Properties

Label 1225.2.a.n.1.2
Level $1225$
Weight $2$
Character 1225.1
Self dual yes
Analytic conductor $9.782$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 1225 = 5^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1225.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(9.78167424761\)
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 1.2
Root \(-0.618034\) of defining polynomial
Character \(\chi\) \(=\) 1225.1

$q$-expansion

\(f(q)\) \(=\) \(q+0.618034 q^{2} -3.23607 q^{3} -1.61803 q^{4} -2.00000 q^{6} -2.23607 q^{8} +7.47214 q^{9} +O(q^{10})\) \(q+0.618034 q^{2} -3.23607 q^{3} -1.61803 q^{4} -2.00000 q^{6} -2.23607 q^{8} +7.47214 q^{9} -0.236068 q^{11} +5.23607 q^{12} +1.23607 q^{13} +1.85410 q^{16} +2.47214 q^{17} +4.61803 q^{18} +4.47214 q^{19} -0.145898 q^{22} -6.23607 q^{23} +7.23607 q^{24} +0.763932 q^{26} -14.4721 q^{27} +5.00000 q^{29} -3.70820 q^{31} +5.61803 q^{32} +0.763932 q^{33} +1.52786 q^{34} -12.0902 q^{36} -3.00000 q^{37} +2.76393 q^{38} -4.00000 q^{39} -4.76393 q^{41} -1.76393 q^{43} +0.381966 q^{44} -3.85410 q^{46} -2.00000 q^{47} -6.00000 q^{48} -8.00000 q^{51} -2.00000 q^{52} -8.47214 q^{53} -8.94427 q^{54} -14.4721 q^{57} +3.09017 q^{58} -11.7082 q^{59} +9.70820 q^{61} -2.29180 q^{62} -0.236068 q^{64} +0.472136 q^{66} +4.23607 q^{67} -4.00000 q^{68} +20.1803 q^{69} +8.70820 q^{71} -16.7082 q^{72} -8.76393 q^{73} -1.85410 q^{74} -7.23607 q^{76} -2.47214 q^{78} -11.1803 q^{79} +24.4164 q^{81} -2.94427 q^{82} -7.70820 q^{83} -1.09017 q^{86} -16.1803 q^{87} +0.527864 q^{88} -17.2361 q^{89} +10.0902 q^{92} +12.0000 q^{93} -1.23607 q^{94} -18.1803 q^{96} +5.23607 q^{97} -1.76393 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - q^{2} - 2q^{3} - q^{4} - 4q^{6} + 6q^{9} + O(q^{10}) \) \( 2q - q^{2} - 2q^{3} - q^{4} - 4q^{6} + 6q^{9} + 4q^{11} + 6q^{12} - 2q^{13} - 3q^{16} - 4q^{17} + 7q^{18} - 7q^{22} - 8q^{23} + 10q^{24} + 6q^{26} - 20q^{27} + 10q^{29} + 6q^{31} + 9q^{32} + 6q^{33} + 12q^{34} - 13q^{36} - 6q^{37} + 10q^{38} - 8q^{39} - 14q^{41} - 8q^{43} + 3q^{44} - q^{46} - 4q^{47} - 12q^{48} - 16q^{51} - 4q^{52} - 8q^{53} - 20q^{57} - 5q^{58} - 10q^{59} + 6q^{61} - 18q^{62} + 4q^{64} - 8q^{66} + 4q^{67} - 8q^{68} + 18q^{69} + 4q^{71} - 20q^{72} - 22q^{73} + 3q^{74} - 10q^{76} + 4q^{78} + 22q^{81} + 12q^{82} - 2q^{83} + 9q^{86} - 10q^{87} + 10q^{88} - 30q^{89} + 9q^{92} + 24q^{93} + 2q^{94} - 14q^{96} + 6q^{97} - 8q^{99} + O(q^{100}) \)

Coefficient data

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

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.618034 0.437016 0.218508 0.975835i \(-0.429881\pi\)
0.218508 + 0.975835i \(0.429881\pi\)
\(3\) −3.23607 −1.86834 −0.934172 0.356822i \(-0.883860\pi\)
−0.934172 + 0.356822i \(0.883860\pi\)
\(4\) −1.61803 −0.809017
\(5\) 0 0
\(6\) −2.00000 −0.816497
\(7\) 0 0
\(8\) −2.23607 −0.790569
\(9\) 7.47214 2.49071
\(10\) 0 0
\(11\) −0.236068 −0.0711772 −0.0355886 0.999367i \(-0.511331\pi\)
−0.0355886 + 0.999367i \(0.511331\pi\)
\(12\) 5.23607 1.51152
\(13\) 1.23607 0.342824 0.171412 0.985199i \(-0.445167\pi\)
0.171412 + 0.985199i \(0.445167\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 1.85410 0.463525
\(17\) 2.47214 0.599581 0.299791 0.954005i \(-0.403083\pi\)
0.299791 + 0.954005i \(0.403083\pi\)
\(18\) 4.61803 1.08848
\(19\) 4.47214 1.02598 0.512989 0.858395i \(-0.328538\pi\)
0.512989 + 0.858395i \(0.328538\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) −0.145898 −0.0311056
\(23\) −6.23607 −1.30031 −0.650155 0.759802i \(-0.725296\pi\)
−0.650155 + 0.759802i \(0.725296\pi\)
\(24\) 7.23607 1.47706
\(25\) 0 0
\(26\) 0.763932 0.149819
\(27\) −14.4721 −2.78516
\(28\) 0 0
\(29\) 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 \(-0.608063\pi\)
−0.333007 + 0.942925i \(0.608063\pi\)
\(32\) 5.61803 0.993137
\(33\) 0.763932 0.132983
\(34\) 1.52786 0.262027
\(35\) 0 0
\(36\) −12.0902 −2.01503
\(37\) −3.00000 −0.493197 −0.246598 0.969118i \(-0.579313\pi\)
−0.246598 + 0.969118i \(0.579313\pi\)
\(38\) 2.76393 0.448369
\(39\) −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
\(43\) −1.76393 −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 \(0.546597\pi\)
\(48\) −6.00000 −0.866025
\(49\) 0 0
\(50\) 0 0
\(51\) −8.00000 −1.12022
\(52\) −2.00000 −0.277350
\(53\) −8.47214 −1.16374 −0.581869 0.813283i \(-0.697678\pi\)
−0.581869 + 0.813283i \(0.697678\pi\)
\(54\) −8.94427 −1.21716
\(55\) 0 0
\(56\) 0 0
\(57\) −14.4721 −1.91688
\(58\) 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 \(-0.286522\pi\)
0.621504 + 0.783411i \(0.286522\pi\)
\(62\) −2.29180 −0.291058
\(63\) 0 0
\(64\) −0.236068 −0.0295085
\(65\) 0 0
\(66\) 0.472136 0.0581159
\(67\) 4.23607 0.517518 0.258759 0.965942i \(-0.416686\pi\)
0.258759 + 0.965942i \(0.416686\pi\)
\(68\) −4.00000 −0.485071
\(69\) 20.1803 2.42943
\(70\) 0 0
\(71\) 8.70820 1.03347 0.516737 0.856144i \(-0.327147\pi\)
0.516737 + 0.856144i \(0.327147\pi\)
\(72\) −16.7082 −1.96908
\(73\) −8.76393 −1.02574 −0.512870 0.858466i \(-0.671418\pi\)
−0.512870 + 0.858466i \(0.671418\pi\)
\(74\) −1.85410 −0.215535
\(75\) 0 0
\(76\) −7.23607 −0.830034
\(77\) 0 0
\(78\) −2.47214 −0.279914
\(79\) −11.1803 −1.25789 −0.628943 0.777451i \(-0.716512\pi\)
−0.628943 + 0.777451i \(0.716512\pi\)
\(80\) 0 0
\(81\) 24.4164 2.71293
\(82\) −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
\(87\) −16.1803 −1.73471
\(88\) 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
\(91\) 0 0
\(92\) 10.0902 1.05197
\(93\) 12.0000 1.24434
\(94\) −1.23607 −0.127491
\(95\) 0 0
\(96\) −18.1803 −1.85552
\(97\) 5.23607 0.531642 0.265821 0.964022i \(-0.414357\pi\)
0.265821 + 0.964022i \(0.414357\pi\)
\(98\) 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 \(0.576169\pi\)
\(102\) −4.94427 −0.489556
\(103\) 8.47214 0.834784 0.417392 0.908726i \(-0.362944\pi\)
0.417392 + 0.908726i \(0.362944\pi\)
\(104\) −2.76393 −0.271026
\(105\) 0 0
\(106\) −5.23607 −0.508572
\(107\) −8.00000 −0.773389 −0.386695 0.922208i \(-0.626383\pi\)
−0.386695 + 0.922208i \(0.626383\pi\)
\(108\) 23.4164 2.25324
\(109\) 8.41641 0.806146 0.403073 0.915168i \(-0.367942\pi\)
0.403073 + 0.915168i \(0.367942\pi\)
\(110\) 0 0
\(111\) 9.70820 0.921462
\(112\) 0 0
\(113\) 14.4164 1.35618 0.678091 0.734978i \(-0.262808\pi\)
0.678091 + 0.734978i \(0.262808\pi\)
\(114\) −8.94427 −0.837708
\(115\) 0 0
\(116\) −8.09017 −0.751153
\(117\) 9.23607 0.853875
\(118\) −7.23607 −0.666134
\(119\) 0 0
\(120\) 0 0
\(121\) −10.9443 −0.994934
\(122\) 6.00000 0.543214
\(123\) 15.4164 1.39005
\(124\) 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
\(129\) 5.70820 0.502579
\(130\) 0 0
\(131\) 16.9443 1.48043 0.740214 0.672371i \(-0.234724\pi\)
0.740214 + 0.672371i \(0.234724\pi\)
\(132\) −1.23607 −0.107586
\(133\) 0 0
\(134\) 2.61803 0.226164
\(135\) 0 0
\(136\) −5.52786 −0.474010
\(137\) 10.9443 0.935032 0.467516 0.883985i \(-0.345149\pi\)
0.467516 + 0.883985i \(0.345149\pi\)
\(138\) 12.4721 1.06170
\(139\) −10.6525 −0.903531 −0.451766 0.892137i \(-0.649206\pi\)
−0.451766 + 0.892137i \(0.649206\pi\)
\(140\) 0 0
\(141\) 6.47214 0.545052
\(142\) 5.38197 0.451645
\(143\) −0.291796 −0.0244012
\(144\) 13.8541 1.15451
\(145\) 0 0
\(146\) −5.41641 −0.448265
\(147\) 0 0
\(148\) 4.85410 0.399005
\(149\) 3.94427 0.323127 0.161564 0.986862i \(-0.448346\pi\)
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 \(0.807922\pi\)
\(152\) −10.0000 −0.811107
\(153\) 18.4721 1.49338
\(154\) 0 0
\(155\) 0 0
\(156\) 6.47214 0.518186
\(157\) 0.763932 0.0609684 0.0304842 0.999535i \(-0.490295\pi\)
0.0304842 + 0.999535i \(0.490295\pi\)
\(158\) −6.90983 −0.549717
\(159\) 27.4164 2.17426
\(160\) 0 0
\(161\) 0 0
\(162\) 15.0902 1.18560
\(163\) 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
\(171\) 33.4164 2.55542
\(172\) 2.85410 0.217623
\(173\) −11.5279 −0.876447 −0.438224 0.898866i \(-0.644392\pi\)
−0.438224 + 0.898866i \(0.644392\pi\)
\(174\) −10.0000 −0.758098
\(175\) 0 0
\(176\) −0.437694 −0.0329924
\(177\) 37.8885 2.84788
\(178\) −10.6525 −0.798437
\(179\) −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 \(-0.598329\pi\)
−0.304020 + 0.952666i \(0.598329\pi\)
\(182\) 0 0
\(183\) −31.4164 −2.32237
\(184\) 13.9443 1.02799
\(185\) 0 0
\(186\) 7.41641 0.543797
\(187\) −0.583592 −0.0426765
\(188\) 3.23607 0.236015
\(189\) 0 0
\(190\) 0 0
\(191\) 6.47214 0.468307 0.234154 0.972200i \(-0.424768\pi\)
0.234154 + 0.972200i \(0.424768\pi\)
\(192\) 0.763932 0.0551320
\(193\) −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
\(196\) 0 0
\(197\) 1.47214 0.104885 0.0524427 0.998624i \(-0.483299\pi\)
0.0524427 + 0.998624i \(0.483299\pi\)
\(198\) −1.09017 −0.0774750
\(199\) −7.23607 −0.512951 −0.256476 0.966551i \(-0.582561\pi\)
−0.256476 + 0.966551i \(0.582561\pi\)
\(200\) 0 0
\(201\) −13.7082 −0.966902
\(202\) −2.94427 −0.207158
\(203\) 0 0
\(204\) 12.9443 0.906280
\(205\) 0 0
\(206\) 5.23607 0.364814
\(207\) −46.5967 −3.23870
\(208\) 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
\(213\) −28.1803 −1.93089
\(214\) −4.94427 −0.337983
\(215\) 0 0
\(216\) 32.3607 2.20187
\(217\) 0 0
\(218\) 5.20163 0.352299
\(219\) 28.3607 1.91644
\(220\) 0 0
\(221\) 3.05573 0.205551
\(222\) 6.00000 0.402694
\(223\) 20.1803 1.35138 0.675688 0.737188i \(-0.263847\pi\)
0.675688 + 0.737188i \(0.263847\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 8.90983 0.592673
\(227\) 21.4164 1.42146 0.710728 0.703466i \(-0.248365\pi\)
0.710728 + 0.703466i \(0.248365\pi\)
\(228\) 23.4164 1.55079
\(229\) 4.47214 0.295527 0.147764 0.989023i \(-0.452793\pi\)
0.147764 + 0.989023i \(0.452793\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) −11.1803 −0.734025
\(233\) −7.94427 −0.520447 −0.260223 0.965548i \(-0.583796\pi\)
−0.260223 + 0.965548i \(0.583796\pi\)
\(234\) 5.70820 0.373157
\(235\) 0 0
\(236\) 18.9443 1.23317
\(237\) 36.1803 2.35017
\(238\) 0 0
\(239\) −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 \(-0.463754\pi\)
0.113625 + 0.993524i \(0.463754\pi\)
\(242\) −6.76393 −0.434802
\(243\) −35.5967 −2.28353
\(244\) −15.7082 −1.00561
\(245\) 0 0
\(246\) 9.52786 0.607474
\(247\) 5.52786 0.351730
\(248\) 8.29180 0.526530
\(249\) 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
\(253\) 1.47214 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 \(0.629124\pi\)
\(258\) 3.52786 0.219635
\(259\) 0 0
\(260\) 0 0
\(261\) 37.3607 2.31257
\(262\) 10.4721 0.646971
\(263\) −16.2361 −1.00116 −0.500579 0.865691i \(-0.666880\pi\)
−0.500579 + 0.865691i \(0.666880\pi\)
\(264\) −1.70820 −0.105133
\(265\) 0 0
\(266\) 0 0
\(267\) 55.7771 3.41350
\(268\) −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 \(-0.755897\pi\)
−0.720085 + 0.693885i \(0.755897\pi\)
\(272\) 4.58359 0.277921
\(273\) 0 0
\(274\) 6.76393 0.408624
\(275\) 0 0
\(276\) −32.6525 −1.96545
\(277\) 19.8885 1.19499 0.597493 0.801874i \(-0.296163\pi\)
0.597493 + 0.801874i \(0.296163\pi\)
\(278\) −6.58359 −0.394858
\(279\) −27.7082 −1.65885
\(280\) 0 0
\(281\) −15.3607 −0.916341 −0.458171 0.888864i \(-0.651495\pi\)
−0.458171 + 0.888864i \(0.651495\pi\)
\(282\) 4.00000 0.238197
\(283\) 17.4164 1.03530 0.517649 0.855593i \(-0.326807\pi\)
0.517649 + 0.855593i \(0.326807\pi\)
\(284\) −14.0902 −0.836098
\(285\) 0 0
\(286\) −0.180340 −0.0106637
\(287\) 0 0
\(288\) 41.9787 2.47362
\(289\) −10.8885 −0.640503
\(290\) 0 0
\(291\) −16.9443 −0.993291
\(292\) 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
\(297\) 3.41641 0.198240
\(298\) 2.43769 0.141212
\(299\) −7.70820 −0.445777
\(300\) 0 0
\(301\) 0 0
\(302\) −12.5066 −0.719673
\(303\) 15.4164 0.885649
\(304\) 8.29180 0.475567
\(305\) 0 0
\(306\) 11.4164 0.652633
\(307\) 4.58359 0.261599 0.130800 0.991409i \(-0.458246\pi\)
0.130800 + 0.991409i \(0.458246\pi\)
\(308\) 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 \(0.742690\pi\)
\(312\) 8.94427 0.506370
\(313\) 19.5279 1.10378 0.551890 0.833917i \(-0.313907\pi\)
0.551890 + 0.833917i \(0.313907\pi\)
\(314\) 0.472136 0.0266442
\(315\) 0 0
\(316\) 18.0902 1.01765
\(317\) −25.3607 −1.42440 −0.712199 0.701978i \(-0.752301\pi\)
−0.712199 + 0.701978i \(0.752301\pi\)
\(318\) 16.9443 0.950188
\(319\) −1.18034 −0.0660863
\(320\) 0 0
\(321\) 25.8885 1.44496
\(322\) 0 0
\(323\) 11.0557 0.615157
\(324\) −39.5066 −2.19481
\(325\) 0 0
\(326\) 0.944272 0.0522984
\(327\) −27.2361 −1.50616
\(328\) 10.6525 0.588185
\(329\) 0 0
\(330\) 0 0
\(331\) −24.7082 −1.35809 −0.679043 0.734099i \(-0.737605\pi\)
−0.679043 + 0.734099i \(0.737605\pi\)
\(332\) 12.4721 0.684497
\(333\) −22.4164 −1.22841
\(334\) 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
\(339\) −46.6525 −2.53381
\(340\) 0 0
\(341\) 0.875388 0.0474049
\(342\) 20.6525 1.11676
\(343\) 0 0
\(344\) 3.94427 0.212661
\(345\) 0 0
\(346\) −7.12461 −0.383022
\(347\) −20.2361 −1.08633 −0.543165 0.839626i \(-0.682774\pi\)
−0.543165 + 0.839626i \(0.682774\pi\)
\(348\) 26.1803 1.40341
\(349\) −4.47214 −0.239388 −0.119694 0.992811i \(-0.538191\pi\)
−0.119694 + 0.992811i \(0.538191\pi\)
\(350\) 0 0
\(351\) −17.8885 −0.954820
\(352\) −1.32624 −0.0706887
\(353\) −2.18034 −0.116048 −0.0580239 0.998315i \(-0.518480\pi\)
−0.0580239 + 0.998315i \(0.518480\pi\)
\(354\) 23.4164 1.24457
\(355\) 0 0
\(356\) 27.8885 1.47809
\(357\) 0 0
\(358\) −14.4721 −0.764876
\(359\) −30.1246 −1.58992 −0.794958 0.606664i \(-0.792507\pi\)
−0.794958 + 0.606664i \(0.792507\pi\)
\(360\) 0 0
\(361\) 1.00000 0.0526316
\(362\) −5.05573 −0.265723
\(363\) 35.4164 1.85888
\(364\) 0 0
\(365\) 0 0
\(366\) −19.4164 −1.01491
\(367\) −37.1246 −1.93789 −0.968944 0.247278i \(-0.920464\pi\)
−0.968944 + 0.247278i \(0.920464\pi\)
\(368\) −11.5623 −0.602727
\(369\) −35.5967 −1.85309
\(370\) 0 0
\(371\) 0 0
\(372\) −19.4164 −1.00669
\(373\) 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
\(381\) 44.1803 2.26343
\(382\) 4.00000 0.204658
\(383\) −33.2361 −1.69828 −0.849142 0.528165i \(-0.822880\pi\)
−0.849142 + 0.528165i \(0.822880\pi\)
\(384\) 36.8328 1.87962
\(385\) 0 0
\(386\) −7.67376 −0.390584
\(387\) −13.1803 −0.669994
\(388\) −8.47214 −0.430108
\(389\) 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
\(392\) 0 0
\(393\) −54.8328 −2.76595
\(394\) 0.909830 0.0458366
\(395\) 0 0
\(396\) 2.85410 0.143424
\(397\) 9.05573 0.454494 0.227247 0.973837i \(-0.427028\pi\)
0.227247 + 0.973837i \(0.427028\pi\)
\(398\) −4.47214 −0.224168
\(399\) 0 0
\(400\) 0 0
\(401\) 2.52786 0.126236 0.0631178 0.998006i \(-0.479896\pi\)
0.0631178 + 0.998006i \(0.479896\pi\)
\(402\) −8.47214 −0.422552
\(403\) −4.58359 −0.228325
\(404\) 7.70820 0.383497
\(405\) 0 0
\(406\) 0 0
\(407\) 0.708204 0.0351044
\(408\) 17.8885 0.885615
\(409\) −24.4721 −1.21007 −0.605035 0.796199i \(-0.706841\pi\)
−0.605035 + 0.796199i \(0.706841\pi\)
\(410\) 0 0
\(411\) −35.4164 −1.74696
\(412\) −13.7082 −0.675355
\(413\) 0 0
\(414\) −28.7984 −1.41536
\(415\) 0 0
\(416\) 6.94427 0.340471
\(417\) 34.4721 1.68811
\(418\) −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
\(423\) −14.9443 −0.726615
\(424\) 18.9443 0.920015
\(425\) 0 0
\(426\) −17.4164 −0.843828
\(427\) 0 0
\(428\) 12.9443 0.625685
\(429\) 0.944272 0.0455899
\(430\) 0 0
\(431\) 17.5279 0.844288 0.422144 0.906529i \(-0.361278\pi\)
0.422144 + 0.906529i \(0.361278\pi\)
\(432\) −26.8328 −1.29099
\(433\) −28.3607 −1.36293 −0.681464 0.731852i \(-0.738656\pi\)
−0.681464 + 0.731852i \(0.738656\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) −13.6180 −0.652186
\(437\) −27.8885 −1.33409
\(438\) 17.5279 0.837514
\(439\) 8.29180 0.395746 0.197873 0.980228i \(-0.436597\pi\)
0.197873 + 0.980228i \(0.436597\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 1.88854 0.0898289
\(443\) 19.4164 0.922501 0.461251 0.887270i \(-0.347401\pi\)
0.461251 + 0.887270i \(0.347401\pi\)
\(444\) −15.7082 −0.745478
\(445\) 0 0
\(446\) 12.4721 0.590573
\(447\) −12.7639 −0.603713
\(448\) 0 0
\(449\) 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
\(452\) −23.3262 −1.09717
\(453\) 65.4853 3.07677
\(454\) 13.2361 0.621199
\(455\) 0 0
\(456\) 32.3607 1.51543
\(457\) 12.5279 0.586029 0.293014 0.956108i \(-0.405342\pi\)
0.293014 + 0.956108i \(0.405342\pi\)
\(458\) 2.76393 0.129150
\(459\) −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
\(463\) 13.8885 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 \(0.448634\pi\)
\(468\) −14.9443 −0.690799
\(469\) 0 0
\(470\) 0 0
\(471\) −2.47214 −0.113910
\(472\) 26.1803 1.20505
\(473\) 0.416408 0.0191465
\(474\) 22.3607 1.02706
\(475\) 0 0
\(476\) 0 0
\(477\) −63.3050 −2.89853
\(478\) −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
\(486\) −22.0000 −0.997940
\(487\) −5.76393 −0.261189 −0.130594 0.991436i \(-0.541689\pi\)
−0.130594 + 0.991436i \(0.541689\pi\)
\(488\) −21.7082 −0.982684
\(489\) −4.94427 −0.223588
\(490\) 0 0
\(491\) −5.76393 −0.260123 −0.130061 0.991506i \(-0.541517\pi\)
−0.130061 + 0.991506i \(0.541517\pi\)
\(492\) −24.9443 −1.12457
\(493\) 12.3607 0.556697
\(494\) 3.41641 0.153711
\(495\) 0 0
\(496\) −6.87539 −0.308714
\(497\) 0 0
\(498\) 15.4164 0.690826
\(499\) 11.0557 0.494922 0.247461 0.968898i \(-0.420404\pi\)
0.247461 + 0.968898i \(0.420404\pi\)
\(500\) 0 0
\(501\) −16.9443 −0.757014
\(502\) −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
\(507\) 37.1246 1.64876
\(508\) 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
\(511\) 0 0
\(512\) 18.7082 0.826794
\(513\) −64.7214 −2.85752
\(514\) −7.81966 −0.344910
\(515\) 0 0
\(516\) −9.23607 −0.406595
\(517\) 0.472136 0.0207645
\(518\) 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 \(0.660060\pi\)
\(522\) 23.0902 1.01063
\(523\) 16.3607 0.715403 0.357701 0.933836i \(-0.383561\pi\)
0.357701 + 0.933836i \(0.383561\pi\)
\(524\) −27.4164 −1.19769
\(525\) 0 0
\(526\) −10.0344 −0.437522
\(527\) −9.16718 −0.399329
\(528\) 1.41641 0.0616412
\(529\) 15.8885 0.690806
\(530\) 0 0
\(531\) −87.4853 −3.79654
\(532\) 0 0
\(533\) −5.88854 −0.255061
\(534\) 34.4721 1.49176
\(535\) 0 0
\(536\) −9.47214 −0.409134
\(537\) 75.7771 3.27002
\(538\) 7.23607 0.311969
\(539\) 0 0
\(540\) 0 0
\(541\) 15.9443 0.685498 0.342749 0.939427i \(-0.388642\pi\)
0.342749 + 0.939427i \(0.388642\pi\)
\(542\) −14.6525 −0.629378
\(543\) 26.4721 1.13603
\(544\) 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
\(549\) 72.5410 3.09598
\(550\) 0 0
\(551\) 22.3607 0.952597
\(552\) −45.1246 −1.92063
\(553\) 0 0
\(554\) 12.2918 0.522228
\(555\) 0 0
\(556\) 17.2361 0.730972
\(557\) 9.11146 0.386065 0.193032 0.981192i \(-0.438168\pi\)
0.193032 + 0.981192i \(0.438168\pi\)
\(558\) −17.1246 −0.724943
\(559\) −2.18034 −0.0922186
\(560\) 0 0
\(561\) 1.88854 0.0797344
\(562\) −9.49342 −0.400456
\(563\) 17.4164 0.734014 0.367007 0.930218i \(-0.380382\pi\)
0.367007 + 0.930218i \(0.380382\pi\)
\(564\) −10.4721 −0.440956
\(565\) 0 0
\(566\) 10.7639 0.452442
\(567\) 0 0
\(568\) −19.4721 −0.817033
\(569\) −3.94427 −0.165352 −0.0826762 0.996576i \(-0.526347\pi\)
−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 \(0.222362\pi\)
\(572\) 0.472136 0.0197410
\(573\) −20.9443 −0.874960
\(574\) 0 0
\(575\) 0 0
\(576\) −1.76393 −0.0734972
\(577\) −2.00000 −0.0832611 −0.0416305 0.999133i \(-0.513255\pi\)
−0.0416305 + 0.999133i \(0.513255\pi\)
\(578\) −6.72949 −0.279910
\(579\) 40.1803 1.66984
\(580\) 0 0
\(581\) 0 0
\(582\) −10.4721 −0.434084
\(583\) 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
\(591\) −4.76393 −0.195962
\(592\) −5.56231 −0.228609
\(593\) −37.3050 −1.53193 −0.765965 0.642882i \(-0.777739\pi\)
−0.765965 + 0.642882i \(0.777739\pi\)
\(594\) 2.11146 0.0866341
\(595\) 0 0
\(596\) −6.38197 −0.261416
\(597\) 23.4164 0.958370
\(598\) −4.76393 −0.194812
\(599\) −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 \(-0.228367\pi\)
0.753494 + 0.657455i \(0.228367\pi\)
\(602\) 0 0
\(603\) 31.6525 1.28899
\(604\) 32.7426 1.33228
\(605\) 0 0
\(606\) 9.52786 0.387043
\(607\) −7.12461 −0.289179 −0.144590 0.989492i \(-0.546186\pi\)
−0.144590 + 0.989492i \(0.546186\pi\)
\(608\) 25.1246 1.01894
\(609\) 0 0
\(610\) 0 0
\(611\) −2.47214 −0.100012
\(612\) −29.8885 −1.20817
\(613\) 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 0
\(617\) 5.94427 0.239307 0.119654 0.992816i \(-0.461822\pi\)
0.119654 + 0.992816i \(0.461822\pi\)
\(618\) −16.9443 −0.681599
\(619\) 11.7082 0.470592 0.235296 0.971924i \(-0.424394\pi\)
0.235296 + 0.971924i \(0.424394\pi\)
\(620\) 0 0
\(621\) 90.2492 3.62158
\(622\) −15.0557 −0.603680
\(623\) 0 0
\(624\) −7.41641 −0.296894
\(625\) 0 0
\(626\) 12.0689 0.482370
\(627\) 3.41641 0.136438
\(628\) −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
\(633\) −38.8328 −1.54347
\(634\) −15.6738 −0.622485
\(635\) 0 0
\(636\) −44.3607 −1.75902
\(637\) 0 0
\(638\) −0.729490 −0.0288808
\(639\) 65.0689 2.57409
\(640\) 0 0
\(641\) 43.8328 1.73129 0.865646 0.500656i \(-0.166908\pi\)
0.865646 + 0.500656i \(0.166908\pi\)
\(642\) 16.0000 0.631470
\(643\) 18.4721 0.728470 0.364235 0.931307i \(-0.381331\pi\)
0.364235 + 0.931307i \(0.381331\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 6.83282 0.268834
\(647\) −19.8885 −0.781899 −0.390950 0.920412i \(-0.627853\pi\)
−0.390950 + 0.920412i \(0.627853\pi\)
\(648\) −54.5967 −2.14476
\(649\) 2.76393 0.108494
\(650\) 0 0
\(651\) 0 0
\(652\) −2.47214 −0.0968163
\(653\) −25.0557 −0.980506 −0.490253 0.871580i \(-0.663096\pi\)
−0.490253 + 0.871580i \(0.663096\pi\)
\(654\) −16.8328 −0.658215
\(655\) 0 0
\(656\) −8.83282 −0.344864
\(657\) −65.4853 −2.55482
\(658\) 0 0
\(659\) 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 \(-0.187864\pi\)
0.830834 + 0.556520i \(0.187864\pi\)
\(662\) −15.2705 −0.593505
\(663\) −9.88854 −0.384039
\(664\) 17.2361 0.668889
\(665\) 0 0
\(666\) −13.8541 −0.536836
\(667\) −31.1803 −1.20731
\(668\) −8.47214 −0.327797
\(669\) −65.3050 −2.52484
\(670\) 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 \(0.588997\pi\)
\(678\) −28.8328 −1.10732
\(679\) 0 0
\(680\) 0 0
\(681\) −69.3050 −2.65577
\(682\) 0.541020 0.0207167
\(683\) −14.1246 −0.540463 −0.270232 0.962795i \(-0.587100\pi\)
−0.270232 + 0.962795i \(0.587100\pi\)
\(684\) −54.0689 −2.06738
\(685\) 0 0
\(686\) 0 0
\(687\) −14.4721 −0.552146
\(688\) −3.27051 −0.124687
\(689\) −10.4721 −0.398957
\(690\) 0 0
\(691\) 4.18034 0.159028 0.0795138 0.996834i \(-0.474663\pi\)
0.0795138 + 0.996834i \(0.474663\pi\)
\(692\) 18.6525 0.709061
\(693\) 0 0
\(694\) −12.5066 −0.474743
\(695\) 0 0
\(696\) 36.1803 1.37141
\(697\) −11.7771 −0.446089
\(698\) −2.76393 −0.104616
\(699\) 25.7082 0.972374
\(700\) 0 0
\(701\) −29.0557 −1.09742 −0.548710 0.836013i \(-0.684881\pi\)
−0.548710 + 0.836013i \(0.684881\pi\)
\(702\) −11.0557 −0.417272
\(703\) −13.4164 −0.506009
\(704\) 0.0557281 0.00210033
\(705\) 0 0
\(706\) −1.34752 −0.0507147
\(707\) 0 0
\(708\) −61.3050 −2.30398
\(709\) −12.1115 −0.454855 −0.227428 0.973795i \(-0.573032\pi\)
−0.227428 + 0.973795i \(0.573032\pi\)
\(710\) 0 0
\(711\) −83.5410 −3.13303
\(712\) 38.5410 1.44439
\(713\) 23.1246 0.866024
\(714\) 0 0
\(715\) 0 0
\(716\) 37.8885 1.41596
\(717\) 17.8885 0.668060
\(718\) −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
\(721\) 0 0
\(722\) 0.618034 0.0230008
\(723\) −11.4164 −0.424581
\(724\) 13.2361 0.491915
\(725\) 0 0
\(726\) 21.8885 0.812360
\(727\) −3.05573 −0.113331 −0.0566653 0.998393i \(-0.518047\pi\)
−0.0566653 + 0.998393i \(0.518047\pi\)
\(728\) 0 0
\(729\) 41.9443 1.55349
\(730\) 0 0
\(731\) −4.36068 −0.161286
\(732\) 50.8328 1.87883
\(733\) 4.00000 0.147743 0.0738717 0.997268i \(-0.476464\pi\)
0.0738717 + 0.997268i \(0.476464\pi\)
\(734\) −22.9443 −0.846889
\(735\) 0 0
\(736\) −35.0344 −1.29139
\(737\) −1.00000 −0.0368355
\(738\) −22.0000 −0.809831
\(739\) 25.6525 0.943642 0.471821 0.881694i \(-0.343597\pi\)
0.471821 + 0.881694i \(0.343597\pi\)
\(740\) 0 0
\(741\) −17.8885 −0.657152
\(742\) 0 0
\(743\) 10.4721 0.384185 0.192093 0.981377i \(-0.438473\pi\)
0.192093 + 0.981377i \(0.438473\pi\)
\(744\) −26.8328 −0.983739
\(745\) 0 0
\(746\) 23.3820 0.856075
\(747\) −57.5967 −2.10735
\(748\) 0.944272 0.0345260
\(749\) 0 0
\(750\) 0 0
\(751\) 3.05573 0.111505 0.0557526 0.998445i \(-0.482244\pi\)
0.0557526 + 0.998445i \(0.482244\pi\)
\(752\) −3.70820 −0.135224
\(753\) 20.9443 0.763252
\(754\) 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
\(759\) −4.76393 −0.172920
\(760\) 0 0
\(761\) −27.7771 −1.00692 −0.503459 0.864019i \(-0.667940\pi\)
−0.503459 + 0.864019i \(0.667940\pi\)
\(762\) 27.3050 0.989154
\(763\) 0 0
\(764\) −10.4721 −0.378869
\(765\) 0 0
\(766\) −20.5410 −0.742177
\(767\) −14.4721 −0.522559
\(768\) 21.2361 0.766291
\(769\) 43.0132 1.55109 0.775547 0.631290i \(-0.217474\pi\)
0.775547 + 0.631290i \(0.217474\pi\)
\(770\) 0 0
\(771\) 40.9443 1.47457
\(772\) 20.0902 0.723061
\(773\) 50.1803 1.80486 0.902431 0.430835i \(-0.141781\pi\)
0.902431 + 0.430835i \(0.141781\pi\)
\(774\) −8.14590 −0.292798
\(775\) 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
\(783\) −72.3607 −2.58596
\(784\) 0 0
\(785\) 0 0
\(786\) −33.8885 −1.20876
\(787\) 40.7639 1.45308 0.726539 0.687126i \(-0.241128\pi\)
0.726539 + 0.687126i \(0.241128\pi\)
\(788\) −2.38197 −0.0848540
\(789\) 52.5410 1.87051
\(790\) 0 0
\(791\) 0 0
\(792\) 3.94427 0.140154
\(793\) 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
\(801\) −128.790 −4.55058
\(802\) 1.56231 0.0551669
\(803\) 2.06888 0.0730093
\(804\) 22.1803 0.782240
\(805\) 0 0
\(806\) −2.83282 −0.0997817
\(807\) −37.8885 −1.33374
\(808\) 10.6525 0.374753
\(809\) 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 \(-0.230017\pi\)
0.750075 + 0.661353i \(0.230017\pi\)
\(812\) 0 0
\(813\) 76.7214 2.69074
\(814\) 0.437694 0.0153412
\(815\) 0 0
\(816\) −14.8328 −0.519252
\(817\) −7.88854 −0.275985
\(818\) −15.1246 −0.528820
\(819\) 0 0
\(820\) 0 0
\(821\) 28.8328 1.00627 0.503136 0.864207i \(-0.332179\pi\)
0.503136 + 0.864207i \(0.332179\pi\)
\(822\) −21.8885 −0.763451
\(823\) 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
\(826\) 0 0
\(827\) −41.5410 −1.44452 −0.722261 0.691620i \(-0.756897\pi\)
−0.722261 + 0.691620i \(0.756897\pi\)
\(828\) 75.3951 2.62016
\(829\) 7.63932 0.265325 0.132662 0.991161i \(-0.457647\pi\)
0.132662 + 0.991161i \(0.457647\pi\)
\(830\) 0 0
\(831\) −64.3607 −2.23265
\(832\) −0.291796 −0.0101162
\(833\) 0 0
\(834\) 21.3050 0.737730
\(835\) 0 0
\(836\) 1.70820 0.0590795
\(837\) 53.6656 1.85496
\(838\) 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
\(843\) 49.7082 1.71204
\(844\) −19.4164 −0.668340
\(845\) 0 0
\(846\) −9.23607 −0.317543
\(847\) 0 0
\(848\) −15.7082 −0.539422
\(849\) −56.3607 −1.93429
\(850\) 0 0
\(851\) 18.7082 0.641309
\(852\) 45.5967 1.56212
\(853\) 27.4164 0.938720 0.469360 0.883007i \(-0.344485\pi\)
0.469360 + 0.883007i \(0.344485\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 17.8885 0.611418
\(857\) −15.8197 −0.540389 −0.270195 0.962806i \(-0.587088\pi\)
−0.270195 + 0.962806i \(0.587088\pi\)
\(858\) 0.583592 0.0199235
\(859\) −22.3607 −0.762937 −0.381468 0.924382i \(-0.624581\pi\)
−0.381468 + 0.924382i \(0.624581\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 10.8328 0.368967
\(863\) −18.3475 −0.624557 −0.312278 0.949991i \(-0.601092\pi\)
−0.312278 + 0.949991i \(0.601092\pi\)
\(864\) −81.3050 −2.76605
\(865\) 0 0
\(866\) −17.5279 −0.595621
\(867\) 35.2361 1.19668
\(868\) 0 0
\(869\) 2.63932 0.0895328
\(870\) 0 0
\(871\) 5.23607 0.177417
\(872\) −18.8197 −0.637314
\(873\) 39.1246 1.32417
\(874\) −17.2361 −0.583019
\(875\) 0 0
\(876\) −45.8885 −1.55043
\(877\) −30.3607 −1.02521 −0.512604 0.858625i \(-0.671319\pi\)
−0.512604 + 0.858625i \(0.671319\pi\)
\(878\) 5.12461 0.172947
\(879\) 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
\(883\) 1.40325 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 \(0.616674\pi\)
\(888\) −21.7082 −0.728480
\(889\) 0 0
\(890\) 0 0
\(891\) −5.76393 −0.193099
\(892\) −32.6525 −1.09329
\(893\) −8.94427 −0.299309
\(894\) −7.88854 −0.263832
\(895\) 0 0
\(896\) 0 0
\(897\) 24.9443 0.832865
\(898\) 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
\(906\) 40.4721 1.34460
\(907\) −34.8328 −1.15660 −0.578302 0.815823i \(-0.696285\pi\)
−0.578302 + 0.815823i \(0.696285\pi\)
\(908\) −34.6525 −1.14998
\(909\) −35.5967 −1.18067
\(910\) 0 0
\(911\) 0.819660 0.0271566 0.0135783 0.999908i \(-0.495678\pi\)
0.0135783 + 0.999908i \(0.495678\pi\)
\(912\) −26.8328 −0.888523
\(913\) 1.81966 0.0602220
\(914\) 7.74265 0.256104
\(915\) 0 0
\(916\) −7.23607 −0.239086
\(917\) 0 0
\(918\) −22.1115 −0.729787
\(919\) −27.7639 −0.915848 −0.457924 0.888991i \(-0.651407\pi\)
−0.457924 + 0.888991i \(0.651407\pi\)
\(920\) 0 0
\(921\) −14.8328 −0.488758
\(922\) 8.76393 0.288625
\(923\) 10.7639 0.354299
\(924\) 0 0
\(925\) 0 0
\(926\) 8.58359 0.282074
\(927\) 63.3050 2.07921
\(928\) 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
\(931\) 0 0
\(932\) 12.8541 0.421050
\(933\) 78.8328 2.58087
\(934\) 4.29180 0.140432
\(935\) 0 0
\(936\) −20.6525 −0.675047
\(937\) 35.2361 1.15111 0.575556 0.817762i \(-0.304786\pi\)
0.575556 + 0.817762i \(0.304786\pi\)
\(938\) 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 \(0.472801\pi\)
\(942\) −1.52786 −0.0497805
\(943\) 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 \(-0.691493\pi\)
−0.565957 + 0.824435i \(0.691493\pi\)
\(948\) −58.5410 −1.90132
\(949\) −10.8328 −0.351648
\(950\) 0 0
\(951\) 82.0689 2.66127
\(952\) 0 0
\(953\) −3.47214 −0.112474 −0.0562368 0.998417i \(-0.517910\pi\)
−0.0562368 + 0.998417i \(0.517910\pi\)
\(954\) −39.1246 −1.26671
\(955\) 0 0
\(956\) 8.94427 0.289278
\(957\) 3.81966 0.123472
\(958\) 16.1803 0.522763
\(959\) 0 0
\(960\) 0 0
\(961\) −17.2492 −0.556427
\(962\) −2.29180 −0.0738905
\(963\) −59.7771 −1.92629
\(964\) −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
\(969\) −35.7771 −1.14933
\(970\) 0 0
\(971\) 18.0000 0.577647 0.288824 0.957382i \(-0.406736\pi\)
0.288824 + 0.957382i \(0.406736\pi\)
\(972\) 57.5967 1.84742
\(973\) 0 0
\(974\) −3.56231 −0.114144
\(975\) 0 0
\(976\) 18.0000 0.576166
\(977\) 11.4721 0.367026 0.183513 0.983017i \(-0.441253\pi\)
0.183513 + 0.983017i \(0.441253\pi\)
\(978\) −3.05573 −0.0977114
\(979\) 4.06888 0.130042
\(980\) 0 0
\(981\) 62.8885 2.00788
\(982\) −3.56231 −0.113678
\(983\) −34.5410 −1.10169 −0.550844 0.834608i \(-0.685694\pi\)
−0.550844 + 0.834608i \(0.685694\pi\)
\(984\) −34.4721 −1.09893
\(985\) 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
\(993\) 79.9574 2.53737
\(994\) 0 0
\(995\) 0 0
\(996\) −40.3607 −1.27888
\(997\) −45.4164 −1.43835 −0.719176 0.694828i \(-0.755481\pi\)
−0.719176 + 0.694828i \(0.755481\pi\)
\(998\) 6.83282 0.216289
\(999\) 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.a.n.1.2 2
5.2 odd 4 1225.2.b.k.99.3 4
5.3 odd 4 1225.2.b.k.99.2 4
5.4 even 2 1225.2.a.u.1.1 2
7.6 odd 2 175.2.a.d.1.2 2
21.20 even 2 1575.2.a.s.1.1 2
28.27 even 2 2800.2.a.bh.1.1 2
35.13 even 4 175.2.b.c.99.2 4
35.27 even 4 175.2.b.c.99.3 4
35.34 odd 2 175.2.a.e.1.1 yes 2
105.62 odd 4 1575.2.d.k.1324.2 4
105.83 odd 4 1575.2.d.k.1324.3 4
105.104 even 2 1575.2.a.n.1.2 2
140.27 odd 4 2800.2.g.s.449.4 4
140.83 odd 4 2800.2.g.s.449.1 4
140.139 even 2 2800.2.a.bp.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
175.2.a.d.1.2 2 7.6 odd 2
175.2.a.e.1.1 yes 2 35.34 odd 2
175.2.b.c.99.2 4 35.13 even 4
175.2.b.c.99.3 4 35.27 even 4
1225.2.a.n.1.2 2 1.1 even 1 trivial
1225.2.a.u.1.1 2 5.4 even 2
1225.2.b.k.99.2 4 5.3 odd 4
1225.2.b.k.99.3 4 5.2 odd 4
1575.2.a.n.1.2 2 105.104 even 2
1575.2.a.s.1.1 2 21.20 even 2
1575.2.d.k.1324.2 4 105.62 odd 4
1575.2.d.k.1324.3 4 105.83 odd 4
2800.2.a.bh.1.1 2 28.27 even 2
2800.2.a.bp.1.2 2 140.139 even 2
2800.2.g.s.449.1 4 140.83 odd 4
2800.2.g.s.449.4 4 140.27 odd 4