Properties

Label 507.2.a.j.1.3
Level $507$
Weight $2$
Character 507.1
Self dual yes
Analytic conductor $4.048$
Analytic rank $1$
Dimension $3$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 507 = 3 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 507.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(4.04841538248\)
Analytic rank: \(1\)
Dimension: \(3\)
Coefficient field: \(\Q(\zeta_{14})^+\)
Defining polynomial: \(x^{3} - x^{2} - 2 x + 1\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(-1.24698\) of defining polynomial
Character \(\chi\) \(=\) 507.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.24698 q^{2} +1.00000 q^{3} -0.445042 q^{4} -2.80194 q^{5} +1.24698 q^{6} -4.80194 q^{7} -3.04892 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+1.24698 q^{2} +1.00000 q^{3} -0.445042 q^{4} -2.80194 q^{5} +1.24698 q^{6} -4.80194 q^{7} -3.04892 q^{8} +1.00000 q^{9} -3.49396 q^{10} +1.46681 q^{11} -0.445042 q^{12} -5.98792 q^{14} -2.80194 q^{15} -2.91185 q^{16} -2.44504 q^{17} +1.24698 q^{18} +2.54288 q^{19} +1.24698 q^{20} -4.80194 q^{21} +1.82908 q^{22} -3.51573 q^{23} -3.04892 q^{24} +2.85086 q^{25} +1.00000 q^{27} +2.13706 q^{28} +1.85086 q^{29} -3.49396 q^{30} -7.63102 q^{31} +2.46681 q^{32} +1.46681 q^{33} -3.04892 q^{34} +13.4547 q^{35} -0.445042 q^{36} -4.55496 q^{37} +3.17092 q^{38} +8.54288 q^{40} +1.24698 q^{41} -5.98792 q^{42} +2.38404 q^{43} -0.652793 q^{44} -2.80194 q^{45} -4.38404 q^{46} +12.8170 q^{47} -2.91185 q^{48} +16.0586 q^{49} +3.55496 q^{50} -2.44504 q^{51} -8.85086 q^{53} +1.24698 q^{54} -4.10992 q^{55} +14.6407 q^{56} +2.54288 q^{57} +2.30798 q^{58} +2.17629 q^{59} +1.24698 q^{60} -7.82908 q^{61} -9.51573 q^{62} -4.80194 q^{63} +8.89977 q^{64} +1.82908 q^{66} -3.58211 q^{67} +1.08815 q^{68} -3.51573 q^{69} +16.7778 q^{70} -8.83877 q^{71} -3.04892 q^{72} -7.69202 q^{73} -5.67994 q^{74} +2.85086 q^{75} -1.13169 q^{76} -7.04354 q^{77} -4.02177 q^{79} +8.15883 q^{80} +1.00000 q^{81} +1.55496 q^{82} -0.652793 q^{83} +2.13706 q^{84} +6.85086 q^{85} +2.97285 q^{86} +1.85086 q^{87} -4.47219 q^{88} +6.29590 q^{89} -3.49396 q^{90} +1.56465 q^{92} -7.63102 q^{93} +15.9825 q^{94} -7.12498 q^{95} +2.46681 q^{96} -10.0315 q^{97} +20.0248 q^{98} +1.46681 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3q - q^{2} + 3q^{3} - q^{4} - 4q^{5} - q^{6} - 10q^{7} + 3q^{9} + O(q^{10}) \) \( 3q - q^{2} + 3q^{3} - q^{4} - 4q^{5} - q^{6} - 10q^{7} + 3q^{9} - q^{10} + q^{11} - q^{12} + q^{14} - 4q^{15} - 5q^{16} - 7q^{17} - q^{18} - 11q^{19} - q^{20} - 10q^{21} - 5q^{22} + 2q^{23} - 5q^{25} + 3q^{27} + q^{28} - 8q^{29} - q^{30} - 8q^{31} + 4q^{32} + q^{33} + 18q^{35} - q^{36} - 14q^{37} + 20q^{38} + 7q^{40} - q^{41} + q^{42} - 3q^{43} + 16q^{44} - 4q^{45} - 3q^{46} + 9q^{47} - 5q^{48} + 17q^{49} + 11q^{50} - 7q^{51} - 13q^{53} - q^{54} - 13q^{55} + 7q^{56} - 11q^{57} + 12q^{58} + 14q^{59} - q^{60} - 13q^{61} - 16q^{62} - 10q^{63} + 4q^{64} - 5q^{66} - 5q^{67} + 7q^{68} + 2q^{69} + 8q^{70} + 6q^{71} - 18q^{73} + 7q^{74} - 5q^{75} - q^{76} - 15q^{77} - 9q^{79} + 16q^{80} + 3q^{81} + 5q^{82} + 16q^{83} + q^{84} + 7q^{85} + 15q^{86} - 8q^{87} - 7q^{88} + 5q^{89} - q^{90} - 17q^{92} - 8q^{93} + 32q^{94} + 3q^{95} + 4q^{96} - 5q^{97} + 13q^{98} + q^{99} + O(q^{100}) \)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.24698 0.881748 0.440874 0.897569i \(-0.354669\pi\)
0.440874 + 0.897569i \(0.354669\pi\)
\(3\) 1.00000 0.577350
\(4\) −0.445042 −0.222521
\(5\) −2.80194 −1.25306 −0.626532 0.779395i \(-0.715526\pi\)
−0.626532 + 0.779395i \(0.715526\pi\)
\(6\) 1.24698 0.509077
\(7\) −4.80194 −1.81496 −0.907481 0.420093i \(-0.861997\pi\)
−0.907481 + 0.420093i \(0.861997\pi\)
\(8\) −3.04892 −1.07796
\(9\) 1.00000 0.333333
\(10\) −3.49396 −1.10489
\(11\) 1.46681 0.442260 0.221130 0.975244i \(-0.429025\pi\)
0.221130 + 0.975244i \(0.429025\pi\)
\(12\) −0.445042 −0.128473
\(13\) 0 0
\(14\) −5.98792 −1.60034
\(15\) −2.80194 −0.723457
\(16\) −2.91185 −0.727963
\(17\) −2.44504 −0.593010 −0.296505 0.955031i \(-0.595821\pi\)
−0.296505 + 0.955031i \(0.595821\pi\)
\(18\) 1.24698 0.293916
\(19\) 2.54288 0.583376 0.291688 0.956514i \(-0.405783\pi\)
0.291688 + 0.956514i \(0.405783\pi\)
\(20\) 1.24698 0.278833
\(21\) −4.80194 −1.04787
\(22\) 1.82908 0.389962
\(23\) −3.51573 −0.733080 −0.366540 0.930402i \(-0.619458\pi\)
−0.366540 + 0.930402i \(0.619458\pi\)
\(24\) −3.04892 −0.622358
\(25\) 2.85086 0.570171
\(26\) 0 0
\(27\) 1.00000 0.192450
\(28\) 2.13706 0.403867
\(29\) 1.85086 0.343695 0.171848 0.985124i \(-0.445026\pi\)
0.171848 + 0.985124i \(0.445026\pi\)
\(30\) −3.49396 −0.637907
\(31\) −7.63102 −1.37057 −0.685286 0.728274i \(-0.740323\pi\)
−0.685286 + 0.728274i \(0.740323\pi\)
\(32\) 2.46681 0.436075
\(33\) 1.46681 0.255339
\(34\) −3.04892 −0.522885
\(35\) 13.4547 2.27426
\(36\) −0.445042 −0.0741736
\(37\) −4.55496 −0.748831 −0.374415 0.927261i \(-0.622157\pi\)
−0.374415 + 0.927261i \(0.622157\pi\)
\(38\) 3.17092 0.514390
\(39\) 0 0
\(40\) 8.54288 1.35075
\(41\) 1.24698 0.194745 0.0973727 0.995248i \(-0.468956\pi\)
0.0973727 + 0.995248i \(0.468956\pi\)
\(42\) −5.98792 −0.923956
\(43\) 2.38404 0.363563 0.181782 0.983339i \(-0.441814\pi\)
0.181782 + 0.983339i \(0.441814\pi\)
\(44\) −0.652793 −0.0984122
\(45\) −2.80194 −0.417688
\(46\) −4.38404 −0.646392
\(47\) 12.8170 1.86955 0.934776 0.355238i \(-0.115600\pi\)
0.934776 + 0.355238i \(0.115600\pi\)
\(48\) −2.91185 −0.420290
\(49\) 16.0586 2.29409
\(50\) 3.55496 0.502747
\(51\) −2.44504 −0.342374
\(52\) 0 0
\(53\) −8.85086 −1.21576 −0.607879 0.794030i \(-0.707980\pi\)
−0.607879 + 0.794030i \(0.707980\pi\)
\(54\) 1.24698 0.169692
\(55\) −4.10992 −0.554181
\(56\) 14.6407 1.95645
\(57\) 2.54288 0.336812
\(58\) 2.30798 0.303052
\(59\) 2.17629 0.283329 0.141665 0.989915i \(-0.454755\pi\)
0.141665 + 0.989915i \(0.454755\pi\)
\(60\) 1.24698 0.160984
\(61\) −7.82908 −1.00241 −0.501206 0.865328i \(-0.667110\pi\)
−0.501206 + 0.865328i \(0.667110\pi\)
\(62\) −9.51573 −1.20850
\(63\) −4.80194 −0.604987
\(64\) 8.89977 1.11247
\(65\) 0 0
\(66\) 1.82908 0.225145
\(67\) −3.58211 −0.437624 −0.218812 0.975767i \(-0.570218\pi\)
−0.218812 + 0.975767i \(0.570218\pi\)
\(68\) 1.08815 0.131957
\(69\) −3.51573 −0.423244
\(70\) 16.7778 2.00533
\(71\) −8.83877 −1.04897 −0.524485 0.851420i \(-0.675742\pi\)
−0.524485 + 0.851420i \(0.675742\pi\)
\(72\) −3.04892 −0.359318
\(73\) −7.69202 −0.900283 −0.450142 0.892957i \(-0.648626\pi\)
−0.450142 + 0.892957i \(0.648626\pi\)
\(74\) −5.67994 −0.660280
\(75\) 2.85086 0.329188
\(76\) −1.13169 −0.129813
\(77\) −7.04354 −0.802686
\(78\) 0 0
\(79\) −4.02177 −0.452485 −0.226242 0.974071i \(-0.572644\pi\)
−0.226242 + 0.974071i \(0.572644\pi\)
\(80\) 8.15883 0.912185
\(81\) 1.00000 0.111111
\(82\) 1.55496 0.171716
\(83\) −0.652793 −0.0716533 −0.0358267 0.999358i \(-0.511406\pi\)
−0.0358267 + 0.999358i \(0.511406\pi\)
\(84\) 2.13706 0.233173
\(85\) 6.85086 0.743080
\(86\) 2.97285 0.320571
\(87\) 1.85086 0.198432
\(88\) −4.47219 −0.476737
\(89\) 6.29590 0.667364 0.333682 0.942686i \(-0.391709\pi\)
0.333682 + 0.942686i \(0.391709\pi\)
\(90\) −3.49396 −0.368296
\(91\) 0 0
\(92\) 1.56465 0.163126
\(93\) −7.63102 −0.791300
\(94\) 15.9825 1.64847
\(95\) −7.12498 −0.731008
\(96\) 2.46681 0.251768
\(97\) −10.0315 −1.01854 −0.509270 0.860607i \(-0.670085\pi\)
−0.509270 + 0.860607i \(0.670085\pi\)
\(98\) 20.0248 2.02281
\(99\) 1.46681 0.147420
\(100\) −1.26875 −0.126875
\(101\) 13.8877 1.38188 0.690938 0.722914i \(-0.257198\pi\)
0.690938 + 0.722914i \(0.257198\pi\)
\(102\) −3.04892 −0.301888
\(103\) −17.4034 −1.71481 −0.857405 0.514642i \(-0.827925\pi\)
−0.857405 + 0.514642i \(0.827925\pi\)
\(104\) 0 0
\(105\) 13.4547 1.31305
\(106\) −11.0368 −1.07199
\(107\) 10.5526 1.02015 0.510077 0.860128i \(-0.329617\pi\)
0.510077 + 0.860128i \(0.329617\pi\)
\(108\) −0.445042 −0.0428242
\(109\) 1.07069 0.102553 0.0512766 0.998684i \(-0.483671\pi\)
0.0512766 + 0.998684i \(0.483671\pi\)
\(110\) −5.12498 −0.488648
\(111\) −4.55496 −0.432337
\(112\) 13.9825 1.32123
\(113\) −16.5308 −1.55509 −0.777543 0.628830i \(-0.783534\pi\)
−0.777543 + 0.628830i \(0.783534\pi\)
\(114\) 3.17092 0.296983
\(115\) 9.85086 0.918597
\(116\) −0.823708 −0.0764794
\(117\) 0 0
\(118\) 2.71379 0.249825
\(119\) 11.7409 1.07629
\(120\) 8.54288 0.779854
\(121\) −8.84846 −0.804406
\(122\) −9.76271 −0.883874
\(123\) 1.24698 0.112436
\(124\) 3.39612 0.304981
\(125\) 6.02177 0.538604
\(126\) −5.98792 −0.533446
\(127\) −9.53750 −0.846316 −0.423158 0.906056i \(-0.639079\pi\)
−0.423158 + 0.906056i \(0.639079\pi\)
\(128\) 6.16421 0.544844
\(129\) 2.38404 0.209903
\(130\) 0 0
\(131\) −5.50902 −0.481326 −0.240663 0.970609i \(-0.577365\pi\)
−0.240663 + 0.970609i \(0.577365\pi\)
\(132\) −0.652793 −0.0568183
\(133\) −12.2107 −1.05880
\(134\) −4.46681 −0.385874
\(135\) −2.80194 −0.241152
\(136\) 7.45473 0.639238
\(137\) −16.1836 −1.38266 −0.691329 0.722540i \(-0.742974\pi\)
−0.691329 + 0.722540i \(0.742974\pi\)
\(138\) −4.38404 −0.373195
\(139\) −10.5090 −0.891364 −0.445682 0.895191i \(-0.647039\pi\)
−0.445682 + 0.895191i \(0.647039\pi\)
\(140\) −5.98792 −0.506071
\(141\) 12.8170 1.07939
\(142\) −11.0218 −0.924926
\(143\) 0 0
\(144\) −2.91185 −0.242654
\(145\) −5.18598 −0.430672
\(146\) −9.59179 −0.793823
\(147\) 16.0586 1.32449
\(148\) 2.02715 0.166630
\(149\) 14.3502 1.17561 0.587807 0.809001i \(-0.299992\pi\)
0.587807 + 0.809001i \(0.299992\pi\)
\(150\) 3.55496 0.290261
\(151\) 1.96615 0.160003 0.0800014 0.996795i \(-0.474508\pi\)
0.0800014 + 0.996795i \(0.474508\pi\)
\(152\) −7.75302 −0.628853
\(153\) −2.44504 −0.197670
\(154\) −8.78315 −0.707767
\(155\) 21.3817 1.71742
\(156\) 0 0
\(157\) 10.7017 0.854089 0.427045 0.904231i \(-0.359555\pi\)
0.427045 + 0.904231i \(0.359555\pi\)
\(158\) −5.01507 −0.398977
\(159\) −8.85086 −0.701918
\(160\) −6.91185 −0.546430
\(161\) 16.8823 1.33051
\(162\) 1.24698 0.0979720
\(163\) 3.89977 0.305454 0.152727 0.988268i \(-0.451195\pi\)
0.152727 + 0.988268i \(0.451195\pi\)
\(164\) −0.554958 −0.0433349
\(165\) −4.10992 −0.319957
\(166\) −0.814019 −0.0631802
\(167\) 21.0194 1.62653 0.813264 0.581895i \(-0.197688\pi\)
0.813264 + 0.581895i \(0.197688\pi\)
\(168\) 14.6407 1.12956
\(169\) 0 0
\(170\) 8.54288 0.655209
\(171\) 2.54288 0.194459
\(172\) −1.06100 −0.0809004
\(173\) 13.2349 1.00623 0.503115 0.864219i \(-0.332187\pi\)
0.503115 + 0.864219i \(0.332187\pi\)
\(174\) 2.30798 0.174967
\(175\) −13.6896 −1.03484
\(176\) −4.27114 −0.321950
\(177\) 2.17629 0.163580
\(178\) 7.85086 0.588446
\(179\) 8.52781 0.637399 0.318699 0.947856i \(-0.396754\pi\)
0.318699 + 0.947856i \(0.396754\pi\)
\(180\) 1.24698 0.0929444
\(181\) −3.63640 −0.270291 −0.135146 0.990826i \(-0.543150\pi\)
−0.135146 + 0.990826i \(0.543150\pi\)
\(182\) 0 0
\(183\) −7.82908 −0.578743
\(184\) 10.7192 0.790228
\(185\) 12.7627 0.938333
\(186\) −9.51573 −0.697727
\(187\) −3.58642 −0.262265
\(188\) −5.70410 −0.416014
\(189\) −4.80194 −0.349290
\(190\) −8.88471 −0.644564
\(191\) 21.3817 1.54712 0.773561 0.633722i \(-0.218474\pi\)
0.773561 + 0.633722i \(0.218474\pi\)
\(192\) 8.89977 0.642286
\(193\) −8.42758 −0.606631 −0.303315 0.952890i \(-0.598094\pi\)
−0.303315 + 0.952890i \(0.598094\pi\)
\(194\) −12.5090 −0.898096
\(195\) 0 0
\(196\) −7.14675 −0.510482
\(197\) −26.4765 −1.88637 −0.943186 0.332264i \(-0.892187\pi\)
−0.943186 + 0.332264i \(0.892187\pi\)
\(198\) 1.82908 0.129987
\(199\) −14.2524 −1.01032 −0.505161 0.863025i \(-0.668567\pi\)
−0.505161 + 0.863025i \(0.668567\pi\)
\(200\) −8.69202 −0.614619
\(201\) −3.58211 −0.252662
\(202\) 17.3177 1.21847
\(203\) −8.88769 −0.623794
\(204\) 1.08815 0.0761855
\(205\) −3.49396 −0.244029
\(206\) −21.7017 −1.51203
\(207\) −3.51573 −0.244360
\(208\) 0 0
\(209\) 3.72992 0.258004
\(210\) 16.7778 1.15778
\(211\) 1.85086 0.127418 0.0637091 0.997969i \(-0.479707\pi\)
0.0637091 + 0.997969i \(0.479707\pi\)
\(212\) 3.93900 0.270532
\(213\) −8.83877 −0.605623
\(214\) 13.1588 0.899519
\(215\) −6.67994 −0.455568
\(216\) −3.04892 −0.207453
\(217\) 36.6437 2.48754
\(218\) 1.33513 0.0904261
\(219\) −7.69202 −0.519779
\(220\) 1.82908 0.123317
\(221\) 0 0
\(222\) −5.67994 −0.381213
\(223\) 18.6504 1.24892 0.624462 0.781056i \(-0.285318\pi\)
0.624462 + 0.781056i \(0.285318\pi\)
\(224\) −11.8455 −0.791459
\(225\) 2.85086 0.190057
\(226\) −20.6136 −1.37119
\(227\) 9.75733 0.647617 0.323808 0.946123i \(-0.395037\pi\)
0.323808 + 0.946123i \(0.395037\pi\)
\(228\) −1.13169 −0.0749478
\(229\) −2.86294 −0.189188 −0.0945941 0.995516i \(-0.530155\pi\)
−0.0945941 + 0.995516i \(0.530155\pi\)
\(230\) 12.2838 0.809971
\(231\) −7.04354 −0.463431
\(232\) −5.64310 −0.370488
\(233\) −5.78554 −0.379024 −0.189512 0.981878i \(-0.560691\pi\)
−0.189512 + 0.981878i \(0.560691\pi\)
\(234\) 0 0
\(235\) −35.9124 −2.34267
\(236\) −0.968541 −0.0630467
\(237\) −4.02177 −0.261242
\(238\) 14.6407 0.949016
\(239\) 7.09246 0.458773 0.229386 0.973335i \(-0.426328\pi\)
0.229386 + 0.973335i \(0.426328\pi\)
\(240\) 8.15883 0.526650
\(241\) 3.89977 0.251206 0.125603 0.992081i \(-0.459913\pi\)
0.125603 + 0.992081i \(0.459913\pi\)
\(242\) −11.0339 −0.709283
\(243\) 1.00000 0.0641500
\(244\) 3.48427 0.223058
\(245\) −44.9952 −2.87464
\(246\) 1.55496 0.0991405
\(247\) 0 0
\(248\) 23.2664 1.47742
\(249\) −0.652793 −0.0413691
\(250\) 7.50902 0.474912
\(251\) −2.44504 −0.154330 −0.0771648 0.997018i \(-0.524587\pi\)
−0.0771648 + 0.997018i \(0.524587\pi\)
\(252\) 2.13706 0.134622
\(253\) −5.15691 −0.324212
\(254\) −11.8931 −0.746237
\(255\) 6.85086 0.429017
\(256\) −10.1129 −0.632056
\(257\) −14.1304 −0.881428 −0.440714 0.897648i \(-0.645275\pi\)
−0.440714 + 0.897648i \(0.645275\pi\)
\(258\) 2.97285 0.185082
\(259\) 21.8726 1.35910
\(260\) 0 0
\(261\) 1.85086 0.114565
\(262\) −6.86964 −0.424408
\(263\) −23.7235 −1.46285 −0.731426 0.681921i \(-0.761145\pi\)
−0.731426 + 0.681921i \(0.761145\pi\)
\(264\) −4.47219 −0.275244
\(265\) 24.7995 1.52342
\(266\) −15.2265 −0.933599
\(267\) 6.29590 0.385303
\(268\) 1.59419 0.0973805
\(269\) −5.91617 −0.360715 −0.180357 0.983601i \(-0.557725\pi\)
−0.180357 + 0.983601i \(0.557725\pi\)
\(270\) −3.49396 −0.212636
\(271\) 3.19029 0.193796 0.0968982 0.995294i \(-0.469108\pi\)
0.0968982 + 0.995294i \(0.469108\pi\)
\(272\) 7.11960 0.431689
\(273\) 0 0
\(274\) −20.1806 −1.21915
\(275\) 4.18167 0.252164
\(276\) 1.56465 0.0941807
\(277\) 21.9366 1.31804 0.659022 0.752124i \(-0.270971\pi\)
0.659022 + 0.752124i \(0.270971\pi\)
\(278\) −13.1045 −0.785958
\(279\) −7.63102 −0.456857
\(280\) −41.0224 −2.45155
\(281\) −11.9903 −0.715282 −0.357641 0.933859i \(-0.616419\pi\)
−0.357641 + 0.933859i \(0.616419\pi\)
\(282\) 15.9825 0.951747
\(283\) −14.3666 −0.854005 −0.427002 0.904250i \(-0.640430\pi\)
−0.427002 + 0.904250i \(0.640430\pi\)
\(284\) 3.93362 0.233418
\(285\) −7.12498 −0.422047
\(286\) 0 0
\(287\) −5.98792 −0.353456
\(288\) 2.46681 0.145358
\(289\) −11.0218 −0.648339
\(290\) −6.46681 −0.379744
\(291\) −10.0315 −0.588055
\(292\) 3.42327 0.200332
\(293\) −18.7584 −1.09588 −0.547939 0.836519i \(-0.684587\pi\)
−0.547939 + 0.836519i \(0.684587\pi\)
\(294\) 20.0248 1.16787
\(295\) −6.09783 −0.355030
\(296\) 13.8877 0.807206
\(297\) 1.46681 0.0851131
\(298\) 17.8944 1.03659
\(299\) 0 0
\(300\) −1.26875 −0.0732513
\(301\) −11.4480 −0.659853
\(302\) 2.45175 0.141082
\(303\) 13.8877 0.797827
\(304\) −7.40449 −0.424676
\(305\) 21.9366 1.25609
\(306\) −3.04892 −0.174295
\(307\) −25.6262 −1.46257 −0.731283 0.682074i \(-0.761078\pi\)
−0.731283 + 0.682074i \(0.761078\pi\)
\(308\) 3.13467 0.178614
\(309\) −17.4034 −0.990046
\(310\) 26.6625 1.51433
\(311\) 11.3817 0.645394 0.322697 0.946502i \(-0.395410\pi\)
0.322697 + 0.946502i \(0.395410\pi\)
\(312\) 0 0
\(313\) 27.5743 1.55859 0.779297 0.626655i \(-0.215576\pi\)
0.779297 + 0.626655i \(0.215576\pi\)
\(314\) 13.3448 0.753091
\(315\) 13.4547 0.758088
\(316\) 1.78986 0.100687
\(317\) 11.2597 0.632405 0.316203 0.948692i \(-0.397592\pi\)
0.316203 + 0.948692i \(0.397592\pi\)
\(318\) −11.0368 −0.618915
\(319\) 2.71486 0.152003
\(320\) −24.9366 −1.39400
\(321\) 10.5526 0.588987
\(322\) 21.0519 1.17318
\(323\) −6.21744 −0.345948
\(324\) −0.445042 −0.0247245
\(325\) 0 0
\(326\) 4.86294 0.269333
\(327\) 1.07069 0.0592092
\(328\) −3.80194 −0.209927
\(329\) −61.5465 −3.39317
\(330\) −5.12498 −0.282121
\(331\) 11.9065 0.654439 0.327220 0.944948i \(-0.393888\pi\)
0.327220 + 0.944948i \(0.393888\pi\)
\(332\) 0.290520 0.0159444
\(333\) −4.55496 −0.249610
\(334\) 26.2107 1.43419
\(335\) 10.0368 0.548371
\(336\) 13.9825 0.762810
\(337\) 17.1672 0.935157 0.467578 0.883952i \(-0.345127\pi\)
0.467578 + 0.883952i \(0.345127\pi\)
\(338\) 0 0
\(339\) −16.5308 −0.897830
\(340\) −3.04892 −0.165351
\(341\) −11.1933 −0.606150
\(342\) 3.17092 0.171463
\(343\) −43.4989 −2.34872
\(344\) −7.26875 −0.391905
\(345\) 9.85086 0.530352
\(346\) 16.5036 0.887242
\(347\) −24.2760 −1.30321 −0.651603 0.758560i \(-0.725903\pi\)
−0.651603 + 0.758560i \(0.725903\pi\)
\(348\) −0.823708 −0.0441554
\(349\) 4.57242 0.244756 0.122378 0.992484i \(-0.460948\pi\)
0.122378 + 0.992484i \(0.460948\pi\)
\(350\) −17.0707 −0.912467
\(351\) 0 0
\(352\) 3.61835 0.192859
\(353\) 6.07606 0.323396 0.161698 0.986840i \(-0.448303\pi\)
0.161698 + 0.986840i \(0.448303\pi\)
\(354\) 2.71379 0.144236
\(355\) 24.7657 1.31443
\(356\) −2.80194 −0.148502
\(357\) 11.7409 0.621396
\(358\) 10.6340 0.562025
\(359\) 14.9661 0.789883 0.394942 0.918706i \(-0.370765\pi\)
0.394942 + 0.918706i \(0.370765\pi\)
\(360\) 8.54288 0.450249
\(361\) −12.5338 −0.659673
\(362\) −4.53452 −0.238329
\(363\) −8.84846 −0.464424
\(364\) 0 0
\(365\) 21.5526 1.12811
\(366\) −9.76271 −0.510305
\(367\) 37.0834 1.93574 0.967868 0.251459i \(-0.0809105\pi\)
0.967868 + 0.251459i \(0.0809105\pi\)
\(368\) 10.2373 0.533656
\(369\) 1.24698 0.0649152
\(370\) 15.9148 0.827373
\(371\) 42.5013 2.20656
\(372\) 3.39612 0.176081
\(373\) −36.5090 −1.89037 −0.945183 0.326542i \(-0.894117\pi\)
−0.945183 + 0.326542i \(0.894117\pi\)
\(374\) −4.47219 −0.231251
\(375\) 6.02177 0.310963
\(376\) −39.0780 −2.01529
\(377\) 0 0
\(378\) −5.98792 −0.307985
\(379\) 26.5851 1.36558 0.682792 0.730613i \(-0.260765\pi\)
0.682792 + 0.730613i \(0.260765\pi\)
\(380\) 3.17092 0.162665
\(381\) −9.53750 −0.488621
\(382\) 26.6625 1.36417
\(383\) 14.3502 0.733261 0.366630 0.930367i \(-0.380511\pi\)
0.366630 + 0.930367i \(0.380511\pi\)
\(384\) 6.16421 0.314566
\(385\) 19.7356 1.00582
\(386\) −10.5090 −0.534895
\(387\) 2.38404 0.121188
\(388\) 4.46442 0.226647
\(389\) −22.6582 −1.14881 −0.574407 0.818570i \(-0.694767\pi\)
−0.574407 + 0.818570i \(0.694767\pi\)
\(390\) 0 0
\(391\) 8.59611 0.434724
\(392\) −48.9614 −2.47292
\(393\) −5.50902 −0.277894
\(394\) −33.0157 −1.66330
\(395\) 11.2687 0.566992
\(396\) −0.652793 −0.0328041
\(397\) −7.90754 −0.396868 −0.198434 0.980114i \(-0.563586\pi\)
−0.198434 + 0.980114i \(0.563586\pi\)
\(398\) −17.7724 −0.890850
\(399\) −12.2107 −0.611301
\(400\) −8.30127 −0.415064
\(401\) −2.93661 −0.146647 −0.0733236 0.997308i \(-0.523361\pi\)
−0.0733236 + 0.997308i \(0.523361\pi\)
\(402\) −4.46681 −0.222784
\(403\) 0 0
\(404\) −6.18060 −0.307497
\(405\) −2.80194 −0.139229
\(406\) −11.0828 −0.550029
\(407\) −6.68127 −0.331178
\(408\) 7.45473 0.369064
\(409\) −11.7549 −0.581244 −0.290622 0.956838i \(-0.593862\pi\)
−0.290622 + 0.956838i \(0.593862\pi\)
\(410\) −4.35690 −0.215172
\(411\) −16.1836 −0.798278
\(412\) 7.74525 0.381581
\(413\) −10.4504 −0.514231
\(414\) −4.38404 −0.215464
\(415\) 1.82908 0.0897862
\(416\) 0 0
\(417\) −10.5090 −0.514629
\(418\) 4.65114 0.227495
\(419\) 7.34183 0.358672 0.179336 0.983788i \(-0.442605\pi\)
0.179336 + 0.983788i \(0.442605\pi\)
\(420\) −5.98792 −0.292181
\(421\) −25.6963 −1.25236 −0.626181 0.779677i \(-0.715383\pi\)
−0.626181 + 0.779677i \(0.715383\pi\)
\(422\) 2.30798 0.112351
\(423\) 12.8170 0.623184
\(424\) 26.9855 1.31053
\(425\) −6.97046 −0.338117
\(426\) −11.0218 −0.534007
\(427\) 37.5948 1.81934
\(428\) −4.69633 −0.227006
\(429\) 0 0
\(430\) −8.32975 −0.401696
\(431\) −8.94198 −0.430720 −0.215360 0.976535i \(-0.569093\pi\)
−0.215360 + 0.976535i \(0.569093\pi\)
\(432\) −2.91185 −0.140097
\(433\) 2.91484 0.140078 0.0700391 0.997544i \(-0.477688\pi\)
0.0700391 + 0.997544i \(0.477688\pi\)
\(434\) 45.6939 2.19338
\(435\) −5.18598 −0.248649
\(436\) −0.476501 −0.0228203
\(437\) −8.94007 −0.427661
\(438\) −9.59179 −0.458314
\(439\) 9.05861 0.432344 0.216172 0.976355i \(-0.430643\pi\)
0.216172 + 0.976355i \(0.430643\pi\)
\(440\) 12.5308 0.597382
\(441\) 16.0586 0.764696
\(442\) 0 0
\(443\) 11.2325 0.533672 0.266836 0.963742i \(-0.414022\pi\)
0.266836 + 0.963742i \(0.414022\pi\)
\(444\) 2.02715 0.0962041
\(445\) −17.6407 −0.836250
\(446\) 23.2567 1.10124
\(447\) 14.3502 0.678741
\(448\) −42.7362 −2.01909
\(449\) −28.7579 −1.35717 −0.678585 0.734522i \(-0.737407\pi\)
−0.678585 + 0.734522i \(0.737407\pi\)
\(450\) 3.55496 0.167582
\(451\) 1.82908 0.0861282
\(452\) 7.35690 0.346039
\(453\) 1.96615 0.0923777
\(454\) 12.1672 0.571035
\(455\) 0 0
\(456\) −7.75302 −0.363068
\(457\) 19.0761 0.892341 0.446170 0.894948i \(-0.352788\pi\)
0.446170 + 0.894948i \(0.352788\pi\)
\(458\) −3.57002 −0.166816
\(459\) −2.44504 −0.114125
\(460\) −4.38404 −0.204407
\(461\) −31.7332 −1.47796 −0.738981 0.673727i \(-0.764692\pi\)
−0.738981 + 0.673727i \(0.764692\pi\)
\(462\) −8.78315 −0.408629
\(463\) 36.4784 1.69530 0.847648 0.530559i \(-0.178018\pi\)
0.847648 + 0.530559i \(0.178018\pi\)
\(464\) −5.38942 −0.250198
\(465\) 21.3817 0.991550
\(466\) −7.21446 −0.334203
\(467\) 13.0000 0.601568 0.300784 0.953692i \(-0.402752\pi\)
0.300784 + 0.953692i \(0.402752\pi\)
\(468\) 0 0
\(469\) 17.2010 0.794271
\(470\) −44.7821 −2.06564
\(471\) 10.7017 0.493109
\(472\) −6.63533 −0.305416
\(473\) 3.49694 0.160790
\(474\) −5.01507 −0.230350
\(475\) 7.24937 0.332624
\(476\) −5.22521 −0.239497
\(477\) −8.85086 −0.405253
\(478\) 8.84415 0.404522
\(479\) −5.61655 −0.256627 −0.128313 0.991734i \(-0.540956\pi\)
−0.128313 + 0.991734i \(0.540956\pi\)
\(480\) −6.91185 −0.315482
\(481\) 0 0
\(482\) 4.86294 0.221501
\(483\) 16.8823 0.768172
\(484\) 3.93794 0.178997
\(485\) 28.1075 1.27630
\(486\) 1.24698 0.0565641
\(487\) 9.75733 0.442147 0.221073 0.975257i \(-0.429044\pi\)
0.221073 + 0.975257i \(0.429044\pi\)
\(488\) 23.8702 1.08055
\(489\) 3.89977 0.176354
\(490\) −56.1081 −2.53471
\(491\) −7.38835 −0.333432 −0.166716 0.986005i \(-0.553316\pi\)
−0.166716 + 0.986005i \(0.553316\pi\)
\(492\) −0.554958 −0.0250194
\(493\) −4.52542 −0.203815
\(494\) 0 0
\(495\) −4.10992 −0.184727
\(496\) 22.2204 0.997726
\(497\) 42.4432 1.90384
\(498\) −0.814019 −0.0364771
\(499\) 43.2814 1.93754 0.968771 0.247956i \(-0.0797589\pi\)
0.968771 + 0.247956i \(0.0797589\pi\)
\(500\) −2.67994 −0.119851
\(501\) 21.0194 0.939077
\(502\) −3.04892 −0.136080
\(503\) 10.7670 0.480078 0.240039 0.970763i \(-0.422840\pi\)
0.240039 + 0.970763i \(0.422840\pi\)
\(504\) 14.6407 0.652149
\(505\) −38.9124 −1.73158
\(506\) −6.43057 −0.285874
\(507\) 0 0
\(508\) 4.24459 0.188323
\(509\) 41.5448 1.84144 0.920720 0.390223i \(-0.127602\pi\)
0.920720 + 0.390223i \(0.127602\pi\)
\(510\) 8.54288 0.378285
\(511\) 36.9366 1.63398
\(512\) −24.9390 −1.10216
\(513\) 2.54288 0.112271
\(514\) −17.6203 −0.777197
\(515\) 48.7633 2.14877
\(516\) −1.06100 −0.0467079
\(517\) 18.8001 0.826829
\(518\) 27.2747 1.19838
\(519\) 13.2349 0.580948
\(520\) 0 0
\(521\) 25.7198 1.12680 0.563402 0.826183i \(-0.309492\pi\)
0.563402 + 0.826183i \(0.309492\pi\)
\(522\) 2.30798 0.101017
\(523\) 8.59286 0.375739 0.187870 0.982194i \(-0.439842\pi\)
0.187870 + 0.982194i \(0.439842\pi\)
\(524\) 2.45175 0.107105
\(525\) −13.6896 −0.597464
\(526\) −29.5827 −1.28987
\(527\) 18.6582 0.812763
\(528\) −4.27114 −0.185878
\(529\) −10.6396 −0.462593
\(530\) 30.9245 1.34328
\(531\) 2.17629 0.0944430
\(532\) 5.43429 0.235606
\(533\) 0 0
\(534\) 7.85086 0.339740
\(535\) −29.5676 −1.27832
\(536\) 10.9215 0.471739
\(537\) 8.52781 0.368002
\(538\) −7.37734 −0.318060
\(539\) 23.5550 1.01458
\(540\) 1.24698 0.0536615
\(541\) −31.3534 −1.34799 −0.673995 0.738736i \(-0.735423\pi\)
−0.673995 + 0.738736i \(0.735423\pi\)
\(542\) 3.97823 0.170880
\(543\) −3.63640 −0.156053
\(544\) −6.03146 −0.258597
\(545\) −3.00000 −0.128506
\(546\) 0 0
\(547\) 19.9342 0.852325 0.426163 0.904647i \(-0.359865\pi\)
0.426163 + 0.904647i \(0.359865\pi\)
\(548\) 7.20237 0.307670
\(549\) −7.82908 −0.334137
\(550\) 5.21446 0.222345
\(551\) 4.70650 0.200503
\(552\) 10.7192 0.456238
\(553\) 19.3123 0.821242
\(554\) 27.3545 1.16218
\(555\) 12.7627 0.541747
\(556\) 4.67696 0.198347
\(557\) −33.2349 −1.40821 −0.704104 0.710097i \(-0.748651\pi\)
−0.704104 + 0.710097i \(0.748651\pi\)
\(558\) −9.51573 −0.402833
\(559\) 0 0
\(560\) −39.1782 −1.65558
\(561\) −3.58642 −0.151419
\(562\) −14.9517 −0.630698
\(563\) 3.87130 0.163156 0.0815779 0.996667i \(-0.474004\pi\)
0.0815779 + 0.996667i \(0.474004\pi\)
\(564\) −5.70410 −0.240186
\(565\) 46.3183 1.94862
\(566\) −17.9148 −0.753017
\(567\) −4.80194 −0.201662
\(568\) 26.9487 1.13074
\(569\) 20.1457 0.844551 0.422276 0.906468i \(-0.361231\pi\)
0.422276 + 0.906468i \(0.361231\pi\)
\(570\) −8.88471 −0.372139
\(571\) −32.1269 −1.34447 −0.672234 0.740338i \(-0.734665\pi\)
−0.672234 + 0.740338i \(0.734665\pi\)
\(572\) 0 0
\(573\) 21.3817 0.893231
\(574\) −7.46681 −0.311659
\(575\) −10.0228 −0.417981
\(576\) 8.89977 0.370824
\(577\) 16.7506 0.697338 0.348669 0.937246i \(-0.386634\pi\)
0.348669 + 0.937246i \(0.386634\pi\)
\(578\) −13.7439 −0.571672
\(579\) −8.42758 −0.350238
\(580\) 2.30798 0.0958336
\(581\) 3.13467 0.130048
\(582\) −12.5090 −0.518516
\(583\) −12.9825 −0.537682
\(584\) 23.4523 0.970465
\(585\) 0 0
\(586\) −23.3913 −0.966287
\(587\) 6.73795 0.278105 0.139053 0.990285i \(-0.455594\pi\)
0.139053 + 0.990285i \(0.455594\pi\)
\(588\) −7.14675 −0.294727
\(589\) −19.4047 −0.799559
\(590\) −7.60388 −0.313047
\(591\) −26.4765 −1.08910
\(592\) 13.2634 0.545121
\(593\) −18.1172 −0.743985 −0.371992 0.928236i \(-0.621325\pi\)
−0.371992 + 0.928236i \(0.621325\pi\)
\(594\) 1.82908 0.0750483
\(595\) −32.8974 −1.34866
\(596\) −6.38644 −0.261599
\(597\) −14.2524 −0.583310
\(598\) 0 0
\(599\) −26.7851 −1.09441 −0.547204 0.836999i \(-0.684308\pi\)
−0.547204 + 0.836999i \(0.684308\pi\)
\(600\) −8.69202 −0.354850
\(601\) −4.70171 −0.191787 −0.0958934 0.995392i \(-0.530571\pi\)
−0.0958934 + 0.995392i \(0.530571\pi\)
\(602\) −14.2755 −0.581824
\(603\) −3.58211 −0.145875
\(604\) −0.875018 −0.0356040
\(605\) 24.7928 1.00797
\(606\) 17.3177 0.703482
\(607\) 27.6396 1.12186 0.560929 0.827864i \(-0.310444\pi\)
0.560929 + 0.827864i \(0.310444\pi\)
\(608\) 6.27280 0.254396
\(609\) −8.88769 −0.360147
\(610\) 27.3545 1.10755
\(611\) 0 0
\(612\) 1.08815 0.0439857
\(613\) −48.1782 −1.94590 −0.972950 0.231017i \(-0.925795\pi\)
−0.972950 + 0.231017i \(0.925795\pi\)
\(614\) −31.9554 −1.28961
\(615\) −3.49396 −0.140890
\(616\) 21.4752 0.865259
\(617\) −30.3043 −1.22000 −0.610002 0.792400i \(-0.708831\pi\)
−0.610002 + 0.792400i \(0.708831\pi\)
\(618\) −21.7017 −0.872971
\(619\) 10.9041 0.438272 0.219136 0.975694i \(-0.429676\pi\)
0.219136 + 0.975694i \(0.429676\pi\)
\(620\) −9.51573 −0.382161
\(621\) −3.51573 −0.141081
\(622\) 14.1927 0.569075
\(623\) −30.2325 −1.21124
\(624\) 0 0
\(625\) −31.1269 −1.24508
\(626\) 34.3846 1.37429
\(627\) 3.72992 0.148959
\(628\) −4.76271 −0.190053
\(629\) 11.1371 0.444064
\(630\) 16.7778 0.668443
\(631\) 10.4523 0.416101 0.208050 0.978118i \(-0.433288\pi\)
0.208050 + 0.978118i \(0.433288\pi\)
\(632\) 12.2620 0.487758
\(633\) 1.85086 0.0735649
\(634\) 14.0406 0.557622
\(635\) 26.7235 1.06049
\(636\) 3.93900 0.156192
\(637\) 0 0
\(638\) 3.38537 0.134028
\(639\) −8.83877 −0.349656
\(640\) −17.2717 −0.682725
\(641\) −17.5942 −0.694929 −0.347464 0.937693i \(-0.612957\pi\)
−0.347464 + 0.937693i \(0.612957\pi\)
\(642\) 13.1588 0.519338
\(643\) −22.6058 −0.891486 −0.445743 0.895161i \(-0.647060\pi\)
−0.445743 + 0.895161i \(0.647060\pi\)
\(644\) −7.51334 −0.296067
\(645\) −6.67994 −0.263022
\(646\) −7.75302 −0.305039
\(647\) −24.7918 −0.974665 −0.487333 0.873216i \(-0.662030\pi\)
−0.487333 + 0.873216i \(0.662030\pi\)
\(648\) −3.04892 −0.119773
\(649\) 3.19221 0.125305
\(650\) 0 0
\(651\) 36.6437 1.43618
\(652\) −1.73556 −0.0679699
\(653\) −21.8106 −0.853513 −0.426757 0.904367i \(-0.640344\pi\)
−0.426757 + 0.904367i \(0.640344\pi\)
\(654\) 1.33513 0.0522075
\(655\) 15.4359 0.603132
\(656\) −3.63102 −0.141768
\(657\) −7.69202 −0.300094
\(658\) −76.7472 −2.99192
\(659\) 16.5526 0.644796 0.322398 0.946604i \(-0.395511\pi\)
0.322398 + 0.946604i \(0.395511\pi\)
\(660\) 1.82908 0.0711970
\(661\) 15.9541 0.620541 0.310271 0.950648i \(-0.399580\pi\)
0.310271 + 0.950648i \(0.399580\pi\)
\(662\) 14.8471 0.577050
\(663\) 0 0
\(664\) 1.99031 0.0772391
\(665\) 34.2137 1.32675
\(666\) −5.67994 −0.220093
\(667\) −6.50711 −0.251956
\(668\) −9.35450 −0.361937
\(669\) 18.6504 0.721066
\(670\) 12.5157 0.483525
\(671\) −11.4838 −0.443327
\(672\) −11.8455 −0.456949
\(673\) −7.38835 −0.284800 −0.142400 0.989809i \(-0.545482\pi\)
−0.142400 + 0.989809i \(0.545482\pi\)
\(674\) 21.4071 0.824572
\(675\) 2.85086 0.109729
\(676\) 0 0
\(677\) −22.1454 −0.851118 −0.425559 0.904931i \(-0.639922\pi\)
−0.425559 + 0.904931i \(0.639922\pi\)
\(678\) −20.6136 −0.791659
\(679\) 48.1704 1.84861
\(680\) −20.8877 −0.801006
\(681\) 9.75733 0.373902
\(682\) −13.9578 −0.534471
\(683\) −9.10023 −0.348211 −0.174105 0.984727i \(-0.555703\pi\)
−0.174105 + 0.984727i \(0.555703\pi\)
\(684\) −1.13169 −0.0432711
\(685\) 45.3454 1.73256
\(686\) −54.2422 −2.07098
\(687\) −2.86294 −0.109228
\(688\) −6.94198 −0.264661
\(689\) 0 0
\(690\) 12.2838 0.467637
\(691\) −13.7711 −0.523876 −0.261938 0.965085i \(-0.584362\pi\)
−0.261938 + 0.965085i \(0.584362\pi\)
\(692\) −5.89008 −0.223907
\(693\) −7.04354 −0.267562
\(694\) −30.2717 −1.14910
\(695\) 29.4456 1.11694
\(696\) −5.64310 −0.213901
\(697\) −3.04892 −0.115486
\(698\) 5.70171 0.215813
\(699\) −5.78554 −0.218829
\(700\) 6.09246 0.230273
\(701\) 46.5090 1.75662 0.878311 0.478090i \(-0.158671\pi\)
0.878311 + 0.478090i \(0.158671\pi\)
\(702\) 0 0
\(703\) −11.5827 −0.436850
\(704\) 13.0543 0.492002
\(705\) −35.9124 −1.35254
\(706\) 7.57673 0.285154
\(707\) −66.6878 −2.50805
\(708\) −0.968541 −0.0364000
\(709\) −7.24565 −0.272116 −0.136058 0.990701i \(-0.543443\pi\)
−0.136058 + 0.990701i \(0.543443\pi\)
\(710\) 30.8823 1.15899
\(711\) −4.02177 −0.150828
\(712\) −19.1957 −0.719388
\(713\) 26.8286 1.00474
\(714\) 14.6407 0.547915
\(715\) 0 0
\(716\) −3.79523 −0.141835
\(717\) 7.09246 0.264873
\(718\) 18.6625 0.696478
\(719\) 25.5147 0.951536 0.475768 0.879571i \(-0.342170\pi\)
0.475768 + 0.879571i \(0.342170\pi\)
\(720\) 8.15883 0.304062
\(721\) 83.5701 3.11231
\(722\) −15.6294 −0.581665
\(723\) 3.89977 0.145034
\(724\) 1.61835 0.0601455
\(725\) 5.27652 0.195965
\(726\) −11.0339 −0.409505
\(727\) −14.4873 −0.537303 −0.268651 0.963238i \(-0.586578\pi\)
−0.268651 + 0.963238i \(0.586578\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 26.8756 0.994711
\(731\) −5.82908 −0.215596
\(732\) 3.48427 0.128782
\(733\) −37.5036 −1.38523 −0.692614 0.721308i \(-0.743541\pi\)
−0.692614 + 0.721308i \(0.743541\pi\)
\(734\) 46.2422 1.70683
\(735\) −44.9952 −1.65967
\(736\) −8.67264 −0.319678
\(737\) −5.25428 −0.193544
\(738\) 1.55496 0.0572388
\(739\) −43.1876 −1.58868 −0.794341 0.607472i \(-0.792184\pi\)
−0.794341 + 0.607472i \(0.792184\pi\)
\(740\) −5.67994 −0.208799
\(741\) 0 0
\(742\) 52.9982 1.94563
\(743\) −13.4765 −0.494405 −0.247202 0.968964i \(-0.579511\pi\)
−0.247202 + 0.968964i \(0.579511\pi\)
\(744\) 23.2664 0.852986
\(745\) −40.2083 −1.47312
\(746\) −45.5260 −1.66683
\(747\) −0.652793 −0.0238844
\(748\) 1.59611 0.0583594
\(749\) −50.6728 −1.85154
\(750\) 7.50902 0.274191
\(751\) 35.5894 1.29868 0.649338 0.760500i \(-0.275046\pi\)
0.649338 + 0.760500i \(0.275046\pi\)
\(752\) −37.3212 −1.36097
\(753\) −2.44504 −0.0891023
\(754\) 0 0
\(755\) −5.50902 −0.200494
\(756\) 2.13706 0.0777242
\(757\) −12.2107 −0.443807 −0.221903 0.975069i \(-0.571227\pi\)
−0.221903 + 0.975069i \(0.571227\pi\)
\(758\) 33.1511 1.20410
\(759\) −5.15691 −0.187184
\(760\) 21.7235 0.787993
\(761\) −30.9071 −1.12038 −0.560190 0.828364i \(-0.689272\pi\)
−0.560190 + 0.828364i \(0.689272\pi\)
\(762\) −11.8931 −0.430840
\(763\) −5.14138 −0.186130
\(764\) −9.51573 −0.344267
\(765\) 6.85086 0.247693
\(766\) 17.8944 0.646551
\(767\) 0 0
\(768\) −10.1129 −0.364918
\(769\) −11.9892 −0.432343 −0.216172 0.976355i \(-0.569357\pi\)
−0.216172 + 0.976355i \(0.569357\pi\)
\(770\) 24.6098 0.886877
\(771\) −14.1304 −0.508892
\(772\) 3.75063 0.134988
\(773\) −43.9661 −1.58135 −0.790676 0.612235i \(-0.790271\pi\)
−0.790676 + 0.612235i \(0.790271\pi\)
\(774\) 2.97285 0.106857
\(775\) −21.7549 −0.781460
\(776\) 30.5851 1.09794
\(777\) 21.8726 0.784676
\(778\) −28.2543 −1.01296
\(779\) 3.17092 0.113610
\(780\) 0 0
\(781\) −12.9648 −0.463918
\(782\) 10.7192 0.383317
\(783\) 1.85086 0.0661442
\(784\) −46.7603 −1.67001
\(785\) −29.9855 −1.07023
\(786\) −6.86964 −0.245032
\(787\) −30.3763 −1.08280 −0.541399 0.840766i \(-0.682105\pi\)
−0.541399 + 0.840766i \(0.682105\pi\)
\(788\) 11.7832 0.419757
\(789\) −23.7235 −0.844578
\(790\) 14.0519 0.499944
\(791\) 79.3798 2.82242
\(792\) −4.47219 −0.158912
\(793\) 0 0
\(794\) −9.86054 −0.349938
\(795\) 24.7995 0.879549
\(796\) 6.34290 0.224818
\(797\) 52.5763 1.86235 0.931173 0.364577i \(-0.118786\pi\)
0.931173 + 0.364577i \(0.118786\pi\)
\(798\) −15.2265 −0.539014
\(799\) −31.3381 −1.10866
\(800\) 7.03252 0.248637
\(801\) 6.29590 0.222455
\(802\) −3.66189 −0.129306
\(803\) −11.2828 −0.398160
\(804\) 1.59419 0.0562226
\(805\) −47.3032 −1.66722
\(806\) 0 0
\(807\) −5.91617 −0.208259
\(808\) −42.3424 −1.48960
\(809\) 49.4215 1.73757 0.868783 0.495193i \(-0.164903\pi\)
0.868783 + 0.495193i \(0.164903\pi\)
\(810\) −3.49396 −0.122765
\(811\) 1.36526 0.0479406 0.0239703 0.999713i \(-0.492369\pi\)
0.0239703 + 0.999713i \(0.492369\pi\)
\(812\) 3.95539 0.138807
\(813\) 3.19029 0.111888
\(814\) −8.33140 −0.292016
\(815\) −10.9269 −0.382753
\(816\) 7.11960 0.249236
\(817\) 6.06233 0.212094
\(818\) −14.6582 −0.512511
\(819\) 0 0
\(820\) 1.55496 0.0543015
\(821\) −0.665939 −0.0232414 −0.0116207 0.999932i \(-0.503699\pi\)
−0.0116207 + 0.999932i \(0.503699\pi\)
\(822\) −20.1806 −0.703879
\(823\) −10.0592 −0.350642 −0.175321 0.984511i \(-0.556096\pi\)
−0.175321 + 0.984511i \(0.556096\pi\)
\(824\) 53.0616 1.84849
\(825\) 4.18167 0.145587
\(826\) −13.0315 −0.453422
\(827\) 37.3038 1.29718 0.648590 0.761138i \(-0.275359\pi\)
0.648590 + 0.761138i \(0.275359\pi\)
\(828\) 1.56465 0.0543752
\(829\) −42.6209 −1.48028 −0.740142 0.672451i \(-0.765242\pi\)
−0.740142 + 0.672451i \(0.765242\pi\)
\(830\) 2.28083 0.0791688
\(831\) 21.9366 0.760973
\(832\) 0 0
\(833\) −39.2640 −1.36042
\(834\) −13.1045 −0.453773
\(835\) −58.8950 −2.03815
\(836\) −1.65997 −0.0574113
\(837\) −7.63102 −0.263767
\(838\) 9.15511 0.316258
\(839\) −4.14005 −0.142930 −0.0714652 0.997443i \(-0.522767\pi\)
−0.0714652 + 0.997443i \(0.522767\pi\)
\(840\) −41.0224 −1.41541
\(841\) −25.5743 −0.881874
\(842\) −32.0428 −1.10427
\(843\) −11.9903 −0.412968
\(844\) −0.823708 −0.0283532
\(845\) 0 0
\(846\) 15.9825 0.549491
\(847\) 42.4898 1.45997
\(848\) 25.7724 0.885028
\(849\) −14.3666 −0.493060
\(850\) −8.69202 −0.298134
\(851\) 16.0140 0.548953
\(852\) 3.93362 0.134764
\(853\) −17.3502 −0.594059 −0.297030 0.954868i \(-0.595996\pi\)
−0.297030 + 0.954868i \(0.595996\pi\)
\(854\) 46.8799 1.60420
\(855\) −7.12498 −0.243669
\(856\) −32.1739 −1.09968
\(857\) 41.0180 1.40115 0.700575 0.713579i \(-0.252927\pi\)
0.700575 + 0.713579i \(0.252927\pi\)
\(858\) 0 0
\(859\) 6.59286 0.224945 0.112473 0.993655i \(-0.464123\pi\)
0.112473 + 0.993655i \(0.464123\pi\)
\(860\) 2.97285 0.101373
\(861\) −5.98792 −0.204068
\(862\) −11.1505 −0.379787
\(863\) 16.6455 0.566619 0.283310 0.959028i \(-0.408568\pi\)
0.283310 + 0.959028i \(0.408568\pi\)
\(864\) 2.46681 0.0839227
\(865\) −37.0834 −1.26087
\(866\) 3.63474 0.123514
\(867\) −11.0218 −0.374319
\(868\) −16.3080 −0.553529
\(869\) −5.89918 −0.200116
\(870\) −6.46681 −0.219245
\(871\) 0 0
\(872\) −3.26444 −0.110548
\(873\) −10.0315 −0.339513
\(874\) −11.1481 −0.377089
\(875\) −28.9162 −0.977545
\(876\) 3.42327 0.115662
\(877\) −54.4965 −1.84022 −0.920108 0.391666i \(-0.871899\pi\)
−0.920108 + 0.391666i \(0.871899\pi\)
\(878\) 11.2959 0.381218
\(879\) −18.7584 −0.632705
\(880\) 11.9675 0.403424
\(881\) −9.00670 −0.303444 −0.151722 0.988423i \(-0.548482\pi\)
−0.151722 + 0.988423i \(0.548482\pi\)
\(882\) 20.0248 0.674269
\(883\) 18.8907 0.635722 0.317861 0.948137i \(-0.397035\pi\)
0.317861 + 0.948137i \(0.397035\pi\)
\(884\) 0 0
\(885\) −6.09783 −0.204976
\(886\) 14.0067 0.470564
\(887\) −46.9124 −1.57517 −0.787583 0.616209i \(-0.788668\pi\)
−0.787583 + 0.616209i \(0.788668\pi\)
\(888\) 13.8877 0.466040
\(889\) 45.7985 1.53603
\(890\) −21.9976 −0.737361
\(891\) 1.46681 0.0491401
\(892\) −8.30021 −0.277912
\(893\) 32.5921 1.09065
\(894\) 17.8944 0.598478
\(895\) −23.8944 −0.798702
\(896\) −29.6002 −0.988872
\(897\) 0 0
\(898\) −35.8605 −1.19668
\(899\) −14.1239 −0.471059
\(900\) −1.26875 −0.0422917
\(901\) 21.6407 0.720957
\(902\) 2.28083 0.0759434
\(903\) −11.4480 −0.380966
\(904\) 50.4010 1.67631
\(905\) 10.1890 0.338693
\(906\) 2.45175 0.0814538
\(907\) −35.3013 −1.17216 −0.586080 0.810253i \(-0.699329\pi\)
−0.586080 + 0.810253i \(0.699329\pi\)
\(908\) −4.34242 −0.144108
\(909\) 13.8877 0.460626
\(910\) 0 0
\(911\) −9.80731 −0.324931 −0.162465 0.986714i \(-0.551945\pi\)
−0.162465 + 0.986714i \(0.551945\pi\)
\(912\) −7.40449 −0.245187
\(913\) −0.957524 −0.0316894
\(914\) 23.7875 0.786819
\(915\) 21.9366 0.725202
\(916\) 1.27413 0.0420983
\(917\) 26.4540 0.873588
\(918\) −3.04892 −0.100629
\(919\) 18.4655 0.609120 0.304560 0.952493i \(-0.401491\pi\)
0.304560 + 0.952493i \(0.401491\pi\)
\(920\) −30.0344 −0.990206
\(921\) −25.6262 −0.844413
\(922\) −39.5706 −1.30319
\(923\) 0 0
\(924\) 3.13467 0.103123
\(925\) −12.9855 −0.426961
\(926\) 45.4878 1.49482
\(927\) −17.4034 −0.571603
\(928\) 4.56571 0.149877
\(929\) 25.6267 0.840785 0.420393 0.907342i \(-0.361892\pi\)
0.420393 + 0.907342i \(0.361892\pi\)
\(930\) 26.6625 0.874297
\(931\) 40.8351 1.33831
\(932\) 2.57481 0.0843407
\(933\) 11.3817 0.372618
\(934\) 16.2107 0.530431
\(935\) 10.0489 0.328635
\(936\) 0 0
\(937\) 7.54932 0.246625 0.123313 0.992368i \(-0.460648\pi\)
0.123313 + 0.992368i \(0.460648\pi\)
\(938\) 21.4494 0.700346
\(939\) 27.5743 0.899854
\(940\) 15.9825 0.521293
\(941\) 12.6418 0.412110 0.206055 0.978540i \(-0.433937\pi\)
0.206055 + 0.978540i \(0.433937\pi\)
\(942\) 13.3448 0.434798
\(943\) −4.38404 −0.142764
\(944\) −6.33704 −0.206253
\(945\) 13.4547 0.437682
\(946\) 4.36062 0.141776
\(947\) −27.9801 −0.909233 −0.454616 0.890687i \(-0.650224\pi\)
−0.454616 + 0.890687i \(0.650224\pi\)
\(948\) 1.78986 0.0581318
\(949\) 0 0
\(950\) 9.03982 0.293290
\(951\) 11.2597 0.365119
\(952\) −35.7972 −1.16019
\(953\) 4.00239 0.129650 0.0648251 0.997897i \(-0.479351\pi\)
0.0648251 + 0.997897i \(0.479351\pi\)
\(954\) −11.0368 −0.357331
\(955\) −59.9101 −1.93864
\(956\) −3.15644 −0.102087
\(957\) 2.71486 0.0877589
\(958\) −7.00372 −0.226280
\(959\) 77.7126 2.50947
\(960\) −24.9366 −0.804826
\(961\) 27.2325 0.878468
\(962\) 0 0
\(963\) 10.5526 0.340052
\(964\) −1.73556 −0.0558987
\(965\) 23.6136 0.760148
\(966\) 21.0519 0.677334
\(967\) −12.2239 −0.393094 −0.196547 0.980494i \(-0.562973\pi\)
−0.196547 + 0.980494i \(0.562973\pi\)
\(968\) 26.9782 0.867113
\(969\) −6.21744 −0.199733
\(970\) 35.0495 1.12537
\(971\) −23.6401 −0.758648 −0.379324 0.925264i \(-0.623843\pi\)
−0.379324 + 0.925264i \(0.623843\pi\)
\(972\) −0.445042 −0.0142747
\(973\) 50.4637 1.61779
\(974\) 12.1672 0.389862
\(975\) 0 0
\(976\) 22.7972 0.729719
\(977\) −18.7313 −0.599266 −0.299633 0.954055i \(-0.596864\pi\)
−0.299633 + 0.954055i \(0.596864\pi\)
\(978\) 4.86294 0.155500
\(979\) 9.23490 0.295149
\(980\) 20.0248 0.639667
\(981\) 1.07069 0.0341844
\(982\) −9.21313 −0.294003
\(983\) −12.0954 −0.385785 −0.192892 0.981220i \(-0.561787\pi\)
−0.192892 + 0.981220i \(0.561787\pi\)
\(984\) −3.80194 −0.121201
\(985\) 74.1855 2.36375
\(986\) −5.64310 −0.179713
\(987\) −61.5465 −1.95905
\(988\) 0 0
\(989\) −8.38165 −0.266521
\(990\) −5.12498 −0.162883
\(991\) −28.5526 −0.907002 −0.453501 0.891256i \(-0.649825\pi\)
−0.453501 + 0.891256i \(0.649825\pi\)
\(992\) −18.8243 −0.597672
\(993\) 11.9065 0.377841
\(994\) 52.9259 1.67871
\(995\) 39.9342 1.26600
\(996\) 0.290520 0.00920548
\(997\) −23.9347 −0.758019 −0.379010 0.925393i \(-0.623735\pi\)
−0.379010 + 0.925393i \(0.623735\pi\)
\(998\) 53.9711 1.70842
\(999\) −4.55496 −0.144112
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 507.2.a.j.1.3 3
3.2 odd 2 1521.2.a.q.1.1 3
4.3 odd 2 8112.2.a.by.1.1 3
13.2 odd 12 507.2.j.h.316.5 12
13.3 even 3 507.2.e.k.22.1 6
13.4 even 6 507.2.e.j.484.3 6
13.5 odd 4 507.2.b.g.337.2 6
13.6 odd 12 507.2.j.h.361.2 12
13.7 odd 12 507.2.j.h.361.5 12
13.8 odd 4 507.2.b.g.337.5 6
13.9 even 3 507.2.e.k.484.1 6
13.10 even 6 507.2.e.j.22.3 6
13.11 odd 12 507.2.j.h.316.2 12
13.12 even 2 507.2.a.k.1.1 yes 3
39.5 even 4 1521.2.b.m.1351.5 6
39.8 even 4 1521.2.b.m.1351.2 6
39.38 odd 2 1521.2.a.p.1.3 3
52.51 odd 2 8112.2.a.cf.1.3 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
507.2.a.j.1.3 3 1.1 even 1 trivial
507.2.a.k.1.1 yes 3 13.12 even 2
507.2.b.g.337.2 6 13.5 odd 4
507.2.b.g.337.5 6 13.8 odd 4
507.2.e.j.22.3 6 13.10 even 6
507.2.e.j.484.3 6 13.4 even 6
507.2.e.k.22.1 6 13.3 even 3
507.2.e.k.484.1 6 13.9 even 3
507.2.j.h.316.2 12 13.11 odd 12
507.2.j.h.316.5 12 13.2 odd 12
507.2.j.h.361.2 12 13.6 odd 12
507.2.j.h.361.5 12 13.7 odd 12
1521.2.a.p.1.3 3 39.38 odd 2
1521.2.a.q.1.1 3 3.2 odd 2
1521.2.b.m.1351.2 6 39.8 even 4
1521.2.b.m.1351.5 6 39.5 even 4
8112.2.a.by.1.1 3 4.3 odd 2
8112.2.a.cf.1.3 3 52.51 odd 2