Properties

Label 7800.2.a.bp.1.3
Level $7800$
Weight $2$
Character 7800.1
Self dual yes
Analytic conductor $62.283$
Analytic rank $1$
Dimension $3$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 7800 = 2^{3} \cdot 3 \cdot 5^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 7800.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(62.2833135766\)
Analytic rank: \(1\)
Dimension: \(3\)
Coefficient field: 3.3.568.1
Defining polynomial: \(x^{3} - x^{2} - 6 x - 2\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 1560)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(-0.363328\) of defining polynomial
Character \(\chi\) \(=\) 7800.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{3} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{3} +1.00000 q^{9} +1.50466 q^{11} +1.00000 q^{13} -2.72666 q^{17} +0.726656 q^{19} -4.72666 q^{23} +1.00000 q^{27} -7.55602 q^{29} -3.00933 q^{31} +1.50466 q^{33} -5.00933 q^{37} +1.00000 q^{39} +5.78734 q^{41} +2.72666 q^{43} -10.2313 q^{47} -7.00000 q^{49} -2.72666 q^{51} -7.55602 q^{53} +0.726656 q^{57} -12.5140 q^{59} +6.28267 q^{61} -12.5653 q^{67} -4.72666 q^{69} +4.77801 q^{71} +12.0187 q^{73} +5.27334 q^{79} +1.00000 q^{81} +7.78734 q^{83} -7.55602 q^{87} +1.78734 q^{89} -3.00933 q^{93} -6.00000 q^{97} +1.50466 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3q + 3q^{3} + 3q^{9} + O(q^{10}) \) \( 3q + 3q^{3} + 3q^{9} - 6q^{11} + 3q^{13} - 4q^{17} - 2q^{19} - 10q^{23} + 3q^{27} - 10q^{29} + 12q^{31} - 6q^{33} + 6q^{37} + 3q^{39} - 10q^{41} + 4q^{43} - 16q^{47} - 21q^{49} - 4q^{51} - 10q^{53} - 2q^{57} - 6q^{59} + 2q^{61} - 4q^{67} - 10q^{69} + 8q^{71} - 6q^{73} + 20q^{79} + 3q^{81} - 4q^{83} - 10q^{87} - 22q^{89} + 12q^{93} - 18q^{97} - 6q^{99} + O(q^{100}) \)

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 0
\(6\) 0 0
\(7\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 1.50466 0.453673 0.226837 0.973933i \(-0.427162\pi\)
0.226837 + 0.973933i \(0.427162\pi\)
\(12\) 0 0
\(13\) 1.00000 0.277350
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −2.72666 −0.661311 −0.330656 0.943751i \(-0.607270\pi\)
−0.330656 + 0.943751i \(0.607270\pi\)
\(18\) 0 0
\(19\) 0.726656 0.166706 0.0833532 0.996520i \(-0.473437\pi\)
0.0833532 + 0.996520i \(0.473437\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −4.72666 −0.985576 −0.492788 0.870149i \(-0.664022\pi\)
−0.492788 + 0.870149i \(0.664022\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 1.00000 0.192450
\(28\) 0 0
\(29\) −7.55602 −1.40312 −0.701558 0.712612i \(-0.747512\pi\)
−0.701558 + 0.712612i \(0.747512\pi\)
\(30\) 0 0
\(31\) −3.00933 −0.540491 −0.270246 0.962791i \(-0.587105\pi\)
−0.270246 + 0.962791i \(0.587105\pi\)
\(32\) 0 0
\(33\) 1.50466 0.261928
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −5.00933 −0.823529 −0.411764 0.911290i \(-0.635087\pi\)
−0.411764 + 0.911290i \(0.635087\pi\)
\(38\) 0 0
\(39\) 1.00000 0.160128
\(40\) 0 0
\(41\) 5.78734 0.903830 0.451915 0.892061i \(-0.350741\pi\)
0.451915 + 0.892061i \(0.350741\pi\)
\(42\) 0 0
\(43\) 2.72666 0.415811 0.207906 0.978149i \(-0.433335\pi\)
0.207906 + 0.978149i \(0.433335\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −10.2313 −1.49239 −0.746196 0.665727i \(-0.768122\pi\)
−0.746196 + 0.665727i \(0.768122\pi\)
\(48\) 0 0
\(49\) −7.00000 −1.00000
\(50\) 0 0
\(51\) −2.72666 −0.381808
\(52\) 0 0
\(53\) −7.55602 −1.03790 −0.518949 0.854805i \(-0.673677\pi\)
−0.518949 + 0.854805i \(0.673677\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 0.726656 0.0962480
\(58\) 0 0
\(59\) −12.5140 −1.62918 −0.814592 0.580035i \(-0.803039\pi\)
−0.814592 + 0.580035i \(0.803039\pi\)
\(60\) 0 0
\(61\) 6.28267 0.804414 0.402207 0.915549i \(-0.368243\pi\)
0.402207 + 0.915549i \(0.368243\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) −12.5653 −1.53510 −0.767551 0.640988i \(-0.778525\pi\)
−0.767551 + 0.640988i \(0.778525\pi\)
\(68\) 0 0
\(69\) −4.72666 −0.569023
\(70\) 0 0
\(71\) 4.77801 0.567045 0.283523 0.958966i \(-0.408497\pi\)
0.283523 + 0.958966i \(0.408497\pi\)
\(72\) 0 0
\(73\) 12.0187 1.40668 0.703339 0.710855i \(-0.251692\pi\)
0.703339 + 0.710855i \(0.251692\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 5.27334 0.593297 0.296649 0.954987i \(-0.404131\pi\)
0.296649 + 0.954987i \(0.404131\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) 7.78734 0.854771 0.427386 0.904069i \(-0.359435\pi\)
0.427386 + 0.904069i \(0.359435\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) −7.55602 −0.810090
\(88\) 0 0
\(89\) 1.78734 0.189457 0.0947286 0.995503i \(-0.469802\pi\)
0.0947286 + 0.995503i \(0.469802\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) −3.00933 −0.312053
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −6.00000 −0.609208 −0.304604 0.952479i \(-0.598524\pi\)
−0.304604 + 0.952479i \(0.598524\pi\)
\(98\) 0 0
\(99\) 1.50466 0.151224
\(100\) 0 0
\(101\) −2.99067 −0.297583 −0.148791 0.988869i \(-0.547538\pi\)
−0.148791 + 0.988869i \(0.547538\pi\)
\(102\) 0 0
\(103\) −0.443984 −0.0437471 −0.0218735 0.999761i \(-0.506963\pi\)
−0.0218735 + 0.999761i \(0.506963\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 7.00933 0.677617 0.338809 0.940855i \(-0.389976\pi\)
0.338809 + 0.940855i \(0.389976\pi\)
\(108\) 0 0
\(109\) −13.8387 −1.32551 −0.662753 0.748838i \(-0.730612\pi\)
−0.662753 + 0.748838i \(0.730612\pi\)
\(110\) 0 0
\(111\) −5.00933 −0.475464
\(112\) 0 0
\(113\) −4.28267 −0.402880 −0.201440 0.979501i \(-0.564562\pi\)
−0.201440 + 0.979501i \(0.564562\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 1.00000 0.0924500
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) −8.73599 −0.794180
\(122\) 0 0
\(123\) 5.78734 0.521827
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) −5.71733 −0.507331 −0.253665 0.967292i \(-0.581636\pi\)
−0.253665 + 0.967292i \(0.581636\pi\)
\(128\) 0 0
\(129\) 2.72666 0.240069
\(130\) 0 0
\(131\) 5.55602 0.485431 0.242716 0.970097i \(-0.421962\pi\)
0.242716 + 0.970097i \(0.421962\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −6.67531 −0.570310 −0.285155 0.958481i \(-0.592045\pi\)
−0.285155 + 0.958481i \(0.592045\pi\)
\(138\) 0 0
\(139\) 19.4720 1.65159 0.825795 0.563970i \(-0.190727\pi\)
0.825795 + 0.563970i \(0.190727\pi\)
\(140\) 0 0
\(141\) −10.2313 −0.861633
\(142\) 0 0
\(143\) 1.50466 0.125826
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) −7.00000 −0.577350
\(148\) 0 0
\(149\) −14.5140 −1.18903 −0.594516 0.804084i \(-0.702656\pi\)
−0.594516 + 0.804084i \(0.702656\pi\)
\(150\) 0 0
\(151\) −4.46264 −0.363165 −0.181582 0.983376i \(-0.558122\pi\)
−0.181582 + 0.983376i \(0.558122\pi\)
\(152\) 0 0
\(153\) −2.72666 −0.220437
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 8.30133 0.662518 0.331259 0.943540i \(-0.392527\pi\)
0.331259 + 0.943540i \(0.392527\pi\)
\(158\) 0 0
\(159\) −7.55602 −0.599231
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) −11.0093 −0.862317 −0.431159 0.902276i \(-0.641895\pi\)
−0.431159 + 0.902276i \(0.641895\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −5.76868 −0.446394 −0.223197 0.974773i \(-0.571649\pi\)
−0.223197 + 0.974773i \(0.571649\pi\)
\(168\) 0 0
\(169\) 1.00000 0.0769231
\(170\) 0 0
\(171\) 0.726656 0.0555688
\(172\) 0 0
\(173\) 4.90663 0.373044 0.186522 0.982451i \(-0.440278\pi\)
0.186522 + 0.982451i \(0.440278\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) −12.5140 −0.940609
\(178\) 0 0
\(179\) 9.45331 0.706574 0.353287 0.935515i \(-0.385064\pi\)
0.353287 + 0.935515i \(0.385064\pi\)
\(180\) 0 0
\(181\) 17.4720 1.29868 0.649341 0.760498i \(-0.275045\pi\)
0.649341 + 0.760498i \(0.275045\pi\)
\(182\) 0 0
\(183\) 6.28267 0.464428
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) −4.10270 −0.300019
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −11.1120 −0.804038 −0.402019 0.915631i \(-0.631691\pi\)
−0.402019 + 0.915631i \(0.631691\pi\)
\(192\) 0 0
\(193\) 6.10270 0.439282 0.219641 0.975581i \(-0.429511\pi\)
0.219641 + 0.975581i \(0.429511\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 21.3620 1.52198 0.760990 0.648764i \(-0.224714\pi\)
0.760990 + 0.648764i \(0.224714\pi\)
\(198\) 0 0
\(199\) 8.38538 0.594423 0.297212 0.954812i \(-0.403943\pi\)
0.297212 + 0.954812i \(0.403943\pi\)
\(200\) 0 0
\(201\) −12.5653 −0.886291
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) −4.72666 −0.328525
\(208\) 0 0
\(209\) 1.09337 0.0756303
\(210\) 0 0
\(211\) −1.27334 −0.0876606 −0.0438303 0.999039i \(-0.513956\pi\)
−0.0438303 + 0.999039i \(0.513956\pi\)
\(212\) 0 0
\(213\) 4.77801 0.327384
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) 12.0187 0.812146
\(220\) 0 0
\(221\) −2.72666 −0.183415
\(222\) 0 0
\(223\) −16.4626 −1.10242 −0.551210 0.834367i \(-0.685834\pi\)
−0.551210 + 0.834367i \(0.685834\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) −12.2500 −0.813060 −0.406530 0.913638i \(-0.633261\pi\)
−0.406530 + 0.913638i \(0.633261\pi\)
\(228\) 0 0
\(229\) −1.27334 −0.0841449 −0.0420725 0.999115i \(-0.513396\pi\)
−0.0420725 + 0.999115i \(0.513396\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −18.3013 −1.19896 −0.599480 0.800390i \(-0.704626\pi\)
−0.599480 + 0.800390i \(0.704626\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 5.27334 0.342540
\(238\) 0 0
\(239\) 6.23132 0.403071 0.201535 0.979481i \(-0.435407\pi\)
0.201535 + 0.979481i \(0.435407\pi\)
\(240\) 0 0
\(241\) 19.5560 1.25971 0.629857 0.776711i \(-0.283114\pi\)
0.629857 + 0.776711i \(0.283114\pi\)
\(242\) 0 0
\(243\) 1.00000 0.0641500
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 0.726656 0.0462360
\(248\) 0 0
\(249\) 7.78734 0.493502
\(250\) 0 0
\(251\) 26.4813 1.67148 0.835742 0.549122i \(-0.185038\pi\)
0.835742 + 0.549122i \(0.185038\pi\)
\(252\) 0 0
\(253\) −7.11203 −0.447130
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −14.8294 −0.925030 −0.462515 0.886611i \(-0.653053\pi\)
−0.462515 + 0.886611i \(0.653053\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) −7.55602 −0.467706
\(262\) 0 0
\(263\) −22.9507 −1.41520 −0.707601 0.706612i \(-0.750223\pi\)
−0.707601 + 0.706612i \(0.750223\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 1.78734 0.109383
\(268\) 0 0
\(269\) −27.9160 −1.70207 −0.851033 0.525112i \(-0.824023\pi\)
−0.851033 + 0.525112i \(0.824023\pi\)
\(270\) 0 0
\(271\) −5.65872 −0.343743 −0.171871 0.985119i \(-0.554981\pi\)
−0.171871 + 0.985119i \(0.554981\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) −7.73599 −0.464810 −0.232405 0.972619i \(-0.574660\pi\)
−0.232405 + 0.972619i \(0.574660\pi\)
\(278\) 0 0
\(279\) −3.00933 −0.180164
\(280\) 0 0
\(281\) 5.11929 0.305391 0.152696 0.988273i \(-0.451205\pi\)
0.152696 + 0.988273i \(0.451205\pi\)
\(282\) 0 0
\(283\) −6.90663 −0.410556 −0.205278 0.978704i \(-0.565810\pi\)
−0.205278 + 0.978704i \(0.565810\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) −9.56534 −0.562667
\(290\) 0 0
\(291\) −6.00000 −0.351726
\(292\) 0 0
\(293\) −26.3527 −1.53954 −0.769770 0.638321i \(-0.779629\pi\)
−0.769770 + 0.638321i \(0.779629\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 1.50466 0.0873095
\(298\) 0 0
\(299\) −4.72666 −0.273350
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) −2.99067 −0.171810
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) −20.0000 −1.14146 −0.570730 0.821138i \(-0.693340\pi\)
−0.570730 + 0.821138i \(0.693340\pi\)
\(308\) 0 0
\(309\) −0.443984 −0.0252574
\(310\) 0 0
\(311\) −11.8973 −0.674634 −0.337317 0.941391i \(-0.609519\pi\)
−0.337317 + 0.941391i \(0.609519\pi\)
\(312\) 0 0
\(313\) −17.7360 −1.00250 −0.501249 0.865303i \(-0.667126\pi\)
−0.501249 + 0.865303i \(0.667126\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 1.32469 0.0744023 0.0372011 0.999308i \(-0.488156\pi\)
0.0372011 + 0.999308i \(0.488156\pi\)
\(318\) 0 0
\(319\) −11.3693 −0.636557
\(320\) 0 0
\(321\) 7.00933 0.391223
\(322\) 0 0
\(323\) −1.98134 −0.110245
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) −13.8387 −0.765281
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) −3.73599 −0.205348 −0.102674 0.994715i \(-0.532740\pi\)
−0.102674 + 0.994715i \(0.532740\pi\)
\(332\) 0 0
\(333\) −5.00933 −0.274510
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) −23.3947 −1.27439 −0.637195 0.770702i \(-0.719906\pi\)
−0.637195 + 0.770702i \(0.719906\pi\)
\(338\) 0 0
\(339\) −4.28267 −0.232603
\(340\) 0 0
\(341\) −4.52803 −0.245207
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 30.5840 1.64184 0.820918 0.571047i \(-0.193462\pi\)
0.820918 + 0.571047i \(0.193462\pi\)
\(348\) 0 0
\(349\) 1.37605 0.0736581 0.0368290 0.999322i \(-0.488274\pi\)
0.0368290 + 0.999322i \(0.488274\pi\)
\(350\) 0 0
\(351\) 1.00000 0.0533761
\(352\) 0 0
\(353\) 19.9087 1.05963 0.529817 0.848112i \(-0.322261\pi\)
0.529817 + 0.848112i \(0.322261\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 26.6940 1.40885 0.704427 0.709777i \(-0.251204\pi\)
0.704427 + 0.709777i \(0.251204\pi\)
\(360\) 0 0
\(361\) −18.4720 −0.972209
\(362\) 0 0
\(363\) −8.73599 −0.458520
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 8.12136 0.423932 0.211966 0.977277i \(-0.432013\pi\)
0.211966 + 0.977277i \(0.432013\pi\)
\(368\) 0 0
\(369\) 5.78734 0.301277
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) 9.47197 0.490440 0.245220 0.969467i \(-0.421140\pi\)
0.245220 + 0.969467i \(0.421140\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −7.55602 −0.389155
\(378\) 0 0
\(379\) 16.7267 0.859191 0.429595 0.903022i \(-0.358656\pi\)
0.429595 + 0.903022i \(0.358656\pi\)
\(380\) 0 0
\(381\) −5.71733 −0.292908
\(382\) 0 0
\(383\) 19.6846 1.00584 0.502919 0.864334i \(-0.332259\pi\)
0.502919 + 0.864334i \(0.332259\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 2.72666 0.138604
\(388\) 0 0
\(389\) −14.0000 −0.709828 −0.354914 0.934899i \(-0.615490\pi\)
−0.354914 + 0.934899i \(0.615490\pi\)
\(390\) 0 0
\(391\) 12.8880 0.651773
\(392\) 0 0
\(393\) 5.55602 0.280264
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) −3.35061 −0.168162 −0.0840812 0.996459i \(-0.526795\pi\)
−0.0840812 + 0.996459i \(0.526795\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −36.3713 −1.81630 −0.908149 0.418647i \(-0.862504\pi\)
−0.908149 + 0.418647i \(0.862504\pi\)
\(402\) 0 0
\(403\) −3.00933 −0.149905
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) −7.53736 −0.373613
\(408\) 0 0
\(409\) 35.5933 1.75998 0.879988 0.474995i \(-0.157550\pi\)
0.879988 + 0.474995i \(0.157550\pi\)
\(410\) 0 0
\(411\) −6.67531 −0.329269
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 19.4720 0.953546
\(418\) 0 0
\(419\) −13.9160 −0.679839 −0.339919 0.940455i \(-0.610400\pi\)
−0.339919 + 0.940455i \(0.610400\pi\)
\(420\) 0 0
\(421\) 4.70800 0.229454 0.114727 0.993397i \(-0.463401\pi\)
0.114727 + 0.993397i \(0.463401\pi\)
\(422\) 0 0
\(423\) −10.2313 −0.497464
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) 1.50466 0.0726459
\(430\) 0 0
\(431\) 7.89004 0.380050 0.190025 0.981779i \(-0.439143\pi\)
0.190025 + 0.981779i \(0.439143\pi\)
\(432\) 0 0
\(433\) −2.30133 −0.110595 −0.0552974 0.998470i \(-0.517611\pi\)
−0.0552974 + 0.998470i \(0.517611\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −3.43466 −0.164302
\(438\) 0 0
\(439\) 41.1307 1.96306 0.981530 0.191307i \(-0.0612725\pi\)
0.981530 + 0.191307i \(0.0612725\pi\)
\(440\) 0 0
\(441\) −7.00000 −0.333333
\(442\) 0 0
\(443\) −29.5933 −1.40602 −0.703011 0.711179i \(-0.748161\pi\)
−0.703011 + 0.711179i \(0.748161\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) −14.5140 −0.686488
\(448\) 0 0
\(449\) −11.4461 −0.540173 −0.270086 0.962836i \(-0.587052\pi\)
−0.270086 + 0.962836i \(0.587052\pi\)
\(450\) 0 0
\(451\) 8.70800 0.410044
\(452\) 0 0
\(453\) −4.46264 −0.209673
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −16.5467 −0.774021 −0.387011 0.922075i \(-0.626492\pi\)
−0.387011 + 0.922075i \(0.626492\pi\)
\(458\) 0 0
\(459\) −2.72666 −0.127269
\(460\) 0 0
\(461\) 3.50466 0.163228 0.0816142 0.996664i \(-0.473992\pi\)
0.0816142 + 0.996664i \(0.473992\pi\)
\(462\) 0 0
\(463\) −16.0000 −0.743583 −0.371792 0.928316i \(-0.621256\pi\)
−0.371792 + 0.928316i \(0.621256\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −6.44398 −0.298192 −0.149096 0.988823i \(-0.547636\pi\)
−0.149096 + 0.988823i \(0.547636\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 8.30133 0.382505
\(472\) 0 0
\(473\) 4.10270 0.188642
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) −7.55602 −0.345966
\(478\) 0 0
\(479\) 22.2313 1.01577 0.507887 0.861423i \(-0.330427\pi\)
0.507887 + 0.861423i \(0.330427\pi\)
\(480\) 0 0
\(481\) −5.00933 −0.228406
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) 13.9160 0.630592 0.315296 0.948993i \(-0.397896\pi\)
0.315296 + 0.948993i \(0.397896\pi\)
\(488\) 0 0
\(489\) −11.0093 −0.497859
\(490\) 0 0
\(491\) −28.1400 −1.26994 −0.634971 0.772536i \(-0.718988\pi\)
−0.634971 + 0.772536i \(0.718988\pi\)
\(492\) 0 0
\(493\) 20.6027 0.927897
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) 38.8480 1.73908 0.869538 0.493866i \(-0.164417\pi\)
0.869538 + 0.493866i \(0.164417\pi\)
\(500\) 0 0
\(501\) −5.76868 −0.257726
\(502\) 0 0
\(503\) −8.19863 −0.365559 −0.182779 0.983154i \(-0.558509\pi\)
−0.182779 + 0.983154i \(0.558509\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 1.00000 0.0444116
\(508\) 0 0
\(509\) 16.0700 0.712291 0.356145 0.934431i \(-0.384091\pi\)
0.356145 + 0.934431i \(0.384091\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) 0.726656 0.0320827
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) −15.3947 −0.677058
\(518\) 0 0
\(519\) 4.90663 0.215377
\(520\) 0 0
\(521\) −23.0280 −1.00887 −0.504437 0.863448i \(-0.668300\pi\)
−0.504437 + 0.863448i \(0.668300\pi\)
\(522\) 0 0
\(523\) −5.47875 −0.239569 −0.119784 0.992800i \(-0.538220\pi\)
−0.119784 + 0.992800i \(0.538220\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 8.20541 0.357433
\(528\) 0 0
\(529\) −0.658719 −0.0286399
\(530\) 0 0
\(531\) −12.5140 −0.543061
\(532\) 0 0
\(533\) 5.78734 0.250677
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 9.45331 0.407941
\(538\) 0 0
\(539\) −10.5327 −0.453673
\(540\) 0 0
\(541\) 17.8387 0.766945 0.383473 0.923552i \(-0.374728\pi\)
0.383473 + 0.923552i \(0.374728\pi\)
\(542\) 0 0
\(543\) 17.4720 0.749794
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) −29.1053 −1.24445 −0.622225 0.782838i \(-0.713771\pi\)
−0.622225 + 0.782838i \(0.713771\pi\)
\(548\) 0 0
\(549\) 6.28267 0.268138
\(550\) 0 0
\(551\) −5.49063 −0.233909
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −44.0114 −1.86482 −0.932411 0.361399i \(-0.882299\pi\)
−0.932411 + 0.361399i \(0.882299\pi\)
\(558\) 0 0
\(559\) 2.72666 0.115325
\(560\) 0 0
\(561\) −4.10270 −0.173216
\(562\) 0 0
\(563\) −7.43466 −0.313333 −0.156667 0.987652i \(-0.550075\pi\)
−0.156667 + 0.987652i \(0.550075\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 8.48130 0.355555 0.177777 0.984071i \(-0.443109\pi\)
0.177777 + 0.984071i \(0.443109\pi\)
\(570\) 0 0
\(571\) 40.7826 1.70670 0.853350 0.521339i \(-0.174567\pi\)
0.853350 + 0.521339i \(0.174567\pi\)
\(572\) 0 0
\(573\) −11.1120 −0.464212
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) −1.57467 −0.0655545 −0.0327773 0.999463i \(-0.510435\pi\)
−0.0327773 + 0.999463i \(0.510435\pi\)
\(578\) 0 0
\(579\) 6.10270 0.253620
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) −11.3693 −0.470867
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −27.8247 −1.14845 −0.574223 0.818699i \(-0.694696\pi\)
−0.574223 + 0.818699i \(0.694696\pi\)
\(588\) 0 0
\(589\) −2.18675 −0.0901034
\(590\) 0 0
\(591\) 21.3620 0.878716
\(592\) 0 0
\(593\) −32.6940 −1.34258 −0.671290 0.741195i \(-0.734260\pi\)
−0.671290 + 0.741195i \(0.734260\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 8.38538 0.343191
\(598\) 0 0
\(599\) −32.1400 −1.31321 −0.656603 0.754237i \(-0.728007\pi\)
−0.656603 + 0.754237i \(0.728007\pi\)
\(600\) 0 0
\(601\) 40.8667 1.66699 0.833493 0.552530i \(-0.186337\pi\)
0.833493 + 0.552530i \(0.186337\pi\)
\(602\) 0 0
\(603\) −12.5653 −0.511700
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) −14.9907 −0.608453 −0.304226 0.952600i \(-0.598398\pi\)
−0.304226 + 0.952600i \(0.598398\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) −10.2313 −0.413915
\(612\) 0 0
\(613\) −13.5747 −0.548276 −0.274138 0.961690i \(-0.588392\pi\)
−0.274138 + 0.961690i \(0.588392\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 4.69396 0.188972 0.0944859 0.995526i \(-0.469879\pi\)
0.0944859 + 0.995526i \(0.469879\pi\)
\(618\) 0 0
\(619\) −20.8667 −0.838702 −0.419351 0.907824i \(-0.637742\pi\)
−0.419351 + 0.907824i \(0.637742\pi\)
\(620\) 0 0
\(621\) −4.72666 −0.189674
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) 1.09337 0.0436652
\(628\) 0 0
\(629\) 13.6587 0.544609
\(630\) 0 0
\(631\) 45.9533 1.82937 0.914685 0.404167i \(-0.132438\pi\)
0.914685 + 0.404167i \(0.132438\pi\)
\(632\) 0 0
\(633\) −1.27334 −0.0506109
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) −7.00000 −0.277350
\(638\) 0 0
\(639\) 4.77801 0.189015
\(640\) 0 0
\(641\) 20.0187 0.790689 0.395345 0.918533i \(-0.370625\pi\)
0.395345 + 0.918533i \(0.370625\pi\)
\(642\) 0 0
\(643\) 26.5840 1.04837 0.524185 0.851604i \(-0.324370\pi\)
0.524185 + 0.851604i \(0.324370\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −9.39470 −0.369344 −0.184672 0.982800i \(-0.559122\pi\)
−0.184672 + 0.982800i \(0.559122\pi\)
\(648\) 0 0
\(649\) −18.8294 −0.739117
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −2.00000 −0.0782660 −0.0391330 0.999234i \(-0.512460\pi\)
−0.0391330 + 0.999234i \(0.512460\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 12.0187 0.468892
\(658\) 0 0
\(659\) 47.6774 1.85725 0.928623 0.371024i \(-0.120993\pi\)
0.928623 + 0.371024i \(0.120993\pi\)
\(660\) 0 0
\(661\) −22.0959 −0.859432 −0.429716 0.902964i \(-0.641386\pi\)
−0.429716 + 0.902964i \(0.641386\pi\)
\(662\) 0 0
\(663\) −2.72666 −0.105895
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 35.7147 1.38288
\(668\) 0 0
\(669\) −16.4626 −0.636482
\(670\) 0 0
\(671\) 9.45331 0.364941
\(672\) 0 0
\(673\) 48.6027 1.87349 0.936747 0.350006i \(-0.113820\pi\)
0.936747 + 0.350006i \(0.113820\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) −10.4253 −0.400678 −0.200339 0.979727i \(-0.564204\pi\)
−0.200339 + 0.979727i \(0.564204\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) −12.2500 −0.469420
\(682\) 0 0
\(683\) 9.13795 0.349654 0.174827 0.984599i \(-0.444063\pi\)
0.174827 + 0.984599i \(0.444063\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) −1.27334 −0.0485811
\(688\) 0 0
\(689\) −7.55602 −0.287861
\(690\) 0 0
\(691\) 33.7173 1.28267 0.641334 0.767262i \(-0.278381\pi\)
0.641334 + 0.767262i \(0.278381\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) −15.7801 −0.597713
\(698\) 0 0
\(699\) −18.3013 −0.692220
\(700\) 0 0
\(701\) −19.9813 −0.754685 −0.377342 0.926074i \(-0.623162\pi\)
−0.377342 + 0.926074i \(0.623162\pi\)
\(702\) 0 0
\(703\) −3.64006 −0.137288
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) −43.3293 −1.62727 −0.813633 0.581378i \(-0.802514\pi\)
−0.813633 + 0.581378i \(0.802514\pi\)
\(710\) 0 0
\(711\) 5.27334 0.197766
\(712\) 0 0
\(713\) 14.2241 0.532695
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 6.23132 0.232713
\(718\) 0 0
\(719\) −34.6867 −1.29360 −0.646798 0.762661i \(-0.723892\pi\)
−0.646798 + 0.762661i \(0.723892\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) 19.5560 0.727296
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) 5.39470 0.200078 0.100039 0.994983i \(-0.468103\pi\)
0.100039 + 0.994983i \(0.468103\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) −7.43466 −0.274981
\(732\) 0 0
\(733\) 9.11203 0.336561 0.168280 0.985739i \(-0.446179\pi\)
0.168280 + 0.985739i \(0.446179\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −18.9066 −0.696435
\(738\) 0 0
\(739\) −4.82936 −0.177651 −0.0888254 0.996047i \(-0.528311\pi\)
−0.0888254 + 0.996047i \(0.528311\pi\)
\(740\) 0 0
\(741\) 0.726656 0.0266944
\(742\) 0 0
\(743\) 50.4087 1.84931 0.924657 0.380801i \(-0.124352\pi\)
0.924657 + 0.380801i \(0.124352\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 7.78734 0.284924
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) 4.88797 0.178365 0.0891823 0.996015i \(-0.471575\pi\)
0.0891823 + 0.996015i \(0.471575\pi\)
\(752\) 0 0
\(753\) 26.4813 0.965032
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 32.2241 1.17120 0.585602 0.810599i \(-0.300858\pi\)
0.585602 + 0.810599i \(0.300858\pi\)
\(758\) 0 0
\(759\) −7.11203 −0.258150
\(760\) 0 0
\(761\) 36.4740 1.32218 0.661091 0.750305i \(-0.270094\pi\)
0.661091 + 0.750305i \(0.270094\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) −12.5140 −0.451854
\(768\) 0 0
\(769\) 13.1120 0.472832 0.236416 0.971652i \(-0.424027\pi\)
0.236416 + 0.971652i \(0.424027\pi\)
\(770\) 0 0
\(771\) −14.8294 −0.534066
\(772\) 0 0
\(773\) −12.5913 −0.452876 −0.226438 0.974026i \(-0.572708\pi\)
−0.226438 + 0.974026i \(0.572708\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 4.20541 0.150674
\(780\) 0 0
\(781\) 7.18930 0.257253
\(782\) 0 0
\(783\) −7.55602 −0.270030
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) 36.9253 1.31624 0.658122 0.752911i \(-0.271351\pi\)
0.658122 + 0.752911i \(0.271351\pi\)
\(788\) 0 0
\(789\) −22.9507 −0.817067
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 6.28267 0.223104
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −25.0466 −0.887198 −0.443599 0.896225i \(-0.646298\pi\)
−0.443599 + 0.896225i \(0.646298\pi\)
\(798\) 0 0
\(799\) 27.8973 0.986935
\(800\) 0 0
\(801\) 1.78734 0.0631524
\(802\) 0 0
\(803\) 18.0840 0.638172
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) −27.9160 −0.982688
\(808\) 0 0
\(809\) 11.2334 0.394945 0.197473 0.980308i \(-0.436727\pi\)
0.197473 + 0.980308i \(0.436727\pi\)
\(810\) 0 0
\(811\) 31.8387 1.11801 0.559004 0.829165i \(-0.311184\pi\)
0.559004 + 0.829165i \(0.311184\pi\)
\(812\) 0 0
\(813\) −5.65872 −0.198460
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 1.98134 0.0693184
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −32.1073 −1.12055 −0.560277 0.828306i \(-0.689305\pi\)
−0.560277 + 0.828306i \(0.689305\pi\)
\(822\) 0 0
\(823\) −46.7054 −1.62805 −0.814023 0.580832i \(-0.802727\pi\)
−0.814023 + 0.580832i \(0.802727\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 26.6940 0.928240 0.464120 0.885772i \(-0.346371\pi\)
0.464120 + 0.885772i \(0.346371\pi\)
\(828\) 0 0
\(829\) −28.8294 −1.00129 −0.500643 0.865654i \(-0.666903\pi\)
−0.500643 + 0.865654i \(0.666903\pi\)
\(830\) 0 0
\(831\) −7.73599 −0.268358
\(832\) 0 0
\(833\) 19.0866 0.661311
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) −3.00933 −0.104018
\(838\) 0 0
\(839\) 7.68463 0.265303 0.132652 0.991163i \(-0.457651\pi\)
0.132652 + 0.991163i \(0.457651\pi\)
\(840\) 0 0
\(841\) 28.0934 0.968737
\(842\) 0 0
\(843\) 5.11929 0.176318
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) −6.90663 −0.237035
\(850\) 0 0
\(851\) 23.6774 0.811650
\(852\) 0 0
\(853\) 18.4626 0.632149 0.316074 0.948734i \(-0.397635\pi\)
0.316074 + 0.948734i \(0.397635\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −32.2827 −1.10276 −0.551378 0.834256i \(-0.685898\pi\)
−0.551378 + 0.834256i \(0.685898\pi\)
\(858\) 0 0
\(859\) −18.9694 −0.647227 −0.323613 0.946189i \(-0.604898\pi\)
−0.323613 + 0.946189i \(0.604898\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 16.8807 0.574626 0.287313 0.957837i \(-0.407238\pi\)
0.287313 + 0.957837i \(0.407238\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) −9.56534 −0.324856
\(868\) 0 0
\(869\) 7.93461 0.269163
\(870\) 0 0
\(871\) −12.5653 −0.425760
\(872\) 0 0
\(873\) −6.00000 −0.203069
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 28.1214 0.949591 0.474795 0.880096i \(-0.342522\pi\)
0.474795 + 0.880096i \(0.342522\pi\)
\(878\) 0 0
\(879\) −26.3527 −0.888854
\(880\) 0 0
\(881\) −49.4066 −1.66455 −0.832275 0.554363i \(-0.812962\pi\)
−0.832275 + 0.554363i \(0.812962\pi\)
\(882\) 0 0
\(883\) −5.65872 −0.190431 −0.0952155 0.995457i \(-0.530354\pi\)
−0.0952155 + 0.995457i \(0.530354\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 28.2640 0.949013 0.474506 0.880252i \(-0.342627\pi\)
0.474506 + 0.880252i \(0.342627\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) 1.50466 0.0504082
\(892\) 0 0
\(893\) −7.43466 −0.248791
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) −4.72666 −0.157818
\(898\) 0 0
\(899\) 22.7385 0.758373
\(900\) 0 0
\(901\) 20.6027 0.686374
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) −14.5840 −0.484254 −0.242127 0.970245i \(-0.577845\pi\)
−0.242127 + 0.970245i \(0.577845\pi\)
\(908\) 0 0
\(909\) −2.99067 −0.0991943
\(910\) 0 0
\(911\) 29.1680 0.966379 0.483190 0.875516i \(-0.339478\pi\)
0.483190 + 0.875516i \(0.339478\pi\)
\(912\) 0 0
\(913\) 11.7173 0.387787
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) 2.36672 0.0780708 0.0390354 0.999238i \(-0.487571\pi\)
0.0390354 + 0.999238i \(0.487571\pi\)
\(920\) 0 0
\(921\) −20.0000 −0.659022
\(922\) 0 0
\(923\) 4.77801 0.157270
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) −0.443984 −0.0145824
\(928\) 0 0
\(929\) 31.1753 1.02283 0.511414 0.859335i \(-0.329122\pi\)
0.511414 + 0.859335i \(0.329122\pi\)
\(930\) 0 0
\(931\) −5.08660 −0.166706
\(932\) 0 0
\(933\) −11.8973 −0.389500
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 6.46942 0.211347 0.105673 0.994401i \(-0.466300\pi\)
0.105673 + 0.994401i \(0.466300\pi\)
\(938\) 0 0
\(939\) −17.7360 −0.578792
\(940\) 0 0
\(941\) −19.4020 −0.632486 −0.316243 0.948678i \(-0.602421\pi\)
−0.316243 + 0.948678i \(0.602421\pi\)
\(942\) 0 0
\(943\) −27.3548 −0.890793
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 50.0632 1.62684 0.813418 0.581679i \(-0.197604\pi\)
0.813418 + 0.581679i \(0.197604\pi\)
\(948\) 0 0
\(949\) 12.0187 0.390142
\(950\) 0 0
\(951\) 1.32469 0.0429562
\(952\) 0 0
\(953\) 6.30133 0.204120 0.102060 0.994778i \(-0.467457\pi\)
0.102060 + 0.994778i \(0.467457\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) −11.3693 −0.367516
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) −21.9439 −0.707869
\(962\) 0 0
\(963\) 7.00933 0.225872
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) −37.2334 −1.19735 −0.598673 0.800994i \(-0.704305\pi\)
−0.598673 + 0.800994i \(0.704305\pi\)
\(968\) 0 0
\(969\) −1.98134 −0.0636499
\(970\) 0 0
\(971\) 13.4533 0.431737 0.215869 0.976422i \(-0.430742\pi\)
0.215869 + 0.976422i \(0.430742\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −37.4647 −1.19860 −0.599301 0.800524i \(-0.704555\pi\)
−0.599301 + 0.800524i \(0.704555\pi\)
\(978\) 0 0
\(979\) 2.68934 0.0859517
\(980\) 0 0
\(981\) −13.8387 −0.441835
\(982\) 0 0
\(983\) 29.2406 0.932632 0.466316 0.884618i \(-0.345581\pi\)
0.466316 + 0.884618i \(0.345581\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) −12.8880 −0.409814
\(990\) 0 0
\(991\) 3.11203 0.0988569 0.0494285 0.998778i \(-0.484260\pi\)
0.0494285 + 0.998778i \(0.484260\pi\)
\(992\) 0 0
\(993\) −3.73599 −0.118558
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) −42.3200 −1.34029 −0.670144 0.742231i \(-0.733768\pi\)
−0.670144 + 0.742231i \(0.733768\pi\)
\(998\) 0 0
\(999\) −5.00933 −0.158488
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 7800.2.a.bp.1.3 3
5.2 odd 4 1560.2.l.c.1249.1 6
5.3 odd 4 1560.2.l.c.1249.4 yes 6
5.4 even 2 7800.2.a.bj.1.3 3
15.2 even 4 4680.2.l.e.2809.5 6
15.8 even 4 4680.2.l.e.2809.6 6
20.3 even 4 3120.2.l.m.1249.1 6
20.7 even 4 3120.2.l.m.1249.4 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1560.2.l.c.1249.1 6 5.2 odd 4
1560.2.l.c.1249.4 yes 6 5.3 odd 4
3120.2.l.m.1249.1 6 20.3 even 4
3120.2.l.m.1249.4 6 20.7 even 4
4680.2.l.e.2809.5 6 15.2 even 4
4680.2.l.e.2809.6 6 15.8 even 4
7800.2.a.bj.1.3 3 5.4 even 2
7800.2.a.bp.1.3 3 1.1 even 1 trivial