Properties

Label 5625.2.a.z.1.7
Level $5625$
Weight $2$
Character 5625.1
Self dual yes
Analytic conductor $44.916$
Analytic rank $1$
Dimension $8$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 5625 = 3^{2} \cdot 5^{4} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5625.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(44.9158511370\)
Analytic rank: \(1\)
Dimension: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
Defining polynomial: \( x^{8} - 15x^{6} + 70x^{4} - 105x^{2} + 45 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.7
Root \(2.37653\) of defining polynomial
Character \(\chi\) \(=\) 5625.1

$q$-expansion

\(f(q)\) \(=\) \(q+2.37653 q^{2} +3.64791 q^{4} -4.26594 q^{7} +3.91630 q^{8} +O(q^{10})\) \(q+2.37653 q^{2} +3.64791 q^{4} -4.26594 q^{7} +3.91630 q^{8} +2.49140 q^{11} -2.02987 q^{13} -10.1381 q^{14} +2.01141 q^{16} -7.24447 q^{17} +3.26594 q^{19} +5.92090 q^{22} +6.15085 q^{23} -4.82406 q^{26} -15.5618 q^{28} -0.951631 q^{29} +2.66637 q^{31} -3.05243 q^{32} -17.2167 q^{34} -9.66637 q^{37} +7.76161 q^{38} -12.1679 q^{41} -7.95077 q^{43} +9.08840 q^{44} +14.6177 q^{46} +2.93756 q^{47} +11.1983 q^{49} -7.40479 q^{52} -12.4437 q^{53} -16.7067 q^{56} -2.26158 q^{58} +8.13677 q^{59} -3.33363 q^{61} +6.33671 q^{62} -11.2770 q^{64} +11.2352 q^{67} -26.4271 q^{68} -9.67655 q^{71} -8.24312 q^{73} -22.9724 q^{74} +11.9138 q^{76} -10.6282 q^{77} +7.65496 q^{79} -28.9175 q^{82} -10.3240 q^{83} -18.8953 q^{86} +9.75709 q^{88} -0.997628 q^{89} +8.65932 q^{91} +22.4377 q^{92} +6.98120 q^{94} +5.31428 q^{97} +26.6130 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 14 q^{4} - 10 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 14 q^{4} - 10 q^{7} - 10 q^{13} + 22 q^{16} + 2 q^{19} - 10 q^{22} - 70 q^{28} - 6 q^{31} - 50 q^{34} - 50 q^{37} + 14 q^{49} - 80 q^{52} + 30 q^{58} - 54 q^{61} + 36 q^{64} - 10 q^{67} - 30 q^{73} + 56 q^{76} + 28 q^{79} - 20 q^{88} + 60 q^{91} - 40 q^{94}+O(q^{100}) \) Copy content Toggle raw display

Coefficient data

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

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.37653 1.68046 0.840231 0.542228i \(-0.182419\pi\)
0.840231 + 0.542228i \(0.182419\pi\)
\(3\) 0 0
\(4\) 3.64791 1.82395
\(5\) 0 0
\(6\) 0 0
\(7\) −4.26594 −1.61237 −0.806187 0.591661i \(-0.798472\pi\)
−0.806187 + 0.591661i \(0.798472\pi\)
\(8\) 3.91630 1.38462
\(9\) 0 0
\(10\) 0 0
\(11\) 2.49140 0.751186 0.375593 0.926785i \(-0.377439\pi\)
0.375593 + 0.926785i \(0.377439\pi\)
\(12\) 0 0
\(13\) −2.02987 −0.562985 −0.281493 0.959563i \(-0.590830\pi\)
−0.281493 + 0.959563i \(0.590830\pi\)
\(14\) −10.1381 −2.70953
\(15\) 0 0
\(16\) 2.01141 0.502853
\(17\) −7.24447 −1.75704 −0.878521 0.477704i \(-0.841469\pi\)
−0.878521 + 0.477704i \(0.841469\pi\)
\(18\) 0 0
\(19\) 3.26594 0.749258 0.374629 0.927175i \(-0.377770\pi\)
0.374629 + 0.927175i \(0.377770\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 5.92090 1.26234
\(23\) 6.15085 1.28254 0.641270 0.767315i \(-0.278408\pi\)
0.641270 + 0.767315i \(0.278408\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) −4.82406 −0.946076
\(27\) 0 0
\(28\) −15.5618 −2.94090
\(29\) −0.951631 −0.176713 −0.0883567 0.996089i \(-0.528162\pi\)
−0.0883567 + 0.996089i \(0.528162\pi\)
\(30\) 0 0
\(31\) 2.66637 0.478894 0.239447 0.970909i \(-0.423034\pi\)
0.239447 + 0.970909i \(0.423034\pi\)
\(32\) −3.05243 −0.539598
\(33\) 0 0
\(34\) −17.2167 −2.95264
\(35\) 0 0
\(36\) 0 0
\(37\) −9.66637 −1.58914 −0.794571 0.607172i \(-0.792304\pi\)
−0.794571 + 0.607172i \(0.792304\pi\)
\(38\) 7.76161 1.25910
\(39\) 0 0
\(40\) 0 0
\(41\) −12.1679 −1.90031 −0.950157 0.311771i \(-0.899078\pi\)
−0.950157 + 0.311771i \(0.899078\pi\)
\(42\) 0 0
\(43\) −7.95077 −1.21248 −0.606241 0.795281i \(-0.707323\pi\)
−0.606241 + 0.795281i \(0.707323\pi\)
\(44\) 9.08840 1.37013
\(45\) 0 0
\(46\) 14.6177 2.15526
\(47\) 2.93756 0.428487 0.214243 0.976780i \(-0.431271\pi\)
0.214243 + 0.976780i \(0.431271\pi\)
\(48\) 0 0
\(49\) 11.1983 1.59975
\(50\) 0 0
\(51\) 0 0
\(52\) −7.40479 −1.02686
\(53\) −12.4437 −1.70927 −0.854636 0.519228i \(-0.826220\pi\)
−0.854636 + 0.519228i \(0.826220\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) −16.7067 −2.23253
\(57\) 0 0
\(58\) −2.26158 −0.296960
\(59\) 8.13677 1.05932 0.529659 0.848211i \(-0.322320\pi\)
0.529659 + 0.848211i \(0.322320\pi\)
\(60\) 0 0
\(61\) −3.33363 −0.426828 −0.213414 0.976962i \(-0.568458\pi\)
−0.213414 + 0.976962i \(0.568458\pi\)
\(62\) 6.33671 0.804763
\(63\) 0 0
\(64\) −11.2770 −1.40963
\(65\) 0 0
\(66\) 0 0
\(67\) 11.2352 1.37260 0.686298 0.727321i \(-0.259235\pi\)
0.686298 + 0.727321i \(0.259235\pi\)
\(68\) −26.4271 −3.20476
\(69\) 0 0
\(70\) 0 0
\(71\) −9.67655 −1.14839 −0.574197 0.818717i \(-0.694686\pi\)
−0.574197 + 0.818717i \(0.694686\pi\)
\(72\) 0 0
\(73\) −8.24312 −0.964784 −0.482392 0.875955i \(-0.660232\pi\)
−0.482392 + 0.875955i \(0.660232\pi\)
\(74\) −22.9724 −2.67049
\(75\) 0 0
\(76\) 11.9138 1.36661
\(77\) −10.6282 −1.21119
\(78\) 0 0
\(79\) 7.65496 0.861250 0.430625 0.902531i \(-0.358293\pi\)
0.430625 + 0.902531i \(0.358293\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) −28.9175 −3.19341
\(83\) −10.3240 −1.13321 −0.566604 0.823990i \(-0.691743\pi\)
−0.566604 + 0.823990i \(0.691743\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) −18.8953 −2.03753
\(87\) 0 0
\(88\) 9.75709 1.04011
\(89\) −0.997628 −0.105748 −0.0528742 0.998601i \(-0.516838\pi\)
−0.0528742 + 0.998601i \(0.516838\pi\)
\(90\) 0 0
\(91\) 8.65932 0.907743
\(92\) 22.4377 2.33929
\(93\) 0 0
\(94\) 6.98120 0.720055
\(95\) 0 0
\(96\) 0 0
\(97\) 5.31428 0.539583 0.269791 0.962919i \(-0.413045\pi\)
0.269791 + 0.962919i \(0.413045\pi\)
\(98\) 26.6130 2.68832
\(99\) 0 0
\(100\) 0 0
\(101\) −12.5774 −1.25150 −0.625751 0.780023i \(-0.715207\pi\)
−0.625751 + 0.780023i \(0.715207\pi\)
\(102\) 0 0
\(103\) −12.4423 −1.22597 −0.612986 0.790094i \(-0.710032\pi\)
−0.612986 + 0.790094i \(0.710032\pi\)
\(104\) −7.94960 −0.779522
\(105\) 0 0
\(106\) −29.5728 −2.87237
\(107\) 0.0593251 0.00573518 0.00286759 0.999996i \(-0.499087\pi\)
0.00286759 + 0.999996i \(0.499087\pi\)
\(108\) 0 0
\(109\) 1.05325 0.100883 0.0504413 0.998727i \(-0.483937\pi\)
0.0504413 + 0.998727i \(0.483937\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) −8.58056 −0.810786
\(113\) −0.817881 −0.0769398 −0.0384699 0.999260i \(-0.512248\pi\)
−0.0384699 + 0.999260i \(0.512248\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) −3.47146 −0.322317
\(117\) 0 0
\(118\) 19.3373 1.78014
\(119\) 30.9045 2.83301
\(120\) 0 0
\(121\) −4.79291 −0.435719
\(122\) −7.92248 −0.717268
\(123\) 0 0
\(124\) 9.72667 0.873480
\(125\) 0 0
\(126\) 0 0
\(127\) 2.39739 0.212734 0.106367 0.994327i \(-0.466078\pi\)
0.106367 + 0.994327i \(0.466078\pi\)
\(128\) −20.6953 −1.82923
\(129\) 0 0
\(130\) 0 0
\(131\) 13.7077 1.19765 0.598825 0.800880i \(-0.295635\pi\)
0.598825 + 0.800880i \(0.295635\pi\)
\(132\) 0 0
\(133\) −13.9323 −1.20808
\(134\) 26.7008 2.30659
\(135\) 0 0
\(136\) −28.3715 −2.43284
\(137\) −3.44303 −0.294158 −0.147079 0.989125i \(-0.546987\pi\)
−0.147079 + 0.989125i \(0.546987\pi\)
\(138\) 0 0
\(139\) −1.97414 −0.167445 −0.0837224 0.996489i \(-0.526681\pi\)
−0.0837224 + 0.996489i \(0.526681\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) −22.9966 −1.92983
\(143\) −5.05723 −0.422907
\(144\) 0 0
\(145\) 0 0
\(146\) −19.5900 −1.62128
\(147\) 0 0
\(148\) −35.2620 −2.89852
\(149\) 6.52258 0.534350 0.267175 0.963648i \(-0.413910\pi\)
0.267175 + 0.963648i \(0.413910\pi\)
\(150\) 0 0
\(151\) −8.11513 −0.660400 −0.330200 0.943911i \(-0.607116\pi\)
−0.330200 + 0.943911i \(0.607116\pi\)
\(152\) 12.7904 1.03744
\(153\) 0 0
\(154\) −25.2582 −2.03536
\(155\) 0 0
\(156\) 0 0
\(157\) 16.0545 1.28129 0.640644 0.767838i \(-0.278667\pi\)
0.640644 + 0.767838i \(0.278667\pi\)
\(158\) 18.1923 1.44730
\(159\) 0 0
\(160\) 0 0
\(161\) −26.2392 −2.06794
\(162\) 0 0
\(163\) 5.78730 0.453297 0.226648 0.973977i \(-0.427223\pi\)
0.226648 + 0.973977i \(0.427223\pi\)
\(164\) −44.3875 −3.46608
\(165\) 0 0
\(166\) −24.5353 −1.90431
\(167\) −12.1313 −0.938747 −0.469373 0.883000i \(-0.655520\pi\)
−0.469373 + 0.883000i \(0.655520\pi\)
\(168\) 0 0
\(169\) −8.87962 −0.683047
\(170\) 0 0
\(171\) 0 0
\(172\) −29.0037 −2.21151
\(173\) 13.7537 1.04568 0.522838 0.852432i \(-0.324873\pi\)
0.522838 + 0.852432i \(0.324873\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 5.01123 0.377736
\(177\) 0 0
\(178\) −2.37090 −0.177706
\(179\) 16.6086 1.24139 0.620693 0.784054i \(-0.286851\pi\)
0.620693 + 0.784054i \(0.286851\pi\)
\(180\) 0 0
\(181\) 5.78586 0.430060 0.215030 0.976607i \(-0.431015\pi\)
0.215030 + 0.976607i \(0.431015\pi\)
\(182\) 20.5791 1.52543
\(183\) 0 0
\(184\) 24.0886 1.77583
\(185\) 0 0
\(186\) 0 0
\(187\) −18.0489 −1.31987
\(188\) 10.7159 0.781539
\(189\) 0 0
\(190\) 0 0
\(191\) 15.5826 1.12751 0.563757 0.825941i \(-0.309355\pi\)
0.563757 + 0.825941i \(0.309355\pi\)
\(192\) 0 0
\(193\) 15.3675 1.10618 0.553089 0.833122i \(-0.313449\pi\)
0.553089 + 0.833122i \(0.313449\pi\)
\(194\) 12.6295 0.906749
\(195\) 0 0
\(196\) 40.8502 2.91787
\(197\) 6.50925 0.463765 0.231882 0.972744i \(-0.425512\pi\)
0.231882 + 0.972744i \(0.425512\pi\)
\(198\) 0 0
\(199\) −15.9970 −1.13399 −0.566997 0.823720i \(-0.691895\pi\)
−0.566997 + 0.823720i \(0.691895\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) −29.8907 −2.10310
\(203\) 4.05960 0.284928
\(204\) 0 0
\(205\) 0 0
\(206\) −29.5694 −2.06020
\(207\) 0 0
\(208\) −4.08291 −0.283099
\(209\) 8.13677 0.562832
\(210\) 0 0
\(211\) −15.7153 −1.08188 −0.540941 0.841060i \(-0.681932\pi\)
−0.540941 + 0.841060i \(0.681932\pi\)
\(212\) −45.3934 −3.11763
\(213\) 0 0
\(214\) 0.140988 0.00963775
\(215\) 0 0
\(216\) 0 0
\(217\) −11.3746 −0.772156
\(218\) 2.50307 0.169529
\(219\) 0 0
\(220\) 0 0
\(221\) 14.7053 0.989189
\(222\) 0 0
\(223\) 14.7597 0.988380 0.494190 0.869354i \(-0.335465\pi\)
0.494190 + 0.869354i \(0.335465\pi\)
\(224\) 13.0215 0.870033
\(225\) 0 0
\(226\) −1.94372 −0.129294
\(227\) 7.24447 0.480832 0.240416 0.970670i \(-0.422716\pi\)
0.240416 + 0.970670i \(0.422716\pi\)
\(228\) 0 0
\(229\) −3.39739 −0.224506 −0.112253 0.993680i \(-0.535807\pi\)
−0.112253 + 0.993680i \(0.535807\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) −3.72688 −0.244681
\(233\) 21.7701 1.42620 0.713102 0.701060i \(-0.247289\pi\)
0.713102 + 0.701060i \(0.247289\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 29.6822 1.93215
\(237\) 0 0
\(238\) 73.4455 4.76076
\(239\) −8.10835 −0.524485 −0.262243 0.965002i \(-0.584462\pi\)
−0.262243 + 0.965002i \(0.584462\pi\)
\(240\) 0 0
\(241\) 20.0963 1.29452 0.647259 0.762270i \(-0.275915\pi\)
0.647259 + 0.762270i \(0.275915\pi\)
\(242\) −11.3905 −0.732210
\(243\) 0 0
\(244\) −12.1608 −0.778514
\(245\) 0 0
\(246\) 0 0
\(247\) −6.62944 −0.421821
\(248\) 10.4423 0.663088
\(249\) 0 0
\(250\) 0 0
\(251\) −20.2303 −1.27693 −0.638463 0.769653i \(-0.720429\pi\)
−0.638463 + 0.769653i \(0.720429\pi\)
\(252\) 0 0
\(253\) 15.3242 0.963427
\(254\) 5.69748 0.357492
\(255\) 0 0
\(256\) −26.6291 −1.66432
\(257\) −1.32336 −0.0825489 −0.0412744 0.999148i \(-0.513142\pi\)
−0.0412744 + 0.999148i \(0.513142\pi\)
\(258\) 0 0
\(259\) 41.2362 2.56229
\(260\) 0 0
\(261\) 0 0
\(262\) 32.5768 2.01260
\(263\) 23.8613 1.47135 0.735676 0.677334i \(-0.236865\pi\)
0.735676 + 0.677334i \(0.236865\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) −33.1106 −2.03014
\(267\) 0 0
\(268\) 40.9849 2.50355
\(269\) −6.49415 −0.395955 −0.197978 0.980207i \(-0.563437\pi\)
−0.197978 + 0.980207i \(0.563437\pi\)
\(270\) 0 0
\(271\) −9.50817 −0.577580 −0.288790 0.957392i \(-0.593253\pi\)
−0.288790 + 0.957392i \(0.593253\pi\)
\(272\) −14.5716 −0.883533
\(273\) 0 0
\(274\) −8.18248 −0.494322
\(275\) 0 0
\(276\) 0 0
\(277\) 9.15047 0.549798 0.274899 0.961473i \(-0.411356\pi\)
0.274899 + 0.961473i \(0.411356\pi\)
\(278\) −4.69162 −0.281385
\(279\) 0 0
\(280\) 0 0
\(281\) 17.1968 1.02587 0.512936 0.858427i \(-0.328558\pi\)
0.512936 + 0.858427i \(0.328558\pi\)
\(282\) 0 0
\(283\) 22.5768 1.34205 0.671027 0.741433i \(-0.265853\pi\)
0.671027 + 0.741433i \(0.265853\pi\)
\(284\) −35.2991 −2.09462
\(285\) 0 0
\(286\) −12.0187 −0.710679
\(287\) 51.9077 3.06402
\(288\) 0 0
\(289\) 35.4823 2.08719
\(290\) 0 0
\(291\) 0 0
\(292\) −30.0701 −1.75972
\(293\) −21.3523 −1.24742 −0.623709 0.781657i \(-0.714375\pi\)
−0.623709 + 0.781657i \(0.714375\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) −37.8564 −2.20036
\(297\) 0 0
\(298\) 15.5011 0.897956
\(299\) −12.4854 −0.722052
\(300\) 0 0
\(301\) 33.9175 1.95497
\(302\) −19.2859 −1.10978
\(303\) 0 0
\(304\) 6.56915 0.376766
\(305\) 0 0
\(306\) 0 0
\(307\) 8.43377 0.481341 0.240670 0.970607i \(-0.422633\pi\)
0.240670 + 0.970607i \(0.422633\pi\)
\(308\) −38.7706 −2.20916
\(309\) 0 0
\(310\) 0 0
\(311\) −9.13440 −0.517964 −0.258982 0.965882i \(-0.583387\pi\)
−0.258982 + 0.965882i \(0.583387\pi\)
\(312\) 0 0
\(313\) −20.9683 −1.18520 −0.592600 0.805497i \(-0.701899\pi\)
−0.592600 + 0.805497i \(0.701899\pi\)
\(314\) 38.1540 2.15316
\(315\) 0 0
\(316\) 27.9246 1.57088
\(317\) −12.7561 −0.716453 −0.358227 0.933635i \(-0.616618\pi\)
−0.358227 + 0.933635i \(0.616618\pi\)
\(318\) 0 0
\(319\) −2.37090 −0.132745
\(320\) 0 0
\(321\) 0 0
\(322\) −62.3582 −3.47509
\(323\) −23.6600 −1.31648
\(324\) 0 0
\(325\) 0 0
\(326\) 13.7537 0.761748
\(327\) 0 0
\(328\) −47.6534 −2.63122
\(329\) −12.5314 −0.690881
\(330\) 0 0
\(331\) −15.8024 −0.868578 −0.434289 0.900774i \(-0.643000\pi\)
−0.434289 + 0.900774i \(0.643000\pi\)
\(332\) −37.6610 −2.06692
\(333\) 0 0
\(334\) −28.8304 −1.57753
\(335\) 0 0
\(336\) 0 0
\(337\) 10.2620 0.559008 0.279504 0.960145i \(-0.409830\pi\)
0.279504 + 0.960145i \(0.409830\pi\)
\(338\) −21.1027 −1.14784
\(339\) 0 0
\(340\) 0 0
\(341\) 6.64300 0.359739
\(342\) 0 0
\(343\) −17.9095 −0.967022
\(344\) −31.1376 −1.67883
\(345\) 0 0
\(346\) 32.6862 1.75722
\(347\) 33.9014 1.81992 0.909960 0.414696i \(-0.136112\pi\)
0.909960 + 0.414696i \(0.136112\pi\)
\(348\) 0 0
\(349\) −35.3728 −1.89346 −0.946731 0.322026i \(-0.895636\pi\)
−0.946731 + 0.322026i \(0.895636\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) −7.60482 −0.405338
\(353\) 20.6764 1.10050 0.550248 0.835001i \(-0.314533\pi\)
0.550248 + 0.835001i \(0.314533\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) −3.63926 −0.192880
\(357\) 0 0
\(358\) 39.4709 2.08610
\(359\) 9.05998 0.478167 0.239084 0.970999i \(-0.423153\pi\)
0.239084 + 0.970999i \(0.423153\pi\)
\(360\) 0 0
\(361\) −8.33363 −0.438612
\(362\) 13.7503 0.722699
\(363\) 0 0
\(364\) 31.5884 1.65568
\(365\) 0 0
\(366\) 0 0
\(367\) −30.1733 −1.57503 −0.787516 0.616294i \(-0.788633\pi\)
−0.787516 + 0.616294i \(0.788633\pi\)
\(368\) 12.3719 0.644929
\(369\) 0 0
\(370\) 0 0
\(371\) 53.0840 2.75599
\(372\) 0 0
\(373\) −21.8418 −1.13093 −0.565463 0.824774i \(-0.691302\pi\)
−0.565463 + 0.824774i \(0.691302\pi\)
\(374\) −42.8938 −2.21798
\(375\) 0 0
\(376\) 11.5044 0.593292
\(377\) 1.93169 0.0994871
\(378\) 0 0
\(379\) 13.9320 0.715637 0.357819 0.933791i \(-0.383521\pi\)
0.357819 + 0.933791i \(0.383521\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 37.0324 1.89474
\(383\) −3.86076 −0.197276 −0.0986378 0.995123i \(-0.531449\pi\)
−0.0986378 + 0.995123i \(0.531449\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 36.5214 1.85889
\(387\) 0 0
\(388\) 19.3860 0.984174
\(389\) 30.3908 1.54087 0.770436 0.637517i \(-0.220038\pi\)
0.770436 + 0.637517i \(0.220038\pi\)
\(390\) 0 0
\(391\) −44.5596 −2.25348
\(392\) 43.8558 2.21505
\(393\) 0 0
\(394\) 15.4694 0.779339
\(395\) 0 0
\(396\) 0 0
\(397\) −6.89048 −0.345823 −0.172912 0.984937i \(-0.555317\pi\)
−0.172912 + 0.984937i \(0.555317\pi\)
\(398\) −38.0173 −1.90564
\(399\) 0 0
\(400\) 0 0
\(401\) −32.5187 −1.62390 −0.811952 0.583724i \(-0.801595\pi\)
−0.811952 + 0.583724i \(0.801595\pi\)
\(402\) 0 0
\(403\) −5.41239 −0.269610
\(404\) −45.8813 −2.28268
\(405\) 0 0
\(406\) 9.64778 0.478811
\(407\) −24.0828 −1.19374
\(408\) 0 0
\(409\) −4.71415 −0.233100 −0.116550 0.993185i \(-0.537184\pi\)
−0.116550 + 0.993185i \(0.537184\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) −45.3882 −2.23612
\(413\) −34.7110 −1.70802
\(414\) 0 0
\(415\) 0 0
\(416\) 6.19604 0.303786
\(417\) 0 0
\(418\) 19.3373 0.945819
\(419\) −23.9724 −1.17113 −0.585564 0.810626i \(-0.699127\pi\)
−0.585564 + 0.810626i \(0.699127\pi\)
\(420\) 0 0
\(421\) 27.6965 1.34984 0.674921 0.737890i \(-0.264178\pi\)
0.674921 + 0.737890i \(0.264178\pi\)
\(422\) −37.3478 −1.81806
\(423\) 0 0
\(424\) −48.7333 −2.36670
\(425\) 0 0
\(426\) 0 0
\(427\) 14.2211 0.688206
\(428\) 0.216413 0.0104607
\(429\) 0 0
\(430\) 0 0
\(431\) −6.14152 −0.295826 −0.147913 0.989000i \(-0.547256\pi\)
−0.147913 + 0.989000i \(0.547256\pi\)
\(432\) 0 0
\(433\) −15.3336 −0.736887 −0.368444 0.929650i \(-0.620109\pi\)
−0.368444 + 0.929650i \(0.620109\pi\)
\(434\) −27.0320 −1.29758
\(435\) 0 0
\(436\) 3.84214 0.184005
\(437\) 20.0883 0.960954
\(438\) 0 0
\(439\) −26.1170 −1.24650 −0.623248 0.782024i \(-0.714187\pi\)
−0.623248 + 0.782024i \(0.714187\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 34.9477 1.66229
\(443\) 15.2702 0.725507 0.362753 0.931885i \(-0.381837\pi\)
0.362753 + 0.931885i \(0.381837\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 35.0768 1.66094
\(447\) 0 0
\(448\) 48.1071 2.27284
\(449\) 34.4395 1.62530 0.812650 0.582752i \(-0.198024\pi\)
0.812650 + 0.582752i \(0.198024\pi\)
\(450\) 0 0
\(451\) −30.3153 −1.42749
\(452\) −2.98355 −0.140335
\(453\) 0 0
\(454\) 17.2167 0.808020
\(455\) 0 0
\(456\) 0 0
\(457\) −8.69321 −0.406651 −0.203326 0.979111i \(-0.565175\pi\)
−0.203326 + 0.979111i \(0.565175\pi\)
\(458\) −8.07402 −0.377274
\(459\) 0 0
\(460\) 0 0
\(461\) −11.7300 −0.546322 −0.273161 0.961968i \(-0.588069\pi\)
−0.273161 + 0.961968i \(0.588069\pi\)
\(462\) 0 0
\(463\) 18.8462 0.875855 0.437928 0.899010i \(-0.355713\pi\)
0.437928 + 0.899010i \(0.355713\pi\)
\(464\) −1.91412 −0.0888608
\(465\) 0 0
\(466\) 51.7373 2.39668
\(467\) −18.3055 −0.847076 −0.423538 0.905878i \(-0.639212\pi\)
−0.423538 + 0.905878i \(0.639212\pi\)
\(468\) 0 0
\(469\) −47.9286 −2.21314
\(470\) 0 0
\(471\) 0 0
\(472\) 31.8661 1.46676
\(473\) −19.8086 −0.910799
\(474\) 0 0
\(475\) 0 0
\(476\) 112.737 5.16727
\(477\) 0 0
\(478\) −19.2697 −0.881378
\(479\) 21.5630 0.985238 0.492619 0.870245i \(-0.336040\pi\)
0.492619 + 0.870245i \(0.336040\pi\)
\(480\) 0 0
\(481\) 19.6215 0.894663
\(482\) 47.7596 2.17539
\(483\) 0 0
\(484\) −17.4841 −0.794732
\(485\) 0 0
\(486\) 0 0
\(487\) −12.2616 −0.555625 −0.277813 0.960635i \(-0.589609\pi\)
−0.277813 + 0.960635i \(0.589609\pi\)
\(488\) −13.0555 −0.590995
\(489\) 0 0
\(490\) 0 0
\(491\) 8.67892 0.391674 0.195837 0.980636i \(-0.437258\pi\)
0.195837 + 0.980636i \(0.437258\pi\)
\(492\) 0 0
\(493\) 6.89406 0.310493
\(494\) −15.7551 −0.708855
\(495\) 0 0
\(496\) 5.36316 0.240813
\(497\) 41.2796 1.85164
\(498\) 0 0
\(499\) −6.77589 −0.303331 −0.151665 0.988432i \(-0.548464\pi\)
−0.151665 + 0.988432i \(0.548464\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) −48.0780 −2.14582
\(503\) −1.13028 −0.0503969 −0.0251984 0.999682i \(-0.508022\pi\)
−0.0251984 + 0.999682i \(0.508022\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 36.4186 1.61900
\(507\) 0 0
\(508\) 8.74547 0.388017
\(509\) 5.88844 0.261000 0.130500 0.991448i \(-0.458342\pi\)
0.130500 + 0.991448i \(0.458342\pi\)
\(510\) 0 0
\(511\) 35.1647 1.55559
\(512\) −21.8943 −0.967599
\(513\) 0 0
\(514\) −3.14501 −0.138720
\(515\) 0 0
\(516\) 0 0
\(517\) 7.31863 0.321873
\(518\) 97.9991 4.30583
\(519\) 0 0
\(520\) 0 0
\(521\) −8.50026 −0.372403 −0.186202 0.982512i \(-0.559618\pi\)
−0.186202 + 0.982512i \(0.559618\pi\)
\(522\) 0 0
\(523\) −9.41273 −0.411590 −0.205795 0.978595i \(-0.565978\pi\)
−0.205795 + 0.978595i \(0.565978\pi\)
\(524\) 50.0045 2.18446
\(525\) 0 0
\(526\) 56.7072 2.47255
\(527\) −19.3164 −0.841437
\(528\) 0 0
\(529\) 14.8329 0.644911
\(530\) 0 0
\(531\) 0 0
\(532\) −50.8238 −2.20349
\(533\) 24.6994 1.06985
\(534\) 0 0
\(535\) 0 0
\(536\) 44.0004 1.90053
\(537\) 0 0
\(538\) −15.4336 −0.665388
\(539\) 27.8994 1.20171
\(540\) 0 0
\(541\) 4.40257 0.189281 0.0946406 0.995512i \(-0.469830\pi\)
0.0946406 + 0.995512i \(0.469830\pi\)
\(542\) −22.5965 −0.970602
\(543\) 0 0
\(544\) 22.1132 0.948096
\(545\) 0 0
\(546\) 0 0
\(547\) 13.2431 0.566235 0.283117 0.959085i \(-0.408631\pi\)
0.283117 + 0.959085i \(0.408631\pi\)
\(548\) −12.5599 −0.536531
\(549\) 0 0
\(550\) 0 0
\(551\) −3.10797 −0.132404
\(552\) 0 0
\(553\) −32.6556 −1.38866
\(554\) 21.7464 0.923915
\(555\) 0 0
\(556\) −7.20150 −0.305411
\(557\) −0.997628 −0.0422709 −0.0211354 0.999777i \(-0.506728\pi\)
−0.0211354 + 0.999777i \(0.506728\pi\)
\(558\) 0 0
\(559\) 16.1391 0.682609
\(560\) 0 0
\(561\) 0 0
\(562\) 40.8686 1.72394
\(563\) −5.45230 −0.229787 −0.114893 0.993378i \(-0.536653\pi\)
−0.114893 + 0.993378i \(0.536653\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 53.6546 2.25527
\(567\) 0 0
\(568\) −37.8963 −1.59009
\(569\) 8.03392 0.336799 0.168400 0.985719i \(-0.446140\pi\)
0.168400 + 0.985719i \(0.446140\pi\)
\(570\) 0 0
\(571\) −21.0647 −0.881528 −0.440764 0.897623i \(-0.645293\pi\)
−0.440764 + 0.897623i \(0.645293\pi\)
\(572\) −18.4483 −0.771362
\(573\) 0 0
\(574\) 123.360 5.14897
\(575\) 0 0
\(576\) 0 0
\(577\) 27.9479 1.16349 0.581744 0.813372i \(-0.302370\pi\)
0.581744 + 0.813372i \(0.302370\pi\)
\(578\) 84.3249 3.50745
\(579\) 0 0
\(580\) 0 0
\(581\) 44.0416 1.82715
\(582\) 0 0
\(583\) −31.0022 −1.28398
\(584\) −32.2826 −1.33586
\(585\) 0 0
\(586\) −50.7445 −2.09624
\(587\) −43.6656 −1.80227 −0.901137 0.433534i \(-0.857266\pi\)
−0.901137 + 0.433534i \(0.857266\pi\)
\(588\) 0 0
\(589\) 8.70820 0.358815
\(590\) 0 0
\(591\) 0 0
\(592\) −19.4430 −0.799104
\(593\) 27.8198 1.14242 0.571212 0.820803i \(-0.306473\pi\)
0.571212 + 0.820803i \(0.306473\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 23.7938 0.974630
\(597\) 0 0
\(598\) −29.6721 −1.21338
\(599\) 46.3084 1.89211 0.946054 0.324008i \(-0.105030\pi\)
0.946054 + 0.324008i \(0.105030\pi\)
\(600\) 0 0
\(601\) −18.1955 −0.742209 −0.371105 0.928591i \(-0.621021\pi\)
−0.371105 + 0.928591i \(0.621021\pi\)
\(602\) 80.6061 3.28526
\(603\) 0 0
\(604\) −29.6032 −1.20454
\(605\) 0 0
\(606\) 0 0
\(607\) −34.6351 −1.40579 −0.702897 0.711292i \(-0.748110\pi\)
−0.702897 + 0.711292i \(0.748110\pi\)
\(608\) −9.96904 −0.404298
\(609\) 0 0
\(610\) 0 0
\(611\) −5.96286 −0.241232
\(612\) 0 0
\(613\) −34.2121 −1.38181 −0.690907 0.722944i \(-0.742788\pi\)
−0.690907 + 0.722944i \(0.742788\pi\)
\(614\) 20.0431 0.808875
\(615\) 0 0
\(616\) −41.6232 −1.67705
\(617\) −12.6968 −0.511152 −0.255576 0.966789i \(-0.582265\pi\)
−0.255576 + 0.966789i \(0.582265\pi\)
\(618\) 0 0
\(619\) 26.8216 1.07805 0.539025 0.842290i \(-0.318793\pi\)
0.539025 + 0.842290i \(0.318793\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) −21.7082 −0.870420
\(623\) 4.25582 0.170506
\(624\) 0 0
\(625\) 0 0
\(626\) −49.8319 −1.99169
\(627\) 0 0
\(628\) 58.5653 2.33701
\(629\) 70.0277 2.79219
\(630\) 0 0
\(631\) −29.6972 −1.18223 −0.591114 0.806588i \(-0.701312\pi\)
−0.591114 + 0.806588i \(0.701312\pi\)
\(632\) 29.9791 1.19251
\(633\) 0 0
\(634\) −30.3153 −1.20397
\(635\) 0 0
\(636\) 0 0
\(637\) −22.7310 −0.900636
\(638\) −5.63451 −0.223072
\(639\) 0 0
\(640\) 0 0
\(641\) −2.85489 −0.112762 −0.0563808 0.998409i \(-0.517956\pi\)
−0.0563808 + 0.998409i \(0.517956\pi\)
\(642\) 0 0
\(643\) −30.0385 −1.18460 −0.592301 0.805717i \(-0.701780\pi\)
−0.592301 + 0.805717i \(0.701780\pi\)
\(644\) −95.7180 −3.77182
\(645\) 0 0
\(646\) −56.2288 −2.21229
\(647\) −2.12791 −0.0836569 −0.0418284 0.999125i \(-0.513318\pi\)
−0.0418284 + 0.999125i \(0.513318\pi\)
\(648\) 0 0
\(649\) 20.2720 0.795745
\(650\) 0 0
\(651\) 0 0
\(652\) 21.1115 0.826792
\(653\) 15.5057 0.606783 0.303392 0.952866i \(-0.401881\pi\)
0.303392 + 0.952866i \(0.401881\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) −24.4747 −0.955578
\(657\) 0 0
\(658\) −29.7814 −1.16100
\(659\) −8.54626 −0.332915 −0.166458 0.986049i \(-0.553233\pi\)
−0.166458 + 0.986049i \(0.553233\pi\)
\(660\) 0 0
\(661\) 27.9246 1.08614 0.543070 0.839687i \(-0.317262\pi\)
0.543070 + 0.839687i \(0.317262\pi\)
\(662\) −37.5549 −1.45961
\(663\) 0 0
\(664\) −40.4320 −1.56906
\(665\) 0 0
\(666\) 0 0
\(667\) −5.85334 −0.226642
\(668\) −44.2538 −1.71223
\(669\) 0 0
\(670\) 0 0
\(671\) −8.30542 −0.320627
\(672\) 0 0
\(673\) −11.1115 −0.428319 −0.214159 0.976799i \(-0.568701\pi\)
−0.214159 + 0.976799i \(0.568701\pi\)
\(674\) 24.3880 0.939391
\(675\) 0 0
\(676\) −32.3920 −1.24585
\(677\) −28.8492 −1.10877 −0.554383 0.832262i \(-0.687046\pi\)
−0.554383 + 0.832262i \(0.687046\pi\)
\(678\) 0 0
\(679\) −22.6704 −0.870010
\(680\) 0 0
\(681\) 0 0
\(682\) 15.7873 0.604527
\(683\) −18.9069 −0.723454 −0.361727 0.932284i \(-0.617813\pi\)
−0.361727 + 0.932284i \(0.617813\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) −42.5625 −1.62504
\(687\) 0 0
\(688\) −15.9923 −0.609699
\(689\) 25.2591 0.962295
\(690\) 0 0
\(691\) 5.03727 0.191627 0.0958133 0.995399i \(-0.469455\pi\)
0.0958133 + 0.995399i \(0.469455\pi\)
\(692\) 50.1723 1.90726
\(693\) 0 0
\(694\) 80.5677 3.05831
\(695\) 0 0
\(696\) 0 0
\(697\) 88.1503 3.33893
\(698\) −84.0646 −3.18189
\(699\) 0 0
\(700\) 0 0
\(701\) 41.3716 1.56258 0.781291 0.624167i \(-0.214561\pi\)
0.781291 + 0.624167i \(0.214561\pi\)
\(702\) 0 0
\(703\) −31.5698 −1.19068
\(704\) −28.0956 −1.05889
\(705\) 0 0
\(706\) 49.1383 1.84934
\(707\) 53.6546 2.01789
\(708\) 0 0
\(709\) −36.1114 −1.35619 −0.678096 0.734973i \(-0.737195\pi\)
−0.678096 + 0.734973i \(0.737195\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) −3.90702 −0.146422
\(713\) 16.4004 0.614201
\(714\) 0 0
\(715\) 0 0
\(716\) 60.5867 2.26423
\(717\) 0 0
\(718\) 21.5313 0.803542
\(719\) 13.2982 0.495940 0.247970 0.968768i \(-0.420236\pi\)
0.247970 + 0.968768i \(0.420236\pi\)
\(720\) 0 0
\(721\) 53.0780 1.97673
\(722\) −19.8051 −0.737071
\(723\) 0 0
\(724\) 21.1063 0.784409
\(725\) 0 0
\(726\) 0 0
\(727\) 13.9598 0.517741 0.258871 0.965912i \(-0.416650\pi\)
0.258871 + 0.965912i \(0.416650\pi\)
\(728\) 33.9125 1.25688
\(729\) 0 0
\(730\) 0 0
\(731\) 57.5991 2.13038
\(732\) 0 0
\(733\) −6.51756 −0.240731 −0.120366 0.992730i \(-0.538407\pi\)
−0.120366 + 0.992730i \(0.538407\pi\)
\(734\) −71.7078 −2.64678
\(735\) 0 0
\(736\) −18.7750 −0.692056
\(737\) 27.9913 1.03107
\(738\) 0 0
\(739\) −46.2414 −1.70102 −0.850509 0.525960i \(-0.823706\pi\)
−0.850509 + 0.525960i \(0.823706\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 126.156 4.63133
\(743\) 39.4507 1.44731 0.723653 0.690163i \(-0.242461\pi\)
0.723653 + 0.690163i \(0.242461\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) −51.9077 −1.90048
\(747\) 0 0
\(748\) −65.8407 −2.40737
\(749\) −0.253078 −0.00924725
\(750\) 0 0
\(751\) −26.9003 −0.981606 −0.490803 0.871271i \(-0.663296\pi\)
−0.490803 + 0.871271i \(0.663296\pi\)
\(752\) 5.90863 0.215466
\(753\) 0 0
\(754\) 4.59072 0.167184
\(755\) 0 0
\(756\) 0 0
\(757\) −26.3433 −0.957462 −0.478731 0.877962i \(-0.658903\pi\)
−0.478731 + 0.877962i \(0.658903\pi\)
\(758\) 33.1098 1.20260
\(759\) 0 0
\(760\) 0 0
\(761\) 22.3582 0.810484 0.405242 0.914209i \(-0.367187\pi\)
0.405242 + 0.914209i \(0.367187\pi\)
\(762\) 0 0
\(763\) −4.49308 −0.162660
\(764\) 56.8437 2.05653
\(765\) 0 0
\(766\) −9.17522 −0.331514
\(767\) −16.5166 −0.596380
\(768\) 0 0
\(769\) −47.9185 −1.72799 −0.863993 0.503504i \(-0.832044\pi\)
−0.863993 + 0.503504i \(0.832044\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 56.0593 2.01762
\(773\) −6.38883 −0.229790 −0.114895 0.993378i \(-0.536653\pi\)
−0.114895 + 0.993378i \(0.536653\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 20.8123 0.747119
\(777\) 0 0
\(778\) 72.2246 2.58938
\(779\) −39.7398 −1.42383
\(780\) 0 0
\(781\) −24.1082 −0.862658
\(782\) −105.897 −3.78688
\(783\) 0 0
\(784\) 22.5243 0.804439
\(785\) 0 0
\(786\) 0 0
\(787\) −23.8410 −0.849839 −0.424920 0.905231i \(-0.639698\pi\)
−0.424920 + 0.905231i \(0.639698\pi\)
\(788\) 23.7451 0.845885
\(789\) 0 0
\(790\) 0 0
\(791\) 3.48903 0.124056
\(792\) 0 0
\(793\) 6.76685 0.240298
\(794\) −16.3754 −0.581143
\(795\) 0 0
\(796\) −58.3554 −2.06835
\(797\) −21.4117 −0.758440 −0.379220 0.925306i \(-0.623808\pi\)
−0.379220 + 0.925306i \(0.623808\pi\)
\(798\) 0 0
\(799\) −21.2810 −0.752869
\(800\) 0 0
\(801\) 0 0
\(802\) −77.2817 −2.72891
\(803\) −20.5369 −0.724733
\(804\) 0 0
\(805\) 0 0
\(806\) −12.8627 −0.453070
\(807\) 0 0
\(808\) −49.2571 −1.73286
\(809\) 21.7118 0.763348 0.381674 0.924297i \(-0.375348\pi\)
0.381674 + 0.924297i \(0.375348\pi\)
\(810\) 0 0
\(811\) 13.5518 0.475868 0.237934 0.971281i \(-0.423530\pi\)
0.237934 + 0.971281i \(0.423530\pi\)
\(812\) 14.8091 0.519696
\(813\) 0 0
\(814\) −57.2336 −2.00604
\(815\) 0 0
\(816\) 0 0
\(817\) −25.9668 −0.908462
\(818\) −11.2033 −0.391716
\(819\) 0 0
\(820\) 0 0
\(821\) 11.9433 0.416824 0.208412 0.978041i \(-0.433171\pi\)
0.208412 + 0.978041i \(0.433171\pi\)
\(822\) 0 0
\(823\) −33.3183 −1.16140 −0.580701 0.814117i \(-0.697222\pi\)
−0.580701 + 0.814117i \(0.697222\pi\)
\(824\) −48.7277 −1.69751
\(825\) 0 0
\(826\) −82.4918 −2.87026
\(827\) −47.6602 −1.65731 −0.828653 0.559763i \(-0.810892\pi\)
−0.828653 + 0.559763i \(0.810892\pi\)
\(828\) 0 0
\(829\) 3.26594 0.113431 0.0567154 0.998390i \(-0.481937\pi\)
0.0567154 + 0.998390i \(0.481937\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 22.8909 0.793599
\(833\) −81.1254 −2.81083
\(834\) 0 0
\(835\) 0 0
\(836\) 29.6822 1.02658
\(837\) 0 0
\(838\) −56.9712 −1.96804
\(839\) 54.3532 1.87648 0.938240 0.345986i \(-0.112456\pi\)
0.938240 + 0.345986i \(0.112456\pi\)
\(840\) 0 0
\(841\) −28.0944 −0.968772
\(842\) 65.8215 2.26836
\(843\) 0 0
\(844\) −57.3278 −1.97330
\(845\) 0 0
\(846\) 0 0
\(847\) 20.4463 0.702543
\(848\) −25.0294 −0.859512
\(849\) 0 0
\(850\) 0 0
\(851\) −59.4564 −2.03814
\(852\) 0 0
\(853\) 42.2955 1.44817 0.724085 0.689711i \(-0.242262\pi\)
0.724085 + 0.689711i \(0.242262\pi\)
\(854\) 33.7968 1.15650
\(855\) 0 0
\(856\) 0.232335 0.00794106
\(857\) −39.7714 −1.35856 −0.679282 0.733877i \(-0.737709\pi\)
−0.679282 + 0.733877i \(0.737709\pi\)
\(858\) 0 0
\(859\) 20.9842 0.715973 0.357986 0.933727i \(-0.383463\pi\)
0.357986 + 0.933727i \(0.383463\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) −14.5955 −0.497125
\(863\) −21.5346 −0.733045 −0.366522 0.930409i \(-0.619452\pi\)
−0.366522 + 0.930409i \(0.619452\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) −36.4409 −1.23831
\(867\) 0 0
\(868\) −41.4934 −1.40838
\(869\) 19.0716 0.646959
\(870\) 0 0
\(871\) −22.8060 −0.772751
\(872\) 4.12483 0.139684
\(873\) 0 0
\(874\) 47.7405 1.61485
\(875\) 0 0
\(876\) 0 0
\(877\) 33.4164 1.12839 0.564196 0.825641i \(-0.309186\pi\)
0.564196 + 0.825641i \(0.309186\pi\)
\(878\) −62.0679 −2.09469
\(879\) 0 0
\(880\) 0 0
\(881\) −3.37862 −0.113829 −0.0569143 0.998379i \(-0.518126\pi\)
−0.0569143 + 0.998379i \(0.518126\pi\)
\(882\) 0 0
\(883\) 8.46965 0.285027 0.142513 0.989793i \(-0.454482\pi\)
0.142513 + 0.989793i \(0.454482\pi\)
\(884\) 53.6437 1.80423
\(885\) 0 0
\(886\) 36.2900 1.21919
\(887\) −0.671898 −0.0225601 −0.0112801 0.999936i \(-0.503591\pi\)
−0.0112801 + 0.999936i \(0.503591\pi\)
\(888\) 0 0
\(889\) −10.2271 −0.343007
\(890\) 0 0
\(891\) 0 0
\(892\) 53.8419 1.80276
\(893\) 9.59388 0.321047
\(894\) 0 0
\(895\) 0 0
\(896\) 88.2851 2.94940
\(897\) 0 0
\(898\) 81.8466 2.73126
\(899\) −2.53740 −0.0846270
\(900\) 0 0
\(901\) 90.1479 3.00326
\(902\) −72.0452 −2.39884
\(903\) 0 0
\(904\) −3.20307 −0.106533
\(905\) 0 0
\(906\) 0 0
\(907\) −38.8659 −1.29052 −0.645260 0.763963i \(-0.723251\pi\)
−0.645260 + 0.763963i \(0.723251\pi\)
\(908\) 26.4271 0.877016
\(909\) 0 0
\(910\) 0 0
\(911\) 10.6457 0.352709 0.176355 0.984327i \(-0.443569\pi\)
0.176355 + 0.984327i \(0.443569\pi\)
\(912\) 0 0
\(913\) −25.7213 −0.851250
\(914\) −20.6597 −0.683362
\(915\) 0 0
\(916\) −12.3934 −0.409489
\(917\) −58.4763 −1.93106
\(918\) 0 0
\(919\) −0.508850 −0.0167854 −0.00839271 0.999965i \(-0.502672\pi\)
−0.00839271 + 0.999965i \(0.502672\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) −27.8768 −0.918074
\(923\) 19.6422 0.646529
\(924\) 0 0
\(925\) 0 0
\(926\) 44.7885 1.47184
\(927\) 0 0
\(928\) 2.90478 0.0953542
\(929\) −36.7185 −1.20469 −0.602347 0.798234i \(-0.705768\pi\)
−0.602347 + 0.798234i \(0.705768\pi\)
\(930\) 0 0
\(931\) 36.5728 1.19863
\(932\) 79.4152 2.60133
\(933\) 0 0
\(934\) −43.5035 −1.42348
\(935\) 0 0
\(936\) 0 0
\(937\) 19.9423 0.651488 0.325744 0.945458i \(-0.394385\pi\)
0.325744 + 0.945458i \(0.394385\pi\)
\(938\) −113.904 −3.71909
\(939\) 0 0
\(940\) 0 0
\(941\) −14.0888 −0.459281 −0.229641 0.973276i \(-0.573755\pi\)
−0.229641 + 0.973276i \(0.573755\pi\)
\(942\) 0 0
\(943\) −74.8432 −2.43723
\(944\) 16.3664 0.532681
\(945\) 0 0
\(946\) −47.0757 −1.53056
\(947\) 17.5552 0.570466 0.285233 0.958458i \(-0.407929\pi\)
0.285233 + 0.958458i \(0.407929\pi\)
\(948\) 0 0
\(949\) 16.7325 0.543159
\(950\) 0 0
\(951\) 0 0
\(952\) 121.031 3.92265
\(953\) 10.5915 0.343093 0.171546 0.985176i \(-0.445124\pi\)
0.171546 + 0.985176i \(0.445124\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) −29.5785 −0.956637
\(957\) 0 0
\(958\) 51.2451 1.65566
\(959\) 14.6878 0.474293
\(960\) 0 0
\(961\) −23.8905 −0.770660
\(962\) 46.6311 1.50345
\(963\) 0 0
\(964\) 73.3095 2.36114
\(965\) 0 0
\(966\) 0 0
\(967\) −46.6537 −1.50028 −0.750141 0.661278i \(-0.770015\pi\)
−0.750141 + 0.661278i \(0.770015\pi\)
\(968\) −18.7705 −0.603307
\(969\) 0 0
\(970\) 0 0
\(971\) −14.7798 −0.474305 −0.237153 0.971472i \(-0.576214\pi\)
−0.237153 + 0.971472i \(0.576214\pi\)
\(972\) 0 0
\(973\) 8.42158 0.269984
\(974\) −29.1400 −0.933707
\(975\) 0 0
\(976\) −6.70530 −0.214631
\(977\) 9.55036 0.305543 0.152771 0.988262i \(-0.451180\pi\)
0.152771 + 0.988262i \(0.451180\pi\)
\(978\) 0 0
\(979\) −2.48549 −0.0794367
\(980\) 0 0
\(981\) 0 0
\(982\) 20.6257 0.658193
\(983\) 33.7083 1.07513 0.537564 0.843223i \(-0.319345\pi\)
0.537564 + 0.843223i \(0.319345\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 16.3840 0.521772
\(987\) 0 0
\(988\) −24.1836 −0.769383
\(989\) −48.9040 −1.55506
\(990\) 0 0
\(991\) 14.5129 0.461016 0.230508 0.973070i \(-0.425961\pi\)
0.230508 + 0.973070i \(0.425961\pi\)
\(992\) −8.13889 −0.258410
\(993\) 0 0
\(994\) 98.1022 3.11161
\(995\) 0 0
\(996\) 0 0
\(997\) −22.6197 −0.716374 −0.358187 0.933650i \(-0.616605\pi\)
−0.358187 + 0.933650i \(0.616605\pi\)
\(998\) −16.1031 −0.509736
\(999\) 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 5625.2.a.z.1.7 yes 8
3.2 odd 2 inner 5625.2.a.z.1.2 8
5.4 even 2 5625.2.a.bb.1.2 yes 8
15.14 odd 2 5625.2.a.bb.1.7 yes 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
5625.2.a.z.1.2 8 3.2 odd 2 inner
5625.2.a.z.1.7 yes 8 1.1 even 1 trivial
5625.2.a.bb.1.2 yes 8 5.4 even 2
5625.2.a.bb.1.7 yes 8 15.14 odd 2