Properties

Label 1521.2.a.p.1.3
Level $1521$
Weight $2$
Character 1521.1
Self dual yes
Analytic conductor $12.145$
Analytic rank $0$
Dimension $3$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 1521 = 3^{2} \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1521.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(12.1452461474\)
Analytic rank: \(0\)
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: no (minimal twist has level 507)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(-1.24698\) of defining polynomial
Character \(\chi\) \(=\) 1521.1

$q$-expansion

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

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.24698 0.881748 0.440874 0.897569i \(-0.354669\pi\)
0.440874 + 0.897569i \(0.354669\pi\)
\(3\) 0 0
\(4\) −0.445042 −0.222521
\(5\) −2.80194 −1.25306 −0.626532 0.779395i \(-0.715526\pi\)
−0.626532 + 0.779395i \(0.715526\pi\)
\(6\) 0 0
\(7\) 4.80194 1.81496 0.907481 0.420093i \(-0.138003\pi\)
0.907481 + 0.420093i \(0.138003\pi\)
\(8\) −3.04892 −1.07796
\(9\) 0 0
\(10\) −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 0
\(13\) 0 0
\(14\) 5.98792 1.60034
\(15\) 0 0
\(16\) −2.91185 −0.727963
\(17\) 2.44504 0.593010 0.296505 0.955031i \(-0.404179\pi\)
0.296505 + 0.955031i \(0.404179\pi\)
\(18\) 0 0
\(19\) −2.54288 −0.583376 −0.291688 0.956514i \(-0.594217\pi\)
−0.291688 + 0.956514i \(0.594217\pi\)
\(20\) 1.24698 0.278833
\(21\) 0 0
\(22\) 1.82908 0.389962
\(23\) 3.51573 0.733080 0.366540 0.930402i \(-0.380542\pi\)
0.366540 + 0.930402i \(0.380542\pi\)
\(24\) 0 0
\(25\) 2.85086 0.570171
\(26\) 0 0
\(27\) 0 0
\(28\) −2.13706 −0.403867
\(29\) −1.85086 −0.343695 −0.171848 0.985124i \(-0.554974\pi\)
−0.171848 + 0.985124i \(0.554974\pi\)
\(30\) 0 0
\(31\) 7.63102 1.37057 0.685286 0.728274i \(-0.259677\pi\)
0.685286 + 0.728274i \(0.259677\pi\)
\(32\) 2.46681 0.436075
\(33\) 0 0
\(34\) 3.04892 0.522885
\(35\) −13.4547 −2.27426
\(36\) 0 0
\(37\) 4.55496 0.748831 0.374415 0.927261i \(-0.377843\pi\)
0.374415 + 0.927261i \(0.377843\pi\)
\(38\) −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 \(0.468956\pi\)
\(42\) 0 0
\(43\) 2.38404 0.363563 0.181782 0.983339i \(-0.441814\pi\)
0.181782 + 0.983339i \(0.441814\pi\)
\(44\) −0.652793 −0.0984122
\(45\) 0 0
\(46\) 4.38404 0.646392
\(47\) 12.8170 1.86955 0.934776 0.355238i \(-0.115600\pi\)
0.934776 + 0.355238i \(0.115600\pi\)
\(48\) 0 0
\(49\) 16.0586 2.29409
\(50\) 3.55496 0.502747
\(51\) 0 0
\(52\) 0 0
\(53\) 8.85086 1.21576 0.607879 0.794030i \(-0.292020\pi\)
0.607879 + 0.794030i \(0.292020\pi\)
\(54\) 0 0
\(55\) −4.10992 −0.554181
\(56\) −14.6407 −1.95645
\(57\) 0 0
\(58\) −2.30798 −0.303052
\(59\) 2.17629 0.283329 0.141665 0.989915i \(-0.454755\pi\)
0.141665 + 0.989915i \(0.454755\pi\)
\(60\) 0 0
\(61\) −7.82908 −1.00241 −0.501206 0.865328i \(-0.667110\pi\)
−0.501206 + 0.865328i \(0.667110\pi\)
\(62\) 9.51573 1.20850
\(63\) 0 0
\(64\) 8.89977 1.11247
\(65\) 0 0
\(66\) 0 0
\(67\) 3.58211 0.437624 0.218812 0.975767i \(-0.429782\pi\)
0.218812 + 0.975767i \(0.429782\pi\)
\(68\) −1.08815 −0.131957
\(69\) 0 0
\(70\) −16.7778 −2.00533
\(71\) −8.83877 −1.04897 −0.524485 0.851420i \(-0.675742\pi\)
−0.524485 + 0.851420i \(0.675742\pi\)
\(72\) 0 0
\(73\) 7.69202 0.900283 0.450142 0.892957i \(-0.351374\pi\)
0.450142 + 0.892957i \(0.351374\pi\)
\(74\) 5.67994 0.660280
\(75\) 0 0
\(76\) 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
\(81\) 0 0
\(82\) 1.55496 0.171716
\(83\) −0.652793 −0.0716533 −0.0358267 0.999358i \(-0.511406\pi\)
−0.0358267 + 0.999358i \(0.511406\pi\)
\(84\) 0 0
\(85\) −6.85086 −0.743080
\(86\) 2.97285 0.320571
\(87\) 0 0
\(88\) −4.47219 −0.476737
\(89\) 6.29590 0.667364 0.333682 0.942686i \(-0.391709\pi\)
0.333682 + 0.942686i \(0.391709\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) −1.56465 −0.163126
\(93\) 0 0
\(94\) 15.9825 1.64847
\(95\) 7.12498 0.731008
\(96\) 0 0
\(97\) 10.0315 1.01854 0.509270 0.860607i \(-0.329915\pi\)
0.509270 + 0.860607i \(0.329915\pi\)
\(98\) 20.0248 2.02281
\(99\) 0 0
\(100\) −1.26875 −0.126875
\(101\) −13.8877 −1.38188 −0.690938 0.722914i \(-0.742802\pi\)
−0.690938 + 0.722914i \(0.742802\pi\)
\(102\) 0 0
\(103\) −17.4034 −1.71481 −0.857405 0.514642i \(-0.827925\pi\)
−0.857405 + 0.514642i \(0.827925\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 11.0368 1.07199
\(107\) −10.5526 −1.02015 −0.510077 0.860128i \(-0.670383\pi\)
−0.510077 + 0.860128i \(0.670383\pi\)
\(108\) 0 0
\(109\) −1.07069 −0.102553 −0.0512766 0.998684i \(-0.516329\pi\)
−0.0512766 + 0.998684i \(0.516329\pi\)
\(110\) −5.12498 −0.488648
\(111\) 0 0
\(112\) −13.9825 −1.32123
\(113\) 16.5308 1.55509 0.777543 0.628830i \(-0.216466\pi\)
0.777543 + 0.628830i \(0.216466\pi\)
\(114\) 0 0
\(115\) −9.85086 −0.918597
\(116\) 0.823708 0.0764794
\(117\) 0 0
\(118\) 2.71379 0.249825
\(119\) 11.7409 1.07629
\(120\) 0 0
\(121\) −8.84846 −0.804406
\(122\) −9.76271 −0.883874
\(123\) 0 0
\(124\) −3.39612 −0.304981
\(125\) 6.02177 0.538604
\(126\) 0 0
\(127\) −9.53750 −0.846316 −0.423158 0.906056i \(-0.639079\pi\)
−0.423158 + 0.906056i \(0.639079\pi\)
\(128\) 6.16421 0.544844
\(129\) 0 0
\(130\) 0 0
\(131\) 5.50902 0.481326 0.240663 0.970609i \(-0.422635\pi\)
0.240663 + 0.970609i \(0.422635\pi\)
\(132\) 0 0
\(133\) −12.2107 −1.05880
\(134\) 4.46681 0.385874
\(135\) 0 0
\(136\) −7.45473 −0.639238
\(137\) −16.1836 −1.38266 −0.691329 0.722540i \(-0.742974\pi\)
−0.691329 + 0.722540i \(0.742974\pi\)
\(138\) 0 0
\(139\) −10.5090 −0.891364 −0.445682 0.895191i \(-0.647039\pi\)
−0.445682 + 0.895191i \(0.647039\pi\)
\(140\) 5.98792 0.506071
\(141\) 0 0
\(142\) −11.0218 −0.924926
\(143\) 0 0
\(144\) 0 0
\(145\) 5.18598 0.430672
\(146\) 9.59179 0.793823
\(147\) 0 0
\(148\) −2.02715 −0.166630
\(149\) 14.3502 1.17561 0.587807 0.809001i \(-0.299992\pi\)
0.587807 + 0.809001i \(0.299992\pi\)
\(150\) 0 0
\(151\) −1.96615 −0.160003 −0.0800014 0.996795i \(-0.525492\pi\)
−0.0800014 + 0.996795i \(0.525492\pi\)
\(152\) 7.75302 0.628853
\(153\) 0 0
\(154\) 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
\(159\) 0 0
\(160\) −6.91185 −0.546430
\(161\) 16.8823 1.33051
\(162\) 0 0
\(163\) −3.89977 −0.305454 −0.152727 0.988268i \(-0.548805\pi\)
−0.152727 + 0.988268i \(0.548805\pi\)
\(164\) −0.554958 −0.0433349
\(165\) 0 0
\(166\) −0.814019 −0.0631802
\(167\) 21.0194 1.62653 0.813264 0.581895i \(-0.197688\pi\)
0.813264 + 0.581895i \(0.197688\pi\)
\(168\) 0 0
\(169\) 0 0
\(170\) −8.54288 −0.655209
\(171\) 0 0
\(172\) −1.06100 −0.0809004
\(173\) −13.2349 −1.00623 −0.503115 0.864219i \(-0.667813\pi\)
−0.503115 + 0.864219i \(0.667813\pi\)
\(174\) 0 0
\(175\) 13.6896 1.03484
\(176\) −4.27114 −0.321950
\(177\) 0 0
\(178\) 7.85086 0.588446
\(179\) −8.52781 −0.637399 −0.318699 0.947856i \(-0.603246\pi\)
−0.318699 + 0.947856i \(0.603246\pi\)
\(180\) 0 0
\(181\) −3.63640 −0.270291 −0.135146 0.990826i \(-0.543150\pi\)
−0.135146 + 0.990826i \(0.543150\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) −10.7192 −0.790228
\(185\) −12.7627 −0.938333
\(186\) 0 0
\(187\) 3.58642 0.262265
\(188\) −5.70410 −0.416014
\(189\) 0 0
\(190\) 8.88471 0.644564
\(191\) −21.3817 −1.54712 −0.773561 0.633722i \(-0.781526\pi\)
−0.773561 + 0.633722i \(0.781526\pi\)
\(192\) 0 0
\(193\) 8.42758 0.606631 0.303315 0.952890i \(-0.401906\pi\)
0.303315 + 0.952890i \(0.401906\pi\)
\(194\) 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 \(0.892187\pi\)
\(198\) 0 0
\(199\) −14.2524 −1.01032 −0.505161 0.863025i \(-0.668567\pi\)
−0.505161 + 0.863025i \(0.668567\pi\)
\(200\) −8.69202 −0.614619
\(201\) 0 0
\(202\) −17.3177 −1.21847
\(203\) −8.88769 −0.623794
\(204\) 0 0
\(205\) −3.49396 −0.244029
\(206\) −21.7017 −1.51203
\(207\) 0 0
\(208\) 0 0
\(209\) −3.72992 −0.258004
\(210\) 0 0
\(211\) 1.85086 0.127418 0.0637091 0.997969i \(-0.479707\pi\)
0.0637091 + 0.997969i \(0.479707\pi\)
\(212\) −3.93900 −0.270532
\(213\) 0 0
\(214\) −13.1588 −0.899519
\(215\) −6.67994 −0.455568
\(216\) 0 0
\(217\) 36.6437 2.48754
\(218\) −1.33513 −0.0904261
\(219\) 0 0
\(220\) 1.82908 0.123317
\(221\) 0 0
\(222\) 0 0
\(223\) −18.6504 −1.24892 −0.624462 0.781056i \(-0.714682\pi\)
−0.624462 + 0.781056i \(0.714682\pi\)
\(224\) 11.8455 0.791459
\(225\) 0 0
\(226\) 20.6136 1.37119
\(227\) 9.75733 0.647617 0.323808 0.946123i \(-0.395037\pi\)
0.323808 + 0.946123i \(0.395037\pi\)
\(228\) 0 0
\(229\) 2.86294 0.189188 0.0945941 0.995516i \(-0.469845\pi\)
0.0945941 + 0.995516i \(0.469845\pi\)
\(230\) −12.2838 −0.809971
\(231\) 0 0
\(232\) 5.64310 0.370488
\(233\) 5.78554 0.379024 0.189512 0.981878i \(-0.439309\pi\)
0.189512 + 0.981878i \(0.439309\pi\)
\(234\) 0 0
\(235\) −35.9124 −2.34267
\(236\) −0.968541 −0.0630467
\(237\) 0 0
\(238\) 14.6407 0.949016
\(239\) 7.09246 0.458773 0.229386 0.973335i \(-0.426328\pi\)
0.229386 + 0.973335i \(0.426328\pi\)
\(240\) 0 0
\(241\) −3.89977 −0.251206 −0.125603 0.992081i \(-0.540087\pi\)
−0.125603 + 0.992081i \(0.540087\pi\)
\(242\) −11.0339 −0.709283
\(243\) 0 0
\(244\) 3.48427 0.223058
\(245\) −44.9952 −2.87464
\(246\) 0 0
\(247\) 0 0
\(248\) −23.2664 −1.47742
\(249\) 0 0
\(250\) 7.50902 0.474912
\(251\) 2.44504 0.154330 0.0771648 0.997018i \(-0.475413\pi\)
0.0771648 + 0.997018i \(0.475413\pi\)
\(252\) 0 0
\(253\) 5.15691 0.324212
\(254\) −11.8931 −0.746237
\(255\) 0 0
\(256\) −10.1129 −0.632056
\(257\) 14.1304 0.881428 0.440714 0.897648i \(-0.354725\pi\)
0.440714 + 0.897648i \(0.354725\pi\)
\(258\) 0 0
\(259\) 21.8726 1.35910
\(260\) 0 0
\(261\) 0 0
\(262\) 6.86964 0.424408
\(263\) 23.7235 1.46285 0.731426 0.681921i \(-0.238855\pi\)
0.731426 + 0.681921i \(0.238855\pi\)
\(264\) 0 0
\(265\) −24.7995 −1.52342
\(266\) −15.2265 −0.933599
\(267\) 0 0
\(268\) −1.59419 −0.0973805
\(269\) 5.91617 0.360715 0.180357 0.983601i \(-0.442275\pi\)
0.180357 + 0.983601i \(0.442275\pi\)
\(270\) 0 0
\(271\) −3.19029 −0.193796 −0.0968982 0.995294i \(-0.530892\pi\)
−0.0968982 + 0.995294i \(0.530892\pi\)
\(272\) −7.11960 −0.431689
\(273\) 0 0
\(274\) −20.1806 −1.21915
\(275\) 4.18167 0.252164
\(276\) 0 0
\(277\) 21.9366 1.31804 0.659022 0.752124i \(-0.270971\pi\)
0.659022 + 0.752124i \(0.270971\pi\)
\(278\) −13.1045 −0.785958
\(279\) 0 0
\(280\) 41.0224 2.45155
\(281\) −11.9903 −0.715282 −0.357641 0.933859i \(-0.616419\pi\)
−0.357641 + 0.933859i \(0.616419\pi\)
\(282\) 0 0
\(283\) −14.3666 −0.854005 −0.427002 0.904250i \(-0.640430\pi\)
−0.427002 + 0.904250i \(0.640430\pi\)
\(284\) 3.93362 0.233418
\(285\) 0 0
\(286\) 0 0
\(287\) 5.98792 0.353456
\(288\) 0 0
\(289\) −11.0218 −0.648339
\(290\) 6.46681 0.379744
\(291\) 0 0
\(292\) −3.42327 −0.200332
\(293\) −18.7584 −1.09588 −0.547939 0.836519i \(-0.684587\pi\)
−0.547939 + 0.836519i \(0.684587\pi\)
\(294\) 0 0
\(295\) −6.09783 −0.355030
\(296\) −13.8877 −0.807206
\(297\) 0 0
\(298\) 17.8944 1.03659
\(299\) 0 0
\(300\) 0 0
\(301\) 11.4480 0.659853
\(302\) −2.45175 −0.141082
\(303\) 0 0
\(304\) 7.40449 0.424676
\(305\) 21.9366 1.25609
\(306\) 0 0
\(307\) 25.6262 1.46257 0.731283 0.682074i \(-0.238922\pi\)
0.731283 + 0.682074i \(0.238922\pi\)
\(308\) −3.13467 −0.178614
\(309\) 0 0
\(310\) −26.6625 −1.51433
\(311\) −11.3817 −0.645394 −0.322697 0.946502i \(-0.604590\pi\)
−0.322697 + 0.946502i \(0.604590\pi\)
\(312\) 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
\(315\) 0 0
\(316\) 1.78986 0.100687
\(317\) 11.2597 0.632405 0.316203 0.948692i \(-0.397592\pi\)
0.316203 + 0.948692i \(0.397592\pi\)
\(318\) 0 0
\(319\) −2.71486 −0.152003
\(320\) −24.9366 −1.39400
\(321\) 0 0
\(322\) 21.0519 1.17318
\(323\) −6.21744 −0.345948
\(324\) 0 0
\(325\) 0 0
\(326\) −4.86294 −0.269333
\(327\) 0 0
\(328\) −3.80194 −0.209927
\(329\) 61.5465 3.39317
\(330\) 0 0
\(331\) −11.9065 −0.654439 −0.327220 0.944948i \(-0.606112\pi\)
−0.327220 + 0.944948i \(0.606112\pi\)
\(332\) 0.290520 0.0159444
\(333\) 0 0
\(334\) 26.2107 1.43419
\(335\) −10.0368 −0.548371
\(336\) 0 0
\(337\) 17.1672 0.935157 0.467578 0.883952i \(-0.345127\pi\)
0.467578 + 0.883952i \(0.345127\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 3.04892 0.165351
\(341\) 11.1933 0.606150
\(342\) 0 0
\(343\) 43.4989 2.34872
\(344\) −7.26875 −0.391905
\(345\) 0 0
\(346\) −16.5036 −0.887242
\(347\) 24.2760 1.30321 0.651603 0.758560i \(-0.274097\pi\)
0.651603 + 0.758560i \(0.274097\pi\)
\(348\) 0 0
\(349\) −4.57242 −0.244756 −0.122378 0.992484i \(-0.539052\pi\)
−0.122378 + 0.992484i \(0.539052\pi\)
\(350\) 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 \(0.448303\pi\)
\(354\) 0 0
\(355\) 24.7657 1.31443
\(356\) −2.80194 −0.148502
\(357\) 0 0
\(358\) −10.6340 −0.562025
\(359\) 14.9661 0.789883 0.394942 0.918706i \(-0.370765\pi\)
0.394942 + 0.918706i \(0.370765\pi\)
\(360\) 0 0
\(361\) −12.5338 −0.659673
\(362\) −4.53452 −0.238329
\(363\) 0 0
\(364\) 0 0
\(365\) −21.5526 −1.12811
\(366\) 0 0
\(367\) 37.0834 1.93574 0.967868 0.251459i \(-0.0809105\pi\)
0.967868 + 0.251459i \(0.0809105\pi\)
\(368\) −10.2373 −0.533656
\(369\) 0 0
\(370\) −15.9148 −0.827373
\(371\) 42.5013 2.20656
\(372\) 0 0
\(373\) −36.5090 −1.89037 −0.945183 0.326542i \(-0.894117\pi\)
−0.945183 + 0.326542i \(0.894117\pi\)
\(374\) 4.47219 0.231251
\(375\) 0 0
\(376\) −39.0780 −2.01529
\(377\) 0 0
\(378\) 0 0
\(379\) −26.5851 −1.36558 −0.682792 0.730613i \(-0.739235\pi\)
−0.682792 + 0.730613i \(0.739235\pi\)
\(380\) −3.17092 −0.162665
\(381\) 0 0
\(382\) −26.6625 −1.36417
\(383\) 14.3502 0.733261 0.366630 0.930367i \(-0.380511\pi\)
0.366630 + 0.930367i \(0.380511\pi\)
\(384\) 0 0
\(385\) −19.7356 −1.00582
\(386\) 10.5090 0.534895
\(387\) 0 0
\(388\) −4.46442 −0.226647
\(389\) 22.6582 1.14881 0.574407 0.818570i \(-0.305233\pi\)
0.574407 + 0.818570i \(0.305233\pi\)
\(390\) 0 0
\(391\) 8.59611 0.434724
\(392\) −48.9614 −2.47292
\(393\) 0 0
\(394\) −33.0157 −1.66330
\(395\) 11.2687 0.566992
\(396\) 0 0
\(397\) 7.90754 0.396868 0.198434 0.980114i \(-0.436414\pi\)
0.198434 + 0.980114i \(0.436414\pi\)
\(398\) −17.7724 −0.890850
\(399\) 0 0
\(400\) −8.30127 −0.415064
\(401\) −2.93661 −0.146647 −0.0733236 0.997308i \(-0.523361\pi\)
−0.0733236 + 0.997308i \(0.523361\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 6.18060 0.307497
\(405\) 0 0
\(406\) −11.0828 −0.550029
\(407\) 6.68127 0.331178
\(408\) 0 0
\(409\) 11.7549 0.581244 0.290622 0.956838i \(-0.406138\pi\)
0.290622 + 0.956838i \(0.406138\pi\)
\(410\) −4.35690 −0.215172
\(411\) 0 0
\(412\) 7.74525 0.381581
\(413\) 10.4504 0.514231
\(414\) 0 0
\(415\) 1.82908 0.0897862
\(416\) 0 0
\(417\) 0 0
\(418\) −4.65114 −0.227495
\(419\) −7.34183 −0.358672 −0.179336 0.983788i \(-0.557395\pi\)
−0.179336 + 0.983788i \(0.557395\pi\)
\(420\) 0 0
\(421\) 25.6963 1.25236 0.626181 0.779677i \(-0.284617\pi\)
0.626181 + 0.779677i \(0.284617\pi\)
\(422\) 2.30798 0.112351
\(423\) 0 0
\(424\) −26.9855 −1.31053
\(425\) 6.97046 0.338117
\(426\) 0 0
\(427\) −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 \(0.569093\pi\)
\(432\) 0 0
\(433\) 2.91484 0.140078 0.0700391 0.997544i \(-0.477688\pi\)
0.0700391 + 0.997544i \(0.477688\pi\)
\(434\) 45.6939 2.19338
\(435\) 0 0
\(436\) 0.476501 0.0228203
\(437\) −8.94007 −0.427661
\(438\) 0 0
\(439\) 9.05861 0.432344 0.216172 0.976355i \(-0.430643\pi\)
0.216172 + 0.976355i \(0.430643\pi\)
\(440\) 12.5308 0.597382
\(441\) 0 0
\(442\) 0 0
\(443\) −11.2325 −0.533672 −0.266836 0.963742i \(-0.585978\pi\)
−0.266836 + 0.963742i \(0.585978\pi\)
\(444\) 0 0
\(445\) −17.6407 −0.836250
\(446\) −23.2567 −1.10124
\(447\) 0 0
\(448\) 42.7362 2.01909
\(449\) −28.7579 −1.35717 −0.678585 0.734522i \(-0.737407\pi\)
−0.678585 + 0.734522i \(0.737407\pi\)
\(450\) 0 0
\(451\) 1.82908 0.0861282
\(452\) −7.35690 −0.346039
\(453\) 0 0
\(454\) 12.1672 0.571035
\(455\) 0 0
\(456\) 0 0
\(457\) −19.0761 −0.892341 −0.446170 0.894948i \(-0.647212\pi\)
−0.446170 + 0.894948i \(0.647212\pi\)
\(458\) 3.57002 0.166816
\(459\) 0 0
\(460\) 4.38404 0.204407
\(461\) −31.7332 −1.47796 −0.738981 0.673727i \(-0.764692\pi\)
−0.738981 + 0.673727i \(0.764692\pi\)
\(462\) 0 0
\(463\) −36.4784 −1.69530 −0.847648 0.530559i \(-0.821982\pi\)
−0.847648 + 0.530559i \(0.821982\pi\)
\(464\) 5.38942 0.250198
\(465\) 0 0
\(466\) 7.21446 0.334203
\(467\) −13.0000 −0.601568 −0.300784 0.953692i \(-0.597248\pi\)
−0.300784 + 0.953692i \(0.597248\pi\)
\(468\) 0 0
\(469\) 17.2010 0.794271
\(470\) −44.7821 −2.06564
\(471\) 0 0
\(472\) −6.63533 −0.305416
\(473\) 3.49694 0.160790
\(474\) 0 0
\(475\) −7.24937 −0.332624
\(476\) −5.22521 −0.239497
\(477\) 0 0
\(478\) 8.84415 0.404522
\(479\) −5.61655 −0.256627 −0.128313 0.991734i \(-0.540956\pi\)
−0.128313 + 0.991734i \(0.540956\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) −4.86294 −0.221501
\(483\) 0 0
\(484\) 3.93794 0.178997
\(485\) −28.1075 −1.27630
\(486\) 0 0
\(487\) −9.75733 −0.442147 −0.221073 0.975257i \(-0.570956\pi\)
−0.221073 + 0.975257i \(0.570956\pi\)
\(488\) 23.8702 1.08055
\(489\) 0 0
\(490\) −56.1081 −2.53471
\(491\) 7.38835 0.333432 0.166716 0.986005i \(-0.446684\pi\)
0.166716 + 0.986005i \(0.446684\pi\)
\(492\) 0 0
\(493\) −4.52542 −0.203815
\(494\) 0 0
\(495\) 0 0
\(496\) −22.2204 −0.997726
\(497\) −42.4432 −1.90384
\(498\) 0 0
\(499\) −43.2814 −1.93754 −0.968771 0.247956i \(-0.920241\pi\)
−0.968771 + 0.247956i \(0.920241\pi\)
\(500\) −2.67994 −0.119851
\(501\) 0 0
\(502\) 3.04892 0.136080
\(503\) −10.7670 −0.480078 −0.240039 0.970763i \(-0.577160\pi\)
−0.240039 + 0.970763i \(0.577160\pi\)
\(504\) 0 0
\(505\) 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 \(0.127602\pi\)
\(510\) 0 0
\(511\) 36.9366 1.63398
\(512\) −24.9390 −1.10216
\(513\) 0 0
\(514\) 17.6203 0.777197
\(515\) 48.7633 2.14877
\(516\) 0 0
\(517\) 18.8001 0.826829
\(518\) 27.2747 1.19838
\(519\) 0 0
\(520\) 0 0
\(521\) −25.7198 −1.12680 −0.563402 0.826183i \(-0.690508\pi\)
−0.563402 + 0.826183i \(0.690508\pi\)
\(522\) 0 0
\(523\) 8.59286 0.375739 0.187870 0.982194i \(-0.439842\pi\)
0.187870 + 0.982194i \(0.439842\pi\)
\(524\) −2.45175 −0.107105
\(525\) 0 0
\(526\) 29.5827 1.28987
\(527\) 18.6582 0.812763
\(528\) 0 0
\(529\) −10.6396 −0.462593
\(530\) −30.9245 −1.34328
\(531\) 0 0
\(532\) 5.43429 0.235606
\(533\) 0 0
\(534\) 0 0
\(535\) 29.5676 1.27832
\(536\) −10.9215 −0.471739
\(537\) 0 0
\(538\) 7.37734 0.318060
\(539\) 23.5550 1.01458
\(540\) 0 0
\(541\) 31.3534 1.34799 0.673995 0.738736i \(-0.264577\pi\)
0.673995 + 0.738736i \(0.264577\pi\)
\(542\) −3.97823 −0.170880
\(543\) 0 0
\(544\) 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
\(549\) 0 0
\(550\) 5.21446 0.222345
\(551\) 4.70650 0.200503
\(552\) 0 0
\(553\) −19.3123 −0.821242
\(554\) 27.3545 1.16218
\(555\) 0 0
\(556\) 4.67696 0.198347
\(557\) −33.2349 −1.40821 −0.704104 0.710097i \(-0.748651\pi\)
−0.704104 + 0.710097i \(0.748651\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 39.1782 1.65558
\(561\) 0 0
\(562\) −14.9517 −0.630698
\(563\) −3.87130 −0.163156 −0.0815779 0.996667i \(-0.525996\pi\)
−0.0815779 + 0.996667i \(0.525996\pi\)
\(564\) 0 0
\(565\) −46.3183 −1.94862
\(566\) −17.9148 −0.753017
\(567\) 0 0
\(568\) 26.9487 1.13074
\(569\) −20.1457 −0.844551 −0.422276 0.906468i \(-0.638769\pi\)
−0.422276 + 0.906468i \(0.638769\pi\)
\(570\) 0 0
\(571\) −32.1269 −1.34447 −0.672234 0.740338i \(-0.734665\pi\)
−0.672234 + 0.740338i \(0.734665\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 7.46681 0.311659
\(575\) 10.0228 0.417981
\(576\) 0 0
\(577\) −16.7506 −0.697338 −0.348669 0.937246i \(-0.613366\pi\)
−0.348669 + 0.937246i \(0.613366\pi\)
\(578\) −13.7439 −0.571672
\(579\) 0 0
\(580\) −2.30798 −0.0958336
\(581\) −3.13467 −0.130048
\(582\) 0 0
\(583\) 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 \(0.455594\pi\)
\(588\) 0 0
\(589\) −19.4047 −0.799559
\(590\) −7.60388 −0.313047
\(591\) 0 0
\(592\) −13.2634 −0.545121
\(593\) −18.1172 −0.743985 −0.371992 0.928236i \(-0.621325\pi\)
−0.371992 + 0.928236i \(0.621325\pi\)
\(594\) 0 0
\(595\) −32.8974 −1.34866
\(596\) −6.38644 −0.261599
\(597\) 0 0
\(598\) 0 0
\(599\) 26.7851 1.09441 0.547204 0.836999i \(-0.315692\pi\)
0.547204 + 0.836999i \(0.315692\pi\)
\(600\) 0 0
\(601\) −4.70171 −0.191787 −0.0958934 0.995392i \(-0.530571\pi\)
−0.0958934 + 0.995392i \(0.530571\pi\)
\(602\) 14.2755 0.581824
\(603\) 0 0
\(604\) 0.875018 0.0356040
\(605\) 24.7928 1.00797
\(606\) 0 0
\(607\) 27.6396 1.12186 0.560929 0.827864i \(-0.310444\pi\)
0.560929 + 0.827864i \(0.310444\pi\)
\(608\) −6.27280 −0.254396
\(609\) 0 0
\(610\) 27.3545 1.10755
\(611\) 0 0
\(612\) 0 0
\(613\) 48.1782 1.94590 0.972950 0.231017i \(-0.0742052\pi\)
0.972950 + 0.231017i \(0.0742052\pi\)
\(614\) 31.9554 1.28961
\(615\) 0 0
\(616\) −21.4752 −0.865259
\(617\) −30.3043 −1.22000 −0.610002 0.792400i \(-0.708831\pi\)
−0.610002 + 0.792400i \(0.708831\pi\)
\(618\) 0 0
\(619\) −10.9041 −0.438272 −0.219136 0.975694i \(-0.570324\pi\)
−0.219136 + 0.975694i \(0.570324\pi\)
\(620\) 9.51573 0.382161
\(621\) 0 0
\(622\) −14.1927 −0.569075
\(623\) 30.2325 1.21124
\(624\) 0 0
\(625\) −31.1269 −1.24508
\(626\) 34.3846 1.37429
\(627\) 0 0
\(628\) −4.76271 −0.190053
\(629\) 11.1371 0.444064
\(630\) 0 0
\(631\) −10.4523 −0.416101 −0.208050 0.978118i \(-0.566712\pi\)
−0.208050 + 0.978118i \(0.566712\pi\)
\(632\) 12.2620 0.487758
\(633\) 0 0
\(634\) 14.0406 0.557622
\(635\) 26.7235 1.06049
\(636\) 0 0
\(637\) 0 0
\(638\) −3.38537 −0.134028
\(639\) 0 0
\(640\) −17.2717 −0.682725
\(641\) 17.5942 0.694929 0.347464 0.937693i \(-0.387043\pi\)
0.347464 + 0.937693i \(0.387043\pi\)
\(642\) 0 0
\(643\) 22.6058 0.891486 0.445743 0.895161i \(-0.352940\pi\)
0.445743 + 0.895161i \(0.352940\pi\)
\(644\) −7.51334 −0.296067
\(645\) 0 0
\(646\) −7.75302 −0.305039
\(647\) 24.7918 0.974665 0.487333 0.873216i \(-0.337970\pi\)
0.487333 + 0.873216i \(0.337970\pi\)
\(648\) 0 0
\(649\) 3.19221 0.125305
\(650\) 0 0
\(651\) 0 0
\(652\) 1.73556 0.0679699
\(653\) 21.8106 0.853513 0.426757 0.904367i \(-0.359656\pi\)
0.426757 + 0.904367i \(0.359656\pi\)
\(654\) 0 0
\(655\) −15.4359 −0.603132
\(656\) −3.63102 −0.141768
\(657\) 0 0
\(658\) 76.7472 2.99192
\(659\) −16.5526 −0.644796 −0.322398 0.946604i \(-0.604489\pi\)
−0.322398 + 0.946604i \(0.604489\pi\)
\(660\) 0 0
\(661\) −15.9541 −0.620541 −0.310271 0.950648i \(-0.600420\pi\)
−0.310271 + 0.950648i \(0.600420\pi\)
\(662\) −14.8471 −0.577050
\(663\) 0 0
\(664\) 1.99031 0.0772391
\(665\) 34.2137 1.32675
\(666\) 0 0
\(667\) −6.50711 −0.251956
\(668\) −9.35450 −0.361937
\(669\) 0 0
\(670\) −12.5157 −0.483525
\(671\) −11.4838 −0.443327
\(672\) 0 0
\(673\) −7.38835 −0.284800 −0.142400 0.989809i \(-0.545482\pi\)
−0.142400 + 0.989809i \(0.545482\pi\)
\(674\) 21.4071 0.824572
\(675\) 0 0
\(676\) 0 0
\(677\) 22.1454 0.851118 0.425559 0.904931i \(-0.360078\pi\)
0.425559 + 0.904931i \(0.360078\pi\)
\(678\) 0 0
\(679\) 48.1704 1.84861
\(680\) 20.8877 0.801006
\(681\) 0 0
\(682\) 13.9578 0.534471
\(683\) −9.10023 −0.348211 −0.174105 0.984727i \(-0.555703\pi\)
−0.174105 + 0.984727i \(0.555703\pi\)
\(684\) 0 0
\(685\) 45.3454 1.73256
\(686\) 54.2422 2.07098
\(687\) 0 0
\(688\) −6.94198 −0.264661
\(689\) 0 0
\(690\) 0 0
\(691\) 13.7711 0.523876 0.261938 0.965085i \(-0.415638\pi\)
0.261938 + 0.965085i \(0.415638\pi\)
\(692\) 5.89008 0.223907
\(693\) 0 0
\(694\) 30.2717 1.14910
\(695\) 29.4456 1.11694
\(696\) 0 0
\(697\) 3.04892 0.115486
\(698\) −5.70171 −0.215813
\(699\) 0 0
\(700\) −6.09246 −0.230273
\(701\) −46.5090 −1.75662 −0.878311 0.478090i \(-0.841329\pi\)
−0.878311 + 0.478090i \(0.841329\pi\)
\(702\) 0 0
\(703\) −11.5827 −0.436850
\(704\) 13.0543 0.492002
\(705\) 0 0
\(706\) 7.57673 0.285154
\(707\) −66.6878 −2.50805
\(708\) 0 0
\(709\) 7.24565 0.272116 0.136058 0.990701i \(-0.456557\pi\)
0.136058 + 0.990701i \(0.456557\pi\)
\(710\) 30.8823 1.15899
\(711\) 0 0
\(712\) −19.1957 −0.719388
\(713\) 26.8286 1.00474
\(714\) 0 0
\(715\) 0 0
\(716\) 3.79523 0.141835
\(717\) 0 0
\(718\) 18.6625 0.696478
\(719\) −25.5147 −0.951536 −0.475768 0.879571i \(-0.657830\pi\)
−0.475768 + 0.879571i \(0.657830\pi\)
\(720\) 0 0
\(721\) −83.5701 −3.11231
\(722\) −15.6294 −0.581665
\(723\) 0 0
\(724\) 1.61835 0.0601455
\(725\) −5.27652 −0.195965
\(726\) 0 0
\(727\) −14.4873 −0.537303 −0.268651 0.963238i \(-0.586578\pi\)
−0.268651 + 0.963238i \(0.586578\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) −26.8756 −0.994711
\(731\) 5.82908 0.215596
\(732\) 0 0
\(733\) 37.5036 1.38523 0.692614 0.721308i \(-0.256459\pi\)
0.692614 + 0.721308i \(0.256459\pi\)
\(734\) 46.2422 1.70683
\(735\) 0 0
\(736\) 8.67264 0.319678
\(737\) 5.25428 0.193544
\(738\) 0 0
\(739\) 43.1876 1.58868 0.794341 0.607472i \(-0.207816\pi\)
0.794341 + 0.607472i \(0.207816\pi\)
\(740\) 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 \(0.579511\pi\)
\(744\) 0 0
\(745\) −40.2083 −1.47312
\(746\) −45.5260 −1.66683
\(747\) 0 0
\(748\) −1.59611 −0.0583594
\(749\) −50.6728 −1.85154
\(750\) 0 0
\(751\) 35.5894 1.29868 0.649338 0.760500i \(-0.275046\pi\)
0.649338 + 0.760500i \(0.275046\pi\)
\(752\) −37.3212 −1.36097
\(753\) 0 0
\(754\) 0 0
\(755\) 5.50902 0.200494
\(756\) 0 0
\(757\) −12.2107 −0.443807 −0.221903 0.975069i \(-0.571227\pi\)
−0.221903 + 0.975069i \(0.571227\pi\)
\(758\) −33.1511 −1.20410
\(759\) 0 0
\(760\) −21.7235 −0.787993
\(761\) −30.9071 −1.12038 −0.560190 0.828364i \(-0.689272\pi\)
−0.560190 + 0.828364i \(0.689272\pi\)
\(762\) 0 0
\(763\) −5.14138 −0.186130
\(764\) 9.51573 0.344267
\(765\) 0 0
\(766\) 17.8944 0.646551
\(767\) 0 0
\(768\) 0 0
\(769\) 11.9892 0.432343 0.216172 0.976355i \(-0.430643\pi\)
0.216172 + 0.976355i \(0.430643\pi\)
\(770\) −24.6098 −0.886877
\(771\) 0 0
\(772\) −3.75063 −0.134988
\(773\) −43.9661 −1.58135 −0.790676 0.612235i \(-0.790271\pi\)
−0.790676 + 0.612235i \(0.790271\pi\)
\(774\) 0 0
\(775\) 21.7549 0.781460
\(776\) −30.5851 −1.09794
\(777\) 0 0
\(778\) 28.2543 1.01296
\(779\) −3.17092 −0.113610
\(780\) 0 0
\(781\) −12.9648 −0.463918
\(782\) 10.7192 0.383317
\(783\) 0 0
\(784\) −46.7603 −1.67001
\(785\) −29.9855 −1.07023
\(786\) 0 0
\(787\) 30.3763 1.08280 0.541399 0.840766i \(-0.317895\pi\)
0.541399 + 0.840766i \(0.317895\pi\)
\(788\) 11.7832 0.419757
\(789\) 0 0
\(790\) 14.0519 0.499944
\(791\) 79.3798 2.82242
\(792\) 0 0
\(793\) 0 0
\(794\) 9.86054 0.349938
\(795\) 0 0
\(796\) 6.34290 0.224818
\(797\) −52.5763 −1.86235 −0.931173 0.364577i \(-0.881214\pi\)
−0.931173 + 0.364577i \(0.881214\pi\)
\(798\) 0 0
\(799\) 31.3381 1.10866
\(800\) 7.03252 0.248637
\(801\) 0 0
\(802\) −3.66189 −0.129306
\(803\) 11.2828 0.398160
\(804\) 0 0
\(805\) −47.3032 −1.66722
\(806\) 0 0
\(807\) 0 0
\(808\) 42.3424 1.48960
\(809\) −49.4215 −1.73757 −0.868783 0.495193i \(-0.835097\pi\)
−0.868783 + 0.495193i \(0.835097\pi\)
\(810\) 0 0
\(811\) −1.36526 −0.0479406 −0.0239703 0.999713i \(-0.507631\pi\)
−0.0239703 + 0.999713i \(0.507631\pi\)
\(812\) 3.95539 0.138807
\(813\) 0 0
\(814\) 8.33140 0.292016
\(815\) 10.9269 0.382753
\(816\) 0 0
\(817\) −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 \(0.503699\pi\)
\(822\) 0 0
\(823\) −10.0592 −0.350642 −0.175321 0.984511i \(-0.556096\pi\)
−0.175321 + 0.984511i \(0.556096\pi\)
\(824\) 53.0616 1.84849
\(825\) 0 0
\(826\) 13.0315 0.453422
\(827\) 37.3038 1.29718 0.648590 0.761138i \(-0.275359\pi\)
0.648590 + 0.761138i \(0.275359\pi\)
\(828\) 0 0
\(829\) −42.6209 −1.48028 −0.740142 0.672451i \(-0.765242\pi\)
−0.740142 + 0.672451i \(0.765242\pi\)
\(830\) 2.28083 0.0791688
\(831\) 0 0
\(832\) 0 0
\(833\) 39.2640 1.36042
\(834\) 0 0
\(835\) −58.8950 −2.03815
\(836\) 1.65997 0.0574113
\(837\) 0 0
\(838\) −9.15511 −0.316258
\(839\) −4.14005 −0.142930 −0.0714652 0.997443i \(-0.522767\pi\)
−0.0714652 + 0.997443i \(0.522767\pi\)
\(840\) 0 0
\(841\) −25.5743 −0.881874
\(842\) 32.0428 1.10427
\(843\) 0 0
\(844\) −0.823708 −0.0283532
\(845\) 0 0
\(846\) 0 0
\(847\) −42.4898 −1.45997
\(848\) −25.7724 −0.885028
\(849\) 0 0
\(850\) 8.69202 0.298134
\(851\) 16.0140 0.548953
\(852\) 0 0
\(853\) 17.3502 0.594059 0.297030 0.954868i \(-0.404004\pi\)
0.297030 + 0.954868i \(0.404004\pi\)
\(854\) −46.8799 −1.60420
\(855\) 0 0
\(856\) 32.1739 1.09968
\(857\) −41.0180 −1.40115 −0.700575 0.713579i \(-0.747073\pi\)
−0.700575 + 0.713579i \(0.747073\pi\)
\(858\) 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
\(861\) 0 0
\(862\) −11.1505 −0.379787
\(863\) 16.6455 0.566619 0.283310 0.959028i \(-0.408568\pi\)
0.283310 + 0.959028i \(0.408568\pi\)
\(864\) 0 0
\(865\) 37.0834 1.26087
\(866\) 3.63474 0.123514
\(867\) 0 0
\(868\) −16.3080 −0.553529
\(869\) −5.89918 −0.200116
\(870\) 0 0
\(871\) 0 0
\(872\) 3.26444 0.110548
\(873\) 0 0
\(874\) −11.1481 −0.377089
\(875\) 28.9162 0.977545
\(876\) 0 0
\(877\) 54.4965 1.84022 0.920108 0.391666i \(-0.128101\pi\)
0.920108 + 0.391666i \(0.128101\pi\)
\(878\) 11.2959 0.381218
\(879\) 0 0
\(880\) 11.9675 0.403424
\(881\) 9.00670 0.303444 0.151722 0.988423i \(-0.451518\pi\)
0.151722 + 0.988423i \(0.451518\pi\)
\(882\) 0 0
\(883\) 18.8907 0.635722 0.317861 0.948137i \(-0.397035\pi\)
0.317861 + 0.948137i \(0.397035\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) −14.0067 −0.470564
\(887\) 46.9124 1.57517 0.787583 0.616209i \(-0.211332\pi\)
0.787583 + 0.616209i \(0.211332\pi\)
\(888\) 0 0
\(889\) −45.7985 −1.53603
\(890\) −21.9976 −0.737361
\(891\) 0 0
\(892\) 8.30021 0.277912
\(893\) −32.5921 −1.09065
\(894\) 0 0
\(895\) 23.8944 0.798702
\(896\) 29.6002 0.988872
\(897\) 0 0
\(898\) −35.8605 −1.19668
\(899\) −14.1239 −0.471059
\(900\) 0 0
\(901\) 21.6407 0.720957
\(902\) 2.28083 0.0759434
\(903\) 0 0
\(904\) −50.4010 −1.67631
\(905\) 10.1890 0.338693
\(906\) 0 0
\(907\) −35.3013 −1.17216 −0.586080 0.810253i \(-0.699329\pi\)
−0.586080 + 0.810253i \(0.699329\pi\)
\(908\) −4.34242 −0.144108
\(909\) 0 0
\(910\) 0 0
\(911\) 9.80731 0.324931 0.162465 0.986714i \(-0.448055\pi\)
0.162465 + 0.986714i \(0.448055\pi\)
\(912\) 0 0
\(913\) −0.957524 −0.0316894
\(914\) −23.7875 −0.786819
\(915\) 0 0
\(916\) −1.27413 −0.0420983
\(917\) 26.4540 0.873588
\(918\) 0 0
\(919\) 18.4655 0.609120 0.304560 0.952493i \(-0.401491\pi\)
0.304560 + 0.952493i \(0.401491\pi\)
\(920\) 30.0344 0.990206
\(921\) 0 0
\(922\) −39.5706 −1.30319
\(923\) 0 0
\(924\) 0 0
\(925\) 12.9855 0.426961
\(926\) −45.4878 −1.49482
\(927\) 0 0
\(928\) −4.56571 −0.149877
\(929\) 25.6267 0.840785 0.420393 0.907342i \(-0.361892\pi\)
0.420393 + 0.907342i \(0.361892\pi\)
\(930\) 0 0
\(931\) −40.8351 −1.33831
\(932\) −2.57481 −0.0843407
\(933\) 0 0
\(934\) −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
\(939\) 0 0
\(940\) 15.9825 0.521293
\(941\) 12.6418 0.412110 0.206055 0.978540i \(-0.433937\pi\)
0.206055 + 0.978540i \(0.433937\pi\)
\(942\) 0 0
\(943\) 4.38404 0.142764
\(944\) −6.33704 −0.206253
\(945\) 0 0
\(946\) 4.36062 0.141776
\(947\) −27.9801 −0.909233 −0.454616 0.890687i \(-0.650224\pi\)
−0.454616 + 0.890687i \(0.650224\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) −9.03982 −0.293290
\(951\) 0 0
\(952\) −35.7972 −1.16019
\(953\) −4.00239 −0.129650 −0.0648251 0.997897i \(-0.520649\pi\)
−0.0648251 + 0.997897i \(0.520649\pi\)
\(954\) 0 0
\(955\) 59.9101 1.93864
\(956\) −3.15644 −0.102087
\(957\) 0 0
\(958\) −7.00372 −0.226280
\(959\) −77.7126 −2.50947
\(960\) 0 0
\(961\) 27.2325 0.878468
\(962\) 0 0
\(963\) 0 0
\(964\) 1.73556 0.0558987
\(965\) −23.6136 −0.760148
\(966\) 0 0
\(967\) 12.2239 0.393094 0.196547 0.980494i \(-0.437027\pi\)
0.196547 + 0.980494i \(0.437027\pi\)
\(968\) 26.9782 0.867113
\(969\) 0 0
\(970\) −35.0495 −1.12537
\(971\) 23.6401 0.758648 0.379324 0.925264i \(-0.376157\pi\)
0.379324 + 0.925264i \(0.376157\pi\)
\(972\) 0 0
\(973\) −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 \(0.596864\pi\)
\(978\) 0 0
\(979\) 9.23490 0.295149
\(980\) 20.0248 0.639667
\(981\) 0 0
\(982\) 9.21313 0.294003
\(983\) −12.0954 −0.385785 −0.192892 0.981220i \(-0.561787\pi\)
−0.192892 + 0.981220i \(0.561787\pi\)
\(984\) 0 0
\(985\) 74.1855 2.36375
\(986\) −5.64310 −0.179713
\(987\) 0 0
\(988\) 0 0
\(989\) 8.38165 0.266521
\(990\) 0 0
\(991\) −28.5526 −0.907002 −0.453501 0.891256i \(-0.649825\pi\)
−0.453501 + 0.891256i \(0.649825\pi\)
\(992\) 18.8243 0.597672
\(993\) 0 0
\(994\) −52.9259 −1.67871
\(995\) 39.9342 1.26600
\(996\) 0 0
\(997\) −23.9347 −0.758019 −0.379010 0.925393i \(-0.623735\pi\)
−0.379010 + 0.925393i \(0.623735\pi\)
\(998\) −53.9711 −1.70842
\(999\) 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 1521.2.a.p.1.3 3
3.2 odd 2 507.2.a.k.1.1 yes 3
12.11 even 2 8112.2.a.cf.1.3 3
13.5 odd 4 1521.2.b.m.1351.2 6
13.8 odd 4 1521.2.b.m.1351.5 6
13.12 even 2 1521.2.a.q.1.1 3
39.2 even 12 507.2.j.h.316.2 12
39.5 even 4 507.2.b.g.337.5 6
39.8 even 4 507.2.b.g.337.2 6
39.11 even 12 507.2.j.h.316.5 12
39.17 odd 6 507.2.e.k.484.1 6
39.20 even 12 507.2.j.h.361.2 12
39.23 odd 6 507.2.e.k.22.1 6
39.29 odd 6 507.2.e.j.22.3 6
39.32 even 12 507.2.j.h.361.5 12
39.35 odd 6 507.2.e.j.484.3 6
39.38 odd 2 507.2.a.j.1.3 3
156.155 even 2 8112.2.a.by.1.1 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
507.2.a.j.1.3 3 39.38 odd 2
507.2.a.k.1.1 yes 3 3.2 odd 2
507.2.b.g.337.2 6 39.8 even 4
507.2.b.g.337.5 6 39.5 even 4
507.2.e.j.22.3 6 39.29 odd 6
507.2.e.j.484.3 6 39.35 odd 6
507.2.e.k.22.1 6 39.23 odd 6
507.2.e.k.484.1 6 39.17 odd 6
507.2.j.h.316.2 12 39.2 even 12
507.2.j.h.316.5 12 39.11 even 12
507.2.j.h.361.2 12 39.20 even 12
507.2.j.h.361.5 12 39.32 even 12
1521.2.a.p.1.3 3 1.1 even 1 trivial
1521.2.a.q.1.1 3 13.12 even 2
1521.2.b.m.1351.2 6 13.5 odd 4
1521.2.b.m.1351.5 6 13.8 odd 4
8112.2.a.by.1.1 3 156.155 even 2
8112.2.a.cf.1.3 3 12.11 even 2