Properties

Label 1225.2.b.k.99.3
Level $1225$
Weight $2$
Character 1225.99
Analytic conductor $9.782$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 1225 = 5^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1225.b (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(9.78167424761\)
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 99.3
Root \(0.618034i\) of defining polynomial
Character \(\chi\) \(=\) 1225.99
Dual form 1225.2.b.k.99.2

$q$-expansion

\(f(q)\) \(=\) \(q+0.618034i q^{2} +3.23607i q^{3} +1.61803 q^{4} -2.00000 q^{6} +2.23607i q^{8} -7.47214 q^{9} +O(q^{10})\) \(q+0.618034i q^{2} +3.23607i q^{3} +1.61803 q^{4} -2.00000 q^{6} +2.23607i q^{8} -7.47214 q^{9} -0.236068 q^{11} +5.23607i q^{12} -1.23607i q^{13} +1.85410 q^{16} +2.47214i q^{17} -4.61803i q^{18} -4.47214 q^{19} -0.145898i q^{22} +6.23607i q^{23} -7.23607 q^{24} +0.763932 q^{26} -14.4721i q^{27} -5.00000 q^{29} -3.70820 q^{31} +5.61803i q^{32} -0.763932i q^{33} -1.52786 q^{34} -12.0902 q^{36} -3.00000i q^{37} -2.76393i q^{38} +4.00000 q^{39} -4.76393 q^{41} +1.76393i q^{43} -0.381966 q^{44} -3.85410 q^{46} -2.00000i q^{47} +6.00000i q^{48} -8.00000 q^{51} -2.00000i q^{52} +8.47214i q^{53} +8.94427 q^{54} -14.4721i q^{57} -3.09017i q^{58} +11.7082 q^{59} +9.70820 q^{61} -2.29180i q^{62} +0.236068 q^{64} +0.472136 q^{66} +4.23607i q^{67} +4.00000i q^{68} -20.1803 q^{69} +8.70820 q^{71} -16.7082i q^{72} +8.76393i q^{73} +1.85410 q^{74} -7.23607 q^{76} +2.47214i q^{78} +11.1803 q^{79} +24.4164 q^{81} -2.94427i q^{82} +7.70820i q^{83} -1.09017 q^{86} -16.1803i q^{87} -0.527864i q^{88} +17.2361 q^{89} +10.0902i q^{92} -12.0000i q^{93} +1.23607 q^{94} -18.1803 q^{96} +5.23607i q^{97} +1.76393 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 2 q^{4} - 8 q^{6} - 12 q^{9} + O(q^{10}) \) \( 4 q + 2 q^{4} - 8 q^{6} - 12 q^{9} + 8 q^{11} - 6 q^{16} - 20 q^{24} + 12 q^{26} - 20 q^{29} + 12 q^{31} - 24 q^{34} - 26 q^{36} + 16 q^{39} - 28 q^{41} - 6 q^{44} - 2 q^{46} - 32 q^{51} + 20 q^{59} + 12 q^{61} - 8 q^{64} - 16 q^{66} - 36 q^{69} + 8 q^{71} - 6 q^{74} - 20 q^{76} + 44 q^{81} + 18 q^{86} + 60 q^{89} - 4 q^{94} - 28 q^{96} + 16 q^{99} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1225\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(1177\)
\(\chi(n)\) \(1\) \(-1\)

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 \(0.929881\pi\)
\(3\) 3.23607i 1.86834i 0.356822 + 0.934172i \(0.383860\pi\)
−0.356822 + 0.934172i \(0.616140\pi\)
\(4\) 1.61803 0.809017
\(5\) 0 0
\(6\) −2.00000 −0.816497
\(7\) 0 0
\(8\) 2.23607i 0.790569i
\(9\) −7.47214 −2.49071
\(10\) 0 0
\(11\) −0.236068 −0.0711772 −0.0355886 0.999367i \(-0.511331\pi\)
−0.0355886 + 0.999367i \(0.511331\pi\)
\(12\) 5.23607i 1.51152i
\(13\) − 1.23607i − 0.342824i −0.985199 0.171412i \(-0.945167\pi\)
0.985199 0.171412i \(-0.0548329\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 1.85410 0.463525
\(17\) 2.47214i 0.599581i 0.954005 + 0.299791i \(0.0969168\pi\)
−0.954005 + 0.299791i \(0.903083\pi\)
\(18\) − 4.61803i − 1.08848i
\(19\) −4.47214 −1.02598 −0.512989 0.858395i \(-0.671462\pi\)
−0.512989 + 0.858395i \(0.671462\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) − 0.145898i − 0.0311056i
\(23\) 6.23607i 1.30031i 0.759802 + 0.650155i \(0.225296\pi\)
−0.759802 + 0.650155i \(0.774704\pi\)
\(24\) −7.23607 −1.47706
\(25\) 0 0
\(26\) 0.763932 0.149819
\(27\) − 14.4721i − 2.78516i
\(28\) 0 0
\(29\) −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 \(-0.608063\pi\)
−0.333007 + 0.942925i \(0.608063\pi\)
\(32\) 5.61803i 0.993137i
\(33\) − 0.763932i − 0.132983i
\(34\) −1.52786 −0.262027
\(35\) 0 0
\(36\) −12.0902 −2.01503
\(37\) − 3.00000i − 0.493197i −0.969118 0.246598i \(-0.920687\pi\)
0.969118 0.246598i \(-0.0793129\pi\)
\(38\) − 2.76393i − 0.448369i
\(39\) 4.00000 0.640513
\(40\) 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 \(-0.0465965\pi\)
\(48\) 6.00000i 0.866025i
\(49\) 0 0
\(50\) 0 0
\(51\) −8.00000 −1.12022
\(52\) − 2.00000i − 0.277350i
\(53\) 8.47214i 1.16374i 0.813283 + 0.581869i \(0.197678\pi\)
−0.813283 + 0.581869i \(0.802322\pi\)
\(54\) 8.94427 1.21716
\(55\) 0 0
\(56\) 0 0
\(57\) − 14.4721i − 1.91688i
\(58\) − 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 \(-0.286522\pi\)
0.621504 + 0.783411i \(0.286522\pi\)
\(62\) − 2.29180i − 0.291058i
\(63\) 0 0
\(64\) 0.236068 0.0295085
\(65\) 0 0
\(66\) 0.472136 0.0581159
\(67\) 4.23607i 0.517518i 0.965942 + 0.258759i \(0.0833136\pi\)
−0.965942 + 0.258759i \(0.916686\pi\)
\(68\) 4.00000i 0.485071i
\(69\) −20.1803 −2.42943
\(70\) 0 0
\(71\) 8.70820 1.03347 0.516737 0.856144i \(-0.327147\pi\)
0.516737 + 0.856144i \(0.327147\pi\)
\(72\) − 16.7082i − 1.96908i
\(73\) 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 0
\(78\) 2.47214i 0.279914i
\(79\) 11.1803 1.25789 0.628943 0.777451i \(-0.283488\pi\)
0.628943 + 0.777451i \(0.283488\pi\)
\(80\) 0 0
\(81\) 24.4164 2.71293
\(82\) − 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
\(87\) − 16.1803i − 1.73471i
\(88\) − 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
\(91\) 0 0
\(92\) 10.0902i 1.05197i
\(93\) − 12.0000i − 1.24434i
\(94\) 1.23607 0.127491
\(95\) 0 0
\(96\) −18.1803 −1.85552
\(97\) 5.23607i 0.531642i 0.964022 + 0.265821i \(0.0856430\pi\)
−0.964022 + 0.265821i \(0.914357\pi\)
\(98\) 0 0
\(99\) 1.76393 0.177282
\(100\) 0 0
\(101\) −4.76393 −0.474029 −0.237014 0.971506i \(-0.576169\pi\)
−0.237014 + 0.971506i \(0.576169\pi\)
\(102\) − 4.94427i − 0.489556i
\(103\) − 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 \(-0.126383\pi\)
\(108\) − 23.4164i − 2.25324i
\(109\) −8.41641 −0.806146 −0.403073 0.915168i \(-0.632058\pi\)
−0.403073 + 0.915168i \(0.632058\pi\)
\(110\) 0 0
\(111\) 9.70820 0.921462
\(112\) 0 0
\(113\) − 14.4164i − 1.35618i −0.734978 0.678091i \(-0.762808\pi\)
0.734978 0.678091i \(-0.237192\pi\)
\(114\) 8.94427 0.837708
\(115\) 0 0
\(116\) −8.09017 −0.751153
\(117\) 9.23607i 0.853875i
\(118\) 7.23607i 0.666134i
\(119\) 0 0
\(120\) 0 0
\(121\) −10.9443 −0.994934
\(122\) 6.00000i 0.543214i
\(123\) − 15.4164i − 1.39005i
\(124\) −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
\(129\) −5.70820 −0.502579
\(130\) 0 0
\(131\) 16.9443 1.48043 0.740214 0.672371i \(-0.234724\pi\)
0.740214 + 0.672371i \(0.234724\pi\)
\(132\) − 1.23607i − 0.107586i
\(133\) 0 0
\(134\) −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 \(0.845149\pi\)
\(138\) − 12.4721i − 1.06170i
\(139\) 10.6525 0.903531 0.451766 0.892137i \(-0.350794\pi\)
0.451766 + 0.892137i \(0.350794\pi\)
\(140\) 0 0
\(141\) 6.47214 0.545052
\(142\) 5.38197i 0.451645i
\(143\) 0.291796i 0.0244012i
\(144\) −13.8541 −1.15451
\(145\) 0 0
\(146\) −5.41641 −0.448265
\(147\) 0 0
\(148\) − 4.85410i − 0.399005i
\(149\) −3.94427 −0.323127 −0.161564 0.986862i \(-0.551654\pi\)
−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 \(0.807922\pi\)
\(152\) − 10.0000i − 0.811107i
\(153\) − 18.4721i − 1.49338i
\(154\) 0 0
\(155\) 0 0
\(156\) 6.47214 0.518186
\(157\) 0.763932i 0.0609684i 0.999535 + 0.0304842i \(0.00970493\pi\)
−0.999535 + 0.0304842i \(0.990295\pi\)
\(158\) 6.90983i 0.549717i
\(159\) −27.4164 −2.17426
\(160\) 0 0
\(161\) 0 0
\(162\) 15.0902i 1.18560i
\(163\) − 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
\(171\) 33.4164 2.55542
\(172\) 2.85410i 0.217623i
\(173\) 11.5279i 0.876447i 0.898866 + 0.438224i \(0.144392\pi\)
−0.898866 + 0.438224i \(0.855608\pi\)
\(174\) 10.0000 0.758098
\(175\) 0 0
\(176\) −0.437694 −0.0329924
\(177\) 37.8885i 2.84788i
\(178\) 10.6525i 0.798437i
\(179\) 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 \(-0.598329\pi\)
−0.304020 + 0.952666i \(0.598329\pi\)
\(182\) 0 0
\(183\) 31.4164i 2.32237i
\(184\) −13.9443 −1.02799
\(185\) 0 0
\(186\) 7.41641 0.543797
\(187\) − 0.583592i − 0.0426765i
\(188\) − 3.23607i − 0.236015i
\(189\) 0 0
\(190\) 0 0
\(191\) 6.47214 0.468307 0.234154 0.972200i \(-0.424768\pi\)
0.234154 + 0.972200i \(0.424768\pi\)
\(192\) 0.763932i 0.0551320i
\(193\) 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
\(196\) 0 0
\(197\) 1.47214i 0.104885i 0.998624 + 0.0524427i \(0.0167007\pi\)
−0.998624 + 0.0524427i \(0.983299\pi\)
\(198\) 1.09017i 0.0774750i
\(199\) 7.23607 0.512951 0.256476 0.966551i \(-0.417439\pi\)
0.256476 + 0.966551i \(0.417439\pi\)
\(200\) 0 0
\(201\) −13.7082 −0.966902
\(202\) − 2.94427i − 0.207158i
\(203\) 0 0
\(204\) −12.9443 −0.906280
\(205\) 0 0
\(206\) 5.23607 0.364814
\(207\) − 46.5967i − 3.23870i
\(208\) − 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
\(213\) 28.1803i 1.93089i
\(214\) 4.94427 0.337983
\(215\) 0 0
\(216\) 32.3607 2.20187
\(217\) 0 0
\(218\) − 5.20163i − 0.352299i
\(219\) −28.3607 −1.91644
\(220\) 0 0
\(221\) 3.05573 0.205551
\(222\) 6.00000i 0.402694i
\(223\) − 20.1803i − 1.35138i −0.737188 0.675688i \(-0.763847\pi\)
0.737188 0.675688i \(-0.236153\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 8.90983 0.592673
\(227\) 21.4164i 1.42146i 0.703466 + 0.710728i \(0.251635\pi\)
−0.703466 + 0.710728i \(0.748365\pi\)
\(228\) − 23.4164i − 1.55079i
\(229\) −4.47214 −0.295527 −0.147764 0.989023i \(-0.547207\pi\)
−0.147764 + 0.989023i \(0.547207\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) − 11.1803i − 0.734025i
\(233\) 7.94427i 0.520447i 0.965548 + 0.260223i \(0.0837962\pi\)
−0.965548 + 0.260223i \(0.916204\pi\)
\(234\) −5.70820 −0.373157
\(235\) 0 0
\(236\) 18.9443 1.23317
\(237\) 36.1803i 2.35017i
\(238\) 0 0
\(239\) 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 \(-0.463754\pi\)
0.113625 + 0.993524i \(0.463754\pi\)
\(242\) − 6.76393i − 0.434802i
\(243\) 35.5967i 2.28353i
\(244\) 15.7082 1.00561
\(245\) 0 0
\(246\) 9.52786 0.607474
\(247\) 5.52786i 0.351730i
\(248\) − 8.29180i − 0.526530i
\(249\) −24.9443 −1.58078
\(250\) 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 \(-0.129124\pi\)
\(258\) − 3.52786i − 0.219635i
\(259\) 0 0
\(260\) 0 0
\(261\) 37.3607 2.31257
\(262\) 10.4721i 0.646971i
\(263\) 16.2361i 1.00116i 0.865691 + 0.500579i \(0.166880\pi\)
−0.865691 + 0.500579i \(0.833120\pi\)
\(264\) 1.70820 0.105133
\(265\) 0 0
\(266\) 0 0
\(267\) 55.7771i 3.41350i
\(268\) 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 \(-0.755897\pi\)
−0.720085 + 0.693885i \(0.755897\pi\)
\(272\) 4.58359i 0.277921i
\(273\) 0 0
\(274\) −6.76393 −0.408624
\(275\) 0 0
\(276\) −32.6525 −1.96545
\(277\) 19.8885i 1.19499i 0.801874 + 0.597493i \(0.203837\pi\)
−0.801874 + 0.597493i \(0.796163\pi\)
\(278\) 6.58359i 0.394858i
\(279\) 27.7082 1.65885
\(280\) 0 0
\(281\) −15.3607 −0.916341 −0.458171 0.888864i \(-0.651495\pi\)
−0.458171 + 0.888864i \(0.651495\pi\)
\(282\) 4.00000i 0.238197i
\(283\) − 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
\(287\) 0 0
\(288\) − 41.9787i − 2.47362i
\(289\) 10.8885 0.640503
\(290\) 0 0
\(291\) −16.9443 −0.993291
\(292\) 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
\(297\) 3.41641i 0.198240i
\(298\) − 2.43769i − 0.141212i
\(299\) 7.70820 0.445777
\(300\) 0 0
\(301\) 0 0
\(302\) − 12.5066i − 0.719673i
\(303\) − 15.4164i − 0.885649i
\(304\) −8.29180 −0.475567
\(305\) 0 0
\(306\) 11.4164 0.652633
\(307\) 4.58359i 0.261599i 0.991409 + 0.130800i \(0.0417545\pi\)
−0.991409 + 0.130800i \(0.958246\pi\)
\(308\) 0 0
\(309\) 27.4164 1.55966
\(310\) 0 0
\(311\) −24.3607 −1.38137 −0.690684 0.723157i \(-0.742690\pi\)
−0.690684 + 0.723157i \(0.742690\pi\)
\(312\) 8.94427i 0.506370i
\(313\) − 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 \(-0.252301\pi\)
\(318\) − 16.9443i − 0.950188i
\(319\) 1.18034 0.0660863
\(320\) 0 0
\(321\) 25.8885 1.44496
\(322\) 0 0
\(323\) − 11.0557i − 0.615157i
\(324\) 39.5066 2.19481
\(325\) 0 0
\(326\) 0.944272 0.0522984
\(327\) − 27.2361i − 1.50616i
\(328\) − 10.6525i − 0.588185i
\(329\) 0 0
\(330\) 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
\(333\) 22.4164i 1.22841i
\(334\) −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
\(339\) 46.6525 2.53381
\(340\) 0 0
\(341\) 0.875388 0.0474049
\(342\) 20.6525i 1.11676i
\(343\) 0 0
\(344\) −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 \(-0.182774\pi\)
\(348\) − 26.1803i − 1.40341i
\(349\) 4.47214 0.239388 0.119694 0.992811i \(-0.461809\pi\)
0.119694 + 0.992811i \(0.461809\pi\)
\(350\) 0 0
\(351\) −17.8885 −0.954820
\(352\) − 1.32624i − 0.0706887i
\(353\) 2.18034i 0.116048i 0.998315 + 0.0580239i \(0.0184800\pi\)
−0.998315 + 0.0580239i \(0.981520\pi\)
\(354\) −23.4164 −1.24457
\(355\) 0 0
\(356\) 27.8885 1.47809
\(357\) 0 0
\(358\) 14.4721i 0.764876i
\(359\) 30.1246 1.58992 0.794958 0.606664i \(-0.207493\pi\)
0.794958 + 0.606664i \(0.207493\pi\)
\(360\) 0 0
\(361\) 1.00000 0.0526316
\(362\) − 5.05573i − 0.265723i
\(363\) − 35.4164i − 1.85888i
\(364\) 0 0
\(365\) 0 0
\(366\) −19.4164 −1.01491
\(367\) − 37.1246i − 1.93789i −0.247278 0.968944i \(-0.579536\pi\)
0.247278 0.968944i \(-0.420464\pi\)
\(368\) 11.5623i 0.602727i
\(369\) 35.5967 1.85309
\(370\) 0 0
\(371\) 0 0
\(372\) − 19.4164i − 1.00669i
\(373\) − 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
\(381\) 44.1803 2.26343
\(382\) 4.00000i 0.204658i
\(383\) 33.2361i 1.69828i 0.528165 + 0.849142i \(0.322880\pi\)
−0.528165 + 0.849142i \(0.677120\pi\)
\(384\) −36.8328 −1.87962
\(385\) 0 0
\(386\) −7.67376 −0.390584
\(387\) − 13.1803i − 0.669994i
\(388\) 8.47214i 0.430108i
\(389\) −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
\(392\) 0 0
\(393\) 54.8328i 2.76595i
\(394\) −0.909830 −0.0458366
\(395\) 0 0
\(396\) 2.85410 0.143424
\(397\) 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 \(-0.479896\pi\)
0.0631178 + 0.998006i \(0.479896\pi\)
\(402\) − 8.47214i − 0.422552i
\(403\) 4.58359i 0.228325i
\(404\) −7.70820 −0.383497
\(405\) 0 0
\(406\) 0 0
\(407\) 0.708204i 0.0351044i
\(408\) − 17.8885i − 0.885615i
\(409\) 24.4721 1.21007 0.605035 0.796199i \(-0.293159\pi\)
0.605035 + 0.796199i \(0.293159\pi\)
\(410\) 0 0
\(411\) −35.4164 −1.74696
\(412\) − 13.7082i − 0.675355i
\(413\) 0 0
\(414\) 28.7984 1.41536
\(415\) 0 0
\(416\) 6.94427 0.340471
\(417\) 34.4721i 1.68811i
\(418\) 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
\(423\) 14.9443i 0.726615i
\(424\) −18.9443 −0.920015
\(425\) 0 0
\(426\) −17.4164 −0.843828
\(427\) 0 0
\(428\) − 12.9443i − 0.625685i
\(429\) −0.944272 −0.0455899
\(430\) 0 0
\(431\) 17.5279 0.844288 0.422144 0.906529i \(-0.361278\pi\)
0.422144 + 0.906529i \(0.361278\pi\)
\(432\) − 26.8328i − 1.29099i
\(433\) 28.3607i 1.36293i 0.731852 + 0.681464i \(0.238656\pi\)
−0.731852 + 0.681464i \(0.761344\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) −13.6180 −0.652186
\(437\) − 27.8885i − 1.33409i
\(438\) − 17.5279i − 0.837514i
\(439\) −8.29180 −0.395746 −0.197873 0.980228i \(-0.563403\pi\)
−0.197873 + 0.980228i \(0.563403\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 1.88854i 0.0898289i
\(443\) − 19.4164i − 0.922501i −0.887270 0.461251i \(-0.847401\pi\)
0.887270 0.461251i \(-0.152599\pi\)
\(444\) 15.7082 0.745478
\(445\) 0 0
\(446\) 12.4721 0.590573
\(447\) − 12.7639i − 0.603713i
\(448\) 0 0
\(449\) −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
\(452\) − 23.3262i − 1.09717i
\(453\) − 65.4853i − 3.07677i
\(454\) −13.2361 −0.621199
\(455\) 0 0
\(456\) 32.3607 1.51543
\(457\) 12.5279i 0.586029i 0.956108 + 0.293014i \(0.0946584\pi\)
−0.956108 + 0.293014i \(0.905342\pi\)
\(458\) − 2.76393i − 0.129150i
\(459\) 35.7771 1.66993
\(460\) 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 \(0.948634\pi\)
\(468\) 14.9443i 0.690799i
\(469\) 0 0
\(470\) 0 0
\(471\) −2.47214 −0.113910
\(472\) 26.1803i 1.20505i
\(473\) − 0.416408i − 0.0191465i
\(474\) −22.3607 −1.02706
\(475\) 0 0
\(476\) 0 0
\(477\) − 63.3050i − 2.89853i
\(478\) 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
\(486\) −22.0000 −0.997940
\(487\) − 5.76393i − 0.261189i −0.991436 0.130594i \(-0.958311\pi\)
0.991436 0.130594i \(-0.0416885\pi\)
\(488\) 21.7082i 0.982684i
\(489\) 4.94427 0.223588
\(490\) 0 0
\(491\) −5.76393 −0.260123 −0.130061 0.991506i \(-0.541517\pi\)
−0.130061 + 0.991506i \(0.541517\pi\)
\(492\) − 24.9443i − 1.12457i
\(493\) − 12.3607i − 0.556697i
\(494\) −3.41641 −0.153711
\(495\) 0 0
\(496\) −6.87539 −0.308714
\(497\) 0 0
\(498\) − 15.4164i − 0.690826i
\(499\) −11.0557 −0.494922 −0.247461 0.968898i \(-0.579596\pi\)
−0.247461 + 0.968898i \(0.579596\pi\)
\(500\) 0 0
\(501\) −16.9443 −0.757014
\(502\) − 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
\(507\) 37.1246i 1.64876i
\(508\) − 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
\(511\) 0 0
\(512\) 18.7082i 0.826794i
\(513\) 64.7214i 2.85752i
\(514\) 7.81966 0.344910
\(515\) 0 0
\(516\) −9.23607 −0.406595
\(517\) 0.472136i 0.0207645i
\(518\) 0 0
\(519\) −37.3050 −1.63751
\(520\) 0 0
\(521\) −22.0000 −0.963837 −0.481919 0.876216i \(-0.660060\pi\)
−0.481919 + 0.876216i \(0.660060\pi\)
\(522\) 23.0902i 1.01063i
\(523\) − 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
\(528\) − 1.41641i − 0.0616412i
\(529\) −15.8885 −0.690806
\(530\) 0 0
\(531\) −87.4853 −3.79654
\(532\) 0 0
\(533\) 5.88854i 0.255061i
\(534\) −34.4721 −1.49176
\(535\) 0 0
\(536\) −9.47214 −0.409134
\(537\) 75.7771i 3.27002i
\(538\) − 7.23607i − 0.311969i
\(539\) 0 0
\(540\) 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
\(543\) − 26.4721i − 1.13603i
\(544\) −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
\(549\) −72.5410 −3.09598
\(550\) 0 0
\(551\) 22.3607 0.952597
\(552\) − 45.1246i − 1.92063i
\(553\) 0 0
\(554\) −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 \(0.938168\pi\)
\(558\) 17.1246i 0.724943i
\(559\) 2.18034 0.0922186
\(560\) 0 0
\(561\) 1.88854 0.0797344
\(562\) − 9.49342i − 0.400456i
\(563\) − 17.4164i − 0.734014i −0.930218 0.367007i \(-0.880382\pi\)
0.930218 0.367007i \(-0.119618\pi\)
\(564\) 10.4721 0.440956
\(565\) 0 0
\(566\) 10.7639 0.452442
\(567\) 0 0
\(568\) 19.4721i 0.817033i
\(569\) 3.94427 0.165352 0.0826762 0.996576i \(-0.473653\pi\)
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 \(0.222362\pi\)
\(572\) 0.472136i 0.0197410i
\(573\) 20.9443i 0.874960i
\(574\) 0 0
\(575\) 0 0
\(576\) −1.76393 −0.0734972
\(577\) − 2.00000i − 0.0832611i −0.999133 0.0416305i \(-0.986745\pi\)
0.999133 0.0416305i \(-0.0132552\pi\)
\(578\) 6.72949i 0.279910i
\(579\) −40.1803 −1.66984
\(580\) 0 0
\(581\) 0 0
\(582\) − 10.4721i − 0.434084i
\(583\) − 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
\(591\) −4.76393 −0.195962
\(592\) − 5.56231i − 0.228609i
\(593\) 37.3050i 1.53193i 0.642882 + 0.765965i \(0.277739\pi\)
−0.642882 + 0.765965i \(0.722261\pi\)
\(594\) −2.11146 −0.0866341
\(595\) 0 0
\(596\) −6.38197 −0.261416
\(597\) 23.4164i 0.958370i
\(598\) 4.76393i 0.194812i
\(599\) 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 \(-0.228367\pi\)
0.753494 + 0.657455i \(0.228367\pi\)
\(602\) 0 0
\(603\) − 31.6525i − 1.28899i
\(604\) −32.7426 −1.33228
\(605\) 0 0
\(606\) 9.52786 0.387043
\(607\) − 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
\(612\) − 29.8885i − 1.20817i
\(613\) − 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 0
\(617\) 5.94427i 0.239307i 0.992816 + 0.119654i \(0.0381784\pi\)
−0.992816 + 0.119654i \(0.961822\pi\)
\(618\) 16.9443i 0.681599i
\(619\) −11.7082 −0.470592 −0.235296 0.971924i \(-0.575606\pi\)
−0.235296 + 0.971924i \(0.575606\pi\)
\(620\) 0 0
\(621\) 90.2492 3.62158
\(622\) − 15.0557i − 0.603680i
\(623\) 0 0
\(624\) 7.41641 0.296894
\(625\) 0 0
\(626\) 12.0689 0.482370
\(627\) 3.41641i 0.136438i
\(628\) 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
\(633\) 38.8328i 1.54347i
\(634\) 15.6738 0.622485
\(635\) 0 0
\(636\) −44.3607 −1.75902
\(637\) 0 0
\(638\) 0.729490i 0.0288808i
\(639\) −65.0689 −2.57409
\(640\) 0 0
\(641\) 43.8328 1.73129 0.865646 0.500656i \(-0.166908\pi\)
0.865646 + 0.500656i \(0.166908\pi\)
\(642\) 16.0000i 0.631470i
\(643\) − 18.4721i − 0.728470i −0.931307 0.364235i \(-0.881331\pi\)
0.931307 0.364235i \(-0.118669\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 6.83282 0.268834
\(647\) − 19.8885i − 0.781899i −0.920412 0.390950i \(-0.872147\pi\)
0.920412 0.390950i \(-0.127853\pi\)
\(648\) 54.5967i 2.14476i
\(649\) −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 \(0.836904\pi\)
\(654\) 16.8328 0.658215
\(655\) 0 0
\(656\) −8.83282 −0.344864
\(657\) − 65.4853i − 2.55482i
\(658\) 0 0
\(659\) −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 \(-0.187864\pi\)
0.830834 + 0.556520i \(0.187864\pi\)
\(662\) − 15.2705i − 0.593505i
\(663\) 9.88854i 0.384039i
\(664\) −17.2361 −0.668889
\(665\) 0 0
\(666\) −13.8541 −0.536836
\(667\) − 31.1803i − 1.20731i
\(668\) 8.47214i 0.327797i
\(669\) 65.3050 2.52484
\(670\) 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 \(-0.0889967\pi\)
\(678\) 28.8328i 1.10732i
\(679\) 0 0
\(680\) 0 0
\(681\) −69.3050 −2.65577
\(682\) 0.541020i 0.0207167i
\(683\) 14.1246i 0.540463i 0.962795 + 0.270232i \(0.0871003\pi\)
−0.962795 + 0.270232i \(0.912900\pi\)
\(684\) 54.0689 2.06738
\(685\) 0 0
\(686\) 0 0
\(687\) − 14.4721i − 0.552146i
\(688\) 3.27051i 0.124687i
\(689\) 10.4721 0.398957
\(690\) 0 0
\(691\) 4.18034 0.159028 0.0795138 0.996834i \(-0.474663\pi\)
0.0795138 + 0.996834i \(0.474663\pi\)
\(692\) 18.6525i 0.709061i
\(693\) 0 0
\(694\) 12.5066 0.474743
\(695\) 0 0
\(696\) 36.1803 1.37141
\(697\) − 11.7771i − 0.446089i
\(698\) 2.76393i 0.104616i
\(699\) −25.7082 −0.972374
\(700\) 0 0
\(701\) −29.0557 −1.09742 −0.548710 0.836013i \(-0.684881\pi\)
−0.548710 + 0.836013i \(0.684881\pi\)
\(702\) − 11.0557i − 0.417272i
\(703\) 13.4164i 0.506009i
\(704\) −0.0557281 −0.00210033
\(705\) 0 0
\(706\) −1.34752 −0.0507147
\(707\) 0 0
\(708\) 61.3050i 2.30398i
\(709\) 12.1115 0.454855 0.227428 0.973795i \(-0.426968\pi\)
0.227428 + 0.973795i \(0.426968\pi\)
\(710\) 0 0
\(711\) −83.5410 −3.13303
\(712\) 38.5410i 1.44439i
\(713\) − 23.1246i − 0.866024i
\(714\) 0 0
\(715\) 0 0
\(716\) 37.8885 1.41596
\(717\) 17.8885i 0.668060i
\(718\) 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
\(721\) 0 0
\(722\) 0.618034i 0.0230008i
\(723\) 11.4164i 0.424581i
\(724\) −13.2361 −0.491915
\(725\) 0 0
\(726\) 21.8885 0.812360
\(727\) − 3.05573i − 0.113331i −0.998393 0.0566653i \(-0.981953\pi\)
0.998393 0.0566653i \(-0.0180468\pi\)
\(728\) 0 0
\(729\) −41.9443 −1.55349
\(730\) 0 0
\(731\) −4.36068 −0.161286
\(732\) 50.8328i 1.87883i
\(733\) − 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
\(738\) 22.0000i 0.809831i
\(739\) −25.6525 −0.943642 −0.471821 0.881694i \(-0.656403\pi\)
−0.471821 + 0.881694i \(0.656403\pi\)
\(740\) 0 0
\(741\) −17.8885 −0.657152
\(742\) 0 0
\(743\) − 10.4721i − 0.384185i −0.981377 0.192093i \(-0.938473\pi\)
0.981377 0.192093i \(-0.0615274\pi\)
\(744\) 26.8328 0.983739
\(745\) 0 0
\(746\) 23.3820 0.856075
\(747\) − 57.5967i − 2.10735i
\(748\) − 0.944272i − 0.0345260i
\(749\) 0 0
\(750\) 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
\(753\) − 20.9443i − 0.763252i
\(754\) −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
\(759\) 4.76393 0.172920
\(760\) 0 0
\(761\) −27.7771 −1.00692 −0.503459 0.864019i \(-0.667940\pi\)
−0.503459 + 0.864019i \(0.667940\pi\)
\(762\) 27.3050i 0.989154i
\(763\) 0 0
\(764\) 10.4721 0.378869
\(765\) 0 0
\(766\) −20.5410 −0.742177
\(767\) − 14.4721i − 0.522559i
\(768\) − 21.2361i − 0.766291i
\(769\) −43.0132 −1.55109 −0.775547 0.631290i \(-0.782526\pi\)
−0.775547 + 0.631290i \(0.782526\pi\)
\(770\) 0 0
\(771\) 40.9443 1.47457
\(772\) 20.0902i 0.723061i
\(773\) − 50.1803i − 1.80486i −0.430835 0.902431i \(-0.641781\pi\)
0.430835 0.902431i \(-0.358219\pi\)
\(774\) 8.14590 0.292798
\(775\) 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
\(783\) 72.3607i 2.58596i
\(784\) 0 0
\(785\) 0 0
\(786\) −33.8885 −1.20876
\(787\) 40.7639i 1.45308i 0.687126 + 0.726539i \(0.258872\pi\)
−0.687126 + 0.726539i \(0.741128\pi\)
\(788\) 2.38197i 0.0848540i
\(789\) −52.5410 −1.87051
\(790\) 0 0
\(791\) 0 0
\(792\) 3.94427i 0.140154i
\(793\) − 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
\(801\) −128.790 −4.55058
\(802\) 1.56231i 0.0551669i
\(803\) − 2.06888i − 0.0730093i
\(804\) −22.1803 −0.782240
\(805\) 0 0
\(806\) −2.83282 −0.0997817
\(807\) − 37.8885i − 1.33374i
\(808\) − 10.6525i − 0.374753i
\(809\) −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 \(-0.230017\pi\)
0.750075 + 0.661353i \(0.230017\pi\)
\(812\) 0 0
\(813\) − 76.7214i − 2.69074i
\(814\) −0.437694 −0.0153412
\(815\) 0 0
\(816\) −14.8328 −0.519252
\(817\) − 7.88854i − 0.275985i
\(818\) 15.1246i 0.528820i
\(819\) 0 0
\(820\) 0 0
\(821\) 28.8328 1.00627 0.503136 0.864207i \(-0.332179\pi\)
0.503136 + 0.864207i \(0.332179\pi\)
\(822\) − 21.8885i − 0.763451i
\(823\) − 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
\(826\) 0 0
\(827\) − 41.5410i − 1.44452i −0.691620 0.722261i \(-0.743103\pi\)
0.691620 0.722261i \(-0.256897\pi\)
\(828\) − 75.3951i − 2.62016i
\(829\) −7.63932 −0.265325 −0.132662 0.991161i \(-0.542353\pi\)
−0.132662 + 0.991161i \(0.542353\pi\)
\(830\) 0 0
\(831\) −64.3607 −2.23265
\(832\) − 0.291796i − 0.0101162i
\(833\) 0 0
\(834\) −21.3050 −0.737730
\(835\) 0 0
\(836\) 1.70820 0.0590795
\(837\) 53.6656i 1.85496i
\(838\) − 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
\(843\) − 49.7082i − 1.71204i
\(844\) 19.4164 0.668340
\(845\) 0 0
\(846\) −9.23607 −0.317543
\(847\) 0 0
\(848\) 15.7082i 0.539422i
\(849\) 56.3607 1.93429
\(850\) 0 0
\(851\) 18.7082 0.641309
\(852\) 45.5967i 1.56212i
\(853\) − 27.4164i − 0.938720i −0.883007 0.469360i \(-0.844485\pi\)
0.883007 0.469360i \(-0.155515\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 17.8885 0.611418
\(857\) − 15.8197i − 0.540389i −0.962806 0.270195i \(-0.912912\pi\)
0.962806 0.270195i \(-0.0870881\pi\)
\(858\) − 0.583592i − 0.0199235i
\(859\) 22.3607 0.762937 0.381468 0.924382i \(-0.375419\pi\)
0.381468 + 0.924382i \(0.375419\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 10.8328i 0.368967i
\(863\) 18.3475i 0.624557i 0.949991 + 0.312278i \(0.101092\pi\)
−0.949991 + 0.312278i \(0.898908\pi\)
\(864\) 81.3050 2.76605
\(865\) 0 0
\(866\) −17.5279 −0.595621
\(867\) 35.2361i 1.19668i
\(868\) 0 0
\(869\) −2.63932 −0.0895328
\(870\) 0 0
\(871\) 5.23607 0.177417
\(872\) − 18.8197i − 0.637314i
\(873\) − 39.1246i − 1.32417i
\(874\) 17.2361 0.583019
\(875\) 0 0
\(876\) −45.8885 −1.55043
\(877\) − 30.3607i − 1.02521i −0.858625 0.512604i \(-0.828681\pi\)
0.858625 0.512604i \(-0.171319\pi\)
\(878\) − 5.12461i − 0.172947i
\(879\) −100.721 −3.39725
\(880\) 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 \(-0.116674\pi\)
\(888\) 21.7082i 0.728480i
\(889\) 0 0
\(890\) 0 0
\(891\) −5.76393 −0.193099
\(892\) − 32.6525i − 1.09329i
\(893\) 8.94427i 0.299309i
\(894\) 7.88854 0.263832
\(895\) 0 0
\(896\) 0 0
\(897\) 24.9443i 0.832865i
\(898\) − 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
\(906\) 40.4721 1.34460
\(907\) − 34.8328i − 1.15660i −0.815823 0.578302i \(-0.803715\pi\)
0.815823 0.578302i \(-0.196285\pi\)
\(908\) 34.6525i 1.14998i
\(909\) 35.5967 1.18067
\(910\) 0 0
\(911\) 0.819660 0.0271566 0.0135783 0.999908i \(-0.495678\pi\)
0.0135783 + 0.999908i \(0.495678\pi\)
\(912\) − 26.8328i − 0.888523i
\(913\) − 1.81966i − 0.0602220i
\(914\) −7.74265 −0.256104
\(915\) 0 0
\(916\) −7.23607 −0.239086
\(917\) 0 0
\(918\) 22.1115i 0.729787i
\(919\) 27.7639 0.915848 0.457924 0.888991i \(-0.348593\pi\)
0.457924 + 0.888991i \(0.348593\pi\)
\(920\) 0 0
\(921\) −14.8328 −0.488758
\(922\) 8.76393i 0.288625i
\(923\) − 10.7639i − 0.354299i
\(924\) 0 0
\(925\) 0 0
\(926\) 8.58359 0.282074
\(927\) 63.3050i 2.07921i
\(928\) − 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
\(931\) 0 0
\(932\) 12.8541i 0.421050i
\(933\) − 78.8328i − 2.58087i
\(934\) −4.29180 −0.140432
\(935\) 0 0
\(936\) −20.6525 −0.675047
\(937\) 35.2361i 1.15111i 0.817762 + 0.575556i \(0.195214\pi\)
−0.817762 + 0.575556i \(0.804786\pi\)
\(938\) 0 0
\(939\) 63.1935 2.06224
\(940\) 0 0
\(941\) 5.23607 0.170691 0.0853455 0.996351i \(-0.472801\pi\)
0.0853455 + 0.996351i \(0.472801\pi\)
\(942\) − 1.52786i − 0.0497805i
\(943\) − 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 \(-0.191493\pi\)
\(948\) 58.5410i 1.90132i
\(949\) 10.8328 0.351648
\(950\) 0 0
\(951\) 82.0689 2.66127
\(952\) 0 0
\(953\) 3.47214i 0.112474i 0.998417 + 0.0562368i \(0.0179102\pi\)
−0.998417 + 0.0562368i \(0.982090\pi\)
\(954\) 39.1246 1.26671
\(955\) 0 0
\(956\) 8.94427 0.289278
\(957\) 3.81966i 0.123472i
\(958\) − 16.1803i − 0.522763i
\(959\) 0 0
\(960\) 0 0
\(961\) −17.2492 −0.556427
\(962\) − 2.29180i − 0.0738905i
\(963\) 59.7771i 1.92629i
\(964\) 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
\(969\) 35.7771 1.14933
\(970\) 0 0
\(971\) 18.0000 0.577647 0.288824 0.957382i \(-0.406736\pi\)
0.288824 + 0.957382i \(0.406736\pi\)
\(972\) 57.5967i 1.84742i
\(973\) 0 0
\(974\) 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 \(0.941253\pi\)
\(978\) 3.05573i 0.0977114i
\(979\) −4.06888 −0.130042
\(980\) 0 0
\(981\) 62.8885 2.00788
\(982\) − 3.56231i − 0.113678i
\(983\) 34.5410i 1.10169i 0.834608 + 0.550844i \(0.185694\pi\)
−0.834608 + 0.550844i \(0.814306\pi\)
\(984\) 34.4721 1.09893
\(985\) 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
\(993\) − 79.9574i − 2.53737i
\(994\) 0 0
\(995\) 0 0
\(996\) −40.3607 −1.27888
\(997\) − 45.4164i − 1.43835i −0.694828 0.719176i \(-0.744519\pi\)
0.694828 0.719176i \(-0.255481\pi\)
\(998\) − 6.83282i − 0.216289i
\(999\) −43.4164 −1.37363
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 1225.2.b.k.99.3 4
5.2 odd 4 1225.2.a.u.1.1 2
5.3 odd 4 1225.2.a.n.1.2 2
5.4 even 2 inner 1225.2.b.k.99.2 4
7.6 odd 2 175.2.b.c.99.3 4
21.20 even 2 1575.2.d.k.1324.2 4
28.27 even 2 2800.2.g.s.449.4 4
35.13 even 4 175.2.a.d.1.2 2
35.27 even 4 175.2.a.e.1.1 yes 2
35.34 odd 2 175.2.b.c.99.2 4
105.62 odd 4 1575.2.a.n.1.2 2
105.83 odd 4 1575.2.a.s.1.1 2
105.104 even 2 1575.2.d.k.1324.3 4
140.27 odd 4 2800.2.a.bp.1.2 2
140.83 odd 4 2800.2.a.bh.1.1 2
140.139 even 2 2800.2.g.s.449.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
175.2.a.d.1.2 2 35.13 even 4
175.2.a.e.1.1 yes 2 35.27 even 4
175.2.b.c.99.2 4 35.34 odd 2
175.2.b.c.99.3 4 7.6 odd 2
1225.2.a.n.1.2 2 5.3 odd 4
1225.2.a.u.1.1 2 5.2 odd 4
1225.2.b.k.99.2 4 5.4 even 2 inner
1225.2.b.k.99.3 4 1.1 even 1 trivial
1575.2.a.n.1.2 2 105.62 odd 4
1575.2.a.s.1.1 2 105.83 odd 4
1575.2.d.k.1324.2 4 21.20 even 2
1575.2.d.k.1324.3 4 105.104 even 2
2800.2.a.bh.1.1 2 140.83 odd 4
2800.2.a.bp.1.2 2 140.27 odd 4
2800.2.g.s.449.1 4 140.139 even 2
2800.2.g.s.449.4 4 28.27 even 2