Properties

Label 8036.2.a.q.1.13
Level 8036
Weight 2
Character 8036.1
Self dual yes
Analytic conductor 64.168
Analytic rank 0
Dimension 15
CM no
Inner twists 1

Related objects

Downloads

Learn more about

Newspace parameters

Level: \( N \) = \( 8036 = 2^{2} \cdot 7^{2} \cdot 41 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 8036.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(64.1677830643\)
Analytic rank: \(0\)
Dimension: \(15\)
Coefficient field: \(\mathbb{Q}[x]/(x^{15} - \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 1148)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.13
Root \(-2.73742\)
Character \(\chi\) = 8036.1

$q$-expansion

\(f(q)\) \(=\) \(q+2.73742 q^{3} -0.863054 q^{5} +4.49344 q^{9} +O(q^{10})\) \(q+2.73742 q^{3} -0.863054 q^{5} +4.49344 q^{9} -5.59068 q^{11} -1.90008 q^{13} -2.36254 q^{15} -4.01928 q^{17} -1.91765 q^{19} +9.06557 q^{23} -4.25514 q^{25} +4.08817 q^{27} +5.61917 q^{29} +9.11195 q^{31} -15.3040 q^{33} +1.45610 q^{37} -5.20129 q^{39} -1.00000 q^{41} +7.83032 q^{43} -3.87808 q^{45} +10.1609 q^{47} -11.0024 q^{51} -1.16468 q^{53} +4.82506 q^{55} -5.24940 q^{57} +6.86817 q^{59} +5.22611 q^{61} +1.63987 q^{65} +14.6443 q^{67} +24.8162 q^{69} +3.14178 q^{71} -4.41735 q^{73} -11.6481 q^{75} -5.30208 q^{79} -2.28930 q^{81} +6.73846 q^{83} +3.46886 q^{85} +15.3820 q^{87} -3.99849 q^{89} +24.9432 q^{93} +1.65503 q^{95} +3.44705 q^{97} -25.1214 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 15q - q^{3} + 3q^{5} + 30q^{9} + O(q^{10}) \) \( 15q - q^{3} + 3q^{5} + 30q^{9} + 9q^{11} + 7q^{13} + 2q^{15} + 3q^{17} + 7q^{19} - q^{23} + 32q^{25} + 11q^{27} + 18q^{29} + 30q^{31} - 16q^{33} + 23q^{37} + 5q^{39} - 15q^{41} + 12q^{43} - 13q^{45} - 16q^{47} + 29q^{51} + 33q^{53} + 37q^{55} + 16q^{57} - 10q^{59} + q^{61} + 16q^{65} + 20q^{67} + 21q^{69} + 5q^{71} - 3q^{73} - 51q^{75} + 25q^{79} + 43q^{81} + 18q^{83} + 36q^{85} - 53q^{87} - 11q^{89} + 65q^{93} - 30q^{95} + 16q^{97} - 18q^{99} + O(q^{100}) \)

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
\(3\) 2.73742 1.58045 0.790224 0.612818i \(-0.209964\pi\)
0.790224 + 0.612818i \(0.209964\pi\)
\(4\) 0 0
\(5\) −0.863054 −0.385970 −0.192985 0.981202i \(-0.561817\pi\)
−0.192985 + 0.981202i \(0.561817\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) 0 0
\(9\) 4.49344 1.49781
\(10\) 0 0
\(11\) −5.59068 −1.68565 −0.842826 0.538186i \(-0.819110\pi\)
−0.842826 + 0.538186i \(0.819110\pi\)
\(12\) 0 0
\(13\) −1.90008 −0.526986 −0.263493 0.964661i \(-0.584875\pi\)
−0.263493 + 0.964661i \(0.584875\pi\)
\(14\) 0 0
\(15\) −2.36254 −0.610005
\(16\) 0 0
\(17\) −4.01928 −0.974819 −0.487409 0.873174i \(-0.662058\pi\)
−0.487409 + 0.873174i \(0.662058\pi\)
\(18\) 0 0
\(19\) −1.91765 −0.439939 −0.219969 0.975507i \(-0.570596\pi\)
−0.219969 + 0.975507i \(0.570596\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 9.06557 1.89030 0.945151 0.326632i \(-0.105914\pi\)
0.945151 + 0.326632i \(0.105914\pi\)
\(24\) 0 0
\(25\) −4.25514 −0.851027
\(26\) 0 0
\(27\) 4.08817 0.786769
\(28\) 0 0
\(29\) 5.61917 1.04345 0.521727 0.853113i \(-0.325288\pi\)
0.521727 + 0.853113i \(0.325288\pi\)
\(30\) 0 0
\(31\) 9.11195 1.63655 0.818277 0.574824i \(-0.194929\pi\)
0.818277 + 0.574824i \(0.194929\pi\)
\(32\) 0 0
\(33\) −15.3040 −2.66409
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 1.45610 0.239382 0.119691 0.992811i \(-0.461810\pi\)
0.119691 + 0.992811i \(0.461810\pi\)
\(38\) 0 0
\(39\) −5.20129 −0.832874
\(40\) 0 0
\(41\) −1.00000 −0.156174
\(42\) 0 0
\(43\) 7.83032 1.19411 0.597056 0.802199i \(-0.296337\pi\)
0.597056 + 0.802199i \(0.296337\pi\)
\(44\) 0 0
\(45\) −3.87808 −0.578111
\(46\) 0 0
\(47\) 10.1609 1.48212 0.741059 0.671440i \(-0.234324\pi\)
0.741059 + 0.671440i \(0.234324\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 0 0
\(51\) −11.0024 −1.54065
\(52\) 0 0
\(53\) −1.16468 −0.159982 −0.0799909 0.996796i \(-0.525489\pi\)
−0.0799909 + 0.996796i \(0.525489\pi\)
\(54\) 0 0
\(55\) 4.82506 0.650611
\(56\) 0 0
\(57\) −5.24940 −0.695300
\(58\) 0 0
\(59\) 6.86817 0.894159 0.447080 0.894494i \(-0.352464\pi\)
0.447080 + 0.894494i \(0.352464\pi\)
\(60\) 0 0
\(61\) 5.22611 0.669135 0.334568 0.942372i \(-0.391410\pi\)
0.334568 + 0.942372i \(0.391410\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 1.63987 0.203401
\(66\) 0 0
\(67\) 14.6443 1.78909 0.894546 0.446977i \(-0.147499\pi\)
0.894546 + 0.446977i \(0.147499\pi\)
\(68\) 0 0
\(69\) 24.8162 2.98752
\(70\) 0 0
\(71\) 3.14178 0.372860 0.186430 0.982468i \(-0.440308\pi\)
0.186430 + 0.982468i \(0.440308\pi\)
\(72\) 0 0
\(73\) −4.41735 −0.517012 −0.258506 0.966010i \(-0.583230\pi\)
−0.258506 + 0.966010i \(0.583230\pi\)
\(74\) 0 0
\(75\) −11.6481 −1.34500
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) −5.30208 −0.596530 −0.298265 0.954483i \(-0.596408\pi\)
−0.298265 + 0.954483i \(0.596408\pi\)
\(80\) 0 0
\(81\) −2.28930 −0.254367
\(82\) 0 0
\(83\) 6.73846 0.739642 0.369821 0.929103i \(-0.379419\pi\)
0.369821 + 0.929103i \(0.379419\pi\)
\(84\) 0 0
\(85\) 3.46886 0.376250
\(86\) 0 0
\(87\) 15.3820 1.64912
\(88\) 0 0
\(89\) −3.99849 −0.423839 −0.211919 0.977287i \(-0.567971\pi\)
−0.211919 + 0.977287i \(0.567971\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 24.9432 2.58649
\(94\) 0 0
\(95\) 1.65503 0.169803
\(96\) 0 0
\(97\) 3.44705 0.349995 0.174997 0.984569i \(-0.444008\pi\)
0.174997 + 0.984569i \(0.444008\pi\)
\(98\) 0 0
\(99\) −25.1214 −2.52479
\(100\) 0 0
\(101\) −17.7081 −1.76202 −0.881011 0.473096i \(-0.843136\pi\)
−0.881011 + 0.473096i \(0.843136\pi\)
\(102\) 0 0
\(103\) 15.8237 1.55915 0.779576 0.626308i \(-0.215435\pi\)
0.779576 + 0.626308i \(0.215435\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 9.15327 0.884880 0.442440 0.896798i \(-0.354113\pi\)
0.442440 + 0.896798i \(0.354113\pi\)
\(108\) 0 0
\(109\) −13.8982 −1.33121 −0.665604 0.746305i \(-0.731826\pi\)
−0.665604 + 0.746305i \(0.731826\pi\)
\(110\) 0 0
\(111\) 3.98596 0.378331
\(112\) 0 0
\(113\) 9.44966 0.888949 0.444475 0.895791i \(-0.353390\pi\)
0.444475 + 0.895791i \(0.353390\pi\)
\(114\) 0 0
\(115\) −7.82408 −0.729599
\(116\) 0 0
\(117\) −8.53788 −0.789327
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) 20.2557 1.84142
\(122\) 0 0
\(123\) −2.73742 −0.246824
\(124\) 0 0
\(125\) 7.98769 0.714440
\(126\) 0 0
\(127\) −1.14798 −0.101866 −0.0509332 0.998702i \(-0.516220\pi\)
−0.0509332 + 0.998702i \(0.516220\pi\)
\(128\) 0 0
\(129\) 21.4348 1.88723
\(130\) 0 0
\(131\) 1.30559 0.114070 0.0570348 0.998372i \(-0.481835\pi\)
0.0570348 + 0.998372i \(0.481835\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) −3.52831 −0.303669
\(136\) 0 0
\(137\) 12.0254 1.02740 0.513698 0.857971i \(-0.328275\pi\)
0.513698 + 0.857971i \(0.328275\pi\)
\(138\) 0 0
\(139\) 6.79439 0.576293 0.288146 0.957586i \(-0.406961\pi\)
0.288146 + 0.957586i \(0.406961\pi\)
\(140\) 0 0
\(141\) 27.8146 2.34241
\(142\) 0 0
\(143\) 10.6227 0.888315
\(144\) 0 0
\(145\) −4.84965 −0.402741
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 19.9624 1.63539 0.817694 0.575653i \(-0.195252\pi\)
0.817694 + 0.575653i \(0.195252\pi\)
\(150\) 0 0
\(151\) −7.91750 −0.644317 −0.322158 0.946686i \(-0.604408\pi\)
−0.322158 + 0.946686i \(0.604408\pi\)
\(152\) 0 0
\(153\) −18.0604 −1.46010
\(154\) 0 0
\(155\) −7.86411 −0.631660
\(156\) 0 0
\(157\) 7.85253 0.626700 0.313350 0.949638i \(-0.398549\pi\)
0.313350 + 0.949638i \(0.398549\pi\)
\(158\) 0 0
\(159\) −3.18823 −0.252843
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) 17.8785 1.40035 0.700174 0.713972i \(-0.253106\pi\)
0.700174 + 0.713972i \(0.253106\pi\)
\(164\) 0 0
\(165\) 13.2082 1.02826
\(166\) 0 0
\(167\) 5.60745 0.433918 0.216959 0.976181i \(-0.430386\pi\)
0.216959 + 0.976181i \(0.430386\pi\)
\(168\) 0 0
\(169\) −9.38971 −0.722286
\(170\) 0 0
\(171\) −8.61684 −0.658946
\(172\) 0 0
\(173\) 7.61552 0.578997 0.289499 0.957178i \(-0.406511\pi\)
0.289499 + 0.957178i \(0.406511\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 18.8010 1.41317
\(178\) 0 0
\(179\) −21.5408 −1.61003 −0.805017 0.593252i \(-0.797844\pi\)
−0.805017 + 0.593252i \(0.797844\pi\)
\(180\) 0 0
\(181\) 4.71251 0.350278 0.175139 0.984544i \(-0.443962\pi\)
0.175139 + 0.984544i \(0.443962\pi\)
\(182\) 0 0
\(183\) 14.3060 1.05753
\(184\) 0 0
\(185\) −1.25670 −0.0923942
\(186\) 0 0
\(187\) 22.4705 1.64321
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −11.1408 −0.806120 −0.403060 0.915174i \(-0.632053\pi\)
−0.403060 + 0.915174i \(0.632053\pi\)
\(192\) 0 0
\(193\) 24.5322 1.76587 0.882933 0.469499i \(-0.155566\pi\)
0.882933 + 0.469499i \(0.155566\pi\)
\(194\) 0 0
\(195\) 4.48900 0.321464
\(196\) 0 0
\(197\) −25.7126 −1.83195 −0.915973 0.401241i \(-0.868579\pi\)
−0.915973 + 0.401241i \(0.868579\pi\)
\(198\) 0 0
\(199\) −0.972796 −0.0689597 −0.0344798 0.999405i \(-0.510977\pi\)
−0.0344798 + 0.999405i \(0.510977\pi\)
\(200\) 0 0
\(201\) 40.0877 2.82756
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 0.863054 0.0602783
\(206\) 0 0
\(207\) 40.7356 2.83132
\(208\) 0 0
\(209\) 10.7209 0.741584
\(210\) 0 0
\(211\) 2.56072 0.176287 0.0881436 0.996108i \(-0.471907\pi\)
0.0881436 + 0.996108i \(0.471907\pi\)
\(212\) 0 0
\(213\) 8.60035 0.589286
\(214\) 0 0
\(215\) −6.75799 −0.460891
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) −12.0921 −0.817110
\(220\) 0 0
\(221\) 7.63694 0.513716
\(222\) 0 0
\(223\) −21.6493 −1.44974 −0.724872 0.688884i \(-0.758101\pi\)
−0.724872 + 0.688884i \(0.758101\pi\)
\(224\) 0 0
\(225\) −19.1202 −1.27468
\(226\) 0 0
\(227\) −0.644605 −0.0427839 −0.0213920 0.999771i \(-0.506810\pi\)
−0.0213920 + 0.999771i \(0.506810\pi\)
\(228\) 0 0
\(229\) −1.31952 −0.0871966 −0.0435983 0.999049i \(-0.513882\pi\)
−0.0435983 + 0.999049i \(0.513882\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −28.3002 −1.85401 −0.927003 0.375053i \(-0.877624\pi\)
−0.927003 + 0.375053i \(0.877624\pi\)
\(234\) 0 0
\(235\) −8.76940 −0.572053
\(236\) 0 0
\(237\) −14.5140 −0.942785
\(238\) 0 0
\(239\) −9.13220 −0.590713 −0.295356 0.955387i \(-0.595438\pi\)
−0.295356 + 0.955387i \(0.595438\pi\)
\(240\) 0 0
\(241\) −14.9952 −0.965924 −0.482962 0.875641i \(-0.660439\pi\)
−0.482962 + 0.875641i \(0.660439\pi\)
\(242\) 0 0
\(243\) −18.5313 −1.18878
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 3.64367 0.231841
\(248\) 0 0
\(249\) 18.4460 1.16897
\(250\) 0 0
\(251\) −19.0748 −1.20399 −0.601994 0.798500i \(-0.705627\pi\)
−0.601994 + 0.798500i \(0.705627\pi\)
\(252\) 0 0
\(253\) −50.6827 −3.18639
\(254\) 0 0
\(255\) 9.49570 0.594644
\(256\) 0 0
\(257\) −11.7626 −0.733734 −0.366867 0.930273i \(-0.619570\pi\)
−0.366867 + 0.930273i \(0.619570\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) 25.2494 1.56290
\(262\) 0 0
\(263\) −7.87248 −0.485438 −0.242719 0.970097i \(-0.578039\pi\)
−0.242719 + 0.970097i \(0.578039\pi\)
\(264\) 0 0
\(265\) 1.00519 0.0617481
\(266\) 0 0
\(267\) −10.9455 −0.669855
\(268\) 0 0
\(269\) −11.5817 −0.706150 −0.353075 0.935595i \(-0.614864\pi\)
−0.353075 + 0.935595i \(0.614864\pi\)
\(270\) 0 0
\(271\) 12.0932 0.734609 0.367305 0.930101i \(-0.380281\pi\)
0.367305 + 0.930101i \(0.380281\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 23.7891 1.43454
\(276\) 0 0
\(277\) 21.6075 1.29827 0.649135 0.760673i \(-0.275131\pi\)
0.649135 + 0.760673i \(0.275131\pi\)
\(278\) 0 0
\(279\) 40.9440 2.45125
\(280\) 0 0
\(281\) 12.2417 0.730278 0.365139 0.930953i \(-0.381022\pi\)
0.365139 + 0.930953i \(0.381022\pi\)
\(282\) 0 0
\(283\) −30.5526 −1.81616 −0.908082 0.418791i \(-0.862454\pi\)
−0.908082 + 0.418791i \(0.862454\pi\)
\(284\) 0 0
\(285\) 4.53052 0.268365
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) −0.845377 −0.0497281
\(290\) 0 0
\(291\) 9.43600 0.553148
\(292\) 0 0
\(293\) 15.2363 0.890117 0.445058 0.895502i \(-0.353183\pi\)
0.445058 + 0.895502i \(0.353183\pi\)
\(294\) 0 0
\(295\) −5.92760 −0.345118
\(296\) 0 0
\(297\) −22.8557 −1.32622
\(298\) 0 0
\(299\) −17.2253 −0.996163
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) −48.4744 −2.78478
\(304\) 0 0
\(305\) −4.51042 −0.258266
\(306\) 0 0
\(307\) −3.85409 −0.219964 −0.109982 0.993934i \(-0.535079\pi\)
−0.109982 + 0.993934i \(0.535079\pi\)
\(308\) 0 0
\(309\) 43.3159 2.46416
\(310\) 0 0
\(311\) 5.97634 0.338887 0.169444 0.985540i \(-0.445803\pi\)
0.169444 + 0.985540i \(0.445803\pi\)
\(312\) 0 0
\(313\) 14.7468 0.833540 0.416770 0.909012i \(-0.363162\pi\)
0.416770 + 0.909012i \(0.363162\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 30.3411 1.70412 0.852062 0.523441i \(-0.175352\pi\)
0.852062 + 0.523441i \(0.175352\pi\)
\(318\) 0 0
\(319\) −31.4150 −1.75890
\(320\) 0 0
\(321\) 25.0563 1.39851
\(322\) 0 0
\(323\) 7.70757 0.428860
\(324\) 0 0
\(325\) 8.08508 0.448480
\(326\) 0 0
\(327\) −38.0452 −2.10391
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) 20.2963 1.11559 0.557793 0.829980i \(-0.311648\pi\)
0.557793 + 0.829980i \(0.311648\pi\)
\(332\) 0 0
\(333\) 6.54292 0.358550
\(334\) 0 0
\(335\) −12.6389 −0.690535
\(336\) 0 0
\(337\) 2.08648 0.113658 0.0568290 0.998384i \(-0.481901\pi\)
0.0568290 + 0.998384i \(0.481901\pi\)
\(338\) 0 0
\(339\) 25.8676 1.40494
\(340\) 0 0
\(341\) −50.9420 −2.75866
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) −21.4178 −1.15309
\(346\) 0 0
\(347\) 0.664742 0.0356852 0.0178426 0.999841i \(-0.494320\pi\)
0.0178426 + 0.999841i \(0.494320\pi\)
\(348\) 0 0
\(349\) −29.1533 −1.56054 −0.780272 0.625441i \(-0.784919\pi\)
−0.780272 + 0.625441i \(0.784919\pi\)
\(350\) 0 0
\(351\) −7.76783 −0.414616
\(352\) 0 0
\(353\) −30.8199 −1.64038 −0.820188 0.572094i \(-0.806131\pi\)
−0.820188 + 0.572094i \(0.806131\pi\)
\(354\) 0 0
\(355\) −2.71152 −0.143913
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −18.9341 −0.999304 −0.499652 0.866226i \(-0.666539\pi\)
−0.499652 + 0.866226i \(0.666539\pi\)
\(360\) 0 0
\(361\) −15.3226 −0.806454
\(362\) 0 0
\(363\) 55.4482 2.91027
\(364\) 0 0
\(365\) 3.81241 0.199551
\(366\) 0 0
\(367\) 8.85703 0.462333 0.231167 0.972914i \(-0.425746\pi\)
0.231167 + 0.972914i \(0.425746\pi\)
\(368\) 0 0
\(369\) −4.49344 −0.233919
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) −3.48349 −0.180368 −0.0901840 0.995925i \(-0.528746\pi\)
−0.0901840 + 0.995925i \(0.528746\pi\)
\(374\) 0 0
\(375\) 21.8656 1.12914
\(376\) 0 0
\(377\) −10.6768 −0.549885
\(378\) 0 0
\(379\) 18.8253 0.966991 0.483496 0.875347i \(-0.339367\pi\)
0.483496 + 0.875347i \(0.339367\pi\)
\(380\) 0 0
\(381\) −3.14248 −0.160994
\(382\) 0 0
\(383\) 0.762543 0.0389641 0.0194821 0.999810i \(-0.493798\pi\)
0.0194821 + 0.999810i \(0.493798\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 35.1851 1.78856
\(388\) 0 0
\(389\) 6.98396 0.354101 0.177050 0.984202i \(-0.443344\pi\)
0.177050 + 0.984202i \(0.443344\pi\)
\(390\) 0 0
\(391\) −36.4371 −1.84270
\(392\) 0 0
\(393\) 3.57393 0.180281
\(394\) 0 0
\(395\) 4.57598 0.230243
\(396\) 0 0
\(397\) −16.9448 −0.850435 −0.425218 0.905091i \(-0.639802\pi\)
−0.425218 + 0.905091i \(0.639802\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −12.5586 −0.627147 −0.313573 0.949564i \(-0.601526\pi\)
−0.313573 + 0.949564i \(0.601526\pi\)
\(402\) 0 0
\(403\) −17.3134 −0.862441
\(404\) 0 0
\(405\) 1.97579 0.0981779
\(406\) 0 0
\(407\) −8.14061 −0.403515
\(408\) 0 0
\(409\) 19.4920 0.963817 0.481909 0.876221i \(-0.339944\pi\)
0.481909 + 0.876221i \(0.339944\pi\)
\(410\) 0 0
\(411\) 32.9184 1.62374
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) −5.81566 −0.285479
\(416\) 0 0
\(417\) 18.5991 0.910801
\(418\) 0 0
\(419\) 34.0935 1.66558 0.832788 0.553593i \(-0.186744\pi\)
0.832788 + 0.553593i \(0.186744\pi\)
\(420\) 0 0
\(421\) 36.1622 1.76244 0.881219 0.472708i \(-0.156723\pi\)
0.881219 + 0.472708i \(0.156723\pi\)
\(422\) 0 0
\(423\) 45.6574 2.21994
\(424\) 0 0
\(425\) 17.1026 0.829598
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) 29.0788 1.40394
\(430\) 0 0
\(431\) 15.5755 0.750245 0.375122 0.926975i \(-0.377601\pi\)
0.375122 + 0.926975i \(0.377601\pi\)
\(432\) 0 0
\(433\) 8.91790 0.428567 0.214283 0.976772i \(-0.431258\pi\)
0.214283 + 0.976772i \(0.431258\pi\)
\(434\) 0 0
\(435\) −13.2755 −0.636511
\(436\) 0 0
\(437\) −17.3846 −0.831617
\(438\) 0 0
\(439\) −32.4555 −1.54901 −0.774507 0.632565i \(-0.782002\pi\)
−0.774507 + 0.632565i \(0.782002\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 6.17519 0.293392 0.146696 0.989182i \(-0.453136\pi\)
0.146696 + 0.989182i \(0.453136\pi\)
\(444\) 0 0
\(445\) 3.45091 0.163589
\(446\) 0 0
\(447\) 54.6455 2.58464
\(448\) 0 0
\(449\) −33.2976 −1.57141 −0.785706 0.618600i \(-0.787700\pi\)
−0.785706 + 0.618600i \(0.787700\pi\)
\(450\) 0 0
\(451\) 5.59068 0.263255
\(452\) 0 0
\(453\) −21.6735 −1.01831
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 10.6002 0.495857 0.247929 0.968778i \(-0.420250\pi\)
0.247929 + 0.968778i \(0.420250\pi\)
\(458\) 0 0
\(459\) −16.4315 −0.766957
\(460\) 0 0
\(461\) −32.3114 −1.50489 −0.752445 0.658655i \(-0.771126\pi\)
−0.752445 + 0.658655i \(0.771126\pi\)
\(462\) 0 0
\(463\) 24.6791 1.14694 0.573469 0.819228i \(-0.305597\pi\)
0.573469 + 0.819228i \(0.305597\pi\)
\(464\) 0 0
\(465\) −21.5273 −0.998306
\(466\) 0 0
\(467\) −2.61652 −0.121078 −0.0605391 0.998166i \(-0.519282\pi\)
−0.0605391 + 0.998166i \(0.519282\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 21.4956 0.990466
\(472\) 0 0
\(473\) −43.7768 −2.01286
\(474\) 0 0
\(475\) 8.15985 0.374400
\(476\) 0 0
\(477\) −5.23344 −0.239623
\(478\) 0 0
\(479\) −21.8559 −0.998621 −0.499311 0.866423i \(-0.666413\pi\)
−0.499311 + 0.866423i \(0.666413\pi\)
\(480\) 0 0
\(481\) −2.76671 −0.126151
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) −2.97499 −0.135087
\(486\) 0 0
\(487\) 23.0302 1.04360 0.521799 0.853068i \(-0.325261\pi\)
0.521799 + 0.853068i \(0.325261\pi\)
\(488\) 0 0
\(489\) 48.9408 2.21318
\(490\) 0 0
\(491\) 19.4215 0.876480 0.438240 0.898858i \(-0.355602\pi\)
0.438240 + 0.898858i \(0.355602\pi\)
\(492\) 0 0
\(493\) −22.5850 −1.01718
\(494\) 0 0
\(495\) 21.6811 0.974494
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) 14.9571 0.669572 0.334786 0.942294i \(-0.391336\pi\)
0.334786 + 0.942294i \(0.391336\pi\)
\(500\) 0 0
\(501\) 15.3499 0.685784
\(502\) 0 0
\(503\) −12.0347 −0.536602 −0.268301 0.963335i \(-0.586462\pi\)
−0.268301 + 0.963335i \(0.586462\pi\)
\(504\) 0 0
\(505\) 15.2831 0.680087
\(506\) 0 0
\(507\) −25.7035 −1.14153
\(508\) 0 0
\(509\) 6.92030 0.306737 0.153368 0.988169i \(-0.450988\pi\)
0.153368 + 0.988169i \(0.450988\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) −7.83967 −0.346130
\(514\) 0 0
\(515\) −13.6567 −0.601785
\(516\) 0 0
\(517\) −56.8063 −2.49834
\(518\) 0 0
\(519\) 20.8468 0.915075
\(520\) 0 0
\(521\) 2.78492 0.122009 0.0610047 0.998137i \(-0.480570\pi\)
0.0610047 + 0.998137i \(0.480570\pi\)
\(522\) 0 0
\(523\) 14.8470 0.649215 0.324607 0.945849i \(-0.394768\pi\)
0.324607 + 0.945849i \(0.394768\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −36.6235 −1.59534
\(528\) 0 0
\(529\) 59.1846 2.57325
\(530\) 0 0
\(531\) 30.8617 1.33928
\(532\) 0 0
\(533\) 1.90008 0.0823014
\(534\) 0 0
\(535\) −7.89977 −0.341537
\(536\) 0 0
\(537\) −58.9661 −2.54457
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) 8.23646 0.354113 0.177057 0.984201i \(-0.443342\pi\)
0.177057 + 0.984201i \(0.443342\pi\)
\(542\) 0 0
\(543\) 12.9001 0.553597
\(544\) 0 0
\(545\) 11.9949 0.513806
\(546\) 0 0
\(547\) 40.5675 1.73454 0.867270 0.497838i \(-0.165872\pi\)
0.867270 + 0.497838i \(0.165872\pi\)
\(548\) 0 0
\(549\) 23.4832 1.00224
\(550\) 0 0
\(551\) −10.7756 −0.459055
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) −3.44010 −0.146024
\(556\) 0 0
\(557\) 22.6289 0.958818 0.479409 0.877592i \(-0.340851\pi\)
0.479409 + 0.877592i \(0.340851\pi\)
\(558\) 0 0
\(559\) −14.8782 −0.629280
\(560\) 0 0
\(561\) 61.5111 2.59700
\(562\) 0 0
\(563\) −17.3463 −0.731061 −0.365531 0.930799i \(-0.619112\pi\)
−0.365531 + 0.930799i \(0.619112\pi\)
\(564\) 0 0
\(565\) −8.15557 −0.343107
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −10.5921 −0.444045 −0.222023 0.975042i \(-0.571266\pi\)
−0.222023 + 0.975042i \(0.571266\pi\)
\(570\) 0 0
\(571\) 10.5451 0.441299 0.220649 0.975353i \(-0.429182\pi\)
0.220649 + 0.975353i \(0.429182\pi\)
\(572\) 0 0
\(573\) −30.4970 −1.27403
\(574\) 0 0
\(575\) −38.5753 −1.60870
\(576\) 0 0
\(577\) −19.4790 −0.810921 −0.405460 0.914113i \(-0.632889\pi\)
−0.405460 + 0.914113i \(0.632889\pi\)
\(578\) 0 0
\(579\) 67.1548 2.79086
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 6.51138 0.269674
\(584\) 0 0
\(585\) 7.36865 0.304656
\(586\) 0 0
\(587\) 16.5937 0.684896 0.342448 0.939537i \(-0.388744\pi\)
0.342448 + 0.939537i \(0.388744\pi\)
\(588\) 0 0
\(589\) −17.4735 −0.719983
\(590\) 0 0
\(591\) −70.3860 −2.89529
\(592\) 0 0
\(593\) −22.6551 −0.930332 −0.465166 0.885224i \(-0.654005\pi\)
−0.465166 + 0.885224i \(0.654005\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) −2.66295 −0.108987
\(598\) 0 0
\(599\) −19.2453 −0.786340 −0.393170 0.919466i \(-0.628622\pi\)
−0.393170 + 0.919466i \(0.628622\pi\)
\(600\) 0 0
\(601\) −21.9736 −0.896322 −0.448161 0.893953i \(-0.647921\pi\)
−0.448161 + 0.893953i \(0.647921\pi\)
\(602\) 0 0
\(603\) 65.8035 2.67973
\(604\) 0 0
\(605\) −17.4817 −0.710734
\(606\) 0 0
\(607\) 34.3757 1.39527 0.697633 0.716455i \(-0.254237\pi\)
0.697633 + 0.716455i \(0.254237\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) −19.3065 −0.781056
\(612\) 0 0
\(613\) −24.5084 −0.989885 −0.494943 0.868926i \(-0.664811\pi\)
−0.494943 + 0.868926i \(0.664811\pi\)
\(614\) 0 0
\(615\) 2.36254 0.0952667
\(616\) 0 0
\(617\) −11.7986 −0.474993 −0.237496 0.971388i \(-0.576327\pi\)
−0.237496 + 0.971388i \(0.576327\pi\)
\(618\) 0 0
\(619\) −32.1597 −1.29261 −0.646304 0.763080i \(-0.723686\pi\)
−0.646304 + 0.763080i \(0.723686\pi\)
\(620\) 0 0
\(621\) 37.0616 1.48723
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) 14.3819 0.575275
\(626\) 0 0
\(627\) 29.3477 1.17203
\(628\) 0 0
\(629\) −5.85249 −0.233354
\(630\) 0 0
\(631\) 37.4091 1.48923 0.744617 0.667492i \(-0.232632\pi\)
0.744617 + 0.667492i \(0.232632\pi\)
\(632\) 0 0
\(633\) 7.00975 0.278613
\(634\) 0 0
\(635\) 0.990765 0.0393173
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) 14.1174 0.558475
\(640\) 0 0
\(641\) 4.05878 0.160312 0.0801560 0.996782i \(-0.474458\pi\)
0.0801560 + 0.996782i \(0.474458\pi\)
\(642\) 0 0
\(643\) −26.5083 −1.04539 −0.522693 0.852521i \(-0.675073\pi\)
−0.522693 + 0.852521i \(0.675073\pi\)
\(644\) 0 0
\(645\) −18.4994 −0.728414
\(646\) 0 0
\(647\) 9.38614 0.369007 0.184504 0.982832i \(-0.440932\pi\)
0.184504 + 0.982832i \(0.440932\pi\)
\(648\) 0 0
\(649\) −38.3977 −1.50724
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −7.32782 −0.286760 −0.143380 0.989668i \(-0.545797\pi\)
−0.143380 + 0.989668i \(0.545797\pi\)
\(654\) 0 0
\(655\) −1.12679 −0.0440274
\(656\) 0 0
\(657\) −19.8491 −0.774387
\(658\) 0 0
\(659\) 5.69956 0.222023 0.111012 0.993819i \(-0.464591\pi\)
0.111012 + 0.993819i \(0.464591\pi\)
\(660\) 0 0
\(661\) 44.0578 1.71365 0.856824 0.515608i \(-0.172434\pi\)
0.856824 + 0.515608i \(0.172434\pi\)
\(662\) 0 0
\(663\) 20.9055 0.811901
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 50.9410 1.97244
\(668\) 0 0
\(669\) −59.2631 −2.29124
\(670\) 0 0
\(671\) −29.2175 −1.12793
\(672\) 0 0
\(673\) 5.92353 0.228335 0.114168 0.993461i \(-0.463580\pi\)
0.114168 + 0.993461i \(0.463580\pi\)
\(674\) 0 0
\(675\) −17.3957 −0.669562
\(676\) 0 0
\(677\) −20.0586 −0.770914 −0.385457 0.922726i \(-0.625956\pi\)
−0.385457 + 0.922726i \(0.625956\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) −1.76455 −0.0676178
\(682\) 0 0
\(683\) 41.5746 1.59081 0.795404 0.606080i \(-0.207259\pi\)
0.795404 + 0.606080i \(0.207259\pi\)
\(684\) 0 0
\(685\) −10.3785 −0.396543
\(686\) 0 0
\(687\) −3.61209 −0.137810
\(688\) 0 0
\(689\) 2.21299 0.0843082
\(690\) 0 0
\(691\) −2.77305 −0.105492 −0.0527459 0.998608i \(-0.516797\pi\)
−0.0527459 + 0.998608i \(0.516797\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −5.86393 −0.222432
\(696\) 0 0
\(697\) 4.01928 0.152241
\(698\) 0 0
\(699\) −77.4693 −2.93016
\(700\) 0 0
\(701\) 32.4197 1.22448 0.612238 0.790674i \(-0.290269\pi\)
0.612238 + 0.790674i \(0.290269\pi\)
\(702\) 0 0
\(703\) −2.79229 −0.105313
\(704\) 0 0
\(705\) −24.0055 −0.904099
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) −10.5705 −0.396984 −0.198492 0.980103i \(-0.563604\pi\)
−0.198492 + 0.980103i \(0.563604\pi\)
\(710\) 0 0
\(711\) −23.8246 −0.893492
\(712\) 0 0
\(713\) 82.6051 3.09358
\(714\) 0 0
\(715\) −9.16797 −0.342863
\(716\) 0 0
\(717\) −24.9986 −0.933590
\(718\) 0 0
\(719\) −33.0291 −1.23178 −0.615889 0.787833i \(-0.711203\pi\)
−0.615889 + 0.787833i \(0.711203\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) −41.0480 −1.52659
\(724\) 0 0
\(725\) −23.9103 −0.888007
\(726\) 0 0
\(727\) 27.4483 1.01800 0.509001 0.860766i \(-0.330015\pi\)
0.509001 + 0.860766i \(0.330015\pi\)
\(728\) 0 0
\(729\) −43.8599 −1.62444
\(730\) 0 0
\(731\) −31.4722 −1.16404
\(732\) 0 0
\(733\) 9.03135 0.333581 0.166790 0.985992i \(-0.446660\pi\)
0.166790 + 0.985992i \(0.446660\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −81.8718 −3.01579
\(738\) 0 0
\(739\) 40.2255 1.47972 0.739859 0.672762i \(-0.234892\pi\)
0.739859 + 0.672762i \(0.234892\pi\)
\(740\) 0 0
\(741\) 9.97425 0.366413
\(742\) 0 0
\(743\) 2.30265 0.0844761 0.0422380 0.999108i \(-0.486551\pi\)
0.0422380 + 0.999108i \(0.486551\pi\)
\(744\) 0 0
\(745\) −17.2287 −0.631210
\(746\) 0 0
\(747\) 30.2789 1.10785
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) 19.5371 0.712921 0.356460 0.934310i \(-0.383984\pi\)
0.356460 + 0.934310i \(0.383984\pi\)
\(752\) 0 0
\(753\) −52.2156 −1.90284
\(754\) 0 0
\(755\) 6.83323 0.248687
\(756\) 0 0
\(757\) 8.28174 0.301005 0.150502 0.988610i \(-0.451911\pi\)
0.150502 + 0.988610i \(0.451911\pi\)
\(758\) 0 0
\(759\) −138.740 −5.03593
\(760\) 0 0
\(761\) 18.7920 0.681210 0.340605 0.940206i \(-0.389368\pi\)
0.340605 + 0.940206i \(0.389368\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) 15.5871 0.563553
\(766\) 0 0
\(767\) −13.0500 −0.471210
\(768\) 0 0
\(769\) −15.1327 −0.545700 −0.272850 0.962057i \(-0.587966\pi\)
−0.272850 + 0.962057i \(0.587966\pi\)
\(770\) 0 0
\(771\) −32.1992 −1.15963
\(772\) 0 0
\(773\) 26.9296 0.968591 0.484295 0.874905i \(-0.339076\pi\)
0.484295 + 0.874905i \(0.339076\pi\)
\(774\) 0 0
\(775\) −38.7726 −1.39275
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 1.91765 0.0687069
\(780\) 0 0
\(781\) −17.5647 −0.628513
\(782\) 0 0
\(783\) 22.9721 0.820957
\(784\) 0 0
\(785\) −6.77716 −0.241887
\(786\) 0 0
\(787\) −2.91892 −0.104048 −0.0520241 0.998646i \(-0.516567\pi\)
−0.0520241 + 0.998646i \(0.516567\pi\)
\(788\) 0 0
\(789\) −21.5502 −0.767209
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) −9.93001 −0.352625
\(794\) 0 0
\(795\) 2.75161 0.0975896
\(796\) 0 0
\(797\) −16.4159 −0.581481 −0.290740 0.956802i \(-0.593902\pi\)
−0.290740 + 0.956802i \(0.593902\pi\)
\(798\) 0 0
\(799\) −40.8395 −1.44480
\(800\) 0 0
\(801\) −17.9670 −0.634832
\(802\) 0 0
\(803\) 24.6960 0.871502
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) −31.7040 −1.11603
\(808\) 0 0
\(809\) 49.3677 1.73568 0.867838 0.496846i \(-0.165509\pi\)
0.867838 + 0.496846i \(0.165509\pi\)
\(810\) 0 0
\(811\) −23.4315 −0.822793 −0.411396 0.911457i \(-0.634959\pi\)
−0.411396 + 0.911457i \(0.634959\pi\)
\(812\) 0 0
\(813\) 33.1041 1.16101
\(814\) 0 0
\(815\) −15.4301 −0.540492
\(816\) 0 0
\(817\) −15.0158 −0.525336
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −4.32744 −0.151029 −0.0755143 0.997145i \(-0.524060\pi\)
−0.0755143 + 0.997145i \(0.524060\pi\)
\(822\) 0 0
\(823\) −13.0805 −0.455956 −0.227978 0.973666i \(-0.573211\pi\)
−0.227978 + 0.973666i \(0.573211\pi\)
\(824\) 0 0
\(825\) 65.1206 2.26721
\(826\) 0 0
\(827\) 6.08156 0.211477 0.105738 0.994394i \(-0.466279\pi\)
0.105738 + 0.994394i \(0.466279\pi\)
\(828\) 0 0
\(829\) 20.9946 0.729172 0.364586 0.931170i \(-0.381211\pi\)
0.364586 + 0.931170i \(0.381211\pi\)
\(830\) 0 0
\(831\) 59.1487 2.05185
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) −4.83953 −0.167479
\(836\) 0 0
\(837\) 37.2512 1.28759
\(838\) 0 0
\(839\) −50.0675 −1.72852 −0.864261 0.503044i \(-0.832213\pi\)
−0.864261 + 0.503044i \(0.832213\pi\)
\(840\) 0 0
\(841\) 2.57505 0.0887948
\(842\) 0 0
\(843\) 33.5106 1.15417
\(844\) 0 0
\(845\) 8.10383 0.278780
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) −83.6352 −2.87035
\(850\) 0 0
\(851\) 13.2004 0.452504
\(852\) 0 0
\(853\) 31.8851 1.09173 0.545863 0.837875i \(-0.316202\pi\)
0.545863 + 0.837875i \(0.316202\pi\)
\(854\) 0 0
\(855\) 7.43680 0.254333
\(856\) 0 0
\(857\) 50.4764 1.72424 0.862121 0.506703i \(-0.169136\pi\)
0.862121 + 0.506703i \(0.169136\pi\)
\(858\) 0 0
\(859\) 54.6739 1.86545 0.932724 0.360591i \(-0.117425\pi\)
0.932724 + 0.360591i \(0.117425\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 31.0631 1.05740 0.528700 0.848809i \(-0.322680\pi\)
0.528700 + 0.848809i \(0.322680\pi\)
\(864\) 0 0
\(865\) −6.57261 −0.223475
\(866\) 0 0
\(867\) −2.31415 −0.0785926
\(868\) 0 0
\(869\) 29.6422 1.00554
\(870\) 0 0
\(871\) −27.8254 −0.942826
\(872\) 0 0
\(873\) 15.4891 0.524227
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) −1.75325 −0.0592031 −0.0296016 0.999562i \(-0.509424\pi\)
−0.0296016 + 0.999562i \(0.509424\pi\)
\(878\) 0 0
\(879\) 41.7082 1.40678
\(880\) 0 0
\(881\) −7.18675 −0.242128 −0.121064 0.992645i \(-0.538631\pi\)
−0.121064 + 0.992645i \(0.538631\pi\)
\(882\) 0 0
\(883\) −22.2021 −0.747159 −0.373579 0.927598i \(-0.621870\pi\)
−0.373579 + 0.927598i \(0.621870\pi\)
\(884\) 0 0
\(885\) −16.2263 −0.545441
\(886\) 0 0
\(887\) −7.26838 −0.244048 −0.122024 0.992527i \(-0.538939\pi\)
−0.122024 + 0.992527i \(0.538939\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) 12.7988 0.428774
\(892\) 0 0
\(893\) −19.4850 −0.652041
\(894\) 0 0
\(895\) 18.5909 0.621424
\(896\) 0 0
\(897\) −47.1527 −1.57438
\(898\) 0 0
\(899\) 51.2016 1.70767
\(900\) 0 0
\(901\) 4.68120 0.155953
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) −4.06715 −0.135197
\(906\) 0 0
\(907\) 53.2523 1.76821 0.884107 0.467285i \(-0.154768\pi\)
0.884107 + 0.467285i \(0.154768\pi\)
\(908\) 0 0
\(909\) −79.5703 −2.63918
\(910\) 0 0
\(911\) −30.2046 −1.00072 −0.500361 0.865817i \(-0.666799\pi\)
−0.500361 + 0.865817i \(0.666799\pi\)
\(912\) 0 0
\(913\) −37.6725 −1.24678
\(914\) 0 0
\(915\) −12.3469 −0.408176
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) −37.1650 −1.22596 −0.612980 0.790099i \(-0.710029\pi\)
−0.612980 + 0.790099i \(0.710029\pi\)
\(920\) 0 0
\(921\) −10.5502 −0.347642
\(922\) 0 0
\(923\) −5.96961 −0.196492
\(924\) 0 0
\(925\) −6.19592 −0.203721
\(926\) 0 0
\(927\) 71.1027 2.33532
\(928\) 0 0
\(929\) −46.2523 −1.51749 −0.758744 0.651389i \(-0.774187\pi\)
−0.758744 + 0.651389i \(0.774187\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 0 0
\(933\) 16.3597 0.535593
\(934\) 0 0
\(935\) −19.3933 −0.634228
\(936\) 0 0
\(937\) 29.1366 0.951850 0.475925 0.879486i \(-0.342113\pi\)
0.475925 + 0.879486i \(0.342113\pi\)
\(938\) 0 0
\(939\) 40.3682 1.31737
\(940\) 0 0
\(941\) 32.7856 1.06878 0.534391 0.845238i \(-0.320541\pi\)
0.534391 + 0.845238i \(0.320541\pi\)
\(942\) 0 0
\(943\) −9.06557 −0.295216
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −23.0635 −0.749465 −0.374732 0.927133i \(-0.622265\pi\)
−0.374732 + 0.927133i \(0.622265\pi\)
\(948\) 0 0
\(949\) 8.39330 0.272458
\(950\) 0 0
\(951\) 83.0561 2.69328
\(952\) 0 0
\(953\) 16.8472 0.545734 0.272867 0.962052i \(-0.412028\pi\)
0.272867 + 0.962052i \(0.412028\pi\)
\(954\) 0 0
\(955\) 9.61512 0.311138
\(956\) 0 0
\(957\) −85.9958 −2.77985
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) 52.0276 1.67831
\(962\) 0 0
\(963\) 41.1297 1.32539
\(964\) 0 0
\(965\) −21.1726 −0.681570
\(966\) 0 0
\(967\) 12.4420 0.400107 0.200053 0.979785i \(-0.435888\pi\)
0.200053 + 0.979785i \(0.435888\pi\)
\(968\) 0 0
\(969\) 21.0988 0.677791
\(970\) 0 0
\(971\) −10.8254 −0.347404 −0.173702 0.984798i \(-0.555573\pi\)
−0.173702 + 0.984798i \(0.555573\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) 22.1322 0.708798
\(976\) 0 0
\(977\) −26.1505 −0.836628 −0.418314 0.908303i \(-0.637379\pi\)
−0.418314 + 0.908303i \(0.637379\pi\)
\(978\) 0 0
\(979\) 22.3543 0.714445
\(980\) 0 0
\(981\) −62.4509 −1.99390
\(982\) 0 0
\(983\) 44.7712 1.42798 0.713990 0.700156i \(-0.246886\pi\)
0.713990 + 0.700156i \(0.246886\pi\)
\(984\) 0 0
\(985\) 22.1913 0.707075
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 70.9863 2.25723
\(990\) 0 0
\(991\) −22.5346 −0.715834 −0.357917 0.933753i \(-0.616513\pi\)
−0.357917 + 0.933753i \(0.616513\pi\)
\(992\) 0 0
\(993\) 55.5594 1.76312
\(994\) 0 0
\(995\) 0.839576 0.0266163
\(996\) 0 0
\(997\) −0.655575 −0.0207623 −0.0103811 0.999946i \(-0.503304\pi\)
−0.0103811 + 0.999946i \(0.503304\pi\)
\(998\) 0 0
\(999\) 5.95280 0.188338
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 8036.2.a.q.1.13 15
7.2 even 3 1148.2.i.e.165.3 30
7.4 even 3 1148.2.i.e.821.3 yes 30
7.6 odd 2 8036.2.a.r.1.3 15
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1148.2.i.e.165.3 30 7.2 even 3
1148.2.i.e.821.3 yes 30 7.4 even 3
8036.2.a.q.1.13 15 1.1 even 1 trivial
8036.2.a.r.1.3 15 7.6 odd 2