Properties

Label 3856.2.a.j.1.3
Level $3856$
Weight $2$
Character 3856.1
Self dual yes
Analytic conductor $30.790$
Analytic rank $0$
Dimension $7$
CM no
Inner twists $1$

Related objects

Downloads

Learn more about

Newspace parameters

Level: \( N \) \(=\) \( 3856 = 2^{4} \cdot 241 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3856.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(30.7903150194\)
Analytic rank: \(0\)
Dimension: \(7\)
Coefficient field: 7.7.31056073.1
Defining polynomial: \(x^{7} - 3 x^{6} - 3 x^{5} + 11 x^{4} + x^{3} - 9 x^{2} + 1\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 241)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(-0.911223\) of defining polynomial
Character \(\chi\) \(=\) 3856.1

$q$-expansion

\(f(q)\) \(=\) \(q+0.186202 q^{3} -2.25110 q^{5} -3.52970 q^{7} -2.96533 q^{9} +O(q^{10})\) \(q+0.186202 q^{3} -2.25110 q^{5} -3.52970 q^{7} -2.96533 q^{9} -0.515564 q^{11} -5.38098 q^{13} -0.419160 q^{15} -4.16566 q^{17} -4.92935 q^{19} -0.657237 q^{21} +7.69193 q^{23} +0.0674429 q^{25} -1.11076 q^{27} -8.93755 q^{29} +4.43182 q^{31} -0.0959992 q^{33} +7.94569 q^{35} +5.99816 q^{37} -1.00195 q^{39} -8.99946 q^{41} +1.66336 q^{43} +6.67525 q^{45} -8.55484 q^{47} +5.45876 q^{49} -0.775656 q^{51} +13.1736 q^{53} +1.16059 q^{55} -0.917857 q^{57} -9.25521 q^{59} -10.4203 q^{61} +10.4667 q^{63} +12.1131 q^{65} +3.91715 q^{67} +1.43226 q^{69} +13.6724 q^{71} +11.5529 q^{73} +0.0125580 q^{75} +1.81979 q^{77} +1.43448 q^{79} +8.68916 q^{81} -1.73047 q^{83} +9.37732 q^{85} -1.66419 q^{87} -1.07999 q^{89} +18.9932 q^{91} +0.825214 q^{93} +11.0965 q^{95} -16.0883 q^{97} +1.52882 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 7q + 3q^{3} - 8q^{5} + 7q^{7} - 2q^{9} + O(q^{10}) \) \( 7q + 3q^{3} - 8q^{5} + 7q^{7} - 2q^{9} + 18q^{11} - q^{13} + 11q^{15} - 2q^{17} + 6q^{19} - 2q^{21} + 22q^{23} + 5q^{25} - 3q^{27} - 16q^{29} + 18q^{31} + 4q^{33} - 7q^{35} + 8q^{37} + 9q^{39} - 15q^{41} - 14q^{43} + 3q^{45} + 10q^{47} + 6q^{49} - 13q^{51} + 15q^{53} - 29q^{55} + 14q^{57} + 18q^{59} + 4q^{61} + 16q^{63} - 7q^{65} - 18q^{67} + 26q^{69} + 50q^{71} - 16q^{75} + 17q^{77} + 15q^{79} - 9q^{81} + 24q^{83} - 2q^{85} - 12q^{87} - 13q^{89} + 12q^{91} + 14q^{93} + 41q^{95} + q^{97} + 20q^{99} + O(q^{100}) \)

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
\(3\) 0.186202 0.107504 0.0537520 0.998554i \(-0.482882\pi\)
0.0537520 + 0.998554i \(0.482882\pi\)
\(4\) 0 0
\(5\) −2.25110 −1.00672 −0.503361 0.864076i \(-0.667903\pi\)
−0.503361 + 0.864076i \(0.667903\pi\)
\(6\) 0 0
\(7\) −3.52970 −1.33410 −0.667050 0.745013i \(-0.732443\pi\)
−0.667050 + 0.745013i \(0.732443\pi\)
\(8\) 0 0
\(9\) −2.96533 −0.988443
\(10\) 0 0
\(11\) −0.515564 −0.155448 −0.0777242 0.996975i \(-0.524765\pi\)
−0.0777242 + 0.996975i \(0.524765\pi\)
\(12\) 0 0
\(13\) −5.38098 −1.49242 −0.746208 0.665712i \(-0.768128\pi\)
−0.746208 + 0.665712i \(0.768128\pi\)
\(14\) 0 0
\(15\) −0.419160 −0.108227
\(16\) 0 0
\(17\) −4.16566 −1.01032 −0.505161 0.863025i \(-0.668567\pi\)
−0.505161 + 0.863025i \(0.668567\pi\)
\(18\) 0 0
\(19\) −4.92935 −1.13087 −0.565436 0.824792i \(-0.691292\pi\)
−0.565436 + 0.824792i \(0.691292\pi\)
\(20\) 0 0
\(21\) −0.657237 −0.143421
\(22\) 0 0
\(23\) 7.69193 1.60388 0.801940 0.597405i \(-0.203802\pi\)
0.801940 + 0.597405i \(0.203802\pi\)
\(24\) 0 0
\(25\) 0.0674429 0.0134886
\(26\) 0 0
\(27\) −1.11076 −0.213765
\(28\) 0 0
\(29\) −8.93755 −1.65966 −0.829831 0.558015i \(-0.811563\pi\)
−0.829831 + 0.558015i \(0.811563\pi\)
\(30\) 0 0
\(31\) 4.43182 0.795978 0.397989 0.917390i \(-0.369708\pi\)
0.397989 + 0.917390i \(0.369708\pi\)
\(32\) 0 0
\(33\) −0.0959992 −0.0167113
\(34\) 0 0
\(35\) 7.94569 1.34307
\(36\) 0 0
\(37\) 5.99816 0.986091 0.493046 0.870003i \(-0.335884\pi\)
0.493046 + 0.870003i \(0.335884\pi\)
\(38\) 0 0
\(39\) −1.00195 −0.160441
\(40\) 0 0
\(41\) −8.99946 −1.40548 −0.702740 0.711447i \(-0.748040\pi\)
−0.702740 + 0.711447i \(0.748040\pi\)
\(42\) 0 0
\(43\) 1.66336 0.253660 0.126830 0.991924i \(-0.459520\pi\)
0.126830 + 0.991924i \(0.459520\pi\)
\(44\) 0 0
\(45\) 6.67525 0.995087
\(46\) 0 0
\(47\) −8.55484 −1.24785 −0.623926 0.781483i \(-0.714463\pi\)
−0.623926 + 0.781483i \(0.714463\pi\)
\(48\) 0 0
\(49\) 5.45876 0.779823
\(50\) 0 0
\(51\) −0.775656 −0.108614
\(52\) 0 0
\(53\) 13.1736 1.80954 0.904769 0.425903i \(-0.140043\pi\)
0.904769 + 0.425903i \(0.140043\pi\)
\(54\) 0 0
\(55\) 1.16059 0.156493
\(56\) 0 0
\(57\) −0.917857 −0.121573
\(58\) 0 0
\(59\) −9.25521 −1.20493 −0.602463 0.798147i \(-0.705814\pi\)
−0.602463 + 0.798147i \(0.705814\pi\)
\(60\) 0 0
\(61\) −10.4203 −1.33418 −0.667090 0.744978i \(-0.732460\pi\)
−0.667090 + 0.744978i \(0.732460\pi\)
\(62\) 0 0
\(63\) 10.4667 1.31868
\(64\) 0 0
\(65\) 12.1131 1.50245
\(66\) 0 0
\(67\) 3.91715 0.478555 0.239278 0.970951i \(-0.423089\pi\)
0.239278 + 0.970951i \(0.423089\pi\)
\(68\) 0 0
\(69\) 1.43226 0.172423
\(70\) 0 0
\(71\) 13.6724 1.62262 0.811308 0.584619i \(-0.198756\pi\)
0.811308 + 0.584619i \(0.198756\pi\)
\(72\) 0 0
\(73\) 11.5529 1.35217 0.676083 0.736825i \(-0.263676\pi\)
0.676083 + 0.736825i \(0.263676\pi\)
\(74\) 0 0
\(75\) 0.0125580 0.00145007
\(76\) 0 0
\(77\) 1.81979 0.207384
\(78\) 0 0
\(79\) 1.43448 0.161391 0.0806956 0.996739i \(-0.474286\pi\)
0.0806956 + 0.996739i \(0.474286\pi\)
\(80\) 0 0
\(81\) 8.68916 0.965462
\(82\) 0 0
\(83\) −1.73047 −0.189944 −0.0949720 0.995480i \(-0.530276\pi\)
−0.0949720 + 0.995480i \(0.530276\pi\)
\(84\) 0 0
\(85\) 9.37732 1.01711
\(86\) 0 0
\(87\) −1.66419 −0.178420
\(88\) 0 0
\(89\) −1.07999 −0.114478 −0.0572392 0.998360i \(-0.518230\pi\)
−0.0572392 + 0.998360i \(0.518230\pi\)
\(90\) 0 0
\(91\) 18.9932 1.99103
\(92\) 0 0
\(93\) 0.825214 0.0855707
\(94\) 0 0
\(95\) 11.0965 1.13847
\(96\) 0 0
\(97\) −16.0883 −1.63352 −0.816759 0.576979i \(-0.804231\pi\)
−0.816759 + 0.576979i \(0.804231\pi\)
\(98\) 0 0
\(99\) 1.52882 0.153652
\(100\) 0 0
\(101\) −1.23922 −0.123307 −0.0616536 0.998098i \(-0.519637\pi\)
−0.0616536 + 0.998098i \(0.519637\pi\)
\(102\) 0 0
\(103\) −5.27633 −0.519892 −0.259946 0.965623i \(-0.583705\pi\)
−0.259946 + 0.965623i \(0.583705\pi\)
\(104\) 0 0
\(105\) 1.47951 0.144385
\(106\) 0 0
\(107\) −10.7822 −1.04236 −0.521178 0.853448i \(-0.674507\pi\)
−0.521178 + 0.853448i \(0.674507\pi\)
\(108\) 0 0
\(109\) −1.15255 −0.110394 −0.0551972 0.998475i \(-0.517579\pi\)
−0.0551972 + 0.998475i \(0.517579\pi\)
\(110\) 0 0
\(111\) 1.11687 0.106009
\(112\) 0 0
\(113\) −8.70999 −0.819367 −0.409684 0.912228i \(-0.634361\pi\)
−0.409684 + 0.912228i \(0.634361\pi\)
\(114\) 0 0
\(115\) −17.3153 −1.61466
\(116\) 0 0
\(117\) 15.9564 1.47517
\(118\) 0 0
\(119\) 14.7035 1.34787
\(120\) 0 0
\(121\) −10.7342 −0.975836
\(122\) 0 0
\(123\) −1.67572 −0.151095
\(124\) 0 0
\(125\) 11.1037 0.993142
\(126\) 0 0
\(127\) 8.22063 0.729463 0.364732 0.931113i \(-0.381161\pi\)
0.364732 + 0.931113i \(0.381161\pi\)
\(128\) 0 0
\(129\) 0.309722 0.0272695
\(130\) 0 0
\(131\) 7.30323 0.638086 0.319043 0.947740i \(-0.396638\pi\)
0.319043 + 0.947740i \(0.396638\pi\)
\(132\) 0 0
\(133\) 17.3991 1.50870
\(134\) 0 0
\(135\) 2.50042 0.215202
\(136\) 0 0
\(137\) 5.95317 0.508614 0.254307 0.967124i \(-0.418153\pi\)
0.254307 + 0.967124i \(0.418153\pi\)
\(138\) 0 0
\(139\) −14.9650 −1.26931 −0.634656 0.772795i \(-0.718858\pi\)
−0.634656 + 0.772795i \(0.718858\pi\)
\(140\) 0 0
\(141\) −1.59293 −0.134149
\(142\) 0 0
\(143\) 2.77424 0.231994
\(144\) 0 0
\(145\) 20.1193 1.67082
\(146\) 0 0
\(147\) 1.01643 0.0838340
\(148\) 0 0
\(149\) −0.130576 −0.0106972 −0.00534860 0.999986i \(-0.501703\pi\)
−0.00534860 + 0.999986i \(0.501703\pi\)
\(150\) 0 0
\(151\) −0.276102 −0.0224688 −0.0112344 0.999937i \(-0.503576\pi\)
−0.0112344 + 0.999937i \(0.503576\pi\)
\(152\) 0 0
\(153\) 12.3526 0.998645
\(154\) 0 0
\(155\) −9.97646 −0.801328
\(156\) 0 0
\(157\) 16.1044 1.28527 0.642636 0.766172i \(-0.277841\pi\)
0.642636 + 0.766172i \(0.277841\pi\)
\(158\) 0 0
\(159\) 2.45296 0.194532
\(160\) 0 0
\(161\) −27.1502 −2.13973
\(162\) 0 0
\(163\) 18.5215 1.45071 0.725357 0.688373i \(-0.241675\pi\)
0.725357 + 0.688373i \(0.241675\pi\)
\(164\) 0 0
\(165\) 0.216104 0.0168236
\(166\) 0 0
\(167\) −15.5017 −1.19956 −0.599779 0.800166i \(-0.704745\pi\)
−0.599779 + 0.800166i \(0.704745\pi\)
\(168\) 0 0
\(169\) 15.9550 1.22731
\(170\) 0 0
\(171\) 14.6172 1.11780
\(172\) 0 0
\(173\) −0.366438 −0.0278597 −0.0139299 0.999903i \(-0.504434\pi\)
−0.0139299 + 0.999903i \(0.504434\pi\)
\(174\) 0 0
\(175\) −0.238053 −0.0179951
\(176\) 0 0
\(177\) −1.72334 −0.129534
\(178\) 0 0
\(179\) −12.8719 −0.962090 −0.481045 0.876696i \(-0.659743\pi\)
−0.481045 + 0.876696i \(0.659743\pi\)
\(180\) 0 0
\(181\) −12.2125 −0.907746 −0.453873 0.891066i \(-0.649958\pi\)
−0.453873 + 0.891066i \(0.649958\pi\)
\(182\) 0 0
\(183\) −1.94028 −0.143429
\(184\) 0 0
\(185\) −13.5024 −0.992719
\(186\) 0 0
\(187\) 2.14767 0.157053
\(188\) 0 0
\(189\) 3.92064 0.285184
\(190\) 0 0
\(191\) 3.92400 0.283930 0.141965 0.989872i \(-0.454658\pi\)
0.141965 + 0.989872i \(0.454658\pi\)
\(192\) 0 0
\(193\) −1.69319 −0.121879 −0.0609393 0.998141i \(-0.519410\pi\)
−0.0609393 + 0.998141i \(0.519410\pi\)
\(194\) 0 0
\(195\) 2.25549 0.161519
\(196\) 0 0
\(197\) −2.20382 −0.157016 −0.0785080 0.996913i \(-0.525016\pi\)
−0.0785080 + 0.996913i \(0.525016\pi\)
\(198\) 0 0
\(199\) 15.5533 1.10255 0.551274 0.834324i \(-0.314142\pi\)
0.551274 + 0.834324i \(0.314142\pi\)
\(200\) 0 0
\(201\) 0.729381 0.0514466
\(202\) 0 0
\(203\) 31.5468 2.21415
\(204\) 0 0
\(205\) 20.2587 1.41493
\(206\) 0 0
\(207\) −22.8091 −1.58534
\(208\) 0 0
\(209\) 2.54140 0.175792
\(210\) 0 0
\(211\) −6.55996 −0.451606 −0.225803 0.974173i \(-0.572501\pi\)
−0.225803 + 0.974173i \(0.572501\pi\)
\(212\) 0 0
\(213\) 2.54583 0.174438
\(214\) 0 0
\(215\) −3.74439 −0.255365
\(216\) 0 0
\(217\) −15.6430 −1.06191
\(218\) 0 0
\(219\) 2.15118 0.145363
\(220\) 0 0
\(221\) 22.4154 1.50782
\(222\) 0 0
\(223\) −6.53800 −0.437817 −0.218908 0.975745i \(-0.570250\pi\)
−0.218908 + 0.975745i \(0.570250\pi\)
\(224\) 0 0
\(225\) −0.199990 −0.0133327
\(226\) 0 0
\(227\) −16.1673 −1.07306 −0.536530 0.843881i \(-0.680265\pi\)
−0.536530 + 0.843881i \(0.680265\pi\)
\(228\) 0 0
\(229\) 0.0208412 0.00137722 0.000688612 1.00000i \(-0.499781\pi\)
0.000688612 1.00000i \(0.499781\pi\)
\(230\) 0 0
\(231\) 0.338848 0.0222946
\(232\) 0 0
\(233\) −0.651845 −0.0427038 −0.0213519 0.999772i \(-0.506797\pi\)
−0.0213519 + 0.999772i \(0.506797\pi\)
\(234\) 0 0
\(235\) 19.2578 1.25624
\(236\) 0 0
\(237\) 0.267103 0.0173502
\(238\) 0 0
\(239\) 8.99997 0.582160 0.291080 0.956699i \(-0.405985\pi\)
0.291080 + 0.956699i \(0.405985\pi\)
\(240\) 0 0
\(241\) −1.00000 −0.0644157
\(242\) 0 0
\(243\) 4.95021 0.317556
\(244\) 0 0
\(245\) −12.2882 −0.785064
\(246\) 0 0
\(247\) 26.5248 1.68773
\(248\) 0 0
\(249\) −0.322218 −0.0204197
\(250\) 0 0
\(251\) −16.4986 −1.04138 −0.520691 0.853745i \(-0.674326\pi\)
−0.520691 + 0.853745i \(0.674326\pi\)
\(252\) 0 0
\(253\) −3.96569 −0.249321
\(254\) 0 0
\(255\) 1.74608 0.109344
\(256\) 0 0
\(257\) 7.99029 0.498420 0.249210 0.968449i \(-0.419829\pi\)
0.249210 + 0.968449i \(0.419829\pi\)
\(258\) 0 0
\(259\) −21.1717 −1.31554
\(260\) 0 0
\(261\) 26.5028 1.64048
\(262\) 0 0
\(263\) 5.47488 0.337595 0.168798 0.985651i \(-0.446012\pi\)
0.168798 + 0.985651i \(0.446012\pi\)
\(264\) 0 0
\(265\) −29.6551 −1.82170
\(266\) 0 0
\(267\) −0.201096 −0.0123069
\(268\) 0 0
\(269\) −2.34697 −0.143097 −0.0715486 0.997437i \(-0.522794\pi\)
−0.0715486 + 0.997437i \(0.522794\pi\)
\(270\) 0 0
\(271\) −1.60301 −0.0973758 −0.0486879 0.998814i \(-0.515504\pi\)
−0.0486879 + 0.998814i \(0.515504\pi\)
\(272\) 0 0
\(273\) 3.53658 0.214044
\(274\) 0 0
\(275\) −0.0347711 −0.00209678
\(276\) 0 0
\(277\) −21.6775 −1.30247 −0.651237 0.758875i \(-0.725750\pi\)
−0.651237 + 0.758875i \(0.725750\pi\)
\(278\) 0 0
\(279\) −13.1418 −0.786779
\(280\) 0 0
\(281\) 26.9010 1.60478 0.802388 0.596802i \(-0.203562\pi\)
0.802388 + 0.596802i \(0.203562\pi\)
\(282\) 0 0
\(283\) −3.89230 −0.231373 −0.115687 0.993286i \(-0.536907\pi\)
−0.115687 + 0.993286i \(0.536907\pi\)
\(284\) 0 0
\(285\) 2.06619 0.122390
\(286\) 0 0
\(287\) 31.7654 1.87505
\(288\) 0 0
\(289\) 0.352752 0.0207501
\(290\) 0 0
\(291\) −2.99567 −0.175610
\(292\) 0 0
\(293\) −5.57389 −0.325630 −0.162815 0.986657i \(-0.552057\pi\)
−0.162815 + 0.986657i \(0.552057\pi\)
\(294\) 0 0
\(295\) 20.8344 1.21302
\(296\) 0 0
\(297\) 0.572667 0.0332295
\(298\) 0 0
\(299\) −41.3902 −2.39366
\(300\) 0 0
\(301\) −5.87116 −0.338408
\(302\) 0 0
\(303\) −0.230746 −0.0132560
\(304\) 0 0
\(305\) 23.4571 1.34315
\(306\) 0 0
\(307\) −23.0795 −1.31722 −0.658608 0.752486i \(-0.728854\pi\)
−0.658608 + 0.752486i \(0.728854\pi\)
\(308\) 0 0
\(309\) −0.982465 −0.0558905
\(310\) 0 0
\(311\) 14.9884 0.849916 0.424958 0.905213i \(-0.360289\pi\)
0.424958 + 0.905213i \(0.360289\pi\)
\(312\) 0 0
\(313\) −2.31082 −0.130615 −0.0653076 0.997865i \(-0.520803\pi\)
−0.0653076 + 0.997865i \(0.520803\pi\)
\(314\) 0 0
\(315\) −23.5616 −1.32755
\(316\) 0 0
\(317\) −18.3630 −1.03137 −0.515684 0.856779i \(-0.672462\pi\)
−0.515684 + 0.856779i \(0.672462\pi\)
\(318\) 0 0
\(319\) 4.60788 0.257992
\(320\) 0 0
\(321\) −2.00767 −0.112057
\(322\) 0 0
\(323\) 20.5340 1.14254
\(324\) 0 0
\(325\) −0.362909 −0.0201306
\(326\) 0 0
\(327\) −0.214608 −0.0118678
\(328\) 0 0
\(329\) 30.1960 1.66476
\(330\) 0 0
\(331\) 6.47933 0.356136 0.178068 0.984018i \(-0.443015\pi\)
0.178068 + 0.984018i \(0.443015\pi\)
\(332\) 0 0
\(333\) −17.7865 −0.974695
\(334\) 0 0
\(335\) −8.81788 −0.481772
\(336\) 0 0
\(337\) −3.54257 −0.192976 −0.0964880 0.995334i \(-0.530761\pi\)
−0.0964880 + 0.995334i \(0.530761\pi\)
\(338\) 0 0
\(339\) −1.62182 −0.0880852
\(340\) 0 0
\(341\) −2.28489 −0.123734
\(342\) 0 0
\(343\) 5.44011 0.293739
\(344\) 0 0
\(345\) −3.22415 −0.173582
\(346\) 0 0
\(347\) 14.4844 0.777566 0.388783 0.921329i \(-0.372896\pi\)
0.388783 + 0.921329i \(0.372896\pi\)
\(348\) 0 0
\(349\) −23.6287 −1.26482 −0.632408 0.774636i \(-0.717933\pi\)
−0.632408 + 0.774636i \(0.717933\pi\)
\(350\) 0 0
\(351\) 5.97697 0.319027
\(352\) 0 0
\(353\) −18.2662 −0.972209 −0.486105 0.873901i \(-0.661583\pi\)
−0.486105 + 0.873901i \(0.661583\pi\)
\(354\) 0 0
\(355\) −30.7779 −1.63352
\(356\) 0 0
\(357\) 2.73783 0.144901
\(358\) 0 0
\(359\) 20.3758 1.07539 0.537697 0.843138i \(-0.319294\pi\)
0.537697 + 0.843138i \(0.319294\pi\)
\(360\) 0 0
\(361\) 5.29852 0.278870
\(362\) 0 0
\(363\) −1.99873 −0.104906
\(364\) 0 0
\(365\) −26.0067 −1.36125
\(366\) 0 0
\(367\) 1.76412 0.0920862 0.0460431 0.998939i \(-0.485339\pi\)
0.0460431 + 0.998939i \(0.485339\pi\)
\(368\) 0 0
\(369\) 26.6864 1.38924
\(370\) 0 0
\(371\) −46.4989 −2.41410
\(372\) 0 0
\(373\) −32.1736 −1.66589 −0.832943 0.553359i \(-0.813346\pi\)
−0.832943 + 0.553359i \(0.813346\pi\)
\(374\) 0 0
\(375\) 2.06753 0.106767
\(376\) 0 0
\(377\) 48.0928 2.47691
\(378\) 0 0
\(379\) 31.3680 1.61126 0.805632 0.592416i \(-0.201826\pi\)
0.805632 + 0.592416i \(0.201826\pi\)
\(380\) 0 0
\(381\) 1.53070 0.0784201
\(382\) 0 0
\(383\) −16.8623 −0.861624 −0.430812 0.902442i \(-0.641773\pi\)
−0.430812 + 0.902442i \(0.641773\pi\)
\(384\) 0 0
\(385\) −4.09651 −0.208778
\(386\) 0 0
\(387\) −4.93242 −0.250729
\(388\) 0 0
\(389\) −0.816001 −0.0413729 −0.0206864 0.999786i \(-0.506585\pi\)
−0.0206864 + 0.999786i \(0.506585\pi\)
\(390\) 0 0
\(391\) −32.0420 −1.62043
\(392\) 0 0
\(393\) 1.35988 0.0685968
\(394\) 0 0
\(395\) −3.22915 −0.162476
\(396\) 0 0
\(397\) −21.8279 −1.09551 −0.547755 0.836639i \(-0.684517\pi\)
−0.547755 + 0.836639i \(0.684517\pi\)
\(398\) 0 0
\(399\) 3.23976 0.162191
\(400\) 0 0
\(401\) −15.4857 −0.773321 −0.386661 0.922222i \(-0.626372\pi\)
−0.386661 + 0.922222i \(0.626372\pi\)
\(402\) 0 0
\(403\) −23.8475 −1.18793
\(404\) 0 0
\(405\) −19.5602 −0.971952
\(406\) 0 0
\(407\) −3.09244 −0.153286
\(408\) 0 0
\(409\) 26.0377 1.28748 0.643740 0.765245i \(-0.277382\pi\)
0.643740 + 0.765245i \(0.277382\pi\)
\(410\) 0 0
\(411\) 1.10849 0.0546780
\(412\) 0 0
\(413\) 32.6681 1.60749
\(414\) 0 0
\(415\) 3.89546 0.191221
\(416\) 0 0
\(417\) −2.78651 −0.136456
\(418\) 0 0
\(419\) 33.2204 1.62292 0.811462 0.584406i \(-0.198672\pi\)
0.811462 + 0.584406i \(0.198672\pi\)
\(420\) 0 0
\(421\) 2.90586 0.141623 0.0708115 0.997490i \(-0.477441\pi\)
0.0708115 + 0.997490i \(0.477441\pi\)
\(422\) 0 0
\(423\) 25.3679 1.23343
\(424\) 0 0
\(425\) −0.280944 −0.0136278
\(426\) 0 0
\(427\) 36.7804 1.77993
\(428\) 0 0
\(429\) 0.516570 0.0249402
\(430\) 0 0
\(431\) 28.4892 1.37228 0.686139 0.727471i \(-0.259304\pi\)
0.686139 + 0.727471i \(0.259304\pi\)
\(432\) 0 0
\(433\) −28.9311 −1.39034 −0.695169 0.718846i \(-0.744671\pi\)
−0.695169 + 0.718846i \(0.744671\pi\)
\(434\) 0 0
\(435\) 3.74626 0.179619
\(436\) 0 0
\(437\) −37.9163 −1.81378
\(438\) 0 0
\(439\) 8.92156 0.425803 0.212901 0.977074i \(-0.431709\pi\)
0.212901 + 0.977074i \(0.431709\pi\)
\(440\) 0 0
\(441\) −16.1870 −0.770810
\(442\) 0 0
\(443\) −24.8216 −1.17931 −0.589656 0.807655i \(-0.700736\pi\)
−0.589656 + 0.807655i \(0.700736\pi\)
\(444\) 0 0
\(445\) 2.43116 0.115248
\(446\) 0 0
\(447\) −0.0243136 −0.00114999
\(448\) 0 0
\(449\) 11.9254 0.562793 0.281397 0.959592i \(-0.409202\pi\)
0.281397 + 0.959592i \(0.409202\pi\)
\(450\) 0 0
\(451\) 4.63980 0.218480
\(452\) 0 0
\(453\) −0.0514107 −0.00241549
\(454\) 0 0
\(455\) −42.7557 −2.00442
\(456\) 0 0
\(457\) 8.44205 0.394902 0.197451 0.980313i \(-0.436734\pi\)
0.197451 + 0.980313i \(0.436734\pi\)
\(458\) 0 0
\(459\) 4.62704 0.215972
\(460\) 0 0
\(461\) 23.2925 1.08484 0.542419 0.840108i \(-0.317509\pi\)
0.542419 + 0.840108i \(0.317509\pi\)
\(462\) 0 0
\(463\) −1.66578 −0.0774152 −0.0387076 0.999251i \(-0.512324\pi\)
−0.0387076 + 0.999251i \(0.512324\pi\)
\(464\) 0 0
\(465\) −1.85764 −0.0861459
\(466\) 0 0
\(467\) 6.66428 0.308386 0.154193 0.988041i \(-0.450722\pi\)
0.154193 + 0.988041i \(0.450722\pi\)
\(468\) 0 0
\(469\) −13.8263 −0.638441
\(470\) 0 0
\(471\) 2.99868 0.138172
\(472\) 0 0
\(473\) −0.857570 −0.0394311
\(474\) 0 0
\(475\) −0.332450 −0.0152538
\(476\) 0 0
\(477\) −39.0642 −1.78863
\(478\) 0 0
\(479\) 13.6902 0.625523 0.312762 0.949832i \(-0.398746\pi\)
0.312762 + 0.949832i \(0.398746\pi\)
\(480\) 0 0
\(481\) −32.2760 −1.47166
\(482\) 0 0
\(483\) −5.05543 −0.230030
\(484\) 0 0
\(485\) 36.2163 1.64450
\(486\) 0 0
\(487\) −31.9634 −1.44840 −0.724200 0.689590i \(-0.757791\pi\)
−0.724200 + 0.689590i \(0.757791\pi\)
\(488\) 0 0
\(489\) 3.44874 0.155957
\(490\) 0 0
\(491\) 10.0550 0.453777 0.226889 0.973921i \(-0.427145\pi\)
0.226889 + 0.973921i \(0.427145\pi\)
\(492\) 0 0
\(493\) 37.2308 1.67679
\(494\) 0 0
\(495\) −3.44152 −0.154685
\(496\) 0 0
\(497\) −48.2594 −2.16473
\(498\) 0 0
\(499\) −9.14295 −0.409295 −0.204647 0.978836i \(-0.565605\pi\)
−0.204647 + 0.978836i \(0.565605\pi\)
\(500\) 0 0
\(501\) −2.88645 −0.128957
\(502\) 0 0
\(503\) 27.2050 1.21301 0.606506 0.795079i \(-0.292571\pi\)
0.606506 + 0.795079i \(0.292571\pi\)
\(504\) 0 0
\(505\) 2.78961 0.124136
\(506\) 0 0
\(507\) 2.97086 0.131940
\(508\) 0 0
\(509\) 7.79598 0.345551 0.172775 0.984961i \(-0.444726\pi\)
0.172775 + 0.984961i \(0.444726\pi\)
\(510\) 0 0
\(511\) −40.7783 −1.80392
\(512\) 0 0
\(513\) 5.47532 0.241741
\(514\) 0 0
\(515\) 11.8775 0.523387
\(516\) 0 0
\(517\) 4.41057 0.193977
\(518\) 0 0
\(519\) −0.0682315 −0.00299503
\(520\) 0 0
\(521\) 32.6959 1.43243 0.716216 0.697878i \(-0.245872\pi\)
0.716216 + 0.697878i \(0.245872\pi\)
\(522\) 0 0
\(523\) 7.80136 0.341129 0.170565 0.985346i \(-0.445441\pi\)
0.170565 + 0.985346i \(0.445441\pi\)
\(524\) 0 0
\(525\) −0.0443260 −0.00193454
\(526\) 0 0
\(527\) −18.4615 −0.804194
\(528\) 0 0
\(529\) 36.1658 1.57243
\(530\) 0 0
\(531\) 27.4447 1.19100
\(532\) 0 0
\(533\) 48.4260 2.09756
\(534\) 0 0
\(535\) 24.2718 1.04936
\(536\) 0 0
\(537\) −2.39677 −0.103428
\(538\) 0 0
\(539\) −2.81434 −0.121222
\(540\) 0 0
\(541\) 22.3971 0.962926 0.481463 0.876467i \(-0.340106\pi\)
0.481463 + 0.876467i \(0.340106\pi\)
\(542\) 0 0
\(543\) −2.27399 −0.0975863
\(544\) 0 0
\(545\) 2.59451 0.111136
\(546\) 0 0
\(547\) −37.2322 −1.59193 −0.795966 0.605341i \(-0.793037\pi\)
−0.795966 + 0.605341i \(0.793037\pi\)
\(548\) 0 0
\(549\) 30.8995 1.31876
\(550\) 0 0
\(551\) 44.0563 1.87686
\(552\) 0 0
\(553\) −5.06326 −0.215312
\(554\) 0 0
\(555\) −2.51419 −0.106721
\(556\) 0 0
\(557\) −27.6457 −1.17139 −0.585693 0.810533i \(-0.699177\pi\)
−0.585693 + 0.810533i \(0.699177\pi\)
\(558\) 0 0
\(559\) −8.95053 −0.378567
\(560\) 0 0
\(561\) 0.399900 0.0168838
\(562\) 0 0
\(563\) 23.3454 0.983891 0.491945 0.870626i \(-0.336286\pi\)
0.491945 + 0.870626i \(0.336286\pi\)
\(564\) 0 0
\(565\) 19.6070 0.824875
\(566\) 0 0
\(567\) −30.6701 −1.28802
\(568\) 0 0
\(569\) 2.61346 0.109562 0.0547811 0.998498i \(-0.482554\pi\)
0.0547811 + 0.998498i \(0.482554\pi\)
\(570\) 0 0
\(571\) −16.9259 −0.708326 −0.354163 0.935184i \(-0.615234\pi\)
−0.354163 + 0.935184i \(0.615234\pi\)
\(572\) 0 0
\(573\) 0.730657 0.0305236
\(574\) 0 0
\(575\) 0.518766 0.0216340
\(576\) 0 0
\(577\) 4.67534 0.194637 0.0973185 0.995253i \(-0.468973\pi\)
0.0973185 + 0.995253i \(0.468973\pi\)
\(578\) 0 0
\(579\) −0.315276 −0.0131024
\(580\) 0 0
\(581\) 6.10804 0.253404
\(582\) 0 0
\(583\) −6.79185 −0.281290
\(584\) 0 0
\(585\) −35.9194 −1.48508
\(586\) 0 0
\(587\) 3.78940 0.156405 0.0782027 0.996937i \(-0.475082\pi\)
0.0782027 + 0.996937i \(0.475082\pi\)
\(588\) 0 0
\(589\) −21.8460 −0.900148
\(590\) 0 0
\(591\) −0.410357 −0.0168798
\(592\) 0 0
\(593\) −6.81434 −0.279831 −0.139916 0.990163i \(-0.544683\pi\)
−0.139916 + 0.990163i \(0.544683\pi\)
\(594\) 0 0
\(595\) −33.0991 −1.35693
\(596\) 0 0
\(597\) 2.89607 0.118528
\(598\) 0 0
\(599\) 39.7739 1.62512 0.812559 0.582879i \(-0.198074\pi\)
0.812559 + 0.582879i \(0.198074\pi\)
\(600\) 0 0
\(601\) 27.4256 1.11871 0.559356 0.828927i \(-0.311048\pi\)
0.559356 + 0.828927i \(0.311048\pi\)
\(602\) 0 0
\(603\) −11.6156 −0.473025
\(604\) 0 0
\(605\) 24.1637 0.982395
\(606\) 0 0
\(607\) −20.6168 −0.836811 −0.418405 0.908260i \(-0.637411\pi\)
−0.418405 + 0.908260i \(0.637411\pi\)
\(608\) 0 0
\(609\) 5.87409 0.238030
\(610\) 0 0
\(611\) 46.0335 1.86232
\(612\) 0 0
\(613\) −12.9127 −0.521540 −0.260770 0.965401i \(-0.583976\pi\)
−0.260770 + 0.965401i \(0.583976\pi\)
\(614\) 0 0
\(615\) 3.77221 0.152110
\(616\) 0 0
\(617\) 26.3856 1.06225 0.531123 0.847295i \(-0.321770\pi\)
0.531123 + 0.847295i \(0.321770\pi\)
\(618\) 0 0
\(619\) 2.30850 0.0927866 0.0463933 0.998923i \(-0.485227\pi\)
0.0463933 + 0.998923i \(0.485227\pi\)
\(620\) 0 0
\(621\) −8.54387 −0.342854
\(622\) 0 0
\(623\) 3.81203 0.152726
\(624\) 0 0
\(625\) −25.3327 −1.01331
\(626\) 0 0
\(627\) 0.473214 0.0188983
\(628\) 0 0
\(629\) −24.9863 −0.996270
\(630\) 0 0
\(631\) 8.60820 0.342687 0.171343 0.985211i \(-0.445189\pi\)
0.171343 + 0.985211i \(0.445189\pi\)
\(632\) 0 0
\(633\) −1.22148 −0.0485494
\(634\) 0 0
\(635\) −18.5054 −0.734366
\(636\) 0 0
\(637\) −29.3735 −1.16382
\(638\) 0 0
\(639\) −40.5432 −1.60386
\(640\) 0 0
\(641\) −11.7479 −0.464014 −0.232007 0.972714i \(-0.574529\pi\)
−0.232007 + 0.972714i \(0.574529\pi\)
\(642\) 0 0
\(643\) −15.0808 −0.594729 −0.297364 0.954764i \(-0.596108\pi\)
−0.297364 + 0.954764i \(0.596108\pi\)
\(644\) 0 0
\(645\) −0.697214 −0.0274528
\(646\) 0 0
\(647\) 5.68014 0.223309 0.111655 0.993747i \(-0.464385\pi\)
0.111655 + 0.993747i \(0.464385\pi\)
\(648\) 0 0
\(649\) 4.77166 0.187304
\(650\) 0 0
\(651\) −2.91276 −0.114160
\(652\) 0 0
\(653\) −48.9941 −1.91729 −0.958644 0.284608i \(-0.908137\pi\)
−0.958644 + 0.284608i \(0.908137\pi\)
\(654\) 0 0
\(655\) −16.4403 −0.642375
\(656\) 0 0
\(657\) −34.2582 −1.33654
\(658\) 0 0
\(659\) 23.5789 0.918505 0.459252 0.888306i \(-0.348117\pi\)
0.459252 + 0.888306i \(0.348117\pi\)
\(660\) 0 0
\(661\) −38.6322 −1.50262 −0.751310 0.659949i \(-0.770578\pi\)
−0.751310 + 0.659949i \(0.770578\pi\)
\(662\) 0 0
\(663\) 4.17379 0.162097
\(664\) 0 0
\(665\) −39.1671 −1.51884
\(666\) 0 0
\(667\) −68.7470 −2.66190
\(668\) 0 0
\(669\) −1.21739 −0.0470670
\(670\) 0 0
\(671\) 5.37232 0.207396
\(672\) 0 0
\(673\) −37.4530 −1.44371 −0.721853 0.692047i \(-0.756709\pi\)
−0.721853 + 0.692047i \(0.756709\pi\)
\(674\) 0 0
\(675\) −0.0749127 −0.00288339
\(676\) 0 0
\(677\) 16.0616 0.617297 0.308649 0.951176i \(-0.400123\pi\)
0.308649 + 0.951176i \(0.400123\pi\)
\(678\) 0 0
\(679\) 56.7868 2.17928
\(680\) 0 0
\(681\) −3.01038 −0.115358
\(682\) 0 0
\(683\) −13.5819 −0.519696 −0.259848 0.965649i \(-0.583673\pi\)
−0.259848 + 0.965649i \(0.583673\pi\)
\(684\) 0 0
\(685\) −13.4012 −0.512032
\(686\) 0 0
\(687\) 0.00388067 0.000148057 0
\(688\) 0 0
\(689\) −70.8871 −2.70058
\(690\) 0 0
\(691\) 5.72328 0.217724 0.108862 0.994057i \(-0.465279\pi\)
0.108862 + 0.994057i \(0.465279\pi\)
\(692\) 0 0
\(693\) −5.39626 −0.204987
\(694\) 0 0
\(695\) 33.6876 1.27784
\(696\) 0 0
\(697\) 37.4887 1.41999
\(698\) 0 0
\(699\) −0.121375 −0.00459083
\(700\) 0 0
\(701\) −20.2921 −0.766421 −0.383211 0.923661i \(-0.625182\pi\)
−0.383211 + 0.923661i \(0.625182\pi\)
\(702\) 0 0
\(703\) −29.5670 −1.11514
\(704\) 0 0
\(705\) 3.58584 0.135051
\(706\) 0 0
\(707\) 4.37408 0.164504
\(708\) 0 0
\(709\) −31.5398 −1.18450 −0.592251 0.805754i \(-0.701761\pi\)
−0.592251 + 0.805754i \(0.701761\pi\)
\(710\) 0 0
\(711\) −4.25369 −0.159526
\(712\) 0 0
\(713\) 34.0892 1.27665
\(714\) 0 0
\(715\) −6.24509 −0.233553
\(716\) 0 0
\(717\) 1.67581 0.0625844
\(718\) 0 0
\(719\) 11.0676 0.412753 0.206376 0.978473i \(-0.433833\pi\)
0.206376 + 0.978473i \(0.433833\pi\)
\(720\) 0 0
\(721\) 18.6238 0.693588
\(722\) 0 0
\(723\) −0.186202 −0.00692494
\(724\) 0 0
\(725\) −0.602774 −0.0223865
\(726\) 0 0
\(727\) 49.5631 1.83819 0.919096 0.394033i \(-0.128921\pi\)
0.919096 + 0.394033i \(0.128921\pi\)
\(728\) 0 0
\(729\) −25.1457 −0.931324
\(730\) 0 0
\(731\) −6.92901 −0.256279
\(732\) 0 0
\(733\) −2.44508 −0.0903110 −0.0451555 0.998980i \(-0.514378\pi\)
−0.0451555 + 0.998980i \(0.514378\pi\)
\(734\) 0 0
\(735\) −2.28809 −0.0843975
\(736\) 0 0
\(737\) −2.01954 −0.0743907
\(738\) 0 0
\(739\) −36.9067 −1.35764 −0.678818 0.734307i \(-0.737508\pi\)
−0.678818 + 0.734307i \(0.737508\pi\)
\(740\) 0 0
\(741\) 4.93897 0.181438
\(742\) 0 0
\(743\) −40.6188 −1.49016 −0.745079 0.666976i \(-0.767588\pi\)
−0.745079 + 0.666976i \(0.767588\pi\)
\(744\) 0 0
\(745\) 0.293940 0.0107691
\(746\) 0 0
\(747\) 5.13142 0.187749
\(748\) 0 0
\(749\) 38.0579 1.39061
\(750\) 0 0
\(751\) 14.0406 0.512349 0.256174 0.966631i \(-0.417538\pi\)
0.256174 + 0.966631i \(0.417538\pi\)
\(752\) 0 0
\(753\) −3.07207 −0.111953
\(754\) 0 0
\(755\) 0.621532 0.0226199
\(756\) 0 0
\(757\) 24.6936 0.897505 0.448752 0.893656i \(-0.351869\pi\)
0.448752 + 0.893656i \(0.351869\pi\)
\(758\) 0 0
\(759\) −0.738419 −0.0268029
\(760\) 0 0
\(761\) 12.6083 0.457051 0.228525 0.973538i \(-0.426610\pi\)
0.228525 + 0.973538i \(0.426610\pi\)
\(762\) 0 0
\(763\) 4.06816 0.147277
\(764\) 0 0
\(765\) −27.8068 −1.00536
\(766\) 0 0
\(767\) 49.8021 1.79825
\(768\) 0 0
\(769\) 42.1413 1.51965 0.759827 0.650125i \(-0.225283\pi\)
0.759827 + 0.650125i \(0.225283\pi\)
\(770\) 0 0
\(771\) 1.48781 0.0535821
\(772\) 0 0
\(773\) −9.62659 −0.346244 −0.173122 0.984900i \(-0.555386\pi\)
−0.173122 + 0.984900i \(0.555386\pi\)
\(774\) 0 0
\(775\) 0.298895 0.0107366
\(776\) 0 0
\(777\) −3.94221 −0.141426
\(778\) 0 0
\(779\) 44.3615 1.58942
\(780\) 0 0
\(781\) −7.04900 −0.252233
\(782\) 0 0
\(783\) 9.92745 0.354778
\(784\) 0 0
\(785\) −36.2526 −1.29391
\(786\) 0 0
\(787\) −50.4413 −1.79804 −0.899019 0.437909i \(-0.855719\pi\)
−0.899019 + 0.437909i \(0.855719\pi\)
\(788\) 0 0
\(789\) 1.01943 0.0362928
\(790\) 0 0
\(791\) 30.7436 1.09312
\(792\) 0 0
\(793\) 56.0713 1.99115
\(794\) 0 0
\(795\) −5.52185 −0.195840
\(796\) 0 0
\(797\) 34.3043 1.21512 0.607561 0.794273i \(-0.292148\pi\)
0.607561 + 0.794273i \(0.292148\pi\)
\(798\) 0 0
\(799\) 35.6366 1.26073
\(800\) 0 0
\(801\) 3.20252 0.113155
\(802\) 0 0
\(803\) −5.95627 −0.210192
\(804\) 0 0
\(805\) 61.1177 2.15412
\(806\) 0 0
\(807\) −0.437011 −0.0153835
\(808\) 0 0
\(809\) −43.9886 −1.54656 −0.773279 0.634066i \(-0.781385\pi\)
−0.773279 + 0.634066i \(0.781385\pi\)
\(810\) 0 0
\(811\) 43.0270 1.51088 0.755440 0.655217i \(-0.227423\pi\)
0.755440 + 0.655217i \(0.227423\pi\)
\(812\) 0 0
\(813\) −0.298484 −0.0104683
\(814\) 0 0
\(815\) −41.6937 −1.46047
\(816\) 0 0
\(817\) −8.19930 −0.286857
\(818\) 0 0
\(819\) −56.3212 −1.96802
\(820\) 0 0
\(821\) 18.7704 0.655090 0.327545 0.944836i \(-0.393779\pi\)
0.327545 + 0.944836i \(0.393779\pi\)
\(822\) 0 0
\(823\) −27.1588 −0.946695 −0.473348 0.880876i \(-0.656955\pi\)
−0.473348 + 0.880876i \(0.656955\pi\)
\(824\) 0 0
\(825\) −0.00647446 −0.000225412 0
\(826\) 0 0
\(827\) −41.5157 −1.44364 −0.721821 0.692079i \(-0.756695\pi\)
−0.721821 + 0.692079i \(0.756695\pi\)
\(828\) 0 0
\(829\) −30.6349 −1.06399 −0.531997 0.846746i \(-0.678558\pi\)
−0.531997 + 0.846746i \(0.678558\pi\)
\(830\) 0 0
\(831\) −4.03640 −0.140021
\(832\) 0 0
\(833\) −22.7394 −0.787872
\(834\) 0 0
\(835\) 34.8959 1.20762
\(836\) 0 0
\(837\) −4.92267 −0.170153
\(838\) 0 0
\(839\) −30.2456 −1.04419 −0.522097 0.852886i \(-0.674850\pi\)
−0.522097 + 0.852886i \(0.674850\pi\)
\(840\) 0 0
\(841\) 50.8798 1.75447
\(842\) 0 0
\(843\) 5.00902 0.172520
\(844\) 0 0
\(845\) −35.9163 −1.23556
\(846\) 0 0
\(847\) 37.8884 1.30186
\(848\) 0 0
\(849\) −0.724755 −0.0248735
\(850\) 0 0
\(851\) 46.1374 1.58157
\(852\) 0 0
\(853\) 42.7734 1.46453 0.732267 0.681018i \(-0.238462\pi\)
0.732267 + 0.681018i \(0.238462\pi\)
\(854\) 0 0
\(855\) −32.9046 −1.12532
\(856\) 0 0
\(857\) −18.8048 −0.642361 −0.321181 0.947018i \(-0.604080\pi\)
−0.321181 + 0.947018i \(0.604080\pi\)
\(858\) 0 0
\(859\) 8.22676 0.280693 0.140347 0.990102i \(-0.455178\pi\)
0.140347 + 0.990102i \(0.455178\pi\)
\(860\) 0 0
\(861\) 5.91478 0.201575
\(862\) 0 0
\(863\) 41.8464 1.42447 0.712235 0.701941i \(-0.247683\pi\)
0.712235 + 0.701941i \(0.247683\pi\)
\(864\) 0 0
\(865\) 0.824887 0.0280470
\(866\) 0 0
\(867\) 0.0656832 0.00223072
\(868\) 0 0
\(869\) −0.739564 −0.0250880
\(870\) 0 0
\(871\) −21.0781 −0.714204
\(872\) 0 0
\(873\) 47.7071 1.61464
\(874\) 0 0
\(875\) −39.1926 −1.32495
\(876\) 0 0
\(877\) −6.11483 −0.206483 −0.103242 0.994656i \(-0.532921\pi\)
−0.103242 + 0.994656i \(0.532921\pi\)
\(878\) 0 0
\(879\) −1.03787 −0.0350065
\(880\) 0 0
\(881\) −13.5157 −0.455355 −0.227678 0.973737i \(-0.573113\pi\)
−0.227678 + 0.973737i \(0.573113\pi\)
\(882\) 0 0
\(883\) −38.5489 −1.29727 −0.648637 0.761098i \(-0.724661\pi\)
−0.648637 + 0.761098i \(0.724661\pi\)
\(884\) 0 0
\(885\) 3.87941 0.130405
\(886\) 0 0
\(887\) −40.8386 −1.37123 −0.685613 0.727966i \(-0.740466\pi\)
−0.685613 + 0.727966i \(0.740466\pi\)
\(888\) 0 0
\(889\) −29.0163 −0.973177
\(890\) 0 0
\(891\) −4.47982 −0.150080
\(892\) 0 0
\(893\) 42.1698 1.41116
\(894\) 0 0
\(895\) 28.9759 0.968557
\(896\) 0 0
\(897\) −7.70694 −0.257327
\(898\) 0 0
\(899\) −39.6096 −1.32105
\(900\) 0 0
\(901\) −54.8769 −1.82822
\(902\) 0 0
\(903\) −1.09322 −0.0363802
\(904\) 0 0
\(905\) 27.4915 0.913848
\(906\) 0 0
\(907\) 34.3321 1.13998 0.569989 0.821652i \(-0.306947\pi\)
0.569989 + 0.821652i \(0.306947\pi\)
\(908\) 0 0
\(909\) 3.67470 0.121882
\(910\) 0 0
\(911\) −49.0410 −1.62480 −0.812400 0.583100i \(-0.801840\pi\)
−0.812400 + 0.583100i \(0.801840\pi\)
\(912\) 0 0
\(913\) 0.892169 0.0295265
\(914\) 0 0
\(915\) 4.36776 0.144394
\(916\) 0 0
\(917\) −25.7782 −0.851271
\(918\) 0 0
\(919\) −48.7329 −1.60755 −0.803775 0.594934i \(-0.797178\pi\)
−0.803775 + 0.594934i \(0.797178\pi\)
\(920\) 0 0
\(921\) −4.29745 −0.141606
\(922\) 0 0
\(923\) −73.5710 −2.42162
\(924\) 0 0
\(925\) 0.404533 0.0133010
\(926\) 0 0
\(927\) 15.6461 0.513884
\(928\) 0 0
\(929\) 52.9031 1.73569 0.867847 0.496832i \(-0.165504\pi\)
0.867847 + 0.496832i \(0.165504\pi\)
\(930\) 0 0
\(931\) −26.9081 −0.881879
\(932\) 0 0
\(933\) 2.79088 0.0913693
\(934\) 0 0
\(935\) −4.83461 −0.158109
\(936\) 0 0
\(937\) −16.1310 −0.526976 −0.263488 0.964663i \(-0.584873\pi\)
−0.263488 + 0.964663i \(0.584873\pi\)
\(938\) 0 0
\(939\) −0.430280 −0.0140416
\(940\) 0 0
\(941\) 10.7225 0.349544 0.174772 0.984609i \(-0.444081\pi\)
0.174772 + 0.984609i \(0.444081\pi\)
\(942\) 0 0
\(943\) −69.2233 −2.25422
\(944\) 0 0
\(945\) −8.82574 −0.287101
\(946\) 0 0
\(947\) 21.9984 0.714851 0.357426 0.933942i \(-0.383655\pi\)
0.357426 + 0.933942i \(0.383655\pi\)
\(948\) 0 0
\(949\) −62.1660 −2.01800
\(950\) 0 0
\(951\) −3.41923 −0.110876
\(952\) 0 0
\(953\) −40.5214 −1.31262 −0.656309 0.754492i \(-0.727883\pi\)
−0.656309 + 0.754492i \(0.727883\pi\)
\(954\) 0 0
\(955\) −8.83330 −0.285839
\(956\) 0 0
\(957\) 0.857998 0.0277351
\(958\) 0 0
\(959\) −21.0129 −0.678542
\(960\) 0 0
\(961\) −11.3590 −0.366419
\(962\) 0 0
\(963\) 31.9728 1.03031
\(964\) 0 0
\(965\) 3.81154 0.122698
\(966\) 0 0
\(967\) 46.5645 1.49741 0.748706 0.662902i \(-0.230676\pi\)
0.748706 + 0.662902i \(0.230676\pi\)
\(968\) 0 0
\(969\) 3.82348 0.122828
\(970\) 0 0
\(971\) −2.03099 −0.0651775 −0.0325887 0.999469i \(-0.510375\pi\)
−0.0325887 + 0.999469i \(0.510375\pi\)
\(972\) 0 0
\(973\) 52.8218 1.69339
\(974\) 0 0
\(975\) −0.0675745 −0.00216412
\(976\) 0 0
\(977\) −45.7170 −1.46262 −0.731308 0.682047i \(-0.761090\pi\)
−0.731308 + 0.682047i \(0.761090\pi\)
\(978\) 0 0
\(979\) 0.556803 0.0177955
\(980\) 0 0
\(981\) 3.41770 0.109119
\(982\) 0 0
\(983\) 47.0190 1.49967 0.749837 0.661623i \(-0.230132\pi\)
0.749837 + 0.661623i \(0.230132\pi\)
\(984\) 0 0
\(985\) 4.96103 0.158071
\(986\) 0 0
\(987\) 5.62256 0.178968
\(988\) 0 0
\(989\) 12.7945 0.406841
\(990\) 0 0
\(991\) −54.4626 −1.73006 −0.865031 0.501718i \(-0.832702\pi\)
−0.865031 + 0.501718i \(0.832702\pi\)
\(992\) 0 0
\(993\) 1.20647 0.0382860
\(994\) 0 0
\(995\) −35.0121 −1.10996
\(996\) 0 0
\(997\) 13.5868 0.430297 0.215149 0.976581i \(-0.430976\pi\)
0.215149 + 0.976581i \(0.430976\pi\)
\(998\) 0 0
\(999\) −6.66250 −0.210792
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 3856.2.a.j.1.3 7
4.3 odd 2 241.2.a.a.1.2 7
12.11 even 2 2169.2.a.e.1.6 7
20.19 odd 2 6025.2.a.f.1.6 7
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
241.2.a.a.1.2 7 4.3 odd 2
2169.2.a.e.1.6 7 12.11 even 2
3856.2.a.j.1.3 7 1.1 even 1 trivial
6025.2.a.f.1.6 7 20.19 odd 2