Properties

Label 175.2.a.d.1.2
Level $175$
Weight $2$
Character 175.1
Self dual yes
Analytic conductor $1.397$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Newspace parameters

Level: \( N \) \(=\) \( 175 = 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 175.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(1.39738203537\)
Analytic rank: \(0\)
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: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-0.618034\) of defining polynomial
Character \(\chi\) \(=\) 175.1

$q$-expansion

\(f(q)\) \(=\) \(q+0.618034 q^{2} +3.23607 q^{3} -1.61803 q^{4} +2.00000 q^{6} -1.00000 q^{7} -2.23607 q^{8} +7.47214 q^{9} +O(q^{10})\) \(q+0.618034 q^{2} +3.23607 q^{3} -1.61803 q^{4} +2.00000 q^{6} -1.00000 q^{7} -2.23607 q^{8} +7.47214 q^{9} -0.236068 q^{11} -5.23607 q^{12} -1.23607 q^{13} -0.618034 q^{14} +1.85410 q^{16} -2.47214 q^{17} +4.61803 q^{18} -4.47214 q^{19} -3.23607 q^{21} -0.145898 q^{22} -6.23607 q^{23} -7.23607 q^{24} -0.763932 q^{26} +14.4721 q^{27} +1.61803 q^{28} +5.00000 q^{29} +3.70820 q^{31} +5.61803 q^{32} -0.763932 q^{33} -1.52786 q^{34} -12.0902 q^{36} -3.00000 q^{37} -2.76393 q^{38} -4.00000 q^{39} +4.76393 q^{41} -2.00000 q^{42} -1.76393 q^{43} +0.381966 q^{44} -3.85410 q^{46} +2.00000 q^{47} +6.00000 q^{48} +1.00000 q^{49} -8.00000 q^{51} +2.00000 q^{52} -8.47214 q^{53} +8.94427 q^{54} +2.23607 q^{56} -14.4721 q^{57} +3.09017 q^{58} +11.7082 q^{59} -9.70820 q^{61} +2.29180 q^{62} -7.47214 q^{63} -0.236068 q^{64} -0.472136 q^{66} +4.23607 q^{67} +4.00000 q^{68} -20.1803 q^{69} +8.70820 q^{71} -16.7082 q^{72} +8.76393 q^{73} -1.85410 q^{74} +7.23607 q^{76} +0.236068 q^{77} -2.47214 q^{78} -11.1803 q^{79} +24.4164 q^{81} +2.94427 q^{82} +7.70820 q^{83} +5.23607 q^{84} -1.09017 q^{86} +16.1803 q^{87} +0.527864 q^{88} +17.2361 q^{89} +1.23607 q^{91} +10.0902 q^{92} +12.0000 q^{93} +1.23607 q^{94} +18.1803 q^{96} -5.23607 q^{97} +0.618034 q^{98} -1.76393 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - q^{2} + 2 q^{3} - q^{4} + 4 q^{6} - 2 q^{7} + 6 q^{9} + O(q^{10}) \) \( 2 q - q^{2} + 2 q^{3} - q^{4} + 4 q^{6} - 2 q^{7} + 6 q^{9} + 4 q^{11} - 6 q^{12} + 2 q^{13} + q^{14} - 3 q^{16} + 4 q^{17} + 7 q^{18} - 2 q^{21} - 7 q^{22} - 8 q^{23} - 10 q^{24} - 6 q^{26} + 20 q^{27} + q^{28} + 10 q^{29} - 6 q^{31} + 9 q^{32} - 6 q^{33} - 12 q^{34} - 13 q^{36} - 6 q^{37} - 10 q^{38} - 8 q^{39} + 14 q^{41} - 4 q^{42} - 8 q^{43} + 3 q^{44} - q^{46} + 4 q^{47} + 12 q^{48} + 2 q^{49} - 16 q^{51} + 4 q^{52} - 8 q^{53} - 20 q^{57} - 5 q^{58} + 10 q^{59} - 6 q^{61} + 18 q^{62} - 6 q^{63} + 4 q^{64} + 8 q^{66} + 4 q^{67} + 8 q^{68} - 18 q^{69} + 4 q^{71} - 20 q^{72} + 22 q^{73} + 3 q^{74} + 10 q^{76} - 4 q^{77} + 4 q^{78} + 22 q^{81} - 12 q^{82} + 2 q^{83} + 6 q^{84} + 9 q^{86} + 10 q^{87} + 10 q^{88} + 30 q^{89} - 2 q^{91} + 9 q^{92} + 24 q^{93} - 2 q^{94} + 14 q^{96} - 6 q^{97} - q^{98} - 8 q^{99} + O(q^{100}) \)

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 \(-0.116140\pi\)
0.934172 + 0.356822i \(0.116140\pi\)
\(4\) −1.61803 −0.809017
\(5\) 0 0
\(6\) 2.00000 0.816497
\(7\) −1.00000 −0.377964
\(8\) −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 \(-0.554833\pi\)
−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 \(-0.596917\pi\)
−0.299791 + 0.954005i \(0.596917\pi\)
\(18\) 4.61803 1.08848
\(19\) −4.47214 −1.02598 −0.512989 0.858395i \(-0.671462\pi\)
−0.512989 + 0.858395i \(0.671462\pi\)
\(20\) 0 0
\(21\) −3.23607 −0.706168
\(22\) −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
\(28\) 1.61803 0.305780
\(29\) 5.00000 0.928477 0.464238 0.885710i \(-0.346328\pi\)
0.464238 + 0.885710i \(0.346328\pi\)
\(30\) 0 0
\(31\) 3.70820 0.666013 0.333007 0.942925i \(-0.391937\pi\)
0.333007 + 0.942925i \(0.391937\pi\)
\(32\) 5.61803 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 \(-0.378672\pi\)
0.372001 + 0.928232i \(0.378672\pi\)
\(42\) −2.00000 −0.308607
\(43\) −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 \(-0.453403\pi\)
0.145865 + 0.989305i \(0.453403\pi\)
\(48\) 6.00000 0.866025
\(49\) 1.00000 0.142857
\(50\) 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
\(56\) 2.23607 0.298807
\(57\) −14.4721 −1.91688
\(58\) 3.09017 0.405759
\(59\) 11.7082 1.52428 0.762139 0.647413i \(-0.224149\pi\)
0.762139 + 0.647413i \(0.224149\pi\)
\(60\) 0 0
\(61\) −9.70820 −1.24301 −0.621504 0.783411i \(-0.713478\pi\)
−0.621504 + 0.783411i \(0.713478\pi\)
\(62\) 2.29180 0.291058
\(63\) −7.47214 −0.941401
\(64\) −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 \(-0.328582\pi\)
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
\(78\) −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 \(-0.360962\pi\)
0.423043 + 0.906110i \(0.360962\pi\)
\(84\) 5.23607 0.571302
\(85\) 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 \(-0.133361\pi\)
0.913510 + 0.406817i \(0.133361\pi\)
\(90\) 0 0
\(91\) 1.23607 0.129575
\(92\) 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 \(-0.585643\pi\)
−0.265821 + 0.964022i \(0.585643\pi\)
\(98\) 0.618034 0.0624309
\(99\) −1.76393 −0.177282
\(100\) 0 0
\(101\) 4.76393 0.474029 0.237014 0.971506i \(-0.423831\pi\)
0.237014 + 0.971506i \(0.423831\pi\)
\(102\) −4.94427 −0.489556
\(103\) −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 \(-0.626383\pi\)
−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
\(112\) −1.85410 −0.175196
\(113\) 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
\(119\) 2.47214 0.226620
\(120\) 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
\(126\) −4.61803 −0.411407
\(127\) −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 \(-0.765276\pi\)
−0.740214 + 0.672371i \(0.765276\pi\)
\(132\) 1.23607 0.107586
\(133\) 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 \(-0.345149\pi\)
0.467516 + 0.883985i \(0.345149\pi\)
\(138\) −12.4721 −1.06170
\(139\) 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
\(142\) 5.38197 0.451645
\(143\) 0.291796 0.0244012
\(144\) 13.8541 1.15451
\(145\) 0 0
\(146\) 5.41641 0.448265
\(147\) 3.23607 0.266906
\(148\) 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
\(154\) 0.145898 0.0117568
\(155\) 0 0
\(156\) 6.47214 0.518186
\(157\) −0.763932 −0.0609684 −0.0304842 0.999535i \(-0.509705\pi\)
−0.0304842 + 0.999535i \(0.509705\pi\)
\(158\) −6.90983 −0.549717
\(159\) −27.4164 −2.17426
\(160\) 0 0
\(161\) 6.23607 0.491471
\(162\) 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 \(-0.564936\pi\)
−0.202590 + 0.979264i \(0.564936\pi\)
\(168\) 7.23607 0.558275
\(169\) −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 \(-0.355608\pi\)
0.438224 + 0.898866i \(0.355608\pi\)
\(174\) 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 \(-0.401671\pi\)
0.304020 + 0.952666i \(0.401671\pi\)
\(182\) 0.763932 0.0566264
\(183\) −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
\(189\) −14.4721 −1.05269
\(190\) 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
\(196\) −1.61803 −0.115574
\(197\) 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 \(-0.417439\pi\)
0.256476 + 0.966551i \(0.417439\pi\)
\(200\) 0 0
\(201\) 13.7082 0.966902
\(202\) 2.94427 0.207158
\(203\) −5.00000 −0.350931
\(204\) 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
\(217\) −3.70820 −0.251729
\(218\) 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 \(-0.736153\pi\)
−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 \(-0.751635\pi\)
−0.710728 + 0.703466i \(0.751635\pi\)
\(228\) 23.4164 1.55079
\(229\) −4.47214 −0.295527 −0.147764 0.989023i \(-0.547207\pi\)
−0.147764 + 0.989023i \(0.547207\pi\)
\(230\) 0 0
\(231\) 0.763932 0.0502630
\(232\) −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
\(238\) 1.52786 0.0990367
\(239\) −5.52786 −0.357568 −0.178784 0.983888i \(-0.557216\pi\)
−0.178784 + 0.983888i \(0.557216\pi\)
\(240\) 0 0
\(241\) −3.52786 −0.227250 −0.113625 0.993524i \(-0.536246\pi\)
−0.113625 + 0.993524i \(0.536246\pi\)
\(242\) −6.76393 −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 \(-0.434522\pi\)
0.204259 + 0.978917i \(0.434522\pi\)
\(252\) 12.0902 0.761609
\(253\) 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 \(-0.370876\pi\)
0.394620 + 0.918844i \(0.370876\pi\)
\(258\) −3.52786 −0.219635
\(259\) 3.00000 0.186411
\(260\) 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
\(266\) 2.76393 0.169468
\(267\) 55.7771 3.41350
\(268\) −6.85410 −0.418681
\(269\) −11.7082 −0.713862 −0.356931 0.934131i \(-0.616177\pi\)
−0.356931 + 0.934131i \(0.616177\pi\)
\(270\) 0 0
\(271\) 23.7082 1.44017 0.720085 0.693885i \(-0.244103\pi\)
0.720085 + 0.693885i \(0.244103\pi\)
\(272\) −4.58359 −0.277921
\(273\) 4.00000 0.242091
\(274\) 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 \(-0.673193\pi\)
−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
\(288\) 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 \(-0.136725\pi\)
0.909160 + 0.416448i \(0.136725\pi\)
\(294\) 2.00000 0.116642
\(295\) 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
\(301\) 1.76393 0.101671
\(302\) −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 \(-0.541754\pi\)
−0.130800 + 0.991409i \(0.541754\pi\)
\(308\) −0.381966 −0.0217645
\(309\) −27.4164 −1.55966
\(310\) 0 0
\(311\) 24.3607 1.38137 0.690684 0.723157i \(-0.257310\pi\)
0.690684 + 0.723157i \(0.257310\pi\)
\(312\) 8.94427 0.506370
\(313\) −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 \(-0.752301\pi\)
−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
\(322\) 3.85410 0.214781
\(323\) 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
\(329\) −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
\(333\) −22.4164 −1.22841
\(334\) −3.23607 −0.177070
\(335\) 0 0
\(336\) −6.00000 −0.327327
\(337\) 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
\(343\) −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 \(-0.682774\pi\)
−0.543165 + 0.839626i \(0.682774\pi\)
\(348\) −26.1803 −1.40341
\(349\) 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
\(352\) −1.32624 −0.0706887
\(353\) 2.18034 0.116048 0.0580239 0.998315i \(-0.481520\pi\)
0.0580239 + 0.998315i \(0.481520\pi\)
\(354\) 23.4164 1.24457
\(355\) 0 0
\(356\) −27.8885 −1.47809
\(357\) 8.00000 0.423405
\(358\) −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
\(364\) −2.00000 −0.104828
\(365\) 0 0
\(366\) −19.4164 −1.01491
\(367\) 37.1246 1.93789 0.968944 0.247278i \(-0.0795362\pi\)
0.968944 + 0.247278i \(0.0795362\pi\)
\(368\) −11.5623 −0.602727
\(369\) 35.5967 1.85309
\(370\) 0 0
\(371\) 8.47214 0.439851
\(372\) −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
\(378\) −8.94427 −0.460044
\(379\) −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 \(-0.177120\pi\)
0.849142 + 0.528165i \(0.177120\pi\)
\(384\) −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
\(392\) −2.23607 −0.112938
\(393\) −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 \(-0.572972\pi\)
−0.227247 + 0.973837i \(0.572972\pi\)
\(398\) 4.47214 0.224168
\(399\) 14.4721 0.724513
\(400\) 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
\(406\) −3.09017 −0.153363
\(407\) 0.708204 0.0351044
\(408\) 17.8885 0.885615
\(409\) 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
\(412\) 13.7082 0.675355
\(413\) −11.7082 −0.576123
\(414\) −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 \(-0.720857\pi\)
−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 \(0.602605\pi\)
\(422\) 7.41641 0.361025
\(423\) 14.9443 0.726615
\(424\) 18.9443 0.920015
\(425\) 0 0
\(426\) 17.4164 0.843828
\(427\) 9.70820 0.469813
\(428\) 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 \(-0.261344\pi\)
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
\(438\) 17.5279 0.837514
\(439\) −8.29180 −0.395746 −0.197873 0.980228i \(-0.563403\pi\)
−0.197873 + 0.980228i \(0.563403\pi\)
\(440\) 0 0
\(441\) 7.47214 0.355816
\(442\) 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
\(448\) 0.236068 0.0111532
\(449\) 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 \(-0.607124\pi\)
−0.330222 + 0.943903i \(0.607124\pi\)
\(462\) 0.472136 0.0219658
\(463\) 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 \(-0.551366\pi\)
−0.160671 + 0.987008i \(0.551366\pi\)
\(468\) 14.9443 0.690799
\(469\) −4.23607 −0.195603
\(470\) 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
\(476\) −4.00000 −0.183340
\(477\) −63.3050 −2.89853
\(478\) −3.41641 −0.156263
\(479\) −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
\(482\) −2.18034 −0.0993118
\(483\) 20.1803 0.918237
\(484\) 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
\(497\) −8.70820 −0.390616
\(498\) 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 \(-0.442120\pi\)
0.180836 + 0.983513i \(0.442120\pi\)
\(504\) 16.7082 0.744243
\(505\) 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 \(-0.142875\pi\)
0.900945 + 0.433934i \(0.142875\pi\)
\(510\) 0 0
\(511\) −8.76393 −0.387694
\(512\) 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
\(518\) 1.85410 0.0814646
\(519\) 37.3050 1.63751
\(520\) 0 0
\(521\) 22.0000 0.963837 0.481919 0.876216i \(-0.339940\pi\)
0.481919 + 0.876216i \(0.339940\pi\)
\(522\) 23.0902 1.01063
\(523\) −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
\(528\) −1.41641 −0.0616412
\(529\) 15.8885 0.690806
\(530\) 0 0
\(531\) 87.4853 3.79654
\(532\) −7.23607 −0.313723
\(533\) −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
\(539\) −0.236068 −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
\(543\) 26.4721 1.13603
\(544\) −13.8885 −0.595466
\(545\) 0 0
\(546\) 2.47214 0.105798
\(547\) 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
\(553\) 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 \(-0.438168\pi\)
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 \(-0.619618\pi\)
−0.367007 + 0.930218i \(0.619618\pi\)
\(564\) −10.4721 −0.440956
\(565\) 0 0
\(566\) −10.7639 −0.452442
\(567\) −24.4164 −1.02539
\(568\) −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
\(574\) −2.94427 −0.122892
\(575\) 0 0
\(576\) −1.76393 −0.0734972
\(577\) 2.00000 0.0832611 0.0416305 0.999133i \(-0.486745\pi\)
0.0416305 + 0.999133i \(0.486745\pi\)
\(578\) −6.72949 −0.279910
\(579\) −40.1803 −1.66984
\(580\) 0 0
\(581\) −7.70820 −0.319790
\(582\) −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 \(-0.329254\pi\)
0.511058 + 0.859546i \(0.329254\pi\)
\(588\) −5.23607 −0.215932
\(589\) −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 \(-0.222261\pi\)
0.765965 + 0.642882i \(0.222261\pi\)
\(594\) −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 \(-0.771633\pi\)
−0.753494 + 0.657455i \(0.771633\pi\)
\(602\) 1.09017 0.0444320
\(603\) 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 \(-0.453814\pi\)
0.144590 + 0.989492i \(0.453814\pi\)
\(608\) −25.1246 −1.01894
\(609\) −16.1803 −0.655660
\(610\) 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
\(616\) −0.527864 −0.0212682
\(617\) 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 \(-0.575606\pi\)
−0.235296 + 0.971924i \(0.575606\pi\)
\(620\) 0 0
\(621\) −90.2492 −3.62158
\(622\) 15.0557 0.603680
\(623\) −17.2361 −0.690548
\(624\) −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
\(637\) −1.23607 −0.0489748
\(638\) −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 \(-0.618669\pi\)
−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 \(-0.372147\pi\)
0.390950 + 0.920412i \(0.372147\pi\)
\(648\) −54.5967 −2.14476
\(649\) −2.76393 −0.108494
\(650\) 0 0
\(651\) −12.0000 −0.470317
\(652\) −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
\(658\) −1.23607 −0.0481869
\(659\) 17.8885 0.696839 0.348419 0.937339i \(-0.386719\pi\)
0.348419 + 0.937339i \(0.386719\pi\)
\(660\) 0 0
\(661\) −42.7214 −1.66167 −0.830834 0.556520i \(-0.812136\pi\)
−0.830834 + 0.556520i \(0.812136\pi\)
\(662\) −15.2705 −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
\(672\) −18.1803 −0.701322
\(673\) −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 \(-0.411003\pi\)
0.275963 + 0.961168i \(0.411003\pi\)
\(678\) 28.8328 1.10732
\(679\) 5.23607 0.200942
\(680\) 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
\(686\) −0.618034 −0.0235966
\(687\) −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 \(-0.525337\pi\)
−0.0795138 + 0.996834i \(0.525337\pi\)
\(692\) −18.6525 −0.709061
\(693\) 1.76393 0.0670062
\(694\) −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
\(707\) −4.76393 −0.179166
\(708\) −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
\(714\) 4.94427 0.185035
\(715\) 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 \(-0.402442\pi\)
0.301712 + 0.953399i \(0.402442\pi\)
\(720\) 0 0
\(721\) 8.47214 0.315519
\(722\) 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 \(-0.481953\pi\)
0.0566653 + 0.998393i \(0.481953\pi\)
\(728\) −2.76393 −0.102438
\(729\) 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 \(-0.523536\pi\)
−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
\(738\) 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
\(742\) 5.23607 0.192222
\(743\) 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
\(749\) 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
\(753\) 20.9443 0.763252
\(754\) −3.81966 −0.139104
\(755\) 0 0
\(756\) 23.4164 0.851647
\(757\) −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 \(-0.332060\pi\)
0.503459 + 0.864019i \(0.332060\pi\)
\(762\) −27.3050 −0.989154
\(763\) −8.41641 −0.304694
\(764\) −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 \(-0.782526\pi\)
−0.775547 + 0.631290i \(0.782526\pi\)
\(770\) 0 0
\(771\) 40.9443 1.47457
\(772\) 20.0902 0.723061
\(773\) −50.1803 −1.80486 −0.902431 0.430835i \(-0.858219\pi\)
−0.902431 + 0.430835i \(0.858219\pi\)
\(774\) −8.14590 −0.292798
\(775\) 0 0
\(776\) 11.7082 0.420300
\(777\) 9.70820 0.348280
\(778\) 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
\(784\) 1.85410 0.0662179
\(785\) 0 0
\(786\) −33.8885 −1.20876
\(787\) −40.7639 −1.45308 −0.726539 0.687126i \(-0.758872\pi\)
−0.726539 + 0.687126i \(0.758872\pi\)
\(788\) −2.38197 −0.0848540
\(789\) −52.5410 −1.87051
\(790\) 0 0
\(791\) −14.4164 −0.512588
\(792\) 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 \(-0.284178\pi\)
0.627257 + 0.778813i \(0.284178\pi\)
\(798\) 8.94427 0.316624
\(799\) −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 \(-0.769983\pi\)
−0.750075 + 0.661353i \(0.769983\pi\)
\(812\) 8.09017 0.283909
\(813\) 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
\(819\) 9.23607 0.322734
\(820\) 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
\(826\) −7.23607 −0.251775
\(827\) −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 \(-0.542353\pi\)
−0.132662 + 0.991161i \(0.542353\pi\)
\(830\) 0 0
\(831\) 64.3607 2.23265
\(832\) 0.291796 0.0101162
\(833\) −2.47214 −0.0856544
\(834\) 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 \(-0.677478\pi\)
−0.529120 + 0.848547i \(0.677478\pi\)
\(840\) 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
\(847\) 10.9443 0.376050
\(848\) −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 \(-0.655515\pi\)
−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 \(-0.412912\pi\)
0.270195 + 0.962806i \(0.412912\pi\)
\(858\) 0.583592 0.0199235
\(859\) 22.3607 0.762937 0.381468 0.924382i \(-0.375419\pi\)
0.381468 + 0.924382i \(0.375419\pi\)
\(860\) 0 0
\(861\) −15.4164 −0.525390
\(862\) 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
\(868\) 6.00000 0.203653
\(869\) 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 \(-0.468744\pi\)
0.0980347 + 0.995183i \(0.468744\pi\)
\(882\) 4.61803 0.155497
\(883\) 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 \(-0.383326\pi\)
0.358390 + 0.933572i \(0.383326\pi\)
\(888\) 21.7082 0.728480
\(889\) 13.6525 0.457889
\(890\) 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
\(896\) 11.3820 0.380245
\(897\) 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
\(903\) 5.70820 0.189957
\(904\) −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
\(917\) 16.9443 0.559549
\(918\) −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
\(924\) −1.23607 −0.0406637
\(925\) 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 \(-0.283810\pi\)
0.628157 + 0.778087i \(0.283810\pi\)
\(930\) 0 0
\(931\) −4.47214 −0.146568
\(932\) 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 \(-0.695214\pi\)
−0.575556 + 0.817762i \(0.695214\pi\)
\(938\) −2.61803 −0.0854818
\(939\) −63.1935 −2.06224
\(940\) 0 0
\(941\) −5.23607 −0.170691 −0.0853455 0.996351i \(-0.527199\pi\)
−0.0853455 + 0.996351i \(0.527199\pi\)
\(942\) −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
\(952\) −5.52786 −0.179159
\(953\) −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
\(959\) −10.9443 −0.353409
\(960\) 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
\(966\) 12.4721 0.401284
\(967\) 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 \(-0.593264\pi\)
−0.288824 + 0.957382i \(0.593264\pi\)
\(972\) −57.5967 −1.84742
\(973\) −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 \(-0.441253\pi\)
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 \(-0.314306\pi\)
0.550844 + 0.834608i \(0.314306\pi\)
\(984\) −34.4721 −1.09893
\(985\) 0 0
\(986\) −7.63932 −0.243286
\(987\) −6.47214 −0.206010
\(988\) −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
\(994\) −5.38197 −0.170706
\(995\) 0 0
\(996\) −40.3607 −1.27888
\(997\) 45.4164 1.43835 0.719176 0.694828i \(-0.244519\pi\)
0.719176 + 0.694828i \(0.244519\pi\)
\(998\) 6.83282 0.216289
\(999\) −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 175.2.a.d.1.2 2
3.2 odd 2 1575.2.a.s.1.1 2
4.3 odd 2 2800.2.a.bh.1.1 2
5.2 odd 4 175.2.b.c.99.3 4
5.3 odd 4 175.2.b.c.99.2 4
5.4 even 2 175.2.a.e.1.1 yes 2
7.6 odd 2 1225.2.a.n.1.2 2
15.2 even 4 1575.2.d.k.1324.2 4
15.8 even 4 1575.2.d.k.1324.3 4
15.14 odd 2 1575.2.a.n.1.2 2
20.3 even 4 2800.2.g.s.449.1 4
20.7 even 4 2800.2.g.s.449.4 4
20.19 odd 2 2800.2.a.bp.1.2 2
35.13 even 4 1225.2.b.k.99.2 4
35.27 even 4 1225.2.b.k.99.3 4
35.34 odd 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 1.1 even 1 trivial
175.2.a.e.1.1 yes 2 5.4 even 2
175.2.b.c.99.2 4 5.3 odd 4
175.2.b.c.99.3 4 5.2 odd 4
1225.2.a.n.1.2 2 7.6 odd 2
1225.2.a.u.1.1 2 35.34 odd 2
1225.2.b.k.99.2 4 35.13 even 4
1225.2.b.k.99.3 4 35.27 even 4
1575.2.a.n.1.2 2 15.14 odd 2
1575.2.a.s.1.1 2 3.2 odd 2
1575.2.d.k.1324.2 4 15.2 even 4
1575.2.d.k.1324.3 4 15.8 even 4
2800.2.a.bh.1.1 2 4.3 odd 2
2800.2.a.bp.1.2 2 20.19 odd 2
2800.2.g.s.449.1 4 20.3 even 4
2800.2.g.s.449.4 4 20.7 even 4