Properties

Label 7728.2.a.w.1.1
Level $7728$
Weight $2$
Character 7728.1
Self dual yes
Analytic conductor $61.708$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 7728 = 2^{4} \cdot 3 \cdot 7 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 7728.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(61.7083906820\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{10})^+\)
Defining polynomial: \(x^{2} - x - 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 1932)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(1.61803\) of defining polynomial
Character \(\chi\) \(=\) 7728.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{3} -3.61803 q^{5} -1.00000 q^{7} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{3} -3.61803 q^{5} -1.00000 q^{7} +1.00000 q^{9} +1.00000 q^{11} +5.09017 q^{13} +3.61803 q^{15} -5.00000 q^{17} +3.47214 q^{19} +1.00000 q^{21} -1.00000 q^{23} +8.09017 q^{25} -1.00000 q^{27} -6.23607 q^{29} +8.70820 q^{31} -1.00000 q^{33} +3.61803 q^{35} +1.47214 q^{37} -5.09017 q^{39} -5.76393 q^{41} -6.32624 q^{43} -3.61803 q^{45} -3.70820 q^{47} +1.00000 q^{49} +5.00000 q^{51} +0.381966 q^{53} -3.61803 q^{55} -3.47214 q^{57} +3.61803 q^{59} -7.56231 q^{61} -1.00000 q^{63} -18.4164 q^{65} +7.32624 q^{67} +1.00000 q^{69} +4.32624 q^{71} -0.527864 q^{73} -8.09017 q^{75} -1.00000 q^{77} +10.7082 q^{79} +1.00000 q^{81} -9.18034 q^{83} +18.0902 q^{85} +6.23607 q^{87} -16.6180 q^{89} -5.09017 q^{91} -8.70820 q^{93} -12.5623 q^{95} -12.2361 q^{97} +1.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{3} - 5 q^{5} - 2 q^{7} + 2 q^{9} + O(q^{10}) \) \( 2 q - 2 q^{3} - 5 q^{5} - 2 q^{7} + 2 q^{9} + 2 q^{11} - q^{13} + 5 q^{15} - 10 q^{17} - 2 q^{19} + 2 q^{21} - 2 q^{23} + 5 q^{25} - 2 q^{27} - 8 q^{29} + 4 q^{31} - 2 q^{33} + 5 q^{35} - 6 q^{37} + q^{39} - 16 q^{41} + 3 q^{43} - 5 q^{45} + 6 q^{47} + 2 q^{49} + 10 q^{51} + 3 q^{53} - 5 q^{55} + 2 q^{57} + 5 q^{59} + 5 q^{61} - 2 q^{63} - 10 q^{65} - q^{67} + 2 q^{69} - 7 q^{71} - 10 q^{73} - 5 q^{75} - 2 q^{77} + 8 q^{79} + 2 q^{81} + 4 q^{83} + 25 q^{85} + 8 q^{87} - 31 q^{89} + q^{91} - 4 q^{93} - 5 q^{95} - 20 q^{97} + 2 q^{99} + O(q^{100}) \)

Coefficient data

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

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) −1.00000 −0.577350
\(4\) 0 0
\(5\) −3.61803 −1.61803 −0.809017 0.587785i \(-0.800000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(6\) 0 0
\(7\) −1.00000 −0.377964
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 1.00000 0.301511 0.150756 0.988571i \(-0.451829\pi\)
0.150756 + 0.988571i \(0.451829\pi\)
\(12\) 0 0
\(13\) 5.09017 1.41176 0.705880 0.708332i \(-0.250552\pi\)
0.705880 + 0.708332i \(0.250552\pi\)
\(14\) 0 0
\(15\) 3.61803 0.934172
\(16\) 0 0
\(17\) −5.00000 −1.21268 −0.606339 0.795206i \(-0.707363\pi\)
−0.606339 + 0.795206i \(0.707363\pi\)
\(18\) 0 0
\(19\) 3.47214 0.796563 0.398281 0.917263i \(-0.369607\pi\)
0.398281 + 0.917263i \(0.369607\pi\)
\(20\) 0 0
\(21\) 1.00000 0.218218
\(22\) 0 0
\(23\) −1.00000 −0.208514
\(24\) 0 0
\(25\) 8.09017 1.61803
\(26\) 0 0
\(27\) −1.00000 −0.192450
\(28\) 0 0
\(29\) −6.23607 −1.15801 −0.579004 0.815324i \(-0.696559\pi\)
−0.579004 + 0.815324i \(0.696559\pi\)
\(30\) 0 0
\(31\) 8.70820 1.56404 0.782020 0.623254i \(-0.214190\pi\)
0.782020 + 0.623254i \(0.214190\pi\)
\(32\) 0 0
\(33\) −1.00000 −0.174078
\(34\) 0 0
\(35\) 3.61803 0.611559
\(36\) 0 0
\(37\) 1.47214 0.242018 0.121009 0.992651i \(-0.461387\pi\)
0.121009 + 0.992651i \(0.461387\pi\)
\(38\) 0 0
\(39\) −5.09017 −0.815080
\(40\) 0 0
\(41\) −5.76393 −0.900175 −0.450087 0.892984i \(-0.648607\pi\)
−0.450087 + 0.892984i \(0.648607\pi\)
\(42\) 0 0
\(43\) −6.32624 −0.964742 −0.482371 0.875967i \(-0.660224\pi\)
−0.482371 + 0.875967i \(0.660224\pi\)
\(44\) 0 0
\(45\) −3.61803 −0.539345
\(46\) 0 0
\(47\) −3.70820 −0.540897 −0.270449 0.962734i \(-0.587172\pi\)
−0.270449 + 0.962734i \(0.587172\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) 5.00000 0.700140
\(52\) 0 0
\(53\) 0.381966 0.0524671 0.0262335 0.999656i \(-0.491649\pi\)
0.0262335 + 0.999656i \(0.491649\pi\)
\(54\) 0 0
\(55\) −3.61803 −0.487856
\(56\) 0 0
\(57\) −3.47214 −0.459896
\(58\) 0 0
\(59\) 3.61803 0.471028 0.235514 0.971871i \(-0.424323\pi\)
0.235514 + 0.971871i \(0.424323\pi\)
\(60\) 0 0
\(61\) −7.56231 −0.968254 −0.484127 0.874998i \(-0.660863\pi\)
−0.484127 + 0.874998i \(0.660863\pi\)
\(62\) 0 0
\(63\) −1.00000 −0.125988
\(64\) 0 0
\(65\) −18.4164 −2.28427
\(66\) 0 0
\(67\) 7.32624 0.895042 0.447521 0.894273i \(-0.352307\pi\)
0.447521 + 0.894273i \(0.352307\pi\)
\(68\) 0 0
\(69\) 1.00000 0.120386
\(70\) 0 0
\(71\) 4.32624 0.513430 0.256715 0.966487i \(-0.417360\pi\)
0.256715 + 0.966487i \(0.417360\pi\)
\(72\) 0 0
\(73\) −0.527864 −0.0617818 −0.0308909 0.999523i \(-0.509834\pi\)
−0.0308909 + 0.999523i \(0.509834\pi\)
\(74\) 0 0
\(75\) −8.09017 −0.934172
\(76\) 0 0
\(77\) −1.00000 −0.113961
\(78\) 0 0
\(79\) 10.7082 1.20477 0.602384 0.798207i \(-0.294218\pi\)
0.602384 + 0.798207i \(0.294218\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) −9.18034 −1.00767 −0.503837 0.863799i \(-0.668079\pi\)
−0.503837 + 0.863799i \(0.668079\pi\)
\(84\) 0 0
\(85\) 18.0902 1.96215
\(86\) 0 0
\(87\) 6.23607 0.668577
\(88\) 0 0
\(89\) −16.6180 −1.76151 −0.880754 0.473574i \(-0.842964\pi\)
−0.880754 + 0.473574i \(0.842964\pi\)
\(90\) 0 0
\(91\) −5.09017 −0.533595
\(92\) 0 0
\(93\) −8.70820 −0.902999
\(94\) 0 0
\(95\) −12.5623 −1.28887
\(96\) 0 0
\(97\) −12.2361 −1.24238 −0.621192 0.783658i \(-0.713351\pi\)
−0.621192 + 0.783658i \(0.713351\pi\)
\(98\) 0 0
\(99\) 1.00000 0.100504
\(100\) 0 0
\(101\) −5.61803 −0.559015 −0.279508 0.960143i \(-0.590171\pi\)
−0.279508 + 0.960143i \(0.590171\pi\)
\(102\) 0 0
\(103\) −2.47214 −0.243587 −0.121793 0.992555i \(-0.538865\pi\)
−0.121793 + 0.992555i \(0.538865\pi\)
\(104\) 0 0
\(105\) −3.61803 −0.353084
\(106\) 0 0
\(107\) 14.6180 1.41318 0.706589 0.707624i \(-0.250233\pi\)
0.706589 + 0.707624i \(0.250233\pi\)
\(108\) 0 0
\(109\) −17.5066 −1.67683 −0.838413 0.545035i \(-0.816516\pi\)
−0.838413 + 0.545035i \(0.816516\pi\)
\(110\) 0 0
\(111\) −1.47214 −0.139729
\(112\) 0 0
\(113\) 3.14590 0.295941 0.147971 0.988992i \(-0.452726\pi\)
0.147971 + 0.988992i \(0.452726\pi\)
\(114\) 0 0
\(115\) 3.61803 0.337383
\(116\) 0 0
\(117\) 5.09017 0.470586
\(118\) 0 0
\(119\) 5.00000 0.458349
\(120\) 0 0
\(121\) −10.0000 −0.909091
\(122\) 0 0
\(123\) 5.76393 0.519716
\(124\) 0 0
\(125\) −11.1803 −1.00000
\(126\) 0 0
\(127\) −4.61803 −0.409784 −0.204892 0.978785i \(-0.565684\pi\)
−0.204892 + 0.978785i \(0.565684\pi\)
\(128\) 0 0
\(129\) 6.32624 0.556994
\(130\) 0 0
\(131\) 9.18034 0.802090 0.401045 0.916058i \(-0.368647\pi\)
0.401045 + 0.916058i \(0.368647\pi\)
\(132\) 0 0
\(133\) −3.47214 −0.301072
\(134\) 0 0
\(135\) 3.61803 0.311391
\(136\) 0 0
\(137\) −4.52786 −0.386842 −0.193421 0.981116i \(-0.561958\pi\)
−0.193421 + 0.981116i \(0.561958\pi\)
\(138\) 0 0
\(139\) −0.673762 −0.0571478 −0.0285739 0.999592i \(-0.509097\pi\)
−0.0285739 + 0.999592i \(0.509097\pi\)
\(140\) 0 0
\(141\) 3.70820 0.312287
\(142\) 0 0
\(143\) 5.09017 0.425661
\(144\) 0 0
\(145\) 22.5623 1.87370
\(146\) 0 0
\(147\) −1.00000 −0.0824786
\(148\) 0 0
\(149\) 19.1246 1.56675 0.783375 0.621550i \(-0.213497\pi\)
0.783375 + 0.621550i \(0.213497\pi\)
\(150\) 0 0
\(151\) −5.23607 −0.426105 −0.213053 0.977041i \(-0.568341\pi\)
−0.213053 + 0.977041i \(0.568341\pi\)
\(152\) 0 0
\(153\) −5.00000 −0.404226
\(154\) 0 0
\(155\) −31.5066 −2.53067
\(156\) 0 0
\(157\) 14.6525 1.16939 0.584697 0.811251i \(-0.301213\pi\)
0.584697 + 0.811251i \(0.301213\pi\)
\(158\) 0 0
\(159\) −0.381966 −0.0302919
\(160\) 0 0
\(161\) 1.00000 0.0788110
\(162\) 0 0
\(163\) 14.0902 1.10363 0.551814 0.833967i \(-0.313936\pi\)
0.551814 + 0.833967i \(0.313936\pi\)
\(164\) 0 0
\(165\) 3.61803 0.281664
\(166\) 0 0
\(167\) 3.18034 0.246102 0.123051 0.992400i \(-0.460732\pi\)
0.123051 + 0.992400i \(0.460732\pi\)
\(168\) 0 0
\(169\) 12.9098 0.993064
\(170\) 0 0
\(171\) 3.47214 0.265521
\(172\) 0 0
\(173\) 9.18034 0.697968 0.348984 0.937129i \(-0.386527\pi\)
0.348984 + 0.937129i \(0.386527\pi\)
\(174\) 0 0
\(175\) −8.09017 −0.611559
\(176\) 0 0
\(177\) −3.61803 −0.271948
\(178\) 0 0
\(179\) −10.2705 −0.767654 −0.383827 0.923405i \(-0.625394\pi\)
−0.383827 + 0.923405i \(0.625394\pi\)
\(180\) 0 0
\(181\) −0.236068 −0.0175468 −0.00877340 0.999962i \(-0.502793\pi\)
−0.00877340 + 0.999962i \(0.502793\pi\)
\(182\) 0 0
\(183\) 7.56231 0.559022
\(184\) 0 0
\(185\) −5.32624 −0.391593
\(186\) 0 0
\(187\) −5.00000 −0.365636
\(188\) 0 0
\(189\) 1.00000 0.0727393
\(190\) 0 0
\(191\) −16.6525 −1.20493 −0.602465 0.798145i \(-0.705815\pi\)
−0.602465 + 0.798145i \(0.705815\pi\)
\(192\) 0 0
\(193\) 21.7082 1.56259 0.781295 0.624161i \(-0.214559\pi\)
0.781295 + 0.624161i \(0.214559\pi\)
\(194\) 0 0
\(195\) 18.4164 1.31883
\(196\) 0 0
\(197\) 22.6180 1.61147 0.805734 0.592277i \(-0.201771\pi\)
0.805734 + 0.592277i \(0.201771\pi\)
\(198\) 0 0
\(199\) −3.79837 −0.269260 −0.134630 0.990896i \(-0.542985\pi\)
−0.134630 + 0.990896i \(0.542985\pi\)
\(200\) 0 0
\(201\) −7.32624 −0.516753
\(202\) 0 0
\(203\) 6.23607 0.437686
\(204\) 0 0
\(205\) 20.8541 1.45651
\(206\) 0 0
\(207\) −1.00000 −0.0695048
\(208\) 0 0
\(209\) 3.47214 0.240173
\(210\) 0 0
\(211\) −18.2361 −1.25542 −0.627711 0.778446i \(-0.716008\pi\)
−0.627711 + 0.778446i \(0.716008\pi\)
\(212\) 0 0
\(213\) −4.32624 −0.296429
\(214\) 0 0
\(215\) 22.8885 1.56099
\(216\) 0 0
\(217\) −8.70820 −0.591151
\(218\) 0 0
\(219\) 0.527864 0.0356697
\(220\) 0 0
\(221\) −25.4508 −1.71201
\(222\) 0 0
\(223\) −10.5623 −0.707304 −0.353652 0.935377i \(-0.615060\pi\)
−0.353652 + 0.935377i \(0.615060\pi\)
\(224\) 0 0
\(225\) 8.09017 0.539345
\(226\) 0 0
\(227\) 16.8541 1.11865 0.559323 0.828950i \(-0.311061\pi\)
0.559323 + 0.828950i \(0.311061\pi\)
\(228\) 0 0
\(229\) −7.79837 −0.515331 −0.257666 0.966234i \(-0.582953\pi\)
−0.257666 + 0.966234i \(0.582953\pi\)
\(230\) 0 0
\(231\) 1.00000 0.0657952
\(232\) 0 0
\(233\) 18.5623 1.21606 0.608029 0.793915i \(-0.291961\pi\)
0.608029 + 0.793915i \(0.291961\pi\)
\(234\) 0 0
\(235\) 13.4164 0.875190
\(236\) 0 0
\(237\) −10.7082 −0.695573
\(238\) 0 0
\(239\) −12.6738 −0.819798 −0.409899 0.912131i \(-0.634436\pi\)
−0.409899 + 0.912131i \(0.634436\pi\)
\(240\) 0 0
\(241\) −25.6525 −1.65242 −0.826211 0.563361i \(-0.809508\pi\)
−0.826211 + 0.563361i \(0.809508\pi\)
\(242\) 0 0
\(243\) −1.00000 −0.0641500
\(244\) 0 0
\(245\) −3.61803 −0.231148
\(246\) 0 0
\(247\) 17.6738 1.12455
\(248\) 0 0
\(249\) 9.18034 0.581780
\(250\) 0 0
\(251\) −3.81966 −0.241095 −0.120547 0.992708i \(-0.538465\pi\)
−0.120547 + 0.992708i \(0.538465\pi\)
\(252\) 0 0
\(253\) −1.00000 −0.0628695
\(254\) 0 0
\(255\) −18.0902 −1.13285
\(256\) 0 0
\(257\) −0.291796 −0.0182017 −0.00910087 0.999959i \(-0.502897\pi\)
−0.00910087 + 0.999959i \(0.502897\pi\)
\(258\) 0 0
\(259\) −1.47214 −0.0914741
\(260\) 0 0
\(261\) −6.23607 −0.386003
\(262\) 0 0
\(263\) 27.1803 1.67601 0.838006 0.545661i \(-0.183721\pi\)
0.838006 + 0.545661i \(0.183721\pi\)
\(264\) 0 0
\(265\) −1.38197 −0.0848935
\(266\) 0 0
\(267\) 16.6180 1.01701
\(268\) 0 0
\(269\) 19.8541 1.21053 0.605263 0.796026i \(-0.293068\pi\)
0.605263 + 0.796026i \(0.293068\pi\)
\(270\) 0 0
\(271\) 6.52786 0.396540 0.198270 0.980147i \(-0.436468\pi\)
0.198270 + 0.980147i \(0.436468\pi\)
\(272\) 0 0
\(273\) 5.09017 0.308071
\(274\) 0 0
\(275\) 8.09017 0.487856
\(276\) 0 0
\(277\) 31.0902 1.86803 0.934014 0.357237i \(-0.116281\pi\)
0.934014 + 0.357237i \(0.116281\pi\)
\(278\) 0 0
\(279\) 8.70820 0.521347
\(280\) 0 0
\(281\) 8.18034 0.487998 0.243999 0.969775i \(-0.421541\pi\)
0.243999 + 0.969775i \(0.421541\pi\)
\(282\) 0 0
\(283\) −2.96556 −0.176284 −0.0881421 0.996108i \(-0.528093\pi\)
−0.0881421 + 0.996108i \(0.528093\pi\)
\(284\) 0 0
\(285\) 12.5623 0.744127
\(286\) 0 0
\(287\) 5.76393 0.340234
\(288\) 0 0
\(289\) 8.00000 0.470588
\(290\) 0 0
\(291\) 12.2361 0.717291
\(292\) 0 0
\(293\) 2.47214 0.144424 0.0722119 0.997389i \(-0.476994\pi\)
0.0722119 + 0.997389i \(0.476994\pi\)
\(294\) 0 0
\(295\) −13.0902 −0.762139
\(296\) 0 0
\(297\) −1.00000 −0.0580259
\(298\) 0 0
\(299\) −5.09017 −0.294372
\(300\) 0 0
\(301\) 6.32624 0.364638
\(302\) 0 0
\(303\) 5.61803 0.322748
\(304\) 0 0
\(305\) 27.3607 1.56667
\(306\) 0 0
\(307\) −29.0689 −1.65905 −0.829524 0.558470i \(-0.811388\pi\)
−0.829524 + 0.558470i \(0.811388\pi\)
\(308\) 0 0
\(309\) 2.47214 0.140635
\(310\) 0 0
\(311\) 15.0902 0.855685 0.427843 0.903853i \(-0.359274\pi\)
0.427843 + 0.903853i \(0.359274\pi\)
\(312\) 0 0
\(313\) −0.583592 −0.0329866 −0.0164933 0.999864i \(-0.505250\pi\)
−0.0164933 + 0.999864i \(0.505250\pi\)
\(314\) 0 0
\(315\) 3.61803 0.203853
\(316\) 0 0
\(317\) 26.3820 1.48176 0.740879 0.671638i \(-0.234409\pi\)
0.740879 + 0.671638i \(0.234409\pi\)
\(318\) 0 0
\(319\) −6.23607 −0.349153
\(320\) 0 0
\(321\) −14.6180 −0.815899
\(322\) 0 0
\(323\) −17.3607 −0.965974
\(324\) 0 0
\(325\) 41.1803 2.28427
\(326\) 0 0
\(327\) 17.5066 0.968116
\(328\) 0 0
\(329\) 3.70820 0.204440
\(330\) 0 0
\(331\) 33.4164 1.83673 0.918366 0.395732i \(-0.129509\pi\)
0.918366 + 0.395732i \(0.129509\pi\)
\(332\) 0 0
\(333\) 1.47214 0.0806726
\(334\) 0 0
\(335\) −26.5066 −1.44821
\(336\) 0 0
\(337\) −12.9787 −0.706996 −0.353498 0.935435i \(-0.615008\pi\)
−0.353498 + 0.935435i \(0.615008\pi\)
\(338\) 0 0
\(339\) −3.14590 −0.170862
\(340\) 0 0
\(341\) 8.70820 0.471576
\(342\) 0 0
\(343\) −1.00000 −0.0539949
\(344\) 0 0
\(345\) −3.61803 −0.194788
\(346\) 0 0
\(347\) 34.3050 1.84159 0.920793 0.390051i \(-0.127543\pi\)
0.920793 + 0.390051i \(0.127543\pi\)
\(348\) 0 0
\(349\) 2.96556 0.158743 0.0793713 0.996845i \(-0.474709\pi\)
0.0793713 + 0.996845i \(0.474709\pi\)
\(350\) 0 0
\(351\) −5.09017 −0.271693
\(352\) 0 0
\(353\) −15.6525 −0.833097 −0.416549 0.909113i \(-0.636760\pi\)
−0.416549 + 0.909113i \(0.636760\pi\)
\(354\) 0 0
\(355\) −15.6525 −0.830747
\(356\) 0 0
\(357\) −5.00000 −0.264628
\(358\) 0 0
\(359\) 16.5623 0.874125 0.437063 0.899431i \(-0.356019\pi\)
0.437063 + 0.899431i \(0.356019\pi\)
\(360\) 0 0
\(361\) −6.94427 −0.365488
\(362\) 0 0
\(363\) 10.0000 0.524864
\(364\) 0 0
\(365\) 1.90983 0.0999651
\(366\) 0 0
\(367\) 27.2705 1.42351 0.711755 0.702428i \(-0.247901\pi\)
0.711755 + 0.702428i \(0.247901\pi\)
\(368\) 0 0
\(369\) −5.76393 −0.300058
\(370\) 0 0
\(371\) −0.381966 −0.0198307
\(372\) 0 0
\(373\) 32.8885 1.70290 0.851452 0.524432i \(-0.175722\pi\)
0.851452 + 0.524432i \(0.175722\pi\)
\(374\) 0 0
\(375\) 11.1803 0.577350
\(376\) 0 0
\(377\) −31.7426 −1.63483
\(378\) 0 0
\(379\) 17.8885 0.918873 0.459436 0.888211i \(-0.348051\pi\)
0.459436 + 0.888211i \(0.348051\pi\)
\(380\) 0 0
\(381\) 4.61803 0.236589
\(382\) 0 0
\(383\) 16.2361 0.829624 0.414812 0.909907i \(-0.363847\pi\)
0.414812 + 0.909907i \(0.363847\pi\)
\(384\) 0 0
\(385\) 3.61803 0.184392
\(386\) 0 0
\(387\) −6.32624 −0.321581
\(388\) 0 0
\(389\) 20.1246 1.02036 0.510179 0.860068i \(-0.329579\pi\)
0.510179 + 0.860068i \(0.329579\pi\)
\(390\) 0 0
\(391\) 5.00000 0.252861
\(392\) 0 0
\(393\) −9.18034 −0.463087
\(394\) 0 0
\(395\) −38.7426 −1.94935
\(396\) 0 0
\(397\) −2.18034 −0.109428 −0.0547141 0.998502i \(-0.517425\pi\)
−0.0547141 + 0.998502i \(0.517425\pi\)
\(398\) 0 0
\(399\) 3.47214 0.173824
\(400\) 0 0
\(401\) −0.708204 −0.0353660 −0.0176830 0.999844i \(-0.505629\pi\)
−0.0176830 + 0.999844i \(0.505629\pi\)
\(402\) 0 0
\(403\) 44.3262 2.20805
\(404\) 0 0
\(405\) −3.61803 −0.179782
\(406\) 0 0
\(407\) 1.47214 0.0729711
\(408\) 0 0
\(409\) 25.1803 1.24509 0.622544 0.782585i \(-0.286099\pi\)
0.622544 + 0.782585i \(0.286099\pi\)
\(410\) 0 0
\(411\) 4.52786 0.223343
\(412\) 0 0
\(413\) −3.61803 −0.178032
\(414\) 0 0
\(415\) 33.2148 1.63045
\(416\) 0 0
\(417\) 0.673762 0.0329943
\(418\) 0 0
\(419\) 3.14590 0.153687 0.0768436 0.997043i \(-0.475516\pi\)
0.0768436 + 0.997043i \(0.475516\pi\)
\(420\) 0 0
\(421\) 38.0344 1.85369 0.926843 0.375450i \(-0.122512\pi\)
0.926843 + 0.375450i \(0.122512\pi\)
\(422\) 0 0
\(423\) −3.70820 −0.180299
\(424\) 0 0
\(425\) −40.4508 −1.96215
\(426\) 0 0
\(427\) 7.56231 0.365966
\(428\) 0 0
\(429\) −5.09017 −0.245756
\(430\) 0 0
\(431\) −18.0344 −0.868688 −0.434344 0.900747i \(-0.643020\pi\)
−0.434344 + 0.900747i \(0.643020\pi\)
\(432\) 0 0
\(433\) −5.70820 −0.274319 −0.137159 0.990549i \(-0.543797\pi\)
−0.137159 + 0.990549i \(0.543797\pi\)
\(434\) 0 0
\(435\) −22.5623 −1.08178
\(436\) 0 0
\(437\) −3.47214 −0.166095
\(438\) 0 0
\(439\) 28.7082 1.37017 0.685084 0.728464i \(-0.259766\pi\)
0.685084 + 0.728464i \(0.259766\pi\)
\(440\) 0 0
\(441\) 1.00000 0.0476190
\(442\) 0 0
\(443\) 8.94427 0.424955 0.212478 0.977166i \(-0.431847\pi\)
0.212478 + 0.977166i \(0.431847\pi\)
\(444\) 0 0
\(445\) 60.1246 2.85018
\(446\) 0 0
\(447\) −19.1246 −0.904563
\(448\) 0 0
\(449\) −19.1459 −0.903551 −0.451775 0.892132i \(-0.649209\pi\)
−0.451775 + 0.892132i \(0.649209\pi\)
\(450\) 0 0
\(451\) −5.76393 −0.271413
\(452\) 0 0
\(453\) 5.23607 0.246012
\(454\) 0 0
\(455\) 18.4164 0.863375
\(456\) 0 0
\(457\) −6.27051 −0.293322 −0.146661 0.989187i \(-0.546853\pi\)
−0.146661 + 0.989187i \(0.546853\pi\)
\(458\) 0 0
\(459\) 5.00000 0.233380
\(460\) 0 0
\(461\) −3.72949 −0.173700 −0.0868498 0.996221i \(-0.527680\pi\)
−0.0868498 + 0.996221i \(0.527680\pi\)
\(462\) 0 0
\(463\) −22.5279 −1.04696 −0.523479 0.852038i \(-0.675366\pi\)
−0.523479 + 0.852038i \(0.675366\pi\)
\(464\) 0 0
\(465\) 31.5066 1.46108
\(466\) 0 0
\(467\) −34.5967 −1.60095 −0.800473 0.599368i \(-0.795418\pi\)
−0.800473 + 0.599368i \(0.795418\pi\)
\(468\) 0 0
\(469\) −7.32624 −0.338294
\(470\) 0 0
\(471\) −14.6525 −0.675150
\(472\) 0 0
\(473\) −6.32624 −0.290881
\(474\) 0 0
\(475\) 28.0902 1.28887
\(476\) 0 0
\(477\) 0.381966 0.0174890
\(478\) 0 0
\(479\) 24.8885 1.13719 0.568593 0.822619i \(-0.307488\pi\)
0.568593 + 0.822619i \(0.307488\pi\)
\(480\) 0 0
\(481\) 7.49342 0.341671
\(482\) 0 0
\(483\) −1.00000 −0.0455016
\(484\) 0 0
\(485\) 44.2705 2.01022
\(486\) 0 0
\(487\) −42.7771 −1.93841 −0.969207 0.246246i \(-0.920803\pi\)
−0.969207 + 0.246246i \(0.920803\pi\)
\(488\) 0 0
\(489\) −14.0902 −0.637180
\(490\) 0 0
\(491\) −5.09017 −0.229716 −0.114858 0.993382i \(-0.536641\pi\)
−0.114858 + 0.993382i \(0.536641\pi\)
\(492\) 0 0
\(493\) 31.1803 1.40429
\(494\) 0 0
\(495\) −3.61803 −0.162619
\(496\) 0 0
\(497\) −4.32624 −0.194058
\(498\) 0 0
\(499\) 21.0344 0.941631 0.470815 0.882232i \(-0.343960\pi\)
0.470815 + 0.882232i \(0.343960\pi\)
\(500\) 0 0
\(501\) −3.18034 −0.142087
\(502\) 0 0
\(503\) 19.3262 0.861714 0.430857 0.902420i \(-0.358211\pi\)
0.430857 + 0.902420i \(0.358211\pi\)
\(504\) 0 0
\(505\) 20.3262 0.904506
\(506\) 0 0
\(507\) −12.9098 −0.573346
\(508\) 0 0
\(509\) 6.76393 0.299806 0.149903 0.988701i \(-0.452104\pi\)
0.149903 + 0.988701i \(0.452104\pi\)
\(510\) 0 0
\(511\) 0.527864 0.0233513
\(512\) 0 0
\(513\) −3.47214 −0.153299
\(514\) 0 0
\(515\) 8.94427 0.394132
\(516\) 0 0
\(517\) −3.70820 −0.163087
\(518\) 0 0
\(519\) −9.18034 −0.402972
\(520\) 0 0
\(521\) 31.3050 1.37149 0.685747 0.727840i \(-0.259475\pi\)
0.685747 + 0.727840i \(0.259475\pi\)
\(522\) 0 0
\(523\) 8.47214 0.370461 0.185230 0.982695i \(-0.440697\pi\)
0.185230 + 0.982695i \(0.440697\pi\)
\(524\) 0 0
\(525\) 8.09017 0.353084
\(526\) 0 0
\(527\) −43.5410 −1.89668
\(528\) 0 0
\(529\) 1.00000 0.0434783
\(530\) 0 0
\(531\) 3.61803 0.157009
\(532\) 0 0
\(533\) −29.3394 −1.27083
\(534\) 0 0
\(535\) −52.8885 −2.28657
\(536\) 0 0
\(537\) 10.2705 0.443205
\(538\) 0 0
\(539\) 1.00000 0.0430730
\(540\) 0 0
\(541\) −5.23607 −0.225116 −0.112558 0.993645i \(-0.535904\pi\)
−0.112558 + 0.993645i \(0.535904\pi\)
\(542\) 0 0
\(543\) 0.236068 0.0101306
\(544\) 0 0
\(545\) 63.3394 2.71316
\(546\) 0 0
\(547\) 32.4508 1.38750 0.693749 0.720217i \(-0.255958\pi\)
0.693749 + 0.720217i \(0.255958\pi\)
\(548\) 0 0
\(549\) −7.56231 −0.322751
\(550\) 0 0
\(551\) −21.6525 −0.922426
\(552\) 0 0
\(553\) −10.7082 −0.455359
\(554\) 0 0
\(555\) 5.32624 0.226086
\(556\) 0 0
\(557\) −24.1803 −1.02455 −0.512277 0.858820i \(-0.671198\pi\)
−0.512277 + 0.858820i \(0.671198\pi\)
\(558\) 0 0
\(559\) −32.2016 −1.36198
\(560\) 0 0
\(561\) 5.00000 0.211100
\(562\) 0 0
\(563\) −37.5623 −1.58306 −0.791531 0.611129i \(-0.790716\pi\)
−0.791531 + 0.611129i \(0.790716\pi\)
\(564\) 0 0
\(565\) −11.3820 −0.478843
\(566\) 0 0
\(567\) −1.00000 −0.0419961
\(568\) 0 0
\(569\) −41.8885 −1.75606 −0.878030 0.478606i \(-0.841142\pi\)
−0.878030 + 0.478606i \(0.841142\pi\)
\(570\) 0 0
\(571\) −21.1803 −0.886370 −0.443185 0.896430i \(-0.646151\pi\)
−0.443185 + 0.896430i \(0.646151\pi\)
\(572\) 0 0
\(573\) 16.6525 0.695667
\(574\) 0 0
\(575\) −8.09017 −0.337383
\(576\) 0 0
\(577\) −5.41641 −0.225488 −0.112744 0.993624i \(-0.535964\pi\)
−0.112744 + 0.993624i \(0.535964\pi\)
\(578\) 0 0
\(579\) −21.7082 −0.902162
\(580\) 0 0
\(581\) 9.18034 0.380865
\(582\) 0 0
\(583\) 0.381966 0.0158194
\(584\) 0 0
\(585\) −18.4164 −0.761425
\(586\) 0 0
\(587\) 42.6180 1.75903 0.879517 0.475867i \(-0.157866\pi\)
0.879517 + 0.475867i \(0.157866\pi\)
\(588\) 0 0
\(589\) 30.2361 1.24586
\(590\) 0 0
\(591\) −22.6180 −0.930382
\(592\) 0 0
\(593\) −35.0132 −1.43782 −0.718909 0.695104i \(-0.755358\pi\)
−0.718909 + 0.695104i \(0.755358\pi\)
\(594\) 0 0
\(595\) −18.0902 −0.741625
\(596\) 0 0
\(597\) 3.79837 0.155457
\(598\) 0 0
\(599\) 4.79837 0.196056 0.0980281 0.995184i \(-0.468746\pi\)
0.0980281 + 0.995184i \(0.468746\pi\)
\(600\) 0 0
\(601\) −29.9098 −1.22005 −0.610024 0.792383i \(-0.708840\pi\)
−0.610024 + 0.792383i \(0.708840\pi\)
\(602\) 0 0
\(603\) 7.32624 0.298347
\(604\) 0 0
\(605\) 36.1803 1.47094
\(606\) 0 0
\(607\) −45.3394 −1.84027 −0.920135 0.391602i \(-0.871921\pi\)
−0.920135 + 0.391602i \(0.871921\pi\)
\(608\) 0 0
\(609\) −6.23607 −0.252698
\(610\) 0 0
\(611\) −18.8754 −0.763616
\(612\) 0 0
\(613\) 37.0689 1.49720 0.748599 0.663023i \(-0.230727\pi\)
0.748599 + 0.663023i \(0.230727\pi\)
\(614\) 0 0
\(615\) −20.8541 −0.840919
\(616\) 0 0
\(617\) −30.0344 −1.20914 −0.604571 0.796552i \(-0.706655\pi\)
−0.604571 + 0.796552i \(0.706655\pi\)
\(618\) 0 0
\(619\) −19.9230 −0.800772 −0.400386 0.916346i \(-0.631124\pi\)
−0.400386 + 0.916346i \(0.631124\pi\)
\(620\) 0 0
\(621\) 1.00000 0.0401286
\(622\) 0 0
\(623\) 16.6180 0.665787
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) −3.47214 −0.138664
\(628\) 0 0
\(629\) −7.36068 −0.293490
\(630\) 0 0
\(631\) −0.527864 −0.0210139 −0.0105070 0.999945i \(-0.503345\pi\)
−0.0105070 + 0.999945i \(0.503345\pi\)
\(632\) 0 0
\(633\) 18.2361 0.724819
\(634\) 0 0
\(635\) 16.7082 0.663045
\(636\) 0 0
\(637\) 5.09017 0.201680
\(638\) 0 0
\(639\) 4.32624 0.171143
\(640\) 0 0
\(641\) −16.6180 −0.656373 −0.328186 0.944613i \(-0.606437\pi\)
−0.328186 + 0.944613i \(0.606437\pi\)
\(642\) 0 0
\(643\) 10.9098 0.430242 0.215121 0.976587i \(-0.430985\pi\)
0.215121 + 0.976587i \(0.430985\pi\)
\(644\) 0 0
\(645\) −22.8885 −0.901236
\(646\) 0 0
\(647\) −15.9098 −0.625480 −0.312740 0.949839i \(-0.601247\pi\)
−0.312740 + 0.949839i \(0.601247\pi\)
\(648\) 0 0
\(649\) 3.61803 0.142020
\(650\) 0 0
\(651\) 8.70820 0.341301
\(652\) 0 0
\(653\) 34.5623 1.35253 0.676264 0.736660i \(-0.263598\pi\)
0.676264 + 0.736660i \(0.263598\pi\)
\(654\) 0 0
\(655\) −33.2148 −1.29781
\(656\) 0 0
\(657\) −0.527864 −0.0205939
\(658\) 0 0
\(659\) 1.00000 0.0389545 0.0194772 0.999810i \(-0.493800\pi\)
0.0194772 + 0.999810i \(0.493800\pi\)
\(660\) 0 0
\(661\) −38.7082 −1.50557 −0.752787 0.658264i \(-0.771291\pi\)
−0.752787 + 0.658264i \(0.771291\pi\)
\(662\) 0 0
\(663\) 25.4508 0.988429
\(664\) 0 0
\(665\) 12.5623 0.487145
\(666\) 0 0
\(667\) 6.23607 0.241462
\(668\) 0 0
\(669\) 10.5623 0.408362
\(670\) 0 0
\(671\) −7.56231 −0.291940
\(672\) 0 0
\(673\) 29.9443 1.15427 0.577133 0.816650i \(-0.304171\pi\)
0.577133 + 0.816650i \(0.304171\pi\)
\(674\) 0 0
\(675\) −8.09017 −0.311391
\(676\) 0 0
\(677\) −0.437694 −0.0168220 −0.00841098 0.999965i \(-0.502677\pi\)
−0.00841098 + 0.999965i \(0.502677\pi\)
\(678\) 0 0
\(679\) 12.2361 0.469577
\(680\) 0 0
\(681\) −16.8541 −0.645851
\(682\) 0 0
\(683\) 15.9443 0.610091 0.305045 0.952338i \(-0.401328\pi\)
0.305045 + 0.952338i \(0.401328\pi\)
\(684\) 0 0
\(685\) 16.3820 0.625923
\(686\) 0 0
\(687\) 7.79837 0.297527
\(688\) 0 0
\(689\) 1.94427 0.0740709
\(690\) 0 0
\(691\) −14.2148 −0.540756 −0.270378 0.962754i \(-0.587149\pi\)
−0.270378 + 0.962754i \(0.587149\pi\)
\(692\) 0 0
\(693\) −1.00000 −0.0379869
\(694\) 0 0
\(695\) 2.43769 0.0924670
\(696\) 0 0
\(697\) 28.8197 1.09162
\(698\) 0 0
\(699\) −18.5623 −0.702091
\(700\) 0 0
\(701\) −25.9098 −0.978601 −0.489300 0.872115i \(-0.662748\pi\)
−0.489300 + 0.872115i \(0.662748\pi\)
\(702\) 0 0
\(703\) 5.11146 0.192782
\(704\) 0 0
\(705\) −13.4164 −0.505291
\(706\) 0 0
\(707\) 5.61803 0.211288
\(708\) 0 0
\(709\) 27.1591 1.01998 0.509990 0.860180i \(-0.329649\pi\)
0.509990 + 0.860180i \(0.329649\pi\)
\(710\) 0 0
\(711\) 10.7082 0.401589
\(712\) 0 0
\(713\) −8.70820 −0.326125
\(714\) 0 0
\(715\) −18.4164 −0.688735
\(716\) 0 0
\(717\) 12.6738 0.473310
\(718\) 0 0
\(719\) 20.7771 0.774855 0.387427 0.921900i \(-0.373364\pi\)
0.387427 + 0.921900i \(0.373364\pi\)
\(720\) 0 0
\(721\) 2.47214 0.0920672
\(722\) 0 0
\(723\) 25.6525 0.954026
\(724\) 0 0
\(725\) −50.4508 −1.87370
\(726\) 0 0
\(727\) 37.2492 1.38150 0.690749 0.723095i \(-0.257281\pi\)
0.690749 + 0.723095i \(0.257281\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) 31.6312 1.16992
\(732\) 0 0
\(733\) 2.29180 0.0846494 0.0423247 0.999104i \(-0.486524\pi\)
0.0423247 + 0.999104i \(0.486524\pi\)
\(734\) 0 0
\(735\) 3.61803 0.133453
\(736\) 0 0
\(737\) 7.32624 0.269865
\(738\) 0 0
\(739\) 20.2361 0.744396 0.372198 0.928153i \(-0.378604\pi\)
0.372198 + 0.928153i \(0.378604\pi\)
\(740\) 0 0
\(741\) −17.6738 −0.649262
\(742\) 0 0
\(743\) −6.85410 −0.251453 −0.125726 0.992065i \(-0.540126\pi\)
−0.125726 + 0.992065i \(0.540126\pi\)
\(744\) 0 0
\(745\) −69.1935 −2.53505
\(746\) 0 0
\(747\) −9.18034 −0.335891
\(748\) 0 0
\(749\) −14.6180 −0.534131
\(750\) 0 0
\(751\) 22.6738 0.827377 0.413689 0.910418i \(-0.364240\pi\)
0.413689 + 0.910418i \(0.364240\pi\)
\(752\) 0 0
\(753\) 3.81966 0.139196
\(754\) 0 0
\(755\) 18.9443 0.689453
\(756\) 0 0
\(757\) 16.4164 0.596664 0.298332 0.954462i \(-0.403570\pi\)
0.298332 + 0.954462i \(0.403570\pi\)
\(758\) 0 0
\(759\) 1.00000 0.0362977
\(760\) 0 0
\(761\) −7.05573 −0.255770 −0.127885 0.991789i \(-0.540819\pi\)
−0.127885 + 0.991789i \(0.540819\pi\)
\(762\) 0 0
\(763\) 17.5066 0.633781
\(764\) 0 0
\(765\) 18.0902 0.654051
\(766\) 0 0
\(767\) 18.4164 0.664978
\(768\) 0 0
\(769\) −13.3475 −0.481324 −0.240662 0.970609i \(-0.577365\pi\)
−0.240662 + 0.970609i \(0.577365\pi\)
\(770\) 0 0
\(771\) 0.291796 0.0105088
\(772\) 0 0
\(773\) −1.29180 −0.0464627 −0.0232313 0.999730i \(-0.507395\pi\)
−0.0232313 + 0.999730i \(0.507395\pi\)
\(774\) 0 0
\(775\) 70.4508 2.53067
\(776\) 0 0
\(777\) 1.47214 0.0528126
\(778\) 0 0
\(779\) −20.0132 −0.717046
\(780\) 0 0
\(781\) 4.32624 0.154805
\(782\) 0 0
\(783\) 6.23607 0.222859
\(784\) 0 0
\(785\) −53.0132 −1.89212
\(786\) 0 0
\(787\) −24.7426 −0.881980 −0.440990 0.897512i \(-0.645373\pi\)
−0.440990 + 0.897512i \(0.645373\pi\)
\(788\) 0 0
\(789\) −27.1803 −0.967646
\(790\) 0 0
\(791\) −3.14590 −0.111855
\(792\) 0 0
\(793\) −38.4934 −1.36694
\(794\) 0 0
\(795\) 1.38197 0.0490133
\(796\) 0 0
\(797\) 40.0689 1.41931 0.709656 0.704548i \(-0.248850\pi\)
0.709656 + 0.704548i \(0.248850\pi\)
\(798\) 0 0
\(799\) 18.5410 0.655934
\(800\) 0 0
\(801\) −16.6180 −0.587169
\(802\) 0 0
\(803\) −0.527864 −0.0186279
\(804\) 0 0
\(805\) −3.61803 −0.127519
\(806\) 0 0
\(807\) −19.8541 −0.698897
\(808\) 0 0
\(809\) 47.3820 1.66586 0.832931 0.553377i \(-0.186661\pi\)
0.832931 + 0.553377i \(0.186661\pi\)
\(810\) 0 0
\(811\) 47.3607 1.66306 0.831529 0.555481i \(-0.187466\pi\)
0.831529 + 0.555481i \(0.187466\pi\)
\(812\) 0 0
\(813\) −6.52786 −0.228942
\(814\) 0 0
\(815\) −50.9787 −1.78571
\(816\) 0 0
\(817\) −21.9656 −0.768478
\(818\) 0 0
\(819\) −5.09017 −0.177865
\(820\) 0 0
\(821\) 7.34752 0.256430 0.128215 0.991746i \(-0.459075\pi\)
0.128215 + 0.991746i \(0.459075\pi\)
\(822\) 0 0
\(823\) 13.7984 0.480981 0.240491 0.970651i \(-0.422692\pi\)
0.240491 + 0.970651i \(0.422692\pi\)
\(824\) 0 0
\(825\) −8.09017 −0.281664
\(826\) 0 0
\(827\) −53.7984 −1.87075 −0.935376 0.353654i \(-0.884939\pi\)
−0.935376 + 0.353654i \(0.884939\pi\)
\(828\) 0 0
\(829\) −4.23607 −0.147125 −0.0735624 0.997291i \(-0.523437\pi\)
−0.0735624 + 0.997291i \(0.523437\pi\)
\(830\) 0 0
\(831\) −31.0902 −1.07851
\(832\) 0 0
\(833\) −5.00000 −0.173240
\(834\) 0 0
\(835\) −11.5066 −0.398202
\(836\) 0 0
\(837\) −8.70820 −0.301000
\(838\) 0 0
\(839\) 7.27051 0.251006 0.125503 0.992093i \(-0.459946\pi\)
0.125503 + 0.992093i \(0.459946\pi\)
\(840\) 0 0
\(841\) 9.88854 0.340984
\(842\) 0 0
\(843\) −8.18034 −0.281746
\(844\) 0 0
\(845\) −46.7082 −1.60681
\(846\) 0 0
\(847\) 10.0000 0.343604
\(848\) 0 0
\(849\) 2.96556 0.101778
\(850\) 0 0
\(851\) −1.47214 −0.0504642
\(852\) 0 0
\(853\) −44.5410 −1.52506 −0.762528 0.646956i \(-0.776042\pi\)
−0.762528 + 0.646956i \(0.776042\pi\)
\(854\) 0 0
\(855\) −12.5623 −0.429622
\(856\) 0 0
\(857\) 22.6525 0.773794 0.386897 0.922123i \(-0.373547\pi\)
0.386897 + 0.922123i \(0.373547\pi\)
\(858\) 0 0
\(859\) −23.8885 −0.815067 −0.407533 0.913190i \(-0.633611\pi\)
−0.407533 + 0.913190i \(0.633611\pi\)
\(860\) 0 0
\(861\) −5.76393 −0.196434
\(862\) 0 0
\(863\) 53.4853 1.82066 0.910330 0.413883i \(-0.135828\pi\)
0.910330 + 0.413883i \(0.135828\pi\)
\(864\) 0 0
\(865\) −33.2148 −1.12934
\(866\) 0 0
\(867\) −8.00000 −0.271694
\(868\) 0 0
\(869\) 10.7082 0.363251
\(870\) 0 0
\(871\) 37.2918 1.26358
\(872\) 0 0
\(873\) −12.2361 −0.414128
\(874\) 0 0
\(875\) 11.1803 0.377964
\(876\) 0 0
\(877\) −57.3050 −1.93505 −0.967525 0.252774i \(-0.918657\pi\)
−0.967525 + 0.252774i \(0.918657\pi\)
\(878\) 0 0
\(879\) −2.47214 −0.0833831
\(880\) 0 0
\(881\) −10.8754 −0.366401 −0.183201 0.983076i \(-0.558646\pi\)
−0.183201 + 0.983076i \(0.558646\pi\)
\(882\) 0 0
\(883\) −26.3262 −0.885948 −0.442974 0.896534i \(-0.646077\pi\)
−0.442974 + 0.896534i \(0.646077\pi\)
\(884\) 0 0
\(885\) 13.0902 0.440021
\(886\) 0 0
\(887\) 14.7295 0.494568 0.247284 0.968943i \(-0.420462\pi\)
0.247284 + 0.968943i \(0.420462\pi\)
\(888\) 0 0
\(889\) 4.61803 0.154884
\(890\) 0 0
\(891\) 1.00000 0.0335013
\(892\) 0 0
\(893\) −12.8754 −0.430858
\(894\) 0 0
\(895\) 37.1591 1.24209
\(896\) 0 0
\(897\) 5.09017 0.169956
\(898\) 0 0
\(899\) −54.3050 −1.81117
\(900\) 0 0
\(901\) −1.90983 −0.0636257
\(902\) 0 0
\(903\) −6.32624 −0.210524
\(904\) 0 0
\(905\) 0.854102 0.0283913
\(906\) 0 0
\(907\) 48.9230 1.62446 0.812231 0.583337i \(-0.198253\pi\)
0.812231 + 0.583337i \(0.198253\pi\)
\(908\) 0 0
\(909\) −5.61803 −0.186338
\(910\) 0 0
\(911\) −42.5410 −1.40945 −0.704723 0.709482i \(-0.748929\pi\)
−0.704723 + 0.709482i \(0.748929\pi\)
\(912\) 0 0
\(913\) −9.18034 −0.303825
\(914\) 0 0
\(915\) −27.3607 −0.904516
\(916\) 0 0
\(917\) −9.18034 −0.303162
\(918\) 0 0
\(919\) −19.0000 −0.626752 −0.313376 0.949629i \(-0.601460\pi\)
−0.313376 + 0.949629i \(0.601460\pi\)
\(920\) 0 0
\(921\) 29.0689 0.957852
\(922\) 0 0
\(923\) 22.0213 0.724839
\(924\) 0 0
\(925\) 11.9098 0.391593
\(926\) 0 0
\(927\) −2.47214 −0.0811956
\(928\) 0 0
\(929\) 39.0344 1.28068 0.640339 0.768092i \(-0.278794\pi\)
0.640339 + 0.768092i \(0.278794\pi\)
\(930\) 0 0
\(931\) 3.47214 0.113795
\(932\) 0 0
\(933\) −15.0902 −0.494030
\(934\) 0 0
\(935\) 18.0902 0.591612
\(936\) 0 0
\(937\) −13.0000 −0.424691 −0.212346 0.977195i \(-0.568110\pi\)
−0.212346 + 0.977195i \(0.568110\pi\)
\(938\) 0 0
\(939\) 0.583592 0.0190448
\(940\) 0 0
\(941\) −10.8328 −0.353140 −0.176570 0.984288i \(-0.556500\pi\)
−0.176570 + 0.984288i \(0.556500\pi\)
\(942\) 0 0
\(943\) 5.76393 0.187699
\(944\) 0 0
\(945\) −3.61803 −0.117695
\(946\) 0 0
\(947\) −2.36068 −0.0767118 −0.0383559 0.999264i \(-0.512212\pi\)
−0.0383559 + 0.999264i \(0.512212\pi\)
\(948\) 0 0
\(949\) −2.68692 −0.0872210
\(950\) 0 0
\(951\) −26.3820 −0.855494
\(952\) 0 0
\(953\) 23.6312 0.765489 0.382745 0.923854i \(-0.374979\pi\)
0.382745 + 0.923854i \(0.374979\pi\)
\(954\) 0 0
\(955\) 60.2492 1.94962
\(956\) 0 0
\(957\) 6.23607 0.201583
\(958\) 0 0
\(959\) 4.52786 0.146212
\(960\) 0 0
\(961\) 44.8328 1.44622
\(962\) 0 0
\(963\) 14.6180 0.471060
\(964\) 0 0
\(965\) −78.5410 −2.52832
\(966\) 0 0
\(967\) 46.9443 1.50963 0.754813 0.655940i \(-0.227728\pi\)
0.754813 + 0.655940i \(0.227728\pi\)
\(968\) 0 0
\(969\) 17.3607 0.557705
\(970\) 0 0
\(971\) −15.7426 −0.505206 −0.252603 0.967570i \(-0.581287\pi\)
−0.252603 + 0.967570i \(0.581287\pi\)
\(972\) 0 0
\(973\) 0.673762 0.0215998
\(974\) 0 0
\(975\) −41.1803 −1.31883
\(976\) 0 0
\(977\) 59.7426 1.91134 0.955668 0.294445i \(-0.0951349\pi\)
0.955668 + 0.294445i \(0.0951349\pi\)
\(978\) 0 0
\(979\) −16.6180 −0.531115
\(980\) 0 0
\(981\) −17.5066 −0.558942
\(982\) 0 0
\(983\) −8.70820 −0.277749 −0.138874 0.990310i \(-0.544348\pi\)
−0.138874 + 0.990310i \(0.544348\pi\)
\(984\) 0 0
\(985\) −81.8328 −2.60741
\(986\) 0 0
\(987\) −3.70820 −0.118033
\(988\) 0 0
\(989\) 6.32624 0.201163
\(990\) 0 0
\(991\) 10.7295 0.340833 0.170417 0.985372i \(-0.445489\pi\)
0.170417 + 0.985372i \(0.445489\pi\)
\(992\) 0 0
\(993\) −33.4164 −1.06044
\(994\) 0 0
\(995\) 13.7426 0.435671
\(996\) 0 0
\(997\) −17.7639 −0.562589 −0.281295 0.959621i \(-0.590764\pi\)
−0.281295 + 0.959621i \(0.590764\pi\)
\(998\) 0 0
\(999\) −1.47214 −0.0465763
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 7728.2.a.w.1.1 2
4.3 odd 2 1932.2.a.f.1.1 2
12.11 even 2 5796.2.a.n.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1932.2.a.f.1.1 2 4.3 odd 2
5796.2.a.n.1.2 2 12.11 even 2
7728.2.a.w.1.1 2 1.1 even 1 trivial