Properties

Label 825.2.a.n.1.1
Level $825$
Weight $2$
Character 825.1
Self dual yes
Analytic conductor $6.588$
Analytic rank $0$
Dimension $3$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 825 = 3 \cdot 5^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 825.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(6.58765816676\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.148.1
Defining polynomial: \( x^{3} - x^{2} - 3x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 165)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(0.311108\) of defining polynomial
Character \(\chi\) \(=\) 825.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.21432 q^{2} -1.00000 q^{3} -0.525428 q^{4} +1.21432 q^{6} +4.90321 q^{7} +3.06668 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-1.21432 q^{2} -1.00000 q^{3} -0.525428 q^{4} +1.21432 q^{6} +4.90321 q^{7} +3.06668 q^{8} +1.00000 q^{9} -1.00000 q^{11} +0.525428 q^{12} +4.14764 q^{13} -5.95407 q^{14} -2.67307 q^{16} -5.33185 q^{17} -1.21432 q^{18} -5.18421 q^{19} -4.90321 q^{21} +1.21432 q^{22} +4.00000 q^{23} -3.06668 q^{24} -5.03657 q^{26} -1.00000 q^{27} -2.57628 q^{28} +1.80642 q^{29} +2.62222 q^{31} -2.88739 q^{32} +1.00000 q^{33} +6.47457 q^{34} -0.525428 q^{36} +5.80642 q^{37} +6.29529 q^{38} -4.14764 q^{39} +1.80642 q^{41} +5.95407 q^{42} +4.90321 q^{43} +0.525428 q^{44} -4.85728 q^{46} +7.05086 q^{47} +2.67307 q^{48} +17.0415 q^{49} +5.33185 q^{51} -2.17929 q^{52} -7.18421 q^{53} +1.21432 q^{54} +15.0366 q^{56} +5.18421 q^{57} -2.19358 q^{58} +1.67307 q^{59} +0.755569 q^{61} -3.18421 q^{62} +4.90321 q^{63} +8.85236 q^{64} -1.21432 q^{66} -4.85728 q^{67} +2.80150 q^{68} -4.00000 q^{69} +0.428639 q^{71} +3.06668 q^{72} +12.7096 q^{73} -7.05086 q^{74} +2.72393 q^{76} -4.90321 q^{77} +5.03657 q^{78} -6.42864 q^{79} +1.00000 q^{81} -2.19358 q^{82} -2.90321 q^{83} +2.57628 q^{84} -5.95407 q^{86} -1.80642 q^{87} -3.06668 q^{88} +0.622216 q^{89} +20.3368 q^{91} -2.10171 q^{92} -2.62222 q^{93} -8.56199 q^{94} +2.88739 q^{96} -2.75557 q^{97} -20.6938 q^{98} -1.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 3 q^{2} - 3 q^{3} + 5 q^{4} - 3 q^{6} + 8 q^{7} + 9 q^{8} + 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q + 3 q^{2} - 3 q^{3} + 5 q^{4} - 3 q^{6} + 8 q^{7} + 9 q^{8} + 3 q^{9} - 3 q^{11} - 5 q^{12} + 6 q^{13} + 2 q^{14} + 5 q^{16} + 4 q^{17} + 3 q^{18} - 2 q^{19} - 8 q^{21} - 3 q^{22} + 12 q^{23} - 9 q^{24} - 8 q^{26} - 3 q^{27} + 12 q^{28} - 8 q^{29} + 8 q^{31} + 11 q^{32} + 3 q^{33} + 26 q^{34} + 5 q^{36} + 4 q^{37} + 6 q^{38} - 6 q^{39} - 8 q^{41} - 2 q^{42} + 8 q^{43} - 5 q^{44} + 12 q^{46} + 8 q^{47} - 5 q^{48} + 11 q^{49} - 4 q^{51} - 26 q^{52} - 8 q^{53} - 3 q^{54} + 38 q^{56} + 2 q^{57} - 20 q^{58} - 8 q^{59} + 2 q^{61} + 4 q^{62} + 8 q^{63} + 33 q^{64} + 3 q^{66} + 12 q^{67} + 28 q^{68} - 12 q^{69} - 12 q^{71} + 9 q^{72} + 18 q^{73} - 8 q^{74} - 18 q^{76} - 8 q^{77} + 8 q^{78} - 6 q^{79} + 3 q^{81} - 20 q^{82} - 2 q^{83} - 12 q^{84} + 2 q^{86} + 8 q^{87} - 9 q^{88} + 2 q^{89} + 8 q^{91} + 20 q^{92} - 8 q^{93} - 12 q^{94} - 11 q^{96} - 8 q^{97} - 29 q^{98} - 3 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.21432 −0.858654 −0.429327 0.903149i \(-0.641249\pi\)
−0.429327 + 0.903149i \(0.641249\pi\)
\(3\) −1.00000 −0.577350
\(4\) −0.525428 −0.262714
\(5\) 0 0
\(6\) 1.21432 0.495744
\(7\) 4.90321 1.85324 0.926620 0.375999i \(-0.122700\pi\)
0.926620 + 0.375999i \(0.122700\pi\)
\(8\) 3.06668 1.08423
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) −1.00000 −0.301511
\(12\) 0.525428 0.151678
\(13\) 4.14764 1.15035 0.575175 0.818031i \(-0.304934\pi\)
0.575175 + 0.818031i \(0.304934\pi\)
\(14\) −5.95407 −1.59129
\(15\) 0 0
\(16\) −2.67307 −0.668268
\(17\) −5.33185 −1.29316 −0.646582 0.762845i \(-0.723802\pi\)
−0.646582 + 0.762845i \(0.723802\pi\)
\(18\) −1.21432 −0.286218
\(19\) −5.18421 −1.18934 −0.594669 0.803970i \(-0.702717\pi\)
−0.594669 + 0.803970i \(0.702717\pi\)
\(20\) 0 0
\(21\) −4.90321 −1.06997
\(22\) 1.21432 0.258894
\(23\) 4.00000 0.834058 0.417029 0.908893i \(-0.363071\pi\)
0.417029 + 0.908893i \(0.363071\pi\)
\(24\) −3.06668 −0.625983
\(25\) 0 0
\(26\) −5.03657 −0.987752
\(27\) −1.00000 −0.192450
\(28\) −2.57628 −0.486872
\(29\) 1.80642 0.335444 0.167722 0.985834i \(-0.446359\pi\)
0.167722 + 0.985834i \(0.446359\pi\)
\(30\) 0 0
\(31\) 2.62222 0.470964 0.235482 0.971879i \(-0.424333\pi\)
0.235482 + 0.971879i \(0.424333\pi\)
\(32\) −2.88739 −0.510423
\(33\) 1.00000 0.174078
\(34\) 6.47457 1.11038
\(35\) 0 0
\(36\) −0.525428 −0.0875713
\(37\) 5.80642 0.954570 0.477285 0.878749i \(-0.341621\pi\)
0.477285 + 0.878749i \(0.341621\pi\)
\(38\) 6.29529 1.02123
\(39\) −4.14764 −0.664154
\(40\) 0 0
\(41\) 1.80642 0.282116 0.141058 0.990001i \(-0.454950\pi\)
0.141058 + 0.990001i \(0.454950\pi\)
\(42\) 5.95407 0.918732
\(43\) 4.90321 0.747733 0.373866 0.927483i \(-0.378032\pi\)
0.373866 + 0.927483i \(0.378032\pi\)
\(44\) 0.525428 0.0792112
\(45\) 0 0
\(46\) −4.85728 −0.716167
\(47\) 7.05086 1.02847 0.514236 0.857648i \(-0.328075\pi\)
0.514236 + 0.857648i \(0.328075\pi\)
\(48\) 2.67307 0.385825
\(49\) 17.0415 2.43450
\(50\) 0 0
\(51\) 5.33185 0.746609
\(52\) −2.17929 −0.302213
\(53\) −7.18421 −0.986827 −0.493413 0.869795i \(-0.664251\pi\)
−0.493413 + 0.869795i \(0.664251\pi\)
\(54\) 1.21432 0.165248
\(55\) 0 0
\(56\) 15.0366 2.00935
\(57\) 5.18421 0.686665
\(58\) −2.19358 −0.288031
\(59\) 1.67307 0.217815 0.108908 0.994052i \(-0.465265\pi\)
0.108908 + 0.994052i \(0.465265\pi\)
\(60\) 0 0
\(61\) 0.755569 0.0967407 0.0483703 0.998829i \(-0.484597\pi\)
0.0483703 + 0.998829i \(0.484597\pi\)
\(62\) −3.18421 −0.404395
\(63\) 4.90321 0.617747
\(64\) 8.85236 1.10654
\(65\) 0 0
\(66\) −1.21432 −0.149472
\(67\) −4.85728 −0.593411 −0.296706 0.954969i \(-0.595888\pi\)
−0.296706 + 0.954969i \(0.595888\pi\)
\(68\) 2.80150 0.339732
\(69\) −4.00000 −0.481543
\(70\) 0 0
\(71\) 0.428639 0.0508701 0.0254351 0.999676i \(-0.491903\pi\)
0.0254351 + 0.999676i \(0.491903\pi\)
\(72\) 3.06668 0.361411
\(73\) 12.7096 1.48755 0.743775 0.668430i \(-0.233033\pi\)
0.743775 + 0.668430i \(0.233033\pi\)
\(74\) −7.05086 −0.819645
\(75\) 0 0
\(76\) 2.72393 0.312456
\(77\) −4.90321 −0.558773
\(78\) 5.03657 0.570279
\(79\) −6.42864 −0.723278 −0.361639 0.932318i \(-0.617783\pi\)
−0.361639 + 0.932318i \(0.617783\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) −2.19358 −0.242240
\(83\) −2.90321 −0.318669 −0.159334 0.987225i \(-0.550935\pi\)
−0.159334 + 0.987225i \(0.550935\pi\)
\(84\) 2.57628 0.281095
\(85\) 0 0
\(86\) −5.95407 −0.642044
\(87\) −1.80642 −0.193669
\(88\) −3.06668 −0.326909
\(89\) 0.622216 0.0659547 0.0329774 0.999456i \(-0.489501\pi\)
0.0329774 + 0.999456i \(0.489501\pi\)
\(90\) 0 0
\(91\) 20.3368 2.13187
\(92\) −2.10171 −0.219118
\(93\) −2.62222 −0.271911
\(94\) −8.56199 −0.883102
\(95\) 0 0
\(96\) 2.88739 0.294693
\(97\) −2.75557 −0.279786 −0.139893 0.990167i \(-0.544676\pi\)
−0.139893 + 0.990167i \(0.544676\pi\)
\(98\) −20.6938 −2.09039
\(99\) −1.00000 −0.100504
\(100\) 0 0
\(101\) −17.8064 −1.77181 −0.885903 0.463871i \(-0.846460\pi\)
−0.885903 + 0.463871i \(0.846460\pi\)
\(102\) −6.47457 −0.641078
\(103\) 4.94914 0.487654 0.243827 0.969819i \(-0.421597\pi\)
0.243827 + 0.969819i \(0.421597\pi\)
\(104\) 12.7195 1.24725
\(105\) 0 0
\(106\) 8.72393 0.847343
\(107\) 11.1985 1.08260 0.541300 0.840830i \(-0.317932\pi\)
0.541300 + 0.840830i \(0.317932\pi\)
\(108\) 0.525428 0.0505593
\(109\) 15.7146 1.50518 0.752591 0.658488i \(-0.228804\pi\)
0.752591 + 0.658488i \(0.228804\pi\)
\(110\) 0 0
\(111\) −5.80642 −0.551121
\(112\) −13.1066 −1.23846
\(113\) −1.76494 −0.166031 −0.0830156 0.996548i \(-0.526455\pi\)
−0.0830156 + 0.996548i \(0.526455\pi\)
\(114\) −6.29529 −0.589608
\(115\) 0 0
\(116\) −0.949145 −0.0881259
\(117\) 4.14764 0.383450
\(118\) −2.03164 −0.187028
\(119\) −26.1432 −2.39654
\(120\) 0 0
\(121\) 1.00000 0.0909091
\(122\) −0.917502 −0.0830667
\(123\) −1.80642 −0.162880
\(124\) −1.37778 −0.123729
\(125\) 0 0
\(126\) −5.95407 −0.530430
\(127\) 18.7096 1.66021 0.830106 0.557606i \(-0.188280\pi\)
0.830106 + 0.557606i \(0.188280\pi\)
\(128\) −4.97481 −0.439715
\(129\) −4.90321 −0.431704
\(130\) 0 0
\(131\) −1.24443 −0.108726 −0.0543632 0.998521i \(-0.517313\pi\)
−0.0543632 + 0.998521i \(0.517313\pi\)
\(132\) −0.525428 −0.0457326
\(133\) −25.4193 −2.20413
\(134\) 5.89829 0.509535
\(135\) 0 0
\(136\) −16.3511 −1.40209
\(137\) 18.7971 1.60594 0.802970 0.596019i \(-0.203252\pi\)
0.802970 + 0.596019i \(0.203252\pi\)
\(138\) 4.85728 0.413479
\(139\) 14.0415 1.19098 0.595492 0.803361i \(-0.296957\pi\)
0.595492 + 0.803361i \(0.296957\pi\)
\(140\) 0 0
\(141\) −7.05086 −0.593789
\(142\) −0.520505 −0.0436798
\(143\) −4.14764 −0.346843
\(144\) −2.67307 −0.222756
\(145\) 0 0
\(146\) −15.4336 −1.27729
\(147\) −17.0415 −1.40556
\(148\) −3.05086 −0.250779
\(149\) −3.05086 −0.249936 −0.124968 0.992161i \(-0.539883\pi\)
−0.124968 + 0.992161i \(0.539883\pi\)
\(150\) 0 0
\(151\) −0.326929 −0.0266051 −0.0133026 0.999912i \(-0.504234\pi\)
−0.0133026 + 0.999912i \(0.504234\pi\)
\(152\) −15.8983 −1.28952
\(153\) −5.33185 −0.431055
\(154\) 5.95407 0.479792
\(155\) 0 0
\(156\) 2.17929 0.174483
\(157\) −19.9081 −1.58884 −0.794421 0.607367i \(-0.792226\pi\)
−0.794421 + 0.607367i \(0.792226\pi\)
\(158\) 7.80642 0.621046
\(159\) 7.18421 0.569745
\(160\) 0 0
\(161\) 19.6128 1.54571
\(162\) −1.21432 −0.0954060
\(163\) 12.1748 0.953607 0.476804 0.879010i \(-0.341795\pi\)
0.476804 + 0.879010i \(0.341795\pi\)
\(164\) −0.949145 −0.0741158
\(165\) 0 0
\(166\) 3.52543 0.273626
\(167\) 13.0049 1.00635 0.503176 0.864184i \(-0.332165\pi\)
0.503176 + 0.864184i \(0.332165\pi\)
\(168\) −15.0366 −1.16010
\(169\) 4.20294 0.323303
\(170\) 0 0
\(171\) −5.18421 −0.396446
\(172\) −2.57628 −0.196440
\(173\) 13.8938 1.05633 0.528165 0.849142i \(-0.322880\pi\)
0.528165 + 0.849142i \(0.322880\pi\)
\(174\) 2.19358 0.166295
\(175\) 0 0
\(176\) 2.67307 0.201490
\(177\) −1.67307 −0.125756
\(178\) −0.755569 −0.0566323
\(179\) 12.8573 0.960998 0.480499 0.876995i \(-0.340456\pi\)
0.480499 + 0.876995i \(0.340456\pi\)
\(180\) 0 0
\(181\) 0.917502 0.0681974 0.0340987 0.999418i \(-0.489144\pi\)
0.0340987 + 0.999418i \(0.489144\pi\)
\(182\) −24.6953 −1.83054
\(183\) −0.755569 −0.0558532
\(184\) 12.2667 0.904314
\(185\) 0 0
\(186\) 3.18421 0.233477
\(187\) 5.33185 0.389904
\(188\) −3.70471 −0.270194
\(189\) −4.90321 −0.356656
\(190\) 0 0
\(191\) 14.3684 1.03966 0.519831 0.854269i \(-0.325995\pi\)
0.519831 + 0.854269i \(0.325995\pi\)
\(192\) −8.85236 −0.638864
\(193\) −11.7605 −0.846539 −0.423269 0.906004i \(-0.639118\pi\)
−0.423269 + 0.906004i \(0.639118\pi\)
\(194\) 3.34614 0.240239
\(195\) 0 0
\(196\) −8.95407 −0.639576
\(197\) −3.82071 −0.272215 −0.136107 0.990694i \(-0.543459\pi\)
−0.136107 + 0.990694i \(0.543459\pi\)
\(198\) 1.21432 0.0862979
\(199\) −13.7146 −0.972199 −0.486100 0.873903i \(-0.661581\pi\)
−0.486100 + 0.873903i \(0.661581\pi\)
\(200\) 0 0
\(201\) 4.85728 0.342606
\(202\) 21.6227 1.52137
\(203\) 8.85728 0.621659
\(204\) −2.80150 −0.196144
\(205\) 0 0
\(206\) −6.00984 −0.418726
\(207\) 4.00000 0.278019
\(208\) −11.0869 −0.768741
\(209\) 5.18421 0.358599
\(210\) 0 0
\(211\) 1.95851 0.134830 0.0674148 0.997725i \(-0.478525\pi\)
0.0674148 + 0.997725i \(0.478525\pi\)
\(212\) 3.77478 0.259253
\(213\) −0.428639 −0.0293699
\(214\) −13.5986 −0.929578
\(215\) 0 0
\(216\) −3.06668 −0.208661
\(217\) 12.8573 0.872809
\(218\) −19.0825 −1.29243
\(219\) −12.7096 −0.858838
\(220\) 0 0
\(221\) −22.1146 −1.48759
\(222\) 7.05086 0.473222
\(223\) −26.0098 −1.74175 −0.870874 0.491506i \(-0.836446\pi\)
−0.870874 + 0.491506i \(0.836446\pi\)
\(224\) −14.1575 −0.945937
\(225\) 0 0
\(226\) 2.14320 0.142563
\(227\) −6.34122 −0.420882 −0.210441 0.977607i \(-0.567490\pi\)
−0.210441 + 0.977607i \(0.567490\pi\)
\(228\) −2.72393 −0.180396
\(229\) −23.3274 −1.54152 −0.770759 0.637127i \(-0.780123\pi\)
−0.770759 + 0.637127i \(0.780123\pi\)
\(230\) 0 0
\(231\) 4.90321 0.322608
\(232\) 5.53972 0.363700
\(233\) −1.42372 −0.0932708 −0.0466354 0.998912i \(-0.514850\pi\)
−0.0466354 + 0.998912i \(0.514850\pi\)
\(234\) −5.03657 −0.329251
\(235\) 0 0
\(236\) −0.879077 −0.0572231
\(237\) 6.42864 0.417585
\(238\) 31.7462 2.05780
\(239\) −18.9590 −1.22636 −0.613178 0.789945i \(-0.710109\pi\)
−0.613178 + 0.789945i \(0.710109\pi\)
\(240\) 0 0
\(241\) −1.34614 −0.0867126 −0.0433563 0.999060i \(-0.513805\pi\)
−0.0433563 + 0.999060i \(0.513805\pi\)
\(242\) −1.21432 −0.0780594
\(243\) −1.00000 −0.0641500
\(244\) −0.396997 −0.0254151
\(245\) 0 0
\(246\) 2.19358 0.139857
\(247\) −21.5022 −1.36816
\(248\) 8.04149 0.510635
\(249\) 2.90321 0.183984
\(250\) 0 0
\(251\) 1.08250 0.0683267 0.0341633 0.999416i \(-0.489123\pi\)
0.0341633 + 0.999416i \(0.489123\pi\)
\(252\) −2.57628 −0.162291
\(253\) −4.00000 −0.251478
\(254\) −22.7195 −1.42555
\(255\) 0 0
\(256\) −11.6637 −0.728981
\(257\) −0.133353 −0.00831834 −0.00415917 0.999991i \(-0.501324\pi\)
−0.00415917 + 0.999991i \(0.501324\pi\)
\(258\) 5.95407 0.370684
\(259\) 28.4701 1.76905
\(260\) 0 0
\(261\) 1.80642 0.111815
\(262\) 1.51114 0.0933584
\(263\) −0.147643 −0.00910407 −0.00455203 0.999990i \(-0.501449\pi\)
−0.00455203 + 0.999990i \(0.501449\pi\)
\(264\) 3.06668 0.188741
\(265\) 0 0
\(266\) 30.8671 1.89258
\(267\) −0.622216 −0.0380790
\(268\) 2.55215 0.155897
\(269\) −26.8573 −1.63752 −0.818759 0.574138i \(-0.805337\pi\)
−0.818759 + 0.574138i \(0.805337\pi\)
\(270\) 0 0
\(271\) 3.08250 0.187248 0.0936242 0.995608i \(-0.470155\pi\)
0.0936242 + 0.995608i \(0.470155\pi\)
\(272\) 14.2524 0.864180
\(273\) −20.3368 −1.23084
\(274\) −22.8256 −1.37895
\(275\) 0 0
\(276\) 2.10171 0.126508
\(277\) −8.70964 −0.523311 −0.261656 0.965161i \(-0.584268\pi\)
−0.261656 + 0.965161i \(0.584268\pi\)
\(278\) −17.0509 −1.02264
\(279\) 2.62222 0.156988
\(280\) 0 0
\(281\) −20.3783 −1.21567 −0.607833 0.794065i \(-0.707961\pi\)
−0.607833 + 0.794065i \(0.707961\pi\)
\(282\) 8.56199 0.509859
\(283\) −6.32248 −0.375833 −0.187916 0.982185i \(-0.560173\pi\)
−0.187916 + 0.982185i \(0.560173\pi\)
\(284\) −0.225219 −0.0133643
\(285\) 0 0
\(286\) 5.03657 0.297818
\(287\) 8.85728 0.522829
\(288\) −2.88739 −0.170141
\(289\) 11.4286 0.672273
\(290\) 0 0
\(291\) 2.75557 0.161534
\(292\) −6.67799 −0.390800
\(293\) 16.6780 0.974339 0.487169 0.873308i \(-0.338029\pi\)
0.487169 + 0.873308i \(0.338029\pi\)
\(294\) 20.6938 1.20689
\(295\) 0 0
\(296\) 17.8064 1.03498
\(297\) 1.00000 0.0580259
\(298\) 3.70471 0.214608
\(299\) 16.5906 0.959458
\(300\) 0 0
\(301\) 24.0415 1.38573
\(302\) 0.396997 0.0228446
\(303\) 17.8064 1.02295
\(304\) 13.8578 0.794797
\(305\) 0 0
\(306\) 6.47457 0.370127
\(307\) −9.58565 −0.547082 −0.273541 0.961860i \(-0.588195\pi\)
−0.273541 + 0.961860i \(0.588195\pi\)
\(308\) 2.57628 0.146797
\(309\) −4.94914 −0.281547
\(310\) 0 0
\(311\) 14.5303 0.823941 0.411970 0.911197i \(-0.364841\pi\)
0.411970 + 0.911197i \(0.364841\pi\)
\(312\) −12.7195 −0.720099
\(313\) −21.0321 −1.18881 −0.594403 0.804167i \(-0.702612\pi\)
−0.594403 + 0.804167i \(0.702612\pi\)
\(314\) 24.1748 1.36427
\(315\) 0 0
\(316\) 3.37778 0.190015
\(317\) −0.990632 −0.0556394 −0.0278197 0.999613i \(-0.508856\pi\)
−0.0278197 + 0.999613i \(0.508856\pi\)
\(318\) −8.72393 −0.489213
\(319\) −1.80642 −0.101140
\(320\) 0 0
\(321\) −11.1985 −0.625039
\(322\) −23.8163 −1.32723
\(323\) 27.6414 1.53801
\(324\) −0.525428 −0.0291904
\(325\) 0 0
\(326\) −14.7841 −0.818818
\(327\) −15.7146 −0.869017
\(328\) 5.53972 0.305880
\(329\) 34.5718 1.90601
\(330\) 0 0
\(331\) −17.5812 −0.966350 −0.483175 0.875524i \(-0.660517\pi\)
−0.483175 + 0.875524i \(0.660517\pi\)
\(332\) 1.52543 0.0837187
\(333\) 5.80642 0.318190
\(334\) −15.7921 −0.864107
\(335\) 0 0
\(336\) 13.1066 0.715025
\(337\) −3.16992 −0.172676 −0.0863382 0.996266i \(-0.527517\pi\)
−0.0863382 + 0.996266i \(0.527517\pi\)
\(338\) −5.10372 −0.277606
\(339\) 1.76494 0.0958582
\(340\) 0 0
\(341\) −2.62222 −0.142001
\(342\) 6.29529 0.340410
\(343\) 49.2355 2.65847
\(344\) 15.0366 0.810717
\(345\) 0 0
\(346\) −16.8716 −0.907021
\(347\) 4.97634 0.267144 0.133572 0.991039i \(-0.457355\pi\)
0.133572 + 0.991039i \(0.457355\pi\)
\(348\) 0.949145 0.0508795
\(349\) 18.2034 0.974407 0.487203 0.873289i \(-0.338017\pi\)
0.487203 + 0.873289i \(0.338017\pi\)
\(350\) 0 0
\(351\) −4.14764 −0.221385
\(352\) 2.88739 0.153898
\(353\) 22.4099 1.19276 0.596379 0.802703i \(-0.296605\pi\)
0.596379 + 0.802703i \(0.296605\pi\)
\(354\) 2.03164 0.107981
\(355\) 0 0
\(356\) −0.326929 −0.0173272
\(357\) 26.1432 1.38364
\(358\) −15.6128 −0.825165
\(359\) 21.3274 1.12562 0.562809 0.826587i \(-0.309721\pi\)
0.562809 + 0.826587i \(0.309721\pi\)
\(360\) 0 0
\(361\) 7.87601 0.414527
\(362\) −1.11414 −0.0585579
\(363\) −1.00000 −0.0524864
\(364\) −10.6855 −0.560072
\(365\) 0 0
\(366\) 0.917502 0.0479586
\(367\) 35.1338 1.83397 0.916985 0.398921i \(-0.130615\pi\)
0.916985 + 0.398921i \(0.130615\pi\)
\(368\) −10.6923 −0.557374
\(369\) 1.80642 0.0940387
\(370\) 0 0
\(371\) −35.2257 −1.82883
\(372\) 1.37778 0.0714348
\(373\) −17.0049 −0.880481 −0.440241 0.897880i \(-0.645107\pi\)
−0.440241 + 0.897880i \(0.645107\pi\)
\(374\) −6.47457 −0.334792
\(375\) 0 0
\(376\) 21.6227 1.11511
\(377\) 7.49240 0.385878
\(378\) 5.95407 0.306244
\(379\) 2.36842 0.121657 0.0608287 0.998148i \(-0.480626\pi\)
0.0608287 + 0.998148i \(0.480626\pi\)
\(380\) 0 0
\(381\) −18.7096 −0.958524
\(382\) −17.4479 −0.892710
\(383\) 1.21585 0.0621271 0.0310635 0.999517i \(-0.490111\pi\)
0.0310635 + 0.999517i \(0.490111\pi\)
\(384\) 4.97481 0.253870
\(385\) 0 0
\(386\) 14.2810 0.726884
\(387\) 4.90321 0.249244
\(388\) 1.44785 0.0735035
\(389\) 2.26671 0.114927 0.0574633 0.998348i \(-0.481699\pi\)
0.0574633 + 0.998348i \(0.481699\pi\)
\(390\) 0 0
\(391\) −21.3274 −1.07857
\(392\) 52.2607 2.63957
\(393\) 1.24443 0.0627733
\(394\) 4.63957 0.233738
\(395\) 0 0
\(396\) 0.525428 0.0264037
\(397\) −18.4889 −0.927929 −0.463965 0.885854i \(-0.653574\pi\)
−0.463965 + 0.885854i \(0.653574\pi\)
\(398\) 16.6539 0.834782
\(399\) 25.4193 1.27256
\(400\) 0 0
\(401\) 17.5625 0.877028 0.438514 0.898724i \(-0.355505\pi\)
0.438514 + 0.898724i \(0.355505\pi\)
\(402\) −5.89829 −0.294180
\(403\) 10.8760 0.541773
\(404\) 9.35599 0.465478
\(405\) 0 0
\(406\) −10.7556 −0.533790
\(407\) −5.80642 −0.287814
\(408\) 16.3511 0.809498
\(409\) −21.3461 −1.05550 −0.527749 0.849400i \(-0.676964\pi\)
−0.527749 + 0.849400i \(0.676964\pi\)
\(410\) 0 0
\(411\) −18.7971 −0.927190
\(412\) −2.60042 −0.128113
\(413\) 8.20342 0.403664
\(414\) −4.85728 −0.238722
\(415\) 0 0
\(416\) −11.9759 −0.587165
\(417\) −14.0415 −0.687615
\(418\) −6.29529 −0.307913
\(419\) 28.8573 1.40977 0.704885 0.709321i \(-0.250999\pi\)
0.704885 + 0.709321i \(0.250999\pi\)
\(420\) 0 0
\(421\) −35.4893 −1.72964 −0.864822 0.502078i \(-0.832569\pi\)
−0.864822 + 0.502078i \(0.832569\pi\)
\(422\) −2.37826 −0.115772
\(423\) 7.05086 0.342824
\(424\) −22.0316 −1.06995
\(425\) 0 0
\(426\) 0.520505 0.0252186
\(427\) 3.70471 0.179284
\(428\) −5.88400 −0.284414
\(429\) 4.14764 0.200250
\(430\) 0 0
\(431\) 9.24443 0.445289 0.222644 0.974900i \(-0.428531\pi\)
0.222644 + 0.974900i \(0.428531\pi\)
\(432\) 2.67307 0.128608
\(433\) −6.28544 −0.302059 −0.151030 0.988529i \(-0.548259\pi\)
−0.151030 + 0.988529i \(0.548259\pi\)
\(434\) −15.6128 −0.749441
\(435\) 0 0
\(436\) −8.25686 −0.395432
\(437\) −20.7368 −0.991977
\(438\) 15.4336 0.737444
\(439\) −36.5303 −1.74350 −0.871749 0.489952i \(-0.837014\pi\)
−0.871749 + 0.489952i \(0.837014\pi\)
\(440\) 0 0
\(441\) 17.0415 0.811499
\(442\) 26.8542 1.27732
\(443\) −38.2766 −1.81857 −0.909287 0.416170i \(-0.863372\pi\)
−0.909287 + 0.416170i \(0.863372\pi\)
\(444\) 3.05086 0.144787
\(445\) 0 0
\(446\) 31.5843 1.49556
\(447\) 3.05086 0.144300
\(448\) 43.4050 2.05069
\(449\) −31.8479 −1.50300 −0.751498 0.659735i \(-0.770668\pi\)
−0.751498 + 0.659735i \(0.770668\pi\)
\(450\) 0 0
\(451\) −1.80642 −0.0850612
\(452\) 0.927346 0.0436187
\(453\) 0.326929 0.0153605
\(454\) 7.70027 0.361391
\(455\) 0 0
\(456\) 15.8983 0.744506
\(457\) 1.39207 0.0651185 0.0325592 0.999470i \(-0.489634\pi\)
0.0325592 + 0.999470i \(0.489634\pi\)
\(458\) 28.3269 1.32363
\(459\) 5.33185 0.248870
\(460\) 0 0
\(461\) −7.70471 −0.358844 −0.179422 0.983772i \(-0.557423\pi\)
−0.179422 + 0.983772i \(0.557423\pi\)
\(462\) −5.95407 −0.277008
\(463\) 4.68244 0.217611 0.108806 0.994063i \(-0.465297\pi\)
0.108806 + 0.994063i \(0.465297\pi\)
\(464\) −4.82870 −0.224167
\(465\) 0 0
\(466\) 1.72885 0.0800873
\(467\) 12.8573 0.594964 0.297482 0.954727i \(-0.403853\pi\)
0.297482 + 0.954727i \(0.403853\pi\)
\(468\) −2.17929 −0.100738
\(469\) −23.8163 −1.09973
\(470\) 0 0
\(471\) 19.9081 0.917318
\(472\) 5.13077 0.236163
\(473\) −4.90321 −0.225450
\(474\) −7.80642 −0.358561
\(475\) 0 0
\(476\) 13.7364 0.629605
\(477\) −7.18421 −0.328942
\(478\) 23.0223 1.05301
\(479\) 8.38715 0.383219 0.191609 0.981471i \(-0.438629\pi\)
0.191609 + 0.981471i \(0.438629\pi\)
\(480\) 0 0
\(481\) 24.0830 1.09809
\(482\) 1.63465 0.0744561
\(483\) −19.6128 −0.892415
\(484\) −0.525428 −0.0238831
\(485\) 0 0
\(486\) 1.21432 0.0550827
\(487\) 9.83500 0.445667 0.222833 0.974857i \(-0.428469\pi\)
0.222833 + 0.974857i \(0.428469\pi\)
\(488\) 2.31708 0.104890
\(489\) −12.1748 −0.550565
\(490\) 0 0
\(491\) −32.9403 −1.48657 −0.743286 0.668973i \(-0.766734\pi\)
−0.743286 + 0.668973i \(0.766734\pi\)
\(492\) 0.949145 0.0427908
\(493\) −9.63158 −0.433785
\(494\) 26.1106 1.17477
\(495\) 0 0
\(496\) −7.00937 −0.314730
\(497\) 2.10171 0.0942746
\(498\) −3.52543 −0.157978
\(499\) −1.63158 −0.0730397 −0.0365199 0.999333i \(-0.511627\pi\)
−0.0365199 + 0.999333i \(0.511627\pi\)
\(500\) 0 0
\(501\) −13.0049 −0.581017
\(502\) −1.31450 −0.0586689
\(503\) 41.8622 1.86654 0.933272 0.359171i \(-0.116941\pi\)
0.933272 + 0.359171i \(0.116941\pi\)
\(504\) 15.0366 0.669782
\(505\) 0 0
\(506\) 4.85728 0.215932
\(507\) −4.20294 −0.186659
\(508\) −9.83056 −0.436160
\(509\) −38.8573 −1.72232 −0.861159 0.508335i \(-0.830261\pi\)
−0.861159 + 0.508335i \(0.830261\pi\)
\(510\) 0 0
\(511\) 62.3180 2.75679
\(512\) 24.1131 1.06566
\(513\) 5.18421 0.228888
\(514\) 0.161933 0.00714257
\(515\) 0 0
\(516\) 2.57628 0.113415
\(517\) −7.05086 −0.310096
\(518\) −34.5718 −1.51900
\(519\) −13.8938 −0.609872
\(520\) 0 0
\(521\) 11.1111 0.486785 0.243393 0.969928i \(-0.421740\pi\)
0.243393 + 0.969928i \(0.421740\pi\)
\(522\) −2.19358 −0.0960102
\(523\) 27.3002 1.19375 0.596877 0.802332i \(-0.296408\pi\)
0.596877 + 0.802332i \(0.296408\pi\)
\(524\) 0.653858 0.0285639
\(525\) 0 0
\(526\) 0.179286 0.00781724
\(527\) −13.9813 −0.609033
\(528\) −2.67307 −0.116330
\(529\) −7.00000 −0.304348
\(530\) 0 0
\(531\) 1.67307 0.0726051
\(532\) 13.3560 0.579055
\(533\) 7.49240 0.324532
\(534\) 0.755569 0.0326967
\(535\) 0 0
\(536\) −14.8957 −0.643396
\(537\) −12.8573 −0.554833
\(538\) 32.6133 1.40606
\(539\) −17.0415 −0.734029
\(540\) 0 0
\(541\) −16.1017 −0.692267 −0.346133 0.938185i \(-0.612506\pi\)
−0.346133 + 0.938185i \(0.612506\pi\)
\(542\) −3.74314 −0.160782
\(543\) −0.917502 −0.0393738
\(544\) 15.3951 0.660061
\(545\) 0 0
\(546\) 24.6953 1.05686
\(547\) 40.0370 1.71186 0.855930 0.517091i \(-0.172985\pi\)
0.855930 + 0.517091i \(0.172985\pi\)
\(548\) −9.87649 −0.421903
\(549\) 0.755569 0.0322469
\(550\) 0 0
\(551\) −9.36488 −0.398957
\(552\) −12.2667 −0.522106
\(553\) −31.5210 −1.34041
\(554\) 10.5763 0.449343
\(555\) 0 0
\(556\) −7.37778 −0.312888
\(557\) −28.2908 −1.19872 −0.599361 0.800479i \(-0.704578\pi\)
−0.599361 + 0.800479i \(0.704578\pi\)
\(558\) −3.18421 −0.134798
\(559\) 20.3368 0.860154
\(560\) 0 0
\(561\) −5.33185 −0.225111
\(562\) 24.7457 1.04384
\(563\) −32.7926 −1.38204 −0.691022 0.722834i \(-0.742839\pi\)
−0.691022 + 0.722834i \(0.742839\pi\)
\(564\) 3.70471 0.155997
\(565\) 0 0
\(566\) 7.67752 0.322710
\(567\) 4.90321 0.205916
\(568\) 1.31450 0.0551551
\(569\) −8.88586 −0.372515 −0.186257 0.982501i \(-0.559636\pi\)
−0.186257 + 0.982501i \(0.559636\pi\)
\(570\) 0 0
\(571\) −10.6953 −0.447586 −0.223793 0.974637i \(-0.571844\pi\)
−0.223793 + 0.974637i \(0.571844\pi\)
\(572\) 2.17929 0.0911205
\(573\) −14.3684 −0.600249
\(574\) −10.7556 −0.448929
\(575\) 0 0
\(576\) 8.85236 0.368848
\(577\) −27.1338 −1.12960 −0.564798 0.825229i \(-0.691046\pi\)
−0.564798 + 0.825229i \(0.691046\pi\)
\(578\) −13.8780 −0.577250
\(579\) 11.7605 0.488749
\(580\) 0 0
\(581\) −14.2351 −0.590570
\(582\) −3.34614 −0.138702
\(583\) 7.18421 0.297540
\(584\) 38.9763 1.61285
\(585\) 0 0
\(586\) −20.2524 −0.836620
\(587\) −10.9590 −0.452326 −0.226163 0.974089i \(-0.572618\pi\)
−0.226163 + 0.974089i \(0.572618\pi\)
\(588\) 8.95407 0.369260
\(589\) −13.5941 −0.560136
\(590\) 0 0
\(591\) 3.82071 0.157163
\(592\) −15.5210 −0.637908
\(593\) −23.7003 −0.973253 −0.486627 0.873610i \(-0.661773\pi\)
−0.486627 + 0.873610i \(0.661773\pi\)
\(594\) −1.21432 −0.0498241
\(595\) 0 0
\(596\) 1.60300 0.0656616
\(597\) 13.7146 0.561299
\(598\) −20.1463 −0.823842
\(599\) 41.7146 1.70441 0.852205 0.523208i \(-0.175265\pi\)
0.852205 + 0.523208i \(0.175265\pi\)
\(600\) 0 0
\(601\) 14.5906 0.595162 0.297581 0.954697i \(-0.403820\pi\)
0.297581 + 0.954697i \(0.403820\pi\)
\(602\) −29.1941 −1.18986
\(603\) −4.85728 −0.197804
\(604\) 0.171778 0.00698953
\(605\) 0 0
\(606\) −21.6227 −0.878362
\(607\) −19.9826 −0.811071 −0.405535 0.914079i \(-0.632915\pi\)
−0.405535 + 0.914079i \(0.632915\pi\)
\(608\) 14.9688 0.607066
\(609\) −8.85728 −0.358915
\(610\) 0 0
\(611\) 29.2444 1.18310
\(612\) 2.80150 0.113244
\(613\) 19.0781 0.770555 0.385278 0.922801i \(-0.374106\pi\)
0.385278 + 0.922801i \(0.374106\pi\)
\(614\) 11.6400 0.469754
\(615\) 0 0
\(616\) −15.0366 −0.605840
\(617\) −39.3590 −1.58454 −0.792268 0.610174i \(-0.791100\pi\)
−0.792268 + 0.610174i \(0.791100\pi\)
\(618\) 6.00984 0.241751
\(619\) −23.0923 −0.928160 −0.464080 0.885793i \(-0.653615\pi\)
−0.464080 + 0.885793i \(0.653615\pi\)
\(620\) 0 0
\(621\) −4.00000 −0.160514
\(622\) −17.6445 −0.707480
\(623\) 3.05086 0.122230
\(624\) 11.0869 0.443833
\(625\) 0 0
\(626\) 25.5397 1.02077
\(627\) −5.18421 −0.207037
\(628\) 10.4603 0.417411
\(629\) −30.9590 −1.23442
\(630\) 0 0
\(631\) −25.5111 −1.01558 −0.507791 0.861480i \(-0.669538\pi\)
−0.507791 + 0.861480i \(0.669538\pi\)
\(632\) −19.7146 −0.784203
\(633\) −1.95851 −0.0778439
\(634\) 1.20294 0.0477750
\(635\) 0 0
\(636\) −3.77478 −0.149680
\(637\) 70.6820 2.80052
\(638\) 2.19358 0.0868445
\(639\) 0.428639 0.0169567
\(640\) 0 0
\(641\) −6.25380 −0.247010 −0.123505 0.992344i \(-0.539414\pi\)
−0.123505 + 0.992344i \(0.539414\pi\)
\(642\) 13.5986 0.536692
\(643\) 6.84743 0.270036 0.135018 0.990843i \(-0.456891\pi\)
0.135018 + 0.990843i \(0.456891\pi\)
\(644\) −10.3051 −0.406079
\(645\) 0 0
\(646\) −33.5655 −1.32062
\(647\) 20.2953 0.797890 0.398945 0.916975i \(-0.369376\pi\)
0.398945 + 0.916975i \(0.369376\pi\)
\(648\) 3.06668 0.120470
\(649\) −1.67307 −0.0656738
\(650\) 0 0
\(651\) −12.8573 −0.503916
\(652\) −6.39700 −0.250526
\(653\) −10.6222 −0.415679 −0.207840 0.978163i \(-0.566643\pi\)
−0.207840 + 0.978163i \(0.566643\pi\)
\(654\) 19.0825 0.746185
\(655\) 0 0
\(656\) −4.82870 −0.188529
\(657\) 12.7096 0.495850
\(658\) −41.9813 −1.63660
\(659\) 10.1017 0.393507 0.196753 0.980453i \(-0.436960\pi\)
0.196753 + 0.980453i \(0.436960\pi\)
\(660\) 0 0
\(661\) 21.6128 0.840642 0.420321 0.907375i \(-0.361917\pi\)
0.420321 + 0.907375i \(0.361917\pi\)
\(662\) 21.3492 0.829760
\(663\) 22.1146 0.858861
\(664\) −8.90321 −0.345512
\(665\) 0 0
\(666\) −7.05086 −0.273215
\(667\) 7.22570 0.279780
\(668\) −6.83314 −0.264382
\(669\) 26.0098 1.00560
\(670\) 0 0
\(671\) −0.755569 −0.0291684
\(672\) 14.1575 0.546137
\(673\) 10.2208 0.393982 0.196991 0.980405i \(-0.436883\pi\)
0.196991 + 0.980405i \(0.436883\pi\)
\(674\) 3.84929 0.148269
\(675\) 0 0
\(676\) −2.20834 −0.0849363
\(677\) 13.9224 0.535082 0.267541 0.963546i \(-0.413789\pi\)
0.267541 + 0.963546i \(0.413789\pi\)
\(678\) −2.14320 −0.0823090
\(679\) −13.5111 −0.518510
\(680\) 0 0
\(681\) 6.34122 0.242996
\(682\) 3.18421 0.121930
\(683\) −10.3970 −0.397830 −0.198915 0.980017i \(-0.563742\pi\)
−0.198915 + 0.980017i \(0.563742\pi\)
\(684\) 2.72393 0.104152
\(685\) 0 0
\(686\) −59.7877 −2.28270
\(687\) 23.3274 0.889996
\(688\) −13.1066 −0.499686
\(689\) −29.7975 −1.13520
\(690\) 0 0
\(691\) −0.977725 −0.0371944 −0.0185972 0.999827i \(-0.505920\pi\)
−0.0185972 + 0.999827i \(0.505920\pi\)
\(692\) −7.30021 −0.277512
\(693\) −4.90321 −0.186258
\(694\) −6.04287 −0.229384
\(695\) 0 0
\(696\) −5.53972 −0.209982
\(697\) −9.63158 −0.364822
\(698\) −22.1048 −0.836678
\(699\) 1.42372 0.0538499
\(700\) 0 0
\(701\) 48.9688 1.84953 0.924764 0.380542i \(-0.124263\pi\)
0.924764 + 0.380542i \(0.124263\pi\)
\(702\) 5.03657 0.190093
\(703\) −30.1017 −1.13531
\(704\) −8.85236 −0.333636
\(705\) 0 0
\(706\) −27.2128 −1.02417
\(707\) −87.3087 −3.28358
\(708\) 0.879077 0.0330378
\(709\) −37.2672 −1.39960 −0.699799 0.714340i \(-0.746727\pi\)
−0.699799 + 0.714340i \(0.746727\pi\)
\(710\) 0 0
\(711\) −6.42864 −0.241093
\(712\) 1.90813 0.0715103
\(713\) 10.4889 0.392811
\(714\) −31.7462 −1.18807
\(715\) 0 0
\(716\) −6.75557 −0.252467
\(717\) 18.9590 0.708036
\(718\) −25.8983 −0.966516
\(719\) 5.83500 0.217609 0.108804 0.994063i \(-0.465298\pi\)
0.108804 + 0.994063i \(0.465298\pi\)
\(720\) 0 0
\(721\) 24.2667 0.903739
\(722\) −9.56400 −0.355935
\(723\) 1.34614 0.0500635
\(724\) −0.482081 −0.0179164
\(725\) 0 0
\(726\) 1.21432 0.0450676
\(727\) −46.8385 −1.73715 −0.868573 0.495562i \(-0.834962\pi\)
−0.868573 + 0.495562i \(0.834962\pi\)
\(728\) 62.3663 2.31145
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) −26.1432 −0.966941
\(732\) 0.396997 0.0146734
\(733\) −45.2083 −1.66981 −0.834904 0.550395i \(-0.814477\pi\)
−0.834904 + 0.550395i \(0.814477\pi\)
\(734\) −42.6637 −1.57475
\(735\) 0 0
\(736\) −11.5496 −0.425723
\(737\) 4.85728 0.178920
\(738\) −2.19358 −0.0807467
\(739\) 5.65433 0.207998 0.103999 0.994577i \(-0.466836\pi\)
0.103999 + 0.994577i \(0.466836\pi\)
\(740\) 0 0
\(741\) 21.5022 0.789905
\(742\) 42.7753 1.57033
\(743\) 4.50622 0.165317 0.0826585 0.996578i \(-0.473659\pi\)
0.0826585 + 0.996578i \(0.473659\pi\)
\(744\) −8.04149 −0.294815
\(745\) 0 0
\(746\) 20.6494 0.756029
\(747\) −2.90321 −0.106223
\(748\) −2.80150 −0.102433
\(749\) 54.9086 2.00632
\(750\) 0 0
\(751\) 47.5121 1.73374 0.866870 0.498534i \(-0.166128\pi\)
0.866870 + 0.498534i \(0.166128\pi\)
\(752\) −18.8474 −0.687295
\(753\) −1.08250 −0.0394484
\(754\) −9.09817 −0.331336
\(755\) 0 0
\(756\) 2.57628 0.0936985
\(757\) −46.6637 −1.69602 −0.848011 0.529979i \(-0.822200\pi\)
−0.848011 + 0.529979i \(0.822200\pi\)
\(758\) −2.87601 −0.104462
\(759\) 4.00000 0.145191
\(760\) 0 0
\(761\) 14.9304 0.541227 0.270613 0.962688i \(-0.412773\pi\)
0.270613 + 0.962688i \(0.412773\pi\)
\(762\) 22.7195 0.823040
\(763\) 77.0518 2.78946
\(764\) −7.54956 −0.273134
\(765\) 0 0
\(766\) −1.47643 −0.0533457
\(767\) 6.93930 0.250564
\(768\) 11.6637 0.420878
\(769\) 38.8573 1.40123 0.700615 0.713540i \(-0.252909\pi\)
0.700615 + 0.713540i \(0.252909\pi\)
\(770\) 0 0
\(771\) 0.133353 0.00480259
\(772\) 6.17929 0.222397
\(773\) −36.3368 −1.30694 −0.653471 0.756951i \(-0.726688\pi\)
−0.653471 + 0.756951i \(0.726688\pi\)
\(774\) −5.95407 −0.214015
\(775\) 0 0
\(776\) −8.45044 −0.303353
\(777\) −28.4701 −1.02136
\(778\) −2.75251 −0.0986821
\(779\) −9.36488 −0.335532
\(780\) 0 0
\(781\) −0.428639 −0.0153379
\(782\) 25.8983 0.926121
\(783\) −1.80642 −0.0645563
\(784\) −45.5531 −1.62690
\(785\) 0 0
\(786\) −1.51114 −0.0539005
\(787\) −33.5482 −1.19586 −0.597932 0.801547i \(-0.704011\pi\)
−0.597932 + 0.801547i \(0.704011\pi\)
\(788\) 2.00751 0.0715145
\(789\) 0.147643 0.00525624
\(790\) 0 0
\(791\) −8.65386 −0.307696
\(792\) −3.06668 −0.108970
\(793\) 3.13383 0.111286
\(794\) 22.4514 0.796770
\(795\) 0 0
\(796\) 7.20601 0.255410
\(797\) −16.1334 −0.571473 −0.285736 0.958308i \(-0.592238\pi\)
−0.285736 + 0.958308i \(0.592238\pi\)
\(798\) −30.8671 −1.09268
\(799\) −37.5941 −1.32998
\(800\) 0 0
\(801\) 0.622216 0.0219849
\(802\) −21.3265 −0.753063
\(803\) −12.7096 −0.448513
\(804\) −2.55215 −0.0900073
\(805\) 0 0
\(806\) −13.2070 −0.465195
\(807\) 26.8573 0.945421
\(808\) −54.6065 −1.92105
\(809\) 25.7431 0.905081 0.452540 0.891744i \(-0.350518\pi\)
0.452540 + 0.891744i \(0.350518\pi\)
\(810\) 0 0
\(811\) 13.4509 0.472325 0.236163 0.971714i \(-0.424110\pi\)
0.236163 + 0.971714i \(0.424110\pi\)
\(812\) −4.65386 −0.163318
\(813\) −3.08250 −0.108108
\(814\) 7.05086 0.247132
\(815\) 0 0
\(816\) −14.2524 −0.498934
\(817\) −25.4193 −0.889308
\(818\) 25.9210 0.906308
\(819\) 20.3368 0.710624
\(820\) 0 0
\(821\) 24.1748 0.843708 0.421854 0.906664i \(-0.361380\pi\)
0.421854 + 0.906664i \(0.361380\pi\)
\(822\) 22.8256 0.796135
\(823\) −40.9117 −1.42609 −0.713046 0.701118i \(-0.752685\pi\)
−0.713046 + 0.701118i \(0.752685\pi\)
\(824\) 15.1774 0.528731
\(825\) 0 0
\(826\) −9.96158 −0.346608
\(827\) −20.1476 −0.700602 −0.350301 0.936637i \(-0.613921\pi\)
−0.350301 + 0.936637i \(0.613921\pi\)
\(828\) −2.10171 −0.0730395
\(829\) 31.4322 1.09168 0.545842 0.837888i \(-0.316210\pi\)
0.545842 + 0.837888i \(0.316210\pi\)
\(830\) 0 0
\(831\) 8.70964 0.302134
\(832\) 36.7164 1.27291
\(833\) −90.8627 −3.14821
\(834\) 17.0509 0.590423
\(835\) 0 0
\(836\) −2.72393 −0.0942089
\(837\) −2.62222 −0.0906370
\(838\) −35.0420 −1.21050
\(839\) −52.8988 −1.82627 −0.913134 0.407659i \(-0.866345\pi\)
−0.913134 + 0.407659i \(0.866345\pi\)
\(840\) 0 0
\(841\) −25.7368 −0.887477
\(842\) 43.0954 1.48517
\(843\) 20.3783 0.701865
\(844\) −1.02906 −0.0354216
\(845\) 0 0
\(846\) −8.56199 −0.294367
\(847\) 4.90321 0.168476
\(848\) 19.2039 0.659465
\(849\) 6.32248 0.216987
\(850\) 0 0
\(851\) 23.2257 0.796167
\(852\) 0.225219 0.00771588
\(853\) −46.9229 −1.60661 −0.803305 0.595568i \(-0.796927\pi\)
−0.803305 + 0.595568i \(0.796927\pi\)
\(854\) −4.49871 −0.153943
\(855\) 0 0
\(856\) 34.3422 1.17379
\(857\) −25.1481 −0.859043 −0.429522 0.903057i \(-0.641318\pi\)
−0.429522 + 0.903057i \(0.641318\pi\)
\(858\) −5.03657 −0.171945
\(859\) 1.84791 0.0630499 0.0315250 0.999503i \(-0.489964\pi\)
0.0315250 + 0.999503i \(0.489964\pi\)
\(860\) 0 0
\(861\) −8.85728 −0.301855
\(862\) −11.2257 −0.382349
\(863\) 32.6824 1.11252 0.556262 0.831007i \(-0.312235\pi\)
0.556262 + 0.831007i \(0.312235\pi\)
\(864\) 2.88739 0.0982310
\(865\) 0 0
\(866\) 7.63254 0.259364
\(867\) −11.4286 −0.388137
\(868\) −6.75557 −0.229299
\(869\) 6.42864 0.218077
\(870\) 0 0
\(871\) −20.1463 −0.682630
\(872\) 48.1915 1.63197
\(873\) −2.75557 −0.0932619
\(874\) 25.1811 0.851765
\(875\) 0 0
\(876\) 6.67799 0.225628
\(877\) 49.1798 1.66068 0.830341 0.557255i \(-0.188145\pi\)
0.830341 + 0.557255i \(0.188145\pi\)
\(878\) 44.3595 1.49706
\(879\) −16.6780 −0.562535
\(880\) 0 0
\(881\) 33.8163 1.13930 0.569650 0.821888i \(-0.307079\pi\)
0.569650 + 0.821888i \(0.307079\pi\)
\(882\) −20.6938 −0.696797
\(883\) 24.7368 0.832461 0.416230 0.909259i \(-0.363351\pi\)
0.416230 + 0.909259i \(0.363351\pi\)
\(884\) 11.6196 0.390810
\(885\) 0 0
\(886\) 46.4800 1.56153
\(887\) 7.64004 0.256528 0.128264 0.991740i \(-0.459060\pi\)
0.128264 + 0.991740i \(0.459060\pi\)
\(888\) −17.8064 −0.597544
\(889\) 91.7373 3.07677
\(890\) 0 0
\(891\) −1.00000 −0.0335013
\(892\) 13.6663 0.457581
\(893\) −36.5531 −1.22320
\(894\) −3.70471 −0.123904
\(895\) 0 0
\(896\) −24.3926 −0.814898
\(897\) −16.5906 −0.553943
\(898\) 38.6735 1.29055
\(899\) 4.73683 0.157982
\(900\) 0 0
\(901\) 38.3051 1.27613
\(902\) 2.19358 0.0730381
\(903\) −24.0415 −0.800051
\(904\) −5.41249 −0.180017
\(905\) 0 0
\(906\) −0.396997 −0.0131893
\(907\) −30.3970 −1.00932 −0.504658 0.863319i \(-0.668381\pi\)
−0.504658 + 0.863319i \(0.668381\pi\)
\(908\) 3.33185 0.110571
\(909\) −17.8064 −0.590602
\(910\) 0 0
\(911\) −45.3274 −1.50176 −0.750882 0.660436i \(-0.770371\pi\)
−0.750882 + 0.660436i \(0.770371\pi\)
\(912\) −13.8578 −0.458876
\(913\) 2.90321 0.0960823
\(914\) −1.69042 −0.0559142
\(915\) 0 0
\(916\) 12.2569 0.404978
\(917\) −6.10171 −0.201496
\(918\) −6.47457 −0.213693
\(919\) −16.3269 −0.538576 −0.269288 0.963060i \(-0.586788\pi\)
−0.269288 + 0.963060i \(0.586788\pi\)
\(920\) 0 0
\(921\) 9.58565 0.315858
\(922\) 9.35599 0.308123
\(923\) 1.77784 0.0585184
\(924\) −2.57628 −0.0847535
\(925\) 0 0
\(926\) −5.68598 −0.186853
\(927\) 4.94914 0.162551
\(928\) −5.21585 −0.171219
\(929\) −29.6128 −0.971566 −0.485783 0.874079i \(-0.661465\pi\)
−0.485783 + 0.874079i \(0.661465\pi\)
\(930\) 0 0
\(931\) −88.3466 −2.89544
\(932\) 0.748060 0.0245035
\(933\) −14.5303 −0.475702
\(934\) −15.6128 −0.510868
\(935\) 0 0
\(936\) 12.7195 0.415749
\(937\) 8.92195 0.291467 0.145734 0.989324i \(-0.453446\pi\)
0.145734 + 0.989324i \(0.453446\pi\)
\(938\) 28.9206 0.944290
\(939\) 21.0321 0.686357
\(940\) 0 0
\(941\) −22.2766 −0.726195 −0.363097 0.931751i \(-0.618281\pi\)
−0.363097 + 0.931751i \(0.618281\pi\)
\(942\) −24.1748 −0.787659
\(943\) 7.22570 0.235301
\(944\) −4.47224 −0.145559
\(945\) 0 0
\(946\) 5.95407 0.193583
\(947\) 44.4612 1.44480 0.722398 0.691477i \(-0.243040\pi\)
0.722398 + 0.691477i \(0.243040\pi\)
\(948\) −3.37778 −0.109705
\(949\) 52.7150 1.71120
\(950\) 0 0
\(951\) 0.990632 0.0321234
\(952\) −80.1727 −2.59841
\(953\) 20.5575 0.665924 0.332962 0.942940i \(-0.391952\pi\)
0.332962 + 0.942940i \(0.391952\pi\)
\(954\) 8.72393 0.282448
\(955\) 0 0
\(956\) 9.96158 0.322180
\(957\) 1.80642 0.0583934
\(958\) −10.1847 −0.329052
\(959\) 92.1659 2.97619
\(960\) 0 0
\(961\) −24.1240 −0.778193
\(962\) −29.2444 −0.942878
\(963\) 11.1985 0.360867
\(964\) 0.707300 0.0227806
\(965\) 0 0
\(966\) 23.8163 0.766276
\(967\) 27.4839 0.883824 0.441912 0.897058i \(-0.354300\pi\)
0.441912 + 0.897058i \(0.354300\pi\)
\(968\) 3.06668 0.0985667
\(969\) −27.6414 −0.887971
\(970\) 0 0
\(971\) −5.81532 −0.186622 −0.0933112 0.995637i \(-0.529745\pi\)
−0.0933112 + 0.995637i \(0.529745\pi\)
\(972\) 0.525428 0.0168531
\(973\) 68.8484 2.20718
\(974\) −11.9428 −0.382673
\(975\) 0 0
\(976\) −2.01969 −0.0646487
\(977\) 48.3912 1.54817 0.774085 0.633081i \(-0.218210\pi\)
0.774085 + 0.633081i \(0.218210\pi\)
\(978\) 14.7841 0.472745
\(979\) −0.622216 −0.0198861
\(980\) 0 0
\(981\) 15.7146 0.501727
\(982\) 40.0000 1.27645
\(983\) −49.9724 −1.59387 −0.796936 0.604064i \(-0.793547\pi\)
−0.796936 + 0.604064i \(0.793547\pi\)
\(984\) −5.53972 −0.176600
\(985\) 0 0
\(986\) 11.6958 0.372471
\(987\) −34.5718 −1.10043
\(988\) 11.2979 0.359433
\(989\) 19.6128 0.623652
\(990\) 0 0
\(991\) −7.35905 −0.233768 −0.116884 0.993146i \(-0.537291\pi\)
−0.116884 + 0.993146i \(0.537291\pi\)
\(992\) −7.57136 −0.240391
\(993\) 17.5812 0.557923
\(994\) −2.55215 −0.0809492
\(995\) 0 0
\(996\) −1.52543 −0.0483350
\(997\) −4.33138 −0.137176 −0.0685880 0.997645i \(-0.521849\pi\)
−0.0685880 + 0.997645i \(0.521849\pi\)
\(998\) 1.98126 0.0627158
\(999\) −5.80642 −0.183707
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 825.2.a.n.1.1 3
3.2 odd 2 2475.2.a.y.1.3 3
5.2 odd 4 165.2.c.a.34.3 6
5.3 odd 4 165.2.c.a.34.4 yes 6
5.4 even 2 825.2.a.h.1.3 3
11.10 odd 2 9075.2.a.cc.1.3 3
15.2 even 4 495.2.c.d.199.4 6
15.8 even 4 495.2.c.d.199.3 6
15.14 odd 2 2475.2.a.be.1.1 3
20.3 even 4 2640.2.d.i.529.5 6
20.7 even 4 2640.2.d.i.529.2 6
55.32 even 4 1815.2.c.d.364.4 6
55.43 even 4 1815.2.c.d.364.3 6
55.54 odd 2 9075.2.a.ck.1.1 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
165.2.c.a.34.3 6 5.2 odd 4
165.2.c.a.34.4 yes 6 5.3 odd 4
495.2.c.d.199.3 6 15.8 even 4
495.2.c.d.199.4 6 15.2 even 4
825.2.a.h.1.3 3 5.4 even 2
825.2.a.n.1.1 3 1.1 even 1 trivial
1815.2.c.d.364.3 6 55.43 even 4
1815.2.c.d.364.4 6 55.32 even 4
2475.2.a.y.1.3 3 3.2 odd 2
2475.2.a.be.1.1 3 15.14 odd 2
2640.2.d.i.529.2 6 20.7 even 4
2640.2.d.i.529.5 6 20.3 even 4
9075.2.a.cc.1.3 3 11.10 odd 2
9075.2.a.ck.1.1 3 55.54 odd 2