Properties

Label 1859.2.a.t.1.8
Level $1859$
Weight $2$
Character 1859.1
Self dual yes
Analytic conductor $14.844$
Analytic rank $0$
Dimension $21$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 1859 = 11 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1859.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(14.8441897358\)
Analytic rank: \(0\)
Dimension: \(21\)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.8
Character \(\chi\) \(=\) 1859.1

$q$-expansion

\(f(q)\) \(=\) \(q-0.930026 q^{2} +2.53271 q^{3} -1.13505 q^{4} -3.38691 q^{5} -2.35548 q^{6} -3.38196 q^{7} +2.91568 q^{8} +3.41460 q^{9} +O(q^{10})\) \(q-0.930026 q^{2} +2.53271 q^{3} -1.13505 q^{4} -3.38691 q^{5} -2.35548 q^{6} -3.38196 q^{7} +2.91568 q^{8} +3.41460 q^{9} +3.14991 q^{10} +1.00000 q^{11} -2.87475 q^{12} +3.14532 q^{14} -8.57804 q^{15} -0.441558 q^{16} -7.19830 q^{17} -3.17567 q^{18} -3.06566 q^{19} +3.84431 q^{20} -8.56552 q^{21} -0.930026 q^{22} +5.94916 q^{23} +7.38456 q^{24} +6.47113 q^{25} +1.05006 q^{27} +3.83870 q^{28} +5.43442 q^{29} +7.97780 q^{30} +9.06534 q^{31} -5.42070 q^{32} +2.53271 q^{33} +6.69461 q^{34} +11.4544 q^{35} -3.87574 q^{36} -4.87541 q^{37} +2.85114 q^{38} -9.87513 q^{40} +0.859934 q^{41} +7.96616 q^{42} +5.81891 q^{43} -1.13505 q^{44} -11.5649 q^{45} -5.53287 q^{46} +8.69187 q^{47} -1.11834 q^{48} +4.43768 q^{49} -6.01832 q^{50} -18.2312 q^{51} -9.91682 q^{53} -0.976585 q^{54} -3.38691 q^{55} -9.86073 q^{56} -7.76441 q^{57} -5.05416 q^{58} +7.38488 q^{59} +9.73651 q^{60} +5.88757 q^{61} -8.43101 q^{62} -11.5481 q^{63} +5.92451 q^{64} -2.35548 q^{66} -1.27537 q^{67} +8.17043 q^{68} +15.0675 q^{69} -10.6529 q^{70} +13.2250 q^{71} +9.95588 q^{72} +1.40595 q^{73} +4.53426 q^{74} +16.3895 q^{75} +3.47968 q^{76} -3.38196 q^{77} +4.47756 q^{79} +1.49552 q^{80} -7.58430 q^{81} -0.799761 q^{82} -2.06037 q^{83} +9.72230 q^{84} +24.3800 q^{85} -5.41174 q^{86} +13.7638 q^{87} +2.91568 q^{88} -0.472173 q^{89} +10.7557 q^{90} -6.75259 q^{92} +22.9599 q^{93} -8.08367 q^{94} +10.3831 q^{95} -13.7290 q^{96} +6.51513 q^{97} -4.12716 q^{98} +3.41460 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 21 q + 6 q^{3} + 32 q^{4} + 7 q^{5} - 19 q^{6} + q^{7} - 3 q^{8} + 33 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 21 q + 6 q^{3} + 32 q^{4} + 7 q^{5} - 19 q^{6} + q^{7} - 3 q^{8} + 33 q^{9} + 18 q^{10} + 21 q^{11} + 23 q^{12} + 20 q^{14} + 16 q^{15} + 50 q^{16} + 16 q^{17} + 3 q^{18} - 11 q^{19} + 24 q^{20} - 5 q^{21} - 9 q^{23} - 54 q^{24} + 36 q^{25} - 11 q^{28} + 28 q^{29} + 21 q^{30} + 15 q^{31} - 61 q^{32} + 6 q^{33} - 6 q^{34} - 3 q^{35} + 45 q^{36} - 12 q^{37} + q^{38} + 55 q^{40} - 4 q^{41} - 34 q^{42} + 17 q^{43} + 32 q^{44} + 9 q^{45} + 11 q^{46} + 36 q^{47} + 24 q^{48} + 72 q^{49} - 9 q^{50} + 2 q^{51} + 19 q^{53} + q^{54} + 7 q^{55} + 44 q^{56} - 4 q^{57} - 33 q^{58} + 54 q^{59} + 64 q^{60} + 98 q^{61} - 29 q^{62} - 81 q^{63} + 63 q^{64} - 19 q^{66} + 25 q^{67} + 4 q^{68} + 89 q^{69} + 65 q^{70} + 37 q^{71} + 55 q^{72} + 8 q^{73} - 11 q^{74} + 24 q^{75} + 13 q^{76} + q^{77} + 24 q^{79} + 26 q^{80} + 81 q^{81} + 26 q^{82} - 34 q^{83} - 103 q^{84} - 11 q^{85} + 30 q^{86} + 32 q^{87} - 3 q^{88} + 6 q^{89} + 47 q^{90} - 80 q^{92} + 41 q^{93} + 40 q^{94} + 20 q^{95} - 98 q^{96} - 5 q^{98} + 33 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.930026 −0.657628 −0.328814 0.944395i \(-0.606649\pi\)
−0.328814 + 0.944395i \(0.606649\pi\)
\(3\) 2.53271 1.46226 0.731129 0.682239i \(-0.238994\pi\)
0.731129 + 0.682239i \(0.238994\pi\)
\(4\) −1.13505 −0.567525
\(5\) −3.38691 −1.51467 −0.757335 0.653026i \(-0.773499\pi\)
−0.757335 + 0.653026i \(0.773499\pi\)
\(6\) −2.35548 −0.961622
\(7\) −3.38196 −1.27826 −0.639131 0.769098i \(-0.720706\pi\)
−0.639131 + 0.769098i \(0.720706\pi\)
\(8\) 2.91568 1.03085
\(9\) 3.41460 1.13820
\(10\) 3.14991 0.996090
\(11\) 1.00000 0.301511
\(12\) −2.87475 −0.829869
\(13\) 0 0
\(14\) 3.14532 0.840621
\(15\) −8.57804 −2.21484
\(16\) −0.441558 −0.110390
\(17\) −7.19830 −1.74584 −0.872922 0.487860i \(-0.837778\pi\)
−0.872922 + 0.487860i \(0.837778\pi\)
\(18\) −3.17567 −0.748512
\(19\) −3.06566 −0.703310 −0.351655 0.936130i \(-0.614381\pi\)
−0.351655 + 0.936130i \(0.614381\pi\)
\(20\) 3.84431 0.859614
\(21\) −8.56552 −1.86915
\(22\) −0.930026 −0.198282
\(23\) 5.94916 1.24048 0.620242 0.784410i \(-0.287034\pi\)
0.620242 + 0.784410i \(0.287034\pi\)
\(24\) 7.38456 1.50737
\(25\) 6.47113 1.29423
\(26\) 0 0
\(27\) 1.05006 0.202084
\(28\) 3.83870 0.725446
\(29\) 5.43442 1.00915 0.504573 0.863369i \(-0.331650\pi\)
0.504573 + 0.863369i \(0.331650\pi\)
\(30\) 7.97780 1.45654
\(31\) 9.06534 1.62818 0.814092 0.580736i \(-0.197235\pi\)
0.814092 + 0.580736i \(0.197235\pi\)
\(32\) −5.42070 −0.958253
\(33\) 2.53271 0.440888
\(34\) 6.69461 1.14812
\(35\) 11.4544 1.93615
\(36\) −3.87574 −0.645957
\(37\) −4.87541 −0.801512 −0.400756 0.916185i \(-0.631253\pi\)
−0.400756 + 0.916185i \(0.631253\pi\)
\(38\) 2.85114 0.462517
\(39\) 0 0
\(40\) −9.87513 −1.56140
\(41\) 0.859934 0.134299 0.0671495 0.997743i \(-0.478610\pi\)
0.0671495 + 0.997743i \(0.478610\pi\)
\(42\) 7.96616 1.22921
\(43\) 5.81891 0.887376 0.443688 0.896181i \(-0.353670\pi\)
0.443688 + 0.896181i \(0.353670\pi\)
\(44\) −1.13505 −0.171115
\(45\) −11.5649 −1.72400
\(46\) −5.53287 −0.815777
\(47\) 8.69187 1.26784 0.633920 0.773399i \(-0.281445\pi\)
0.633920 + 0.773399i \(0.281445\pi\)
\(48\) −1.11834 −0.161418
\(49\) 4.43768 0.633954
\(50\) −6.01832 −0.851119
\(51\) −18.2312 −2.55287
\(52\) 0 0
\(53\) −9.91682 −1.36218 −0.681090 0.732200i \(-0.738494\pi\)
−0.681090 + 0.732200i \(0.738494\pi\)
\(54\) −0.976585 −0.132896
\(55\) −3.38691 −0.456690
\(56\) −9.86073 −1.31769
\(57\) −7.76441 −1.02842
\(58\) −5.05416 −0.663643
\(59\) 7.38488 0.961429 0.480715 0.876877i \(-0.340377\pi\)
0.480715 + 0.876877i \(0.340377\pi\)
\(60\) 9.73651 1.25698
\(61\) 5.88757 0.753827 0.376913 0.926249i \(-0.376985\pi\)
0.376913 + 0.926249i \(0.376985\pi\)
\(62\) −8.43101 −1.07074
\(63\) −11.5481 −1.45492
\(64\) 5.92451 0.740564
\(65\) 0 0
\(66\) −2.35548 −0.289940
\(67\) −1.27537 −0.155812 −0.0779058 0.996961i \(-0.524823\pi\)
−0.0779058 + 0.996961i \(0.524823\pi\)
\(68\) 8.17043 0.990810
\(69\) 15.0675 1.81391
\(70\) −10.6529 −1.27326
\(71\) 13.2250 1.56952 0.784758 0.619803i \(-0.212787\pi\)
0.784758 + 0.619803i \(0.212787\pi\)
\(72\) 9.95588 1.17331
\(73\) 1.40595 0.164554 0.0822771 0.996609i \(-0.473781\pi\)
0.0822771 + 0.996609i \(0.473781\pi\)
\(74\) 4.53426 0.527097
\(75\) 16.3895 1.89249
\(76\) 3.47968 0.399147
\(77\) −3.38196 −0.385411
\(78\) 0 0
\(79\) 4.47756 0.503765 0.251883 0.967758i \(-0.418950\pi\)
0.251883 + 0.967758i \(0.418950\pi\)
\(80\) 1.49552 0.167204
\(81\) −7.58430 −0.842701
\(82\) −0.799761 −0.0883188
\(83\) −2.06037 −0.226155 −0.113078 0.993586i \(-0.536071\pi\)
−0.113078 + 0.993586i \(0.536071\pi\)
\(84\) 9.72230 1.06079
\(85\) 24.3800 2.64438
\(86\) −5.41174 −0.583563
\(87\) 13.7638 1.47563
\(88\) 2.91568 0.310813
\(89\) −0.472173 −0.0500503 −0.0250251 0.999687i \(-0.507967\pi\)
−0.0250251 + 0.999687i \(0.507967\pi\)
\(90\) 10.7557 1.13375
\(91\) 0 0
\(92\) −6.75259 −0.704007
\(93\) 22.9599 2.38083
\(94\) −8.08367 −0.833767
\(95\) 10.3831 1.06528
\(96\) −13.7290 −1.40121
\(97\) 6.51513 0.661511 0.330756 0.943716i \(-0.392696\pi\)
0.330756 + 0.943716i \(0.392696\pi\)
\(98\) −4.12716 −0.416906
\(99\) 3.41460 0.343180
\(100\) −7.34506 −0.734506
\(101\) 2.64800 0.263486 0.131743 0.991284i \(-0.457943\pi\)
0.131743 + 0.991284i \(0.457943\pi\)
\(102\) 16.9555 1.67884
\(103\) 1.41401 0.139327 0.0696634 0.997571i \(-0.477807\pi\)
0.0696634 + 0.997571i \(0.477807\pi\)
\(104\) 0 0
\(105\) 29.0106 2.83115
\(106\) 9.22290 0.895808
\(107\) 3.62837 0.350768 0.175384 0.984500i \(-0.443883\pi\)
0.175384 + 0.984500i \(0.443883\pi\)
\(108\) −1.19187 −0.114688
\(109\) 9.42565 0.902813 0.451407 0.892318i \(-0.350922\pi\)
0.451407 + 0.892318i \(0.350922\pi\)
\(110\) 3.14991 0.300332
\(111\) −12.3480 −1.17202
\(112\) 1.49333 0.141107
\(113\) −11.8576 −1.11547 −0.557734 0.830020i \(-0.688329\pi\)
−0.557734 + 0.830020i \(0.688329\pi\)
\(114\) 7.22111 0.676319
\(115\) −20.1492 −1.87893
\(116\) −6.16834 −0.572716
\(117\) 0 0
\(118\) −6.86813 −0.632263
\(119\) 24.3444 2.23165
\(120\) −25.0108 −2.28316
\(121\) 1.00000 0.0909091
\(122\) −5.47560 −0.495737
\(123\) 2.17796 0.196380
\(124\) −10.2896 −0.924036
\(125\) −4.98257 −0.445655
\(126\) 10.7400 0.956795
\(127\) 8.65803 0.768276 0.384138 0.923276i \(-0.374499\pi\)
0.384138 + 0.923276i \(0.374499\pi\)
\(128\) 5.33145 0.471238
\(129\) 14.7376 1.29757
\(130\) 0 0
\(131\) 8.57478 0.749182 0.374591 0.927190i \(-0.377783\pi\)
0.374591 + 0.927190i \(0.377783\pi\)
\(132\) −2.87475 −0.250215
\(133\) 10.3679 0.899015
\(134\) 1.18613 0.102466
\(135\) −3.55646 −0.306091
\(136\) −20.9879 −1.79970
\(137\) −12.7244 −1.08712 −0.543559 0.839371i \(-0.682924\pi\)
−0.543559 + 0.839371i \(0.682924\pi\)
\(138\) −14.0131 −1.19288
\(139\) 7.49251 0.635506 0.317753 0.948173i \(-0.397072\pi\)
0.317753 + 0.948173i \(0.397072\pi\)
\(140\) −13.0013 −1.09881
\(141\) 22.0140 1.85391
\(142\) −12.2996 −1.03216
\(143\) 0 0
\(144\) −1.50775 −0.125645
\(145\) −18.4059 −1.52852
\(146\) −1.30757 −0.108215
\(147\) 11.2393 0.927005
\(148\) 5.53384 0.454879
\(149\) 7.25497 0.594350 0.297175 0.954823i \(-0.403955\pi\)
0.297175 + 0.954823i \(0.403955\pi\)
\(150\) −15.2426 −1.24456
\(151\) −14.1259 −1.14955 −0.574775 0.818312i \(-0.694910\pi\)
−0.574775 + 0.818312i \(0.694910\pi\)
\(152\) −8.93848 −0.725007
\(153\) −24.5793 −1.98712
\(154\) 3.14532 0.253457
\(155\) −30.7035 −2.46616
\(156\) 0 0
\(157\) 8.79175 0.701658 0.350829 0.936440i \(-0.385900\pi\)
0.350829 + 0.936440i \(0.385900\pi\)
\(158\) −4.16425 −0.331290
\(159\) −25.1164 −1.99186
\(160\) 18.3594 1.45144
\(161\) −20.1198 −1.58566
\(162\) 7.05360 0.554184
\(163\) 19.3644 1.51674 0.758368 0.651826i \(-0.225997\pi\)
0.758368 + 0.651826i \(0.225997\pi\)
\(164\) −0.976068 −0.0762181
\(165\) −8.57804 −0.667799
\(166\) 1.91620 0.148726
\(167\) −14.5289 −1.12428 −0.562140 0.827042i \(-0.690022\pi\)
−0.562140 + 0.827042i \(0.690022\pi\)
\(168\) −24.9743 −1.92681
\(169\) 0 0
\(170\) −22.6740 −1.73902
\(171\) −10.4680 −0.800508
\(172\) −6.60476 −0.503608
\(173\) −23.5645 −1.79158 −0.895789 0.444480i \(-0.853389\pi\)
−0.895789 + 0.444480i \(0.853389\pi\)
\(174\) −12.8007 −0.970418
\(175\) −21.8851 −1.65436
\(176\) −0.441558 −0.0332837
\(177\) 18.7037 1.40586
\(178\) 0.439134 0.0329144
\(179\) 9.01232 0.673612 0.336806 0.941574i \(-0.390653\pi\)
0.336806 + 0.941574i \(0.390653\pi\)
\(180\) 13.1268 0.978413
\(181\) 10.6542 0.791917 0.395958 0.918268i \(-0.370412\pi\)
0.395958 + 0.918268i \(0.370412\pi\)
\(182\) 0 0
\(183\) 14.9115 1.10229
\(184\) 17.3458 1.27875
\(185\) 16.5126 1.21403
\(186\) −21.3533 −1.56570
\(187\) −7.19830 −0.526392
\(188\) −9.86572 −0.719531
\(189\) −3.55127 −0.258317
\(190\) −9.65656 −0.700560
\(191\) 20.5050 1.48369 0.741846 0.670570i \(-0.233951\pi\)
0.741846 + 0.670570i \(0.233951\pi\)
\(192\) 15.0050 1.08290
\(193\) −11.1839 −0.805035 −0.402518 0.915412i \(-0.631865\pi\)
−0.402518 + 0.915412i \(0.631865\pi\)
\(194\) −6.05925 −0.435028
\(195\) 0 0
\(196\) −5.03699 −0.359785
\(197\) 14.8983 1.06146 0.530729 0.847542i \(-0.321918\pi\)
0.530729 + 0.847542i \(0.321918\pi\)
\(198\) −3.17567 −0.225685
\(199\) −6.74457 −0.478110 −0.239055 0.971006i \(-0.576838\pi\)
−0.239055 + 0.971006i \(0.576838\pi\)
\(200\) 18.8677 1.33415
\(201\) −3.23015 −0.227837
\(202\) −2.46271 −0.173276
\(203\) −18.3790 −1.28995
\(204\) 20.6933 1.44882
\(205\) −2.91251 −0.203419
\(206\) −1.31507 −0.0916253
\(207\) 20.3140 1.41192
\(208\) 0 0
\(209\) −3.06566 −0.212056
\(210\) −26.9806 −1.86184
\(211\) −24.1718 −1.66406 −0.832029 0.554732i \(-0.812821\pi\)
−0.832029 + 0.554732i \(0.812821\pi\)
\(212\) 11.2561 0.773072
\(213\) 33.4950 2.29504
\(214\) −3.37448 −0.230675
\(215\) −19.7081 −1.34408
\(216\) 3.06164 0.208318
\(217\) −30.6587 −2.08125
\(218\) −8.76610 −0.593715
\(219\) 3.56086 0.240621
\(220\) 3.84431 0.259183
\(221\) 0 0
\(222\) 11.4839 0.770752
\(223\) 17.3564 1.16227 0.581134 0.813808i \(-0.302609\pi\)
0.581134 + 0.813808i \(0.302609\pi\)
\(224\) 18.3326 1.22490
\(225\) 22.0963 1.47309
\(226\) 11.0279 0.733563
\(227\) −10.7311 −0.712247 −0.356123 0.934439i \(-0.615902\pi\)
−0.356123 + 0.934439i \(0.615902\pi\)
\(228\) 8.81300 0.583655
\(229\) 22.4942 1.48646 0.743229 0.669037i \(-0.233293\pi\)
0.743229 + 0.669037i \(0.233293\pi\)
\(230\) 18.7393 1.23563
\(231\) −8.56552 −0.563570
\(232\) 15.8450 1.04028
\(233\) −7.41928 −0.486053 −0.243027 0.970020i \(-0.578140\pi\)
−0.243027 + 0.970020i \(0.578140\pi\)
\(234\) 0 0
\(235\) −29.4386 −1.92036
\(236\) −8.38221 −0.545635
\(237\) 11.3404 0.736635
\(238\) −22.6409 −1.46759
\(239\) −1.77240 −0.114647 −0.0573235 0.998356i \(-0.518257\pi\)
−0.0573235 + 0.998356i \(0.518257\pi\)
\(240\) 3.78770 0.244495
\(241\) −11.4199 −0.735622 −0.367811 0.929901i \(-0.619893\pi\)
−0.367811 + 0.929901i \(0.619893\pi\)
\(242\) −0.930026 −0.0597844
\(243\) −22.3590 −1.43433
\(244\) −6.68269 −0.427816
\(245\) −15.0300 −0.960232
\(246\) −2.02556 −0.129145
\(247\) 0 0
\(248\) 26.4316 1.67841
\(249\) −5.21832 −0.330697
\(250\) 4.63393 0.293075
\(251\) 18.1455 1.14533 0.572667 0.819788i \(-0.305909\pi\)
0.572667 + 0.819788i \(0.305909\pi\)
\(252\) 13.1076 0.825703
\(253\) 5.94916 0.374020
\(254\) −8.05220 −0.505240
\(255\) 61.7473 3.86676
\(256\) −16.8074 −1.05046
\(257\) −7.34137 −0.457942 −0.228971 0.973433i \(-0.573536\pi\)
−0.228971 + 0.973433i \(0.573536\pi\)
\(258\) −13.7064 −0.853320
\(259\) 16.4885 1.02454
\(260\) 0 0
\(261\) 18.5564 1.14861
\(262\) −7.97477 −0.492683
\(263\) −23.5615 −1.45286 −0.726431 0.687239i \(-0.758822\pi\)
−0.726431 + 0.687239i \(0.758822\pi\)
\(264\) 7.38456 0.454488
\(265\) 33.5873 2.06325
\(266\) −9.64247 −0.591218
\(267\) −1.19588 −0.0731864
\(268\) 1.44761 0.0884271
\(269\) −12.3713 −0.754291 −0.377146 0.926154i \(-0.623094\pi\)
−0.377146 + 0.926154i \(0.623094\pi\)
\(270\) 3.30760 0.201294
\(271\) 21.6867 1.31737 0.658686 0.752418i \(-0.271113\pi\)
0.658686 + 0.752418i \(0.271113\pi\)
\(272\) 3.17847 0.192723
\(273\) 0 0
\(274\) 11.8340 0.714920
\(275\) 6.47113 0.390224
\(276\) −17.1023 −1.02944
\(277\) −24.6590 −1.48162 −0.740809 0.671716i \(-0.765558\pi\)
−0.740809 + 0.671716i \(0.765558\pi\)
\(278\) −6.96823 −0.417927
\(279\) 30.9545 1.85320
\(280\) 33.3973 1.99587
\(281\) −31.3448 −1.86988 −0.934938 0.354810i \(-0.884545\pi\)
−0.934938 + 0.354810i \(0.884545\pi\)
\(282\) −20.4736 −1.21918
\(283\) 20.4209 1.21390 0.606949 0.794741i \(-0.292393\pi\)
0.606949 + 0.794741i \(0.292393\pi\)
\(284\) −15.0110 −0.890740
\(285\) 26.2973 1.55772
\(286\) 0 0
\(287\) −2.90826 −0.171669
\(288\) −18.5095 −1.09068
\(289\) 34.8155 2.04797
\(290\) 17.1179 1.00520
\(291\) 16.5009 0.967301
\(292\) −1.59583 −0.0933887
\(293\) 10.9817 0.641555 0.320778 0.947155i \(-0.396056\pi\)
0.320778 + 0.947155i \(0.396056\pi\)
\(294\) −10.4529 −0.609625
\(295\) −25.0119 −1.45625
\(296\) −14.2151 −0.826238
\(297\) 1.05006 0.0609307
\(298\) −6.74732 −0.390861
\(299\) 0 0
\(300\) −18.6029 −1.07404
\(301\) −19.6793 −1.13430
\(302\) 13.1375 0.755976
\(303\) 6.70661 0.385285
\(304\) 1.35367 0.0776381
\(305\) −19.9407 −1.14180
\(306\) 22.8594 1.30679
\(307\) 15.5836 0.889402 0.444701 0.895679i \(-0.353310\pi\)
0.444701 + 0.895679i \(0.353310\pi\)
\(308\) 3.83870 0.218730
\(309\) 3.58128 0.203732
\(310\) 28.5550 1.62182
\(311\) 8.37476 0.474889 0.237444 0.971401i \(-0.423690\pi\)
0.237444 + 0.971401i \(0.423690\pi\)
\(312\) 0 0
\(313\) 15.9360 0.900757 0.450378 0.892838i \(-0.351289\pi\)
0.450378 + 0.892838i \(0.351289\pi\)
\(314\) −8.17656 −0.461430
\(315\) 39.1122 2.20372
\(316\) −5.08226 −0.285900
\(317\) −15.0857 −0.847295 −0.423647 0.905827i \(-0.639250\pi\)
−0.423647 + 0.905827i \(0.639250\pi\)
\(318\) 23.3589 1.30990
\(319\) 5.43442 0.304269
\(320\) −20.0658 −1.12171
\(321\) 9.18959 0.512913
\(322\) 18.7120 1.04278
\(323\) 22.0675 1.22787
\(324\) 8.60857 0.478254
\(325\) 0 0
\(326\) −18.0094 −0.997449
\(327\) 23.8724 1.32015
\(328\) 2.50729 0.138442
\(329\) −29.3956 −1.62063
\(330\) 7.97780 0.439163
\(331\) 1.05736 0.0581180 0.0290590 0.999578i \(-0.490749\pi\)
0.0290590 + 0.999578i \(0.490749\pi\)
\(332\) 2.33863 0.128349
\(333\) −16.6476 −0.912282
\(334\) 13.5123 0.739358
\(335\) 4.31957 0.236003
\(336\) 3.78218 0.206335
\(337\) −5.81233 −0.316618 −0.158309 0.987390i \(-0.550604\pi\)
−0.158309 + 0.987390i \(0.550604\pi\)
\(338\) 0 0
\(339\) −30.0318 −1.63110
\(340\) −27.6725 −1.50075
\(341\) 9.06534 0.490916
\(342\) 9.73552 0.526437
\(343\) 8.66567 0.467902
\(344\) 16.9661 0.914750
\(345\) −51.0321 −2.74747
\(346\) 21.9156 1.17819
\(347\) −16.8264 −0.903289 −0.451644 0.892198i \(-0.649162\pi\)
−0.451644 + 0.892198i \(0.649162\pi\)
\(348\) −15.6226 −0.837459
\(349\) 5.50086 0.294454 0.147227 0.989103i \(-0.452965\pi\)
0.147227 + 0.989103i \(0.452965\pi\)
\(350\) 20.3537 1.08795
\(351\) 0 0
\(352\) −5.42070 −0.288924
\(353\) −25.6484 −1.36513 −0.682563 0.730826i \(-0.739135\pi\)
−0.682563 + 0.730826i \(0.739135\pi\)
\(354\) −17.3950 −0.924532
\(355\) −44.7917 −2.37730
\(356\) 0.535940 0.0284048
\(357\) 61.6572 3.26324
\(358\) −8.38170 −0.442986
\(359\) −5.58116 −0.294562 −0.147281 0.989095i \(-0.547052\pi\)
−0.147281 + 0.989095i \(0.547052\pi\)
\(360\) −33.7196 −1.77718
\(361\) −9.60173 −0.505354
\(362\) −9.90864 −0.520787
\(363\) 2.53271 0.132933
\(364\) 0 0
\(365\) −4.76183 −0.249245
\(366\) −13.8681 −0.724896
\(367\) 26.0854 1.36165 0.680823 0.732448i \(-0.261622\pi\)
0.680823 + 0.732448i \(0.261622\pi\)
\(368\) −2.62690 −0.136937
\(369\) 2.93633 0.152859
\(370\) −15.3571 −0.798378
\(371\) 33.5383 1.74122
\(372\) −26.0606 −1.35118
\(373\) 30.8984 1.59986 0.799930 0.600094i \(-0.204870\pi\)
0.799930 + 0.600094i \(0.204870\pi\)
\(374\) 6.69461 0.346170
\(375\) −12.6194 −0.651663
\(376\) 25.3427 1.30695
\(377\) 0 0
\(378\) 3.30277 0.169876
\(379\) −11.8349 −0.607920 −0.303960 0.952685i \(-0.598309\pi\)
−0.303960 + 0.952685i \(0.598309\pi\)
\(380\) −11.7853 −0.604575
\(381\) 21.9282 1.12342
\(382\) −19.0702 −0.975717
\(383\) 13.0913 0.668935 0.334468 0.942407i \(-0.391443\pi\)
0.334468 + 0.942407i \(0.391443\pi\)
\(384\) 13.5030 0.689072
\(385\) 11.4544 0.583770
\(386\) 10.4013 0.529414
\(387\) 19.8693 1.01001
\(388\) −7.39501 −0.375425
\(389\) −1.62816 −0.0825508 −0.0412754 0.999148i \(-0.513142\pi\)
−0.0412754 + 0.999148i \(0.513142\pi\)
\(390\) 0 0
\(391\) −42.8238 −2.16569
\(392\) 12.9389 0.653511
\(393\) 21.7174 1.09550
\(394\) −13.8558 −0.698045
\(395\) −15.1651 −0.763038
\(396\) −3.87574 −0.194764
\(397\) −9.43035 −0.473296 −0.236648 0.971596i \(-0.576049\pi\)
−0.236648 + 0.971596i \(0.576049\pi\)
\(398\) 6.27263 0.314419
\(399\) 26.2590 1.31459
\(400\) −2.85738 −0.142869
\(401\) 9.65745 0.482270 0.241135 0.970492i \(-0.422480\pi\)
0.241135 + 0.970492i \(0.422480\pi\)
\(402\) 3.00412 0.149832
\(403\) 0 0
\(404\) −3.00561 −0.149535
\(405\) 25.6873 1.27641
\(406\) 17.0930 0.848310
\(407\) −4.87541 −0.241665
\(408\) −53.1563 −2.63163
\(409\) 6.56175 0.324458 0.162229 0.986753i \(-0.448132\pi\)
0.162229 + 0.986753i \(0.448132\pi\)
\(410\) 2.70871 0.133774
\(411\) −32.2272 −1.58965
\(412\) −1.60498 −0.0790715
\(413\) −24.9754 −1.22896
\(414\) −18.8925 −0.928518
\(415\) 6.97829 0.342551
\(416\) 0 0
\(417\) 18.9763 0.929274
\(418\) 2.85114 0.139454
\(419\) −25.7883 −1.25984 −0.629921 0.776659i \(-0.716913\pi\)
−0.629921 + 0.776659i \(0.716913\pi\)
\(420\) −32.9285 −1.60675
\(421\) 2.65368 0.129332 0.0646661 0.997907i \(-0.479402\pi\)
0.0646661 + 0.997907i \(0.479402\pi\)
\(422\) 22.4805 1.09433
\(423\) 29.6793 1.44306
\(424\) −28.9143 −1.40420
\(425\) −46.5811 −2.25952
\(426\) −31.1512 −1.50928
\(427\) −19.9116 −0.963588
\(428\) −4.11838 −0.199070
\(429\) 0 0
\(430\) 18.3291 0.883906
\(431\) 21.4843 1.03486 0.517430 0.855725i \(-0.326889\pi\)
0.517430 + 0.855725i \(0.326889\pi\)
\(432\) −0.463663 −0.0223080
\(433\) 8.87391 0.426453 0.213226 0.977003i \(-0.431603\pi\)
0.213226 + 0.977003i \(0.431603\pi\)
\(434\) 28.5134 1.36869
\(435\) −46.6167 −2.23510
\(436\) −10.6986 −0.512370
\(437\) −18.2381 −0.872446
\(438\) −3.31170 −0.158239
\(439\) 32.2696 1.54014 0.770072 0.637957i \(-0.220220\pi\)
0.770072 + 0.637957i \(0.220220\pi\)
\(440\) −9.87513 −0.470779
\(441\) 15.1529 0.721567
\(442\) 0 0
\(443\) 21.3864 1.01610 0.508050 0.861327i \(-0.330366\pi\)
0.508050 + 0.861327i \(0.330366\pi\)
\(444\) 14.0156 0.665150
\(445\) 1.59921 0.0758096
\(446\) −16.1419 −0.764340
\(447\) 18.3747 0.869094
\(448\) −20.0365 −0.946635
\(449\) −16.7306 −0.789564 −0.394782 0.918775i \(-0.629180\pi\)
−0.394782 + 0.918775i \(0.629180\pi\)
\(450\) −20.5502 −0.968744
\(451\) 0.859934 0.0404927
\(452\) 13.4590 0.633057
\(453\) −35.7768 −1.68094
\(454\) 9.98019 0.468393
\(455\) 0 0
\(456\) −22.6385 −1.06015
\(457\) −9.78889 −0.457905 −0.228953 0.973438i \(-0.573530\pi\)
−0.228953 + 0.973438i \(0.573530\pi\)
\(458\) −20.9202 −0.977537
\(459\) −7.55865 −0.352808
\(460\) 22.8704 1.06634
\(461\) −18.7517 −0.873352 −0.436676 0.899619i \(-0.643844\pi\)
−0.436676 + 0.899619i \(0.643844\pi\)
\(462\) 7.96616 0.370619
\(463\) 23.2559 1.08079 0.540396 0.841411i \(-0.318274\pi\)
0.540396 + 0.841411i \(0.318274\pi\)
\(464\) −2.39961 −0.111399
\(465\) −77.7628 −3.60617
\(466\) 6.90013 0.319642
\(467\) 0.0987954 0.00457171 0.00228585 0.999997i \(-0.499272\pi\)
0.00228585 + 0.999997i \(0.499272\pi\)
\(468\) 0 0
\(469\) 4.31327 0.199168
\(470\) 27.3786 1.26288
\(471\) 22.2669 1.02601
\(472\) 21.5319 0.991088
\(473\) 5.81891 0.267554
\(474\) −10.5468 −0.484432
\(475\) −19.8383 −0.910242
\(476\) −27.6321 −1.26652
\(477\) −33.8620 −1.55043
\(478\) 1.64838 0.0753951
\(479\) −23.4409 −1.07104 −0.535520 0.844523i \(-0.679884\pi\)
−0.535520 + 0.844523i \(0.679884\pi\)
\(480\) 46.4990 2.12238
\(481\) 0 0
\(482\) 10.6208 0.483766
\(483\) −50.9576 −2.31865
\(484\) −1.13505 −0.0515932
\(485\) −22.0661 −1.00197
\(486\) 20.7945 0.943256
\(487\) 1.58984 0.0720424 0.0360212 0.999351i \(-0.488532\pi\)
0.0360212 + 0.999351i \(0.488532\pi\)
\(488\) 17.1663 0.777081
\(489\) 49.0443 2.21786
\(490\) 13.9783 0.631475
\(491\) 3.34966 0.151168 0.0755841 0.997139i \(-0.475918\pi\)
0.0755841 + 0.997139i \(0.475918\pi\)
\(492\) −2.47209 −0.111451
\(493\) −39.1186 −1.76181
\(494\) 0 0
\(495\) −11.5649 −0.519805
\(496\) −4.00288 −0.179734
\(497\) −44.7264 −2.00625
\(498\) 4.85317 0.217476
\(499\) 33.5789 1.50320 0.751600 0.659619i \(-0.229282\pi\)
0.751600 + 0.659619i \(0.229282\pi\)
\(500\) 5.65547 0.252921
\(501\) −36.7974 −1.64399
\(502\) −16.8758 −0.753204
\(503\) −9.93290 −0.442886 −0.221443 0.975173i \(-0.571077\pi\)
−0.221443 + 0.975173i \(0.571077\pi\)
\(504\) −33.6704 −1.49980
\(505\) −8.96853 −0.399094
\(506\) −5.53287 −0.245966
\(507\) 0 0
\(508\) −9.82730 −0.436016
\(509\) −0.669939 −0.0296945 −0.0148473 0.999890i \(-0.504726\pi\)
−0.0148473 + 0.999890i \(0.504726\pi\)
\(510\) −57.4266 −2.54289
\(511\) −4.75488 −0.210343
\(512\) 4.96844 0.219576
\(513\) −3.21913 −0.142128
\(514\) 6.82767 0.301155
\(515\) −4.78913 −0.211034
\(516\) −16.7279 −0.736405
\(517\) 8.69187 0.382268
\(518\) −15.3347 −0.673768
\(519\) −59.6820 −2.61975
\(520\) 0 0
\(521\) 35.5156 1.55597 0.777983 0.628286i \(-0.216243\pi\)
0.777983 + 0.628286i \(0.216243\pi\)
\(522\) −17.2579 −0.755359
\(523\) 12.2650 0.536310 0.268155 0.963376i \(-0.413586\pi\)
0.268155 + 0.963376i \(0.413586\pi\)
\(524\) −9.73281 −0.425180
\(525\) −55.4286 −2.41910
\(526\) 21.9128 0.955443
\(527\) −65.2550 −2.84255
\(528\) −1.11834 −0.0486694
\(529\) 12.3925 0.538802
\(530\) −31.2371 −1.35685
\(531\) 25.2164 1.09430
\(532\) −11.7681 −0.510214
\(533\) 0 0
\(534\) 1.11220 0.0481294
\(535\) −12.2889 −0.531297
\(536\) −3.71858 −0.160618
\(537\) 22.8256 0.984995
\(538\) 11.5056 0.496043
\(539\) 4.43768 0.191144
\(540\) 4.03676 0.173715
\(541\) −34.5002 −1.48328 −0.741639 0.670799i \(-0.765951\pi\)
−0.741639 + 0.670799i \(0.765951\pi\)
\(542\) −20.1692 −0.866341
\(543\) 26.9838 1.15799
\(544\) 39.0198 1.67296
\(545\) −31.9238 −1.36746
\(546\) 0 0
\(547\) 23.6264 1.01019 0.505096 0.863063i \(-0.331457\pi\)
0.505096 + 0.863063i \(0.331457\pi\)
\(548\) 14.4428 0.616967
\(549\) 20.1037 0.858006
\(550\) −6.01832 −0.256622
\(551\) −16.6601 −0.709743
\(552\) 43.9319 1.86987
\(553\) −15.1430 −0.643944
\(554\) 22.9336 0.974354
\(555\) 41.8214 1.77522
\(556\) −8.50438 −0.360666
\(557\) 24.7328 1.04796 0.523981 0.851730i \(-0.324446\pi\)
0.523981 + 0.851730i \(0.324446\pi\)
\(558\) −28.7885 −1.21872
\(559\) 0 0
\(560\) −5.05778 −0.213730
\(561\) −18.2312 −0.769721
\(562\) 29.1515 1.22968
\(563\) 25.1410 1.05957 0.529784 0.848133i \(-0.322273\pi\)
0.529784 + 0.848133i \(0.322273\pi\)
\(564\) −24.9870 −1.05214
\(565\) 40.1605 1.68957
\(566\) −18.9920 −0.798293
\(567\) 25.6498 1.07719
\(568\) 38.5598 1.61793
\(569\) 3.84060 0.161006 0.0805031 0.996754i \(-0.474347\pi\)
0.0805031 + 0.996754i \(0.474347\pi\)
\(570\) −24.4572 −1.02440
\(571\) −11.0865 −0.463954 −0.231977 0.972721i \(-0.574519\pi\)
−0.231977 + 0.972721i \(0.574519\pi\)
\(572\) 0 0
\(573\) 51.9332 2.16954
\(574\) 2.70476 0.112895
\(575\) 38.4978 1.60547
\(576\) 20.2298 0.842910
\(577\) −15.9929 −0.665794 −0.332897 0.942963i \(-0.608026\pi\)
−0.332897 + 0.942963i \(0.608026\pi\)
\(578\) −32.3793 −1.34680
\(579\) −28.3255 −1.17717
\(580\) 20.8916 0.867476
\(581\) 6.96810 0.289086
\(582\) −15.3463 −0.636124
\(583\) −9.91682 −0.410713
\(584\) 4.09931 0.169630
\(585\) 0 0
\(586\) −10.2132 −0.421905
\(587\) 3.37192 0.139174 0.0695869 0.997576i \(-0.477832\pi\)
0.0695869 + 0.997576i \(0.477832\pi\)
\(588\) −12.7572 −0.526099
\(589\) −27.7913 −1.14512
\(590\) 23.2617 0.957670
\(591\) 37.7330 1.55213
\(592\) 2.15278 0.0884786
\(593\) 46.2935 1.90105 0.950523 0.310654i \(-0.100548\pi\)
0.950523 + 0.310654i \(0.100548\pi\)
\(594\) −0.976585 −0.0400698
\(595\) −82.4521 −3.38021
\(596\) −8.23476 −0.337309
\(597\) −17.0820 −0.699120
\(598\) 0 0
\(599\) −33.4089 −1.36505 −0.682524 0.730863i \(-0.739118\pi\)
−0.682524 + 0.730863i \(0.739118\pi\)
\(600\) 47.7864 1.95087
\(601\) 8.82058 0.359799 0.179900 0.983685i \(-0.442423\pi\)
0.179900 + 0.983685i \(0.442423\pi\)
\(602\) 18.3023 0.745947
\(603\) −4.35489 −0.177345
\(604\) 16.0336 0.652399
\(605\) −3.38691 −0.137697
\(606\) −6.23732 −0.253374
\(607\) 40.4703 1.64264 0.821319 0.570469i \(-0.193239\pi\)
0.821319 + 0.570469i \(0.193239\pi\)
\(608\) 16.6180 0.673950
\(609\) −46.5486 −1.88625
\(610\) 18.5453 0.750879
\(611\) 0 0
\(612\) 27.8988 1.12774
\(613\) −37.4564 −1.51285 −0.756425 0.654081i \(-0.773056\pi\)
−0.756425 + 0.654081i \(0.773056\pi\)
\(614\) −14.4931 −0.584896
\(615\) −7.37654 −0.297451
\(616\) −9.86073 −0.397300
\(617\) 10.7790 0.433946 0.216973 0.976178i \(-0.430382\pi\)
0.216973 + 0.976178i \(0.430382\pi\)
\(618\) −3.33069 −0.133980
\(619\) −18.0458 −0.725321 −0.362660 0.931921i \(-0.618131\pi\)
−0.362660 + 0.931921i \(0.618131\pi\)
\(620\) 34.8500 1.39961
\(621\) 6.24698 0.250683
\(622\) −7.78875 −0.312300
\(623\) 1.59687 0.0639773
\(624\) 0 0
\(625\) −15.4801 −0.619205
\(626\) −14.8209 −0.592363
\(627\) −7.76441 −0.310081
\(628\) −9.97908 −0.398209
\(629\) 35.0946 1.39932
\(630\) −36.3754 −1.44923
\(631\) 27.7527 1.10482 0.552409 0.833573i \(-0.313709\pi\)
0.552409 + 0.833573i \(0.313709\pi\)
\(632\) 13.0551 0.519306
\(633\) −61.2202 −2.43328
\(634\) 14.0301 0.557205
\(635\) −29.3239 −1.16368
\(636\) 28.5084 1.13043
\(637\) 0 0
\(638\) −5.05416 −0.200096
\(639\) 45.1580 1.78642
\(640\) −18.0571 −0.713770
\(641\) 12.4477 0.491653 0.245827 0.969314i \(-0.420941\pi\)
0.245827 + 0.969314i \(0.420941\pi\)
\(642\) −8.54657 −0.337306
\(643\) 8.59509 0.338957 0.169479 0.985534i \(-0.445792\pi\)
0.169479 + 0.985534i \(0.445792\pi\)
\(644\) 22.8370 0.899905
\(645\) −49.9148 −1.96539
\(646\) −20.5234 −0.807482
\(647\) 28.0626 1.10325 0.551627 0.834091i \(-0.314007\pi\)
0.551627 + 0.834091i \(0.314007\pi\)
\(648\) −22.1134 −0.868697
\(649\) 7.38488 0.289882
\(650\) 0 0
\(651\) −77.6494 −3.04332
\(652\) −21.9796 −0.860787
\(653\) −40.9904 −1.60408 −0.802040 0.597270i \(-0.796252\pi\)
−0.802040 + 0.597270i \(0.796252\pi\)
\(654\) −22.2020 −0.868165
\(655\) −29.0420 −1.13476
\(656\) −0.379711 −0.0148252
\(657\) 4.80076 0.187296
\(658\) 27.3387 1.06577
\(659\) −41.7306 −1.62559 −0.812797 0.582547i \(-0.802056\pi\)
−0.812797 + 0.582547i \(0.802056\pi\)
\(660\) 9.73651 0.378993
\(661\) −16.5881 −0.645201 −0.322601 0.946535i \(-0.604557\pi\)
−0.322601 + 0.946535i \(0.604557\pi\)
\(662\) −0.983377 −0.0382200
\(663\) 0 0
\(664\) −6.00739 −0.233132
\(665\) −35.1153 −1.36171
\(666\) 15.4827 0.599942
\(667\) 32.3302 1.25183
\(668\) 16.4910 0.638057
\(669\) 43.9586 1.69954
\(670\) −4.01731 −0.155202
\(671\) 5.88757 0.227287
\(672\) 46.4311 1.79112
\(673\) 29.8674 1.15130 0.575652 0.817695i \(-0.304748\pi\)
0.575652 + 0.817695i \(0.304748\pi\)
\(674\) 5.40562 0.208217
\(675\) 6.79508 0.261543
\(676\) 0 0
\(677\) 20.8295 0.800542 0.400271 0.916397i \(-0.368916\pi\)
0.400271 + 0.916397i \(0.368916\pi\)
\(678\) 27.9304 1.07266
\(679\) −22.0339 −0.845585
\(680\) 71.0841 2.72595
\(681\) −27.1787 −1.04149
\(682\) −8.43101 −0.322840
\(683\) 34.2842 1.31185 0.655924 0.754827i \(-0.272279\pi\)
0.655924 + 0.754827i \(0.272279\pi\)
\(684\) 11.8817 0.454309
\(685\) 43.0963 1.64663
\(686\) −8.05930 −0.307706
\(687\) 56.9712 2.17359
\(688\) −2.56939 −0.0979570
\(689\) 0 0
\(690\) 47.4612 1.80682
\(691\) 8.39056 0.319192 0.159596 0.987182i \(-0.448981\pi\)
0.159596 + 0.987182i \(0.448981\pi\)
\(692\) 26.7469 1.01677
\(693\) −11.5481 −0.438674
\(694\) 15.6490 0.594028
\(695\) −25.3764 −0.962582
\(696\) 40.1308 1.52115
\(697\) −6.19006 −0.234465
\(698\) −5.11594 −0.193641
\(699\) −18.7909 −0.710735
\(700\) 24.8407 0.938891
\(701\) −33.0702 −1.24904 −0.624522 0.781007i \(-0.714706\pi\)
−0.624522 + 0.781007i \(0.714706\pi\)
\(702\) 0 0
\(703\) 14.9463 0.563712
\(704\) 5.92451 0.223288
\(705\) −74.5592 −2.80806
\(706\) 23.8537 0.897746
\(707\) −8.95544 −0.336804
\(708\) −21.2297 −0.797860
\(709\) −0.364882 −0.0137034 −0.00685171 0.999977i \(-0.502181\pi\)
−0.00685171 + 0.999977i \(0.502181\pi\)
\(710\) 41.6575 1.56338
\(711\) 15.2891 0.573386
\(712\) −1.37671 −0.0515942
\(713\) 53.9311 2.01974
\(714\) −57.3428 −2.14600
\(715\) 0 0
\(716\) −10.2294 −0.382292
\(717\) −4.48897 −0.167644
\(718\) 5.19063 0.193713
\(719\) −18.6303 −0.694794 −0.347397 0.937718i \(-0.612934\pi\)
−0.347397 + 0.937718i \(0.612934\pi\)
\(720\) 5.10659 0.190311
\(721\) −4.78214 −0.178096
\(722\) 8.92987 0.332335
\(723\) −28.9233 −1.07567
\(724\) −12.0930 −0.449433
\(725\) 35.1668 1.30606
\(726\) −2.35548 −0.0874202
\(727\) 15.2502 0.565599 0.282800 0.959179i \(-0.408737\pi\)
0.282800 + 0.959179i \(0.408737\pi\)
\(728\) 0 0
\(729\) −33.8759 −1.25466
\(730\) 4.42862 0.163911
\(731\) −41.8863 −1.54922
\(732\) −16.9253 −0.625577
\(733\) 1.47710 0.0545581 0.0272791 0.999628i \(-0.491316\pi\)
0.0272791 + 0.999628i \(0.491316\pi\)
\(734\) −24.2601 −0.895456
\(735\) −38.0666 −1.40411
\(736\) −32.2486 −1.18870
\(737\) −1.27537 −0.0469790
\(738\) −2.73086 −0.100524
\(739\) 31.5462 1.16045 0.580223 0.814457i \(-0.302965\pi\)
0.580223 + 0.814457i \(0.302965\pi\)
\(740\) −18.7426 −0.688991
\(741\) 0 0
\(742\) −31.1915 −1.14508
\(743\) −22.0189 −0.807796 −0.403898 0.914804i \(-0.632345\pi\)
−0.403898 + 0.914804i \(0.632345\pi\)
\(744\) 66.9436 2.45427
\(745\) −24.5719 −0.900245
\(746\) −28.7363 −1.05211
\(747\) −7.03535 −0.257410
\(748\) 8.17043 0.298741
\(749\) −12.2710 −0.448373
\(750\) 11.7364 0.428552
\(751\) 31.5506 1.15130 0.575649 0.817697i \(-0.304749\pi\)
0.575649 + 0.817697i \(0.304749\pi\)
\(752\) −3.83797 −0.139956
\(753\) 45.9572 1.67477
\(754\) 0 0
\(755\) 47.8431 1.74119
\(756\) 4.03087 0.146601
\(757\) −15.4522 −0.561618 −0.280809 0.959764i \(-0.590603\pi\)
−0.280809 + 0.959764i \(0.590603\pi\)
\(758\) 11.0068 0.399785
\(759\) 15.0675 0.546914
\(760\) 30.2738 1.09815
\(761\) 15.1957 0.550842 0.275421 0.961324i \(-0.411183\pi\)
0.275421 + 0.961324i \(0.411183\pi\)
\(762\) −20.3938 −0.738791
\(763\) −31.8772 −1.15403
\(764\) −23.2742 −0.842033
\(765\) 83.2478 3.00983
\(766\) −12.1753 −0.439911
\(767\) 0 0
\(768\) −42.5682 −1.53605
\(769\) 19.6669 0.709207 0.354603 0.935017i \(-0.384616\pi\)
0.354603 + 0.935017i \(0.384616\pi\)
\(770\) −10.6529 −0.383903
\(771\) −18.5935 −0.669630
\(772\) 12.6943 0.456878
\(773\) 4.04584 0.145519 0.0727594 0.997350i \(-0.476819\pi\)
0.0727594 + 0.997350i \(0.476819\pi\)
\(774\) −18.4789 −0.664212
\(775\) 58.6630 2.10724
\(776\) 18.9960 0.681918
\(777\) 41.7604 1.49815
\(778\) 1.51423 0.0542877
\(779\) −2.63626 −0.0944539
\(780\) 0 0
\(781\) 13.2250 0.473227
\(782\) 39.8273 1.42422
\(783\) 5.70647 0.203933
\(784\) −1.95949 −0.0699819
\(785\) −29.7768 −1.06278
\(786\) −20.1977 −0.720430
\(787\) 9.62403 0.343060 0.171530 0.985179i \(-0.445129\pi\)
0.171530 + 0.985179i \(0.445129\pi\)
\(788\) −16.9103 −0.602404
\(789\) −59.6743 −2.12446
\(790\) 14.1039 0.501795
\(791\) 40.1020 1.42586
\(792\) 9.95588 0.353767
\(793\) 0 0
\(794\) 8.77047 0.311252
\(795\) 85.0668 3.01701
\(796\) 7.65543 0.271340
\(797\) 12.4277 0.440213 0.220106 0.975476i \(-0.429360\pi\)
0.220106 + 0.975476i \(0.429360\pi\)
\(798\) −24.4215 −0.864513
\(799\) −62.5667 −2.21345
\(800\) −35.0780 −1.24020
\(801\) −1.61228 −0.0569672
\(802\) −8.98169 −0.317154
\(803\) 1.40595 0.0496150
\(804\) 3.66638 0.129303
\(805\) 68.1440 2.40176
\(806\) 0 0
\(807\) −31.3329 −1.10297
\(808\) 7.72072 0.271614
\(809\) −0.111775 −0.00392978 −0.00196489 0.999998i \(-0.500625\pi\)
−0.00196489 + 0.999998i \(0.500625\pi\)
\(810\) −23.8899 −0.839405
\(811\) −3.40641 −0.119615 −0.0598076 0.998210i \(-0.519049\pi\)
−0.0598076 + 0.998210i \(0.519049\pi\)
\(812\) 20.8611 0.732082
\(813\) 54.9260 1.92634
\(814\) 4.53426 0.158926
\(815\) −65.5854 −2.29736
\(816\) 8.05012 0.281811
\(817\) −17.8388 −0.624101
\(818\) −6.10260 −0.213372
\(819\) 0 0
\(820\) 3.30585 0.115445
\(821\) −26.9560 −0.940770 −0.470385 0.882461i \(-0.655885\pi\)
−0.470385 + 0.882461i \(0.655885\pi\)
\(822\) 29.9721 1.04540
\(823\) −47.1110 −1.64219 −0.821093 0.570795i \(-0.806635\pi\)
−0.821093 + 0.570795i \(0.806635\pi\)
\(824\) 4.12281 0.143625
\(825\) 16.3895 0.570608
\(826\) 23.2278 0.808198
\(827\) 2.81708 0.0979595 0.0489798 0.998800i \(-0.484403\pi\)
0.0489798 + 0.998800i \(0.484403\pi\)
\(828\) −23.0574 −0.801300
\(829\) 39.1854 1.36097 0.680483 0.732764i \(-0.261770\pi\)
0.680483 + 0.732764i \(0.261770\pi\)
\(830\) −6.48999 −0.225271
\(831\) −62.4541 −2.16651
\(832\) 0 0
\(833\) −31.9437 −1.10679
\(834\) −17.6485 −0.611117
\(835\) 49.2080 1.70291
\(836\) 3.47968 0.120347
\(837\) 9.51917 0.329030
\(838\) 23.9838 0.828507
\(839\) −6.19239 −0.213785 −0.106893 0.994271i \(-0.534090\pi\)
−0.106893 + 0.994271i \(0.534090\pi\)
\(840\) 84.5857 2.91848
\(841\) 0.532928 0.0183768
\(842\) −2.46799 −0.0850525
\(843\) −79.3873 −2.73424
\(844\) 27.4363 0.944395
\(845\) 0 0
\(846\) −27.6025 −0.948994
\(847\) −3.38196 −0.116206
\(848\) 4.37885 0.150370
\(849\) 51.7202 1.77503
\(850\) 43.3217 1.48592
\(851\) −29.0046 −0.994264
\(852\) −38.0185 −1.30249
\(853\) −34.7550 −1.18999 −0.594995 0.803729i \(-0.702846\pi\)
−0.594995 + 0.803729i \(0.702846\pi\)
\(854\) 18.5183 0.633682
\(855\) 35.4541 1.21251
\(856\) 10.5792 0.361588
\(857\) 37.6326 1.28551 0.642753 0.766073i \(-0.277792\pi\)
0.642753 + 0.766073i \(0.277792\pi\)
\(858\) 0 0
\(859\) 19.8890 0.678606 0.339303 0.940677i \(-0.389809\pi\)
0.339303 + 0.940677i \(0.389809\pi\)
\(860\) 22.3697 0.762800
\(861\) −7.36578 −0.251025
\(862\) −19.9809 −0.680553
\(863\) 14.1149 0.480478 0.240239 0.970714i \(-0.422774\pi\)
0.240239 + 0.970714i \(0.422774\pi\)
\(864\) −5.69207 −0.193648
\(865\) 79.8108 2.71365
\(866\) −8.25297 −0.280447
\(867\) 88.1774 2.99466
\(868\) 34.7991 1.18116
\(869\) 4.47756 0.151891
\(870\) 43.3547 1.46986
\(871\) 0 0
\(872\) 27.4822 0.930664
\(873\) 22.2466 0.752932
\(874\) 16.9619 0.573745
\(875\) 16.8509 0.569664
\(876\) −4.04176 −0.136558
\(877\) −54.6173 −1.84429 −0.922147 0.386839i \(-0.873567\pi\)
−0.922147 + 0.386839i \(0.873567\pi\)
\(878\) −30.0116 −1.01284
\(879\) 27.8133 0.938120
\(880\) 1.49552 0.0504138
\(881\) 16.5871 0.558835 0.279417 0.960170i \(-0.409859\pi\)
0.279417 + 0.960170i \(0.409859\pi\)
\(882\) −14.0926 −0.474523
\(883\) 8.16157 0.274659 0.137329 0.990525i \(-0.456148\pi\)
0.137329 + 0.990525i \(0.456148\pi\)
\(884\) 0 0
\(885\) −63.3478 −2.12941
\(886\) −19.8900 −0.668216
\(887\) 28.9794 0.973033 0.486516 0.873672i \(-0.338267\pi\)
0.486516 + 0.873672i \(0.338267\pi\)
\(888\) −36.0028 −1.20817
\(889\) −29.2811 −0.982058
\(890\) −1.48730 −0.0498545
\(891\) −7.58430 −0.254084
\(892\) −19.7004 −0.659617
\(893\) −26.6463 −0.891685
\(894\) −17.0890 −0.571541
\(895\) −30.5239 −1.02030
\(896\) −18.0308 −0.602366
\(897\) 0 0
\(898\) 15.5599 0.519239
\(899\) 49.2649 1.64308
\(900\) −25.0804 −0.836015
\(901\) 71.3842 2.37815
\(902\) −0.799761 −0.0266291
\(903\) −49.8420 −1.65864
\(904\) −34.5729 −1.14988
\(905\) −36.0846 −1.19949
\(906\) 33.2733 1.10543
\(907\) 40.7378 1.35268 0.676339 0.736591i \(-0.263566\pi\)
0.676339 + 0.736591i \(0.263566\pi\)
\(908\) 12.1803 0.404218
\(909\) 9.04186 0.299900
\(910\) 0 0
\(911\) 31.8194 1.05422 0.527111 0.849796i \(-0.323275\pi\)
0.527111 + 0.849796i \(0.323275\pi\)
\(912\) 3.42844 0.113527
\(913\) −2.06037 −0.0681884
\(914\) 9.10393 0.301131
\(915\) −50.5038 −1.66960
\(916\) −25.5321 −0.843603
\(917\) −28.9996 −0.957651
\(918\) 7.02975 0.232016
\(919\) −31.5844 −1.04187 −0.520937 0.853595i \(-0.674417\pi\)
−0.520937 + 0.853595i \(0.674417\pi\)
\(920\) −58.7487 −1.93689
\(921\) 39.4686 1.30054
\(922\) 17.4396 0.574341
\(923\) 0 0
\(924\) 9.72230 0.319840
\(925\) −31.5494 −1.03734
\(926\) −21.6286 −0.710759
\(927\) 4.82829 0.158582
\(928\) −29.4584 −0.967018
\(929\) −28.6916 −0.941340 −0.470670 0.882309i \(-0.655988\pi\)
−0.470670 + 0.882309i \(0.655988\pi\)
\(930\) 72.3215 2.37152
\(931\) −13.6044 −0.445867
\(932\) 8.42126 0.275848
\(933\) 21.2108 0.694410
\(934\) −0.0918824 −0.00300648
\(935\) 24.3800 0.797310
\(936\) 0 0
\(937\) 13.3518 0.436184 0.218092 0.975928i \(-0.430017\pi\)
0.218092 + 0.975928i \(0.430017\pi\)
\(938\) −4.01145 −0.130979
\(939\) 40.3612 1.31714
\(940\) 33.4143 1.08985
\(941\) −26.2375 −0.855320 −0.427660 0.903940i \(-0.640662\pi\)
−0.427660 + 0.903940i \(0.640662\pi\)
\(942\) −20.7088 −0.674730
\(943\) 5.11588 0.166596
\(944\) −3.26085 −0.106132
\(945\) 12.0278 0.391265
\(946\) −5.41174 −0.175951
\(947\) −18.0926 −0.587929 −0.293965 0.955816i \(-0.594975\pi\)
−0.293965 + 0.955816i \(0.594975\pi\)
\(948\) −12.8719 −0.418059
\(949\) 0 0
\(950\) 18.4501 0.598601
\(951\) −38.2075 −1.23896
\(952\) 70.9804 2.30049
\(953\) 52.7655 1.70924 0.854622 0.519251i \(-0.173789\pi\)
0.854622 + 0.519251i \(0.173789\pi\)
\(954\) 31.4925 1.01961
\(955\) −69.4486 −2.24730
\(956\) 2.01176 0.0650651
\(957\) 13.7638 0.444920
\(958\) 21.8006 0.704346
\(959\) 43.0335 1.38962
\(960\) −50.8207 −1.64023
\(961\) 51.1804 1.65098
\(962\) 0 0
\(963\) 12.3894 0.399244
\(964\) 12.9622 0.417484
\(965\) 37.8788 1.21936
\(966\) 47.3919 1.52481
\(967\) 40.4348 1.30030 0.650148 0.759808i \(-0.274707\pi\)
0.650148 + 0.759808i \(0.274707\pi\)
\(968\) 2.91568 0.0937135
\(969\) 55.8906 1.79546
\(970\) 20.5221 0.658925
\(971\) 46.5419 1.49360 0.746801 0.665048i \(-0.231589\pi\)
0.746801 + 0.665048i \(0.231589\pi\)
\(972\) 25.3786 0.814019
\(973\) −25.3394 −0.812344
\(974\) −1.47859 −0.0473771
\(975\) 0 0
\(976\) −2.59971 −0.0832146
\(977\) −42.0365 −1.34487 −0.672433 0.740158i \(-0.734751\pi\)
−0.672433 + 0.740158i \(0.734751\pi\)
\(978\) −45.6125 −1.45853
\(979\) −0.472173 −0.0150907
\(980\) 17.0598 0.544956
\(981\) 32.1848 1.02758
\(982\) −3.11528 −0.0994124
\(983\) 15.2856 0.487534 0.243767 0.969834i \(-0.421617\pi\)
0.243767 + 0.969834i \(0.421617\pi\)
\(984\) 6.35023 0.202438
\(985\) −50.4591 −1.60776
\(986\) 36.3813 1.15862
\(987\) −74.4504 −2.36978
\(988\) 0 0
\(989\) 34.6176 1.10078
\(990\) 10.7557 0.341838
\(991\) 23.5932 0.749463 0.374732 0.927133i \(-0.377735\pi\)
0.374732 + 0.927133i \(0.377735\pi\)
\(992\) −49.1405 −1.56021
\(993\) 2.67799 0.0849835
\(994\) 41.5967 1.31937
\(995\) 22.8432 0.724179
\(996\) 5.92305 0.187679
\(997\) 52.3098 1.65667 0.828334 0.560235i \(-0.189289\pi\)
0.828334 + 0.560235i \(0.189289\pi\)
\(998\) −31.2293 −0.988546
\(999\) −5.11948 −0.161973
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 1859.2.a.t.1.8 yes 21
13.12 even 2 1859.2.a.s.1.14 21
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1859.2.a.s.1.14 21 13.12 even 2
1859.2.a.t.1.8 yes 21 1.1 even 1 trivial