Properties

Label 8004.2.a.i.1.7
Level $8004$
Weight $2$
Character 8004.1
Self dual yes
Analytic conductor $63.912$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 8004 = 2^{2} \cdot 3 \cdot 23 \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8004.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(63.9122617778\)
Analytic rank: \(0\)
Dimension: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
Defining polynomial: \(x^{16} - 5 x^{15} - 43 x^{14} + 234 x^{13} + 634 x^{12} - 4048 x^{11} - 3483 x^{10} + 32512 x^{9} - 137 x^{8} - 121665 x^{7} + 60168 x^{6} + 172218 x^{5} - 138024 x^{4} - 17844 x^{3} + 14352 x^{2} + 72 x - 208\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.7
Root \(-0.123882\) of defining polynomial
Character \(\chi\) \(=\) 8004.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{3} -0.123882 q^{5} +3.12124 q^{7} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{3} -0.123882 q^{5} +3.12124 q^{7} +1.00000 q^{9} -3.34537 q^{11} -4.64227 q^{13} +0.123882 q^{15} -4.27779 q^{17} -2.36905 q^{19} -3.12124 q^{21} +1.00000 q^{23} -4.98465 q^{25} -1.00000 q^{27} +1.00000 q^{29} -4.48181 q^{31} +3.34537 q^{33} -0.386665 q^{35} -5.19624 q^{37} +4.64227 q^{39} -7.96483 q^{41} +3.81430 q^{43} -0.123882 q^{45} +2.73638 q^{47} +2.74217 q^{49} +4.27779 q^{51} +11.8801 q^{53} +0.414430 q^{55} +2.36905 q^{57} +11.8045 q^{59} +1.29173 q^{61} +3.12124 q^{63} +0.575092 q^{65} +13.6476 q^{67} -1.00000 q^{69} -14.5132 q^{71} -1.55686 q^{73} +4.98465 q^{75} -10.4417 q^{77} +9.29227 q^{79} +1.00000 q^{81} +16.0061 q^{83} +0.529940 q^{85} -1.00000 q^{87} +9.47785 q^{89} -14.4897 q^{91} +4.48181 q^{93} +0.293481 q^{95} +11.4680 q^{97} -3.34537 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16q - 16q^{3} + 5q^{5} - 4q^{7} + 16q^{9} + O(q^{10}) \) \( 16q - 16q^{3} + 5q^{5} - 4q^{7} + 16q^{9} + 5q^{11} + 8q^{13} - 5q^{15} + 7q^{17} + q^{19} + 4q^{21} + 16q^{23} + 31q^{25} - 16q^{27} + 16q^{29} - 2q^{31} - 5q^{33} + 5q^{35} + 14q^{37} - 8q^{39} - q^{41} - 13q^{43} + 5q^{45} - 4q^{47} + 30q^{49} - 7q^{51} + 19q^{53} - 37q^{55} - q^{57} + 12q^{59} + 21q^{61} - 4q^{63} + 26q^{65} - 11q^{67} - 16q^{69} + 7q^{71} - 13q^{73} - 31q^{75} + 4q^{77} - 18q^{79} + 16q^{81} + 25q^{83} + 48q^{85} - 16q^{87} + 12q^{89} - 11q^{91} + 2q^{93} - q^{95} + 5q^{97} + 5q^{99} + O(q^{100}) \)

Coefficient data

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

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) −1.00000 −0.577350
\(4\) 0 0
\(5\) −0.123882 −0.0554016 −0.0277008 0.999616i \(-0.508819\pi\)
−0.0277008 + 0.999616i \(0.508819\pi\)
\(6\) 0 0
\(7\) 3.12124 1.17972 0.589860 0.807506i \(-0.299183\pi\)
0.589860 + 0.807506i \(0.299183\pi\)
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) −3.34537 −1.00867 −0.504333 0.863509i \(-0.668262\pi\)
−0.504333 + 0.863509i \(0.668262\pi\)
\(12\) 0 0
\(13\) −4.64227 −1.28753 −0.643767 0.765221i \(-0.722630\pi\)
−0.643767 + 0.765221i \(0.722630\pi\)
\(14\) 0 0
\(15\) 0.123882 0.0319861
\(16\) 0 0
\(17\) −4.27779 −1.03752 −0.518759 0.854921i \(-0.673606\pi\)
−0.518759 + 0.854921i \(0.673606\pi\)
\(18\) 0 0
\(19\) −2.36905 −0.543497 −0.271748 0.962368i \(-0.587602\pi\)
−0.271748 + 0.962368i \(0.587602\pi\)
\(20\) 0 0
\(21\) −3.12124 −0.681111
\(22\) 0 0
\(23\) 1.00000 0.208514
\(24\) 0 0
\(25\) −4.98465 −0.996931
\(26\) 0 0
\(27\) −1.00000 −0.192450
\(28\) 0 0
\(29\) 1.00000 0.185695
\(30\) 0 0
\(31\) −4.48181 −0.804956 −0.402478 0.915430i \(-0.631851\pi\)
−0.402478 + 0.915430i \(0.631851\pi\)
\(32\) 0 0
\(33\) 3.34537 0.582354
\(34\) 0 0
\(35\) −0.386665 −0.0653583
\(36\) 0 0
\(37\) −5.19624 −0.854257 −0.427128 0.904191i \(-0.640475\pi\)
−0.427128 + 0.904191i \(0.640475\pi\)
\(38\) 0 0
\(39\) 4.64227 0.743358
\(40\) 0 0
\(41\) −7.96483 −1.24390 −0.621949 0.783058i \(-0.713659\pi\)
−0.621949 + 0.783058i \(0.713659\pi\)
\(42\) 0 0
\(43\) 3.81430 0.581676 0.290838 0.956772i \(-0.406066\pi\)
0.290838 + 0.956772i \(0.406066\pi\)
\(44\) 0 0
\(45\) −0.123882 −0.0184672
\(46\) 0 0
\(47\) 2.73638 0.399143 0.199571 0.979883i \(-0.436045\pi\)
0.199571 + 0.979883i \(0.436045\pi\)
\(48\) 0 0
\(49\) 2.74217 0.391738
\(50\) 0 0
\(51\) 4.27779 0.599011
\(52\) 0 0
\(53\) 11.8801 1.63186 0.815928 0.578153i \(-0.196226\pi\)
0.815928 + 0.578153i \(0.196226\pi\)
\(54\) 0 0
\(55\) 0.414430 0.0558817
\(56\) 0 0
\(57\) 2.36905 0.313788
\(58\) 0 0
\(59\) 11.8045 1.53682 0.768410 0.639957i \(-0.221048\pi\)
0.768410 + 0.639957i \(0.221048\pi\)
\(60\) 0 0
\(61\) 1.29173 0.165389 0.0826945 0.996575i \(-0.473647\pi\)
0.0826945 + 0.996575i \(0.473647\pi\)
\(62\) 0 0
\(63\) 3.12124 0.393240
\(64\) 0 0
\(65\) 0.575092 0.0713314
\(66\) 0 0
\(67\) 13.6476 1.66732 0.833662 0.552274i \(-0.186240\pi\)
0.833662 + 0.552274i \(0.186240\pi\)
\(68\) 0 0
\(69\) −1.00000 −0.120386
\(70\) 0 0
\(71\) −14.5132 −1.72240 −0.861201 0.508264i \(-0.830287\pi\)
−0.861201 + 0.508264i \(0.830287\pi\)
\(72\) 0 0
\(73\) −1.55686 −0.182217 −0.0911083 0.995841i \(-0.529041\pi\)
−0.0911083 + 0.995841i \(0.529041\pi\)
\(74\) 0 0
\(75\) 4.98465 0.575578
\(76\) 0 0
\(77\) −10.4417 −1.18994
\(78\) 0 0
\(79\) 9.29227 1.04546 0.522731 0.852498i \(-0.324913\pi\)
0.522731 + 0.852498i \(0.324913\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) 16.0061 1.75690 0.878450 0.477834i \(-0.158578\pi\)
0.878450 + 0.477834i \(0.158578\pi\)
\(84\) 0 0
\(85\) 0.529940 0.0574801
\(86\) 0 0
\(87\) −1.00000 −0.107211
\(88\) 0 0
\(89\) 9.47785 1.00465 0.502325 0.864679i \(-0.332478\pi\)
0.502325 + 0.864679i \(0.332478\pi\)
\(90\) 0 0
\(91\) −14.4897 −1.51893
\(92\) 0 0
\(93\) 4.48181 0.464742
\(94\) 0 0
\(95\) 0.293481 0.0301106
\(96\) 0 0
\(97\) 11.4680 1.16439 0.582197 0.813048i \(-0.302193\pi\)
0.582197 + 0.813048i \(0.302193\pi\)
\(98\) 0 0
\(99\) −3.34537 −0.336222
\(100\) 0 0
\(101\) 5.00291 0.497808 0.248904 0.968528i \(-0.419930\pi\)
0.248904 + 0.968528i \(0.419930\pi\)
\(102\) 0 0
\(103\) 6.43917 0.634471 0.317235 0.948347i \(-0.397245\pi\)
0.317235 + 0.948347i \(0.397245\pi\)
\(104\) 0 0
\(105\) 0.386665 0.0377346
\(106\) 0 0
\(107\) 4.23096 0.409022 0.204511 0.978864i \(-0.434440\pi\)
0.204511 + 0.978864i \(0.434440\pi\)
\(108\) 0 0
\(109\) 3.01374 0.288664 0.144332 0.989529i \(-0.453897\pi\)
0.144332 + 0.989529i \(0.453897\pi\)
\(110\) 0 0
\(111\) 5.19624 0.493205
\(112\) 0 0
\(113\) 14.5321 1.36707 0.683534 0.729919i \(-0.260442\pi\)
0.683534 + 0.729919i \(0.260442\pi\)
\(114\) 0 0
\(115\) −0.123882 −0.0115520
\(116\) 0 0
\(117\) −4.64227 −0.429178
\(118\) 0 0
\(119\) −13.3520 −1.22398
\(120\) 0 0
\(121\) 0.191499 0.0174090
\(122\) 0 0
\(123\) 7.96483 0.718165
\(124\) 0 0
\(125\) 1.23692 0.110633
\(126\) 0 0
\(127\) −14.1554 −1.25609 −0.628043 0.778179i \(-0.716144\pi\)
−0.628043 + 0.778179i \(0.716144\pi\)
\(128\) 0 0
\(129\) −3.81430 −0.335831
\(130\) 0 0
\(131\) −13.8029 −1.20596 −0.602981 0.797756i \(-0.706021\pi\)
−0.602981 + 0.797756i \(0.706021\pi\)
\(132\) 0 0
\(133\) −7.39437 −0.641173
\(134\) 0 0
\(135\) 0.123882 0.0106620
\(136\) 0 0
\(137\) 15.5918 1.33209 0.666047 0.745910i \(-0.267985\pi\)
0.666047 + 0.745910i \(0.267985\pi\)
\(138\) 0 0
\(139\) 1.46483 0.124245 0.0621226 0.998069i \(-0.480213\pi\)
0.0621226 + 0.998069i \(0.480213\pi\)
\(140\) 0 0
\(141\) −2.73638 −0.230445
\(142\) 0 0
\(143\) 15.5301 1.29869
\(144\) 0 0
\(145\) −0.123882 −0.0102878
\(146\) 0 0
\(147\) −2.74217 −0.226170
\(148\) 0 0
\(149\) 20.0755 1.64465 0.822323 0.569021i \(-0.192678\pi\)
0.822323 + 0.569021i \(0.192678\pi\)
\(150\) 0 0
\(151\) −15.8842 −1.29263 −0.646317 0.763069i \(-0.723692\pi\)
−0.646317 + 0.763069i \(0.723692\pi\)
\(152\) 0 0
\(153\) −4.27779 −0.345839
\(154\) 0 0
\(155\) 0.555214 0.0445958
\(156\) 0 0
\(157\) −12.6535 −1.00986 −0.504931 0.863159i \(-0.668482\pi\)
−0.504931 + 0.863159i \(0.668482\pi\)
\(158\) 0 0
\(159\) −11.8801 −0.942153
\(160\) 0 0
\(161\) 3.12124 0.245989
\(162\) 0 0
\(163\) −15.0684 −1.18025 −0.590126 0.807311i \(-0.700922\pi\)
−0.590126 + 0.807311i \(0.700922\pi\)
\(164\) 0 0
\(165\) −0.414430 −0.0322633
\(166\) 0 0
\(167\) 21.1441 1.63618 0.818090 0.575091i \(-0.195033\pi\)
0.818090 + 0.575091i \(0.195033\pi\)
\(168\) 0 0
\(169\) 8.55069 0.657745
\(170\) 0 0
\(171\) −2.36905 −0.181166
\(172\) 0 0
\(173\) 1.19786 0.0910714 0.0455357 0.998963i \(-0.485501\pi\)
0.0455357 + 0.998963i \(0.485501\pi\)
\(174\) 0 0
\(175\) −15.5583 −1.17610
\(176\) 0 0
\(177\) −11.8045 −0.887284
\(178\) 0 0
\(179\) −3.76867 −0.281684 −0.140842 0.990032i \(-0.544981\pi\)
−0.140842 + 0.990032i \(0.544981\pi\)
\(180\) 0 0
\(181\) 26.2502 1.95116 0.975581 0.219642i \(-0.0704889\pi\)
0.975581 + 0.219642i \(0.0704889\pi\)
\(182\) 0 0
\(183\) −1.29173 −0.0954874
\(184\) 0 0
\(185\) 0.643719 0.0473272
\(186\) 0 0
\(187\) 14.3108 1.04651
\(188\) 0 0
\(189\) −3.12124 −0.227037
\(190\) 0 0
\(191\) 9.44833 0.683657 0.341828 0.939762i \(-0.388954\pi\)
0.341828 + 0.939762i \(0.388954\pi\)
\(192\) 0 0
\(193\) −19.5334 −1.40604 −0.703022 0.711168i \(-0.748166\pi\)
−0.703022 + 0.711168i \(0.748166\pi\)
\(194\) 0 0
\(195\) −0.575092 −0.0411832
\(196\) 0 0
\(197\) −16.1253 −1.14888 −0.574438 0.818548i \(-0.694779\pi\)
−0.574438 + 0.818548i \(0.694779\pi\)
\(198\) 0 0
\(199\) −0.786826 −0.0557766 −0.0278883 0.999611i \(-0.508878\pi\)
−0.0278883 + 0.999611i \(0.508878\pi\)
\(200\) 0 0
\(201\) −13.6476 −0.962630
\(202\) 0 0
\(203\) 3.12124 0.219068
\(204\) 0 0
\(205\) 0.986697 0.0689139
\(206\) 0 0
\(207\) 1.00000 0.0695048
\(208\) 0 0
\(209\) 7.92534 0.548207
\(210\) 0 0
\(211\) 17.5450 1.20785 0.603924 0.797042i \(-0.293603\pi\)
0.603924 + 0.797042i \(0.293603\pi\)
\(212\) 0 0
\(213\) 14.5132 0.994430
\(214\) 0 0
\(215\) −0.472522 −0.0322257
\(216\) 0 0
\(217\) −13.9888 −0.949622
\(218\) 0 0
\(219\) 1.55686 0.105203
\(220\) 0 0
\(221\) 19.8587 1.33584
\(222\) 0 0
\(223\) 14.1765 0.949326 0.474663 0.880168i \(-0.342570\pi\)
0.474663 + 0.880168i \(0.342570\pi\)
\(224\) 0 0
\(225\) −4.98465 −0.332310
\(226\) 0 0
\(227\) −23.0451 −1.52955 −0.764777 0.644295i \(-0.777151\pi\)
−0.764777 + 0.644295i \(0.777151\pi\)
\(228\) 0 0
\(229\) 9.81224 0.648411 0.324206 0.945987i \(-0.394903\pi\)
0.324206 + 0.945987i \(0.394903\pi\)
\(230\) 0 0
\(231\) 10.4417 0.687015
\(232\) 0 0
\(233\) 27.3650 1.79274 0.896371 0.443304i \(-0.146194\pi\)
0.896371 + 0.443304i \(0.146194\pi\)
\(234\) 0 0
\(235\) −0.338988 −0.0221131
\(236\) 0 0
\(237\) −9.29227 −0.603598
\(238\) 0 0
\(239\) 12.3849 0.801112 0.400556 0.916272i \(-0.368817\pi\)
0.400556 + 0.916272i \(0.368817\pi\)
\(240\) 0 0
\(241\) −28.5715 −1.84045 −0.920226 0.391388i \(-0.871995\pi\)
−0.920226 + 0.391388i \(0.871995\pi\)
\(242\) 0 0
\(243\) −1.00000 −0.0641500
\(244\) 0 0
\(245\) −0.339704 −0.0217029
\(246\) 0 0
\(247\) 10.9978 0.699771
\(248\) 0 0
\(249\) −16.0061 −1.01435
\(250\) 0 0
\(251\) −13.2296 −0.835047 −0.417523 0.908666i \(-0.637102\pi\)
−0.417523 + 0.908666i \(0.637102\pi\)
\(252\) 0 0
\(253\) −3.34537 −0.210322
\(254\) 0 0
\(255\) −0.529940 −0.0331862
\(256\) 0 0
\(257\) −2.30623 −0.143858 −0.0719292 0.997410i \(-0.522916\pi\)
−0.0719292 + 0.997410i \(0.522916\pi\)
\(258\) 0 0
\(259\) −16.2187 −1.00778
\(260\) 0 0
\(261\) 1.00000 0.0618984
\(262\) 0 0
\(263\) 15.0755 0.929595 0.464797 0.885417i \(-0.346127\pi\)
0.464797 + 0.885417i \(0.346127\pi\)
\(264\) 0 0
\(265\) −1.47173 −0.0904074
\(266\) 0 0
\(267\) −9.47785 −0.580035
\(268\) 0 0
\(269\) 10.7910 0.657938 0.328969 0.944341i \(-0.393299\pi\)
0.328969 + 0.944341i \(0.393299\pi\)
\(270\) 0 0
\(271\) −5.46497 −0.331973 −0.165987 0.986128i \(-0.553081\pi\)
−0.165987 + 0.986128i \(0.553081\pi\)
\(272\) 0 0
\(273\) 14.4897 0.876954
\(274\) 0 0
\(275\) 16.6755 1.00557
\(276\) 0 0
\(277\) 19.5186 1.17276 0.586379 0.810037i \(-0.300553\pi\)
0.586379 + 0.810037i \(0.300553\pi\)
\(278\) 0 0
\(279\) −4.48181 −0.268319
\(280\) 0 0
\(281\) −22.9300 −1.36789 −0.683945 0.729533i \(-0.739737\pi\)
−0.683945 + 0.729533i \(0.739737\pi\)
\(282\) 0 0
\(283\) 25.4692 1.51399 0.756994 0.653422i \(-0.226667\pi\)
0.756994 + 0.653422i \(0.226667\pi\)
\(284\) 0 0
\(285\) −0.293481 −0.0173843
\(286\) 0 0
\(287\) −24.8602 −1.46745
\(288\) 0 0
\(289\) 1.29952 0.0764423
\(290\) 0 0
\(291\) −11.4680 −0.672263
\(292\) 0 0
\(293\) −14.1564 −0.827026 −0.413513 0.910498i \(-0.635698\pi\)
−0.413513 + 0.910498i \(0.635698\pi\)
\(294\) 0 0
\(295\) −1.46237 −0.0851423
\(296\) 0 0
\(297\) 3.34537 0.194118
\(298\) 0 0
\(299\) −4.64227 −0.268470
\(300\) 0 0
\(301\) 11.9054 0.686214
\(302\) 0 0
\(303\) −5.00291 −0.287410
\(304\) 0 0
\(305\) −0.160022 −0.00916281
\(306\) 0 0
\(307\) −24.3935 −1.39221 −0.696106 0.717939i \(-0.745085\pi\)
−0.696106 + 0.717939i \(0.745085\pi\)
\(308\) 0 0
\(309\) −6.43917 −0.366312
\(310\) 0 0
\(311\) −6.36759 −0.361073 −0.180536 0.983568i \(-0.557783\pi\)
−0.180536 + 0.983568i \(0.557783\pi\)
\(312\) 0 0
\(313\) −28.0338 −1.58457 −0.792283 0.610154i \(-0.791108\pi\)
−0.792283 + 0.610154i \(0.791108\pi\)
\(314\) 0 0
\(315\) −0.386665 −0.0217861
\(316\) 0 0
\(317\) 8.70755 0.489065 0.244532 0.969641i \(-0.421366\pi\)
0.244532 + 0.969641i \(0.421366\pi\)
\(318\) 0 0
\(319\) −3.34537 −0.187305
\(320\) 0 0
\(321\) −4.23096 −0.236149
\(322\) 0 0
\(323\) 10.1343 0.563887
\(324\) 0 0
\(325\) 23.1401 1.28358
\(326\) 0 0
\(327\) −3.01374 −0.166660
\(328\) 0 0
\(329\) 8.54092 0.470876
\(330\) 0 0
\(331\) −24.5067 −1.34701 −0.673506 0.739181i \(-0.735213\pi\)
−0.673506 + 0.739181i \(0.735213\pi\)
\(332\) 0 0
\(333\) −5.19624 −0.284752
\(334\) 0 0
\(335\) −1.69069 −0.0923724
\(336\) 0 0
\(337\) 0.131842 0.00718190 0.00359095 0.999994i \(-0.498857\pi\)
0.00359095 + 0.999994i \(0.498857\pi\)
\(338\) 0 0
\(339\) −14.5321 −0.789277
\(340\) 0 0
\(341\) 14.9933 0.811933
\(342\) 0 0
\(343\) −13.2897 −0.717578
\(344\) 0 0
\(345\) 0.123882 0.00666957
\(346\) 0 0
\(347\) −2.80296 −0.150471 −0.0752354 0.997166i \(-0.523971\pi\)
−0.0752354 + 0.997166i \(0.523971\pi\)
\(348\) 0 0
\(349\) 8.54612 0.457464 0.228732 0.973489i \(-0.426542\pi\)
0.228732 + 0.973489i \(0.426542\pi\)
\(350\) 0 0
\(351\) 4.64227 0.247786
\(352\) 0 0
\(353\) −2.12642 −0.113178 −0.0565889 0.998398i \(-0.518022\pi\)
−0.0565889 + 0.998398i \(0.518022\pi\)
\(354\) 0 0
\(355\) 1.79792 0.0954238
\(356\) 0 0
\(357\) 13.3520 0.706665
\(358\) 0 0
\(359\) 8.06665 0.425741 0.212871 0.977080i \(-0.431719\pi\)
0.212871 + 0.977080i \(0.431719\pi\)
\(360\) 0 0
\(361\) −13.3876 −0.704612
\(362\) 0 0
\(363\) −0.191499 −0.0100511
\(364\) 0 0
\(365\) 0.192866 0.0100951
\(366\) 0 0
\(367\) 16.9668 0.885658 0.442829 0.896606i \(-0.353975\pi\)
0.442829 + 0.896606i \(0.353975\pi\)
\(368\) 0 0
\(369\) −7.96483 −0.414633
\(370\) 0 0
\(371\) 37.0807 1.92513
\(372\) 0 0
\(373\) −5.91353 −0.306191 −0.153095 0.988211i \(-0.548924\pi\)
−0.153095 + 0.988211i \(0.548924\pi\)
\(374\) 0 0
\(375\) −1.23692 −0.0638741
\(376\) 0 0
\(377\) −4.64227 −0.239089
\(378\) 0 0
\(379\) 15.6879 0.805831 0.402916 0.915237i \(-0.367997\pi\)
0.402916 + 0.915237i \(0.367997\pi\)
\(380\) 0 0
\(381\) 14.1554 0.725201
\(382\) 0 0
\(383\) −17.3356 −0.885808 −0.442904 0.896569i \(-0.646052\pi\)
−0.442904 + 0.896569i \(0.646052\pi\)
\(384\) 0 0
\(385\) 1.29354 0.0659248
\(386\) 0 0
\(387\) 3.81430 0.193892
\(388\) 0 0
\(389\) −0.709337 −0.0359648 −0.0179824 0.999838i \(-0.505724\pi\)
−0.0179824 + 0.999838i \(0.505724\pi\)
\(390\) 0 0
\(391\) −4.27779 −0.216337
\(392\) 0 0
\(393\) 13.8029 0.696262
\(394\) 0 0
\(395\) −1.15114 −0.0579203
\(396\) 0 0
\(397\) 29.5867 1.48491 0.742456 0.669895i \(-0.233661\pi\)
0.742456 + 0.669895i \(0.233661\pi\)
\(398\) 0 0
\(399\) 7.39437 0.370182
\(400\) 0 0
\(401\) 32.6563 1.63078 0.815389 0.578914i \(-0.196523\pi\)
0.815389 + 0.578914i \(0.196523\pi\)
\(402\) 0 0
\(403\) 20.8058 1.03641
\(404\) 0 0
\(405\) −0.123882 −0.00615573
\(406\) 0 0
\(407\) 17.3833 0.861661
\(408\) 0 0
\(409\) 6.65832 0.329233 0.164616 0.986358i \(-0.447361\pi\)
0.164616 + 0.986358i \(0.447361\pi\)
\(410\) 0 0
\(411\) −15.5918 −0.769085
\(412\) 0 0
\(413\) 36.8449 1.81302
\(414\) 0 0
\(415\) −1.98287 −0.0973350
\(416\) 0 0
\(417\) −1.46483 −0.0717330
\(418\) 0 0
\(419\) −37.8148 −1.84738 −0.923688 0.383146i \(-0.874841\pi\)
−0.923688 + 0.383146i \(0.874841\pi\)
\(420\) 0 0
\(421\) 29.1024 1.41837 0.709183 0.705025i \(-0.249064\pi\)
0.709183 + 0.705025i \(0.249064\pi\)
\(422\) 0 0
\(423\) 2.73638 0.133048
\(424\) 0 0
\(425\) 21.3233 1.03433
\(426\) 0 0
\(427\) 4.03180 0.195113
\(428\) 0 0
\(429\) −15.5301 −0.749801
\(430\) 0 0
\(431\) −12.8846 −0.620628 −0.310314 0.950634i \(-0.600434\pi\)
−0.310314 + 0.950634i \(0.600434\pi\)
\(432\) 0 0
\(433\) −16.0657 −0.772069 −0.386034 0.922484i \(-0.626155\pi\)
−0.386034 + 0.922484i \(0.626155\pi\)
\(434\) 0 0
\(435\) 0.123882 0.00593967
\(436\) 0 0
\(437\) −2.36905 −0.113327
\(438\) 0 0
\(439\) −0.642410 −0.0306605 −0.0153303 0.999882i \(-0.504880\pi\)
−0.0153303 + 0.999882i \(0.504880\pi\)
\(440\) 0 0
\(441\) 2.74217 0.130579
\(442\) 0 0
\(443\) 20.6330 0.980304 0.490152 0.871637i \(-0.336941\pi\)
0.490152 + 0.871637i \(0.336941\pi\)
\(444\) 0 0
\(445\) −1.17413 −0.0556592
\(446\) 0 0
\(447\) −20.0755 −0.949537
\(448\) 0 0
\(449\) 16.1685 0.763037 0.381519 0.924361i \(-0.375401\pi\)
0.381519 + 0.924361i \(0.375401\pi\)
\(450\) 0 0
\(451\) 26.6453 1.25468
\(452\) 0 0
\(453\) 15.8842 0.746303
\(454\) 0 0
\(455\) 1.79500 0.0841511
\(456\) 0 0
\(457\) −21.8185 −1.02063 −0.510313 0.859989i \(-0.670470\pi\)
−0.510313 + 0.859989i \(0.670470\pi\)
\(458\) 0 0
\(459\) 4.27779 0.199670
\(460\) 0 0
\(461\) −13.2053 −0.615034 −0.307517 0.951543i \(-0.599498\pi\)
−0.307517 + 0.951543i \(0.599498\pi\)
\(462\) 0 0
\(463\) −33.1684 −1.54147 −0.770734 0.637157i \(-0.780110\pi\)
−0.770734 + 0.637157i \(0.780110\pi\)
\(464\) 0 0
\(465\) −0.555214 −0.0257474
\(466\) 0 0
\(467\) 37.1989 1.72136 0.860679 0.509148i \(-0.170039\pi\)
0.860679 + 0.509148i \(0.170039\pi\)
\(468\) 0 0
\(469\) 42.5976 1.96698
\(470\) 0 0
\(471\) 12.6535 0.583045
\(472\) 0 0
\(473\) −12.7603 −0.586717
\(474\) 0 0
\(475\) 11.8089 0.541828
\(476\) 0 0
\(477\) 11.8801 0.543952
\(478\) 0 0
\(479\) 15.4721 0.706938 0.353469 0.935446i \(-0.385002\pi\)
0.353469 + 0.935446i \(0.385002\pi\)
\(480\) 0 0
\(481\) 24.1224 1.09989
\(482\) 0 0
\(483\) −3.12124 −0.142022
\(484\) 0 0
\(485\) −1.42067 −0.0645093
\(486\) 0 0
\(487\) 18.2873 0.828676 0.414338 0.910123i \(-0.364013\pi\)
0.414338 + 0.910123i \(0.364013\pi\)
\(488\) 0 0
\(489\) 15.0684 0.681418
\(490\) 0 0
\(491\) −15.4741 −0.698338 −0.349169 0.937060i \(-0.613536\pi\)
−0.349169 + 0.937060i \(0.613536\pi\)
\(492\) 0 0
\(493\) −4.27779 −0.192662
\(494\) 0 0
\(495\) 0.414430 0.0186272
\(496\) 0 0
\(497\) −45.2993 −2.03195
\(498\) 0 0
\(499\) 32.6480 1.46153 0.730763 0.682632i \(-0.239165\pi\)
0.730763 + 0.682632i \(0.239165\pi\)
\(500\) 0 0
\(501\) −21.1441 −0.944649
\(502\) 0 0
\(503\) −25.1863 −1.12300 −0.561501 0.827476i \(-0.689776\pi\)
−0.561501 + 0.827476i \(0.689776\pi\)
\(504\) 0 0
\(505\) −0.619769 −0.0275794
\(506\) 0 0
\(507\) −8.55069 −0.379749
\(508\) 0 0
\(509\) 12.4673 0.552605 0.276303 0.961071i \(-0.410891\pi\)
0.276303 + 0.961071i \(0.410891\pi\)
\(510\) 0 0
\(511\) −4.85934 −0.214964
\(512\) 0 0
\(513\) 2.36905 0.104596
\(514\) 0 0
\(515\) −0.797696 −0.0351507
\(516\) 0 0
\(517\) −9.15422 −0.402602
\(518\) 0 0
\(519\) −1.19786 −0.0525801
\(520\) 0 0
\(521\) 35.7753 1.56734 0.783672 0.621175i \(-0.213344\pi\)
0.783672 + 0.621175i \(0.213344\pi\)
\(522\) 0 0
\(523\) −12.4454 −0.544199 −0.272099 0.962269i \(-0.587718\pi\)
−0.272099 + 0.962269i \(0.587718\pi\)
\(524\) 0 0
\(525\) 15.5583 0.679021
\(526\) 0 0
\(527\) 19.1722 0.835156
\(528\) 0 0
\(529\) 1.00000 0.0434783
\(530\) 0 0
\(531\) 11.8045 0.512274
\(532\) 0 0
\(533\) 36.9749 1.60156
\(534\) 0 0
\(535\) −0.524138 −0.0226605
\(536\) 0 0
\(537\) 3.76867 0.162630
\(538\) 0 0
\(539\) −9.17356 −0.395133
\(540\) 0 0
\(541\) 27.6453 1.18856 0.594282 0.804257i \(-0.297436\pi\)
0.594282 + 0.804257i \(0.297436\pi\)
\(542\) 0 0
\(543\) −26.2502 −1.12650
\(544\) 0 0
\(545\) −0.373347 −0.0159924
\(546\) 0 0
\(547\) 17.4057 0.744215 0.372108 0.928190i \(-0.378635\pi\)
0.372108 + 0.928190i \(0.378635\pi\)
\(548\) 0 0
\(549\) 1.29173 0.0551297
\(550\) 0 0
\(551\) −2.36905 −0.100925
\(552\) 0 0
\(553\) 29.0035 1.23335
\(554\) 0 0
\(555\) −0.643719 −0.0273244
\(556\) 0 0
\(557\) −12.9472 −0.548592 −0.274296 0.961645i \(-0.588445\pi\)
−0.274296 + 0.961645i \(0.588445\pi\)
\(558\) 0 0
\(559\) −17.7070 −0.748927
\(560\) 0 0
\(561\) −14.3108 −0.604203
\(562\) 0 0
\(563\) −18.5120 −0.780189 −0.390094 0.920775i \(-0.627558\pi\)
−0.390094 + 0.920775i \(0.627558\pi\)
\(564\) 0 0
\(565\) −1.80027 −0.0757377
\(566\) 0 0
\(567\) 3.12124 0.131080
\(568\) 0 0
\(569\) 19.1094 0.801108 0.400554 0.916273i \(-0.368818\pi\)
0.400554 + 0.916273i \(0.368818\pi\)
\(570\) 0 0
\(571\) 15.9646 0.668099 0.334050 0.942555i \(-0.391585\pi\)
0.334050 + 0.942555i \(0.391585\pi\)
\(572\) 0 0
\(573\) −9.44833 −0.394709
\(574\) 0 0
\(575\) −4.98465 −0.207874
\(576\) 0 0
\(577\) −37.3694 −1.55571 −0.777855 0.628444i \(-0.783692\pi\)
−0.777855 + 0.628444i \(0.783692\pi\)
\(578\) 0 0
\(579\) 19.5334 0.811780
\(580\) 0 0
\(581\) 49.9590 2.07265
\(582\) 0 0
\(583\) −39.7433 −1.64600
\(584\) 0 0
\(585\) 0.575092 0.0237771
\(586\) 0 0
\(587\) 19.4465 0.802645 0.401322 0.915937i \(-0.368551\pi\)
0.401322 + 0.915937i \(0.368551\pi\)
\(588\) 0 0
\(589\) 10.6176 0.437491
\(590\) 0 0
\(591\) 16.1253 0.663304
\(592\) 0 0
\(593\) 8.79417 0.361133 0.180567 0.983563i \(-0.442207\pi\)
0.180567 + 0.983563i \(0.442207\pi\)
\(594\) 0 0
\(595\) 1.65407 0.0678104
\(596\) 0 0
\(597\) 0.786826 0.0322026
\(598\) 0 0
\(599\) −37.0331 −1.51313 −0.756565 0.653919i \(-0.773124\pi\)
−0.756565 + 0.653919i \(0.773124\pi\)
\(600\) 0 0
\(601\) −28.5398 −1.16416 −0.582080 0.813131i \(-0.697761\pi\)
−0.582080 + 0.813131i \(0.697761\pi\)
\(602\) 0 0
\(603\) 13.6476 0.555775
\(604\) 0 0
\(605\) −0.0237232 −0.000964487 0
\(606\) 0 0
\(607\) −5.79933 −0.235388 −0.117694 0.993050i \(-0.537550\pi\)
−0.117694 + 0.993050i \(0.537550\pi\)
\(608\) 0 0
\(609\) −3.12124 −0.126479
\(610\) 0 0
\(611\) −12.7030 −0.513910
\(612\) 0 0
\(613\) 20.0488 0.809763 0.404881 0.914369i \(-0.367313\pi\)
0.404881 + 0.914369i \(0.367313\pi\)
\(614\) 0 0
\(615\) −0.986697 −0.0397875
\(616\) 0 0
\(617\) 6.23955 0.251195 0.125597 0.992081i \(-0.459915\pi\)
0.125597 + 0.992081i \(0.459915\pi\)
\(618\) 0 0
\(619\) −4.24672 −0.170690 −0.0853450 0.996351i \(-0.527199\pi\)
−0.0853450 + 0.996351i \(0.527199\pi\)
\(620\) 0 0
\(621\) −1.00000 −0.0401286
\(622\) 0 0
\(623\) 29.5827 1.18521
\(624\) 0 0
\(625\) 24.7700 0.990801
\(626\) 0 0
\(627\) −7.92534 −0.316507
\(628\) 0 0
\(629\) 22.2285 0.886306
\(630\) 0 0
\(631\) 22.1747 0.882762 0.441381 0.897320i \(-0.354489\pi\)
0.441381 + 0.897320i \(0.354489\pi\)
\(632\) 0 0
\(633\) −17.5450 −0.697352
\(634\) 0 0
\(635\) 1.75359 0.0695891
\(636\) 0 0
\(637\) −12.7299 −0.504376
\(638\) 0 0
\(639\) −14.5132 −0.574134
\(640\) 0 0
\(641\) 37.6945 1.48884 0.744421 0.667710i \(-0.232725\pi\)
0.744421 + 0.667710i \(0.232725\pi\)
\(642\) 0 0
\(643\) 28.7934 1.13550 0.567750 0.823201i \(-0.307814\pi\)
0.567750 + 0.823201i \(0.307814\pi\)
\(644\) 0 0
\(645\) 0.472522 0.0186055
\(646\) 0 0
\(647\) 25.4215 0.999424 0.499712 0.866192i \(-0.333439\pi\)
0.499712 + 0.866192i \(0.333439\pi\)
\(648\) 0 0
\(649\) −39.4906 −1.55014
\(650\) 0 0
\(651\) 13.9888 0.548265
\(652\) 0 0
\(653\) 18.6157 0.728487 0.364244 0.931304i \(-0.381328\pi\)
0.364244 + 0.931304i \(0.381328\pi\)
\(654\) 0 0
\(655\) 1.70992 0.0668122
\(656\) 0 0
\(657\) −1.55686 −0.0607389
\(658\) 0 0
\(659\) 22.0488 0.858899 0.429449 0.903091i \(-0.358708\pi\)
0.429449 + 0.903091i \(0.358708\pi\)
\(660\) 0 0
\(661\) 21.3375 0.829931 0.414966 0.909837i \(-0.363794\pi\)
0.414966 + 0.909837i \(0.363794\pi\)
\(662\) 0 0
\(663\) −19.8587 −0.771247
\(664\) 0 0
\(665\) 0.916027 0.0355220
\(666\) 0 0
\(667\) 1.00000 0.0387202
\(668\) 0 0
\(669\) −14.1765 −0.548093
\(670\) 0 0
\(671\) −4.32131 −0.166822
\(672\) 0 0
\(673\) 7.24529 0.279285 0.139643 0.990202i \(-0.455405\pi\)
0.139643 + 0.990202i \(0.455405\pi\)
\(674\) 0 0
\(675\) 4.98465 0.191859
\(676\) 0 0
\(677\) 2.27849 0.0875696 0.0437848 0.999041i \(-0.486058\pi\)
0.0437848 + 0.999041i \(0.486058\pi\)
\(678\) 0 0
\(679\) 35.7943 1.37366
\(680\) 0 0
\(681\) 23.0451 0.883088
\(682\) 0 0
\(683\) −7.55361 −0.289031 −0.144515 0.989503i \(-0.546162\pi\)
−0.144515 + 0.989503i \(0.546162\pi\)
\(684\) 0 0
\(685\) −1.93153 −0.0738001
\(686\) 0 0
\(687\) −9.81224 −0.374360
\(688\) 0 0
\(689\) −55.1506 −2.10107
\(690\) 0 0
\(691\) −14.0336 −0.533862 −0.266931 0.963716i \(-0.586010\pi\)
−0.266931 + 0.963716i \(0.586010\pi\)
\(692\) 0 0
\(693\) −10.4417 −0.396648
\(694\) 0 0
\(695\) −0.181466 −0.00688338
\(696\) 0 0
\(697\) 34.0719 1.29057
\(698\) 0 0
\(699\) −27.3650 −1.03504
\(700\) 0 0
\(701\) 23.2228 0.877112 0.438556 0.898704i \(-0.355490\pi\)
0.438556 + 0.898704i \(0.355490\pi\)
\(702\) 0 0
\(703\) 12.3101 0.464286
\(704\) 0 0
\(705\) 0.338988 0.0127670
\(706\) 0 0
\(707\) 15.6153 0.587274
\(708\) 0 0
\(709\) −13.7848 −0.517699 −0.258850 0.965918i \(-0.583343\pi\)
−0.258850 + 0.965918i \(0.583343\pi\)
\(710\) 0 0
\(711\) 9.29227 0.348488
\(712\) 0 0
\(713\) −4.48181 −0.167845
\(714\) 0 0
\(715\) −1.92390 −0.0719497
\(716\) 0 0
\(717\) −12.3849 −0.462522
\(718\) 0 0
\(719\) −7.06763 −0.263578 −0.131789 0.991278i \(-0.542072\pi\)
−0.131789 + 0.991278i \(0.542072\pi\)
\(720\) 0 0
\(721\) 20.0982 0.748498
\(722\) 0 0
\(723\) 28.5715 1.06259
\(724\) 0 0
\(725\) −4.98465 −0.185125
\(726\) 0 0
\(727\) −48.8742 −1.81264 −0.906322 0.422589i \(-0.861122\pi\)
−0.906322 + 0.422589i \(0.861122\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) −16.3168 −0.603499
\(732\) 0 0
\(733\) 0.544142 0.0200984 0.0100492 0.999950i \(-0.496801\pi\)
0.0100492 + 0.999950i \(0.496801\pi\)
\(734\) 0 0
\(735\) 0.339704 0.0125302
\(736\) 0 0
\(737\) −45.6564 −1.68178
\(738\) 0 0
\(739\) 0.446724 0.0164330 0.00821650 0.999966i \(-0.497385\pi\)
0.00821650 + 0.999966i \(0.497385\pi\)
\(740\) 0 0
\(741\) −10.9978 −0.404013
\(742\) 0 0
\(743\) −44.9734 −1.64992 −0.824958 0.565194i \(-0.808801\pi\)
−0.824958 + 0.565194i \(0.808801\pi\)
\(744\) 0 0
\(745\) −2.48698 −0.0911160
\(746\) 0 0
\(747\) 16.0061 0.585633
\(748\) 0 0
\(749\) 13.2059 0.482531
\(750\) 0 0
\(751\) 52.7829 1.92608 0.963038 0.269365i \(-0.0868138\pi\)
0.963038 + 0.269365i \(0.0868138\pi\)
\(752\) 0 0
\(753\) 13.2296 0.482114
\(754\) 0 0
\(755\) 1.96776 0.0716140
\(756\) 0 0
\(757\) 52.7256 1.91634 0.958172 0.286193i \(-0.0923900\pi\)
0.958172 + 0.286193i \(0.0923900\pi\)
\(758\) 0 0
\(759\) 3.34537 0.121429
\(760\) 0 0
\(761\) 17.9115 0.649293 0.324646 0.945835i \(-0.394755\pi\)
0.324646 + 0.945835i \(0.394755\pi\)
\(762\) 0 0
\(763\) 9.40661 0.340542
\(764\) 0 0
\(765\) 0.529940 0.0191600
\(766\) 0 0
\(767\) −54.7999 −1.97871
\(768\) 0 0
\(769\) −6.93378 −0.250039 −0.125019 0.992154i \(-0.539899\pi\)
−0.125019 + 0.992154i \(0.539899\pi\)
\(770\) 0 0
\(771\) 2.30623 0.0830567
\(772\) 0 0
\(773\) 44.3154 1.59391 0.796956 0.604037i \(-0.206442\pi\)
0.796956 + 0.604037i \(0.206442\pi\)
\(774\) 0 0
\(775\) 22.3402 0.802485
\(776\) 0 0
\(777\) 16.2187 0.581844
\(778\) 0 0
\(779\) 18.8691 0.676054
\(780\) 0 0
\(781\) 48.5521 1.73733
\(782\) 0 0
\(783\) −1.00000 −0.0357371
\(784\) 0 0
\(785\) 1.56754 0.0559480
\(786\) 0 0
\(787\) 50.0015 1.78236 0.891181 0.453647i \(-0.149877\pi\)
0.891181 + 0.453647i \(0.149877\pi\)
\(788\) 0 0
\(789\) −15.0755 −0.536702
\(790\) 0 0
\(791\) 45.3583 1.61276
\(792\) 0 0
\(793\) −5.99656 −0.212944
\(794\) 0 0
\(795\) 1.47173 0.0521968
\(796\) 0 0
\(797\) −28.0461 −0.993446 −0.496723 0.867909i \(-0.665463\pi\)
−0.496723 + 0.867909i \(0.665463\pi\)
\(798\) 0 0
\(799\) −11.7057 −0.414117
\(800\) 0 0
\(801\) 9.47785 0.334884
\(802\) 0 0
\(803\) 5.20827 0.183796
\(804\) 0 0
\(805\) −0.386665 −0.0136282
\(806\) 0 0
\(807\) −10.7910 −0.379860
\(808\) 0 0
\(809\) 45.4593 1.59826 0.799132 0.601155i \(-0.205293\pi\)
0.799132 + 0.601155i \(0.205293\pi\)
\(810\) 0 0
\(811\) −41.6591 −1.46285 −0.731425 0.681922i \(-0.761144\pi\)
−0.731425 + 0.681922i \(0.761144\pi\)
\(812\) 0 0
\(813\) 5.46497 0.191665
\(814\) 0 0
\(815\) 1.86670 0.0653878
\(816\) 0 0
\(817\) −9.03626 −0.316139
\(818\) 0 0
\(819\) −14.4897 −0.506310
\(820\) 0 0
\(821\) −28.1607 −0.982815 −0.491407 0.870930i \(-0.663517\pi\)
−0.491407 + 0.870930i \(0.663517\pi\)
\(822\) 0 0
\(823\) −35.5358 −1.23870 −0.619350 0.785115i \(-0.712604\pi\)
−0.619350 + 0.785115i \(0.712604\pi\)
\(824\) 0 0
\(825\) −16.6755 −0.580567
\(826\) 0 0
\(827\) −1.01185 −0.0351856 −0.0175928 0.999845i \(-0.505600\pi\)
−0.0175928 + 0.999845i \(0.505600\pi\)
\(828\) 0 0
\(829\) −49.5249 −1.72007 −0.860036 0.510234i \(-0.829559\pi\)
−0.860036 + 0.510234i \(0.829559\pi\)
\(830\) 0 0
\(831\) −19.5186 −0.677093
\(832\) 0 0
\(833\) −11.7304 −0.406435
\(834\) 0 0
\(835\) −2.61937 −0.0906469
\(836\) 0 0
\(837\) 4.48181 0.154914
\(838\) 0 0
\(839\) −22.5539 −0.778649 −0.389324 0.921101i \(-0.627291\pi\)
−0.389324 + 0.921101i \(0.627291\pi\)
\(840\) 0 0
\(841\) 1.00000 0.0344828
\(842\) 0 0
\(843\) 22.9300 0.789752
\(844\) 0 0
\(845\) −1.05927 −0.0364401
\(846\) 0 0
\(847\) 0.597716 0.0205378
\(848\) 0 0
\(849\) −25.4692 −0.874101
\(850\) 0 0
\(851\) −5.19624 −0.178125
\(852\) 0 0
\(853\) −4.68276 −0.160335 −0.0801673 0.996781i \(-0.525545\pi\)
−0.0801673 + 0.996781i \(0.525545\pi\)
\(854\) 0 0
\(855\) 0.293481 0.0100369
\(856\) 0 0
\(857\) 31.1764 1.06497 0.532483 0.846441i \(-0.321259\pi\)
0.532483 + 0.846441i \(0.321259\pi\)
\(858\) 0 0
\(859\) −54.1840 −1.84873 −0.924367 0.381504i \(-0.875406\pi\)
−0.924367 + 0.381504i \(0.875406\pi\)
\(860\) 0 0
\(861\) 24.8602 0.847233
\(862\) 0 0
\(863\) −28.3776 −0.965986 −0.482993 0.875624i \(-0.660450\pi\)
−0.482993 + 0.875624i \(0.660450\pi\)
\(864\) 0 0
\(865\) −0.148393 −0.00504550
\(866\) 0 0
\(867\) −1.29952 −0.0441340
\(868\) 0 0
\(869\) −31.0861 −1.05452
\(870\) 0 0
\(871\) −63.3561 −2.14674
\(872\) 0 0
\(873\) 11.4680 0.388131
\(874\) 0 0
\(875\) 3.86072 0.130516
\(876\) 0 0
\(877\) 34.9027 1.17858 0.589290 0.807921i \(-0.299407\pi\)
0.589290 + 0.807921i \(0.299407\pi\)
\(878\) 0 0
\(879\) 14.1564 0.477483
\(880\) 0 0
\(881\) 1.50481 0.0506983 0.0253491 0.999679i \(-0.491930\pi\)
0.0253491 + 0.999679i \(0.491930\pi\)
\(882\) 0 0
\(883\) −52.3603 −1.76206 −0.881031 0.473058i \(-0.843150\pi\)
−0.881031 + 0.473058i \(0.843150\pi\)
\(884\) 0 0
\(885\) 1.46237 0.0491569
\(886\) 0 0
\(887\) 45.6573 1.53302 0.766510 0.642232i \(-0.221991\pi\)
0.766510 + 0.642232i \(0.221991\pi\)
\(888\) 0 0
\(889\) −44.1824 −1.48183
\(890\) 0 0
\(891\) −3.34537 −0.112074
\(892\) 0 0
\(893\) −6.48262 −0.216933
\(894\) 0 0
\(895\) 0.466869 0.0156057
\(896\) 0 0
\(897\) 4.64227 0.155001
\(898\) 0 0
\(899\) −4.48181 −0.149477
\(900\) 0 0
\(901\) −50.8206 −1.69308
\(902\) 0 0
\(903\) −11.9054 −0.396186
\(904\) 0 0
\(905\) −3.25192 −0.108097
\(906\) 0 0
\(907\) −42.4726 −1.41028 −0.705140 0.709068i \(-0.749116\pi\)
−0.705140 + 0.709068i \(0.749116\pi\)
\(908\) 0 0
\(909\) 5.00291 0.165936
\(910\) 0 0
\(911\) −13.9406 −0.461874 −0.230937 0.972969i \(-0.574179\pi\)
−0.230937 + 0.972969i \(0.574179\pi\)
\(912\) 0 0
\(913\) −53.5464 −1.77213
\(914\) 0 0
\(915\) 0.160022 0.00529015
\(916\) 0 0
\(917\) −43.0821 −1.42270
\(918\) 0 0
\(919\) −16.5900 −0.547253 −0.273627 0.961836i \(-0.588223\pi\)
−0.273627 + 0.961836i \(0.588223\pi\)
\(920\) 0 0
\(921\) 24.3935 0.803793
\(922\) 0 0
\(923\) 67.3743 2.21765
\(924\) 0 0
\(925\) 25.9015 0.851635
\(926\) 0 0
\(927\) 6.43917 0.211490
\(928\) 0 0
\(929\) −34.0242 −1.11630 −0.558148 0.829741i \(-0.688488\pi\)
−0.558148 + 0.829741i \(0.688488\pi\)
\(930\) 0 0
\(931\) −6.49632 −0.212908
\(932\) 0 0
\(933\) 6.36759 0.208466
\(934\) 0 0
\(935\) −1.77285 −0.0579783
\(936\) 0 0
\(937\) −32.1050 −1.04882 −0.524412 0.851465i \(-0.675715\pi\)
−0.524412 + 0.851465i \(0.675715\pi\)
\(938\) 0 0
\(939\) 28.0338 0.914849
\(940\) 0 0
\(941\) 8.01141 0.261165 0.130582 0.991437i \(-0.458315\pi\)
0.130582 + 0.991437i \(0.458315\pi\)
\(942\) 0 0
\(943\) −7.96483 −0.259371
\(944\) 0 0
\(945\) 0.386665 0.0125782
\(946\) 0 0
\(947\) −18.6384 −0.605668 −0.302834 0.953043i \(-0.597933\pi\)
−0.302834 + 0.953043i \(0.597933\pi\)
\(948\) 0 0
\(949\) 7.22736 0.234610
\(950\) 0 0
\(951\) −8.70755 −0.282362
\(952\) 0 0
\(953\) −24.8735 −0.805733 −0.402867 0.915259i \(-0.631986\pi\)
−0.402867 + 0.915259i \(0.631986\pi\)
\(954\) 0 0
\(955\) −1.17047 −0.0378757
\(956\) 0 0
\(957\) 3.34537 0.108140
\(958\) 0 0
\(959\) 48.6657 1.57150
\(960\) 0 0
\(961\) −10.9134 −0.352046
\(962\) 0 0
\(963\) 4.23096 0.136341
\(964\) 0 0
\(965\) 2.41983 0.0778970
\(966\) 0 0
\(967\) 20.7833 0.668347 0.334173 0.942512i \(-0.391543\pi\)
0.334173 + 0.942512i \(0.391543\pi\)
\(968\) 0 0
\(969\) −10.1343 −0.325560
\(970\) 0 0
\(971\) −25.5827 −0.820988 −0.410494 0.911863i \(-0.634644\pi\)
−0.410494 + 0.911863i \(0.634644\pi\)
\(972\) 0 0
\(973\) 4.57209 0.146575
\(974\) 0 0
\(975\) −23.1401 −0.741077
\(976\) 0 0
\(977\) 50.3427 1.61060 0.805302 0.592865i \(-0.202003\pi\)
0.805302 + 0.592865i \(0.202003\pi\)
\(978\) 0 0
\(979\) −31.7069 −1.01336
\(980\) 0 0
\(981\) 3.01374 0.0962213
\(982\) 0 0
\(983\) −4.80568 −0.153277 −0.0766387 0.997059i \(-0.524419\pi\)
−0.0766387 + 0.997059i \(0.524419\pi\)
\(984\) 0 0
\(985\) 1.99762 0.0636496
\(986\) 0 0
\(987\) −8.54092 −0.271861
\(988\) 0 0
\(989\) 3.81430 0.121288
\(990\) 0 0
\(991\) 55.5752 1.76540 0.882701 0.469934i \(-0.155722\pi\)
0.882701 + 0.469934i \(0.155722\pi\)
\(992\) 0 0
\(993\) 24.5067 0.777698
\(994\) 0 0
\(995\) 0.0974733 0.00309011
\(996\) 0 0
\(997\) 39.5059 1.25117 0.625583 0.780158i \(-0.284861\pi\)
0.625583 + 0.780158i \(0.284861\pi\)
\(998\) 0 0
\(999\) 5.19624 0.164402
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 8004.2.a.i.1.7 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
8004.2.a.i.1.7 16 1.1 even 1 trivial