Properties

Label 756.2.b.a.55.3
Level $756$
Weight $2$
Character 756.55
Analytic conductor $6.037$
Analytic rank $0$
Dimension $4$
CM discriminant -84
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 756 = 2^{2} \cdot 3^{3} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 756.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(6.03669039281\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{-2}, \sqrt{7})\)
Defining polynomial: \(x^{4} + 8 x^{2} + 9\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label 55.3
Root \(-1.16372i\) of defining polynomial
Character \(\chi\) \(=\) 756.55
Dual form 756.2.b.a.55.2

$q$-expansion

\(f(q)\) \(=\) \(q+1.41421i q^{2} -2.00000 q^{4} +0.250492i q^{5} -2.64575 q^{7} -2.82843i q^{8} +O(q^{10})\) \(q+1.41421i q^{2} -2.00000 q^{4} +0.250492i q^{5} -2.64575 q^{7} -2.82843i q^{8} -0.354249 q^{10} -4.90538i q^{11} -3.74166i q^{14} +4.00000 q^{16} +3.74166i q^{17} +8.64575 q^{19} -0.500983i q^{20} +6.93725 q^{22} -9.14802i q^{23} +4.93725 q^{25} +5.29150 q^{28} -8.29150 q^{31} +5.65685i q^{32} -5.29150 q^{34} -0.662739i q^{35} +3.93725 q^{37} +12.2269i q^{38} +0.708497 q^{40} -12.4774i q^{41} +9.81076i q^{44} +12.9373 q^{46} +7.00000 q^{49} +6.98233i q^{50} +1.22876 q^{55} +7.48331i q^{56} -11.7260i q^{62} -8.00000 q^{64} -7.48331i q^{68} +0.937254 q^{70} -13.3907i q^{71} +5.56812i q^{74} -17.2915 q^{76} +12.9784i q^{77} +1.00197i q^{80} +17.6458 q^{82} -0.937254 q^{85} -13.8745 q^{88} +11.4755i q^{89} +18.2960i q^{92} +2.16569i q^{95} +9.89949i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 8 q^{4} + O(q^{10}) \) \( 4 q - 8 q^{4} - 12 q^{10} + 16 q^{16} + 24 q^{19} - 4 q^{22} - 12 q^{25} - 12 q^{31} - 16 q^{37} + 24 q^{40} + 20 q^{46} + 28 q^{49} - 48 q^{55} - 32 q^{64} - 28 q^{70} - 48 q^{76} + 60 q^{82} + 28 q^{85} + 8 q^{88} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/756\mathbb{Z}\right)^\times\).

\(n\) \(29\) \(325\) \(379\)
\(\chi(n)\) \(1\) \(-1\) \(-1\)

Coefficient data

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

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.41421i 1.00000i
\(3\) 0 0
\(4\) −2.00000 −1.00000
\(5\) 0.250492i 0.112023i 0.998430 + 0.0560116i \(0.0178384\pi\)
−0.998430 + 0.0560116i \(0.982162\pi\)
\(6\) 0 0
\(7\) −2.64575 −1.00000
\(8\) − 2.82843i − 1.00000i
\(9\) 0 0
\(10\) −0.354249 −0.112023
\(11\) − 4.90538i − 1.47903i −0.673141 0.739514i \(-0.735055\pi\)
0.673141 0.739514i \(-0.264945\pi\)
\(12\) 0 0
\(13\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(14\) − 3.74166i − 1.00000i
\(15\) 0 0
\(16\) 4.00000 1.00000
\(17\) 3.74166i 0.907485i 0.891133 + 0.453743i \(0.149911\pi\)
−0.891133 + 0.453743i \(0.850089\pi\)
\(18\) 0 0
\(19\) 8.64575 1.98347 0.991736 0.128298i \(-0.0409513\pi\)
0.991736 + 0.128298i \(0.0409513\pi\)
\(20\) − 0.500983i − 0.112023i
\(21\) 0 0
\(22\) 6.93725 1.47903
\(23\) − 9.14802i − 1.90749i −0.300610 0.953747i \(-0.597190\pi\)
0.300610 0.953747i \(-0.402810\pi\)
\(24\) 0 0
\(25\) 4.93725 0.987451
\(26\) 0 0
\(27\) 0 0
\(28\) 5.29150 1.00000
\(29\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(30\) 0 0
\(31\) −8.29150 −1.48920 −0.744599 0.667512i \(-0.767359\pi\)
−0.744599 + 0.667512i \(0.767359\pi\)
\(32\) 5.65685i 1.00000i
\(33\) 0 0
\(34\) −5.29150 −0.907485
\(35\) − 0.662739i − 0.112023i
\(36\) 0 0
\(37\) 3.93725 0.647281 0.323640 0.946180i \(-0.395093\pi\)
0.323640 + 0.946180i \(0.395093\pi\)
\(38\) 12.2269i 1.98347i
\(39\) 0 0
\(40\) 0.708497 0.112023
\(41\) − 12.4774i − 1.94865i −0.225152 0.974324i \(-0.572288\pi\)
0.225152 0.974324i \(-0.427712\pi\)
\(42\) 0 0
\(43\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(44\) 9.81076i 1.47903i
\(45\) 0 0
\(46\) 12.9373 1.90749
\(47\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(48\) 0 0
\(49\) 7.00000 1.00000
\(50\) 6.98233i 0.987451i
\(51\) 0 0
\(52\) 0 0
\(53\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(54\) 0 0
\(55\) 1.22876 0.165686
\(56\) 7.48331i 1.00000i
\(57\) 0 0
\(58\) 0 0
\(59\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(60\) 0 0
\(61\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(62\) − 11.7260i − 1.48920i
\(63\) 0 0
\(64\) −8.00000 −1.00000
\(65\) 0 0
\(66\) 0 0
\(67\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(68\) − 7.48331i − 0.907485i
\(69\) 0 0
\(70\) 0.937254 0.112023
\(71\) − 13.3907i − 1.58918i −0.607147 0.794590i \(-0.707686\pi\)
0.607147 0.794590i \(-0.292314\pi\)
\(72\) 0 0
\(73\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(74\) 5.56812i 0.647281i
\(75\) 0 0
\(76\) −17.2915 −1.98347
\(77\) 12.9784i 1.47903i
\(78\) 0 0
\(79\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(80\) 1.00197i 0.112023i
\(81\) 0 0
\(82\) 17.6458 1.94865
\(83\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(84\) 0 0
\(85\) −0.937254 −0.101659
\(86\) 0 0
\(87\) 0 0
\(88\) −13.8745 −1.47903
\(89\) 11.4755i 1.21640i 0.793785 + 0.608198i \(0.208107\pi\)
−0.793785 + 0.608198i \(0.791893\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 18.2960i 1.90749i
\(93\) 0 0
\(94\) 0 0
\(95\) 2.16569i 0.222195i
\(96\) 0 0
\(97\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(98\) 9.89949i 1.00000i
\(99\) 0 0
\(100\) −9.87451 −0.987451
\(101\) − 18.7083i − 1.86154i −0.365600 0.930772i \(-0.619136\pi\)
0.365600 0.930772i \(-0.380864\pi\)
\(102\) 0 0
\(103\) 9.70850 0.956607 0.478303 0.878195i \(-0.341252\pi\)
0.478303 + 0.878195i \(0.341252\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 7.07107i 0.683586i 0.939775 + 0.341793i \(0.111034\pi\)
−0.939775 + 0.341793i \(0.888966\pi\)
\(108\) 0 0
\(109\) −10.8745 −1.04159 −0.520794 0.853682i \(-0.674364\pi\)
−0.520794 + 0.853682i \(0.674364\pi\)
\(110\) 1.73772i 0.165686i
\(111\) 0 0
\(112\) −10.5830 −1.00000
\(113\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(114\) 0 0
\(115\) 2.29150 0.213684
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) − 9.89949i − 0.907485i
\(120\) 0 0
\(121\) −13.0627 −1.18752
\(122\) 0 0
\(123\) 0 0
\(124\) 16.5830 1.48920
\(125\) 2.48920i 0.222641i
\(126\) 0 0
\(127\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(128\) − 11.3137i − 1.00000i
\(129\) 0 0
\(130\) 0 0
\(131\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(132\) 0 0
\(133\) −22.8745 −1.98347
\(134\) 0 0
\(135\) 0 0
\(136\) 10.5830 0.907485
\(137\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(138\) 0 0
\(139\) −21.1660 −1.79528 −0.897639 0.440732i \(-0.854719\pi\)
−0.897639 + 0.440732i \(0.854719\pi\)
\(140\) 1.32548i 0.112023i
\(141\) 0 0
\(142\) 18.9373 1.58918
\(143\) 0 0
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 0 0
\(148\) −7.87451 −0.647281
\(149\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(150\) 0 0
\(151\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(152\) − 24.4539i − 1.98347i
\(153\) 0 0
\(154\) −18.3542 −1.47903
\(155\) − 2.07695i − 0.166825i
\(156\) 0 0
\(157\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) −1.41699 −0.112023
\(161\) 24.2034i 1.90749i
\(162\) 0 0
\(163\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(164\) 24.9549i 1.94865i
\(165\) 0 0
\(166\) 0 0
\(167\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(168\) 0 0
\(169\) 13.0000 1.00000
\(170\) − 1.32548i − 0.101659i
\(171\) 0 0
\(172\) 0 0
\(173\) − 10.9745i − 0.834374i −0.908821 0.417187i \(-0.863016\pi\)
0.908821 0.417187i \(-0.136984\pi\)
\(174\) 0 0
\(175\) −13.0627 −0.987451
\(176\) − 19.6215i − 1.47903i
\(177\) 0 0
\(178\) −16.2288 −1.21640
\(179\) − 18.3848i − 1.37414i −0.726590 0.687071i \(-0.758896\pi\)
0.726590 0.687071i \(-0.241104\pi\)
\(180\) 0 0
\(181\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) −25.8745 −1.90749
\(185\) 0.986249i 0.0725105i
\(186\) 0 0
\(187\) 18.3542 1.34220
\(188\) 0 0
\(189\) 0 0
\(190\) −3.06275 −0.222195
\(191\) 7.82254i 0.566019i 0.959117 + 0.283010i \(0.0913329\pi\)
−0.959117 + 0.283010i \(0.908667\pi\)
\(192\) 0 0
\(193\) −4.00000 −0.287926 −0.143963 0.989583i \(-0.545985\pi\)
−0.143963 + 0.989583i \(0.545985\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) −14.0000 −1.00000
\(197\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(198\) 0 0
\(199\) 26.6458 1.88887 0.944434 0.328702i \(-0.106611\pi\)
0.944434 + 0.328702i \(0.106611\pi\)
\(200\) − 13.9647i − 0.987451i
\(201\) 0 0
\(202\) 26.4575 1.86154
\(203\) 0 0
\(204\) 0 0
\(205\) 3.12549 0.218294
\(206\) 13.7299i 0.956607i
\(207\) 0 0
\(208\) 0 0
\(209\) − 42.4107i − 2.93361i
\(210\) 0 0
\(211\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) −10.0000 −0.683586
\(215\) 0 0
\(216\) 0 0
\(217\) 21.9373 1.48920
\(218\) − 15.3789i − 1.04159i
\(219\) 0 0
\(220\) −2.45751 −0.165686
\(221\) 0 0
\(222\) 0 0
\(223\) −25.2288 −1.68944 −0.844721 0.535207i \(-0.820234\pi\)
−0.844721 + 0.535207i \(0.820234\pi\)
\(224\) − 14.9666i − 1.00000i
\(225\) 0 0
\(226\) 0 0
\(227\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(228\) 0 0
\(229\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(230\) 3.24067i 0.213684i
\(231\) 0 0
\(232\) 0 0
\(233\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 0 0
\(238\) 14.0000 0.907485
\(239\) 24.0416i 1.55512i 0.628806 + 0.777562i \(0.283544\pi\)
−0.628806 + 0.777562i \(0.716456\pi\)
\(240\) 0 0
\(241\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(242\) − 18.4735i − 1.18752i
\(243\) 0 0
\(244\) 0 0
\(245\) 1.75344i 0.112023i
\(246\) 0 0
\(247\) 0 0
\(248\) 23.4519i 1.48920i
\(249\) 0 0
\(250\) −3.52026 −0.222641
\(251\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(252\) 0 0
\(253\) −44.8745 −2.82124
\(254\) 0 0
\(255\) 0 0
\(256\) 16.0000 1.00000
\(257\) 25.7063i 1.60352i 0.597648 + 0.801759i \(0.296102\pi\)
−0.597648 + 0.801759i \(0.703898\pi\)
\(258\) 0 0
\(259\) −10.4170 −0.647281
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 28.7695i 1.77401i 0.461764 + 0.887003i \(0.347217\pi\)
−0.461764 + 0.887003i \(0.652783\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) − 32.3494i − 1.98347i
\(267\) 0 0
\(268\) 0 0
\(269\) − 13.9804i − 0.852399i −0.904629 0.426199i \(-0.859852\pi\)
0.904629 0.426199i \(-0.140148\pi\)
\(270\) 0 0
\(271\) 10.5830 0.642872 0.321436 0.946931i \(-0.395835\pi\)
0.321436 + 0.946931i \(0.395835\pi\)
\(272\) 14.9666i 0.907485i
\(273\) 0 0
\(274\) 0 0
\(275\) − 24.2191i − 1.46047i
\(276\) 0 0
\(277\) −8.06275 −0.484443 −0.242222 0.970221i \(-0.577876\pi\)
−0.242222 + 0.970221i \(0.577876\pi\)
\(278\) − 29.9333i − 1.79528i
\(279\) 0 0
\(280\) −1.87451 −0.112023
\(281\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(282\) 0 0
\(283\) 26.4575 1.57274 0.786368 0.617758i \(-0.211959\pi\)
0.786368 + 0.617758i \(0.211959\pi\)
\(284\) 26.7813i 1.58918i
\(285\) 0 0
\(286\) 0 0
\(287\) 33.0122i 1.94865i
\(288\) 0 0
\(289\) 3.00000 0.176471
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 26.1916i 1.53013i 0.643953 + 0.765065i \(0.277293\pi\)
−0.643953 + 0.765065i \(0.722707\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) − 11.1362i − 0.647281i
\(297\) 0 0
\(298\) 0 0
\(299\) 0 0
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) 0 0
\(304\) 34.5830 1.98347
\(305\) 0 0
\(306\) 0 0
\(307\) −27.3542 −1.56119 −0.780595 0.625038i \(-0.785084\pi\)
−0.780595 + 0.625038i \(0.785084\pi\)
\(308\) − 25.9568i − 1.47903i
\(309\) 0 0
\(310\) 2.93725 0.166825
\(311\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(312\) 0 0
\(313\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) − 2.00393i − 0.112023i
\(321\) 0 0
\(322\) −34.2288 −1.90749
\(323\) 32.3494i 1.79997i
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) 0 0
\(328\) −35.2915 −1.94865
\(329\) 0 0
\(330\) 0 0
\(331\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 30.7490 1.67501 0.837503 0.546433i \(-0.184015\pi\)
0.837503 + 0.546433i \(0.184015\pi\)
\(338\) 18.3848i 1.00000i
\(339\) 0 0
\(340\) 1.87451 0.101659
\(341\) 40.6730i 2.20256i
\(342\) 0 0
\(343\) −18.5203 −1.00000
\(344\) 0 0
\(345\) 0 0
\(346\) 15.5203 0.834374
\(347\) 37.2548i 1.99994i 0.00752011 + 0.999972i \(0.497606\pi\)
−0.00752011 + 0.999972i \(0.502394\pi\)
\(348\) 0 0
\(349\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(350\) − 18.4735i − 0.987451i
\(351\) 0 0
\(352\) 27.7490 1.47903
\(353\) − 36.4303i − 1.93899i −0.245110 0.969495i \(-0.578824\pi\)
0.245110 0.969495i \(-0.421176\pi\)
\(354\) 0 0
\(355\) 3.35425 0.178025
\(356\) − 22.9509i − 1.21640i
\(357\) 0 0
\(358\) 26.0000 1.37414
\(359\) 32.5269i 1.71670i 0.513061 + 0.858352i \(0.328512\pi\)
−0.513061 + 0.858352i \(0.671488\pi\)
\(360\) 0 0
\(361\) 55.7490 2.93416
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 10.7712 0.562254 0.281127 0.959671i \(-0.409292\pi\)
0.281127 + 0.959671i \(0.409292\pi\)
\(368\) − 36.5921i − 1.90749i
\(369\) 0 0
\(370\) −1.39477 −0.0725105
\(371\) 0 0
\(372\) 0 0
\(373\) 1.12549 0.0582758 0.0291379 0.999575i \(-0.490724\pi\)
0.0291379 + 0.999575i \(0.490724\pi\)
\(374\) 25.9568i 1.34220i
\(375\) 0 0
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(380\) − 4.33138i − 0.222195i
\(381\) 0 0
\(382\) −11.0627 −0.566019
\(383\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(384\) 0 0
\(385\) −3.25098 −0.165686
\(386\) − 5.65685i − 0.287926i
\(387\) 0 0
\(388\) 0 0
\(389\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(390\) 0 0
\(391\) 34.2288 1.73102
\(392\) − 19.7990i − 1.00000i
\(393\) 0 0
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(398\) 37.6828i 1.88887i
\(399\) 0 0
\(400\) 19.7490 0.987451
\(401\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 37.4166i 1.86154i
\(405\) 0 0
\(406\) 0 0
\(407\) − 19.3137i − 0.957346i
\(408\) 0 0
\(409\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(410\) 4.42011i 0.218294i
\(411\) 0 0
\(412\) −19.4170 −0.956607
\(413\) 0 0
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 0 0
\(418\) 59.9778 2.93361
\(419\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(420\) 0 0
\(421\) 27.9373 1.36158 0.680789 0.732479i \(-0.261637\pi\)
0.680789 + 0.732479i \(0.261637\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 18.4735i 0.896097i
\(426\) 0 0
\(427\) 0 0
\(428\) − 14.1421i − 0.683586i
\(429\) 0 0
\(430\) 0 0
\(431\) − 26.1186i − 1.25809i −0.777370 0.629044i \(-0.783447\pi\)
0.777370 0.629044i \(-0.216553\pi\)
\(432\) 0 0
\(433\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(434\) 31.0240i 1.48920i
\(435\) 0 0
\(436\) 21.7490 1.04159
\(437\) − 79.0915i − 3.78346i
\(438\) 0 0
\(439\) −5.29150 −0.252550 −0.126275 0.991995i \(-0.540302\pi\)
−0.126275 + 0.991995i \(0.540302\pi\)
\(440\) − 3.47545i − 0.165686i
\(441\) 0 0
\(442\) 0 0
\(443\) 41.4975i 1.97160i 0.167913 + 0.985802i \(0.446297\pi\)
−0.167913 + 0.985802i \(0.553703\pi\)
\(444\) 0 0
\(445\) −2.87451 −0.136265
\(446\) − 35.6788i − 1.68944i
\(447\) 0 0
\(448\) 21.1660 1.00000
\(449\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(450\) 0 0
\(451\) −61.2065 −2.88210
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 42.7490 1.99971 0.999857 0.0168929i \(-0.00537742\pi\)
0.999857 + 0.0168929i \(0.00537742\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) −4.58301 −0.213684
\(461\) 9.97251i 0.464466i 0.972660 + 0.232233i \(0.0746032\pi\)
−0.972660 + 0.232233i \(0.925397\pi\)
\(462\) 0 0
\(463\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) 42.6863 1.95858
\(476\) 19.7990i 0.907485i
\(477\) 0 0
\(478\) −34.0000 −1.55512
\(479\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 0 0
\(483\) 0 0
\(484\) 26.1255 1.18752
\(485\) 0 0
\(486\) 0 0
\(487\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) −2.47974 −0.112023
\(491\) 16.3078i 0.735962i 0.929833 + 0.367981i \(0.119951\pi\)
−0.929833 + 0.367981i \(0.880049\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) 0 0
\(496\) −33.1660 −1.48920
\(497\) 35.4284i 1.58918i
\(498\) 0 0
\(499\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(500\) − 4.97840i − 0.222641i
\(501\) 0 0
\(502\) 0 0
\(503\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(504\) 0 0
\(505\) 4.68627 0.208536
\(506\) − 63.4621i − 2.82124i
\(507\) 0 0
\(508\) 0 0
\(509\) 26.1916i 1.16092i 0.814288 + 0.580461i \(0.197128\pi\)
−0.814288 + 0.580461i \(0.802872\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 22.6274i 1.00000i
\(513\) 0 0
\(514\) −36.3542 −1.60352
\(515\) 2.43190i 0.107162i
\(516\) 0 0
\(517\) 0 0
\(518\) − 14.7319i − 0.647281i
\(519\) 0 0
\(520\) 0 0
\(521\) − 26.7083i − 1.17011i −0.810993 0.585056i \(-0.801073\pi\)
0.810993 0.585056i \(-0.198927\pi\)
\(522\) 0 0
\(523\) −6.16601 −0.269621 −0.134810 0.990871i \(-0.543043\pi\)
−0.134810 + 0.990871i \(0.543043\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) −40.6863 −1.77401
\(527\) − 31.0240i − 1.35143i
\(528\) 0 0
\(529\) −60.6863 −2.63853
\(530\) 0 0
\(531\) 0 0
\(532\) 45.7490 1.98347
\(533\) 0 0
\(534\) 0 0
\(535\) −1.77124 −0.0765775
\(536\) 0 0
\(537\) 0 0
\(538\) 19.7712 0.852399
\(539\) − 34.3377i − 1.47903i
\(540\) 0 0
\(541\) −43.6863 −1.87822 −0.939110 0.343617i \(-0.888348\pi\)
−0.939110 + 0.343617i \(0.888348\pi\)
\(542\) 14.9666i 0.642872i
\(543\) 0 0
\(544\) −21.1660 −0.907485
\(545\) − 2.72397i − 0.116682i
\(546\) 0 0
\(547\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 34.2510 1.46047
\(551\) 0 0
\(552\) 0 0
\(553\) 0 0
\(554\) − 11.4024i − 0.484443i
\(555\) 0 0
\(556\) 42.3320 1.79528
\(557\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) − 2.65095i − 0.112023i
\(561\) 0 0
\(562\) 0 0
\(563\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 37.4166i 1.57274i
\(567\) 0 0
\(568\) −37.8745 −1.58918
\(569\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(570\) 0 0
\(571\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) −46.6863 −1.94865
\(575\) − 45.1661i − 1.88356i
\(576\) 0 0
\(577\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(578\) 4.24264i 0.176471i
\(579\) 0 0
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) 0 0
\(586\) −37.0405 −1.53013
\(587\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(588\) 0 0
\(589\) −71.6863 −2.95378
\(590\) 0 0
\(591\) 0 0
\(592\) 15.7490 0.647281
\(593\) − 22.1995i − 0.911622i −0.890077 0.455811i \(-0.849349\pi\)
0.890077 0.455811i \(-0.150651\pi\)
\(594\) 0 0
\(595\) 2.47974 0.101659
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) − 30.0950i − 1.22965i −0.788664 0.614824i \(-0.789227\pi\)
0.788664 0.614824i \(-0.210773\pi\)
\(600\) 0 0
\(601\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) − 3.27211i − 0.133030i
\(606\) 0 0
\(607\) 26.4575 1.07388 0.536939 0.843621i \(-0.319581\pi\)
0.536939 + 0.843621i \(0.319581\pi\)
\(608\) 48.9078i 1.98347i
\(609\) 0 0
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) 7.12549 0.287796 0.143898 0.989593i \(-0.454036\pi\)
0.143898 + 0.989593i \(0.454036\pi\)
\(614\) − 38.6847i − 1.56119i
\(615\) 0 0
\(616\) 36.7085 1.47903
\(617\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(618\) 0 0
\(619\) −28.4170 −1.14218 −0.571088 0.820889i \(-0.693478\pi\)
−0.571088 + 0.820889i \(0.693478\pi\)
\(620\) 4.15390i 0.166825i
\(621\) 0 0
\(622\) 0 0
\(623\) − 30.3612i − 1.21640i
\(624\) 0 0
\(625\) 24.0627 0.962510
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 14.7319i 0.587398i
\(630\) 0 0
\(631\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) 0 0
\(640\) 2.83399 0.112023
\(641\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(642\) 0 0
\(643\) −11.4797 −0.452717 −0.226358 0.974044i \(-0.572682\pi\)
−0.226358 + 0.974044i \(0.572682\pi\)
\(644\) − 48.4068i − 1.90749i
\(645\) 0 0
\(646\) −45.7490 −1.79997
\(647\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) − 49.9097i − 1.94865i
\(657\) 0 0
\(658\) 0 0
\(659\) − 51.3082i − 1.99868i −0.0362652 0.999342i \(-0.511546\pi\)
0.0362652 0.999342i \(-0.488454\pi\)
\(660\) 0 0
\(661\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) − 5.72987i − 0.222195i
\(666\) 0 0
\(667\) 0 0
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) 44.0000 1.69608 0.848038 0.529936i \(-0.177784\pi\)
0.848038 + 0.529936i \(0.177784\pi\)
\(674\) 43.4857i 1.67501i
\(675\) 0 0
\(676\) −26.0000 −1.00000
\(677\) 48.1563i 1.85080i 0.378997 + 0.925398i \(0.376269\pi\)
−0.378997 + 0.925398i \(0.623731\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 2.65095i 0.101659i
\(681\) 0 0
\(682\) −57.5203 −2.20256
\(683\) 7.55633i 0.289135i 0.989495 + 0.144568i \(0.0461791\pi\)
−0.989495 + 0.144568i \(0.953821\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) − 26.1916i − 1.00000i
\(687\) 0 0
\(688\) 0 0
\(689\) 0 0
\(690\) 0 0
\(691\) 42.3320 1.61039 0.805193 0.593013i \(-0.202062\pi\)
0.805193 + 0.593013i \(0.202062\pi\)
\(692\) 21.9490i 0.834374i
\(693\) 0 0
\(694\) −52.6863 −1.99994
\(695\) − 5.30191i − 0.201113i
\(696\) 0 0
\(697\) 46.6863 1.76837
\(698\) 0 0
\(699\) 0 0
\(700\) 26.1255 0.987451
\(701\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(702\) 0 0
\(703\) 34.0405 1.28386
\(704\) 39.2430i 1.47903i
\(705\) 0 0
\(706\) 51.5203 1.93899
\(707\) 49.4975i 1.86154i
\(708\) 0 0
\(709\) −40.8745 −1.53507 −0.767537 0.641004i \(-0.778518\pi\)
−0.767537 + 0.641004i \(0.778518\pi\)
\(710\) 4.74362i 0.178025i
\(711\) 0 0
\(712\) 32.4575 1.21640
\(713\) 75.8508i 2.84064i
\(714\) 0 0
\(715\) 0 0
\(716\) 36.7696i 1.37414i
\(717\) 0 0
\(718\) −46.0000 −1.71670
\(719\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(720\) 0 0
\(721\) −25.6863 −0.956607
\(722\) 78.8410i 2.93416i
\(723\) 0 0
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) −52.9150 −1.96251 −0.981255 0.192715i \(-0.938271\pi\)
−0.981255 + 0.192715i \(0.938271\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 0 0
\(732\) 0 0
\(733\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(734\) 15.2328i 0.562254i
\(735\) 0 0
\(736\) 51.7490 1.90749
\(737\) 0 0
\(738\) 0 0
\(739\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(740\) − 1.97250i − 0.0725105i
\(741\) 0 0
\(742\) 0 0
\(743\) 49.9827i 1.83369i 0.399245 + 0.916844i \(0.369272\pi\)
−0.399245 + 0.916844i \(0.630728\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 1.59169i 0.0582758i
\(747\) 0 0
\(748\) −36.7085 −1.34220
\(749\) − 18.7083i − 0.683586i
\(750\) 0 0
\(751\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 2.00000 0.0726912 0.0363456 0.999339i \(-0.488428\pi\)
0.0363456 + 0.999339i \(0.488428\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 6.12549 0.222195
\(761\) − 41.1582i − 1.49198i −0.665955 0.745992i \(-0.731976\pi\)
0.665955 0.745992i \(-0.268024\pi\)
\(762\) 0 0
\(763\) 28.7712 1.04159
\(764\) − 15.6451i − 0.566019i
\(765\) 0 0
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(770\) − 4.59759i − 0.165686i
\(771\) 0 0
\(772\) 8.00000 0.287926
\(773\) − 47.6553i − 1.71404i −0.515282 0.857021i \(-0.672313\pi\)
0.515282 0.857021i \(-0.327687\pi\)
\(774\) 0 0
\(775\) −40.9373 −1.47051
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) − 107.877i − 3.86509i
\(780\) 0 0
\(781\) −65.6863 −2.35044
\(782\) 48.4068i 1.73102i
\(783\) 0 0
\(784\) 28.0000 1.00000
\(785\) 0 0
\(786\) 0 0
\(787\) −21.1660 −0.754487 −0.377243 0.926114i \(-0.623128\pi\)
−0.377243 + 0.926114i \(0.623128\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 0 0
\(794\) 0 0
\(795\) 0 0
\(796\) −53.2915 −1.88887
\(797\) − 50.6612i − 1.79451i −0.441511 0.897256i \(-0.645557\pi\)
0.441511 0.897256i \(-0.354443\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 27.9293i 0.987451i
\(801\) 0 0
\(802\) 0 0
\(803\) 0 0
\(804\) 0 0
\(805\) −6.06275 −0.213684
\(806\) 0 0
\(807\) 0 0
\(808\) −52.9150 −1.86154
\(809\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(810\) 0 0
\(811\) 11.8340 0.415548 0.207774 0.978177i \(-0.433378\pi\)
0.207774 + 0.978177i \(0.433378\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 27.3137 0.957346
\(815\) 0 0
\(816\) 0 0
\(817\) 0 0
\(818\) 0 0
\(819\) 0 0
\(820\) −6.25098 −0.218294
\(821\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(822\) 0 0
\(823\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(824\) − 27.4598i − 0.956607i
\(825\) 0 0
\(826\) 0 0
\(827\) − 21.6097i − 0.751444i −0.926732 0.375722i \(-0.877395\pi\)
0.926732 0.375722i \(-0.122605\pi\)
\(828\) 0 0
\(829\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 26.1916i 0.907485i
\(834\) 0 0
\(835\) 0 0
\(836\) 84.8214i 2.93361i
\(837\) 0 0
\(838\) 0 0
\(839\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(840\) 0 0
\(841\) −29.0000 −1.00000
\(842\) 39.5092i 1.36158i
\(843\) 0 0
\(844\) 0 0
\(845\) 3.25639i 0.112023i
\(846\) 0 0
\(847\) 34.5608 1.18752
\(848\) 0 0
\(849\) 0 0
\(850\) −26.1255 −0.896097
\(851\) − 36.0181i − 1.23468i
\(852\) 0 0
\(853\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 20.0000 0.683586
\(857\) − 15.4833i − 0.528900i −0.964399 0.264450i \(-0.914810\pi\)
0.964399 0.264450i \(-0.0851905\pi\)
\(858\) 0 0
\(859\) −23.1033 −0.788273 −0.394137 0.919052i \(-0.628956\pi\)
−0.394137 + 0.919052i \(0.628956\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 36.9373 1.25809
\(863\) 7.07107i 0.240702i 0.992731 + 0.120351i \(0.0384020\pi\)
−0.992731 + 0.120351i \(0.961598\pi\)
\(864\) 0 0
\(865\) 2.74902 0.0934693
\(866\) 0 0
\(867\) 0 0
\(868\) −43.8745 −1.48920
\(869\) 0 0
\(870\) 0 0
\(871\) 0 0
\(872\) 30.7578i 1.04159i
\(873\) 0 0
\(874\) 111.852 3.78346
\(875\) − 6.58580i − 0.222641i
\(876\) 0 0
\(877\) −22.0000 −0.742887 −0.371444 0.928456i \(-0.621137\pi\)
−0.371444 + 0.928456i \(0.621137\pi\)
\(878\) − 7.48331i − 0.252550i
\(879\) 0 0
\(880\) 4.91503 0.165686
\(881\) − 39.4362i − 1.32864i −0.747448 0.664320i \(-0.768721\pi\)
0.747448 0.664320i \(-0.231279\pi\)
\(882\) 0 0
\(883\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) −58.6863 −1.97160
\(887\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) − 4.06517i − 0.136265i
\(891\) 0 0
\(892\) 50.4575 1.68944
\(893\) 0 0
\(894\) 0 0
\(895\) 4.60523 0.153936
\(896\) 29.9333i 1.00000i
\(897\) 0 0
\(898\) 0 0
\(899\) 0 0
\(900\) 0 0
\(901\) 0 0
\(902\) − 86.5591i − 2.88210i
\(903\) 0 0
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 15.5563i 0.515405i 0.966224 + 0.257702i \(0.0829654\pi\)
−0.966224 + 0.257702i \(0.917035\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) 60.4562i 1.99971i
\(915\) 0 0
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(920\) − 6.48135i − 0.213684i
\(921\) 0 0
\(922\) −14.1033 −0.464466
\(923\) 0 0
\(924\) 0 0
\(925\) 19.4392 0.639158
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 48.6415i 1.59588i 0.602739 + 0.797939i \(0.294076\pi\)
−0.602739 + 0.797939i \(0.705924\pi\)
\(930\) 0 0
\(931\) 60.5203 1.98347
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 4.59759i 0.150357i
\(936\) 0 0
\(937\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 51.1622i 1.66784i 0.551886 + 0.833920i \(0.313908\pi\)
−0.551886 + 0.833920i \(0.686092\pi\)
\(942\) 0 0
\(943\) −114.144 −3.71703
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 54.2254i 1.76209i 0.473034 + 0.881044i \(0.343159\pi\)
−0.473034 + 0.881044i \(0.656841\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) 60.3675i 1.95858i
\(951\) 0 0
\(952\) −28.0000 −0.907485
\(953\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(954\) 0 0
\(955\) −1.95948 −0.0634073
\(956\) − 48.0833i − 1.55512i
\(957\) 0 0
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) 37.7490 1.21771
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) − 1.00197i − 0.0322544i
\(966\) 0 0
\(967\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(968\) 36.9470i 1.18752i
\(969\) 0 0
\(970\) 0 0
\(971\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(972\) 0 0
\(973\) 56.0000 1.79528
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(978\) 0 0
\(979\) 56.2915 1.79908
\(980\) − 3.50688i − 0.112023i
\(981\) 0 0
\(982\) −23.0627 −0.735962
\(983\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 0 0
\(990\) 0 0
\(991\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(992\) − 46.9038i − 1.48920i
\(993\) 0 0
\(994\) −50.1033 −1.58918
\(995\) 6.67454i 0.211597i
\(996\) 0 0
\(997\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(998\) 0 0
\(999\) 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 756.2.b.a.55.3 yes 4
3.2 odd 2 inner 756.2.b.a.55.2 4
4.3 odd 2 756.2.b.b.55.1 yes 4
7.6 odd 2 756.2.b.b.55.4 yes 4
12.11 even 2 756.2.b.b.55.4 yes 4
21.20 even 2 756.2.b.b.55.1 yes 4
28.27 even 2 inner 756.2.b.a.55.2 4
84.83 odd 2 CM 756.2.b.a.55.3 yes 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
756.2.b.a.55.2 4 3.2 odd 2 inner
756.2.b.a.55.2 4 28.27 even 2 inner
756.2.b.a.55.3 yes 4 1.1 even 1 trivial
756.2.b.a.55.3 yes 4 84.83 odd 2 CM
756.2.b.b.55.1 yes 4 4.3 odd 2
756.2.b.b.55.1 yes 4 21.20 even 2
756.2.b.b.55.4 yes 4 7.6 odd 2
756.2.b.b.55.4 yes 4 12.11 even 2