Properties

Label 7728.2.a.bb.1.2
Level $7728$
Weight $2$
Character 7728.1
Self dual yes
Analytic conductor $61.708$
Analytic rank $1$
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: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{10})^+\)
Defining polynomial: \(x^{2} - x - 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
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 1.2
Root \(1.61803\) of defining polynomial
Character \(\chi\) \(=\) 7728.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{3} +3.85410 q^{5} -1.00000 q^{7} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{3} +3.85410 q^{5} -1.00000 q^{7} +1.00000 q^{9} -3.00000 q^{11} -0.381966 q^{13} -3.85410 q^{15} -1.47214 q^{17} -0.236068 q^{19} +1.00000 q^{21} +1.00000 q^{23} +9.85410 q^{25} -1.00000 q^{27} -5.00000 q^{29} -5.76393 q^{31} +3.00000 q^{33} -3.85410 q^{35} -4.23607 q^{37} +0.381966 q^{39} +3.00000 q^{41} +1.61803 q^{43} +3.85410 q^{45} +3.23607 q^{47} +1.00000 q^{49} +1.47214 q^{51} -2.32624 q^{53} -11.5623 q^{55} +0.236068 q^{57} +13.5623 q^{59} -4.85410 q^{61} -1.00000 q^{63} -1.47214 q^{65} -2.32624 q^{67} -1.00000 q^{69} -7.61803 q^{71} -5.94427 q^{73} -9.85410 q^{75} +3.00000 q^{77} +9.47214 q^{79} +1.00000 q^{81} -12.2361 q^{83} -5.67376 q^{85} +5.00000 q^{87} -6.56231 q^{89} +0.381966 q^{91} +5.76393 q^{93} -0.909830 q^{95} -1.94427 q^{97} -3.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{3} + q^{5} - 2 q^{7} + 2 q^{9} + O(q^{10}) \) \( 2 q - 2 q^{3} + q^{5} - 2 q^{7} + 2 q^{9} - 6 q^{11} - 3 q^{13} - q^{15} + 6 q^{17} + 4 q^{19} + 2 q^{21} + 2 q^{23} + 13 q^{25} - 2 q^{27} - 10 q^{29} - 16 q^{31} + 6 q^{33} - q^{35} - 4 q^{37} + 3 q^{39} + 6 q^{41} + q^{43} + q^{45} + 2 q^{47} + 2 q^{49} - 6 q^{51} + 11 q^{53} - 3 q^{55} - 4 q^{57} + 7 q^{59} - 3 q^{61} - 2 q^{63} + 6 q^{65} + 11 q^{67} - 2 q^{69} - 13 q^{71} + 6 q^{73} - 13 q^{75} + 6 q^{77} + 10 q^{79} + 2 q^{81} - 20 q^{83} - 27 q^{85} + 10 q^{87} + 7 q^{89} + 3 q^{91} + 16 q^{93} - 13 q^{95} + 14 q^{97} - 6 q^{99} + O(q^{100}) \)

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
\(5\) 3.85410 1.72361 0.861803 0.507242i \(-0.169335\pi\)
0.861803 + 0.507242i \(0.169335\pi\)
\(6\) 0 0
\(7\) −1.00000 −0.377964
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) −3.00000 −0.904534 −0.452267 0.891883i \(-0.649385\pi\)
−0.452267 + 0.891883i \(0.649385\pi\)
\(12\) 0 0
\(13\) −0.381966 −0.105938 −0.0529692 0.998596i \(-0.516869\pi\)
−0.0529692 + 0.998596i \(0.516869\pi\)
\(14\) 0 0
\(15\) −3.85410 −0.995125
\(16\) 0 0
\(17\) −1.47214 −0.357045 −0.178523 0.983936i \(-0.557132\pi\)
−0.178523 + 0.983936i \(0.557132\pi\)
\(18\) 0 0
\(19\) −0.236068 −0.0541577 −0.0270789 0.999633i \(-0.508621\pi\)
−0.0270789 + 0.999633i \(0.508621\pi\)
\(20\) 0 0
\(21\) 1.00000 0.218218
\(22\) 0 0
\(23\) 1.00000 0.208514
\(24\) 0 0
\(25\) 9.85410 1.97082
\(26\) 0 0
\(27\) −1.00000 −0.192450
\(28\) 0 0
\(29\) −5.00000 −0.928477 −0.464238 0.885710i \(-0.653672\pi\)
−0.464238 + 0.885710i \(0.653672\pi\)
\(30\) 0 0
\(31\) −5.76393 −1.03523 −0.517616 0.855613i \(-0.673181\pi\)
−0.517616 + 0.855613i \(0.673181\pi\)
\(32\) 0 0
\(33\) 3.00000 0.522233
\(34\) 0 0
\(35\) −3.85410 −0.651462
\(36\) 0 0
\(37\) −4.23607 −0.696405 −0.348203 0.937419i \(-0.613208\pi\)
−0.348203 + 0.937419i \(0.613208\pi\)
\(38\) 0 0
\(39\) 0.381966 0.0611635
\(40\) 0 0
\(41\) 3.00000 0.468521 0.234261 0.972174i \(-0.424733\pi\)
0.234261 + 0.972174i \(0.424733\pi\)
\(42\) 0 0
\(43\) 1.61803 0.246748 0.123374 0.992360i \(-0.460629\pi\)
0.123374 + 0.992360i \(0.460629\pi\)
\(44\) 0 0
\(45\) 3.85410 0.574536
\(46\) 0 0
\(47\) 3.23607 0.472029 0.236015 0.971750i \(-0.424159\pi\)
0.236015 + 0.971750i \(0.424159\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) 1.47214 0.206140
\(52\) 0 0
\(53\) −2.32624 −0.319533 −0.159767 0.987155i \(-0.551074\pi\)
−0.159767 + 0.987155i \(0.551074\pi\)
\(54\) 0 0
\(55\) −11.5623 −1.55906
\(56\) 0 0
\(57\) 0.236068 0.0312680
\(58\) 0 0
\(59\) 13.5623 1.76566 0.882831 0.469691i \(-0.155635\pi\)
0.882831 + 0.469691i \(0.155635\pi\)
\(60\) 0 0
\(61\) −4.85410 −0.621504 −0.310752 0.950491i \(-0.600581\pi\)
−0.310752 + 0.950491i \(0.600581\pi\)
\(62\) 0 0
\(63\) −1.00000 −0.125988
\(64\) 0 0
\(65\) −1.47214 −0.182596
\(66\) 0 0
\(67\) −2.32624 −0.284195 −0.142098 0.989853i \(-0.545385\pi\)
−0.142098 + 0.989853i \(0.545385\pi\)
\(68\) 0 0
\(69\) −1.00000 −0.120386
\(70\) 0 0
\(71\) −7.61803 −0.904094 −0.452047 0.891994i \(-0.649306\pi\)
−0.452047 + 0.891994i \(0.649306\pi\)
\(72\) 0 0
\(73\) −5.94427 −0.695724 −0.347862 0.937546i \(-0.613092\pi\)
−0.347862 + 0.937546i \(0.613092\pi\)
\(74\) 0 0
\(75\) −9.85410 −1.13785
\(76\) 0 0
\(77\) 3.00000 0.341882
\(78\) 0 0
\(79\) 9.47214 1.06570 0.532849 0.846210i \(-0.321121\pi\)
0.532849 + 0.846210i \(0.321121\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) −12.2361 −1.34308 −0.671541 0.740967i \(-0.734367\pi\)
−0.671541 + 0.740967i \(0.734367\pi\)
\(84\) 0 0
\(85\) −5.67376 −0.615406
\(86\) 0 0
\(87\) 5.00000 0.536056
\(88\) 0 0
\(89\) −6.56231 −0.695603 −0.347802 0.937568i \(-0.613072\pi\)
−0.347802 + 0.937568i \(0.613072\pi\)
\(90\) 0 0
\(91\) 0.381966 0.0400409
\(92\) 0 0
\(93\) 5.76393 0.597692
\(94\) 0 0
\(95\) −0.909830 −0.0933466
\(96\) 0 0
\(97\) −1.94427 −0.197411 −0.0987055 0.995117i \(-0.531470\pi\)
−0.0987055 + 0.995117i \(0.531470\pi\)
\(98\) 0 0
\(99\) −3.00000 −0.301511
\(100\) 0 0
\(101\) −7.56231 −0.752478 −0.376239 0.926523i \(-0.622783\pi\)
−0.376239 + 0.926523i \(0.622783\pi\)
\(102\) 0 0
\(103\) −12.9443 −1.27544 −0.637719 0.770270i \(-0.720122\pi\)
−0.637719 + 0.770270i \(0.720122\pi\)
\(104\) 0 0
\(105\) 3.85410 0.376122
\(106\) 0 0
\(107\) −12.5623 −1.21444 −0.607222 0.794532i \(-0.707716\pi\)
−0.607222 + 0.794532i \(0.707716\pi\)
\(108\) 0 0
\(109\) −4.90983 −0.470276 −0.235138 0.971962i \(-0.575554\pi\)
−0.235138 + 0.971962i \(0.575554\pi\)
\(110\) 0 0
\(111\) 4.23607 0.402070
\(112\) 0 0
\(113\) −8.03444 −0.755817 −0.377908 0.925843i \(-0.623357\pi\)
−0.377908 + 0.925843i \(0.623357\pi\)
\(114\) 0 0
\(115\) 3.85410 0.359397
\(116\) 0 0
\(117\) −0.381966 −0.0353128
\(118\) 0 0
\(119\) 1.47214 0.134950
\(120\) 0 0
\(121\) −2.00000 −0.181818
\(122\) 0 0
\(123\) −3.00000 −0.270501
\(124\) 0 0
\(125\) 18.7082 1.67331
\(126\) 0 0
\(127\) −3.32624 −0.295156 −0.147578 0.989050i \(-0.547148\pi\)
−0.147578 + 0.989050i \(0.547148\pi\)
\(128\) 0 0
\(129\) −1.61803 −0.142460
\(130\) 0 0
\(131\) −3.00000 −0.262111 −0.131056 0.991375i \(-0.541837\pi\)
−0.131056 + 0.991375i \(0.541837\pi\)
\(132\) 0 0
\(133\) 0.236068 0.0204697
\(134\) 0 0
\(135\) −3.85410 −0.331708
\(136\) 0 0
\(137\) 8.41641 0.719062 0.359531 0.933133i \(-0.382937\pi\)
0.359531 + 0.933133i \(0.382937\pi\)
\(138\) 0 0
\(139\) 10.7984 0.915906 0.457953 0.888976i \(-0.348583\pi\)
0.457953 + 0.888976i \(0.348583\pi\)
\(140\) 0 0
\(141\) −3.23607 −0.272526
\(142\) 0 0
\(143\) 1.14590 0.0958248
\(144\) 0 0
\(145\) −19.2705 −1.60033
\(146\) 0 0
\(147\) −1.00000 −0.0824786
\(148\) 0 0
\(149\) 16.1803 1.32555 0.662773 0.748821i \(-0.269380\pi\)
0.662773 + 0.748821i \(0.269380\pi\)
\(150\) 0 0
\(151\) 22.6525 1.84343 0.921716 0.387865i \(-0.126787\pi\)
0.921716 + 0.387865i \(0.126787\pi\)
\(152\) 0 0
\(153\) −1.47214 −0.119015
\(154\) 0 0
\(155\) −22.2148 −1.78433
\(156\) 0 0
\(157\) −6.76393 −0.539821 −0.269910 0.962885i \(-0.586994\pi\)
−0.269910 + 0.962885i \(0.586994\pi\)
\(158\) 0 0
\(159\) 2.32624 0.184483
\(160\) 0 0
\(161\) −1.00000 −0.0788110
\(162\) 0 0
\(163\) −5.67376 −0.444403 −0.222202 0.975001i \(-0.571324\pi\)
−0.222202 + 0.975001i \(0.571324\pi\)
\(164\) 0 0
\(165\) 11.5623 0.900124
\(166\) 0 0
\(167\) −8.41641 −0.651281 −0.325641 0.945494i \(-0.605580\pi\)
−0.325641 + 0.945494i \(0.605580\pi\)
\(168\) 0 0
\(169\) −12.8541 −0.988777
\(170\) 0 0
\(171\) −0.236068 −0.0180526
\(172\) 0 0
\(173\) −17.4721 −1.32838 −0.664191 0.747563i \(-0.731224\pi\)
−0.664191 + 0.747563i \(0.731224\pi\)
\(174\) 0 0
\(175\) −9.85410 −0.744900
\(176\) 0 0
\(177\) −13.5623 −1.01941
\(178\) 0 0
\(179\) −21.5623 −1.61164 −0.805821 0.592159i \(-0.798276\pi\)
−0.805821 + 0.592159i \(0.798276\pi\)
\(180\) 0 0
\(181\) 6.41641 0.476928 0.238464 0.971151i \(-0.423356\pi\)
0.238464 + 0.971151i \(0.423356\pi\)
\(182\) 0 0
\(183\) 4.85410 0.358826
\(184\) 0 0
\(185\) −16.3262 −1.20033
\(186\) 0 0
\(187\) 4.41641 0.322960
\(188\) 0 0
\(189\) 1.00000 0.0727393
\(190\) 0 0
\(191\) −12.7639 −0.923566 −0.461783 0.886993i \(-0.652790\pi\)
−0.461783 + 0.886993i \(0.652790\pi\)
\(192\) 0 0
\(193\) 24.6525 1.77452 0.887262 0.461266i \(-0.152605\pi\)
0.887262 + 0.461266i \(0.152605\pi\)
\(194\) 0 0
\(195\) 1.47214 0.105422
\(196\) 0 0
\(197\) 4.09017 0.291413 0.145706 0.989328i \(-0.453455\pi\)
0.145706 + 0.989328i \(0.453455\pi\)
\(198\) 0 0
\(199\) −10.7984 −0.765476 −0.382738 0.923857i \(-0.625019\pi\)
−0.382738 + 0.923857i \(0.625019\pi\)
\(200\) 0 0
\(201\) 2.32624 0.164080
\(202\) 0 0
\(203\) 5.00000 0.350931
\(204\) 0 0
\(205\) 11.5623 0.807546
\(206\) 0 0
\(207\) 1.00000 0.0695048
\(208\) 0 0
\(209\) 0.708204 0.0489875
\(210\) 0 0
\(211\) −4.23607 −0.291623 −0.145811 0.989312i \(-0.546579\pi\)
−0.145811 + 0.989312i \(0.546579\pi\)
\(212\) 0 0
\(213\) 7.61803 0.521979
\(214\) 0 0
\(215\) 6.23607 0.425296
\(216\) 0 0
\(217\) 5.76393 0.391281
\(218\) 0 0
\(219\) 5.94427 0.401677
\(220\) 0 0
\(221\) 0.562306 0.0378248
\(222\) 0 0
\(223\) 4.32624 0.289706 0.144853 0.989453i \(-0.453729\pi\)
0.144853 + 0.989453i \(0.453729\pi\)
\(224\) 0 0
\(225\) 9.85410 0.656940
\(226\) 0 0
\(227\) −0.618034 −0.0410204 −0.0205102 0.999790i \(-0.506529\pi\)
−0.0205102 + 0.999790i \(0.506529\pi\)
\(228\) 0 0
\(229\) −2.43769 −0.161087 −0.0805437 0.996751i \(-0.525666\pi\)
−0.0805437 + 0.996751i \(0.525666\pi\)
\(230\) 0 0
\(231\) −3.00000 −0.197386
\(232\) 0 0
\(233\) −16.1459 −1.05775 −0.528876 0.848699i \(-0.677387\pi\)
−0.528876 + 0.848699i \(0.677387\pi\)
\(234\) 0 0
\(235\) 12.4721 0.813592
\(236\) 0 0
\(237\) −9.47214 −0.615281
\(238\) 0 0
\(239\) 5.67376 0.367005 0.183503 0.983019i \(-0.441256\pi\)
0.183503 + 0.983019i \(0.441256\pi\)
\(240\) 0 0
\(241\) −2.05573 −0.132421 −0.0662105 0.997806i \(-0.521091\pi\)
−0.0662105 + 0.997806i \(0.521091\pi\)
\(242\) 0 0
\(243\) −1.00000 −0.0641500
\(244\) 0 0
\(245\) 3.85410 0.246230
\(246\) 0 0
\(247\) 0.0901699 0.00573738
\(248\) 0 0
\(249\) 12.2361 0.775429
\(250\) 0 0
\(251\) −13.2361 −0.835453 −0.417727 0.908573i \(-0.637173\pi\)
−0.417727 + 0.908573i \(0.637173\pi\)
\(252\) 0 0
\(253\) −3.00000 −0.188608
\(254\) 0 0
\(255\) 5.67376 0.355305
\(256\) 0 0
\(257\) −0.763932 −0.0476528 −0.0238264 0.999716i \(-0.507585\pi\)
−0.0238264 + 0.999716i \(0.507585\pi\)
\(258\) 0 0
\(259\) 4.23607 0.263216
\(260\) 0 0
\(261\) −5.00000 −0.309492
\(262\) 0 0
\(263\) −17.2918 −1.06626 −0.533129 0.846034i \(-0.678984\pi\)
−0.533129 + 0.846034i \(0.678984\pi\)
\(264\) 0 0
\(265\) −8.96556 −0.550750
\(266\) 0 0
\(267\) 6.56231 0.401607
\(268\) 0 0
\(269\) 8.67376 0.528849 0.264424 0.964406i \(-0.414818\pi\)
0.264424 + 0.964406i \(0.414818\pi\)
\(270\) 0 0
\(271\) −24.4164 −1.48319 −0.741596 0.670847i \(-0.765931\pi\)
−0.741596 + 0.670847i \(0.765931\pi\)
\(272\) 0 0
\(273\) −0.381966 −0.0231176
\(274\) 0 0
\(275\) −29.5623 −1.78267
\(276\) 0 0
\(277\) 3.32624 0.199854 0.0999271 0.994995i \(-0.468139\pi\)
0.0999271 + 0.994995i \(0.468139\pi\)
\(278\) 0 0
\(279\) −5.76393 −0.345078
\(280\) 0 0
\(281\) −19.1246 −1.14088 −0.570439 0.821340i \(-0.693227\pi\)
−0.570439 + 0.821340i \(0.693227\pi\)
\(282\) 0 0
\(283\) 26.0344 1.54759 0.773793 0.633438i \(-0.218357\pi\)
0.773793 + 0.633438i \(0.218357\pi\)
\(284\) 0 0
\(285\) 0.909830 0.0538937
\(286\) 0 0
\(287\) −3.00000 −0.177084
\(288\) 0 0
\(289\) −14.8328 −0.872519
\(290\) 0 0
\(291\) 1.94427 0.113975
\(292\) 0 0
\(293\) 12.9443 0.756212 0.378106 0.925762i \(-0.376575\pi\)
0.378106 + 0.925762i \(0.376575\pi\)
\(294\) 0 0
\(295\) 52.2705 3.04331
\(296\) 0 0
\(297\) 3.00000 0.174078
\(298\) 0 0
\(299\) −0.381966 −0.0220897
\(300\) 0 0
\(301\) −1.61803 −0.0932619
\(302\) 0 0
\(303\) 7.56231 0.434443
\(304\) 0 0
\(305\) −18.7082 −1.07123
\(306\) 0 0
\(307\) 23.6525 1.34992 0.674959 0.737855i \(-0.264161\pi\)
0.674959 + 0.737855i \(0.264161\pi\)
\(308\) 0 0
\(309\) 12.9443 0.736374
\(310\) 0 0
\(311\) 16.3820 0.928936 0.464468 0.885590i \(-0.346246\pi\)
0.464468 + 0.885590i \(0.346246\pi\)
\(312\) 0 0
\(313\) −12.0000 −0.678280 −0.339140 0.940736i \(-0.610136\pi\)
−0.339140 + 0.940736i \(0.610136\pi\)
\(314\) 0 0
\(315\) −3.85410 −0.217154
\(316\) 0 0
\(317\) 30.0344 1.68690 0.843451 0.537206i \(-0.180520\pi\)
0.843451 + 0.537206i \(0.180520\pi\)
\(318\) 0 0
\(319\) 15.0000 0.839839
\(320\) 0 0
\(321\) 12.5623 0.701160
\(322\) 0 0
\(323\) 0.347524 0.0193368
\(324\) 0 0
\(325\) −3.76393 −0.208785
\(326\) 0 0
\(327\) 4.90983 0.271514
\(328\) 0 0
\(329\) −3.23607 −0.178410
\(330\) 0 0
\(331\) 18.3607 1.00919 0.504597 0.863355i \(-0.331641\pi\)
0.504597 + 0.863355i \(0.331641\pi\)
\(332\) 0 0
\(333\) −4.23607 −0.232135
\(334\) 0 0
\(335\) −8.96556 −0.489841
\(336\) 0 0
\(337\) −4.20163 −0.228877 −0.114439 0.993430i \(-0.536507\pi\)
−0.114439 + 0.993430i \(0.536507\pi\)
\(338\) 0 0
\(339\) 8.03444 0.436371
\(340\) 0 0
\(341\) 17.2918 0.936403
\(342\) 0 0
\(343\) −1.00000 −0.0539949
\(344\) 0 0
\(345\) −3.85410 −0.207498
\(346\) 0 0
\(347\) 28.5967 1.53515 0.767577 0.640957i \(-0.221462\pi\)
0.767577 + 0.640957i \(0.221462\pi\)
\(348\) 0 0
\(349\) −27.4508 −1.46941 −0.734705 0.678387i \(-0.762679\pi\)
−0.734705 + 0.678387i \(0.762679\pi\)
\(350\) 0 0
\(351\) 0.381966 0.0203878
\(352\) 0 0
\(353\) 30.4164 1.61890 0.809451 0.587187i \(-0.199765\pi\)
0.809451 + 0.587187i \(0.199765\pi\)
\(354\) 0 0
\(355\) −29.3607 −1.55830
\(356\) 0 0
\(357\) −1.47214 −0.0779137
\(358\) 0 0
\(359\) 23.8541 1.25897 0.629486 0.777012i \(-0.283266\pi\)
0.629486 + 0.777012i \(0.283266\pi\)
\(360\) 0 0
\(361\) −18.9443 −0.997067
\(362\) 0 0
\(363\) 2.00000 0.104973
\(364\) 0 0
\(365\) −22.9098 −1.19916
\(366\) 0 0
\(367\) −1.43769 −0.0750470 −0.0375235 0.999296i \(-0.511947\pi\)
−0.0375235 + 0.999296i \(0.511947\pi\)
\(368\) 0 0
\(369\) 3.00000 0.156174
\(370\) 0 0
\(371\) 2.32624 0.120772
\(372\) 0 0
\(373\) −16.2361 −0.840672 −0.420336 0.907369i \(-0.638088\pi\)
−0.420336 + 0.907369i \(0.638088\pi\)
\(374\) 0 0
\(375\) −18.7082 −0.966087
\(376\) 0 0
\(377\) 1.90983 0.0983613
\(378\) 0 0
\(379\) 12.0000 0.616399 0.308199 0.951322i \(-0.400274\pi\)
0.308199 + 0.951322i \(0.400274\pi\)
\(380\) 0 0
\(381\) 3.32624 0.170408
\(382\) 0 0
\(383\) 2.70820 0.138383 0.0691914 0.997603i \(-0.477958\pi\)
0.0691914 + 0.997603i \(0.477958\pi\)
\(384\) 0 0
\(385\) 11.5623 0.589270
\(386\) 0 0
\(387\) 1.61803 0.0822493
\(388\) 0 0
\(389\) −15.1803 −0.769674 −0.384837 0.922985i \(-0.625742\pi\)
−0.384837 + 0.922985i \(0.625742\pi\)
\(390\) 0 0
\(391\) −1.47214 −0.0744491
\(392\) 0 0
\(393\) 3.00000 0.151330
\(394\) 0 0
\(395\) 36.5066 1.83685
\(396\) 0 0
\(397\) 4.76393 0.239095 0.119547 0.992828i \(-0.461856\pi\)
0.119547 + 0.992828i \(0.461856\pi\)
\(398\) 0 0
\(399\) −0.236068 −0.0118182
\(400\) 0 0
\(401\) 35.5410 1.77483 0.887417 0.460968i \(-0.152498\pi\)
0.887417 + 0.460968i \(0.152498\pi\)
\(402\) 0 0
\(403\) 2.20163 0.109671
\(404\) 0 0
\(405\) 3.85410 0.191512
\(406\) 0 0
\(407\) 12.7082 0.629922
\(408\) 0 0
\(409\) 1.18034 0.0583641 0.0291820 0.999574i \(-0.490710\pi\)
0.0291820 + 0.999574i \(0.490710\pi\)
\(410\) 0 0
\(411\) −8.41641 −0.415151
\(412\) 0 0
\(413\) −13.5623 −0.667357
\(414\) 0 0
\(415\) −47.1591 −2.31495
\(416\) 0 0
\(417\) −10.7984 −0.528799
\(418\) 0 0
\(419\) −28.6869 −1.40145 −0.700724 0.713433i \(-0.747139\pi\)
−0.700724 + 0.713433i \(0.747139\pi\)
\(420\) 0 0
\(421\) −36.2705 −1.76772 −0.883858 0.467755i \(-0.845063\pi\)
−0.883858 + 0.467755i \(0.845063\pi\)
\(422\) 0 0
\(423\) 3.23607 0.157343
\(424\) 0 0
\(425\) −14.5066 −0.703672
\(426\) 0 0
\(427\) 4.85410 0.234906
\(428\) 0 0
\(429\) −1.14590 −0.0553245
\(430\) 0 0
\(431\) 31.0344 1.49488 0.747438 0.664331i \(-0.231284\pi\)
0.747438 + 0.664331i \(0.231284\pi\)
\(432\) 0 0
\(433\) −15.7082 −0.754888 −0.377444 0.926032i \(-0.623197\pi\)
−0.377444 + 0.926032i \(0.623197\pi\)
\(434\) 0 0
\(435\) 19.2705 0.923950
\(436\) 0 0
\(437\) −0.236068 −0.0112927
\(438\) 0 0
\(439\) −17.1803 −0.819973 −0.409986 0.912092i \(-0.634467\pi\)
−0.409986 + 0.912092i \(0.634467\pi\)
\(440\) 0 0
\(441\) 1.00000 0.0476190
\(442\) 0 0
\(443\) −27.4164 −1.30259 −0.651296 0.758823i \(-0.725775\pi\)
−0.651296 + 0.758823i \(0.725775\pi\)
\(444\) 0 0
\(445\) −25.2918 −1.19895
\(446\) 0 0
\(447\) −16.1803 −0.765304
\(448\) 0 0
\(449\) −27.2705 −1.28697 −0.643487 0.765457i \(-0.722513\pi\)
−0.643487 + 0.765457i \(0.722513\pi\)
\(450\) 0 0
\(451\) −9.00000 −0.423793
\(452\) 0 0
\(453\) −22.6525 −1.06431
\(454\) 0 0
\(455\) 1.47214 0.0690148
\(456\) 0 0
\(457\) −4.97871 −0.232894 −0.116447 0.993197i \(-0.537151\pi\)
−0.116447 + 0.993197i \(0.537151\pi\)
\(458\) 0 0
\(459\) 1.47214 0.0687134
\(460\) 0 0
\(461\) −3.56231 −0.165913 −0.0829566 0.996553i \(-0.526436\pi\)
−0.0829566 + 0.996553i \(0.526436\pi\)
\(462\) 0 0
\(463\) 19.3607 0.899767 0.449884 0.893087i \(-0.351465\pi\)
0.449884 + 0.893087i \(0.351465\pi\)
\(464\) 0 0
\(465\) 22.2148 1.03019
\(466\) 0 0
\(467\) −17.6525 −0.816859 −0.408430 0.912790i \(-0.633923\pi\)
−0.408430 + 0.912790i \(0.633923\pi\)
\(468\) 0 0
\(469\) 2.32624 0.107416
\(470\) 0 0
\(471\) 6.76393 0.311666
\(472\) 0 0
\(473\) −4.85410 −0.223192
\(474\) 0 0
\(475\) −2.32624 −0.106735
\(476\) 0 0
\(477\) −2.32624 −0.106511
\(478\) 0 0
\(479\) 20.0557 0.916370 0.458185 0.888857i \(-0.348500\pi\)
0.458185 + 0.888857i \(0.348500\pi\)
\(480\) 0 0
\(481\) 1.61803 0.0737760
\(482\) 0 0
\(483\) 1.00000 0.0455016
\(484\) 0 0
\(485\) −7.49342 −0.340259
\(486\) 0 0
\(487\) 23.8328 1.07997 0.539984 0.841675i \(-0.318430\pi\)
0.539984 + 0.841675i \(0.318430\pi\)
\(488\) 0 0
\(489\) 5.67376 0.256576
\(490\) 0 0
\(491\) −9.03444 −0.407719 −0.203859 0.979000i \(-0.565349\pi\)
−0.203859 + 0.979000i \(0.565349\pi\)
\(492\) 0 0
\(493\) 7.36068 0.331508
\(494\) 0 0
\(495\) −11.5623 −0.519687
\(496\) 0 0
\(497\) 7.61803 0.341716
\(498\) 0 0
\(499\) −35.4508 −1.58700 −0.793499 0.608572i \(-0.791743\pi\)
−0.793499 + 0.608572i \(0.791743\pi\)
\(500\) 0 0
\(501\) 8.41641 0.376017
\(502\) 0 0
\(503\) −0.0344419 −0.00153569 −0.000767843 1.00000i \(-0.500244\pi\)
−0.000767843 1.00000i \(0.500244\pi\)
\(504\) 0 0
\(505\) −29.1459 −1.29698
\(506\) 0 0
\(507\) 12.8541 0.570871
\(508\) 0 0
\(509\) −25.7082 −1.13950 −0.569748 0.821819i \(-0.692959\pi\)
−0.569748 + 0.821819i \(0.692959\pi\)
\(510\) 0 0
\(511\) 5.94427 0.262959
\(512\) 0 0
\(513\) 0.236068 0.0104227
\(514\) 0 0
\(515\) −49.8885 −2.19835
\(516\) 0 0
\(517\) −9.70820 −0.426966
\(518\) 0 0
\(519\) 17.4721 0.766942
\(520\) 0 0
\(521\) −41.4164 −1.81449 −0.907243 0.420607i \(-0.861817\pi\)
−0.907243 + 0.420607i \(0.861817\pi\)
\(522\) 0 0
\(523\) −28.4721 −1.24500 −0.622500 0.782620i \(-0.713883\pi\)
−0.622500 + 0.782620i \(0.713883\pi\)
\(524\) 0 0
\(525\) 9.85410 0.430068
\(526\) 0 0
\(527\) 8.48529 0.369625
\(528\) 0 0
\(529\) 1.00000 0.0434783
\(530\) 0 0
\(531\) 13.5623 0.588554
\(532\) 0 0
\(533\) −1.14590 −0.0496344
\(534\) 0 0
\(535\) −48.4164 −2.09322
\(536\) 0 0
\(537\) 21.5623 0.930482
\(538\) 0 0
\(539\) −3.00000 −0.129219
\(540\) 0 0
\(541\) −6.29180 −0.270505 −0.135253 0.990811i \(-0.543185\pi\)
−0.135253 + 0.990811i \(0.543185\pi\)
\(542\) 0 0
\(543\) −6.41641 −0.275354
\(544\) 0 0
\(545\) −18.9230 −0.810572
\(546\) 0 0
\(547\) 17.8541 0.763386 0.381693 0.924289i \(-0.375341\pi\)
0.381693 + 0.924289i \(0.375341\pi\)
\(548\) 0 0
\(549\) −4.85410 −0.207168
\(550\) 0 0
\(551\) 1.18034 0.0502842
\(552\) 0 0
\(553\) −9.47214 −0.402796
\(554\) 0 0
\(555\) 16.3262 0.693010
\(556\) 0 0
\(557\) −25.2361 −1.06929 −0.534643 0.845078i \(-0.679554\pi\)
−0.534643 + 0.845078i \(0.679554\pi\)
\(558\) 0 0
\(559\) −0.618034 −0.0261401
\(560\) 0 0
\(561\) −4.41641 −0.186461
\(562\) 0 0
\(563\) −3.50658 −0.147785 −0.0738923 0.997266i \(-0.523542\pi\)
−0.0738923 + 0.997266i \(0.523542\pi\)
\(564\) 0 0
\(565\) −30.9656 −1.30273
\(566\) 0 0
\(567\) −1.00000 −0.0419961
\(568\) 0 0
\(569\) 21.8885 0.917615 0.458808 0.888536i \(-0.348277\pi\)
0.458808 + 0.888536i \(0.348277\pi\)
\(570\) 0 0
\(571\) −24.5279 −1.02646 −0.513230 0.858251i \(-0.671551\pi\)
−0.513230 + 0.858251i \(0.671551\pi\)
\(572\) 0 0
\(573\) 12.7639 0.533221
\(574\) 0 0
\(575\) 9.85410 0.410944
\(576\) 0 0
\(577\) 10.3607 0.431321 0.215660 0.976468i \(-0.430810\pi\)
0.215660 + 0.976468i \(0.430810\pi\)
\(578\) 0 0
\(579\) −24.6525 −1.02452
\(580\) 0 0
\(581\) 12.2361 0.507638
\(582\) 0 0
\(583\) 6.97871 0.289029
\(584\) 0 0
\(585\) −1.47214 −0.0608653
\(586\) 0 0
\(587\) −16.6738 −0.688200 −0.344100 0.938933i \(-0.611816\pi\)
−0.344100 + 0.938933i \(0.611816\pi\)
\(588\) 0 0
\(589\) 1.36068 0.0560658
\(590\) 0 0
\(591\) −4.09017 −0.168247
\(592\) 0 0
\(593\) 39.1246 1.60666 0.803328 0.595537i \(-0.203061\pi\)
0.803328 + 0.595537i \(0.203061\pi\)
\(594\) 0 0
\(595\) 5.67376 0.232602
\(596\) 0 0
\(597\) 10.7984 0.441948
\(598\) 0 0
\(599\) 18.7426 0.765804 0.382902 0.923789i \(-0.374925\pi\)
0.382902 + 0.923789i \(0.374925\pi\)
\(600\) 0 0
\(601\) 31.5623 1.28745 0.643727 0.765256i \(-0.277387\pi\)
0.643727 + 0.765256i \(0.277387\pi\)
\(602\) 0 0
\(603\) −2.32624 −0.0947317
\(604\) 0 0
\(605\) −7.70820 −0.313383
\(606\) 0 0
\(607\) −35.5066 −1.44117 −0.720584 0.693368i \(-0.756126\pi\)
−0.720584 + 0.693368i \(0.756126\pi\)
\(608\) 0 0
\(609\) −5.00000 −0.202610
\(610\) 0 0
\(611\) −1.23607 −0.0500060
\(612\) 0 0
\(613\) −38.4164 −1.55162 −0.775812 0.630964i \(-0.782660\pi\)
−0.775812 + 0.630964i \(0.782660\pi\)
\(614\) 0 0
\(615\) −11.5623 −0.466237
\(616\) 0 0
\(617\) 4.56231 0.183672 0.0918358 0.995774i \(-0.470727\pi\)
0.0918358 + 0.995774i \(0.470727\pi\)
\(618\) 0 0
\(619\) −3.32624 −0.133693 −0.0668464 0.997763i \(-0.521294\pi\)
−0.0668464 + 0.997763i \(0.521294\pi\)
\(620\) 0 0
\(621\) −1.00000 −0.0401286
\(622\) 0 0
\(623\) 6.56231 0.262913
\(624\) 0 0
\(625\) 22.8328 0.913313
\(626\) 0 0
\(627\) −0.708204 −0.0282829
\(628\) 0 0
\(629\) 6.23607 0.248648
\(630\) 0 0
\(631\) 11.6525 0.463878 0.231939 0.972730i \(-0.425493\pi\)
0.231939 + 0.972730i \(0.425493\pi\)
\(632\) 0 0
\(633\) 4.23607 0.168369
\(634\) 0 0
\(635\) −12.8197 −0.508733
\(636\) 0 0
\(637\) −0.381966 −0.0151340
\(638\) 0 0
\(639\) −7.61803 −0.301365
\(640\) 0 0
\(641\) 13.9787 0.552126 0.276063 0.961140i \(-0.410970\pi\)
0.276063 + 0.961140i \(0.410970\pi\)
\(642\) 0 0
\(643\) −10.7426 −0.423649 −0.211824 0.977308i \(-0.567940\pi\)
−0.211824 + 0.977308i \(0.567940\pi\)
\(644\) 0 0
\(645\) −6.23607 −0.245545
\(646\) 0 0
\(647\) −2.32624 −0.0914538 −0.0457269 0.998954i \(-0.514560\pi\)
−0.0457269 + 0.998954i \(0.514560\pi\)
\(648\) 0 0
\(649\) −40.6869 −1.59710
\(650\) 0 0
\(651\) −5.76393 −0.225906
\(652\) 0 0
\(653\) −11.5623 −0.452468 −0.226234 0.974073i \(-0.572641\pi\)
−0.226234 + 0.974073i \(0.572641\pi\)
\(654\) 0 0
\(655\) −11.5623 −0.451777
\(656\) 0 0
\(657\) −5.94427 −0.231908
\(658\) 0 0
\(659\) 13.0000 0.506408 0.253204 0.967413i \(-0.418516\pi\)
0.253204 + 0.967413i \(0.418516\pi\)
\(660\) 0 0
\(661\) 34.4164 1.33864 0.669322 0.742973i \(-0.266585\pi\)
0.669322 + 0.742973i \(0.266585\pi\)
\(662\) 0 0
\(663\) −0.562306 −0.0218382
\(664\) 0 0
\(665\) 0.909830 0.0352817
\(666\) 0 0
\(667\) −5.00000 −0.193601
\(668\) 0 0
\(669\) −4.32624 −0.167262
\(670\) 0 0
\(671\) 14.5623 0.562172
\(672\) 0 0
\(673\) −22.4164 −0.864089 −0.432045 0.901852i \(-0.642208\pi\)
−0.432045 + 0.901852i \(0.642208\pi\)
\(674\) 0 0
\(675\) −9.85410 −0.379285
\(676\) 0 0
\(677\) −29.9098 −1.14953 −0.574764 0.818319i \(-0.694906\pi\)
−0.574764 + 0.818319i \(0.694906\pi\)
\(678\) 0 0
\(679\) 1.94427 0.0746143
\(680\) 0 0
\(681\) 0.618034 0.0236831
\(682\) 0 0
\(683\) 33.0689 1.26535 0.632673 0.774419i \(-0.281958\pi\)
0.632673 + 0.774419i \(0.281958\pi\)
\(684\) 0 0
\(685\) 32.4377 1.23938
\(686\) 0 0
\(687\) 2.43769 0.0930038
\(688\) 0 0
\(689\) 0.888544 0.0338508
\(690\) 0 0
\(691\) −15.3262 −0.583038 −0.291519 0.956565i \(-0.594161\pi\)
−0.291519 + 0.956565i \(0.594161\pi\)
\(692\) 0 0
\(693\) 3.00000 0.113961
\(694\) 0 0
\(695\) 41.6180 1.57866
\(696\) 0 0
\(697\) −4.41641 −0.167283
\(698\) 0 0
\(699\) 16.1459 0.610694
\(700\) 0 0
\(701\) 0.437694 0.0165315 0.00826574 0.999966i \(-0.497369\pi\)
0.00826574 + 0.999966i \(0.497369\pi\)
\(702\) 0 0
\(703\) 1.00000 0.0377157
\(704\) 0 0
\(705\) −12.4721 −0.469728
\(706\) 0 0
\(707\) 7.56231 0.284410
\(708\) 0 0
\(709\) −34.6738 −1.30220 −0.651100 0.758992i \(-0.725692\pi\)
−0.651100 + 0.758992i \(0.725692\pi\)
\(710\) 0 0
\(711\) 9.47214 0.355233
\(712\) 0 0
\(713\) −5.76393 −0.215861
\(714\) 0 0
\(715\) 4.41641 0.165164
\(716\) 0 0
\(717\) −5.67376 −0.211891
\(718\) 0 0
\(719\) −28.2361 −1.05303 −0.526514 0.850167i \(-0.676501\pi\)
−0.526514 + 0.850167i \(0.676501\pi\)
\(720\) 0 0
\(721\) 12.9443 0.482070
\(722\) 0 0
\(723\) 2.05573 0.0764534
\(724\) 0 0
\(725\) −49.2705 −1.82986
\(726\) 0 0
\(727\) −17.0689 −0.633050 −0.316525 0.948584i \(-0.602516\pi\)
−0.316525 + 0.948584i \(0.602516\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) −2.38197 −0.0881002
\(732\) 0 0
\(733\) −9.59675 −0.354464 −0.177232 0.984169i \(-0.556714\pi\)
−0.177232 + 0.984169i \(0.556714\pi\)
\(734\) 0 0
\(735\) −3.85410 −0.142161
\(736\) 0 0
\(737\) 6.97871 0.257064
\(738\) 0 0
\(739\) 35.1803 1.29413 0.647065 0.762435i \(-0.275996\pi\)
0.647065 + 0.762435i \(0.275996\pi\)
\(740\) 0 0
\(741\) −0.0901699 −0.00331248
\(742\) 0 0
\(743\) −8.14590 −0.298844 −0.149422 0.988774i \(-0.547741\pi\)
−0.149422 + 0.988774i \(0.547741\pi\)
\(744\) 0 0
\(745\) 62.3607 2.28472
\(746\) 0 0
\(747\) −12.2361 −0.447694
\(748\) 0 0
\(749\) 12.5623 0.459017
\(750\) 0 0
\(751\) −2.50658 −0.0914663 −0.0457332 0.998954i \(-0.514562\pi\)
−0.0457332 + 0.998954i \(0.514562\pi\)
\(752\) 0 0
\(753\) 13.2361 0.482349
\(754\) 0 0
\(755\) 87.3050 3.17735
\(756\) 0 0
\(757\) −3.18034 −0.115591 −0.0577957 0.998328i \(-0.518407\pi\)
−0.0577957 + 0.998328i \(0.518407\pi\)
\(758\) 0 0
\(759\) 3.00000 0.108893
\(760\) 0 0
\(761\) 37.5279 1.36038 0.680192 0.733034i \(-0.261896\pi\)
0.680192 + 0.733034i \(0.261896\pi\)
\(762\) 0 0
\(763\) 4.90983 0.177748
\(764\) 0 0
\(765\) −5.67376 −0.205135
\(766\) 0 0
\(767\) −5.18034 −0.187051
\(768\) 0 0
\(769\) −36.0689 −1.30068 −0.650339 0.759644i \(-0.725373\pi\)
−0.650339 + 0.759644i \(0.725373\pi\)
\(770\) 0 0
\(771\) 0.763932 0.0275123
\(772\) 0 0
\(773\) 4.23607 0.152361 0.0761804 0.997094i \(-0.475728\pi\)
0.0761804 + 0.997094i \(0.475728\pi\)
\(774\) 0 0
\(775\) −56.7984 −2.04026
\(776\) 0 0
\(777\) −4.23607 −0.151968
\(778\) 0 0
\(779\) −0.708204 −0.0253740
\(780\) 0 0
\(781\) 22.8541 0.817784
\(782\) 0 0
\(783\) 5.00000 0.178685
\(784\) 0 0
\(785\) −26.0689 −0.930438
\(786\) 0 0
\(787\) −22.4377 −0.799817 −0.399909 0.916555i \(-0.630958\pi\)
−0.399909 + 0.916555i \(0.630958\pi\)
\(788\) 0 0
\(789\) 17.2918 0.615604
\(790\) 0 0
\(791\) 8.03444 0.285672
\(792\) 0 0
\(793\) 1.85410 0.0658411
\(794\) 0 0
\(795\) 8.96556 0.317976
\(796\) 0 0
\(797\) 18.2918 0.647929 0.323964 0.946069i \(-0.394984\pi\)
0.323964 + 0.946069i \(0.394984\pi\)
\(798\) 0 0
\(799\) −4.76393 −0.168536
\(800\) 0 0
\(801\) −6.56231 −0.231868
\(802\) 0 0
\(803\) 17.8328 0.629306
\(804\) 0 0
\(805\) −3.85410 −0.135839
\(806\) 0 0
\(807\) −8.67376 −0.305331
\(808\) 0 0
\(809\) 43.7984 1.53987 0.769934 0.638123i \(-0.220289\pi\)
0.769934 + 0.638123i \(0.220289\pi\)
\(810\) 0 0
\(811\) 28.7771 1.01050 0.505250 0.862973i \(-0.331400\pi\)
0.505250 + 0.862973i \(0.331400\pi\)
\(812\) 0 0
\(813\) 24.4164 0.856321
\(814\) 0 0
\(815\) −21.8673 −0.765977
\(816\) 0 0
\(817\) −0.381966 −0.0133633
\(818\) 0 0
\(819\) 0.381966 0.0133470
\(820\) 0 0
\(821\) 6.29180 0.219585 0.109793 0.993955i \(-0.464981\pi\)
0.109793 + 0.993955i \(0.464981\pi\)
\(822\) 0 0
\(823\) −55.7426 −1.94307 −0.971533 0.236903i \(-0.923868\pi\)
−0.971533 + 0.236903i \(0.923868\pi\)
\(824\) 0 0
\(825\) 29.5623 1.02923
\(826\) 0 0
\(827\) −35.0902 −1.22020 −0.610102 0.792323i \(-0.708872\pi\)
−0.610102 + 0.792323i \(0.708872\pi\)
\(828\) 0 0
\(829\) 17.2918 0.600569 0.300284 0.953850i \(-0.402918\pi\)
0.300284 + 0.953850i \(0.402918\pi\)
\(830\) 0 0
\(831\) −3.32624 −0.115386
\(832\) 0 0
\(833\) −1.47214 −0.0510065
\(834\) 0 0
\(835\) −32.4377 −1.12255
\(836\) 0 0
\(837\) 5.76393 0.199231
\(838\) 0 0
\(839\) 37.6869 1.30110 0.650548 0.759465i \(-0.274539\pi\)
0.650548 + 0.759465i \(0.274539\pi\)
\(840\) 0 0
\(841\) −4.00000 −0.137931
\(842\) 0 0
\(843\) 19.1246 0.658687
\(844\) 0 0
\(845\) −49.5410 −1.70426
\(846\) 0 0
\(847\) 2.00000 0.0687208
\(848\) 0 0
\(849\) −26.0344 −0.893500
\(850\) 0 0
\(851\) −4.23607 −0.145211
\(852\) 0 0
\(853\) −17.5967 −0.602501 −0.301251 0.953545i \(-0.597404\pi\)
−0.301251 + 0.953545i \(0.597404\pi\)
\(854\) 0 0
\(855\) −0.909830 −0.0311155
\(856\) 0 0
\(857\) −20.0689 −0.685540 −0.342770 0.939419i \(-0.611365\pi\)
−0.342770 + 0.939419i \(0.611365\pi\)
\(858\) 0 0
\(859\) −52.4721 −1.79033 −0.895163 0.445739i \(-0.852941\pi\)
−0.895163 + 0.445739i \(0.852941\pi\)
\(860\) 0 0
\(861\) 3.00000 0.102240
\(862\) 0 0
\(863\) −6.76393 −0.230247 −0.115123 0.993351i \(-0.536726\pi\)
−0.115123 + 0.993351i \(0.536726\pi\)
\(864\) 0 0
\(865\) −67.3394 −2.28961
\(866\) 0 0
\(867\) 14.8328 0.503749
\(868\) 0 0
\(869\) −28.4164 −0.963961
\(870\) 0 0
\(871\) 0.888544 0.0301072
\(872\) 0 0
\(873\) −1.94427 −0.0658036
\(874\) 0 0
\(875\) −18.7082 −0.632453
\(876\) 0 0
\(877\) −4.94427 −0.166956 −0.0834781 0.996510i \(-0.526603\pi\)
−0.0834781 + 0.996510i \(0.526603\pi\)
\(878\) 0 0
\(879\) −12.9443 −0.436599
\(880\) 0 0
\(881\) 33.5967 1.13190 0.565952 0.824438i \(-0.308509\pi\)
0.565952 + 0.824438i \(0.308509\pi\)
\(882\) 0 0
\(883\) −3.97871 −0.133894 −0.0669472 0.997757i \(-0.521326\pi\)
−0.0669472 + 0.997757i \(0.521326\pi\)
\(884\) 0 0
\(885\) −52.2705 −1.75705
\(886\) 0 0
\(887\) 20.7426 0.696470 0.348235 0.937407i \(-0.386781\pi\)
0.348235 + 0.937407i \(0.386781\pi\)
\(888\) 0 0
\(889\) 3.32624 0.111558
\(890\) 0 0
\(891\) −3.00000 −0.100504
\(892\) 0 0
\(893\) −0.763932 −0.0255640
\(894\) 0 0
\(895\) −83.1033 −2.77784
\(896\) 0 0
\(897\) 0.381966 0.0127535
\(898\) 0 0
\(899\) 28.8197 0.961189
\(900\) 0 0
\(901\) 3.42454 0.114088
\(902\) 0 0
\(903\) 1.61803 0.0538448
\(904\) 0 0
\(905\) 24.7295 0.822036
\(906\) 0 0
\(907\) 3.38197 0.112296 0.0561482 0.998422i \(-0.482118\pi\)
0.0561482 + 0.998422i \(0.482118\pi\)
\(908\) 0 0
\(909\) −7.56231 −0.250826
\(910\) 0 0
\(911\) −35.5967 −1.17937 −0.589686 0.807632i \(-0.700749\pi\)
−0.589686 + 0.807632i \(0.700749\pi\)
\(912\) 0 0
\(913\) 36.7082 1.21486
\(914\) 0 0
\(915\) 18.7082 0.618474
\(916\) 0 0
\(917\) 3.00000 0.0990687
\(918\) 0 0
\(919\) 52.5967 1.73501 0.867503 0.497431i \(-0.165723\pi\)
0.867503 + 0.497431i \(0.165723\pi\)
\(920\) 0 0
\(921\) −23.6525 −0.779376
\(922\) 0 0
\(923\) 2.90983 0.0957782
\(924\) 0 0
\(925\) −41.7426 −1.37249
\(926\) 0 0
\(927\) −12.9443 −0.425146
\(928\) 0 0
\(929\) 41.5623 1.36362 0.681808 0.731532i \(-0.261194\pi\)
0.681808 + 0.731532i \(0.261194\pi\)
\(930\) 0 0
\(931\) −0.236068 −0.00773682
\(932\) 0 0
\(933\) −16.3820 −0.536321
\(934\) 0 0
\(935\) 17.0213 0.556656
\(936\) 0 0
\(937\) 45.5410 1.48776 0.743880 0.668313i \(-0.232983\pi\)
0.743880 + 0.668313i \(0.232983\pi\)
\(938\) 0 0
\(939\) 12.0000 0.391605
\(940\) 0 0
\(941\) −16.5836 −0.540610 −0.270305 0.962775i \(-0.587124\pi\)
−0.270305 + 0.962775i \(0.587124\pi\)
\(942\) 0 0
\(943\) 3.00000 0.0976934
\(944\) 0 0
\(945\) 3.85410 0.125374
\(946\) 0 0
\(947\) −44.4721 −1.44515 −0.722575 0.691292i \(-0.757042\pi\)
−0.722575 + 0.691292i \(0.757042\pi\)
\(948\) 0 0
\(949\) 2.27051 0.0737039
\(950\) 0 0
\(951\) −30.0344 −0.973934
\(952\) 0 0
\(953\) −4.96556 −0.160850 −0.0804251 0.996761i \(-0.525628\pi\)
−0.0804251 + 0.996761i \(0.525628\pi\)
\(954\) 0 0
\(955\) −49.1935 −1.59186
\(956\) 0 0
\(957\) −15.0000 −0.484881
\(958\) 0 0
\(959\) −8.41641 −0.271780
\(960\) 0 0
\(961\) 2.22291 0.0717069
\(962\) 0 0
\(963\) −12.5623 −0.404815
\(964\) 0 0
\(965\) 95.0132 3.05858
\(966\) 0 0
\(967\) 22.0000 0.707472 0.353736 0.935345i \(-0.384911\pi\)
0.353736 + 0.935345i \(0.384911\pi\)
\(968\) 0 0
\(969\) −0.347524 −0.0111641
\(970\) 0 0
\(971\) −32.2705 −1.03561 −0.517805 0.855499i \(-0.673251\pi\)
−0.517805 + 0.855499i \(0.673251\pi\)
\(972\) 0 0
\(973\) −10.7984 −0.346180
\(974\) 0 0
\(975\) 3.76393 0.120542
\(976\) 0 0
\(977\) −9.43769 −0.301939 −0.150969 0.988538i \(-0.548239\pi\)
−0.150969 + 0.988538i \(0.548239\pi\)
\(978\) 0 0
\(979\) 19.6869 0.629197
\(980\) 0 0
\(981\) −4.90983 −0.156759
\(982\) 0 0
\(983\) 40.5967 1.29484 0.647418 0.762135i \(-0.275849\pi\)
0.647418 + 0.762135i \(0.275849\pi\)
\(984\) 0 0
\(985\) 15.7639 0.502281
\(986\) 0 0
\(987\) 3.23607 0.103005
\(988\) 0 0
\(989\) 1.61803 0.0514505
\(990\) 0 0
\(991\) −55.8673 −1.77468 −0.887341 0.461114i \(-0.847450\pi\)
−0.887341 + 0.461114i \(0.847450\pi\)
\(992\) 0 0
\(993\) −18.3607 −0.582659
\(994\) 0 0
\(995\) −41.6180 −1.31938
\(996\) 0 0
\(997\) −24.5967 −0.778987 −0.389493 0.921029i \(-0.627350\pi\)
−0.389493 + 0.921029i \(0.627350\pi\)
\(998\) 0 0
\(999\) 4.23607 0.134023
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.bb.1.2 2
4.3 odd 2 1932.2.a.h.1.2 2
12.11 even 2 5796.2.a.j.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1932.2.a.h.1.2 2 4.3 odd 2
5796.2.a.j.1.1 2 12.11 even 2
7728.2.a.bb.1.2 2 1.1 even 1 trivial