) −17.2361 −0.668889
\(665\) 0 0
\(666\) 0 0
\(667\) −31.1803 −1.20731
\(668\) −8.47214 −0.327797
\(669\) 0 0
\(670\) 0 0
\(671\) −2.29180 −0.0884738
\(672\) 0 0
\(673\) −19.5279 −0.752744 −0.376372 0.926469i \(-0.622829\pi\)
−0.376372 + 0.926469i \(0.622829\pi\)
\(674\) −10.1803 −0.392132
\(675\) 0 0
\(676\) 18.5623 0.713935
\(677\) −14.3607 −0.551926 −0.275963 0.961168i \(-0.588997\pi\)
−0.275963 + 0.961168i \(0.588997\pi\)
\(678\) 0 0
\(679\) 5.23607 0.200942
\(680\) 0 0
\(681\) 0 0
\(682\) −0.541020 −0.0207167
\(683\) 14.1246 0.540463 0.270232 0.962795i \(-0.412900\pi\)
0.270232 + 0.962795i \(0.412900\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0.618034 0.0235966
\(687\) 0 0
\(688\) −3.27051 −0.124687
\(689\) −10.4721 −0.398957
\(690\) 0 0
\(691\) −4.18034 −0.159028 −0.0795138 0.996834i \(-0.525337\pi\)
−0.0795138 + 0.996834i \(0.525337\pi\)
\(692\) 18.6525 0.709061
\(693\) 0 0
\(694\) −12.5066 −0.474743
\(695\) 0 0
\(696\) 0 0
\(697\) −11.7771 −0.446089
\(698\) −2.76393 −0.104616
\(699\) 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
\(703\) 13.4164 0.506009
\(704\) −0.0557281 −0.00210033
\(705\) 0 0
\(706\) 1.34752 0.0507147
\(707\) 4.76393 0.179166
\(708\) 0 0
\(709\) −12.1115 −0.454855 −0.227428 0.973795i \(-0.573032\pi\)
−0.227428 + 0.973795i \(0.573032\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) −38.5410 −1.44439
\(713\) 23.1246 0.866024
\(714\) 0 0
\(715\) 0 0
\(716\) −37.8885 −1.41596
\(717\) 0 0
\(718\) −18.6180 −0.694819
\(719\) −16.1803 −0.603425 −0.301712 0.953399i \(-0.597558\pi\)
−0.301712 + 0.953399i \(0.597558\pi\)
\(720\) 0 0
\(721\) 8.47214 0.315519
\(722\) −0.618034 −0.0230008
\(723\) 0 0
\(724\) −13.2361 −0.491915
\(725\) 0 0
\(726\) 0 0
\(727\) 3.05573 0.113331 0.0566653 0.998393i \(-0.481953\pi\)
0.0566653 + 0.998393i \(0.481953\pi\)
\(728\) 2.76393 0.102438
\(729\) 0 0
\(730\) 0 0
\(731\) −4.36068 −0.161286
\(732\) 0 0
\(733\) −4.00000 −0.147743 −0.0738717 0.997268i \(-0.523536\pi\)
−0.0738717 + 0.997268i \(0.523536\pi\)
\(734\) −22.9443 −0.846889
\(735\) 0 0
\(736\) −35.0344 −1.29139
\(737\) 1.00000 0.0368355
\(738\) 0 0
\(739\) 25.6525 0.943642 0.471821 0.881694i \(-0.343597\pi\)
0.471821 + 0.881694i \(0.343597\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 5.23607 0.192222
\(743\) −10.4721 −0.384185 −0.192093 0.981377i \(-0.561527\pi\)
−0.192093 + 0.981377i \(0.561527\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) −23.3820 −0.856075
\(747\) 0 0
\(748\) −0.944272 −0.0345260
\(749\) −8.00000 −0.292314
\(750\) 0 0
\(751\) 3.05573 0.111505 0.0557526 0.998445i \(-0.482244\pi\)
0.0557526 + 0.998445i \(0.482244\pi\)
\(752\) −3.70820 −0.135224
\(753\) 0 0
\(754\) −3.81966 −0.139104
\(755\) 0 0
\(756\) 0 0
\(757\) −19.5836 −0.711778 −0.355889 0.934528i \(-0.615822\pi\)
−0.355889 + 0.934528i \(0.615822\pi\)
\(758\) 6.90983 0.250976
\(759\) 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
\(763\) −8.41641 −0.304694
\(764\) 10.4721 0.378869
\(765\) 0 0
\(766\) 20.5410 0.742177
\(767\) 14.4721 0.522559
\(768\) 0 0
\(769\) −43.0132 −1.55109 −0.775547 0.631290i \(-0.782526\pi\)
−0.775547 + 0.631290i \(0.782526\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 20.0902 0.723061
\(773\) 50.1803 1.80486 0.902431 0.430835i \(-0.141781\pi\)
0.902431 + 0.430835i \(0.141781\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) −11.7082 −0.420300
\(777\) 0 0
\(778\) 1.78522 0.0640032
\(779\) 21.3050 0.763329
\(780\) 0 0
\(781\) −2.05573 −0.0735597
\(782\) −9.52786 −0.340716
\(783\) 0 0
\(784\) 1.85410 0.0662179
\(785\) 0 0
\(786\) 0 0
\(787\) −40.7639 −1.45308 −0.726539 0.687126i \(-0.758872\pi\)
−0.726539 + 0.687126i \(0.758872\pi\)
\(788\) 2.38197 0.0848540
\(789\) 0 0
\(790\) 0 0
\(791\) 14.4164 0.512588
\(792\) 0 0
\(793\) 12.0000 0.426132
\(794\) 5.59675 0.198621
\(795\) 0 0
\(796\) −11.7082 −0.414986
\(797\) −35.4164 −1.25451 −0.627257 0.778813i \(-0.715822\pi\)
−0.627257 + 0.778813i \(0.715822\pi\)
\(798\) 0 0
\(799\) −4.94427 −0.174916
\(800\) 0 0
\(801\) 0 0
\(802\) 1.56231 0.0551669
\(803\) 2.06888 0.0730093
\(804\) 0 0
\(805\) 0 0
\(806\) 2.83282 0.0997817
\(807\) 0 0
\(808\) −10.6525 −0.374753
\(809\) −29.4721 −1.03619 −0.518093 0.855325i \(-0.673358\pi\)
−0.518093 + 0.855325i \(0.673358\pi\)
\(810\) 0 0
\(811\) −42.7214 −1.50015 −0.750075 0.661353i \(-0.769983\pi\)
−0.750075 + 0.661353i \(0.769983\pi\)
\(812\) −8.09017 −0.283909
\(813\) 0 0
\(814\) 0.437694 0.0153412
\(815\) 0 0
\(816\) 0 0
\(817\) 7.88854 0.275985
\(818\) −15.1246 −0.528820
\(819\) 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
\(823\) 31.6525 1.10334 0.551668 0.834064i \(-0.313992\pi\)
0.551668 + 0.834064i \(0.313992\pi\)
\(824\) −18.9443 −0.659955
\(825\) 0 0
\(826\) −7.23607 −0.251775
\(827\) 41.5410 1.44452 0.722261 0.691620i \(-0.243103\pi\)
0.722261 + 0.691620i \(0.243103\pi\)
\(828\) 0 0
\(829\) −7.63932 −0.265325 −0.132662 0.991161i \(-0.542353\pi\)
−0.132662 + 0.991161i \(0.542353\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0.291796 0.0101162
\(833\) 2.47214 0.0856544
\(834\) 0 0
\(835\) 0 0
\(836\) 1.70820 0.0590795
\(837\) 0 0
\(838\) −16.1803 −0.558941
\(839\) 30.6525 1.05824 0.529120 0.848547i \(-0.322522\pi\)
0.529120 + 0.848547i \(0.322522\pi\)
\(840\) 0 0
\(841\) −4.00000 −0.137931
\(842\) 8.03444 0.276885
\(843\) 0 0
\(844\) −19.4164 −0.668340
\(845\) 0 0
\(846\) 0 0
\(847\) 10.9443 0.376050
\(848\) 15.7082 0.539422
\(849\) 0 0
\(850\) 0 0
\(851\) −18.7082 −0.641309
\(852\) 0 0
\(853\) −27.4164 −0.938720 −0.469360 0.883007i \(-0.655515\pi\)
−0.469360 + 0.883007i \(0.655515\pi\)
\(854\) −6.00000 −0.205316
\(855\) 0 0
\(856\) 17.8885 0.611418
\(857\) −15.8197 −0.540389 −0.270195 0.962806i \(-0.587088\pi\)
−0.270195 + 0.962806i \(0.587088\pi\)
\(858\) 0 0
\(859\) 22.3607 0.762937 0.381468 0.924382i \(-0.375419\pi\)
0.381468 + 0.924382i \(0.375419\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 10.8328 0.368967
\(863\) 18.3475 0.624557 0.312278 0.949991i \(-0.398908\pi\)
0.312278 + 0.949991i \(0.398908\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) −17.5279 −0.595621
\(867\) 0 0
\(868\) 6.00000 0.203653
\(869\) −2.63932 −0.0895328
\(870\) 0 0
\(871\) −5.23607 −0.177417
\(872\) 18.8197 0.637314
\(873\) 0 0
\(874\) 17.2361 0.583019
\(875\) 0 0
\(876\) 0 0
\(877\) −30.3607 −1.02521 −0.512604 0.858625i \(-0.671319\pi\)
−0.512604 + 0.858625i \(0.671319\pi\)
\(878\) 5.12461 0.172947
\(879\) 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
\(883\) 1.40325 0.0472232 0.0236116 0.999721i \(-0.492483\pi\)
0.0236116 + 0.999721i \(0.492483\pi\)
\(884\) 4.94427 0.166294
\(885\) 0 0
\(886\) 12.0000 0.403148
\(887\) −21.3475 −0.716780 −0.358390 0.933572i \(-0.616674\pi\)
−0.358390 + 0.933572i \(0.616674\pi\)
\(888\) 0 0
\(889\) 13.6525 0.457889
\(890\) 0 0
\(891\) 0 0
\(892\) 32.6525 1.09329
\(893\) 8.94427 0.299309
\(894\) 0 0
\(895\) 0 0
\(896\) −11.3820 −0.380245
\(897\) 0 0
\(898\) 12.6869 0.423368
\(899\) −18.5410 −0.618378
\(900\) 0 0
\(901\) 20.9443 0.697755
\(902\) 0.695048 0.0231426
\(903\) 0 0
\(904\) −32.2361 −1.07216
\(905\) 0 0
\(906\) 0 0
\(907\) −34.8328 −1.15660 −0.578302 0.815823i \(-0.696285\pi\)
−0.578302 + 0.815823i \(0.696285\pi\)
\(908\) −34.6525 −1.14998
\(909\) 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
\(913\) −1.81966 −0.0602220
\(914\) −7.74265 −0.256104
\(915\) 0 0
\(916\) 7.23607 0.239086
\(917\) −16.9443 −0.559549
\(918\) 0 0
\(919\) −27.7639 −0.915848 −0.457924 0.888991i \(-0.651407\pi\)
−0.457924 + 0.888991i \(0.651407\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) −8.76393 −0.288625
\(923\) 10.7639 0.354299
\(924\) 0 0
\(925\) 0 0
\(926\) −8.58359 −0.282074
\(927\) 0 0
\(928\) 28.0902 0.922105
\(929\) −38.2918 −1.25631 −0.628157 0.778087i \(-0.716190\pi\)
−0.628157 + 0.778087i \(0.716190\pi\)
\(930\) 0 0
\(931\) −4.47214 −0.146568
\(932\) −12.8541 −0.421050
\(933\) 0 0
\(934\) −4.29180 −0.140432
\(935\) 0 0
\(936\) 0 0
\(937\) −35.2361 −1.15111 −0.575556 0.817762i \(-0.695214\pi\)
−0.575556 + 0.817762i \(0.695214\pi\)
\(938\) 2.61803 0.0854818
\(939\) 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
\(943\) −29.7082 −0.967432
\(944\) −21.7082 −0.706542
\(945\) 0 0
\(946\) 0.257354 0.00836731
\(947\) 34.8328 1.13191 0.565957 0.824435i \(-0.308507\pi\)
0.565957 + 0.824435i \(0.308507\pi\)
\(948\) 0 0
\(949\) −10.8328 −0.351648
\(950\) 0 0
\(951\) 0 0
\(952\) −5.52786 −0.179159
\(953\) 3.47214 0.112474 0.0562368 0.998417i \(-0.482090\pi\)
0.0562368 + 0.998417i \(0.482090\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) −8.94427 −0.289278
\(957\) 0 0
\(958\) −16.1803 −0.522763
\(959\) 10.9443 0.353409
\(960\) 0 0
\(961\) −17.2492 −0.556427
\(962\) −2.29180 −0.0738905
\(963\) 0 0
\(964\) 5.70820 0.183849
\(965\) 0 0
\(966\) 0 0
\(967\) 14.1115 0.453794 0.226897 0.973919i \(-0.427142\pi\)
0.226897 + 0.973919i \(0.427142\pi\)
\(968\) −24.4721 −0.786564
\(969\) 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
\(973\) −10.6525 −0.341503
\(974\) 3.56231 0.114144
\(975\) 0 0
\(976\) −18.0000 −0.576166
\(977\) −11.4721 −0.367026 −0.183513 0.983017i \(-0.558747\pi\)
−0.183513 + 0.983017i \(0.558747\pi\)
\(978\) 0 0
\(979\) −4.06888 −0.130042
\(980\) 0 0
\(981\) 0 0
\(982\) −3.56231 −0.113678
\(983\) −34.5410 −1.10169 −0.550844 0.834608i \(-0.685694\pi\)
−0.550844 + 0.834608i \(0.685694\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 7.63932 0.243286
\(987\) 0 0
\(988\) −8.94427 −0.284555
\(989\) −11.0000 −0.349780
\(990\) 0 0
\(991\) 13.1803 0.418687 0.209344 0.977842i \(-0.432867\pi\)
0.209344 + 0.977842i \(0.432867\pi\)
\(992\) −20.8328 −0.661443
\(993\) 0 0
\(994\) −5.38197 −0.170706
\(995\) 0 0
\(996\) 0 0
\(997\) 45.4164 1.43835 0.719176 0.694828i \(-0.244519\pi\)
0.719176 + 0.694828i \(0.244519\pi\)
\(998\) −6.83282 −0.216289
\(999\) 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.a.s.1.1 2
3.2 odd 2 175.2.a.d.1.2 2
5.2 odd 4 1575.2.d.k.1324.2 4
5.3 odd 4 1575.2.d.k.1324.3 4
5.4 even 2 1575.2.a.n.1.2 2
12.11 even 2 2800.2.a.bh.1.1 2
15.2 even 4 175.2.b.c.99.3 4
15.8 even 4 175.2.b.c.99.2 4
15.14 odd 2 175.2.a.e.1.1 yes 2
21.20 even 2 1225.2.a.n.1.2 2
60.23 odd 4 2800.2.g.s.449.1 4
60.47 odd 4 2800.2.g.s.449.4 4
60.59 even 2 2800.2.a.bp.1.2 2
105.62 odd 4 1225.2.b.k.99.3 4
105.83 odd 4 1225.2.b.k.99.2 4
105.104 even 2 1225.2.a.u.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
175.2.a.d.1.2 2 3.2 odd 2
175.2.a.e.1.1 yes 2 15.14 odd 2
175.2.b.c.99.2 4 15.8 even 4
175.2.b.c.99.3 4 15.2 even 4
1225.2.a.n.1.2 2 21.20 even 2
1225.2.a.u.1.1 2 105.104 even 2
1225.2.b.k.99.2 4 105.83 odd 4
1225.2.b.k.99.3 4 105.62 odd 4
1575.2.a.n.1.2 2 5.4 even 2
1575.2.a.s.1.1 2 1.1 even 1 trivial
1575.2.d.k.1324.2 4 5.2 odd 4
1575.2.d.k.1324.3 4 5.3 odd 4
2800.2.a.bh.1.1 2 12.11 even 2
2800.2.a.bp.1.2 2 60.59 even 2
2800.2.g.s.449.1 4 60.23 odd 4
2800.2.g.s.449.4 4 60.47 odd 4