Properties

Label 3640.2.a.s.1.1
Level $3640$
Weight $2$
Character 3640.1
Self dual yes
Analytic conductor $29.066$
Analytic rank $1$
Dimension $4$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 3640 = 2^{3} \cdot 5 \cdot 7 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3640.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(29.0655463357\)
Analytic rank: \(1\)
Dimension: \(4\)
Coefficient field: 4.4.2225.1
Defining polynomial: \( x^{4} - x^{3} - 5x^{2} + 2x + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(2.43828\) of defining polynomial
Character \(\chi\) \(=\) 3640.1

$q$-expansion

\(f(q)\) \(=\) \(q-2.43828 q^{3} +1.00000 q^{5} -1.00000 q^{7} +2.94523 q^{9} +O(q^{10})\) \(q-2.43828 q^{3} +1.00000 q^{5} -1.00000 q^{7} +2.94523 q^{9} +1.23607 q^{11} +1.00000 q^{13} -2.43828 q^{15} +0.931341 q^{17} -7.18129 q^{19} +2.43828 q^{21} +2.65438 q^{23} +1.00000 q^{25} +0.133557 q^{27} +2.66047 q^{29} -7.04774 q^{31} -3.01388 q^{33} -1.00000 q^{35} -0.797785 q^{37} -2.43828 q^{39} -2.52691 q^{41} -3.64050 q^{43} +2.94523 q^{45} +1.05382 q^{47} +1.00000 q^{49} -2.27087 q^{51} +5.89045 q^{53} +1.23607 q^{55} +17.5100 q^{57} +12.3565 q^{59} +1.01388 q^{61} -2.94523 q^{63} +1.00000 q^{65} +0.349656 q^{67} -6.47214 q^{69} +0.249952 q^{71} +8.90433 q^{73} -2.43828 q^{75} -1.23607 q^{77} -6.75409 q^{79} -9.16132 q^{81} -4.43220 q^{83} +0.931341 q^{85} -6.48697 q^{87} -6.64830 q^{89} -1.00000 q^{91} +17.1844 q^{93} -7.18129 q^{95} -4.29488 q^{97} +3.64050 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - q^{3} + 4 q^{5} - 4 q^{7} - q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - q^{3} + 4 q^{5} - 4 q^{7} - q^{9} - 4 q^{11} + 4 q^{13} - q^{15} - q^{17} - 7 q^{19} + q^{21} - 6 q^{23} + 4 q^{25} - 4 q^{27} + q^{29} - 11 q^{31} - 4 q^{33} - 4 q^{35} - 3 q^{37} - q^{39} - 5 q^{41} - 6 q^{43} - q^{45} - 6 q^{47} + 4 q^{49} - 14 q^{51} - 2 q^{53} - 4 q^{55} + 14 q^{57} - q^{59} - 4 q^{61} + q^{63} + 4 q^{65} - 11 q^{67} - 8 q^{69} - 16 q^{71} + 2 q^{73} - q^{75} + 4 q^{77} - 15 q^{79} - 16 q^{81} - 2 q^{83} - q^{85} - 23 q^{87} - 3 q^{89} - 4 q^{91} - 5 q^{93} - 7 q^{95} + 8 q^{97} + 6 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Coefficient data

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

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) −2.43828 −1.40774 −0.703872 0.710327i \(-0.748547\pi\)
−0.703872 + 0.710327i \(0.748547\pi\)
\(4\) 0 0
\(5\) 1.00000 0.447214
\(6\) 0 0
\(7\) −1.00000 −0.377964
\(8\) 0 0
\(9\) 2.94523 0.981742
\(10\) 0 0
\(11\) 1.23607 0.372689 0.186344 0.982485i \(-0.440336\pi\)
0.186344 + 0.982485i \(0.440336\pi\)
\(12\) 0 0
\(13\) 1.00000 0.277350
\(14\) 0 0
\(15\) −2.43828 −0.629562
\(16\) 0 0
\(17\) 0.931341 0.225883 0.112942 0.993602i \(-0.463973\pi\)
0.112942 + 0.993602i \(0.463973\pi\)
\(18\) 0 0
\(19\) −7.18129 −1.64750 −0.823751 0.566952i \(-0.808122\pi\)
−0.823751 + 0.566952i \(0.808122\pi\)
\(20\) 0 0
\(21\) 2.43828 0.532077
\(22\) 0 0
\(23\) 2.65438 0.553477 0.276738 0.960945i \(-0.410746\pi\)
0.276738 + 0.960945i \(0.410746\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) 0.133557 0.0257030
\(28\) 0 0
\(29\) 2.66047 0.494036 0.247018 0.969011i \(-0.420549\pi\)
0.247018 + 0.969011i \(0.420549\pi\)
\(30\) 0 0
\(31\) −7.04774 −1.26581 −0.632905 0.774229i \(-0.718138\pi\)
−0.632905 + 0.774229i \(0.718138\pi\)
\(32\) 0 0
\(33\) −3.01388 −0.524650
\(34\) 0 0
\(35\) −1.00000 −0.169031
\(36\) 0 0
\(37\) −0.797785 −0.131155 −0.0655775 0.997847i \(-0.520889\pi\)
−0.0655775 + 0.997847i \(0.520889\pi\)
\(38\) 0 0
\(39\) −2.43828 −0.390438
\(40\) 0 0
\(41\) −2.52691 −0.394637 −0.197319 0.980339i \(-0.563223\pi\)
−0.197319 + 0.980339i \(0.563223\pi\)
\(42\) 0 0
\(43\) −3.64050 −0.555171 −0.277585 0.960701i \(-0.589534\pi\)
−0.277585 + 0.960701i \(0.589534\pi\)
\(44\) 0 0
\(45\) 2.94523 0.439048
\(46\) 0 0
\(47\) 1.05382 0.153716 0.0768578 0.997042i \(-0.475511\pi\)
0.0768578 + 0.997042i \(0.475511\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) −2.27087 −0.317986
\(52\) 0 0
\(53\) 5.89045 0.809116 0.404558 0.914512i \(-0.367425\pi\)
0.404558 + 0.914512i \(0.367425\pi\)
\(54\) 0 0
\(55\) 1.23607 0.166671
\(56\) 0 0
\(57\) 17.5100 2.31926
\(58\) 0 0
\(59\) 12.3565 1.60868 0.804340 0.594170i \(-0.202519\pi\)
0.804340 + 0.594170i \(0.202519\pi\)
\(60\) 0 0
\(61\) 1.01388 0.129815 0.0649073 0.997891i \(-0.479325\pi\)
0.0649073 + 0.997891i \(0.479325\pi\)
\(62\) 0 0
\(63\) −2.94523 −0.371063
\(64\) 0 0
\(65\) 1.00000 0.124035
\(66\) 0 0
\(67\) 0.349656 0.0427172 0.0213586 0.999772i \(-0.493201\pi\)
0.0213586 + 0.999772i \(0.493201\pi\)
\(68\) 0 0
\(69\) −6.47214 −0.779154
\(70\) 0 0
\(71\) 0.249952 0.0296638 0.0148319 0.999890i \(-0.495279\pi\)
0.0148319 + 0.999890i \(0.495279\pi\)
\(72\) 0 0
\(73\) 8.90433 1.04217 0.521087 0.853504i \(-0.325527\pi\)
0.521087 + 0.853504i \(0.325527\pi\)
\(74\) 0 0
\(75\) −2.43828 −0.281549
\(76\) 0 0
\(77\) −1.23607 −0.140863
\(78\) 0 0
\(79\) −6.75409 −0.759894 −0.379947 0.925008i \(-0.624058\pi\)
−0.379947 + 0.925008i \(0.624058\pi\)
\(80\) 0 0
\(81\) −9.16132 −1.01792
\(82\) 0 0
\(83\) −4.43220 −0.486497 −0.243248 0.969964i \(-0.578213\pi\)
−0.243248 + 0.969964i \(0.578213\pi\)
\(84\) 0 0
\(85\) 0.931341 0.101018
\(86\) 0 0
\(87\) −6.48697 −0.695477
\(88\) 0 0
\(89\) −6.64830 −0.704718 −0.352359 0.935865i \(-0.614620\pi\)
−0.352359 + 0.935865i \(0.614620\pi\)
\(90\) 0 0
\(91\) −1.00000 −0.104828
\(92\) 0 0
\(93\) 17.1844 1.78194
\(94\) 0 0
\(95\) −7.18129 −0.736785
\(96\) 0 0
\(97\) −4.29488 −0.436079 −0.218040 0.975940i \(-0.569966\pi\)
−0.218040 + 0.975940i \(0.569966\pi\)
\(98\) 0 0
\(99\) 3.64050 0.365884
\(100\) 0 0
\(101\) −5.19114 −0.516538 −0.258269 0.966073i \(-0.583152\pi\)
−0.258269 + 0.966073i \(0.583152\pi\)
\(102\) 0 0
\(103\) −19.1394 −1.88587 −0.942933 0.332983i \(-0.891945\pi\)
−0.942933 + 0.332983i \(0.891945\pi\)
\(104\) 0 0
\(105\) 2.43828 0.237952
\(106\) 0 0
\(107\) 13.8935 1.34314 0.671569 0.740942i \(-0.265621\pi\)
0.671569 + 0.740942i \(0.265621\pi\)
\(108\) 0 0
\(109\) −1.45825 −0.139675 −0.0698376 0.997558i \(-0.522248\pi\)
−0.0698376 + 0.997558i \(0.522248\pi\)
\(110\) 0 0
\(111\) 1.94523 0.184633
\(112\) 0 0
\(113\) −3.86268 −0.363371 −0.181685 0.983357i \(-0.558155\pi\)
−0.181685 + 0.983357i \(0.558155\pi\)
\(114\) 0 0
\(115\) 2.65438 0.247522
\(116\) 0 0
\(117\) 2.94523 0.272286
\(118\) 0 0
\(119\) −0.931341 −0.0853759
\(120\) 0 0
\(121\) −9.47214 −0.861103
\(122\) 0 0
\(123\) 6.16132 0.555548
\(124\) 0 0
\(125\) 1.00000 0.0894427
\(126\) 0 0
\(127\) 1.23607 0.109683 0.0548416 0.998495i \(-0.482535\pi\)
0.0548416 + 0.998495i \(0.482535\pi\)
\(128\) 0 0
\(129\) 8.87657 0.781538
\(130\) 0 0
\(131\) −10.9165 −0.953779 −0.476890 0.878963i \(-0.658236\pi\)
−0.476890 + 0.878963i \(0.658236\pi\)
\(132\) 0 0
\(133\) 7.18129 0.622697
\(134\) 0 0
\(135\) 0.133557 0.0114947
\(136\) 0 0
\(137\) 19.8597 1.69673 0.848364 0.529414i \(-0.177588\pi\)
0.848364 + 0.529414i \(0.177588\pi\)
\(138\) 0 0
\(139\) −15.9582 −1.35355 −0.676777 0.736188i \(-0.736624\pi\)
−0.676777 + 0.736188i \(0.736624\pi\)
\(140\) 0 0
\(141\) −2.56952 −0.216392
\(142\) 0 0
\(143\) 1.23607 0.103365
\(144\) 0 0
\(145\) 2.66047 0.220940
\(146\) 0 0
\(147\) −2.43828 −0.201106
\(148\) 0 0
\(149\) −9.10374 −0.745808 −0.372904 0.927870i \(-0.621638\pi\)
−0.372904 + 0.927870i \(0.621638\pi\)
\(150\) 0 0
\(151\) 17.4492 1.41999 0.709997 0.704205i \(-0.248696\pi\)
0.709997 + 0.704205i \(0.248696\pi\)
\(152\) 0 0
\(153\) 2.74301 0.221759
\(154\) 0 0
\(155\) −7.04774 −0.566088
\(156\) 0 0
\(157\) 12.5909 1.00486 0.502430 0.864618i \(-0.332440\pi\)
0.502430 + 0.864618i \(0.332440\pi\)
\(158\) 0 0
\(159\) −14.3626 −1.13903
\(160\) 0 0
\(161\) −2.65438 −0.209195
\(162\) 0 0
\(163\) −5.74396 −0.449902 −0.224951 0.974370i \(-0.572222\pi\)
−0.224951 + 0.974370i \(0.572222\pi\)
\(164\) 0 0
\(165\) −3.01388 −0.234631
\(166\) 0 0
\(167\) 2.56952 0.198835 0.0994175 0.995046i \(-0.468302\pi\)
0.0994175 + 0.995046i \(0.468302\pi\)
\(168\) 0 0
\(169\) 1.00000 0.0769231
\(170\) 0 0
\(171\) −21.1505 −1.61742
\(172\) 0 0
\(173\) −6.48321 −0.492910 −0.246455 0.969154i \(-0.579266\pi\)
−0.246455 + 0.969154i \(0.579266\pi\)
\(174\) 0 0
\(175\) −1.00000 −0.0755929
\(176\) 0 0
\(177\) −30.1286 −2.26461
\(178\) 0 0
\(179\) −15.1444 −1.13195 −0.565974 0.824423i \(-0.691500\pi\)
−0.565974 + 0.824423i \(0.691500\pi\)
\(180\) 0 0
\(181\) −3.41079 −0.253522 −0.126761 0.991933i \(-0.540458\pi\)
−0.126761 + 0.991933i \(0.540458\pi\)
\(182\) 0 0
\(183\) −2.47214 −0.182746
\(184\) 0 0
\(185\) −0.797785 −0.0586543
\(186\) 0 0
\(187\) 1.15120 0.0841842
\(188\) 0 0
\(189\) −0.133557 −0.00971481
\(190\) 0 0
\(191\) −8.32565 −0.602423 −0.301211 0.953557i \(-0.597391\pi\)
−0.301211 + 0.953557i \(0.597391\pi\)
\(192\) 0 0
\(193\) −18.8199 −1.35468 −0.677342 0.735668i \(-0.736868\pi\)
−0.677342 + 0.735668i \(0.736868\pi\)
\(194\) 0 0
\(195\) −2.43828 −0.174609
\(196\) 0 0
\(197\) 4.88764 0.348230 0.174115 0.984725i \(-0.444294\pi\)
0.174115 + 0.984725i \(0.444294\pi\)
\(198\) 0 0
\(199\) −24.3151 −1.72365 −0.861827 0.507203i \(-0.830680\pi\)
−0.861827 + 0.507203i \(0.830680\pi\)
\(200\) 0 0
\(201\) −0.852560 −0.0601349
\(202\) 0 0
\(203\) −2.66047 −0.186728
\(204\) 0 0
\(205\) −2.52691 −0.176487
\(206\) 0 0
\(207\) 7.81775 0.543371
\(208\) 0 0
\(209\) −8.87657 −0.614005
\(210\) 0 0
\(211\) 3.08890 0.212649 0.106324 0.994331i \(-0.466092\pi\)
0.106324 + 0.994331i \(0.466092\pi\)
\(212\) 0 0
\(213\) −0.609453 −0.0417591
\(214\) 0 0
\(215\) −3.64050 −0.248280
\(216\) 0 0
\(217\) 7.04774 0.478432
\(218\) 0 0
\(219\) −21.7113 −1.46711
\(220\) 0 0
\(221\) 0.931341 0.0626488
\(222\) 0 0
\(223\) −17.8885 −1.19791 −0.598953 0.800784i \(-0.704416\pi\)
−0.598953 + 0.800784i \(0.704416\pi\)
\(224\) 0 0
\(225\) 2.94523 0.196348
\(226\) 0 0
\(227\) −2.31457 −0.153624 −0.0768118 0.997046i \(-0.524474\pi\)
−0.0768118 + 0.997046i \(0.524474\pi\)
\(228\) 0 0
\(229\) 3.50086 0.231343 0.115672 0.993288i \(-0.463098\pi\)
0.115672 + 0.993288i \(0.463098\pi\)
\(230\) 0 0
\(231\) 3.01388 0.198299
\(232\) 0 0
\(233\) −23.9662 −1.57008 −0.785040 0.619445i \(-0.787358\pi\)
−0.785040 + 0.619445i \(0.787358\pi\)
\(234\) 0 0
\(235\) 1.05382 0.0687437
\(236\) 0 0
\(237\) 16.4684 1.06974
\(238\) 0 0
\(239\) −17.0847 −1.10512 −0.552558 0.833475i \(-0.686348\pi\)
−0.552558 + 0.833475i \(0.686348\pi\)
\(240\) 0 0
\(241\) −4.30972 −0.277613 −0.138807 0.990319i \(-0.544327\pi\)
−0.138807 + 0.990319i \(0.544327\pi\)
\(242\) 0 0
\(243\) 21.9372 1.40727
\(244\) 0 0
\(245\) 1.00000 0.0638877
\(246\) 0 0
\(247\) −7.18129 −0.456935
\(248\) 0 0
\(249\) 10.8070 0.684863
\(250\) 0 0
\(251\) −17.8430 −1.12624 −0.563120 0.826375i \(-0.690399\pi\)
−0.563120 + 0.826375i \(0.690399\pi\)
\(252\) 0 0
\(253\) 3.28100 0.206275
\(254\) 0 0
\(255\) −2.27087 −0.142208
\(256\) 0 0
\(257\) 8.83663 0.551214 0.275607 0.961270i \(-0.411121\pi\)
0.275607 + 0.961270i \(0.411121\pi\)
\(258\) 0 0
\(259\) 0.797785 0.0495719
\(260\) 0 0
\(261\) 7.83568 0.485016
\(262\) 0 0
\(263\) −5.49101 −0.338590 −0.169295 0.985565i \(-0.554149\pi\)
−0.169295 + 0.985565i \(0.554149\pi\)
\(264\) 0 0
\(265\) 5.89045 0.361847
\(266\) 0 0
\(267\) 16.2104 0.992062
\(268\) 0 0
\(269\) −31.5119 −1.92131 −0.960657 0.277739i \(-0.910415\pi\)
−0.960657 + 0.277739i \(0.910415\pi\)
\(270\) 0 0
\(271\) −21.2164 −1.28880 −0.644402 0.764687i \(-0.722893\pi\)
−0.644402 + 0.764687i \(0.722893\pi\)
\(272\) 0 0
\(273\) 2.43828 0.147572
\(274\) 0 0
\(275\) 1.23607 0.0745377
\(276\) 0 0
\(277\) −5.51379 −0.331291 −0.165646 0.986185i \(-0.552971\pi\)
−0.165646 + 0.986185i \(0.552971\pi\)
\(278\) 0 0
\(279\) −20.7572 −1.24270
\(280\) 0 0
\(281\) −9.10374 −0.543084 −0.271542 0.962427i \(-0.587534\pi\)
−0.271542 + 0.962427i \(0.587534\pi\)
\(282\) 0 0
\(283\) 25.3370 1.50613 0.753063 0.657949i \(-0.228576\pi\)
0.753063 + 0.657949i \(0.228576\pi\)
\(284\) 0 0
\(285\) 17.5100 1.03720
\(286\) 0 0
\(287\) 2.52691 0.149159
\(288\) 0 0
\(289\) −16.1326 −0.948977
\(290\) 0 0
\(291\) 10.4721 0.613887
\(292\) 0 0
\(293\) −8.73925 −0.510552 −0.255276 0.966868i \(-0.582166\pi\)
−0.255276 + 0.966868i \(0.582166\pi\)
\(294\) 0 0
\(295\) 12.3565 0.719423
\(296\) 0 0
\(297\) 0.165085 0.00957920
\(298\) 0 0
\(299\) 2.65438 0.153507
\(300\) 0 0
\(301\) 3.64050 0.209835
\(302\) 0 0
\(303\) 12.6575 0.727152
\(304\) 0 0
\(305\) 1.01388 0.0580548
\(306\) 0 0
\(307\) −3.08159 −0.175876 −0.0879378 0.996126i \(-0.528028\pi\)
−0.0879378 + 0.996126i \(0.528028\pi\)
\(308\) 0 0
\(309\) 46.6674 2.65482
\(310\) 0 0
\(311\) 1.19921 0.0680012 0.0340006 0.999422i \(-0.489175\pi\)
0.0340006 + 0.999422i \(0.489175\pi\)
\(312\) 0 0
\(313\) −25.8875 −1.46325 −0.731623 0.681710i \(-0.761237\pi\)
−0.731623 + 0.681710i \(0.761237\pi\)
\(314\) 0 0
\(315\) −2.94523 −0.165945
\(316\) 0 0
\(317\) −8.78726 −0.493542 −0.246771 0.969074i \(-0.579370\pi\)
−0.246771 + 0.969074i \(0.579370\pi\)
\(318\) 0 0
\(319\) 3.28852 0.184122
\(320\) 0 0
\(321\) −33.8764 −1.89079
\(322\) 0 0
\(323\) −6.68824 −0.372143
\(324\) 0 0
\(325\) 1.00000 0.0554700
\(326\) 0 0
\(327\) 3.55563 0.196627
\(328\) 0 0
\(329\) −1.05382 −0.0580991
\(330\) 0 0
\(331\) −24.0505 −1.32194 −0.660969 0.750413i \(-0.729854\pi\)
−0.660969 + 0.750413i \(0.729854\pi\)
\(332\) 0 0
\(333\) −2.34966 −0.128760
\(334\) 0 0
\(335\) 0.349656 0.0191037
\(336\) 0 0
\(337\) 2.21910 0.120882 0.0604410 0.998172i \(-0.480749\pi\)
0.0604410 + 0.998172i \(0.480749\pi\)
\(338\) 0 0
\(339\) 9.41831 0.511533
\(340\) 0 0
\(341\) −8.71148 −0.471753
\(342\) 0 0
\(343\) −1.00000 −0.0539949
\(344\) 0 0
\(345\) −6.47214 −0.348448
\(346\) 0 0
\(347\) 5.32593 0.285911 0.142955 0.989729i \(-0.454339\pi\)
0.142955 + 0.989729i \(0.454339\pi\)
\(348\) 0 0
\(349\) 19.2159 1.02860 0.514302 0.857609i \(-0.328051\pi\)
0.514302 + 0.857609i \(0.328051\pi\)
\(350\) 0 0
\(351\) 0.133557 0.00712872
\(352\) 0 0
\(353\) 8.60755 0.458133 0.229067 0.973411i \(-0.426433\pi\)
0.229067 + 0.973411i \(0.426433\pi\)
\(354\) 0 0
\(355\) 0.249952 0.0132661
\(356\) 0 0
\(357\) 2.27087 0.120187
\(358\) 0 0
\(359\) 12.3055 0.649459 0.324729 0.945807i \(-0.394727\pi\)
0.324729 + 0.945807i \(0.394727\pi\)
\(360\) 0 0
\(361\) 32.5710 1.71426
\(362\) 0 0
\(363\) 23.0958 1.21221
\(364\) 0 0
\(365\) 8.90433 0.466074
\(366\) 0 0
\(367\) 16.1884 0.845028 0.422514 0.906356i \(-0.361148\pi\)
0.422514 + 0.906356i \(0.361148\pi\)
\(368\) 0 0
\(369\) −7.44232 −0.387432
\(370\) 0 0
\(371\) −5.89045 −0.305817
\(372\) 0 0
\(373\) −32.0683 −1.66043 −0.830216 0.557442i \(-0.811783\pi\)
−0.830216 + 0.557442i \(0.811783\pi\)
\(374\) 0 0
\(375\) −2.43828 −0.125912
\(376\) 0 0
\(377\) 2.66047 0.137021
\(378\) 0 0
\(379\) 20.3101 1.04326 0.521631 0.853171i \(-0.325324\pi\)
0.521631 + 0.853171i \(0.325324\pi\)
\(380\) 0 0
\(381\) −3.01388 −0.154406
\(382\) 0 0
\(383\) −31.3967 −1.60430 −0.802149 0.597124i \(-0.796310\pi\)
−0.802149 + 0.597124i \(0.796310\pi\)
\(384\) 0 0
\(385\) −1.23607 −0.0629959
\(386\) 0 0
\(387\) −10.7221 −0.545034
\(388\) 0 0
\(389\) 1.58264 0.0802430 0.0401215 0.999195i \(-0.487226\pi\)
0.0401215 + 0.999195i \(0.487226\pi\)
\(390\) 0 0
\(391\) 2.47214 0.125021
\(392\) 0 0
\(393\) 26.6175 1.34268
\(394\) 0 0
\(395\) −6.75409 −0.339835
\(396\) 0 0
\(397\) 10.8210 0.543092 0.271546 0.962425i \(-0.412465\pi\)
0.271546 + 0.962425i \(0.412465\pi\)
\(398\) 0 0
\(399\) −17.5100 −0.876598
\(400\) 0 0
\(401\) −10.3470 −0.516704 −0.258352 0.966051i \(-0.583179\pi\)
−0.258352 + 0.966051i \(0.583179\pi\)
\(402\) 0 0
\(403\) −7.04774 −0.351073
\(404\) 0 0
\(405\) −9.16132 −0.455230
\(406\) 0 0
\(407\) −0.986116 −0.0488800
\(408\) 0 0
\(409\) −3.43801 −0.169998 −0.0849992 0.996381i \(-0.527089\pi\)
−0.0849992 + 0.996381i \(0.527089\pi\)
\(410\) 0 0
\(411\) −48.4235 −2.38856
\(412\) 0 0
\(413\) −12.3565 −0.608024
\(414\) 0 0
\(415\) −4.43220 −0.217568
\(416\) 0 0
\(417\) 38.9105 1.90546
\(418\) 0 0
\(419\) 24.0062 1.17278 0.586389 0.810030i \(-0.300549\pi\)
0.586389 + 0.810030i \(0.300549\pi\)
\(420\) 0 0
\(421\) 18.0000 0.877266 0.438633 0.898666i \(-0.355463\pi\)
0.438633 + 0.898666i \(0.355463\pi\)
\(422\) 0 0
\(423\) 3.10374 0.150909
\(424\) 0 0
\(425\) 0.931341 0.0451767
\(426\) 0 0
\(427\) −1.01388 −0.0490653
\(428\) 0 0
\(429\) −3.01388 −0.145512
\(430\) 0 0
\(431\) 38.5632 1.85752 0.928761 0.370678i \(-0.120875\pi\)
0.928761 + 0.370678i \(0.120875\pi\)
\(432\) 0 0
\(433\) −24.2725 −1.16646 −0.583231 0.812306i \(-0.698212\pi\)
−0.583231 + 0.812306i \(0.698212\pi\)
\(434\) 0 0
\(435\) −6.48697 −0.311027
\(436\) 0 0
\(437\) −19.0619 −0.911854
\(438\) 0 0
\(439\) −4.67716 −0.223229 −0.111614 0.993752i \(-0.535602\pi\)
−0.111614 + 0.993752i \(0.535602\pi\)
\(440\) 0 0
\(441\) 2.94523 0.140249
\(442\) 0 0
\(443\) 6.09294 0.289484 0.144742 0.989469i \(-0.453765\pi\)
0.144742 + 0.989469i \(0.453765\pi\)
\(444\) 0 0
\(445\) −6.64830 −0.315160
\(446\) 0 0
\(447\) 22.1975 1.04991
\(448\) 0 0
\(449\) 6.10955 0.288327 0.144164 0.989554i \(-0.453951\pi\)
0.144164 + 0.989554i \(0.453951\pi\)
\(450\) 0 0
\(451\) −3.12343 −0.147077
\(452\) 0 0
\(453\) −42.5460 −1.99899
\(454\) 0 0
\(455\) −1.00000 −0.0468807
\(456\) 0 0
\(457\) 12.0788 0.565022 0.282511 0.959264i \(-0.408833\pi\)
0.282511 + 0.959264i \(0.408833\pi\)
\(458\) 0 0
\(459\) 0.124387 0.00580588
\(460\) 0 0
\(461\) −32.1046 −1.49526 −0.747631 0.664114i \(-0.768809\pi\)
−0.747631 + 0.664114i \(0.768809\pi\)
\(462\) 0 0
\(463\) 3.48526 0.161974 0.0809869 0.996715i \(-0.474193\pi\)
0.0809869 + 0.996715i \(0.474193\pi\)
\(464\) 0 0
\(465\) 17.1844 0.796906
\(466\) 0 0
\(467\) 14.0004 0.647862 0.323931 0.946081i \(-0.394995\pi\)
0.323931 + 0.946081i \(0.394995\pi\)
\(468\) 0 0
\(469\) −0.349656 −0.0161456
\(470\) 0 0
\(471\) −30.7001 −1.41458
\(472\) 0 0
\(473\) −4.49990 −0.206906
\(474\) 0 0
\(475\) −7.18129 −0.329500
\(476\) 0 0
\(477\) 17.3487 0.794343
\(478\) 0 0
\(479\) −17.3351 −0.792061 −0.396030 0.918237i \(-0.629613\pi\)
−0.396030 + 0.918237i \(0.629613\pi\)
\(480\) 0 0
\(481\) −0.797785 −0.0363759
\(482\) 0 0
\(483\) 6.47214 0.294492
\(484\) 0 0
\(485\) −4.29488 −0.195020
\(486\) 0 0
\(487\) −13.2502 −0.600425 −0.300213 0.953872i \(-0.597058\pi\)
−0.300213 + 0.953872i \(0.597058\pi\)
\(488\) 0 0
\(489\) 14.0054 0.633346
\(490\) 0 0
\(491\) 32.3213 1.45864 0.729320 0.684173i \(-0.239837\pi\)
0.729320 + 0.684173i \(0.239837\pi\)
\(492\) 0 0
\(493\) 2.47780 0.111595
\(494\) 0 0
\(495\) 3.64050 0.163628
\(496\) 0 0
\(497\) −0.249952 −0.0112119
\(498\) 0 0
\(499\) 14.0783 0.630232 0.315116 0.949053i \(-0.397957\pi\)
0.315116 + 0.949053i \(0.397957\pi\)
\(500\) 0 0
\(501\) −6.26521 −0.279909
\(502\) 0 0
\(503\) −2.57451 −0.114792 −0.0573958 0.998352i \(-0.518280\pi\)
−0.0573958 + 0.998352i \(0.518280\pi\)
\(504\) 0 0
\(505\) −5.19114 −0.231003
\(506\) 0 0
\(507\) −2.43828 −0.108288
\(508\) 0 0
\(509\) 32.7603 1.45207 0.726036 0.687656i \(-0.241360\pi\)
0.726036 + 0.687656i \(0.241360\pi\)
\(510\) 0 0
\(511\) −8.90433 −0.393905
\(512\) 0 0
\(513\) −0.959109 −0.0423457
\(514\) 0 0
\(515\) −19.1394 −0.843385
\(516\) 0 0
\(517\) 1.30260 0.0572881
\(518\) 0 0
\(519\) 15.8079 0.693890
\(520\) 0 0
\(521\) −21.9878 −0.963304 −0.481652 0.876362i \(-0.659963\pi\)
−0.481652 + 0.876362i \(0.659963\pi\)
\(522\) 0 0
\(523\) 16.0187 0.700448 0.350224 0.936666i \(-0.386105\pi\)
0.350224 + 0.936666i \(0.386105\pi\)
\(524\) 0 0
\(525\) 2.43828 0.106415
\(526\) 0 0
\(527\) −6.56385 −0.285926
\(528\) 0 0
\(529\) −15.9543 −0.693663
\(530\) 0 0
\(531\) 36.3927 1.57931
\(532\) 0 0
\(533\) −2.52691 −0.109453
\(534\) 0 0
\(535\) 13.8935 0.600670
\(536\) 0 0
\(537\) 36.9264 1.59349
\(538\) 0 0
\(539\) 1.23607 0.0532412
\(540\) 0 0
\(541\) −10.8569 −0.466774 −0.233387 0.972384i \(-0.574981\pi\)
−0.233387 + 0.972384i \(0.574981\pi\)
\(542\) 0 0
\(543\) 8.31648 0.356894
\(544\) 0 0
\(545\) −1.45825 −0.0624647
\(546\) 0 0
\(547\) 11.4557 0.489811 0.244906 0.969547i \(-0.421243\pi\)
0.244906 + 0.969547i \(0.421243\pi\)
\(548\) 0 0
\(549\) 2.98612 0.127444
\(550\) 0 0
\(551\) −19.1056 −0.813926
\(552\) 0 0
\(553\) 6.75409 0.287213
\(554\) 0 0
\(555\) 1.94523 0.0825702
\(556\) 0 0
\(557\) 42.6760 1.80824 0.904120 0.427278i \(-0.140527\pi\)
0.904120 + 0.427278i \(0.140527\pi\)
\(558\) 0 0
\(559\) −3.64050 −0.153977
\(560\) 0 0
\(561\) −2.80695 −0.118510
\(562\) 0 0
\(563\) −29.2471 −1.23262 −0.616309 0.787504i \(-0.711373\pi\)
−0.616309 + 0.787504i \(0.711373\pi\)
\(564\) 0 0
\(565\) −3.86268 −0.162504
\(566\) 0 0
\(567\) 9.16132 0.384739
\(568\) 0 0
\(569\) 8.84580 0.370835 0.185418 0.982660i \(-0.440636\pi\)
0.185418 + 0.982660i \(0.440636\pi\)
\(570\) 0 0
\(571\) 30.3596 1.27051 0.635255 0.772303i \(-0.280895\pi\)
0.635255 + 0.772303i \(0.280895\pi\)
\(572\) 0 0
\(573\) 20.3003 0.848057
\(574\) 0 0
\(575\) 2.65438 0.110695
\(576\) 0 0
\(577\) −4.17535 −0.173822 −0.0869110 0.996216i \(-0.527700\pi\)
−0.0869110 + 0.996216i \(0.527700\pi\)
\(578\) 0 0
\(579\) 45.8882 1.90705
\(580\) 0 0
\(581\) 4.43220 0.183879
\(582\) 0 0
\(583\) 7.28100 0.301548
\(584\) 0 0
\(585\) 2.94523 0.121770
\(586\) 0 0
\(587\) −12.3828 −0.511094 −0.255547 0.966797i \(-0.582256\pi\)
−0.255547 + 0.966797i \(0.582256\pi\)
\(588\) 0 0
\(589\) 50.6119 2.08543
\(590\) 0 0
\(591\) −11.9175 −0.490219
\(592\) 0 0
\(593\) 4.46749 0.183458 0.0917289 0.995784i \(-0.470761\pi\)
0.0917289 + 0.995784i \(0.470761\pi\)
\(594\) 0 0
\(595\) −0.931341 −0.0381813
\(596\) 0 0
\(597\) 59.2872 2.42646
\(598\) 0 0
\(599\) −7.19731 −0.294074 −0.147037 0.989131i \(-0.546974\pi\)
−0.147037 + 0.989131i \(0.546974\pi\)
\(600\) 0 0
\(601\) −13.9126 −0.567507 −0.283753 0.958897i \(-0.591580\pi\)
−0.283753 + 0.958897i \(0.591580\pi\)
\(602\) 0 0
\(603\) 1.02981 0.0419373
\(604\) 0 0
\(605\) −9.47214 −0.385097
\(606\) 0 0
\(607\) 1.11735 0.0453518 0.0226759 0.999743i \(-0.492781\pi\)
0.0226759 + 0.999743i \(0.492781\pi\)
\(608\) 0 0
\(609\) 6.48697 0.262865
\(610\) 0 0
\(611\) 1.05382 0.0426331
\(612\) 0 0
\(613\) 13.4507 0.543270 0.271635 0.962400i \(-0.412436\pi\)
0.271635 + 0.962400i \(0.412436\pi\)
\(614\) 0 0
\(615\) 6.16132 0.248449
\(616\) 0 0
\(617\) −0.741916 −0.0298684 −0.0149342 0.999888i \(-0.504754\pi\)
−0.0149342 + 0.999888i \(0.504754\pi\)
\(618\) 0 0
\(619\) 0.655476 0.0263458 0.0131729 0.999913i \(-0.495807\pi\)
0.0131729 + 0.999913i \(0.495807\pi\)
\(620\) 0 0
\(621\) 0.354510 0.0142260
\(622\) 0 0
\(623\) 6.64830 0.266358
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) 21.6436 0.864361
\(628\) 0 0
\(629\) −0.743010 −0.0296257
\(630\) 0 0
\(631\) 18.7221 0.745315 0.372657 0.927969i \(-0.378447\pi\)
0.372657 + 0.927969i \(0.378447\pi\)
\(632\) 0 0
\(633\) −7.53162 −0.299355
\(634\) 0 0
\(635\) 1.23607 0.0490519
\(636\) 0 0
\(637\) 1.00000 0.0396214
\(638\) 0 0
\(639\) 0.736164 0.0291222
\(640\) 0 0
\(641\) −36.1046 −1.42605 −0.713024 0.701140i \(-0.752675\pi\)
−0.713024 + 0.701140i \(0.752675\pi\)
\(642\) 0 0
\(643\) 40.2390 1.58687 0.793435 0.608655i \(-0.208291\pi\)
0.793435 + 0.608655i \(0.208291\pi\)
\(644\) 0 0
\(645\) 8.87657 0.349514
\(646\) 0 0
\(647\) −46.2747 −1.81925 −0.909623 0.415434i \(-0.863630\pi\)
−0.909623 + 0.415434i \(0.863630\pi\)
\(648\) 0 0
\(649\) 15.2735 0.599536
\(650\) 0 0
\(651\) −17.1844 −0.673509
\(652\) 0 0
\(653\) −17.0201 −0.666046 −0.333023 0.942919i \(-0.608069\pi\)
−0.333023 + 0.942919i \(0.608069\pi\)
\(654\) 0 0
\(655\) −10.9165 −0.426543
\(656\) 0 0
\(657\) 26.2253 1.02315
\(658\) 0 0
\(659\) 42.5053 1.65577 0.827886 0.560896i \(-0.189543\pi\)
0.827886 + 0.560896i \(0.189543\pi\)
\(660\) 0 0
\(661\) −8.36525 −0.325371 −0.162685 0.986678i \(-0.552016\pi\)
−0.162685 + 0.986678i \(0.552016\pi\)
\(662\) 0 0
\(663\) −2.27087 −0.0881934
\(664\) 0 0
\(665\) 7.18129 0.278479
\(666\) 0 0
\(667\) 7.06190 0.273438
\(668\) 0 0
\(669\) 43.6173 1.68634
\(670\) 0 0
\(671\) 1.25323 0.0483804
\(672\) 0 0
\(673\) −17.9385 −0.691477 −0.345738 0.938331i \(-0.612372\pi\)
−0.345738 + 0.938331i \(0.612372\pi\)
\(674\) 0 0
\(675\) 0.133557 0.00514060
\(676\) 0 0
\(677\) −30.5061 −1.17244 −0.586222 0.810151i \(-0.699385\pi\)
−0.586222 + 0.810151i \(0.699385\pi\)
\(678\) 0 0
\(679\) 4.29488 0.164822
\(680\) 0 0
\(681\) 5.64358 0.216263
\(682\) 0 0
\(683\) 9.23839 0.353497 0.176749 0.984256i \(-0.443442\pi\)
0.176749 + 0.984256i \(0.443442\pi\)
\(684\) 0 0
\(685\) 19.8597 0.758799
\(686\) 0 0
\(687\) −8.53608 −0.325672
\(688\) 0 0
\(689\) 5.89045 0.224408
\(690\) 0 0
\(691\) −20.8564 −0.793415 −0.396708 0.917945i \(-0.629847\pi\)
−0.396708 + 0.917945i \(0.629847\pi\)
\(692\) 0 0
\(693\) −3.64050 −0.138291
\(694\) 0 0
\(695\) −15.9582 −0.605327
\(696\) 0 0
\(697\) −2.35342 −0.0891420
\(698\) 0 0
\(699\) 58.4365 2.21027
\(700\) 0 0
\(701\) 24.3097 0.918165 0.459083 0.888394i \(-0.348178\pi\)
0.459083 + 0.888394i \(0.348178\pi\)
\(702\) 0 0
\(703\) 5.72913 0.216078
\(704\) 0 0
\(705\) −2.56952 −0.0967736
\(706\) 0 0
\(707\) 5.19114 0.195233
\(708\) 0 0
\(709\) −31.0478 −1.16603 −0.583013 0.812463i \(-0.698126\pi\)
−0.583013 + 0.812463i \(0.698126\pi\)
\(710\) 0 0
\(711\) −19.8923 −0.746020
\(712\) 0 0
\(713\) −18.7074 −0.700597
\(714\) 0 0
\(715\) 1.23607 0.0462263
\(716\) 0 0
\(717\) 41.6573 1.55572
\(718\) 0 0
\(719\) −16.8644 −0.628936 −0.314468 0.949268i \(-0.601826\pi\)
−0.314468 + 0.949268i \(0.601826\pi\)
\(720\) 0 0
\(721\) 19.1394 0.712790
\(722\) 0 0
\(723\) 10.5083 0.390808
\(724\) 0 0
\(725\) 2.66047 0.0988073
\(726\) 0 0
\(727\) −0.348980 −0.0129429 −0.00647147 0.999979i \(-0.502060\pi\)
−0.00647147 + 0.999979i \(0.502060\pi\)
\(728\) 0 0
\(729\) −26.0052 −0.963156
\(730\) 0 0
\(731\) −3.39055 −0.125404
\(732\) 0 0
\(733\) 3.97975 0.146996 0.0734978 0.997295i \(-0.476584\pi\)
0.0734978 + 0.997295i \(0.476584\pi\)
\(734\) 0 0
\(735\) −2.43828 −0.0899374
\(736\) 0 0
\(737\) 0.432198 0.0159202
\(738\) 0 0
\(739\) −23.5249 −0.865379 −0.432690 0.901543i \(-0.642435\pi\)
−0.432690 + 0.901543i \(0.642435\pi\)
\(740\) 0 0
\(741\) 17.5100 0.643247
\(742\) 0 0
\(743\) 48.4912 1.77897 0.889485 0.456963i \(-0.151063\pi\)
0.889485 + 0.456963i \(0.151063\pi\)
\(744\) 0 0
\(745\) −9.10374 −0.333535
\(746\) 0 0
\(747\) −13.0538 −0.477614
\(748\) 0 0
\(749\) −13.8935 −0.507659
\(750\) 0 0
\(751\) 38.6192 1.40923 0.704617 0.709588i \(-0.251119\pi\)
0.704617 + 0.709588i \(0.251119\pi\)
\(752\) 0 0
\(753\) 43.5063 1.58546
\(754\) 0 0
\(755\) 17.4492 0.635040
\(756\) 0 0
\(757\) 4.62353 0.168045 0.0840225 0.996464i \(-0.473223\pi\)
0.0840225 + 0.996464i \(0.473223\pi\)
\(758\) 0 0
\(759\) −8.00000 −0.290382
\(760\) 0 0
\(761\) 18.8614 0.683725 0.341863 0.939750i \(-0.388942\pi\)
0.341863 + 0.939750i \(0.388942\pi\)
\(762\) 0 0
\(763\) 1.45825 0.0527923
\(764\) 0 0
\(765\) 2.74301 0.0991737
\(766\) 0 0
\(767\) 12.3565 0.446167
\(768\) 0 0
\(769\) −6.94427 −0.250417 −0.125208 0.992130i \(-0.539960\pi\)
−0.125208 + 0.992130i \(0.539960\pi\)
\(770\) 0 0
\(771\) −21.5462 −0.775968
\(772\) 0 0
\(773\) 27.4307 0.986612 0.493306 0.869856i \(-0.335788\pi\)
0.493306 + 0.869856i \(0.335788\pi\)
\(774\) 0 0
\(775\) −7.04774 −0.253162
\(776\) 0 0
\(777\) −1.94523 −0.0697846
\(778\) 0 0
\(779\) 18.1465 0.650165
\(780\) 0 0
\(781\) 0.308957 0.0110554
\(782\) 0 0
\(783\) 0.355323 0.0126982
\(784\) 0 0
\(785\) 12.5909 0.449387
\(786\) 0 0
\(787\) −9.80867 −0.349641 −0.174821 0.984600i \(-0.555935\pi\)
−0.174821 + 0.984600i \(0.555935\pi\)
\(788\) 0 0
\(789\) 13.3886 0.476648
\(790\) 0 0
\(791\) 3.86268 0.137341
\(792\) 0 0
\(793\) 1.01388 0.0360041
\(794\) 0 0
\(795\) −14.3626 −0.509388
\(796\) 0 0
\(797\) −13.9673 −0.494748 −0.247374 0.968920i \(-0.579568\pi\)
−0.247374 + 0.968920i \(0.579568\pi\)
\(798\) 0 0
\(799\) 0.981468 0.0347218
\(800\) 0 0
\(801\) −19.5807 −0.691851
\(802\) 0 0
\(803\) 11.0064 0.388406
\(804\) 0 0
\(805\) −2.65438 −0.0935547
\(806\) 0 0
\(807\) 76.8349 2.70472
\(808\) 0 0
\(809\) −6.10688 −0.214707 −0.107353 0.994221i \(-0.534238\pi\)
−0.107353 + 0.994221i \(0.534238\pi\)
\(810\) 0 0
\(811\) 26.0252 0.913870 0.456935 0.889500i \(-0.348947\pi\)
0.456935 + 0.889500i \(0.348947\pi\)
\(812\) 0 0
\(813\) 51.7315 1.81430
\(814\) 0 0
\(815\) −5.74396 −0.201202
\(816\) 0 0
\(817\) 26.1435 0.914645
\(818\) 0 0
\(819\) −2.94523 −0.102914
\(820\) 0 0
\(821\) 24.8328 0.866671 0.433336 0.901233i \(-0.357336\pi\)
0.433336 + 0.901233i \(0.357336\pi\)
\(822\) 0 0
\(823\) −17.7460 −0.618585 −0.309293 0.950967i \(-0.600092\pi\)
−0.309293 + 0.950967i \(0.600092\pi\)
\(824\) 0 0
\(825\) −3.01388 −0.104930
\(826\) 0 0
\(827\) −4.02434 −0.139940 −0.0699700 0.997549i \(-0.522290\pi\)
−0.0699700 + 0.997549i \(0.522290\pi\)
\(828\) 0 0
\(829\) −11.9060 −0.413514 −0.206757 0.978392i \(-0.566291\pi\)
−0.206757 + 0.978392i \(0.566291\pi\)
\(830\) 0 0
\(831\) 13.4442 0.466373
\(832\) 0 0
\(833\) 0.931341 0.0322691
\(834\) 0 0
\(835\) 2.56952 0.0889218
\(836\) 0 0
\(837\) −0.941272 −0.0325351
\(838\) 0 0
\(839\) 19.4876 0.672786 0.336393 0.941722i \(-0.390793\pi\)
0.336393 + 0.941722i \(0.390793\pi\)
\(840\) 0 0
\(841\) −21.9219 −0.755928
\(842\) 0 0
\(843\) 22.1975 0.764523
\(844\) 0 0
\(845\) 1.00000 0.0344010
\(846\) 0 0
\(847\) 9.47214 0.325466
\(848\) 0 0
\(849\) −61.7787 −2.12024
\(850\) 0 0
\(851\) −2.11763 −0.0725913
\(852\) 0 0
\(853\) −42.6218 −1.45934 −0.729671 0.683798i \(-0.760327\pi\)
−0.729671 + 0.683798i \(0.760327\pi\)
\(854\) 0 0
\(855\) −21.1505 −0.723333
\(856\) 0 0
\(857\) 20.3718 0.695886 0.347943 0.937516i \(-0.386880\pi\)
0.347943 + 0.937516i \(0.386880\pi\)
\(858\) 0 0
\(859\) 4.16973 0.142269 0.0711347 0.997467i \(-0.477338\pi\)
0.0711347 + 0.997467i \(0.477338\pi\)
\(860\) 0 0
\(861\) −6.16132 −0.209977
\(862\) 0 0
\(863\) 14.7678 0.502701 0.251350 0.967896i \(-0.419125\pi\)
0.251350 + 0.967896i \(0.419125\pi\)
\(864\) 0 0
\(865\) −6.48321 −0.220436
\(866\) 0 0
\(867\) 39.3359 1.33592
\(868\) 0 0
\(869\) −8.34851 −0.283204
\(870\) 0 0
\(871\) 0.349656 0.0118476
\(872\) 0 0
\(873\) −12.6494 −0.428117
\(874\) 0 0
\(875\) −1.00000 −0.0338062
\(876\) 0 0
\(877\) −6.08119 −0.205347 −0.102674 0.994715i \(-0.532740\pi\)
−0.102674 + 0.994715i \(0.532740\pi\)
\(878\) 0 0
\(879\) 21.3088 0.718727
\(880\) 0 0
\(881\) −26.2868 −0.885625 −0.442812 0.896614i \(-0.646019\pi\)
−0.442812 + 0.896614i \(0.646019\pi\)
\(882\) 0 0
\(883\) 31.4092 1.05700 0.528502 0.848932i \(-0.322754\pi\)
0.528502 + 0.848932i \(0.322754\pi\)
\(884\) 0 0
\(885\) −30.1286 −1.01276
\(886\) 0 0
\(887\) −18.2685 −0.613398 −0.306699 0.951807i \(-0.599224\pi\)
−0.306699 + 0.951807i \(0.599224\pi\)
\(888\) 0 0
\(889\) −1.23607 −0.0414564
\(890\) 0 0
\(891\) −11.3240 −0.379369
\(892\) 0 0
\(893\) −7.56780 −0.253247
\(894\) 0 0
\(895\) −15.1444 −0.506223
\(896\) 0 0
\(897\) −6.47214 −0.216098
\(898\) 0 0
\(899\) −18.7503 −0.625357
\(900\) 0 0
\(901\) 5.48602 0.182766
\(902\) 0 0
\(903\) −8.87657 −0.295394
\(904\) 0 0
\(905\) −3.41079 −0.113379
\(906\) 0 0
\(907\) 28.8297 0.957276 0.478638 0.878012i \(-0.341131\pi\)
0.478638 + 0.878012i \(0.341131\pi\)
\(908\) 0 0
\(909\) −15.2891 −0.507107
\(910\) 0 0
\(911\) −19.1444 −0.634284 −0.317142 0.948378i \(-0.602723\pi\)
−0.317142 + 0.948378i \(0.602723\pi\)
\(912\) 0 0
\(913\) −5.47850 −0.181312
\(914\) 0 0
\(915\) −2.47214 −0.0817263
\(916\) 0 0
\(917\) 10.9165 0.360495
\(918\) 0 0
\(919\) 15.8197 0.521845 0.260923 0.965360i \(-0.415973\pi\)
0.260923 + 0.965360i \(0.415973\pi\)
\(920\) 0 0
\(921\) 7.51379 0.247588
\(922\) 0 0
\(923\) 0.249952 0.00822726
\(924\) 0 0
\(925\) −0.797785 −0.0262310
\(926\) 0 0
\(927\) −56.3700 −1.85143
\(928\) 0 0
\(929\) 32.3399 1.06104 0.530519 0.847673i \(-0.321997\pi\)
0.530519 + 0.847673i \(0.321997\pi\)
\(930\) 0 0
\(931\) −7.18129 −0.235357
\(932\) 0 0
\(933\) −2.92403 −0.0957283
\(934\) 0 0
\(935\) 1.15120 0.0376483
\(936\) 0 0
\(937\) 57.4929 1.87821 0.939106 0.343626i \(-0.111655\pi\)
0.939106 + 0.343626i \(0.111655\pi\)
\(938\) 0 0
\(939\) 63.1209 2.05987
\(940\) 0 0
\(941\) 13.9236 0.453897 0.226949 0.973907i \(-0.427125\pi\)
0.226949 + 0.973907i \(0.427125\pi\)
\(942\) 0 0
\(943\) −6.70739 −0.218423
\(944\) 0 0
\(945\) −0.133557 −0.00434460
\(946\) 0 0
\(947\) −18.3836 −0.597387 −0.298693 0.954349i \(-0.596551\pi\)
−0.298693 + 0.954349i \(0.596551\pi\)
\(948\) 0 0
\(949\) 8.90433 0.289047
\(950\) 0 0
\(951\) 21.4258 0.694780
\(952\) 0 0
\(953\) −28.8625 −0.934948 −0.467474 0.884007i \(-0.654836\pi\)
−0.467474 + 0.884007i \(0.654836\pi\)
\(954\) 0 0
\(955\) −8.32565 −0.269412
\(956\) 0 0
\(957\) −8.01834 −0.259196
\(958\) 0 0
\(959\) −19.8597 −0.641303
\(960\) 0 0
\(961\) 18.6706 0.602277
\(962\) 0 0
\(963\) 40.9196 1.31862
\(964\) 0 0
\(965\) −18.8199 −0.605834
\(966\) 0 0
\(967\) −28.5989 −0.919680 −0.459840 0.888002i \(-0.652093\pi\)
−0.459840 + 0.888002i \(0.652093\pi\)
\(968\) 0 0
\(969\) 16.3078 0.523882
\(970\) 0 0
\(971\) 43.6432 1.40058 0.700288 0.713860i \(-0.253055\pi\)
0.700288 + 0.713860i \(0.253055\pi\)
\(972\) 0 0
\(973\) 15.9582 0.511595
\(974\) 0 0
\(975\) −2.43828 −0.0780876
\(976\) 0 0
\(977\) −31.7186 −1.01477 −0.507384 0.861720i \(-0.669387\pi\)
−0.507384 + 0.861720i \(0.669387\pi\)
\(978\) 0 0
\(979\) −8.21775 −0.262640
\(980\) 0 0
\(981\) −4.29488 −0.137125
\(982\) 0 0
\(983\) 53.7533 1.71446 0.857232 0.514930i \(-0.172182\pi\)
0.857232 + 0.514930i \(0.172182\pi\)
\(984\) 0 0
\(985\) 4.88764 0.155733
\(986\) 0 0
\(987\) 2.56952 0.0817886
\(988\) 0 0
\(989\) −9.66327 −0.307274
\(990\) 0 0
\(991\) 4.98126 0.158235 0.0791175 0.996865i \(-0.474790\pi\)
0.0791175 + 0.996865i \(0.474790\pi\)
\(992\) 0 0
\(993\) 58.6420 1.86095
\(994\) 0 0
\(995\) −24.3151 −0.770841
\(996\) 0 0
\(997\) 34.5764 1.09505 0.547524 0.836790i \(-0.315571\pi\)
0.547524 + 0.836790i \(0.315571\pi\)
\(998\) 0 0
\(999\) −0.106549 −0.00337107
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 3640.2.a.s.1.1 4
4.3 odd 2 7280.2.a.ca.1.4 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
3640.2.a.s.1.1 4 1.1 even 1 trivial
7280.2.a.ca.1.4 4 4.3 odd 2