Properties

Label 175.2.a.e.1.1
Level $175$
Weight $2$
Character 175.1
Self dual yes
Analytic conductor $1.397$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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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 1.1
Root \(-0.618034\) of defining polynomial
Character \(\chi\) \(=\) 175.1

$q$-expansion

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

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 \(-0.570119\pi\)
−0.218508 + 0.975835i \(0.570119\pi\)
\(3\) −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
\(7\) 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 \(-0.445167\pi\)
0.171412 + 0.985199i \(0.445167\pi\)
\(14\) −0.618034 −0.165177
\(15\) 0 0
\(16\) 1.85410 0.463525
\(17\) 2.47214 0.599581 0.299791 0.954005i \(-0.403083\pi\)
0.299791 + 0.954005i \(0.403083\pi\)
\(18\) −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 \(-0.274704\pi\)
0.650155 + 0.759802i \(0.274704\pi\)
\(24\) −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 \(-0.420687\pi\)
0.246598 + 0.969118i \(0.420687\pi\)
\(38\) 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 \(-0.457058\pi\)
0.134499 + 0.990914i \(0.457058\pi\)
\(44\) 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
\(49\) 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 \(-0.302322\pi\)
0.581869 + 0.813283i \(0.302322\pi\)
\(54\) 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 \(-0.583314\pi\)
−0.258759 + 0.965942i \(0.583314\pi\)
\(68\) −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
\(77\) −0.236068 −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 \(-0.639038\pi\)
−0.423043 + 0.906110i \(0.639038\pi\)
\(84\) 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 \(-0.414357\pi\)
0.265821 + 0.964022i \(0.414357\pi\)
\(98\) −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 \(-0.362944\pi\)
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 \(-0.373617\pi\)
0.386695 + 0.922208i \(0.373617\pi\)
\(108\) 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 \(-0.737192\pi\)
−0.678091 + 0.734978i \(0.737192\pi\)
\(114\) −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 \(-0.292881\pi\)
0.605731 + 0.795670i \(0.292881\pi\)
\(128\) 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 \(-0.654851\pi\)
−0.467516 + 0.883985i \(0.654851\pi\)
\(138\) 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 \(-0.490295\pi\)
0.0304842 + 0.999535i \(0.490295\pi\)
\(158\) 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 \(-0.519058\pi\)
−0.0598358 + 0.998208i \(0.519058\pi\)
\(164\) −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 \(0.435064\pi\)
\(168\) −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 \(-0.644392\pi\)
−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 \(-0.401671\pi\)
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 \(-0.352536\pi\)
0.446876 + 0.894596i \(0.352536\pi\)
\(194\) −3.23607 −0.232336
\(195\) 0 0
\(196\) −1.61803 −0.115574
\(197\) −1.47214 −0.104885 −0.0524427 0.998624i \(-0.516701\pi\)
−0.0524427 + 0.998624i \(0.516701\pi\)
\(198\) 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 \(-0.263847\pi\)
0.675688 + 0.737188i \(0.263847\pi\)
\(224\) −5.61803 −0.375371
\(225\) 0 0
\(226\) 8.90983 0.592673
\(227\) 21.4164 1.42146 0.710728 0.703466i \(-0.248365\pi\)
0.710728 + 0.703466i \(0.248365\pi\)
\(228\) −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 \(-0.416204\pi\)
0.260223 + 0.965548i \(0.416204\pi\)
\(234\) −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 \(-0.629124\pi\)
−0.394620 + 0.918844i \(0.629124\pi\)
\(258\) 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 \(-0.333120\pi\)
0.500579 + 0.865691i \(0.333120\pi\)
\(264\) 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 \(-0.703837\pi\)
−0.597493 + 0.801874i \(0.703837\pi\)
\(278\) −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
\(287\) 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 \(-0.863275\pi\)
−0.909160 + 0.416448i \(0.863275\pi\)
\(294\) 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 \(-0.458246\pi\)
0.130800 + 0.991409i \(0.458246\pi\)
\(308\) 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 \(-0.313907\pi\)
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 \(-0.247699\pi\)
0.712199 + 0.701978i \(0.247699\pi\)
\(318\) 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 \(-0.648094\pi\)
−0.448647 + 0.893709i \(0.648094\pi\)
\(338\) 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 \(-0.317226\pi\)
0.543165 + 0.839626i \(0.317226\pi\)
\(348\) 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 \(-0.518480\pi\)
−0.0580239 + 0.998315i \(0.518480\pi\)
\(354\) 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 \(-0.920464\pi\)
−0.968944 + 0.247278i \(0.920464\pi\)
\(368\) 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 \(-0.935365\pi\)
−0.979454 + 0.201665i \(0.935365\pi\)
\(374\) 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 \(-0.822880\pi\)
−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
\(392\) 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 \(-0.427028\pi\)
0.227247 + 0.973837i \(0.427028\pi\)
\(398\) −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 \(-0.738656\pi\)
−0.681464 + 0.731852i \(0.738656\pi\)
\(434\) −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 \(-0.652599\pi\)
−0.461251 + 0.887270i \(0.652599\pi\)
\(444\) 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 \(-0.594658\pi\)
−0.293014 + 0.956108i \(0.594658\pi\)
\(458\) 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 \(-0.604600\pi\)
−0.322728 + 0.946492i \(0.604600\pi\)
\(464\) 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
\(469\) −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 \(-0.458311\pi\)
0.130594 + 0.991436i \(0.458311\pi\)
\(488\) −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 \(-0.557880\pi\)
−0.180836 + 0.983513i \(0.557880\pi\)
\(504\) 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 \(-0.383561\pi\)
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
\(532\) 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 \(-0.566936\pi\)
−0.208738 + 0.977972i \(0.566936\pi\)
\(548\) 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 \(-0.561832\pi\)
−0.193032 + 0.981192i \(0.561832\pi\)
\(558\) −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
\(567\) 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 \(-0.513255\pi\)
−0.0416305 + 0.999133i \(0.513255\pi\)
\(578\) 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 \(-0.670746\pi\)
−0.511058 + 0.859546i \(0.670746\pi\)
\(588\) 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 \(-0.777739\pi\)
−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 \(-0.771633\pi\)
−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 \(-0.546186\pi\)
−0.144590 + 0.989492i \(0.546186\pi\)
\(608\) 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 \(-0.854245\pi\)
−0.896981 + 0.442069i \(0.854245\pi\)
\(614\) −2.83282 −0.114323
\(615\) 0 0
\(616\) −0.527864 −0.0212682
\(617\) −5.94427 −0.239307 −0.119654 0.992816i \(-0.538178\pi\)
−0.119654 + 0.992816i \(0.538178\pi\)
\(618\) 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 \(-0.381331\pi\)
0.364235 + 0.931307i \(0.381331\pi\)
\(644\) −10.0902 −0.397608
\(645\) 0 0
\(646\) 6.83282 0.268834
\(647\) −19.8885 −0.781899 −0.390950 0.920412i \(-0.627853\pi\)
−0.390950 + 0.920412i \(0.627853\pi\)
\(648\) 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 \(-0.336904\pi\)
0.490253 + 0.871580i \(0.336904\pi\)
\(654\) 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 \(-0.377171\pi\)
0.376372 + 0.926469i \(0.377171\pi\)
\(674\) 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
\(679\) 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 \(-0.412900\pi\)
0.270232 + 0.962795i \(0.412900\pi\)
\(684\) 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 \(-0.518047\pi\)
−0.0566653 + 0.998393i \(0.518047\pi\)
\(728\) 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 \(-0.476464\pi\)
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
\(742\) −5.23607 −0.192222
\(743\) −10.4721 −0.384185 −0.192093 0.981377i \(-0.561527\pi\)
−0.192093 + 0.981377i \(0.561527\pi\)
\(744\) −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 \(-0.384178\pi\)
0.355889 + 0.934528i \(0.384178\pi\)
\(758\) 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 \(-0.141781\pi\)
0.902431 + 0.430835i \(0.141781\pi\)
\(774\) −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 \(-0.241128\pi\)
0.726539 + 0.687126i \(0.241128\pi\)
\(788\) 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 \(-0.715822\pi\)
−0.627257 + 0.778813i \(0.715822\pi\)
\(798\) −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 \(-0.686008\pi\)
−0.551668 + 0.834064i \(0.686008\pi\)
\(824\) 18.9443 0.659955
\(825\) 0 0
\(826\) −7.23607 −0.251775
\(827\) 41.5410 1.44452 0.722261 0.691620i \(-0.243103\pi\)
0.722261 + 0.691620i \(0.243103\pi\)
\(828\) −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 \(-0.344485\pi\)
0.469360 + 0.883007i \(0.344485\pi\)
\(854\) 6.00000 0.205316
\(855\) 0 0
\(856\) 17.8885 0.611418
\(857\) −15.8197 −0.540389 −0.270195 0.962806i \(-0.587088\pi\)
−0.270195 + 0.962806i \(0.587088\pi\)
\(858\) −0.583592 −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 \(-0.398908\pi\)
0.312278 + 0.949991i \(0.398908\pi\)
\(864\) 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 \(-0.328681\pi\)
0.512604 + 0.858625i \(0.328681\pi\)
\(878\) 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 \(-0.507517\pi\)
−0.0236116 + 0.999721i \(0.507517\pi\)
\(884\) −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
\(889\) 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 \(-0.303715\pi\)
0.578302 + 0.815823i \(0.303715\pi\)
\(908\) −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 \(-0.304786\pi\)
0.575556 + 0.817762i \(0.304786\pi\)
\(938\) 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 \(-0.308507\pi\)
0.565957 + 0.824435i \(0.308507\pi\)
\(948\) −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 \(-0.482090\pi\)
0.0562368 + 0.998417i \(0.482090\pi\)
\(954\) −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 \(-0.572858\pi\)
−0.226897 + 0.973919i \(0.572858\pi\)
\(968\) −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 \(-0.558747\pi\)
−0.183513 + 0.983017i \(0.558747\pi\)
\(978\) −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
\(987\) 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 \(-0.755481\pi\)
−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 175.2.a.e.1.1 yes 2
3.2 odd 2 1575.2.a.n.1.2 2
4.3 odd 2 2800.2.a.bp.1.2 2
5.2 odd 4 175.2.b.c.99.2 4
5.3 odd 4 175.2.b.c.99.3 4
5.4 even 2 175.2.a.d.1.2 2
7.6 odd 2 1225.2.a.u.1.1 2
15.2 even 4 1575.2.d.k.1324.3 4
15.8 even 4 1575.2.d.k.1324.2 4
15.14 odd 2 1575.2.a.s.1.1 2
20.3 even 4 2800.2.g.s.449.4 4
20.7 even 4 2800.2.g.s.449.1 4
20.19 odd 2 2800.2.a.bh.1.1 2
35.13 even 4 1225.2.b.k.99.3 4
35.27 even 4 1225.2.b.k.99.2 4
35.34 odd 2 1225.2.a.n.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
175.2.a.d.1.2 2 5.4 even 2
175.2.a.e.1.1 yes 2 1.1 even 1 trivial
175.2.b.c.99.2 4 5.2 odd 4
175.2.b.c.99.3 4 5.3 odd 4
1225.2.a.n.1.2 2 35.34 odd 2
1225.2.a.u.1.1 2 7.6 odd 2
1225.2.b.k.99.2 4 35.27 even 4
1225.2.b.k.99.3 4 35.13 even 4
1575.2.a.n.1.2 2 3.2 odd 2
1575.2.a.s.1.1 2 15.14 odd 2
1575.2.d.k.1324.2 4 15.8 even 4
1575.2.d.k.1324.3 4 15.2 even 4
2800.2.a.bh.1.1 2 20.19 odd 2
2800.2.a.bp.1.2 2 4.3 odd 2
2800.2.g.s.449.1 4 20.7 even 4
2800.2.g.s.449.4 4 20.3 even 4