Properties

Label 2151.2.a.j.1.4
Level $2151$
Weight $2$
Character 2151.1
Self dual yes
Analytic conductor $17.176$
Analytic rank $1$
Dimension $20$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 2151 = 3^{2} \cdot 239 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2151.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(17.1758214748\)
Analytic rank: \(1\)
Dimension: \(20\)
Coefficient field: \(\mathbb{Q}[x]/(x^{20} - \cdots)\)
Defining polynomial: \(x^{20} - 4 x^{19} - 21 x^{18} + 96 x^{17} + 164 x^{16} - 936 x^{15} - 540 x^{14} + 4804 x^{13} + 229 x^{12} - 14020 x^{11} + 3356 x^{10} + 23404 x^{9} - 9429 x^{8} - 21252 x^{7} + 10479 x^{6} + 9108 x^{5} - 4844 x^{4} - 1184 x^{3} + 640 x^{2} - 56 x + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.4
Root \(2.39589\) of defining polynomial
Character \(\chi\) \(=\) 2151.1

$q$-expansion

\(f(q)\) \(=\) \(q-2.39589 q^{2} +3.74030 q^{4} -4.10699 q^{5} -3.62464 q^{7} -4.16958 q^{8} +O(q^{10})\) \(q-2.39589 q^{2} +3.74030 q^{4} -4.10699 q^{5} -3.62464 q^{7} -4.16958 q^{8} +9.83991 q^{10} -1.86664 q^{11} +3.35969 q^{13} +8.68425 q^{14} +2.50926 q^{16} -0.173143 q^{17} -1.09786 q^{19} -15.3614 q^{20} +4.47226 q^{22} +5.06374 q^{23} +11.8674 q^{25} -8.04946 q^{26} -13.5573 q^{28} -6.94337 q^{29} +6.11145 q^{31} +2.32723 q^{32} +0.414831 q^{34} +14.8864 q^{35} +3.44771 q^{37} +2.63035 q^{38} +17.1244 q^{40} +5.66600 q^{41} -5.01704 q^{43} -6.98178 q^{44} -12.1322 q^{46} +5.07918 q^{47} +6.13803 q^{49} -28.4330 q^{50} +12.5663 q^{52} +4.55248 q^{53} +7.66626 q^{55} +15.1132 q^{56} +16.6356 q^{58} +4.23002 q^{59} +8.14825 q^{61} -14.6424 q^{62} -10.5943 q^{64} -13.7982 q^{65} -8.96008 q^{67} -0.647606 q^{68} -35.6661 q^{70} -13.9037 q^{71} -6.34022 q^{73} -8.26034 q^{74} -4.10632 q^{76} +6.76589 q^{77} +1.38423 q^{79} -10.3055 q^{80} -13.5751 q^{82} +14.9390 q^{83} +0.711095 q^{85} +12.0203 q^{86} +7.78309 q^{88} -16.3643 q^{89} -12.1777 q^{91} +18.9399 q^{92} -12.1692 q^{94} +4.50888 q^{95} -17.5839 q^{97} -14.7061 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20q - 4q^{2} + 18q^{4} - 16q^{5} - 4q^{7} - 12q^{8} + O(q^{10}) \) \( 20q - 4q^{2} + 18q^{4} - 16q^{5} - 4q^{7} - 12q^{8} + 4q^{10} - 12q^{11} - 4q^{13} - 20q^{14} + 22q^{16} - 24q^{17} - 4q^{19} - 40q^{20} - 6q^{22} - 12q^{23} + 22q^{25} - 30q^{26} - 12q^{28} - 24q^{29} - 4q^{31} - 28q^{32} + 8q^{34} - 20q^{35} - 10q^{37} - 26q^{38} + 6q^{40} - 66q^{41} + 8q^{43} - 36q^{44} - 12q^{46} - 28q^{47} + 18q^{49} - 28q^{50} - 18q^{52} - 28q^{53} - 4q^{55} - 60q^{56} - 54q^{59} - 4q^{61} - 20q^{62} + 22q^{64} - 42q^{65} + 12q^{67} - 12q^{68} + 20q^{70} - 36q^{71} + 14q^{73} - 50q^{76} - 8q^{77} - 12q^{79} - 88q^{80} - 8q^{82} - 20q^{83} + 4q^{85} - 18q^{86} - 10q^{88} - 130q^{89} - 6q^{91} + 46q^{92} - 26q^{94} - 2q^{97} - 12q^{98} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.39589 −1.69415 −0.847076 0.531472i \(-0.821639\pi\)
−0.847076 + 0.531472i \(0.821639\pi\)
\(3\) 0 0
\(4\) 3.74030 1.87015
\(5\) −4.10699 −1.83670 −0.918351 0.395767i \(-0.870479\pi\)
−0.918351 + 0.395767i \(0.870479\pi\)
\(6\) 0 0
\(7\) −3.62464 −1.36999 −0.684993 0.728550i \(-0.740195\pi\)
−0.684993 + 0.728550i \(0.740195\pi\)
\(8\) −4.16958 −1.47417
\(9\) 0 0
\(10\) 9.83991 3.11165
\(11\) −1.86664 −0.562812 −0.281406 0.959589i \(-0.590801\pi\)
−0.281406 + 0.959589i \(0.590801\pi\)
\(12\) 0 0
\(13\) 3.35969 0.931811 0.465905 0.884835i \(-0.345729\pi\)
0.465905 + 0.884835i \(0.345729\pi\)
\(14\) 8.68425 2.32096
\(15\) 0 0
\(16\) 2.50926 0.627316
\(17\) −0.173143 −0.0419933 −0.0209966 0.999780i \(-0.506684\pi\)
−0.0209966 + 0.999780i \(0.506684\pi\)
\(18\) 0 0
\(19\) −1.09786 −0.251865 −0.125933 0.992039i \(-0.540192\pi\)
−0.125933 + 0.992039i \(0.540192\pi\)
\(20\) −15.3614 −3.43491
\(21\) 0 0
\(22\) 4.47226 0.953489
\(23\) 5.06374 1.05586 0.527932 0.849287i \(-0.322968\pi\)
0.527932 + 0.849287i \(0.322968\pi\)
\(24\) 0 0
\(25\) 11.8674 2.37347
\(26\) −8.04946 −1.57863
\(27\) 0 0
\(28\) −13.5573 −2.56208
\(29\) −6.94337 −1.28935 −0.644676 0.764456i \(-0.723008\pi\)
−0.644676 + 0.764456i \(0.723008\pi\)
\(30\) 0 0
\(31\) 6.11145 1.09765 0.548824 0.835938i \(-0.315076\pi\)
0.548824 + 0.835938i \(0.315076\pi\)
\(32\) 2.32723 0.411401
\(33\) 0 0
\(34\) 0.414831 0.0711430
\(35\) 14.8864 2.51626
\(36\) 0 0
\(37\) 3.44771 0.566800 0.283400 0.959002i \(-0.408538\pi\)
0.283400 + 0.959002i \(0.408538\pi\)
\(38\) 2.63035 0.426698
\(39\) 0 0
\(40\) 17.1244 2.70761
\(41\) 5.66600 0.884880 0.442440 0.896798i \(-0.354113\pi\)
0.442440 + 0.896798i \(0.354113\pi\)
\(42\) 0 0
\(43\) −5.01704 −0.765091 −0.382545 0.923937i \(-0.624952\pi\)
−0.382545 + 0.923937i \(0.624952\pi\)
\(44\) −6.98178 −1.05254
\(45\) 0 0
\(46\) −12.1322 −1.78879
\(47\) 5.07918 0.740875 0.370437 0.928857i \(-0.379208\pi\)
0.370437 + 0.928857i \(0.379208\pi\)
\(48\) 0 0
\(49\) 6.13803 0.876861
\(50\) −28.4330 −4.02103
\(51\) 0 0
\(52\) 12.5663 1.74263
\(53\) 4.55248 0.625331 0.312666 0.949863i \(-0.398778\pi\)
0.312666 + 0.949863i \(0.398778\pi\)
\(54\) 0 0
\(55\) 7.66626 1.03372
\(56\) 15.1132 2.01959
\(57\) 0 0
\(58\) 16.6356 2.18436
\(59\) 4.23002 0.550701 0.275351 0.961344i \(-0.411206\pi\)
0.275351 + 0.961344i \(0.411206\pi\)
\(60\) 0 0
\(61\) 8.14825 1.04328 0.521639 0.853167i \(-0.325321\pi\)
0.521639 + 0.853167i \(0.325321\pi\)
\(62\) −14.6424 −1.85958
\(63\) 0 0
\(64\) −10.5943 −1.32429
\(65\) −13.7982 −1.71146
\(66\) 0 0
\(67\) −8.96008 −1.09465 −0.547324 0.836921i \(-0.684353\pi\)
−0.547324 + 0.836921i \(0.684353\pi\)
\(68\) −0.647606 −0.0785338
\(69\) 0 0
\(70\) −35.6661 −4.26292
\(71\) −13.9037 −1.65006 −0.825032 0.565086i \(-0.808843\pi\)
−0.825032 + 0.565086i \(0.808843\pi\)
\(72\) 0 0
\(73\) −6.34022 −0.742066 −0.371033 0.928620i \(-0.620996\pi\)
−0.371033 + 0.928620i \(0.620996\pi\)
\(74\) −8.26034 −0.960245
\(75\) 0 0
\(76\) −4.10632 −0.471027
\(77\) 6.76589 0.771044
\(78\) 0 0
\(79\) 1.38423 0.155738 0.0778688 0.996964i \(-0.475188\pi\)
0.0778688 + 0.996964i \(0.475188\pi\)
\(80\) −10.3055 −1.15219
\(81\) 0 0
\(82\) −13.5751 −1.49912
\(83\) 14.9390 1.63977 0.819886 0.572527i \(-0.194037\pi\)
0.819886 + 0.572527i \(0.194037\pi\)
\(84\) 0 0
\(85\) 0.711095 0.0771291
\(86\) 12.0203 1.29618
\(87\) 0 0
\(88\) 7.78309 0.829680
\(89\) −16.3643 −1.73461 −0.867307 0.497774i \(-0.834151\pi\)
−0.867307 + 0.497774i \(0.834151\pi\)
\(90\) 0 0
\(91\) −12.1777 −1.27657
\(92\) 18.9399 1.97463
\(93\) 0 0
\(94\) −12.1692 −1.25515
\(95\) 4.50888 0.462602
\(96\) 0 0
\(97\) −17.5839 −1.78537 −0.892687 0.450677i \(-0.851183\pi\)
−0.892687 + 0.450677i \(0.851183\pi\)
\(98\) −14.7061 −1.48554
\(99\) 0 0
\(100\) 44.3876 4.43876
\(101\) 9.02895 0.898414 0.449207 0.893428i \(-0.351707\pi\)
0.449207 + 0.893428i \(0.351707\pi\)
\(102\) 0 0
\(103\) 2.28713 0.225358 0.112679 0.993631i \(-0.464057\pi\)
0.112679 + 0.993631i \(0.464057\pi\)
\(104\) −14.0085 −1.37365
\(105\) 0 0
\(106\) −10.9073 −1.05941
\(107\) 9.34465 0.903381 0.451691 0.892175i \(-0.350821\pi\)
0.451691 + 0.892175i \(0.350821\pi\)
\(108\) 0 0
\(109\) −0.314568 −0.0301302 −0.0150651 0.999887i \(-0.504796\pi\)
−0.0150651 + 0.999887i \(0.504796\pi\)
\(110\) −18.3675 −1.75128
\(111\) 0 0
\(112\) −9.09518 −0.859414
\(113\) −2.33072 −0.219255 −0.109628 0.993973i \(-0.534966\pi\)
−0.109628 + 0.993973i \(0.534966\pi\)
\(114\) 0 0
\(115\) −20.7967 −1.93931
\(116\) −25.9703 −2.41128
\(117\) 0 0
\(118\) −10.1347 −0.932972
\(119\) 0.627580 0.0575302
\(120\) 0 0
\(121\) −7.51567 −0.683243
\(122\) −19.5223 −1.76747
\(123\) 0 0
\(124\) 22.8587 2.05277
\(125\) −28.2042 −2.52266
\(126\) 0 0
\(127\) −0.790729 −0.0701658 −0.0350829 0.999384i \(-0.511170\pi\)
−0.0350829 + 0.999384i \(0.511170\pi\)
\(128\) 20.7284 1.83215
\(129\) 0 0
\(130\) 33.0591 2.89947
\(131\) 16.5237 1.44368 0.721842 0.692058i \(-0.243296\pi\)
0.721842 + 0.692058i \(0.243296\pi\)
\(132\) 0 0
\(133\) 3.97933 0.345052
\(134\) 21.4674 1.85450
\(135\) 0 0
\(136\) 0.721932 0.0619052
\(137\) 15.7754 1.34778 0.673891 0.738831i \(-0.264622\pi\)
0.673891 + 0.738831i \(0.264622\pi\)
\(138\) 0 0
\(139\) 14.9098 1.26463 0.632317 0.774710i \(-0.282104\pi\)
0.632317 + 0.774710i \(0.282104\pi\)
\(140\) 55.6795 4.70578
\(141\) 0 0
\(142\) 33.3118 2.79546
\(143\) −6.27132 −0.524434
\(144\) 0 0
\(145\) 28.5164 2.36815
\(146\) 15.1905 1.25717
\(147\) 0 0
\(148\) 12.8955 1.06000
\(149\) −16.5686 −1.35735 −0.678677 0.734437i \(-0.737446\pi\)
−0.678677 + 0.734437i \(0.737446\pi\)
\(150\) 0 0
\(151\) −11.4842 −0.934575 −0.467288 0.884105i \(-0.654769\pi\)
−0.467288 + 0.884105i \(0.654769\pi\)
\(152\) 4.57760 0.371292
\(153\) 0 0
\(154\) −16.2103 −1.30627
\(155\) −25.0997 −2.01605
\(156\) 0 0
\(157\) −10.5567 −0.842520 −0.421260 0.906940i \(-0.638412\pi\)
−0.421260 + 0.906940i \(0.638412\pi\)
\(158\) −3.31646 −0.263843
\(159\) 0 0
\(160\) −9.55793 −0.755621
\(161\) −18.3543 −1.44652
\(162\) 0 0
\(163\) 19.0184 1.48963 0.744816 0.667270i \(-0.232537\pi\)
0.744816 + 0.667270i \(0.232537\pi\)
\(164\) 21.1925 1.65486
\(165\) 0 0
\(166\) −35.7923 −2.77802
\(167\) −1.53829 −0.119036 −0.0595181 0.998227i \(-0.518956\pi\)
−0.0595181 + 0.998227i \(0.518956\pi\)
\(168\) 0 0
\(169\) −1.71247 −0.131729
\(170\) −1.70371 −0.130668
\(171\) 0 0
\(172\) −18.7652 −1.43084
\(173\) 16.0374 1.21930 0.609650 0.792671i \(-0.291310\pi\)
0.609650 + 0.792671i \(0.291310\pi\)
\(174\) 0 0
\(175\) −43.0150 −3.25163
\(176\) −4.68388 −0.353061
\(177\) 0 0
\(178\) 39.2071 2.93870
\(179\) −8.92422 −0.667027 −0.333514 0.942745i \(-0.608234\pi\)
−0.333514 + 0.942745i \(0.608234\pi\)
\(180\) 0 0
\(181\) 10.5224 0.782125 0.391062 0.920364i \(-0.372108\pi\)
0.391062 + 0.920364i \(0.372108\pi\)
\(182\) 29.1764 2.16270
\(183\) 0 0
\(184\) −21.1137 −1.55652
\(185\) −14.1597 −1.04104
\(186\) 0 0
\(187\) 0.323194 0.0236343
\(188\) 18.9977 1.38555
\(189\) 0 0
\(190\) −10.8028 −0.783718
\(191\) −21.1875 −1.53307 −0.766537 0.642200i \(-0.778022\pi\)
−0.766537 + 0.642200i \(0.778022\pi\)
\(192\) 0 0
\(193\) −5.55065 −0.399545 −0.199772 0.979842i \(-0.564020\pi\)
−0.199772 + 0.979842i \(0.564020\pi\)
\(194\) 42.1291 3.02470
\(195\) 0 0
\(196\) 22.9581 1.63986
\(197\) −16.4993 −1.17552 −0.587761 0.809034i \(-0.699991\pi\)
−0.587761 + 0.809034i \(0.699991\pi\)
\(198\) 0 0
\(199\) 20.6187 1.46162 0.730811 0.682580i \(-0.239142\pi\)
0.730811 + 0.682580i \(0.239142\pi\)
\(200\) −49.4820 −3.49890
\(201\) 0 0
\(202\) −21.6324 −1.52205
\(203\) 25.1672 1.76639
\(204\) 0 0
\(205\) −23.2702 −1.62526
\(206\) −5.47973 −0.381791
\(207\) 0 0
\(208\) 8.43035 0.584540
\(209\) 2.04930 0.141753
\(210\) 0 0
\(211\) −28.3629 −1.95258 −0.976290 0.216466i \(-0.930547\pi\)
−0.976290 + 0.216466i \(0.930547\pi\)
\(212\) 17.0277 1.16946
\(213\) 0 0
\(214\) −22.3888 −1.53047
\(215\) 20.6049 1.40524
\(216\) 0 0
\(217\) −22.1518 −1.50376
\(218\) 0.753672 0.0510451
\(219\) 0 0
\(220\) 28.6741 1.93321
\(221\) −0.581706 −0.0391298
\(222\) 0 0
\(223\) 2.14038 0.143331 0.0716653 0.997429i \(-0.477169\pi\)
0.0716653 + 0.997429i \(0.477169\pi\)
\(224\) −8.43539 −0.563613
\(225\) 0 0
\(226\) 5.58415 0.371452
\(227\) −21.4911 −1.42641 −0.713207 0.700953i \(-0.752758\pi\)
−0.713207 + 0.700953i \(0.752758\pi\)
\(228\) 0 0
\(229\) −17.5694 −1.16102 −0.580511 0.814253i \(-0.697147\pi\)
−0.580511 + 0.814253i \(0.697147\pi\)
\(230\) 49.8268 3.28548
\(231\) 0 0
\(232\) 28.9510 1.90072
\(233\) 12.7594 0.835898 0.417949 0.908470i \(-0.362749\pi\)
0.417949 + 0.908470i \(0.362749\pi\)
\(234\) 0 0
\(235\) −20.8601 −1.36077
\(236\) 15.8215 1.02990
\(237\) 0 0
\(238\) −1.50361 −0.0974648
\(239\) −1.00000 −0.0646846
\(240\) 0 0
\(241\) −12.5216 −0.806589 −0.403294 0.915070i \(-0.632135\pi\)
−0.403294 + 0.915070i \(0.632135\pi\)
\(242\) 18.0067 1.15752
\(243\) 0 0
\(244\) 30.4769 1.95109
\(245\) −25.2088 −1.61053
\(246\) 0 0
\(247\) −3.68846 −0.234691
\(248\) −25.4822 −1.61812
\(249\) 0 0
\(250\) 67.5743 4.27378
\(251\) −14.0327 −0.885739 −0.442869 0.896586i \(-0.646039\pi\)
−0.442869 + 0.896586i \(0.646039\pi\)
\(252\) 0 0
\(253\) −9.45216 −0.594252
\(254\) 1.89450 0.118872
\(255\) 0 0
\(256\) −28.4744 −1.77965
\(257\) −5.67848 −0.354214 −0.177107 0.984192i \(-0.556674\pi\)
−0.177107 + 0.984192i \(0.556674\pi\)
\(258\) 0 0
\(259\) −12.4967 −0.776508
\(260\) −51.6095 −3.20069
\(261\) 0 0
\(262\) −39.5890 −2.44582
\(263\) 5.16054 0.318212 0.159106 0.987261i \(-0.449139\pi\)
0.159106 + 0.987261i \(0.449139\pi\)
\(264\) 0 0
\(265\) −18.6970 −1.14855
\(266\) −9.53406 −0.584571
\(267\) 0 0
\(268\) −33.5134 −2.04716
\(269\) −1.69229 −0.103181 −0.0515903 0.998668i \(-0.516429\pi\)
−0.0515903 + 0.998668i \(0.516429\pi\)
\(270\) 0 0
\(271\) −11.7243 −0.712203 −0.356101 0.934447i \(-0.615894\pi\)
−0.356101 + 0.934447i \(0.615894\pi\)
\(272\) −0.434461 −0.0263430
\(273\) 0 0
\(274\) −37.7961 −2.28335
\(275\) −22.1521 −1.33582
\(276\) 0 0
\(277\) 25.2439 1.51676 0.758380 0.651813i \(-0.225991\pi\)
0.758380 + 0.651813i \(0.225991\pi\)
\(278\) −35.7223 −2.14248
\(279\) 0 0
\(280\) −62.0699 −3.70939
\(281\) −23.7234 −1.41522 −0.707609 0.706604i \(-0.750226\pi\)
−0.707609 + 0.706604i \(0.750226\pi\)
\(282\) 0 0
\(283\) 28.7226 1.70738 0.853690 0.520781i \(-0.174359\pi\)
0.853690 + 0.520781i \(0.174359\pi\)
\(284\) −52.0040 −3.08587
\(285\) 0 0
\(286\) 15.0254 0.888471
\(287\) −20.5372 −1.21227
\(288\) 0 0
\(289\) −16.9700 −0.998237
\(290\) −68.3221 −4.01201
\(291\) 0 0
\(292\) −23.7143 −1.38778
\(293\) 5.66344 0.330862 0.165431 0.986221i \(-0.447099\pi\)
0.165431 + 0.986221i \(0.447099\pi\)
\(294\) 0 0
\(295\) −17.3726 −1.01147
\(296\) −14.3755 −0.835559
\(297\) 0 0
\(298\) 39.6966 2.29956
\(299\) 17.0126 0.983865
\(300\) 0 0
\(301\) 18.1850 1.04816
\(302\) 27.5150 1.58331
\(303\) 0 0
\(304\) −2.75481 −0.157999
\(305\) −33.4648 −1.91619
\(306\) 0 0
\(307\) −17.7678 −1.01406 −0.507032 0.861927i \(-0.669257\pi\)
−0.507032 + 0.861927i \(0.669257\pi\)
\(308\) 25.3065 1.44197
\(309\) 0 0
\(310\) 60.1361 3.41550
\(311\) −24.8432 −1.40873 −0.704365 0.709838i \(-0.748768\pi\)
−0.704365 + 0.709838i \(0.748768\pi\)
\(312\) 0 0
\(313\) −29.3512 −1.65903 −0.829514 0.558485i \(-0.811383\pi\)
−0.829514 + 0.558485i \(0.811383\pi\)
\(314\) 25.2928 1.42736
\(315\) 0 0
\(316\) 5.17742 0.291253
\(317\) 22.7860 1.27979 0.639894 0.768463i \(-0.278978\pi\)
0.639894 + 0.768463i \(0.278978\pi\)
\(318\) 0 0
\(319\) 12.9607 0.725662
\(320\) 43.5108 2.43233
\(321\) 0 0
\(322\) 43.9748 2.45062
\(323\) 0.190086 0.0105766
\(324\) 0 0
\(325\) 39.8707 2.21163
\(326\) −45.5659 −2.52366
\(327\) 0 0
\(328\) −23.6248 −1.30446
\(329\) −18.4102 −1.01499
\(330\) 0 0
\(331\) 23.2727 1.27918 0.639591 0.768715i \(-0.279104\pi\)
0.639591 + 0.768715i \(0.279104\pi\)
\(332\) 55.8765 3.06662
\(333\) 0 0
\(334\) 3.68557 0.201666
\(335\) 36.7989 2.01054
\(336\) 0 0
\(337\) 32.4686 1.76868 0.884338 0.466846i \(-0.154610\pi\)
0.884338 + 0.466846i \(0.154610\pi\)
\(338\) 4.10290 0.223169
\(339\) 0 0
\(340\) 2.65971 0.144243
\(341\) −11.4078 −0.617769
\(342\) 0 0
\(343\) 3.12435 0.168699
\(344\) 20.9189 1.12787
\(345\) 0 0
\(346\) −38.4238 −2.06568
\(347\) 8.61985 0.462738 0.231369 0.972866i \(-0.425680\pi\)
0.231369 + 0.972866i \(0.425680\pi\)
\(348\) 0 0
\(349\) −20.5649 −1.10081 −0.550406 0.834897i \(-0.685527\pi\)
−0.550406 + 0.834897i \(0.685527\pi\)
\(350\) 103.059 5.50875
\(351\) 0 0
\(352\) −4.34410 −0.231541
\(353\) −28.1526 −1.49841 −0.749206 0.662337i \(-0.769565\pi\)
−0.749206 + 0.662337i \(0.769565\pi\)
\(354\) 0 0
\(355\) 57.1023 3.03068
\(356\) −61.2075 −3.24399
\(357\) 0 0
\(358\) 21.3815 1.13005
\(359\) 13.5612 0.715735 0.357867 0.933772i \(-0.383504\pi\)
0.357867 + 0.933772i \(0.383504\pi\)
\(360\) 0 0
\(361\) −17.7947 −0.936564
\(362\) −25.2106 −1.32504
\(363\) 0 0
\(364\) −45.5482 −2.38737
\(365\) 26.0392 1.36295
\(366\) 0 0
\(367\) −25.7987 −1.34668 −0.673340 0.739333i \(-0.735141\pi\)
−0.673340 + 0.739333i \(0.735141\pi\)
\(368\) 12.7063 0.662360
\(369\) 0 0
\(370\) 33.9252 1.76368
\(371\) −16.5011 −0.856695
\(372\) 0 0
\(373\) 35.5762 1.84207 0.921033 0.389484i \(-0.127347\pi\)
0.921033 + 0.389484i \(0.127347\pi\)
\(374\) −0.774339 −0.0400401
\(375\) 0 0
\(376\) −21.1781 −1.09217
\(377\) −23.3276 −1.20143
\(378\) 0 0
\(379\) −2.52878 −0.129894 −0.0649472 0.997889i \(-0.520688\pi\)
−0.0649472 + 0.997889i \(0.520688\pi\)
\(380\) 16.8646 0.865136
\(381\) 0 0
\(382\) 50.7630 2.59726
\(383\) 13.7606 0.703134 0.351567 0.936163i \(-0.385649\pi\)
0.351567 + 0.936163i \(0.385649\pi\)
\(384\) 0 0
\(385\) −27.7874 −1.41618
\(386\) 13.2988 0.676890
\(387\) 0 0
\(388\) −65.7691 −3.33892
\(389\) 23.0695 1.16967 0.584836 0.811152i \(-0.301159\pi\)
0.584836 + 0.811152i \(0.301159\pi\)
\(390\) 0 0
\(391\) −0.876750 −0.0443391
\(392\) −25.5930 −1.29264
\(393\) 0 0
\(394\) 39.5304 1.99151
\(395\) −5.68500 −0.286043
\(396\) 0 0
\(397\) 16.1634 0.811218 0.405609 0.914047i \(-0.367059\pi\)
0.405609 + 0.914047i \(0.367059\pi\)
\(398\) −49.4003 −2.47621
\(399\) 0 0
\(400\) 29.7784 1.48892
\(401\) 13.9016 0.694215 0.347107 0.937825i \(-0.387164\pi\)
0.347107 + 0.937825i \(0.387164\pi\)
\(402\) 0 0
\(403\) 20.5326 1.02280
\(404\) 33.7710 1.68017
\(405\) 0 0
\(406\) −60.2980 −2.99254
\(407\) −6.43562 −0.319002
\(408\) 0 0
\(409\) −7.79717 −0.385545 −0.192773 0.981243i \(-0.561748\pi\)
−0.192773 + 0.981243i \(0.561748\pi\)
\(410\) 55.7529 2.75344
\(411\) 0 0
\(412\) 8.55457 0.421454
\(413\) −15.3323 −0.754453
\(414\) 0 0
\(415\) −61.3545 −3.01177
\(416\) 7.81879 0.383348
\(417\) 0 0
\(418\) −4.90990 −0.240151
\(419\) −4.44671 −0.217236 −0.108618 0.994084i \(-0.534643\pi\)
−0.108618 + 0.994084i \(0.534643\pi\)
\(420\) 0 0
\(421\) −15.8360 −0.771798 −0.385899 0.922541i \(-0.626109\pi\)
−0.385899 + 0.922541i \(0.626109\pi\)
\(422\) 67.9544 3.30797
\(423\) 0 0
\(424\) −18.9819 −0.921844
\(425\) −2.05475 −0.0996699
\(426\) 0 0
\(427\) −29.5345 −1.42927
\(428\) 34.9518 1.68946
\(429\) 0 0
\(430\) −49.3672 −2.38070
\(431\) 19.9233 0.959674 0.479837 0.877358i \(-0.340696\pi\)
0.479837 + 0.877358i \(0.340696\pi\)
\(432\) 0 0
\(433\) 5.88331 0.282734 0.141367 0.989957i \(-0.454850\pi\)
0.141367 + 0.989957i \(0.454850\pi\)
\(434\) 53.0733 2.54760
\(435\) 0 0
\(436\) −1.17658 −0.0563480
\(437\) −5.55926 −0.265936
\(438\) 0 0
\(439\) −22.6432 −1.08070 −0.540351 0.841440i \(-0.681708\pi\)
−0.540351 + 0.841440i \(0.681708\pi\)
\(440\) −31.9651 −1.52388
\(441\) 0 0
\(442\) 1.39370 0.0662918
\(443\) −19.5044 −0.926684 −0.463342 0.886180i \(-0.653350\pi\)
−0.463342 + 0.886180i \(0.653350\pi\)
\(444\) 0 0
\(445\) 67.2081 3.18597
\(446\) −5.12812 −0.242824
\(447\) 0 0
\(448\) 38.4007 1.81426
\(449\) 15.3358 0.723741 0.361870 0.932228i \(-0.382138\pi\)
0.361870 + 0.932228i \(0.382138\pi\)
\(450\) 0 0
\(451\) −10.5764 −0.498021
\(452\) −8.71759 −0.410041
\(453\) 0 0
\(454\) 51.4904 2.41656
\(455\) 50.0136 2.34467
\(456\) 0 0
\(457\) −1.69351 −0.0792192 −0.0396096 0.999215i \(-0.512611\pi\)
−0.0396096 + 0.999215i \(0.512611\pi\)
\(458\) 42.0945 1.96695
\(459\) 0 0
\(460\) −77.7861 −3.62680
\(461\) −23.4627 −1.09277 −0.546384 0.837535i \(-0.683996\pi\)
−0.546384 + 0.837535i \(0.683996\pi\)
\(462\) 0 0
\(463\) 17.1364 0.796395 0.398197 0.917300i \(-0.369636\pi\)
0.398197 + 0.917300i \(0.369636\pi\)
\(464\) −17.4228 −0.808831
\(465\) 0 0
\(466\) −30.5702 −1.41614
\(467\) −24.1942 −1.11958 −0.559788 0.828636i \(-0.689117\pi\)
−0.559788 + 0.828636i \(0.689117\pi\)
\(468\) 0 0
\(469\) 32.4771 1.49965
\(470\) 49.9787 2.30534
\(471\) 0 0
\(472\) −17.6374 −0.811827
\(473\) 9.36498 0.430602
\(474\) 0 0
\(475\) −13.0287 −0.597796
\(476\) 2.34734 0.107590
\(477\) 0 0
\(478\) 2.39589 0.109586
\(479\) −5.10566 −0.233284 −0.116642 0.993174i \(-0.537213\pi\)
−0.116642 + 0.993174i \(0.537213\pi\)
\(480\) 0 0
\(481\) 11.5832 0.528150
\(482\) 30.0005 1.36648
\(483\) 0 0
\(484\) −28.1109 −1.27777
\(485\) 72.2169 3.27920
\(486\) 0 0
\(487\) 4.69712 0.212847 0.106423 0.994321i \(-0.466060\pi\)
0.106423 + 0.994321i \(0.466060\pi\)
\(488\) −33.9748 −1.53797
\(489\) 0 0
\(490\) 60.3976 2.72849
\(491\) 32.8000 1.48024 0.740122 0.672473i \(-0.234768\pi\)
0.740122 + 0.672473i \(0.234768\pi\)
\(492\) 0 0
\(493\) 1.20219 0.0541441
\(494\) 8.83715 0.397602
\(495\) 0 0
\(496\) 15.3352 0.688572
\(497\) 50.3959 2.26056
\(498\) 0 0
\(499\) −18.9668 −0.849071 −0.424536 0.905411i \(-0.639563\pi\)
−0.424536 + 0.905411i \(0.639563\pi\)
\(500\) −105.492 −4.71776
\(501\) 0 0
\(502\) 33.6209 1.50058
\(503\) 15.7855 0.703840 0.351920 0.936030i \(-0.385529\pi\)
0.351920 + 0.936030i \(0.385529\pi\)
\(504\) 0 0
\(505\) −37.0818 −1.65012
\(506\) 22.6464 1.00675
\(507\) 0 0
\(508\) −2.95757 −0.131221
\(509\) 9.35294 0.414562 0.207281 0.978281i \(-0.433539\pi\)
0.207281 + 0.978281i \(0.433539\pi\)
\(510\) 0 0
\(511\) 22.9810 1.01662
\(512\) 26.7648 1.18285
\(513\) 0 0
\(514\) 13.6050 0.600092
\(515\) −9.39323 −0.413915
\(516\) 0 0
\(517\) −9.48098 −0.416973
\(518\) 29.9408 1.31552
\(519\) 0 0
\(520\) 57.5328 2.52298
\(521\) −11.6796 −0.511691 −0.255845 0.966718i \(-0.582354\pi\)
−0.255845 + 0.966718i \(0.582354\pi\)
\(522\) 0 0
\(523\) −25.0355 −1.09473 −0.547363 0.836895i \(-0.684368\pi\)
−0.547363 + 0.836895i \(0.684368\pi\)
\(524\) 61.8037 2.69991
\(525\) 0 0
\(526\) −12.3641 −0.539100
\(527\) −1.05815 −0.0460938
\(528\) 0 0
\(529\) 2.64150 0.114848
\(530\) 44.7960 1.94581
\(531\) 0 0
\(532\) 14.8839 0.645300
\(533\) 19.0360 0.824541
\(534\) 0 0
\(535\) −38.3784 −1.65924
\(536\) 37.3598 1.61370
\(537\) 0 0
\(538\) 4.05454 0.174804
\(539\) −11.4575 −0.493508
\(540\) 0 0
\(541\) 22.3624 0.961433 0.480717 0.876876i \(-0.340377\pi\)
0.480717 + 0.876876i \(0.340377\pi\)
\(542\) 28.0903 1.20658
\(543\) 0 0
\(544\) −0.402943 −0.0172761
\(545\) 1.29193 0.0553402
\(546\) 0 0
\(547\) 16.6583 0.712256 0.356128 0.934437i \(-0.384097\pi\)
0.356128 + 0.934437i \(0.384097\pi\)
\(548\) 59.0047 2.52056
\(549\) 0 0
\(550\) 53.0740 2.26308
\(551\) 7.62282 0.324743
\(552\) 0 0
\(553\) −5.01732 −0.213358
\(554\) −60.4817 −2.56962
\(555\) 0 0
\(556\) 55.7672 2.36506
\(557\) −41.6534 −1.76491 −0.882456 0.470394i \(-0.844112\pi\)
−0.882456 + 0.470394i \(0.844112\pi\)
\(558\) 0 0
\(559\) −16.8557 −0.712920
\(560\) 37.3538 1.57849
\(561\) 0 0
\(562\) 56.8387 2.39760
\(563\) −42.9853 −1.81161 −0.905807 0.423691i \(-0.860734\pi\)
−0.905807 + 0.423691i \(0.860734\pi\)
\(564\) 0 0
\(565\) 9.57223 0.402707
\(566\) −68.8162 −2.89256
\(567\) 0 0
\(568\) 57.9726 2.43247
\(569\) −26.8803 −1.12688 −0.563440 0.826157i \(-0.690523\pi\)
−0.563440 + 0.826157i \(0.690523\pi\)
\(570\) 0 0
\(571\) −21.5622 −0.902351 −0.451176 0.892435i \(-0.648995\pi\)
−0.451176 + 0.892435i \(0.648995\pi\)
\(572\) −23.4566 −0.980771
\(573\) 0 0
\(574\) 49.2049 2.05377
\(575\) 60.0933 2.50607
\(576\) 0 0
\(577\) 42.1498 1.75472 0.877359 0.479835i \(-0.159304\pi\)
0.877359 + 0.479835i \(0.159304\pi\)
\(578\) 40.6584 1.69116
\(579\) 0 0
\(580\) 106.660 4.42881
\(581\) −54.1486 −2.24646
\(582\) 0 0
\(583\) −8.49782 −0.351944
\(584\) 26.4360 1.09393
\(585\) 0 0
\(586\) −13.5690 −0.560530
\(587\) −18.8112 −0.776420 −0.388210 0.921571i \(-0.626906\pi\)
−0.388210 + 0.921571i \(0.626906\pi\)
\(588\) 0 0
\(589\) −6.70949 −0.276460
\(590\) 41.6230 1.71359
\(591\) 0 0
\(592\) 8.65121 0.355563
\(593\) −17.1975 −0.706218 −0.353109 0.935582i \(-0.614875\pi\)
−0.353109 + 0.935582i \(0.614875\pi\)
\(594\) 0 0
\(595\) −2.57746 −0.105666
\(596\) −61.9717 −2.53846
\(597\) 0 0
\(598\) −40.7604 −1.66682
\(599\) 7.02859 0.287181 0.143590 0.989637i \(-0.454135\pi\)
0.143590 + 0.989637i \(0.454135\pi\)
\(600\) 0 0
\(601\) −32.4567 −1.32394 −0.661968 0.749532i \(-0.730279\pi\)
−0.661968 + 0.749532i \(0.730279\pi\)
\(602\) −43.5692 −1.77575
\(603\) 0 0
\(604\) −42.9546 −1.74780
\(605\) 30.8668 1.25491
\(606\) 0 0
\(607\) 11.6545 0.473040 0.236520 0.971627i \(-0.423993\pi\)
0.236520 + 0.971627i \(0.423993\pi\)
\(608\) −2.55497 −0.103618
\(609\) 0 0
\(610\) 80.1781 3.24632
\(611\) 17.0645 0.690355
\(612\) 0 0
\(613\) −2.82297 −0.114019 −0.0570093 0.998374i \(-0.518156\pi\)
−0.0570093 + 0.998374i \(0.518156\pi\)
\(614\) 42.5698 1.71798
\(615\) 0 0
\(616\) −28.2109 −1.13665
\(617\) −23.5462 −0.947936 −0.473968 0.880542i \(-0.657179\pi\)
−0.473968 + 0.880542i \(0.657179\pi\)
\(618\) 0 0
\(619\) 38.2730 1.53832 0.769162 0.639054i \(-0.220674\pi\)
0.769162 + 0.639054i \(0.220674\pi\)
\(620\) −93.8803 −3.77032
\(621\) 0 0
\(622\) 59.5217 2.38660
\(623\) 59.3147 2.37640
\(624\) 0 0
\(625\) 56.4977 2.25991
\(626\) 70.3224 2.81065
\(627\) 0 0
\(628\) −39.4854 −1.57564
\(629\) −0.596945 −0.0238018
\(630\) 0 0
\(631\) 14.2574 0.567576 0.283788 0.958887i \(-0.408409\pi\)
0.283788 + 0.958887i \(0.408409\pi\)
\(632\) −5.77164 −0.229584
\(633\) 0 0
\(634\) −54.5928 −2.16816
\(635\) 3.24752 0.128874
\(636\) 0 0
\(637\) 20.6219 0.817068
\(638\) −31.0526 −1.22938
\(639\) 0 0
\(640\) −85.1314 −3.36512
\(641\) 31.3172 1.23696 0.618478 0.785802i \(-0.287750\pi\)
0.618478 + 0.785802i \(0.287750\pi\)
\(642\) 0 0
\(643\) 50.4477 1.98946 0.994732 0.102512i \(-0.0326880\pi\)
0.994732 + 0.102512i \(0.0326880\pi\)
\(644\) −68.6505 −2.70521
\(645\) 0 0
\(646\) −0.455425 −0.0179185
\(647\) −2.82116 −0.110911 −0.0554555 0.998461i \(-0.517661\pi\)
−0.0554555 + 0.998461i \(0.517661\pi\)
\(648\) 0 0
\(649\) −7.89590 −0.309941
\(650\) −95.5260 −3.74684
\(651\) 0 0
\(652\) 71.1344 2.78584
\(653\) 43.4096 1.69875 0.849374 0.527792i \(-0.176980\pi\)
0.849374 + 0.527792i \(0.176980\pi\)
\(654\) 0 0
\(655\) −67.8627 −2.65162
\(656\) 14.2175 0.555099
\(657\) 0 0
\(658\) 44.1089 1.71954
\(659\) −36.4540 −1.42005 −0.710024 0.704178i \(-0.751316\pi\)
−0.710024 + 0.704178i \(0.751316\pi\)
\(660\) 0 0
\(661\) 13.0915 0.509201 0.254600 0.967046i \(-0.418056\pi\)
0.254600 + 0.967046i \(0.418056\pi\)
\(662\) −55.7589 −2.16713
\(663\) 0 0
\(664\) −62.2895 −2.41730
\(665\) −16.3431 −0.633758
\(666\) 0 0
\(667\) −35.1594 −1.36138
\(668\) −5.75366 −0.222616
\(669\) 0 0
\(670\) −88.1663 −3.40616
\(671\) −15.2098 −0.587169
\(672\) 0 0
\(673\) 3.94420 0.152038 0.0760188 0.997106i \(-0.475779\pi\)
0.0760188 + 0.997106i \(0.475779\pi\)
\(674\) −77.7913 −2.99641
\(675\) 0 0
\(676\) −6.40517 −0.246353
\(677\) −34.7858 −1.33693 −0.668464 0.743744i \(-0.733048\pi\)
−0.668464 + 0.743744i \(0.733048\pi\)
\(678\) 0 0
\(679\) 63.7353 2.44594
\(680\) −2.96497 −0.113701
\(681\) 0 0
\(682\) 27.3320 1.04660
\(683\) 32.7697 1.25390 0.626949 0.779060i \(-0.284303\pi\)
0.626949 + 0.779060i \(0.284303\pi\)
\(684\) 0 0
\(685\) −64.7893 −2.47547
\(686\) −7.48561 −0.285802
\(687\) 0 0
\(688\) −12.5891 −0.479954
\(689\) 15.2949 0.582690
\(690\) 0 0
\(691\) −27.6403 −1.05149 −0.525744 0.850643i \(-0.676213\pi\)
−0.525744 + 0.850643i \(0.676213\pi\)
\(692\) 59.9847 2.28027
\(693\) 0 0
\(694\) −20.6522 −0.783948
\(695\) −61.2345 −2.32276
\(696\) 0 0
\(697\) −0.981025 −0.0371590
\(698\) 49.2712 1.86494
\(699\) 0 0
\(700\) −160.889 −6.08103
\(701\) 17.8186 0.673000 0.336500 0.941683i \(-0.390757\pi\)
0.336500 + 0.941683i \(0.390757\pi\)
\(702\) 0 0
\(703\) −3.78509 −0.142757
\(704\) 19.7758 0.745327
\(705\) 0 0
\(706\) 67.4506 2.53854
\(707\) −32.7267 −1.23081
\(708\) 0 0
\(709\) 28.5417 1.07190 0.535952 0.844248i \(-0.319953\pi\)
0.535952 + 0.844248i \(0.319953\pi\)
\(710\) −136.811 −5.13443
\(711\) 0 0
\(712\) 68.2323 2.55711
\(713\) 30.9468 1.15897
\(714\) 0 0
\(715\) 25.7563 0.963229
\(716\) −33.3793 −1.24744
\(717\) 0 0
\(718\) −32.4913 −1.21256
\(719\) −16.9208 −0.631040 −0.315520 0.948919i \(-0.602179\pi\)
−0.315520 + 0.948919i \(0.602179\pi\)
\(720\) 0 0
\(721\) −8.29004 −0.308737
\(722\) 42.6342 1.58668
\(723\) 0 0
\(724\) 39.3570 1.46269
\(725\) −82.3996 −3.06024
\(726\) 0 0
\(727\) −3.88604 −0.144125 −0.0720627 0.997400i \(-0.522958\pi\)
−0.0720627 + 0.997400i \(0.522958\pi\)
\(728\) 50.7758 1.88188
\(729\) 0 0
\(730\) −62.3872 −2.30905
\(731\) 0.868663 0.0321287
\(732\) 0 0
\(733\) 11.3841 0.420480 0.210240 0.977650i \(-0.432575\pi\)
0.210240 + 0.977650i \(0.432575\pi\)
\(734\) 61.8109 2.28148
\(735\) 0 0
\(736\) 11.7845 0.434383
\(737\) 16.7252 0.616081
\(738\) 0 0
\(739\) 31.2138 1.14822 0.574109 0.818779i \(-0.305349\pi\)
0.574109 + 0.818779i \(0.305349\pi\)
\(740\) −52.9616 −1.94691
\(741\) 0 0
\(742\) 39.5349 1.45137
\(743\) 29.8800 1.09619 0.548096 0.836416i \(-0.315353\pi\)
0.548096 + 0.836416i \(0.315353\pi\)
\(744\) 0 0
\(745\) 68.0472 2.49306
\(746\) −85.2368 −3.12074
\(747\) 0 0
\(748\) 1.20884 0.0441997
\(749\) −33.8710 −1.23762
\(750\) 0 0
\(751\) 0.548324 0.0200086 0.0100043 0.999950i \(-0.496815\pi\)
0.0100043 + 0.999950i \(0.496815\pi\)
\(752\) 12.7450 0.464763
\(753\) 0 0
\(754\) 55.8904 2.03541
\(755\) 47.1657 1.71654
\(756\) 0 0
\(757\) −38.9526 −1.41576 −0.707878 0.706335i \(-0.750347\pi\)
−0.707878 + 0.706335i \(0.750347\pi\)
\(758\) 6.05868 0.220061
\(759\) 0 0
\(760\) −18.8002 −0.681954
\(761\) −6.47569 −0.234743 −0.117372 0.993088i \(-0.537447\pi\)
−0.117372 + 0.993088i \(0.537447\pi\)
\(762\) 0 0
\(763\) 1.14020 0.0412779
\(764\) −79.2477 −2.86708
\(765\) 0 0
\(766\) −32.9690 −1.19122
\(767\) 14.2116 0.513149
\(768\) 0 0
\(769\) 9.89337 0.356764 0.178382 0.983961i \(-0.442914\pi\)
0.178382 + 0.983961i \(0.442914\pi\)
\(770\) 66.5757 2.39922
\(771\) 0 0
\(772\) −20.7611 −0.747210
\(773\) −12.2179 −0.439447 −0.219723 0.975562i \(-0.570515\pi\)
−0.219723 + 0.975562i \(0.570515\pi\)
\(774\) 0 0
\(775\) 72.5268 2.60524
\(776\) 73.3175 2.63195
\(777\) 0 0
\(778\) −55.2722 −1.98160
\(779\) −6.22045 −0.222871
\(780\) 0 0
\(781\) 25.9531 0.928676
\(782\) 2.10060 0.0751173
\(783\) 0 0
\(784\) 15.4019 0.550069
\(785\) 43.3565 1.54746
\(786\) 0 0
\(787\) −32.8255 −1.17010 −0.585052 0.810996i \(-0.698926\pi\)
−0.585052 + 0.810996i \(0.698926\pi\)
\(788\) −61.7122 −2.19841
\(789\) 0 0
\(790\) 13.6207 0.484601
\(791\) 8.44801 0.300377
\(792\) 0 0
\(793\) 27.3756 0.972137
\(794\) −38.7258 −1.37433
\(795\) 0 0
\(796\) 77.1203 2.73346
\(797\) −52.9963 −1.87723 −0.938613 0.344972i \(-0.887889\pi\)
−0.938613 + 0.344972i \(0.887889\pi\)
\(798\) 0 0
\(799\) −0.879423 −0.0311117
\(800\) 27.6182 0.976449
\(801\) 0 0
\(802\) −33.3068 −1.17611
\(803\) 11.8349 0.417644
\(804\) 0 0
\(805\) 75.3808 2.65682
\(806\) −49.1939 −1.73278
\(807\) 0 0
\(808\) −37.6469 −1.32441
\(809\) −8.64210 −0.303840 −0.151920 0.988393i \(-0.548546\pi\)
−0.151920 + 0.988393i \(0.548546\pi\)
\(810\) 0 0
\(811\) 15.2237 0.534575 0.267287 0.963617i \(-0.413873\pi\)
0.267287 + 0.963617i \(0.413873\pi\)
\(812\) 94.1331 3.30342
\(813\) 0 0
\(814\) 15.4191 0.540438
\(815\) −78.1082 −2.73601
\(816\) 0 0
\(817\) 5.50798 0.192700
\(818\) 18.6812 0.653172
\(819\) 0 0
\(820\) −87.0376 −3.03948
\(821\) 12.0526 0.420637 0.210318 0.977633i \(-0.432550\pi\)
0.210318 + 0.977633i \(0.432550\pi\)
\(822\) 0 0
\(823\) −48.1644 −1.67891 −0.839453 0.543432i \(-0.817124\pi\)
−0.839453 + 0.543432i \(0.817124\pi\)
\(824\) −9.53639 −0.332216
\(825\) 0 0
\(826\) 36.7345 1.27816
\(827\) 1.62044 0.0563481 0.0281740 0.999603i \(-0.491031\pi\)
0.0281740 + 0.999603i \(0.491031\pi\)
\(828\) 0 0
\(829\) 29.6369 1.02933 0.514666 0.857391i \(-0.327916\pi\)
0.514666 + 0.857391i \(0.327916\pi\)
\(830\) 146.999 5.10240
\(831\) 0 0
\(832\) −35.5937 −1.23399
\(833\) −1.06275 −0.0368222
\(834\) 0 0
\(835\) 6.31773 0.218634
\(836\) 7.66499 0.265099
\(837\) 0 0
\(838\) 10.6538 0.368031
\(839\) −39.2976 −1.35671 −0.678353 0.734737i \(-0.737306\pi\)
−0.678353 + 0.734737i \(0.737306\pi\)
\(840\) 0 0
\(841\) 19.2104 0.662428
\(842\) 37.9413 1.30754
\(843\) 0 0
\(844\) −106.086 −3.65162
\(845\) 7.03311 0.241946
\(846\) 0 0
\(847\) 27.2416 0.936033
\(848\) 11.4234 0.392280
\(849\) 0 0
\(850\) 4.92296 0.168856
\(851\) 17.4583 0.598463
\(852\) 0 0
\(853\) −3.53146 −0.120915 −0.0604574 0.998171i \(-0.519256\pi\)
−0.0604574 + 0.998171i \(0.519256\pi\)
\(854\) 70.7615 2.42141
\(855\) 0 0
\(856\) −38.9633 −1.33174
\(857\) −22.3714 −0.764191 −0.382096 0.924123i \(-0.624797\pi\)
−0.382096 + 0.924123i \(0.624797\pi\)
\(858\) 0 0
\(859\) 6.24869 0.213203 0.106601 0.994302i \(-0.466003\pi\)
0.106601 + 0.994302i \(0.466003\pi\)
\(860\) 77.0687 2.62802
\(861\) 0 0
\(862\) −47.7342 −1.62583
\(863\) −23.4800 −0.799270 −0.399635 0.916674i \(-0.630863\pi\)
−0.399635 + 0.916674i \(0.630863\pi\)
\(864\) 0 0
\(865\) −65.8653 −2.23949
\(866\) −14.0958 −0.478994
\(867\) 0 0
\(868\) −82.8545 −2.81226
\(869\) −2.58384 −0.0876509
\(870\) 0 0
\(871\) −30.1031 −1.02000
\(872\) 1.31162 0.0444170
\(873\) 0 0
\(874\) 13.3194 0.450535
\(875\) 102.230 3.45601
\(876\) 0 0
\(877\) −22.1035 −0.746382 −0.373191 0.927755i \(-0.621736\pi\)
−0.373191 + 0.927755i \(0.621736\pi\)
\(878\) 54.2507 1.83087
\(879\) 0 0
\(880\) 19.2367 0.648468
\(881\) −18.1078 −0.610068 −0.305034 0.952341i \(-0.598668\pi\)
−0.305034 + 0.952341i \(0.598668\pi\)
\(882\) 0 0
\(883\) 1.96256 0.0660455 0.0330227 0.999455i \(-0.489487\pi\)
0.0330227 + 0.999455i \(0.489487\pi\)
\(884\) −2.17576 −0.0731786
\(885\) 0 0
\(886\) 46.7306 1.56994
\(887\) 48.6192 1.63247 0.816236 0.577718i \(-0.196057\pi\)
0.816236 + 0.577718i \(0.196057\pi\)
\(888\) 0 0
\(889\) 2.86611 0.0961262
\(890\) −161.023 −5.39751
\(891\) 0 0
\(892\) 8.00568 0.268050
\(893\) −5.57621 −0.186601
\(894\) 0 0
\(895\) 36.6517 1.22513
\(896\) −75.1331 −2.51002
\(897\) 0 0
\(898\) −36.7429 −1.22613
\(899\) −42.4340 −1.41525
\(900\) 0 0
\(901\) −0.788228 −0.0262597
\(902\) 25.3398 0.843723
\(903\) 0 0
\(904\) 9.71812 0.323220
\(905\) −43.2155 −1.43653
\(906\) 0 0
\(907\) 17.3975 0.577675 0.288837 0.957378i \(-0.406731\pi\)
0.288837 + 0.957378i \(0.406731\pi\)
\(908\) −80.3832 −2.66761
\(909\) 0 0
\(910\) −119.827 −3.97223
\(911\) −9.19217 −0.304550 −0.152275 0.988338i \(-0.548660\pi\)
−0.152275 + 0.988338i \(0.548660\pi\)
\(912\) 0 0
\(913\) −27.8857 −0.922883
\(914\) 4.05748 0.134209
\(915\) 0 0
\(916\) −65.7150 −2.17129
\(917\) −59.8925 −1.97783
\(918\) 0 0
\(919\) 47.2057 1.55717 0.778587 0.627537i \(-0.215937\pi\)
0.778587 + 0.627537i \(0.215937\pi\)
\(920\) 86.7137 2.85887
\(921\) 0 0
\(922\) 56.2142 1.85132
\(923\) −46.7121 −1.53755
\(924\) 0 0
\(925\) 40.9153 1.34529
\(926\) −41.0569 −1.34921
\(927\) 0 0
\(928\) −16.1588 −0.530440
\(929\) −15.7119 −0.515490 −0.257745 0.966213i \(-0.582979\pi\)
−0.257745 + 0.966213i \(0.582979\pi\)
\(930\) 0 0
\(931\) −6.73867 −0.220851
\(932\) 47.7242 1.56326
\(933\) 0 0
\(934\) 57.9668 1.89673
\(935\) −1.32736 −0.0434092
\(936\) 0 0
\(937\) −39.2075 −1.28085 −0.640427 0.768019i \(-0.721242\pi\)
−0.640427 + 0.768019i \(0.721242\pi\)
\(938\) −77.8116 −2.54064
\(939\) 0 0
\(940\) −78.0233 −2.54484
\(941\) −13.4619 −0.438844 −0.219422 0.975630i \(-0.570417\pi\)
−0.219422 + 0.975630i \(0.570417\pi\)
\(942\) 0 0
\(943\) 28.6912 0.934312
\(944\) 10.6142 0.345464
\(945\) 0 0
\(946\) −22.4375 −0.729506
\(947\) 9.05254 0.294168 0.147084 0.989124i \(-0.453011\pi\)
0.147084 + 0.989124i \(0.453011\pi\)
\(948\) 0 0
\(949\) −21.3012 −0.691465
\(950\) 31.2153 1.01276
\(951\) 0 0
\(952\) −2.61675 −0.0848092
\(953\) 29.5658 0.957729 0.478864 0.877889i \(-0.341049\pi\)
0.478864 + 0.877889i \(0.341049\pi\)
\(954\) 0 0
\(955\) 87.0169 2.81580
\(956\) −3.74030 −0.120970
\(957\) 0 0
\(958\) 12.2326 0.395218
\(959\) −57.1801 −1.84644
\(960\) 0 0
\(961\) 6.34978 0.204831
\(962\) −27.7522 −0.894767
\(963\) 0 0
\(964\) −46.8347 −1.50844
\(965\) 22.7965 0.733845
\(966\) 0 0
\(967\) 20.9682 0.674290 0.337145 0.941453i \(-0.390539\pi\)
0.337145 + 0.941453i \(0.390539\pi\)
\(968\) 31.3372 1.00722
\(969\) 0 0
\(970\) −173.024 −5.55547
\(971\) −14.5417 −0.466664 −0.233332 0.972397i \(-0.574963\pi\)
−0.233332 + 0.972397i \(0.574963\pi\)
\(972\) 0 0
\(973\) −54.0427 −1.73253
\(974\) −11.2538 −0.360595
\(975\) 0 0
\(976\) 20.4461 0.654464
\(977\) −42.9015 −1.37254 −0.686270 0.727347i \(-0.740753\pi\)
−0.686270 + 0.727347i \(0.740753\pi\)
\(978\) 0 0
\(979\) 30.5462 0.976261
\(980\) −94.2886 −3.01194
\(981\) 0 0
\(982\) −78.5853 −2.50776
\(983\) −8.10974 −0.258661 −0.129330 0.991602i \(-0.541283\pi\)
−0.129330 + 0.991602i \(0.541283\pi\)
\(984\) 0 0
\(985\) 67.7623 2.15909
\(986\) −2.88033 −0.0917283
\(987\) 0 0
\(988\) −13.7960 −0.438908
\(989\) −25.4050 −0.807831
\(990\) 0 0
\(991\) 39.5827 1.25738 0.628692 0.777654i \(-0.283591\pi\)
0.628692 + 0.777654i \(0.283591\pi\)
\(992\) 14.2228 0.451573
\(993\) 0 0
\(994\) −120.743 −3.82974
\(995\) −84.6809 −2.68456
\(996\) 0 0
\(997\) 35.5573 1.12611 0.563056 0.826419i \(-0.309626\pi\)
0.563056 + 0.826419i \(0.309626\pi\)
\(998\) 45.4425 1.43846
\(999\) 0 0
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 2151.2.a.j.1.4 20
3.2 odd 2 2151.2.a.k.1.17 yes 20
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2151.2.a.j.1.4 20 1.1 even 1 trivial
2151.2.a.k.1.17 yes 20 3.2 odd 2