Properties

Label 3920.2.a.bm.1.1
Level $3920$
Weight $2$
Character 3920.1
Self dual yes
Analytic conductor $31.301$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 3920 = 2^{4} \cdot 5 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3920.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(31.3013575923\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{8})^+\)
Defining polynomial: \(x^{2} - 2\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 490)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-1.41421\) of defining polynomial
Character \(\chi\) \(=\) 3920.1

$q$-expansion

\(f(q)\) \(=\) \(q-3.41421 q^{3} +1.00000 q^{5} +8.65685 q^{9} +O(q^{10})\) \(q-3.41421 q^{3} +1.00000 q^{5} +8.65685 q^{9} +0.828427 q^{11} -4.82843 q^{13} -3.41421 q^{15} +2.58579 q^{17} -0.585786 q^{19} +1.17157 q^{23} +1.00000 q^{25} -19.3137 q^{27} -4.82843 q^{29} +2.82843 q^{31} -2.82843 q^{33} -7.65685 q^{37} +16.4853 q^{39} -3.07107 q^{41} +8.82843 q^{43} +8.65685 q^{45} -5.17157 q^{47} -8.82843 q^{51} +6.48528 q^{53} +0.828427 q^{55} +2.00000 q^{57} -8.58579 q^{59} +9.31371 q^{61} -4.82843 q^{65} -1.65685 q^{67} -4.00000 q^{69} +4.48528 q^{71} -9.41421 q^{73} -3.41421 q^{75} +6.82843 q^{79} +39.9706 q^{81} +2.24264 q^{83} +2.58579 q^{85} +16.4853 q^{87} +12.7279 q^{89} -9.65685 q^{93} -0.585786 q^{95} -7.75736 q^{97} +7.17157 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 4q^{3} + 2q^{5} + 6q^{9} + O(q^{10}) \) \( 2q - 4q^{3} + 2q^{5} + 6q^{9} - 4q^{11} - 4q^{13} - 4q^{15} + 8q^{17} - 4q^{19} + 8q^{23} + 2q^{25} - 16q^{27} - 4q^{29} - 4q^{37} + 16q^{39} + 8q^{41} + 12q^{43} + 6q^{45} - 16q^{47} - 12q^{51} - 4q^{53} - 4q^{55} + 4q^{57} - 20q^{59} - 4q^{61} - 4q^{65} + 8q^{67} - 8q^{69} - 8q^{71} - 16q^{73} - 4q^{75} + 8q^{79} + 46q^{81} - 4q^{83} + 8q^{85} + 16q^{87} - 8q^{93} - 4q^{95} - 24q^{97} + 20q^{99} + O(q^{100}) \)

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
\(3\) −3.41421 −1.97120 −0.985599 0.169102i \(-0.945913\pi\)
−0.985599 + 0.169102i \(0.945913\pi\)
\(4\) 0 0
\(5\) 1.00000 0.447214
\(6\) 0 0
\(7\) 0 0
\(8\) 0 0
\(9\) 8.65685 2.88562
\(10\) 0 0
\(11\) 0.828427 0.249780 0.124890 0.992171i \(-0.460142\pi\)
0.124890 + 0.992171i \(0.460142\pi\)
\(12\) 0 0
\(13\) −4.82843 −1.33916 −0.669582 0.742738i \(-0.733527\pi\)
−0.669582 + 0.742738i \(0.733527\pi\)
\(14\) 0 0
\(15\) −3.41421 −0.881546
\(16\) 0 0
\(17\) 2.58579 0.627145 0.313573 0.949564i \(-0.398474\pi\)
0.313573 + 0.949564i \(0.398474\pi\)
\(18\) 0 0
\(19\) −0.585786 −0.134389 −0.0671943 0.997740i \(-0.521405\pi\)
−0.0671943 + 0.997740i \(0.521405\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 1.17157 0.244290 0.122145 0.992512i \(-0.461023\pi\)
0.122145 + 0.992512i \(0.461023\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) −19.3137 −3.71692
\(28\) 0 0
\(29\) −4.82843 −0.896616 −0.448308 0.893879i \(-0.647973\pi\)
−0.448308 + 0.893879i \(0.647973\pi\)
\(30\) 0 0
\(31\) 2.82843 0.508001 0.254000 0.967204i \(-0.418254\pi\)
0.254000 + 0.967204i \(0.418254\pi\)
\(32\) 0 0
\(33\) −2.82843 −0.492366
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −7.65685 −1.25878 −0.629390 0.777090i \(-0.716695\pi\)
−0.629390 + 0.777090i \(0.716695\pi\)
\(38\) 0 0
\(39\) 16.4853 2.63976
\(40\) 0 0
\(41\) −3.07107 −0.479620 −0.239810 0.970820i \(-0.577085\pi\)
−0.239810 + 0.970820i \(0.577085\pi\)
\(42\) 0 0
\(43\) 8.82843 1.34632 0.673161 0.739496i \(-0.264936\pi\)
0.673161 + 0.739496i \(0.264936\pi\)
\(44\) 0 0
\(45\) 8.65685 1.29049
\(46\) 0 0
\(47\) −5.17157 −0.754351 −0.377176 0.926142i \(-0.623105\pi\)
−0.377176 + 0.926142i \(0.623105\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 0 0
\(51\) −8.82843 −1.23623
\(52\) 0 0
\(53\) 6.48528 0.890822 0.445411 0.895326i \(-0.353058\pi\)
0.445411 + 0.895326i \(0.353058\pi\)
\(54\) 0 0
\(55\) 0.828427 0.111705
\(56\) 0 0
\(57\) 2.00000 0.264906
\(58\) 0 0
\(59\) −8.58579 −1.11777 −0.558887 0.829244i \(-0.688771\pi\)
−0.558887 + 0.829244i \(0.688771\pi\)
\(60\) 0 0
\(61\) 9.31371 1.19250 0.596249 0.802799i \(-0.296657\pi\)
0.596249 + 0.802799i \(0.296657\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −4.82843 −0.598893
\(66\) 0 0
\(67\) −1.65685 −0.202417 −0.101208 0.994865i \(-0.532271\pi\)
−0.101208 + 0.994865i \(0.532271\pi\)
\(68\) 0 0
\(69\) −4.00000 −0.481543
\(70\) 0 0
\(71\) 4.48528 0.532305 0.266152 0.963931i \(-0.414248\pi\)
0.266152 + 0.963931i \(0.414248\pi\)
\(72\) 0 0
\(73\) −9.41421 −1.10185 −0.550925 0.834555i \(-0.685725\pi\)
−0.550925 + 0.834555i \(0.685725\pi\)
\(74\) 0 0
\(75\) −3.41421 −0.394239
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 6.82843 0.768258 0.384129 0.923279i \(-0.374502\pi\)
0.384129 + 0.923279i \(0.374502\pi\)
\(80\) 0 0
\(81\) 39.9706 4.44117
\(82\) 0 0
\(83\) 2.24264 0.246162 0.123081 0.992397i \(-0.460723\pi\)
0.123081 + 0.992397i \(0.460723\pi\)
\(84\) 0 0
\(85\) 2.58579 0.280468
\(86\) 0 0
\(87\) 16.4853 1.76741
\(88\) 0 0
\(89\) 12.7279 1.34916 0.674579 0.738203i \(-0.264325\pi\)
0.674579 + 0.738203i \(0.264325\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) −9.65685 −1.00137
\(94\) 0 0
\(95\) −0.585786 −0.0601004
\(96\) 0 0
\(97\) −7.75736 −0.787641 −0.393820 0.919187i \(-0.628847\pi\)
−0.393820 + 0.919187i \(0.628847\pi\)
\(98\) 0 0
\(99\) 7.17157 0.720770
\(100\) 0 0
\(101\) 13.3137 1.32476 0.662382 0.749166i \(-0.269546\pi\)
0.662382 + 0.749166i \(0.269546\pi\)
\(102\) 0 0
\(103\) 14.8284 1.46109 0.730544 0.682865i \(-0.239266\pi\)
0.730544 + 0.682865i \(0.239266\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −9.65685 −0.933563 −0.466782 0.884373i \(-0.654587\pi\)
−0.466782 + 0.884373i \(0.654587\pi\)
\(108\) 0 0
\(109\) 2.48528 0.238047 0.119023 0.992891i \(-0.462024\pi\)
0.119023 + 0.992891i \(0.462024\pi\)
\(110\) 0 0
\(111\) 26.1421 2.48130
\(112\) 0 0
\(113\) −15.3137 −1.44059 −0.720296 0.693667i \(-0.755994\pi\)
−0.720296 + 0.693667i \(0.755994\pi\)
\(114\) 0 0
\(115\) 1.17157 0.109250
\(116\) 0 0
\(117\) −41.7990 −3.86432
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) −10.3137 −0.937610
\(122\) 0 0
\(123\) 10.4853 0.945426
\(124\) 0 0
\(125\) 1.00000 0.0894427
\(126\) 0 0
\(127\) −2.82843 −0.250982 −0.125491 0.992095i \(-0.540051\pi\)
−0.125491 + 0.992095i \(0.540051\pi\)
\(128\) 0 0
\(129\) −30.1421 −2.65387
\(130\) 0 0
\(131\) 6.24264 0.545422 0.272711 0.962096i \(-0.412080\pi\)
0.272711 + 0.962096i \(0.412080\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) −19.3137 −1.66226
\(136\) 0 0
\(137\) −16.0000 −1.36697 −0.683486 0.729964i \(-0.739537\pi\)
−0.683486 + 0.729964i \(0.739537\pi\)
\(138\) 0 0
\(139\) −19.8995 −1.68785 −0.843927 0.536459i \(-0.819762\pi\)
−0.843927 + 0.536459i \(0.819762\pi\)
\(140\) 0 0
\(141\) 17.6569 1.48698
\(142\) 0 0
\(143\) −4.00000 −0.334497
\(144\) 0 0
\(145\) −4.82843 −0.400979
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) −6.00000 −0.491539 −0.245770 0.969328i \(-0.579041\pi\)
−0.245770 + 0.969328i \(0.579041\pi\)
\(150\) 0 0
\(151\) 11.3137 0.920697 0.460348 0.887738i \(-0.347725\pi\)
0.460348 + 0.887738i \(0.347725\pi\)
\(152\) 0 0
\(153\) 22.3848 1.80970
\(154\) 0 0
\(155\) 2.82843 0.227185
\(156\) 0 0
\(157\) 6.48528 0.517582 0.258791 0.965933i \(-0.416676\pi\)
0.258791 + 0.965933i \(0.416676\pi\)
\(158\) 0 0
\(159\) −22.1421 −1.75599
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) 20.1421 1.57765 0.788827 0.614615i \(-0.210689\pi\)
0.788827 + 0.614615i \(0.210689\pi\)
\(164\) 0 0
\(165\) −2.82843 −0.220193
\(166\) 0 0
\(167\) −15.7990 −1.22256 −0.611281 0.791413i \(-0.709346\pi\)
−0.611281 + 0.791413i \(0.709346\pi\)
\(168\) 0 0
\(169\) 10.3137 0.793362
\(170\) 0 0
\(171\) −5.07107 −0.387794
\(172\) 0 0
\(173\) −8.82843 −0.671213 −0.335606 0.942002i \(-0.608941\pi\)
−0.335606 + 0.942002i \(0.608941\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 29.3137 2.20335
\(178\) 0 0
\(179\) −4.00000 −0.298974 −0.149487 0.988764i \(-0.547762\pi\)
−0.149487 + 0.988764i \(0.547762\pi\)
\(180\) 0 0
\(181\) −2.48528 −0.184730 −0.0923648 0.995725i \(-0.529443\pi\)
−0.0923648 + 0.995725i \(0.529443\pi\)
\(182\) 0 0
\(183\) −31.7990 −2.35065
\(184\) 0 0
\(185\) −7.65685 −0.562943
\(186\) 0 0
\(187\) 2.14214 0.156648
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −10.1421 −0.733859 −0.366930 0.930249i \(-0.619591\pi\)
−0.366930 + 0.930249i \(0.619591\pi\)
\(192\) 0 0
\(193\) 5.65685 0.407189 0.203595 0.979055i \(-0.434738\pi\)
0.203595 + 0.979055i \(0.434738\pi\)
\(194\) 0 0
\(195\) 16.4853 1.18054
\(196\) 0 0
\(197\) −25.7990 −1.83810 −0.919051 0.394139i \(-0.871043\pi\)
−0.919051 + 0.394139i \(0.871043\pi\)
\(198\) 0 0
\(199\) −16.4853 −1.16861 −0.584305 0.811534i \(-0.698633\pi\)
−0.584305 + 0.811534i \(0.698633\pi\)
\(200\) 0 0
\(201\) 5.65685 0.399004
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) −3.07107 −0.214493
\(206\) 0 0
\(207\) 10.1421 0.704927
\(208\) 0 0
\(209\) −0.485281 −0.0335676
\(210\) 0 0
\(211\) −18.6274 −1.28236 −0.641182 0.767389i \(-0.721556\pi\)
−0.641182 + 0.767389i \(0.721556\pi\)
\(212\) 0 0
\(213\) −15.3137 −1.04928
\(214\) 0 0
\(215\) 8.82843 0.602094
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) 32.1421 2.17196
\(220\) 0 0
\(221\) −12.4853 −0.839851
\(222\) 0 0
\(223\) 7.31371 0.489762 0.244881 0.969553i \(-0.421251\pi\)
0.244881 + 0.969553i \(0.421251\pi\)
\(224\) 0 0
\(225\) 8.65685 0.577124
\(226\) 0 0
\(227\) −18.2426 −1.21081 −0.605403 0.795919i \(-0.706988\pi\)
−0.605403 + 0.795919i \(0.706988\pi\)
\(228\) 0 0
\(229\) −16.1421 −1.06670 −0.533351 0.845894i \(-0.679068\pi\)
−0.533351 + 0.845894i \(0.679068\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 23.3137 1.52733 0.763666 0.645612i \(-0.223397\pi\)
0.763666 + 0.645612i \(0.223397\pi\)
\(234\) 0 0
\(235\) −5.17157 −0.337356
\(236\) 0 0
\(237\) −23.3137 −1.51439
\(238\) 0 0
\(239\) −1.65685 −0.107173 −0.0535865 0.998563i \(-0.517065\pi\)
−0.0535865 + 0.998563i \(0.517065\pi\)
\(240\) 0 0
\(241\) −13.4142 −0.864085 −0.432043 0.901853i \(-0.642207\pi\)
−0.432043 + 0.901853i \(0.642207\pi\)
\(242\) 0 0
\(243\) −78.5269 −5.03750
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 2.82843 0.179969
\(248\) 0 0
\(249\) −7.65685 −0.485233
\(250\) 0 0
\(251\) −0.585786 −0.0369745 −0.0184873 0.999829i \(-0.505885\pi\)
−0.0184873 + 0.999829i \(0.505885\pi\)
\(252\) 0 0
\(253\) 0.970563 0.0610188
\(254\) 0 0
\(255\) −8.82843 −0.552858
\(256\) 0 0
\(257\) −9.89949 −0.617514 −0.308757 0.951141i \(-0.599913\pi\)
−0.308757 + 0.951141i \(0.599913\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) −41.7990 −2.58729
\(262\) 0 0
\(263\) −28.0000 −1.72655 −0.863277 0.504730i \(-0.831592\pi\)
−0.863277 + 0.504730i \(0.831592\pi\)
\(264\) 0 0
\(265\) 6.48528 0.398388
\(266\) 0 0
\(267\) −43.4558 −2.65945
\(268\) 0 0
\(269\) 18.4853 1.12707 0.563534 0.826093i \(-0.309441\pi\)
0.563534 + 0.826093i \(0.309441\pi\)
\(270\) 0 0
\(271\) −12.0000 −0.728948 −0.364474 0.931214i \(-0.618751\pi\)
−0.364474 + 0.931214i \(0.618751\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 0.828427 0.0499560
\(276\) 0 0
\(277\) −8.14214 −0.489214 −0.244607 0.969622i \(-0.578659\pi\)
−0.244607 + 0.969622i \(0.578659\pi\)
\(278\) 0 0
\(279\) 24.4853 1.46590
\(280\) 0 0
\(281\) 8.00000 0.477240 0.238620 0.971113i \(-0.423305\pi\)
0.238620 + 0.971113i \(0.423305\pi\)
\(282\) 0 0
\(283\) 2.24264 0.133311 0.0666556 0.997776i \(-0.478767\pi\)
0.0666556 + 0.997776i \(0.478767\pi\)
\(284\) 0 0
\(285\) 2.00000 0.118470
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) −10.3137 −0.606689
\(290\) 0 0
\(291\) 26.4853 1.55259
\(292\) 0 0
\(293\) −8.34315 −0.487412 −0.243706 0.969849i \(-0.578363\pi\)
−0.243706 + 0.969849i \(0.578363\pi\)
\(294\) 0 0
\(295\) −8.58579 −0.499884
\(296\) 0 0
\(297\) −16.0000 −0.928414
\(298\) 0 0
\(299\) −5.65685 −0.327144
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) −45.4558 −2.61137
\(304\) 0 0
\(305\) 9.31371 0.533301
\(306\) 0 0
\(307\) −14.9289 −0.852039 −0.426020 0.904714i \(-0.640085\pi\)
−0.426020 + 0.904714i \(0.640085\pi\)
\(308\) 0 0
\(309\) −50.6274 −2.88009
\(310\) 0 0
\(311\) −4.00000 −0.226819 −0.113410 0.993548i \(-0.536177\pi\)
−0.113410 + 0.993548i \(0.536177\pi\)
\(312\) 0 0
\(313\) 14.3848 0.813076 0.406538 0.913634i \(-0.366736\pi\)
0.406538 + 0.913634i \(0.366736\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 10.4853 0.588912 0.294456 0.955665i \(-0.404862\pi\)
0.294456 + 0.955665i \(0.404862\pi\)
\(318\) 0 0
\(319\) −4.00000 −0.223957
\(320\) 0 0
\(321\) 32.9706 1.84024
\(322\) 0 0
\(323\) −1.51472 −0.0842812
\(324\) 0 0
\(325\) −4.82843 −0.267833
\(326\) 0 0
\(327\) −8.48528 −0.469237
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) −33.7990 −1.85776 −0.928880 0.370380i \(-0.879227\pi\)
−0.928880 + 0.370380i \(0.879227\pi\)
\(332\) 0 0
\(333\) −66.2843 −3.63236
\(334\) 0 0
\(335\) −1.65685 −0.0905236
\(336\) 0 0
\(337\) 6.00000 0.326841 0.163420 0.986557i \(-0.447747\pi\)
0.163420 + 0.986557i \(0.447747\pi\)
\(338\) 0 0
\(339\) 52.2843 2.83969
\(340\) 0 0
\(341\) 2.34315 0.126888
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) −4.00000 −0.215353
\(346\) 0 0
\(347\) 3.17157 0.170259 0.0851295 0.996370i \(-0.472870\pi\)
0.0851295 + 0.996370i \(0.472870\pi\)
\(348\) 0 0
\(349\) 2.48528 0.133034 0.0665170 0.997785i \(-0.478811\pi\)
0.0665170 + 0.997785i \(0.478811\pi\)
\(350\) 0 0
\(351\) 93.2548 4.97757
\(352\) 0 0
\(353\) −2.38478 −0.126929 −0.0634644 0.997984i \(-0.520215\pi\)
−0.0634644 + 0.997984i \(0.520215\pi\)
\(354\) 0 0
\(355\) 4.48528 0.238054
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −28.2843 −1.49279 −0.746393 0.665505i \(-0.768216\pi\)
−0.746393 + 0.665505i \(0.768216\pi\)
\(360\) 0 0
\(361\) −18.6569 −0.981940
\(362\) 0 0
\(363\) 35.2132 1.84821
\(364\) 0 0
\(365\) −9.41421 −0.492762
\(366\) 0 0
\(367\) −24.9706 −1.30345 −0.651726 0.758454i \(-0.725955\pi\)
−0.651726 + 0.758454i \(0.725955\pi\)
\(368\) 0 0
\(369\) −26.5858 −1.38400
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) −30.4853 −1.57847 −0.789234 0.614093i \(-0.789522\pi\)
−0.789234 + 0.614093i \(0.789522\pi\)
\(374\) 0 0
\(375\) −3.41421 −0.176309
\(376\) 0 0
\(377\) 23.3137 1.20072
\(378\) 0 0
\(379\) −34.4853 −1.77139 −0.885695 0.464268i \(-0.846318\pi\)
−0.885695 + 0.464268i \(0.846318\pi\)
\(380\) 0 0
\(381\) 9.65685 0.494736
\(382\) 0 0
\(383\) 32.4853 1.65992 0.829960 0.557823i \(-0.188363\pi\)
0.829960 + 0.557823i \(0.188363\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 76.4264 3.88497
\(388\) 0 0
\(389\) 28.1421 1.42686 0.713431 0.700725i \(-0.247140\pi\)
0.713431 + 0.700725i \(0.247140\pi\)
\(390\) 0 0
\(391\) 3.02944 0.153205
\(392\) 0 0
\(393\) −21.3137 −1.07513
\(394\) 0 0
\(395\) 6.82843 0.343575
\(396\) 0 0
\(397\) 33.7990 1.69632 0.848161 0.529738i \(-0.177710\pi\)
0.848161 + 0.529738i \(0.177710\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −6.00000 −0.299626 −0.149813 0.988714i \(-0.547867\pi\)
−0.149813 + 0.988714i \(0.547867\pi\)
\(402\) 0 0
\(403\) −13.6569 −0.680296
\(404\) 0 0
\(405\) 39.9706 1.98615
\(406\) 0 0
\(407\) −6.34315 −0.314418
\(408\) 0 0
\(409\) −10.5858 −0.523433 −0.261717 0.965145i \(-0.584289\pi\)
−0.261717 + 0.965145i \(0.584289\pi\)
\(410\) 0 0
\(411\) 54.6274 2.69457
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) 2.24264 0.110087
\(416\) 0 0
\(417\) 67.9411 3.32709
\(418\) 0 0
\(419\) 20.8701 1.01957 0.509785 0.860302i \(-0.329725\pi\)
0.509785 + 0.860302i \(0.329725\pi\)
\(420\) 0 0
\(421\) 17.3137 0.843819 0.421909 0.906638i \(-0.361360\pi\)
0.421909 + 0.906638i \(0.361360\pi\)
\(422\) 0 0
\(423\) −44.7696 −2.17677
\(424\) 0 0
\(425\) 2.58579 0.125429
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) 13.6569 0.659359
\(430\) 0 0
\(431\) 22.3431 1.07623 0.538116 0.842871i \(-0.319136\pi\)
0.538116 + 0.842871i \(0.319136\pi\)
\(432\) 0 0
\(433\) 10.5858 0.508720 0.254360 0.967110i \(-0.418135\pi\)
0.254360 + 0.967110i \(0.418135\pi\)
\(434\) 0 0
\(435\) 16.4853 0.790409
\(436\) 0 0
\(437\) −0.686292 −0.0328298
\(438\) 0 0
\(439\) 24.9706 1.19178 0.595890 0.803066i \(-0.296799\pi\)
0.595890 + 0.803066i \(0.296799\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −3.02944 −0.143933 −0.0719665 0.997407i \(-0.522927\pi\)
−0.0719665 + 0.997407i \(0.522927\pi\)
\(444\) 0 0
\(445\) 12.7279 0.603361
\(446\) 0 0
\(447\) 20.4853 0.968921
\(448\) 0 0
\(449\) −16.6274 −0.784696 −0.392348 0.919817i \(-0.628337\pi\)
−0.392348 + 0.919817i \(0.628337\pi\)
\(450\) 0 0
\(451\) −2.54416 −0.119800
\(452\) 0 0
\(453\) −38.6274 −1.81487
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 21.6569 1.01306 0.506532 0.862221i \(-0.330927\pi\)
0.506532 + 0.862221i \(0.330927\pi\)
\(458\) 0 0
\(459\) −49.9411 −2.33105
\(460\) 0 0
\(461\) −12.8284 −0.597479 −0.298740 0.954335i \(-0.596566\pi\)
−0.298740 + 0.954335i \(0.596566\pi\)
\(462\) 0 0
\(463\) 16.9706 0.788689 0.394344 0.918963i \(-0.370972\pi\)
0.394344 + 0.918963i \(0.370972\pi\)
\(464\) 0 0
\(465\) −9.65685 −0.447826
\(466\) 0 0
\(467\) 15.8995 0.735741 0.367870 0.929877i \(-0.380087\pi\)
0.367870 + 0.929877i \(0.380087\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) −22.1421 −1.02026
\(472\) 0 0
\(473\) 7.31371 0.336285
\(474\) 0 0
\(475\) −0.585786 −0.0268777
\(476\) 0 0
\(477\) 56.1421 2.57057
\(478\) 0 0
\(479\) 17.1716 0.784589 0.392295 0.919840i \(-0.371681\pi\)
0.392295 + 0.919840i \(0.371681\pi\)
\(480\) 0 0
\(481\) 36.9706 1.68571
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) −7.75736 −0.352244
\(486\) 0 0
\(487\) −31.7990 −1.44095 −0.720475 0.693481i \(-0.756076\pi\)
−0.720475 + 0.693481i \(0.756076\pi\)
\(488\) 0 0
\(489\) −68.7696 −3.10987
\(490\) 0 0
\(491\) −32.2843 −1.45697 −0.728484 0.685062i \(-0.759775\pi\)
−0.728484 + 0.685062i \(0.759775\pi\)
\(492\) 0 0
\(493\) −12.4853 −0.562309
\(494\) 0 0
\(495\) 7.17157 0.322338
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) −30.3431 −1.35835 −0.679173 0.733978i \(-0.737661\pi\)
−0.679173 + 0.733978i \(0.737661\pi\)
\(500\) 0 0
\(501\) 53.9411 2.40991
\(502\) 0 0
\(503\) −17.6569 −0.787280 −0.393640 0.919265i \(-0.628784\pi\)
−0.393640 + 0.919265i \(0.628784\pi\)
\(504\) 0 0
\(505\) 13.3137 0.592452
\(506\) 0 0
\(507\) −35.2132 −1.56387
\(508\) 0 0
\(509\) 5.79899 0.257036 0.128518 0.991707i \(-0.458978\pi\)
0.128518 + 0.991707i \(0.458978\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) 11.3137 0.499512
\(514\) 0 0
\(515\) 14.8284 0.653419
\(516\) 0 0
\(517\) −4.28427 −0.188422
\(518\) 0 0
\(519\) 30.1421 1.32309
\(520\) 0 0
\(521\) 19.0711 0.835519 0.417759 0.908558i \(-0.362816\pi\)
0.417759 + 0.908558i \(0.362816\pi\)
\(522\) 0 0
\(523\) −23.8995 −1.04505 −0.522526 0.852623i \(-0.675010\pi\)
−0.522526 + 0.852623i \(0.675010\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 7.31371 0.318590
\(528\) 0 0
\(529\) −21.6274 −0.940322
\(530\) 0 0
\(531\) −74.3259 −3.22547
\(532\) 0 0
\(533\) 14.8284 0.642290
\(534\) 0 0
\(535\) −9.65685 −0.417502
\(536\) 0 0
\(537\) 13.6569 0.589337
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) −14.9706 −0.643635 −0.321817 0.946802i \(-0.604294\pi\)
−0.321817 + 0.946802i \(0.604294\pi\)
\(542\) 0 0
\(543\) 8.48528 0.364138
\(544\) 0 0
\(545\) 2.48528 0.106458
\(546\) 0 0
\(547\) 10.4853 0.448318 0.224159 0.974553i \(-0.428036\pi\)
0.224159 + 0.974553i \(0.428036\pi\)
\(548\) 0 0
\(549\) 80.6274 3.44109
\(550\) 0 0
\(551\) 2.82843 0.120495
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) 26.1421 1.10967
\(556\) 0 0
\(557\) 15.1716 0.642840 0.321420 0.946937i \(-0.395840\pi\)
0.321420 + 0.946937i \(0.395840\pi\)
\(558\) 0 0
\(559\) −42.6274 −1.80295
\(560\) 0 0
\(561\) −7.31371 −0.308785
\(562\) 0 0
\(563\) 36.5858 1.54191 0.770954 0.636891i \(-0.219780\pi\)
0.770954 + 0.636891i \(0.219780\pi\)
\(564\) 0 0
\(565\) −15.3137 −0.644253
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −29.3137 −1.22889 −0.614447 0.788958i \(-0.710621\pi\)
−0.614447 + 0.788958i \(0.710621\pi\)
\(570\) 0 0
\(571\) 2.20101 0.0921094 0.0460547 0.998939i \(-0.485335\pi\)
0.0460547 + 0.998939i \(0.485335\pi\)
\(572\) 0 0
\(573\) 34.6274 1.44658
\(574\) 0 0
\(575\) 1.17157 0.0488580
\(576\) 0 0
\(577\) 6.10051 0.253967 0.126984 0.991905i \(-0.459470\pi\)
0.126984 + 0.991905i \(0.459470\pi\)
\(578\) 0 0
\(579\) −19.3137 −0.802650
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 5.37258 0.222510
\(584\) 0 0
\(585\) −41.7990 −1.72818
\(586\) 0 0
\(587\) −17.0711 −0.704598 −0.352299 0.935887i \(-0.614600\pi\)
−0.352299 + 0.935887i \(0.614600\pi\)
\(588\) 0 0
\(589\) −1.65685 −0.0682695
\(590\) 0 0
\(591\) 88.0833 3.62326
\(592\) 0 0
\(593\) 3.27208 0.134368 0.0671841 0.997741i \(-0.478599\pi\)
0.0671841 + 0.997741i \(0.478599\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 56.2843 2.30356
\(598\) 0 0
\(599\) 10.8284 0.442438 0.221219 0.975224i \(-0.428997\pi\)
0.221219 + 0.975224i \(0.428997\pi\)
\(600\) 0 0
\(601\) 6.58579 0.268640 0.134320 0.990938i \(-0.457115\pi\)
0.134320 + 0.990938i \(0.457115\pi\)
\(602\) 0 0
\(603\) −14.3431 −0.584098
\(604\) 0 0
\(605\) −10.3137 −0.419312
\(606\) 0 0
\(607\) −16.2843 −0.660958 −0.330479 0.943813i \(-0.607210\pi\)
−0.330479 + 0.943813i \(0.607210\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 24.9706 1.01020
\(612\) 0 0
\(613\) −12.3431 −0.498535 −0.249267 0.968435i \(-0.580190\pi\)
−0.249267 + 0.968435i \(0.580190\pi\)
\(614\) 0 0
\(615\) 10.4853 0.422807
\(616\) 0 0
\(617\) −33.3137 −1.34116 −0.670580 0.741837i \(-0.733955\pi\)
−0.670580 + 0.741837i \(0.733955\pi\)
\(618\) 0 0
\(619\) −29.0711 −1.16846 −0.584232 0.811586i \(-0.698604\pi\)
−0.584232 + 0.811586i \(0.698604\pi\)
\(620\) 0 0
\(621\) −22.6274 −0.908007
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) 1.65685 0.0661684
\(628\) 0 0
\(629\) −19.7990 −0.789437
\(630\) 0 0
\(631\) 12.4853 0.497031 0.248516 0.968628i \(-0.420057\pi\)
0.248516 + 0.968628i \(0.420057\pi\)
\(632\) 0 0
\(633\) 63.5980 2.52779
\(634\) 0 0
\(635\) −2.82843 −0.112243
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) 38.8284 1.53603
\(640\) 0 0
\(641\) −24.6274 −0.972724 −0.486362 0.873757i \(-0.661676\pi\)
−0.486362 + 0.873757i \(0.661676\pi\)
\(642\) 0 0
\(643\) −4.78680 −0.188773 −0.0943864 0.995536i \(-0.530089\pi\)
−0.0943864 + 0.995536i \(0.530089\pi\)
\(644\) 0 0
\(645\) −30.1421 −1.18685
\(646\) 0 0
\(647\) 23.1127 0.908654 0.454327 0.890835i \(-0.349880\pi\)
0.454327 + 0.890835i \(0.349880\pi\)
\(648\) 0 0
\(649\) −7.11270 −0.279198
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 4.34315 0.169960 0.0849802 0.996383i \(-0.472917\pi\)
0.0849802 + 0.996383i \(0.472917\pi\)
\(654\) 0 0
\(655\) 6.24264 0.243920
\(656\) 0 0
\(657\) −81.4975 −3.17952
\(658\) 0 0
\(659\) 27.1716 1.05845 0.529227 0.848480i \(-0.322482\pi\)
0.529227 + 0.848480i \(0.322482\pi\)
\(660\) 0 0
\(661\) −38.2843 −1.48909 −0.744543 0.667575i \(-0.767332\pi\)
−0.744543 + 0.667575i \(0.767332\pi\)
\(662\) 0 0
\(663\) 42.6274 1.65551
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) −5.65685 −0.219034
\(668\) 0 0
\(669\) −24.9706 −0.965418
\(670\) 0 0
\(671\) 7.71573 0.297862
\(672\) 0 0
\(673\) −48.0000 −1.85026 −0.925132 0.379646i \(-0.876046\pi\)
−0.925132 + 0.379646i \(0.876046\pi\)
\(674\) 0 0
\(675\) −19.3137 −0.743385
\(676\) 0 0
\(677\) 39.4558 1.51641 0.758206 0.652015i \(-0.226076\pi\)
0.758206 + 0.652015i \(0.226076\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 62.2843 2.38674
\(682\) 0 0
\(683\) −33.6569 −1.28784 −0.643922 0.765091i \(-0.722694\pi\)
−0.643922 + 0.765091i \(0.722694\pi\)
\(684\) 0 0
\(685\) −16.0000 −0.611329
\(686\) 0 0
\(687\) 55.1127 2.10268
\(688\) 0 0
\(689\) −31.3137 −1.19296
\(690\) 0 0
\(691\) −1.75736 −0.0668531 −0.0334265 0.999441i \(-0.510642\pi\)
−0.0334265 + 0.999441i \(0.510642\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −19.8995 −0.754831
\(696\) 0 0
\(697\) −7.94113 −0.300792
\(698\) 0 0
\(699\) −79.5980 −3.01067
\(700\) 0 0
\(701\) 2.48528 0.0938678 0.0469339 0.998898i \(-0.485055\pi\)
0.0469339 + 0.998898i \(0.485055\pi\)
\(702\) 0 0
\(703\) 4.48528 0.169166
\(704\) 0 0
\(705\) 17.6569 0.664996
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) 45.1127 1.69424 0.847121 0.531399i \(-0.178334\pi\)
0.847121 + 0.531399i \(0.178334\pi\)
\(710\) 0 0
\(711\) 59.1127 2.21690
\(712\) 0 0
\(713\) 3.31371 0.124099
\(714\) 0 0
\(715\) −4.00000 −0.149592
\(716\) 0 0
\(717\) 5.65685 0.211259
\(718\) 0 0
\(719\) −41.4558 −1.54604 −0.773021 0.634380i \(-0.781255\pi\)
−0.773021 + 0.634380i \(0.781255\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) 45.7990 1.70328
\(724\) 0 0
\(725\) −4.82843 −0.179323
\(726\) 0 0
\(727\) 3.51472 0.130354 0.0651768 0.997874i \(-0.479239\pi\)
0.0651768 + 0.997874i \(0.479239\pi\)
\(728\) 0 0
\(729\) 148.196 5.48874
\(730\) 0 0
\(731\) 22.8284 0.844340
\(732\) 0 0
\(733\) 34.0000 1.25582 0.627909 0.778287i \(-0.283911\pi\)
0.627909 + 0.778287i \(0.283911\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −1.37258 −0.0505597
\(738\) 0 0
\(739\) −3.17157 −0.116668 −0.0583341 0.998297i \(-0.518579\pi\)
−0.0583341 + 0.998297i \(0.518579\pi\)
\(740\) 0 0
\(741\) −9.65685 −0.354753
\(742\) 0 0
\(743\) −51.7990 −1.90032 −0.950160 0.311762i \(-0.899081\pi\)
−0.950160 + 0.311762i \(0.899081\pi\)
\(744\) 0 0
\(745\) −6.00000 −0.219823
\(746\) 0 0
\(747\) 19.4142 0.710329
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) 39.3137 1.43458 0.717289 0.696776i \(-0.245383\pi\)
0.717289 + 0.696776i \(0.245383\pi\)
\(752\) 0 0
\(753\) 2.00000 0.0728841
\(754\) 0 0
\(755\) 11.3137 0.411748
\(756\) 0 0
\(757\) 3.65685 0.132911 0.0664553 0.997789i \(-0.478831\pi\)
0.0664553 + 0.997789i \(0.478831\pi\)
\(758\) 0 0
\(759\) −3.31371 −0.120280
\(760\) 0 0
\(761\) 22.3848 0.811448 0.405724 0.913996i \(-0.367020\pi\)
0.405724 + 0.913996i \(0.367020\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) 22.3848 0.809323
\(766\) 0 0
\(767\) 41.4558 1.49688
\(768\) 0 0
\(769\) 19.5563 0.705220 0.352610 0.935770i \(-0.385294\pi\)
0.352610 + 0.935770i \(0.385294\pi\)
\(770\) 0 0
\(771\) 33.7990 1.21724
\(772\) 0 0
\(773\) −2.00000 −0.0719350 −0.0359675 0.999353i \(-0.511451\pi\)
−0.0359675 + 0.999353i \(0.511451\pi\)
\(774\) 0 0
\(775\) 2.82843 0.101600
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 1.79899 0.0644555
\(780\) 0 0
\(781\) 3.71573 0.132959
\(782\) 0 0
\(783\) 93.2548 3.33266
\(784\) 0 0
\(785\) 6.48528 0.231470
\(786\) 0 0
\(787\) 1.27208 0.0453447 0.0226723 0.999743i \(-0.492783\pi\)
0.0226723 + 0.999743i \(0.492783\pi\)
\(788\) 0 0
\(789\) 95.5980 3.40338
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) −44.9706 −1.59695
\(794\) 0 0
\(795\) −22.1421 −0.785301
\(796\) 0 0
\(797\) −41.7990 −1.48060 −0.740298 0.672279i \(-0.765316\pi\)
−0.740298 + 0.672279i \(0.765316\pi\)
\(798\) 0 0
\(799\) −13.3726 −0.473088
\(800\) 0 0
\(801\) 110.184 3.89315
\(802\) 0 0
\(803\) −7.79899 −0.275220
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) −63.1127 −2.22167
\(808\) 0 0
\(809\) 3.02944 0.106509 0.0532547 0.998581i \(-0.483040\pi\)
0.0532547 + 0.998581i \(0.483040\pi\)
\(810\) 0 0
\(811\) −32.5858 −1.14424 −0.572121 0.820169i \(-0.693879\pi\)
−0.572121 + 0.820169i \(0.693879\pi\)
\(812\) 0 0
\(813\) 40.9706 1.43690
\(814\) 0 0
\(815\) 20.1421 0.705548
\(816\) 0 0
\(817\) −5.17157 −0.180930
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 17.3137 0.604253 0.302126 0.953268i \(-0.402304\pi\)
0.302126 + 0.953268i \(0.402304\pi\)
\(822\) 0 0
\(823\) −20.2843 −0.707065 −0.353533 0.935422i \(-0.615020\pi\)
−0.353533 + 0.935422i \(0.615020\pi\)
\(824\) 0 0
\(825\) −2.82843 −0.0984732
\(826\) 0 0
\(827\) 5.37258 0.186823 0.0934115 0.995628i \(-0.470223\pi\)
0.0934115 + 0.995628i \(0.470223\pi\)
\(828\) 0 0
\(829\) −5.02944 −0.174680 −0.0873398 0.996179i \(-0.527837\pi\)
−0.0873398 + 0.996179i \(0.527837\pi\)
\(830\) 0 0
\(831\) 27.7990 0.964336
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) −15.7990 −0.546747
\(836\) 0 0
\(837\) −54.6274 −1.88820
\(838\) 0 0
\(839\) 42.1421 1.45491 0.727454 0.686156i \(-0.240703\pi\)
0.727454 + 0.686156i \(0.240703\pi\)
\(840\) 0 0
\(841\) −5.68629 −0.196079
\(842\) 0 0
\(843\) −27.3137 −0.940734
\(844\) 0 0
\(845\) 10.3137 0.354802
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) −7.65685 −0.262783
\(850\) 0 0
\(851\) −8.97056 −0.307507
\(852\) 0 0
\(853\) −43.1716 −1.47817 −0.739083 0.673614i \(-0.764741\pi\)
−0.739083 + 0.673614i \(0.764741\pi\)
\(854\) 0 0
\(855\) −5.07107 −0.173427
\(856\) 0 0
\(857\) −4.92893 −0.168369 −0.0841846 0.996450i \(-0.526829\pi\)
−0.0841846 + 0.996450i \(0.526829\pi\)
\(858\) 0 0
\(859\) 7.21320 0.246111 0.123056 0.992400i \(-0.460731\pi\)
0.123056 + 0.992400i \(0.460731\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 4.97056 0.169200 0.0846000 0.996415i \(-0.473039\pi\)
0.0846000 + 0.996415i \(0.473039\pi\)
\(864\) 0 0
\(865\) −8.82843 −0.300176
\(866\) 0 0
\(867\) 35.2132 1.19590
\(868\) 0 0
\(869\) 5.65685 0.191896
\(870\) 0 0
\(871\) 8.00000 0.271070
\(872\) 0 0
\(873\) −67.1543 −2.27283
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) −30.2843 −1.02263 −0.511314 0.859394i \(-0.670841\pi\)
−0.511314 + 0.859394i \(0.670841\pi\)
\(878\) 0 0
\(879\) 28.4853 0.960785
\(880\) 0 0
\(881\) −2.38478 −0.0803452 −0.0401726 0.999193i \(-0.512791\pi\)
−0.0401726 + 0.999193i \(0.512791\pi\)
\(882\) 0 0
\(883\) 41.6569 1.40186 0.700932 0.713228i \(-0.252767\pi\)
0.700932 + 0.713228i \(0.252767\pi\)
\(884\) 0 0
\(885\) 29.3137 0.985370
\(886\) 0 0
\(887\) 55.1127 1.85050 0.925252 0.379354i \(-0.123854\pi\)
0.925252 + 0.379354i \(0.123854\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) 33.1127 1.10932
\(892\) 0 0
\(893\) 3.02944 0.101376
\(894\) 0 0
\(895\) −4.00000 −0.133705
\(896\) 0 0
\(897\) 19.3137 0.644866
\(898\) 0 0
\(899\) −13.6569 −0.455482
\(900\) 0 0
\(901\) 16.7696 0.558675
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) −2.48528 −0.0826135
\(906\) 0 0
\(907\) −0.284271 −0.00943907 −0.00471954 0.999989i \(-0.501502\pi\)
−0.00471954 + 0.999989i \(0.501502\pi\)
\(908\) 0 0
\(909\) 115.255 3.82276
\(910\) 0 0
\(911\) −36.2843 −1.20215 −0.601076 0.799192i \(-0.705261\pi\)
−0.601076 + 0.799192i \(0.705261\pi\)
\(912\) 0 0
\(913\) 1.85786 0.0614863
\(914\) 0 0
\(915\) −31.7990 −1.05124
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) −15.5147 −0.511783 −0.255892 0.966705i \(-0.582369\pi\)
−0.255892 + 0.966705i \(0.582369\pi\)
\(920\) 0 0
\(921\) 50.9706 1.67954
\(922\) 0 0
\(923\) −21.6569 −0.712844
\(924\) 0 0
\(925\) −7.65685 −0.251756
\(926\) 0 0
\(927\) 128.368 4.21614
\(928\) 0 0
\(929\) −17.2132 −0.564747 −0.282373 0.959305i \(-0.591122\pi\)
−0.282373 + 0.959305i \(0.591122\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 0 0
\(933\) 13.6569 0.447105
\(934\) 0 0
\(935\) 2.14214 0.0700553
\(936\) 0 0
\(937\) 20.2426 0.661298 0.330649 0.943754i \(-0.392732\pi\)
0.330649 + 0.943754i \(0.392732\pi\)
\(938\) 0 0
\(939\) −49.1127 −1.60273
\(940\) 0 0
\(941\) −50.0000 −1.62995 −0.814977 0.579494i \(-0.803250\pi\)
−0.814977 + 0.579494i \(0.803250\pi\)
\(942\) 0 0
\(943\) −3.59798 −0.117166
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 4.82843 0.156903 0.0784514 0.996918i \(-0.475002\pi\)
0.0784514 + 0.996918i \(0.475002\pi\)
\(948\) 0 0
\(949\) 45.4558 1.47556
\(950\) 0 0
\(951\) −35.7990 −1.16086
\(952\) 0 0
\(953\) 0.343146 0.0111156 0.00555779 0.999985i \(-0.498231\pi\)
0.00555779 + 0.999985i \(0.498231\pi\)
\(954\) 0 0
\(955\) −10.1421 −0.328192
\(956\) 0 0
\(957\) 13.6569 0.441463
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) −23.0000 −0.741935
\(962\) 0 0
\(963\) −83.5980 −2.69391
\(964\) 0 0
\(965\) 5.65685 0.182101
\(966\) 0 0
\(967\) 37.4558 1.20450 0.602249 0.798308i \(-0.294271\pi\)
0.602249 + 0.798308i \(0.294271\pi\)
\(968\) 0 0
\(969\) 5.17157 0.166135
\(970\) 0 0
\(971\) 33.3553 1.07042 0.535212 0.844718i \(-0.320232\pi\)
0.535212 + 0.844718i \(0.320232\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) 16.4853 0.527952
\(976\) 0 0
\(977\) −12.6863 −0.405870 −0.202935 0.979192i \(-0.565048\pi\)
−0.202935 + 0.979192i \(0.565048\pi\)
\(978\) 0 0
\(979\) 10.5442 0.336993
\(980\) 0 0
\(981\) 21.5147 0.686912
\(982\) 0 0
\(983\) 12.2010 0.389152 0.194576 0.980887i \(-0.437667\pi\)
0.194576 + 0.980887i \(0.437667\pi\)
\(984\) 0 0
\(985\) −25.7990 −0.822024
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 10.3431 0.328893
\(990\) 0 0
\(991\) −44.7696 −1.42215 −0.711076 0.703115i \(-0.751792\pi\)
−0.711076 + 0.703115i \(0.751792\pi\)
\(992\) 0 0
\(993\) 115.397 3.66201
\(994\) 0 0
\(995\) −16.4853 −0.522619
\(996\) 0 0
\(997\) −18.2843 −0.579069 −0.289534 0.957168i \(-0.593500\pi\)
−0.289534 + 0.957168i \(0.593500\pi\)
\(998\) 0 0
\(999\) 147.882 4.67879
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 3920.2.a.bm.1.1 2
4.3 odd 2 490.2.a.m.1.2 yes 2
7.6 odd 2 3920.2.a.ca.1.2 2
12.11 even 2 4410.2.a.bt.1.2 2
20.3 even 4 2450.2.c.t.99.4 4
20.7 even 4 2450.2.c.t.99.1 4
20.19 odd 2 2450.2.a.bn.1.1 2
28.3 even 6 490.2.e.j.471.2 4
28.11 odd 6 490.2.e.i.471.1 4
28.19 even 6 490.2.e.j.361.2 4
28.23 odd 6 490.2.e.i.361.1 4
28.27 even 2 490.2.a.l.1.1 2
84.83 odd 2 4410.2.a.by.1.2 2
140.27 odd 4 2450.2.c.w.99.2 4
140.83 odd 4 2450.2.c.w.99.3 4
140.139 even 2 2450.2.a.bs.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
490.2.a.l.1.1 2 28.27 even 2
490.2.a.m.1.2 yes 2 4.3 odd 2
490.2.e.i.361.1 4 28.23 odd 6
490.2.e.i.471.1 4 28.11 odd 6
490.2.e.j.361.2 4 28.19 even 6
490.2.e.j.471.2 4 28.3 even 6
2450.2.a.bn.1.1 2 20.19 odd 2
2450.2.a.bs.1.2 2 140.139 even 2
2450.2.c.t.99.1 4 20.7 even 4
2450.2.c.t.99.4 4 20.3 even 4
2450.2.c.w.99.2 4 140.27 odd 4
2450.2.c.w.99.3 4 140.83 odd 4
3920.2.a.bm.1.1 2 1.1 even 1 trivial
3920.2.a.ca.1.2 2 7.6 odd 2
4410.2.a.bt.1.2 2 12.11 even 2
4410.2.a.by.1.2 2 84.83 odd 2