Properties

Label 4864.2.a.bt.1.3
Level $4864$
Weight $2$
Character 4864.1
Self dual yes
Analytic conductor $38.839$
Analytic rank $0$
Dimension $10$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 4864 = 2^{8} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4864.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(38.8392355432\)
Analytic rank: \(0\)
Dimension: \(10\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} - \cdots)\)
Defining polynomial: \(x^{10} - 2 x^{9} - 23 x^{8} + 44 x^{7} + 167 x^{6} - 266 x^{5} - 491 x^{4} + 460 x^{3} + 546 x^{2} + 56 x - 8\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{6} \)
Twist minimal: no (minimal twist has level 2432)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(-3.58141\) of defining polynomial
Character \(\chi\) \(=\) 4864.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.15181 q^{3} -1.07587 q^{5} +0.238565 q^{7} -1.67333 q^{9} +O(q^{10})\) \(q-1.15181 q^{3} -1.07587 q^{5} +0.238565 q^{7} -1.67333 q^{9} -2.84249 q^{11} +1.31444 q^{13} +1.23921 q^{15} -4.47279 q^{17} +1.00000 q^{19} -0.274782 q^{21} +1.94338 q^{23} -3.84249 q^{25} +5.38280 q^{27} +8.83944 q^{29} -6.64877 q^{31} +3.27402 q^{33} -0.256666 q^{35} -4.06763 q^{37} -1.51399 q^{39} -2.78211 q^{41} -1.21363 q^{43} +1.80029 q^{45} -10.2695 q^{47} -6.94309 q^{49} +5.15181 q^{51} -8.36231 q^{53} +3.05817 q^{55} -1.15181 q^{57} +9.25916 q^{59} -4.66035 q^{61} -0.399197 q^{63} -1.41417 q^{65} +4.31849 q^{67} -2.23841 q^{69} +2.78069 q^{71} -0.134460 q^{73} +4.42583 q^{75} -0.678119 q^{77} +12.7841 q^{79} -1.17998 q^{81} +13.4028 q^{83} +4.81216 q^{85} -10.1814 q^{87} +9.20650 q^{89} +0.313579 q^{91} +7.65814 q^{93} -1.07587 q^{95} -0.247499 q^{97} +4.75643 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10q + 4q^{3} + 14q^{9} + O(q^{10}) \) \( 10q + 4q^{3} + 14q^{9} + 20q^{11} + 4q^{17} + 10q^{19} + 10q^{25} + 28q^{27} - 8q^{33} + 36q^{35} - 12q^{41} - 4q^{43} + 26q^{49} + 36q^{51} + 4q^{57} + 52q^{59} - 24q^{65} + 12q^{67} + 12q^{73} - 12q^{75} + 34q^{81} + 16q^{83} - 20q^{89} + 60q^{91} - 28q^{97} + 60q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) −1.15181 −0.664999 −0.332500 0.943103i \(-0.607892\pi\)
−0.332500 + 0.943103i \(0.607892\pi\)
\(4\) 0 0
\(5\) −1.07587 −0.481146 −0.240573 0.970631i \(-0.577335\pi\)
−0.240573 + 0.970631i \(0.577335\pi\)
\(6\) 0 0
\(7\) 0.238565 0.0901690 0.0450845 0.998983i \(-0.485644\pi\)
0.0450845 + 0.998983i \(0.485644\pi\)
\(8\) 0 0
\(9\) −1.67333 −0.557776
\(10\) 0 0
\(11\) −2.84249 −0.857044 −0.428522 0.903531i \(-0.640966\pi\)
−0.428522 + 0.903531i \(0.640966\pi\)
\(12\) 0 0
\(13\) 1.31444 0.364560 0.182280 0.983247i \(-0.441652\pi\)
0.182280 + 0.983247i \(0.441652\pi\)
\(14\) 0 0
\(15\) 1.23921 0.319962
\(16\) 0 0
\(17\) −4.47279 −1.08481 −0.542405 0.840117i \(-0.682486\pi\)
−0.542405 + 0.840117i \(0.682486\pi\)
\(18\) 0 0
\(19\) 1.00000 0.229416
\(20\) 0 0
\(21\) −0.274782 −0.0599623
\(22\) 0 0
\(23\) 1.94338 0.405222 0.202611 0.979259i \(-0.435057\pi\)
0.202611 + 0.979259i \(0.435057\pi\)
\(24\) 0 0
\(25\) −3.84249 −0.768499
\(26\) 0 0
\(27\) 5.38280 1.03592
\(28\) 0 0
\(29\) 8.83944 1.64144 0.820721 0.571329i \(-0.193572\pi\)
0.820721 + 0.571329i \(0.193572\pi\)
\(30\) 0 0
\(31\) −6.64877 −1.19415 −0.597077 0.802184i \(-0.703671\pi\)
−0.597077 + 0.802184i \(0.703671\pi\)
\(32\) 0 0
\(33\) 3.27402 0.569933
\(34\) 0 0
\(35\) −0.256666 −0.0433845
\(36\) 0 0
\(37\) −4.06763 −0.668715 −0.334357 0.942446i \(-0.608519\pi\)
−0.334357 + 0.942446i \(0.608519\pi\)
\(38\) 0 0
\(39\) −1.51399 −0.242432
\(40\) 0 0
\(41\) −2.78211 −0.434492 −0.217246 0.976117i \(-0.569707\pi\)
−0.217246 + 0.976117i \(0.569707\pi\)
\(42\) 0 0
\(43\) −1.21363 −0.185077 −0.0925386 0.995709i \(-0.529498\pi\)
−0.0925386 + 0.995709i \(0.529498\pi\)
\(44\) 0 0
\(45\) 1.80029 0.268372
\(46\) 0 0
\(47\) −10.2695 −1.49796 −0.748978 0.662595i \(-0.769455\pi\)
−0.748978 + 0.662595i \(0.769455\pi\)
\(48\) 0 0
\(49\) −6.94309 −0.991870
\(50\) 0 0
\(51\) 5.15181 0.721398
\(52\) 0 0
\(53\) −8.36231 −1.14865 −0.574326 0.818627i \(-0.694736\pi\)
−0.574326 + 0.818627i \(0.694736\pi\)
\(54\) 0 0
\(55\) 3.05817 0.412363
\(56\) 0 0
\(57\) −1.15181 −0.152561
\(58\) 0 0
\(59\) 9.25916 1.20544 0.602720 0.797953i \(-0.294084\pi\)
0.602720 + 0.797953i \(0.294084\pi\)
\(60\) 0 0
\(61\) −4.66035 −0.596697 −0.298349 0.954457i \(-0.596436\pi\)
−0.298349 + 0.954457i \(0.596436\pi\)
\(62\) 0 0
\(63\) −0.399197 −0.0502941
\(64\) 0 0
\(65\) −1.41417 −0.175407
\(66\) 0 0
\(67\) 4.31849 0.527587 0.263794 0.964579i \(-0.415026\pi\)
0.263794 + 0.964579i \(0.415026\pi\)
\(68\) 0 0
\(69\) −2.23841 −0.269472
\(70\) 0 0
\(71\) 2.78069 0.330007 0.165003 0.986293i \(-0.447236\pi\)
0.165003 + 0.986293i \(0.447236\pi\)
\(72\) 0 0
\(73\) −0.134460 −0.0157373 −0.00786866 0.999969i \(-0.502505\pi\)
−0.00786866 + 0.999969i \(0.502505\pi\)
\(74\) 0 0
\(75\) 4.42583 0.511051
\(76\) 0 0
\(77\) −0.678119 −0.0772788
\(78\) 0 0
\(79\) 12.7841 1.43832 0.719162 0.694843i \(-0.244526\pi\)
0.719162 + 0.694843i \(0.244526\pi\)
\(80\) 0 0
\(81\) −1.17998 −0.131109
\(82\) 0 0
\(83\) 13.4028 1.47115 0.735573 0.677445i \(-0.236913\pi\)
0.735573 + 0.677445i \(0.236913\pi\)
\(84\) 0 0
\(85\) 4.81216 0.521952
\(86\) 0 0
\(87\) −10.1814 −1.09156
\(88\) 0 0
\(89\) 9.20650 0.975887 0.487944 0.872875i \(-0.337747\pi\)
0.487944 + 0.872875i \(0.337747\pi\)
\(90\) 0 0
\(91\) 0.313579 0.0328720
\(92\) 0 0
\(93\) 7.65814 0.794112
\(94\) 0 0
\(95\) −1.07587 −0.110382
\(96\) 0 0
\(97\) −0.247499 −0.0251297 −0.0125648 0.999921i \(-0.504000\pi\)
−0.0125648 + 0.999921i \(0.504000\pi\)
\(98\) 0 0
\(99\) 4.75643 0.478039
\(100\) 0 0
\(101\) −16.8026 −1.67193 −0.835963 0.548786i \(-0.815090\pi\)
−0.835963 + 0.548786i \(0.815090\pi\)
\(102\) 0 0
\(103\) 5.40957 0.533020 0.266510 0.963832i \(-0.414129\pi\)
0.266510 + 0.963832i \(0.414129\pi\)
\(104\) 0 0
\(105\) 0.295631 0.0288506
\(106\) 0 0
\(107\) 6.39113 0.617853 0.308927 0.951086i \(-0.400030\pi\)
0.308927 + 0.951086i \(0.400030\pi\)
\(108\) 0 0
\(109\) 14.8779 1.42505 0.712524 0.701648i \(-0.247552\pi\)
0.712524 + 0.701648i \(0.247552\pi\)
\(110\) 0 0
\(111\) 4.68515 0.444695
\(112\) 0 0
\(113\) 8.59073 0.808148 0.404074 0.914726i \(-0.367594\pi\)
0.404074 + 0.914726i \(0.367594\pi\)
\(114\) 0 0
\(115\) −2.09083 −0.194971
\(116\) 0 0
\(117\) −2.19949 −0.203343
\(118\) 0 0
\(119\) −1.06705 −0.0978163
\(120\) 0 0
\(121\) −2.92023 −0.265476
\(122\) 0 0
\(123\) 3.20446 0.288937
\(124\) 0 0
\(125\) 9.51342 0.850906
\(126\) 0 0
\(127\) −1.15983 −0.102918 −0.0514592 0.998675i \(-0.516387\pi\)
−0.0514592 + 0.998675i \(0.516387\pi\)
\(128\) 0 0
\(129\) 1.39788 0.123076
\(130\) 0 0
\(131\) −3.85559 −0.336864 −0.168432 0.985713i \(-0.553870\pi\)
−0.168432 + 0.985713i \(0.553870\pi\)
\(132\) 0 0
\(133\) 0.238565 0.0206862
\(134\) 0 0
\(135\) −5.79122 −0.498428
\(136\) 0 0
\(137\) −1.42976 −0.122152 −0.0610761 0.998133i \(-0.519453\pi\)
−0.0610761 + 0.998133i \(0.519453\pi\)
\(138\) 0 0
\(139\) 3.58190 0.303813 0.151907 0.988395i \(-0.451459\pi\)
0.151907 + 0.988395i \(0.451459\pi\)
\(140\) 0 0
\(141\) 11.8285 0.996139
\(142\) 0 0
\(143\) −3.73629 −0.312444
\(144\) 0 0
\(145\) −9.51013 −0.789773
\(146\) 0 0
\(147\) 7.99713 0.659592
\(148\) 0 0
\(149\) −16.2763 −1.33341 −0.666705 0.745322i \(-0.732296\pi\)
−0.666705 + 0.745322i \(0.732296\pi\)
\(150\) 0 0
\(151\) −14.3242 −1.16569 −0.582845 0.812583i \(-0.698061\pi\)
−0.582845 + 0.812583i \(0.698061\pi\)
\(152\) 0 0
\(153\) 7.48445 0.605082
\(154\) 0 0
\(155\) 7.15325 0.574563
\(156\) 0 0
\(157\) −3.73495 −0.298081 −0.149041 0.988831i \(-0.547619\pi\)
−0.149041 + 0.988831i \(0.547619\pi\)
\(158\) 0 0
\(159\) 9.63181 0.763852
\(160\) 0 0
\(161\) 0.463621 0.0365385
\(162\) 0 0
\(163\) −3.98861 −0.312412 −0.156206 0.987724i \(-0.549926\pi\)
−0.156206 + 0.987724i \(0.549926\pi\)
\(164\) 0 0
\(165\) −3.52243 −0.274221
\(166\) 0 0
\(167\) 8.17158 0.632336 0.316168 0.948703i \(-0.397604\pi\)
0.316168 + 0.948703i \(0.397604\pi\)
\(168\) 0 0
\(169\) −11.2722 −0.867096
\(170\) 0 0
\(171\) −1.67333 −0.127963
\(172\) 0 0
\(173\) −0.351356 −0.0267131 −0.0133565 0.999911i \(-0.504252\pi\)
−0.0133565 + 0.999911i \(0.504252\pi\)
\(174\) 0 0
\(175\) −0.916684 −0.0692948
\(176\) 0 0
\(177\) −10.6648 −0.801616
\(178\) 0 0
\(179\) 15.2740 1.14163 0.570817 0.821077i \(-0.306627\pi\)
0.570817 + 0.821077i \(0.306627\pi\)
\(180\) 0 0
\(181\) −8.94630 −0.664974 −0.332487 0.943108i \(-0.607888\pi\)
−0.332487 + 0.943108i \(0.607888\pi\)
\(182\) 0 0
\(183\) 5.36785 0.396803
\(184\) 0 0
\(185\) 4.37626 0.321749
\(186\) 0 0
\(187\) 12.7139 0.929730
\(188\) 0 0
\(189\) 1.28415 0.0934079
\(190\) 0 0
\(191\) 18.3946 1.33098 0.665492 0.746405i \(-0.268222\pi\)
0.665492 + 0.746405i \(0.268222\pi\)
\(192\) 0 0
\(193\) −5.09902 −0.367035 −0.183518 0.983016i \(-0.558748\pi\)
−0.183518 + 0.983016i \(0.558748\pi\)
\(194\) 0 0
\(195\) 1.62886 0.116645
\(196\) 0 0
\(197\) 20.2916 1.44572 0.722860 0.690995i \(-0.242827\pi\)
0.722860 + 0.690995i \(0.242827\pi\)
\(198\) 0 0
\(199\) 0.310999 0.0220461 0.0110231 0.999939i \(-0.496491\pi\)
0.0110231 + 0.999939i \(0.496491\pi\)
\(200\) 0 0
\(201\) −4.97408 −0.350845
\(202\) 0 0
\(203\) 2.10878 0.148007
\(204\) 0 0
\(205\) 2.99320 0.209054
\(206\) 0 0
\(207\) −3.25191 −0.226023
\(208\) 0 0
\(209\) −2.84249 −0.196619
\(210\) 0 0
\(211\) 14.3284 0.986406 0.493203 0.869914i \(-0.335826\pi\)
0.493203 + 0.869914i \(0.335826\pi\)
\(212\) 0 0
\(213\) −3.20283 −0.219454
\(214\) 0 0
\(215\) 1.30572 0.0890491
\(216\) 0 0
\(217\) −1.58616 −0.107676
\(218\) 0 0
\(219\) 0.154872 0.0104653
\(220\) 0 0
\(221\) −5.87921 −0.395479
\(222\) 0 0
\(223\) −12.8621 −0.861312 −0.430656 0.902516i \(-0.641718\pi\)
−0.430656 + 0.902516i \(0.641718\pi\)
\(224\) 0 0
\(225\) 6.42976 0.428650
\(226\) 0 0
\(227\) 18.4491 1.22451 0.612256 0.790659i \(-0.290262\pi\)
0.612256 + 0.790659i \(0.290262\pi\)
\(228\) 0 0
\(229\) 4.73279 0.312751 0.156376 0.987698i \(-0.450019\pi\)
0.156376 + 0.987698i \(0.450019\pi\)
\(230\) 0 0
\(231\) 0.781066 0.0513903
\(232\) 0 0
\(233\) −4.27809 −0.280267 −0.140133 0.990133i \(-0.544753\pi\)
−0.140133 + 0.990133i \(0.544753\pi\)
\(234\) 0 0
\(235\) 11.0487 0.720735
\(236\) 0 0
\(237\) −14.7249 −0.956483
\(238\) 0 0
\(239\) 20.3905 1.31895 0.659474 0.751727i \(-0.270779\pi\)
0.659474 + 0.751727i \(0.270779\pi\)
\(240\) 0 0
\(241\) 25.5877 1.64825 0.824123 0.566410i \(-0.191668\pi\)
0.824123 + 0.566410i \(0.191668\pi\)
\(242\) 0 0
\(243\) −14.7893 −0.948732
\(244\) 0 0
\(245\) 7.46989 0.477234
\(246\) 0 0
\(247\) 1.31444 0.0836358
\(248\) 0 0
\(249\) −15.4375 −0.978311
\(250\) 0 0
\(251\) 28.9847 1.82950 0.914749 0.404024i \(-0.132389\pi\)
0.914749 + 0.404024i \(0.132389\pi\)
\(252\) 0 0
\(253\) −5.52404 −0.347293
\(254\) 0 0
\(255\) −5.54270 −0.347098
\(256\) 0 0
\(257\) 5.10203 0.318256 0.159128 0.987258i \(-0.449132\pi\)
0.159128 + 0.987258i \(0.449132\pi\)
\(258\) 0 0
\(259\) −0.970394 −0.0602974
\(260\) 0 0
\(261\) −14.7913 −0.915558
\(262\) 0 0
\(263\) 11.6953 0.721160 0.360580 0.932728i \(-0.382579\pi\)
0.360580 + 0.932728i \(0.382579\pi\)
\(264\) 0 0
\(265\) 8.99680 0.552669
\(266\) 0 0
\(267\) −10.6042 −0.648964
\(268\) 0 0
\(269\) −0.163433 −0.00996466 −0.00498233 0.999988i \(-0.501586\pi\)
−0.00498233 + 0.999988i \(0.501586\pi\)
\(270\) 0 0
\(271\) 1.01216 0.0614846 0.0307423 0.999527i \(-0.490213\pi\)
0.0307423 + 0.999527i \(0.490213\pi\)
\(272\) 0 0
\(273\) −0.361184 −0.0218599
\(274\) 0 0
\(275\) 10.9223 0.658637
\(276\) 0 0
\(277\) 3.14825 0.189160 0.0945800 0.995517i \(-0.469849\pi\)
0.0945800 + 0.995517i \(0.469849\pi\)
\(278\) 0 0
\(279\) 11.1256 0.666071
\(280\) 0 0
\(281\) 24.6734 1.47189 0.735946 0.677040i \(-0.236738\pi\)
0.735946 + 0.677040i \(0.236738\pi\)
\(282\) 0 0
\(283\) −13.8095 −0.820889 −0.410445 0.911886i \(-0.634626\pi\)
−0.410445 + 0.911886i \(0.634626\pi\)
\(284\) 0 0
\(285\) 1.23921 0.0734042
\(286\) 0 0
\(287\) −0.663713 −0.0391777
\(288\) 0 0
\(289\) 3.00584 0.176814
\(290\) 0 0
\(291\) 0.285072 0.0167112
\(292\) 0 0
\(293\) 8.24682 0.481784 0.240892 0.970552i \(-0.422560\pi\)
0.240892 + 0.970552i \(0.422560\pi\)
\(294\) 0 0
\(295\) −9.96169 −0.579992
\(296\) 0 0
\(297\) −15.3006 −0.887829
\(298\) 0 0
\(299\) 2.55445 0.147728
\(300\) 0 0
\(301\) −0.289530 −0.0166882
\(302\) 0 0
\(303\) 19.3535 1.11183
\(304\) 0 0
\(305\) 5.01396 0.287098
\(306\) 0 0
\(307\) 17.4540 0.996153 0.498076 0.867133i \(-0.334040\pi\)
0.498076 + 0.867133i \(0.334040\pi\)
\(308\) 0 0
\(309\) −6.23080 −0.354458
\(310\) 0 0
\(311\) 10.5256 0.596851 0.298425 0.954433i \(-0.403539\pi\)
0.298425 + 0.954433i \(0.403539\pi\)
\(312\) 0 0
\(313\) 14.3498 0.811098 0.405549 0.914073i \(-0.367080\pi\)
0.405549 + 0.914073i \(0.367080\pi\)
\(314\) 0 0
\(315\) 0.429486 0.0241988
\(316\) 0 0
\(317\) 16.3470 0.918138 0.459069 0.888401i \(-0.348183\pi\)
0.459069 + 0.888401i \(0.348183\pi\)
\(318\) 0 0
\(319\) −25.1260 −1.40679
\(320\) 0 0
\(321\) −7.36138 −0.410872
\(322\) 0 0
\(323\) −4.47279 −0.248873
\(324\) 0 0
\(325\) −5.05073 −0.280164
\(326\) 0 0
\(327\) −17.1366 −0.947656
\(328\) 0 0
\(329\) −2.44993 −0.135069
\(330\) 0 0
\(331\) −15.0118 −0.825123 −0.412562 0.910930i \(-0.635366\pi\)
−0.412562 + 0.910930i \(0.635366\pi\)
\(332\) 0 0
\(333\) 6.80649 0.372993
\(334\) 0 0
\(335\) −4.64615 −0.253846
\(336\) 0 0
\(337\) 0.462187 0.0251769 0.0125885 0.999921i \(-0.495993\pi\)
0.0125885 + 0.999921i \(0.495993\pi\)
\(338\) 0 0
\(339\) −9.89491 −0.537418
\(340\) 0 0
\(341\) 18.8991 1.02344
\(342\) 0 0
\(343\) −3.32633 −0.179605
\(344\) 0 0
\(345\) 2.40824 0.129655
\(346\) 0 0
\(347\) 17.2669 0.926935 0.463468 0.886114i \(-0.346605\pi\)
0.463468 + 0.886114i \(0.346605\pi\)
\(348\) 0 0
\(349\) −26.1466 −1.39960 −0.699798 0.714341i \(-0.746727\pi\)
−0.699798 + 0.714341i \(0.746727\pi\)
\(350\) 0 0
\(351\) 7.07536 0.377655
\(352\) 0 0
\(353\) 12.3930 0.659614 0.329807 0.944048i \(-0.393016\pi\)
0.329807 + 0.944048i \(0.393016\pi\)
\(354\) 0 0
\(355\) −2.99167 −0.158781
\(356\) 0 0
\(357\) 1.22904 0.0650478
\(358\) 0 0
\(359\) −14.1664 −0.747673 −0.373837 0.927495i \(-0.621958\pi\)
−0.373837 + 0.927495i \(0.621958\pi\)
\(360\) 0 0
\(361\) 1.00000 0.0526316
\(362\) 0 0
\(363\) 3.36356 0.176541
\(364\) 0 0
\(365\) 0.144662 0.00757195
\(366\) 0 0
\(367\) 3.00328 0.156770 0.0783850 0.996923i \(-0.475024\pi\)
0.0783850 + 0.996923i \(0.475024\pi\)
\(368\) 0 0
\(369\) 4.65538 0.242349
\(370\) 0 0
\(371\) −1.99495 −0.103573
\(372\) 0 0
\(373\) 38.0432 1.96980 0.984901 0.173119i \(-0.0553845\pi\)
0.984901 + 0.173119i \(0.0553845\pi\)
\(374\) 0 0
\(375\) −10.9577 −0.565852
\(376\) 0 0
\(377\) 11.6189 0.598404
\(378\) 0 0
\(379\) 28.6291 1.47058 0.735290 0.677753i \(-0.237046\pi\)
0.735290 + 0.677753i \(0.237046\pi\)
\(380\) 0 0
\(381\) 1.33591 0.0684407
\(382\) 0 0
\(383\) −24.1407 −1.23353 −0.616767 0.787146i \(-0.711558\pi\)
−0.616767 + 0.787146i \(0.711558\pi\)
\(384\) 0 0
\(385\) 0.729571 0.0371824
\(386\) 0 0
\(387\) 2.03081 0.103232
\(388\) 0 0
\(389\) −16.2953 −0.826206 −0.413103 0.910684i \(-0.635555\pi\)
−0.413103 + 0.910684i \(0.635555\pi\)
\(390\) 0 0
\(391\) −8.69231 −0.439589
\(392\) 0 0
\(393\) 4.44091 0.224014
\(394\) 0 0
\(395\) −13.7541 −0.692043
\(396\) 0 0
\(397\) −36.8139 −1.84764 −0.923819 0.382829i \(-0.874950\pi\)
−0.923819 + 0.382829i \(0.874950\pi\)
\(398\) 0 0
\(399\) −0.274782 −0.0137563
\(400\) 0 0
\(401\) −23.6899 −1.18302 −0.591508 0.806299i \(-0.701467\pi\)
−0.591508 + 0.806299i \(0.701467\pi\)
\(402\) 0 0
\(403\) −8.73941 −0.435341
\(404\) 0 0
\(405\) 1.26951 0.0630827
\(406\) 0 0
\(407\) 11.5622 0.573118
\(408\) 0 0
\(409\) −0.102363 −0.00506154 −0.00253077 0.999997i \(-0.500806\pi\)
−0.00253077 + 0.999997i \(0.500806\pi\)
\(410\) 0 0
\(411\) 1.64681 0.0812311
\(412\) 0 0
\(413\) 2.20891 0.108693
\(414\) 0 0
\(415\) −14.4197 −0.707836
\(416\) 0 0
\(417\) −4.12568 −0.202035
\(418\) 0 0
\(419\) 25.7326 1.25712 0.628560 0.777761i \(-0.283645\pi\)
0.628560 + 0.777761i \(0.283645\pi\)
\(420\) 0 0
\(421\) 18.0014 0.877334 0.438667 0.898650i \(-0.355451\pi\)
0.438667 + 0.898650i \(0.355451\pi\)
\(422\) 0 0
\(423\) 17.1842 0.835524
\(424\) 0 0
\(425\) 17.1867 0.833675
\(426\) 0 0
\(427\) −1.11180 −0.0538036
\(428\) 0 0
\(429\) 4.30350 0.207775
\(430\) 0 0
\(431\) −27.6392 −1.33133 −0.665667 0.746249i \(-0.731853\pi\)
−0.665667 + 0.746249i \(0.731853\pi\)
\(432\) 0 0
\(433\) 38.5219 1.85124 0.925621 0.378451i \(-0.123543\pi\)
0.925621 + 0.378451i \(0.123543\pi\)
\(434\) 0 0
\(435\) 10.9539 0.525198
\(436\) 0 0
\(437\) 1.94338 0.0929643
\(438\) 0 0
\(439\) 23.1302 1.10394 0.551971 0.833863i \(-0.313876\pi\)
0.551971 + 0.833863i \(0.313876\pi\)
\(440\) 0 0
\(441\) 11.6181 0.553241
\(442\) 0 0
\(443\) 19.2436 0.914290 0.457145 0.889392i \(-0.348872\pi\)
0.457145 + 0.889392i \(0.348872\pi\)
\(444\) 0 0
\(445\) −9.90505 −0.469544
\(446\) 0 0
\(447\) 18.7473 0.886716
\(448\) 0 0
\(449\) 11.0578 0.521851 0.260926 0.965359i \(-0.415972\pi\)
0.260926 + 0.965359i \(0.415972\pi\)
\(450\) 0 0
\(451\) 7.90812 0.372379
\(452\) 0 0
\(453\) 16.4988 0.775183
\(454\) 0 0
\(455\) −0.337372 −0.0158162
\(456\) 0 0
\(457\) −4.08310 −0.190999 −0.0954996 0.995429i \(-0.530445\pi\)
−0.0954996 + 0.995429i \(0.530445\pi\)
\(458\) 0 0
\(459\) −24.0761 −1.12378
\(460\) 0 0
\(461\) −7.59017 −0.353509 −0.176755 0.984255i \(-0.556560\pi\)
−0.176755 + 0.984255i \(0.556560\pi\)
\(462\) 0 0
\(463\) −37.2637 −1.73179 −0.865894 0.500227i \(-0.833250\pi\)
−0.865894 + 0.500227i \(0.833250\pi\)
\(464\) 0 0
\(465\) −8.23920 −0.382084
\(466\) 0 0
\(467\) −8.92480 −0.412991 −0.206495 0.978448i \(-0.566206\pi\)
−0.206495 + 0.978448i \(0.566206\pi\)
\(468\) 0 0
\(469\) 1.03024 0.0475720
\(470\) 0 0
\(471\) 4.30196 0.198224
\(472\) 0 0
\(473\) 3.44974 0.158619
\(474\) 0 0
\(475\) −3.84249 −0.176306
\(476\) 0 0
\(477\) 13.9929 0.640690
\(478\) 0 0
\(479\) 19.6241 0.896650 0.448325 0.893871i \(-0.352021\pi\)
0.448325 + 0.893871i \(0.352021\pi\)
\(480\) 0 0
\(481\) −5.34666 −0.243787
\(482\) 0 0
\(483\) −0.534005 −0.0242981
\(484\) 0 0
\(485\) 0.266277 0.0120910
\(486\) 0 0
\(487\) 13.5448 0.613775 0.306887 0.951746i \(-0.400712\pi\)
0.306887 + 0.951746i \(0.400712\pi\)
\(488\) 0 0
\(489\) 4.59413 0.207754
\(490\) 0 0
\(491\) −9.58255 −0.432454 −0.216227 0.976343i \(-0.569375\pi\)
−0.216227 + 0.976343i \(0.569375\pi\)
\(492\) 0 0
\(493\) −39.5369 −1.78065
\(494\) 0 0
\(495\) −5.11732 −0.230006
\(496\) 0 0
\(497\) 0.663374 0.0297564
\(498\) 0 0
\(499\) 7.24834 0.324480 0.162240 0.986751i \(-0.448128\pi\)
0.162240 + 0.986751i \(0.448128\pi\)
\(500\) 0 0
\(501\) −9.41213 −0.420503
\(502\) 0 0
\(503\) −2.19200 −0.0977367 −0.0488683 0.998805i \(-0.515561\pi\)
−0.0488683 + 0.998805i \(0.515561\pi\)
\(504\) 0 0
\(505\) 18.0775 0.804440
\(506\) 0 0
\(507\) 12.9835 0.576618
\(508\) 0 0
\(509\) −13.5765 −0.601766 −0.300883 0.953661i \(-0.597281\pi\)
−0.300883 + 0.953661i \(0.597281\pi\)
\(510\) 0 0
\(511\) −0.0320774 −0.00141902
\(512\) 0 0
\(513\) 5.38280 0.237656
\(514\) 0 0
\(515\) −5.82002 −0.256461
\(516\) 0 0
\(517\) 29.1909 1.28381
\(518\) 0 0
\(519\) 0.404696 0.0177642
\(520\) 0 0
\(521\) −6.57594 −0.288097 −0.144049 0.989571i \(-0.546012\pi\)
−0.144049 + 0.989571i \(0.546012\pi\)
\(522\) 0 0
\(523\) −22.2161 −0.971443 −0.485722 0.874114i \(-0.661443\pi\)
−0.485722 + 0.874114i \(0.661443\pi\)
\(524\) 0 0
\(525\) 1.05585 0.0460810
\(526\) 0 0
\(527\) 29.7386 1.29543
\(528\) 0 0
\(529\) −19.2233 −0.835795
\(530\) 0 0
\(531\) −15.4936 −0.672366
\(532\) 0 0
\(533\) −3.65691 −0.158398
\(534\) 0 0
\(535\) −6.87605 −0.297278
\(536\) 0 0
\(537\) −17.5928 −0.759185
\(538\) 0 0
\(539\) 19.7357 0.850076
\(540\) 0 0
\(541\) −22.5004 −0.967368 −0.483684 0.875243i \(-0.660702\pi\)
−0.483684 + 0.875243i \(0.660702\pi\)
\(542\) 0 0
\(543\) 10.3045 0.442207
\(544\) 0 0
\(545\) −16.0068 −0.685656
\(546\) 0 0
\(547\) −24.1551 −1.03280 −0.516399 0.856348i \(-0.672728\pi\)
−0.516399 + 0.856348i \(0.672728\pi\)
\(548\) 0 0
\(549\) 7.79830 0.332823
\(550\) 0 0
\(551\) 8.83944 0.376573
\(552\) 0 0
\(553\) 3.04984 0.129692
\(554\) 0 0
\(555\) −5.04063 −0.213963
\(556\) 0 0
\(557\) 21.4469 0.908734 0.454367 0.890815i \(-0.349865\pi\)
0.454367 + 0.890815i \(0.349865\pi\)
\(558\) 0 0
\(559\) −1.59525 −0.0674717
\(560\) 0 0
\(561\) −14.6440 −0.618270
\(562\) 0 0
\(563\) 15.3594 0.647322 0.323661 0.946173i \(-0.395086\pi\)
0.323661 + 0.946173i \(0.395086\pi\)
\(564\) 0 0
\(565\) −9.24256 −0.388837
\(566\) 0 0
\(567\) −0.281503 −0.0118220
\(568\) 0 0
\(569\) 2.05613 0.0861973 0.0430986 0.999071i \(-0.486277\pi\)
0.0430986 + 0.999071i \(0.486277\pi\)
\(570\) 0 0
\(571\) 35.4718 1.48445 0.742225 0.670151i \(-0.233771\pi\)
0.742225 + 0.670151i \(0.233771\pi\)
\(572\) 0 0
\(573\) −21.1871 −0.885103
\(574\) 0 0
\(575\) −7.46741 −0.311413
\(576\) 0 0
\(577\) 14.6910 0.611595 0.305798 0.952097i \(-0.401077\pi\)
0.305798 + 0.952097i \(0.401077\pi\)
\(578\) 0 0
\(579\) 5.87311 0.244078
\(580\) 0 0
\(581\) 3.19743 0.132652
\(582\) 0 0
\(583\) 23.7698 0.984445
\(584\) 0 0
\(585\) 2.36638 0.0978376
\(586\) 0 0
\(587\) 1.71339 0.0707193 0.0353596 0.999375i \(-0.488742\pi\)
0.0353596 + 0.999375i \(0.488742\pi\)
\(588\) 0 0
\(589\) −6.64877 −0.273958
\(590\) 0 0
\(591\) −23.3722 −0.961402
\(592\) 0 0
\(593\) 20.3947 0.837509 0.418754 0.908100i \(-0.362467\pi\)
0.418754 + 0.908100i \(0.362467\pi\)
\(594\) 0 0
\(595\) 1.14801 0.0470639
\(596\) 0 0
\(597\) −0.358212 −0.0146607
\(598\) 0 0
\(599\) −21.9447 −0.896637 −0.448318 0.893874i \(-0.647977\pi\)
−0.448318 + 0.893874i \(0.647977\pi\)
\(600\) 0 0
\(601\) −3.41894 −0.139461 −0.0697306 0.997566i \(-0.522214\pi\)
−0.0697306 + 0.997566i \(0.522214\pi\)
\(602\) 0 0
\(603\) −7.22625 −0.294276
\(604\) 0 0
\(605\) 3.14180 0.127732
\(606\) 0 0
\(607\) 23.1235 0.938555 0.469278 0.883051i \(-0.344514\pi\)
0.469278 + 0.883051i \(0.344514\pi\)
\(608\) 0 0
\(609\) −2.42892 −0.0984247
\(610\) 0 0
\(611\) −13.4986 −0.546095
\(612\) 0 0
\(613\) 46.3763 1.87312 0.936561 0.350504i \(-0.113990\pi\)
0.936561 + 0.350504i \(0.113990\pi\)
\(614\) 0 0
\(615\) −3.44760 −0.139021
\(616\) 0 0
\(617\) −25.2123 −1.01501 −0.507504 0.861649i \(-0.669432\pi\)
−0.507504 + 0.861649i \(0.669432\pi\)
\(618\) 0 0
\(619\) −37.2031 −1.49532 −0.747660 0.664082i \(-0.768822\pi\)
−0.747660 + 0.664082i \(0.768822\pi\)
\(620\) 0 0
\(621\) 10.4608 0.419778
\(622\) 0 0
\(623\) 2.19635 0.0879948
\(624\) 0 0
\(625\) 8.97722 0.359089
\(626\) 0 0
\(627\) 3.27402 0.130752
\(628\) 0 0
\(629\) 18.1937 0.725429
\(630\) 0 0
\(631\) −40.8412 −1.62586 −0.812931 0.582360i \(-0.802129\pi\)
−0.812931 + 0.582360i \(0.802129\pi\)
\(632\) 0 0
\(633\) −16.5036 −0.655959
\(634\) 0 0
\(635\) 1.24783 0.0495188
\(636\) 0 0
\(637\) −9.12627 −0.361596
\(638\) 0 0
\(639\) −4.65300 −0.184070
\(640\) 0 0
\(641\) 5.87446 0.232027 0.116014 0.993248i \(-0.462988\pi\)
0.116014 + 0.993248i \(0.462988\pi\)
\(642\) 0 0
\(643\) −21.3344 −0.841347 −0.420673 0.907212i \(-0.638206\pi\)
−0.420673 + 0.907212i \(0.638206\pi\)
\(644\) 0 0
\(645\) −1.50394 −0.0592176
\(646\) 0 0
\(647\) −48.1625 −1.89346 −0.946732 0.322024i \(-0.895637\pi\)
−0.946732 + 0.322024i \(0.895637\pi\)
\(648\) 0 0
\(649\) −26.3191 −1.03311
\(650\) 0 0
\(651\) 1.82696 0.0716043
\(652\) 0 0
\(653\) −11.3441 −0.443929 −0.221965 0.975055i \(-0.571247\pi\)
−0.221965 + 0.975055i \(0.571247\pi\)
\(654\) 0 0
\(655\) 4.14813 0.162081
\(656\) 0 0
\(657\) 0.224995 0.00877791
\(658\) 0 0
\(659\) 24.6320 0.959526 0.479763 0.877398i \(-0.340723\pi\)
0.479763 + 0.877398i \(0.340723\pi\)
\(660\) 0 0
\(661\) 47.3409 1.84135 0.920673 0.390335i \(-0.127641\pi\)
0.920673 + 0.390335i \(0.127641\pi\)
\(662\) 0 0
\(663\) 6.77175 0.262993
\(664\) 0 0
\(665\) −0.256666 −0.00995308
\(666\) 0 0
\(667\) 17.1784 0.665149
\(668\) 0 0
\(669\) 14.8148 0.572772
\(670\) 0 0
\(671\) 13.2470 0.511396
\(672\) 0 0
\(673\) −35.1767 −1.35596 −0.677981 0.735079i \(-0.737145\pi\)
−0.677981 + 0.735079i \(0.737145\pi\)
\(674\) 0 0
\(675\) −20.6834 −0.796103
\(676\) 0 0
\(677\) −2.71085 −0.104186 −0.0520932 0.998642i \(-0.516589\pi\)
−0.0520932 + 0.998642i \(0.516589\pi\)
\(678\) 0 0
\(679\) −0.0590444 −0.00226592
\(680\) 0 0
\(681\) −21.2499 −0.814300
\(682\) 0 0
\(683\) −48.4820 −1.85511 −0.927557 0.373682i \(-0.878095\pi\)
−0.927557 + 0.373682i \(0.878095\pi\)
\(684\) 0 0
\(685\) 1.53824 0.0587730
\(686\) 0 0
\(687\) −5.45128 −0.207979
\(688\) 0 0
\(689\) −10.9917 −0.418752
\(690\) 0 0
\(691\) 41.3295 1.57225 0.786124 0.618068i \(-0.212085\pi\)
0.786124 + 0.618068i \(0.212085\pi\)
\(692\) 0 0
\(693\) 1.13472 0.0431043
\(694\) 0 0
\(695\) −3.85368 −0.146178
\(696\) 0 0
\(697\) 12.4438 0.471342
\(698\) 0 0
\(699\) 4.92755 0.186377
\(700\) 0 0
\(701\) −15.9692 −0.603147 −0.301573 0.953443i \(-0.597512\pi\)
−0.301573 + 0.953443i \(0.597512\pi\)
\(702\) 0 0
\(703\) −4.06763 −0.153414
\(704\) 0 0
\(705\) −12.7260 −0.479288
\(706\) 0 0
\(707\) −4.00852 −0.150756
\(708\) 0 0
\(709\) 1.27532 0.0478957 0.0239478 0.999713i \(-0.492376\pi\)
0.0239478 + 0.999713i \(0.492376\pi\)
\(710\) 0 0
\(711\) −21.3920 −0.802263
\(712\) 0 0
\(713\) −12.9211 −0.483898
\(714\) 0 0
\(715\) 4.01978 0.150331
\(716\) 0 0
\(717\) −23.4860 −0.877100
\(718\) 0 0
\(719\) −38.6245 −1.44045 −0.720225 0.693740i \(-0.755962\pi\)
−0.720225 + 0.693740i \(0.755962\pi\)
\(720\) 0 0
\(721\) 1.29053 0.0480619
\(722\) 0 0
\(723\) −29.4722 −1.09608
\(724\) 0 0
\(725\) −33.9655 −1.26145
\(726\) 0 0
\(727\) −3.96945 −0.147219 −0.0736095 0.997287i \(-0.523452\pi\)
−0.0736095 + 0.997287i \(0.523452\pi\)
\(728\) 0 0
\(729\) 20.5744 0.762015
\(730\) 0 0
\(731\) 5.42832 0.200774
\(732\) 0 0
\(733\) −23.9821 −0.885800 −0.442900 0.896571i \(-0.646050\pi\)
−0.442900 + 0.896571i \(0.646050\pi\)
\(734\) 0 0
\(735\) −8.60391 −0.317360
\(736\) 0 0
\(737\) −12.2753 −0.452165
\(738\) 0 0
\(739\) −34.2553 −1.26010 −0.630051 0.776554i \(-0.716966\pi\)
−0.630051 + 0.776554i \(0.716966\pi\)
\(740\) 0 0
\(741\) −1.51399 −0.0556177
\(742\) 0 0
\(743\) 31.0058 1.13749 0.568746 0.822513i \(-0.307429\pi\)
0.568746 + 0.822513i \(0.307429\pi\)
\(744\) 0 0
\(745\) 17.5113 0.641564
\(746\) 0 0
\(747\) −22.4273 −0.820571
\(748\) 0 0
\(749\) 1.52470 0.0557112
\(750\) 0 0
\(751\) 36.0417 1.31518 0.657590 0.753376i \(-0.271576\pi\)
0.657590 + 0.753376i \(0.271576\pi\)
\(752\) 0 0
\(753\) −33.3849 −1.21661
\(754\) 0 0
\(755\) 15.4111 0.560867
\(756\) 0 0
\(757\) −26.6253 −0.967712 −0.483856 0.875148i \(-0.660764\pi\)
−0.483856 + 0.875148i \(0.660764\pi\)
\(758\) 0 0
\(759\) 6.36265 0.230950
\(760\) 0 0
\(761\) −40.0064 −1.45023 −0.725115 0.688628i \(-0.758213\pi\)
−0.725115 + 0.688628i \(0.758213\pi\)
\(762\) 0 0
\(763\) 3.54935 0.128495
\(764\) 0 0
\(765\) −8.05233 −0.291133
\(766\) 0 0
\(767\) 12.1706 0.439455
\(768\) 0 0
\(769\) 19.1147 0.689295 0.344648 0.938732i \(-0.387998\pi\)
0.344648 + 0.938732i \(0.387998\pi\)
\(770\) 0 0
\(771\) −5.87658 −0.211640
\(772\) 0 0
\(773\) −42.2244 −1.51871 −0.759353 0.650679i \(-0.774484\pi\)
−0.759353 + 0.650679i \(0.774484\pi\)
\(774\) 0 0
\(775\) 25.5479 0.917706
\(776\) 0 0
\(777\) 1.11771 0.0400977
\(778\) 0 0
\(779\) −2.78211 −0.0996793
\(780\) 0 0
\(781\) −7.90408 −0.282830
\(782\) 0 0
\(783\) 47.5809 1.70040
\(784\) 0 0
\(785\) 4.01833 0.143421
\(786\) 0 0
\(787\) −43.6009 −1.55420 −0.777102 0.629375i \(-0.783311\pi\)
−0.777102 + 0.629375i \(0.783311\pi\)
\(788\) 0 0
\(789\) −13.4707 −0.479571
\(790\) 0 0
\(791\) 2.04945 0.0728699
\(792\) 0 0
\(793\) −6.12575 −0.217532
\(794\) 0 0
\(795\) −10.3626 −0.367524
\(796\) 0 0
\(797\) 14.6901 0.520351 0.260176 0.965561i \(-0.416220\pi\)
0.260176 + 0.965561i \(0.416220\pi\)
\(798\) 0 0
\(799\) 45.9332 1.62500
\(800\) 0 0
\(801\) −15.4055 −0.544327
\(802\) 0 0
\(803\) 0.382201 0.0134876
\(804\) 0 0
\(805\) −0.498799 −0.0175803
\(806\) 0 0
\(807\) 0.188244 0.00662649
\(808\) 0 0
\(809\) −4.76004 −0.167354 −0.0836771 0.996493i \(-0.526666\pi\)
−0.0836771 + 0.996493i \(0.526666\pi\)
\(810\) 0 0
\(811\) −14.6172 −0.513279 −0.256640 0.966507i \(-0.582615\pi\)
−0.256640 + 0.966507i \(0.582615\pi\)
\(812\) 0 0
\(813\) −1.16582 −0.0408872
\(814\) 0 0
\(815\) 4.29125 0.150316
\(816\) 0 0
\(817\) −1.21363 −0.0424596
\(818\) 0 0
\(819\) −0.524721 −0.0183352
\(820\) 0 0
\(821\) 32.3079 1.12755 0.563776 0.825928i \(-0.309348\pi\)
0.563776 + 0.825928i \(0.309348\pi\)
\(822\) 0 0
\(823\) −9.79982 −0.341600 −0.170800 0.985306i \(-0.554635\pi\)
−0.170800 + 0.985306i \(0.554635\pi\)
\(824\) 0 0
\(825\) −12.5804 −0.437993
\(826\) 0 0
\(827\) −11.2066 −0.389691 −0.194845 0.980834i \(-0.562421\pi\)
−0.194845 + 0.980834i \(0.562421\pi\)
\(828\) 0 0
\(829\) 45.4323 1.57793 0.788965 0.614438i \(-0.210617\pi\)
0.788965 + 0.614438i \(0.210617\pi\)
\(830\) 0 0
\(831\) −3.62619 −0.125791
\(832\) 0 0
\(833\) 31.0550 1.07599
\(834\) 0 0
\(835\) −8.79160 −0.304246
\(836\) 0 0
\(837\) −35.7890 −1.23705
\(838\) 0 0
\(839\) −16.8997 −0.583441 −0.291721 0.956504i \(-0.594228\pi\)
−0.291721 + 0.956504i \(0.594228\pi\)
\(840\) 0 0
\(841\) 49.1356 1.69433
\(842\) 0 0
\(843\) −28.4191 −0.978807
\(844\) 0 0
\(845\) 12.1275 0.417200
\(846\) 0 0
\(847\) −0.696664 −0.0239377
\(848\) 0 0
\(849\) 15.9059 0.545890
\(850\) 0 0
\(851\) −7.90494 −0.270978
\(852\) 0 0
\(853\) 37.7943 1.29405 0.647027 0.762467i \(-0.276012\pi\)
0.647027 + 0.762467i \(0.276012\pi\)
\(854\) 0 0
\(855\) 1.80029 0.0615687
\(856\) 0 0
\(857\) −38.8619 −1.32750 −0.663749 0.747956i \(-0.731036\pi\)
−0.663749 + 0.747956i \(0.731036\pi\)
\(858\) 0 0
\(859\) −41.1795 −1.40503 −0.702513 0.711671i \(-0.747939\pi\)
−0.702513 + 0.711671i \(0.747939\pi\)
\(860\) 0 0
\(861\) 0.764473 0.0260532
\(862\) 0 0
\(863\) −25.5909 −0.871125 −0.435562 0.900159i \(-0.643450\pi\)
−0.435562 + 0.900159i \(0.643450\pi\)
\(864\) 0 0
\(865\) 0.378015 0.0128529
\(866\) 0 0
\(867\) −3.46216 −0.117581
\(868\) 0 0
\(869\) −36.3387 −1.23271
\(870\) 0 0
\(871\) 5.67639 0.192337
\(872\) 0 0
\(873\) 0.414146 0.0140167
\(874\) 0 0
\(875\) 2.26957 0.0767253
\(876\) 0 0
\(877\) −14.4790 −0.488920 −0.244460 0.969659i \(-0.578611\pi\)
−0.244460 + 0.969659i \(0.578611\pi\)
\(878\) 0 0
\(879\) −9.49879 −0.320386
\(880\) 0 0
\(881\) 48.3407 1.62864 0.814320 0.580416i \(-0.197110\pi\)
0.814320 + 0.580416i \(0.197110\pi\)
\(882\) 0 0
\(883\) −17.2669 −0.581077 −0.290539 0.956863i \(-0.593834\pi\)
−0.290539 + 0.956863i \(0.593834\pi\)
\(884\) 0 0
\(885\) 11.4740 0.385694
\(886\) 0 0
\(887\) 51.0650 1.71460 0.857298 0.514820i \(-0.172141\pi\)
0.857298 + 0.514820i \(0.172141\pi\)
\(888\) 0 0
\(889\) −0.276695 −0.00928006
\(890\) 0 0
\(891\) 3.35410 0.112366
\(892\) 0 0
\(893\) −10.2695 −0.343655
\(894\) 0 0
\(895\) −16.4329 −0.549292
\(896\) 0 0
\(897\) −2.94225 −0.0982388
\(898\) 0 0
\(899\) −58.7714 −1.96014
\(900\) 0 0
\(901\) 37.4028 1.24607
\(902\) 0 0
\(903\) 0.333484 0.0110977
\(904\) 0 0
\(905\) 9.62510 0.319949
\(906\) 0 0
\(907\) −30.0974 −0.999367 −0.499684 0.866208i \(-0.666550\pi\)
−0.499684 + 0.866208i \(0.666550\pi\)
\(908\) 0 0
\(909\) 28.1164 0.932561
\(910\) 0 0
\(911\) −9.15413 −0.303290 −0.151645 0.988435i \(-0.548457\pi\)
−0.151645 + 0.988435i \(0.548457\pi\)
\(912\) 0 0
\(913\) −38.0973 −1.26084
\(914\) 0 0
\(915\) −5.77514 −0.190920
\(916\) 0 0
\(917\) −0.919807 −0.0303747
\(918\) 0 0
\(919\) −7.49270 −0.247161 −0.123581 0.992335i \(-0.539438\pi\)
−0.123581 + 0.992335i \(0.539438\pi\)
\(920\) 0 0
\(921\) −20.1037 −0.662440
\(922\) 0 0
\(923\) 3.65505 0.120307
\(924\) 0 0
\(925\) 15.6299 0.513906
\(926\) 0 0
\(927\) −9.05198 −0.297306
\(928\) 0 0
\(929\) −12.0792 −0.396306 −0.198153 0.980171i \(-0.563494\pi\)
−0.198153 + 0.980171i \(0.563494\pi\)
\(930\) 0 0
\(931\) −6.94309 −0.227550
\(932\) 0 0
\(933\) −12.1235 −0.396905
\(934\) 0 0
\(935\) −13.6785 −0.447336
\(936\) 0 0
\(937\) 16.6732 0.544690 0.272345 0.962200i \(-0.412201\pi\)
0.272345 + 0.962200i \(0.412201\pi\)
\(938\) 0 0
\(939\) −16.5283 −0.539380
\(940\) 0 0
\(941\) 29.5263 0.962529 0.481265 0.876575i \(-0.340178\pi\)
0.481265 + 0.876575i \(0.340178\pi\)
\(942\) 0 0
\(943\) −5.40668 −0.176066
\(944\) 0 0
\(945\) −1.38158 −0.0449428
\(946\) 0 0
\(947\) 24.2099 0.786717 0.393358 0.919385i \(-0.371313\pi\)
0.393358 + 0.919385i \(0.371313\pi\)
\(948\) 0 0
\(949\) −0.176739 −0.00573720
\(950\) 0 0
\(951\) −18.8287 −0.610561
\(952\) 0 0
\(953\) 18.5304 0.600258 0.300129 0.953899i \(-0.402970\pi\)
0.300129 + 0.953899i \(0.402970\pi\)
\(954\) 0 0
\(955\) −19.7903 −0.640398
\(956\) 0 0
\(957\) 28.9405 0.935513
\(958\) 0 0
\(959\) −0.341089 −0.0110143
\(960\) 0 0
\(961\) 13.2062 0.426005
\(962\) 0 0
\(963\) −10.6945 −0.344624
\(964\) 0 0
\(965\) 5.48590 0.176597
\(966\) 0 0
\(967\) 54.2821 1.74559 0.872797 0.488084i \(-0.162304\pi\)
0.872797 + 0.488084i \(0.162304\pi\)
\(968\) 0 0
\(969\) 5.15181 0.165500
\(970\) 0 0
\(971\) 7.69284 0.246875 0.123437 0.992352i \(-0.460608\pi\)
0.123437 + 0.992352i \(0.460608\pi\)
\(972\) 0 0
\(973\) 0.854516 0.0273945
\(974\) 0 0
\(975\) 5.81749 0.186309
\(976\) 0 0
\(977\) −32.8587 −1.05124 −0.525621 0.850719i \(-0.676167\pi\)
−0.525621 + 0.850719i \(0.676167\pi\)
\(978\) 0 0
\(979\) −26.1694 −0.836378
\(980\) 0 0
\(981\) −24.8957 −0.794858
\(982\) 0 0
\(983\) −10.4144 −0.332169 −0.166084 0.986112i \(-0.553112\pi\)
−0.166084 + 0.986112i \(0.553112\pi\)
\(984\) 0 0
\(985\) −21.8313 −0.695602
\(986\) 0 0
\(987\) 2.82186 0.0898209
\(988\) 0 0
\(989\) −2.35855 −0.0749974
\(990\) 0 0
\(991\) 62.1164 1.97319 0.986596 0.163183i \(-0.0521761\pi\)
0.986596 + 0.163183i \(0.0521761\pi\)
\(992\) 0 0
\(993\) 17.2908 0.548706
\(994\) 0 0
\(995\) −0.334596 −0.0106074
\(996\) 0 0
\(997\) −5.60362 −0.177468 −0.0887341 0.996055i \(-0.528282\pi\)
−0.0887341 + 0.996055i \(0.528282\pi\)
\(998\) 0 0
\(999\) −21.8952 −0.692735
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 4864.2.a.bt.1.3 10
4.3 odd 2 4864.2.a.bs.1.7 10
8.3 odd 2 inner 4864.2.a.bt.1.4 10
8.5 even 2 4864.2.a.bs.1.8 10
16.3 odd 4 2432.2.c.j.1217.14 yes 20
16.5 even 4 2432.2.c.j.1217.13 yes 20
16.11 odd 4 2432.2.c.j.1217.7 20
16.13 even 4 2432.2.c.j.1217.8 yes 20
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2432.2.c.j.1217.7 20 16.11 odd 4
2432.2.c.j.1217.8 yes 20 16.13 even 4
2432.2.c.j.1217.13 yes 20 16.5 even 4
2432.2.c.j.1217.14 yes 20 16.3 odd 4
4864.2.a.bs.1.7 10 4.3 odd 2
4864.2.a.bs.1.8 10 8.5 even 2
4864.2.a.bt.1.3 10 1.1 even 1 trivial
4864.2.a.bt.1.4 10 8.3 odd 2 inner