Properties

Label 2646.2.d.e.2645.4
Level $2646$
Weight $2$
Character 2646.2645
Analytic conductor $21.128$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 2646 = 2 \cdot 3^{3} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2646.d (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(21.1284163748\)
Analytic rank: \(0\)
Dimension: \(16\)
Coefficient field: \(\Q(\zeta_{48})\)
Defining polynomial: \(x^{16} - x^{8} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{14} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 2645.4
Root \(-0.608761 + 0.793353i\) of defining polynomial
Character \(\chi\) \(=\) 2646.2645
Dual form 2646.2.d.e.2645.12

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000i q^{2} -1.00000 q^{4} -0.601731 q^{5} +1.00000i q^{8} +O(q^{10})\) \(q-1.00000i q^{2} -1.00000 q^{4} -0.601731 q^{5} +1.00000i q^{8} +0.601731i q^{10} +1.20041i q^{11} -0.649659i q^{13} +1.00000 q^{16} -0.564050 q^{17} +0.249245i q^{19} +0.601731 q^{20} +1.20041 q^{22} -3.95367i q^{23} -4.63792 q^{25} -0.649659 q^{26} +1.37199i q^{29} +7.11169i q^{31} -1.00000i q^{32} +0.564050i q^{34} +5.17738 q^{37} +0.249245 q^{38} -0.601731i q^{40} -7.32326 q^{41} +9.36414 q^{43} -1.20041i q^{44} -3.95367 q^{46} -9.12845 q^{47} +4.63792i q^{50} +0.649659i q^{52} +10.3178i q^{53} -0.722325i q^{55} +1.37199 q^{58} -7.26508 q^{59} -0.865467i q^{61} +7.11169 q^{62} -1.00000 q^{64} +0.390920i q^{65} -9.37779 q^{67} +0.564050 q^{68} +4.28897i q^{71} +15.7896i q^{73} -5.17738i q^{74} -0.249245i q^{76} +12.1462 q^{79} -0.601731 q^{80} +7.32326i q^{82} -5.58632 q^{83} +0.339406 q^{85} -9.36414i q^{86} -1.20041 q^{88} -7.62781 q^{89} +3.95367i q^{92} +9.12845i q^{94} -0.149978i q^{95} +13.2012i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 16 q^{4} + O(q^{10}) \) \( 16 q - 16 q^{4} + 16 q^{16} - 32 q^{22} + 48 q^{25} - 32 q^{37} + 16 q^{43} - 48 q^{46} + 16 q^{58} - 16 q^{64} + 16 q^{67} + 32 q^{88} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2646\mathbb{Z}\right)^\times\).

\(n\) \(785\) \(1081\)
\(\chi(n)\) \(-1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) − 1.00000i − 0.707107i
\(3\) 0 0
\(4\) −1.00000 −0.500000
\(5\) −0.601731 −0.269102 −0.134551 0.990907i \(-0.542959\pi\)
−0.134551 + 0.990907i \(0.542959\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) 1.00000i 0.353553i
\(9\) 0 0
\(10\) 0.601731i 0.190284i
\(11\) 1.20041i 0.361938i 0.983489 + 0.180969i \(0.0579234\pi\)
−0.983489 + 0.180969i \(0.942077\pi\)
\(12\) 0 0
\(13\) − 0.649659i − 0.180183i −0.995933 0.0900914i \(-0.971284\pi\)
0.995933 0.0900914i \(-0.0287159\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) −0.564050 −0.136802 −0.0684011 0.997658i \(-0.521790\pi\)
−0.0684011 + 0.997658i \(0.521790\pi\)
\(18\) 0 0
\(19\) 0.249245i 0.0571807i 0.999591 + 0.0285904i \(0.00910184\pi\)
−0.999591 + 0.0285904i \(0.990898\pi\)
\(20\) 0.601731 0.134551
\(21\) 0 0
\(22\) 1.20041 0.255929
\(23\) − 3.95367i − 0.824397i −0.911094 0.412198i \(-0.864761\pi\)
0.911094 0.412198i \(-0.135239\pi\)
\(24\) 0 0
\(25\) −4.63792 −0.927584
\(26\) −0.649659 −0.127409
\(27\) 0 0
\(28\) 0 0
\(29\) 1.37199i 0.254771i 0.991853 + 0.127386i \(0.0406586\pi\)
−0.991853 + 0.127386i \(0.959341\pi\)
\(30\) 0 0
\(31\) 7.11169i 1.27730i 0.769498 + 0.638649i \(0.220506\pi\)
−0.769498 + 0.638649i \(0.779494\pi\)
\(32\) − 1.00000i − 0.176777i
\(33\) 0 0
\(34\) 0.564050i 0.0967337i
\(35\) 0 0
\(36\) 0 0
\(37\) 5.17738 0.851155 0.425578 0.904922i \(-0.360071\pi\)
0.425578 + 0.904922i \(0.360071\pi\)
\(38\) 0.249245 0.0404329
\(39\) 0 0
\(40\) − 0.601731i − 0.0951420i
\(41\) −7.32326 −1.14370 −0.571850 0.820358i \(-0.693774\pi\)
−0.571850 + 0.820358i \(0.693774\pi\)
\(42\) 0 0
\(43\) 9.36414 1.42802 0.714009 0.700136i \(-0.246877\pi\)
0.714009 + 0.700136i \(0.246877\pi\)
\(44\) − 1.20041i − 0.180969i
\(45\) 0 0
\(46\) −3.95367 −0.582937
\(47\) −9.12845 −1.33152 −0.665760 0.746166i \(-0.731893\pi\)
−0.665760 + 0.746166i \(0.731893\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 4.63792i 0.655901i
\(51\) 0 0
\(52\) 0.649659i 0.0900914i
\(53\) 10.3178i 1.41726i 0.705580 + 0.708630i \(0.250687\pi\)
−0.705580 + 0.708630i \(0.749313\pi\)
\(54\) 0 0
\(55\) − 0.722325i − 0.0973983i
\(56\) 0 0
\(57\) 0 0
\(58\) 1.37199 0.180151
\(59\) −7.26508 −0.945833 −0.472916 0.881107i \(-0.656799\pi\)
−0.472916 + 0.881107i \(0.656799\pi\)
\(60\) 0 0
\(61\) − 0.865467i − 0.110812i −0.998464 0.0554059i \(-0.982355\pi\)
0.998464 0.0554059i \(-0.0176453\pi\)
\(62\) 7.11169 0.903186
\(63\) 0 0
\(64\) −1.00000 −0.125000
\(65\) 0.390920i 0.0484876i
\(66\) 0 0
\(67\) −9.37779 −1.14568 −0.572840 0.819668i \(-0.694158\pi\)
−0.572840 + 0.819668i \(0.694158\pi\)
\(68\) 0.564050 0.0684011
\(69\) 0 0
\(70\) 0 0
\(71\) 4.28897i 0.509007i 0.967072 + 0.254504i \(0.0819121\pi\)
−0.967072 + 0.254504i \(0.918088\pi\)
\(72\) 0 0
\(73\) 15.7896i 1.84803i 0.382356 + 0.924015i \(0.375113\pi\)
−0.382356 + 0.924015i \(0.624887\pi\)
\(74\) − 5.17738i − 0.601858i
\(75\) 0 0
\(76\) − 0.249245i − 0.0285904i
\(77\) 0 0
\(78\) 0 0
\(79\) 12.1462 1.36656 0.683279 0.730157i \(-0.260553\pi\)
0.683279 + 0.730157i \(0.260553\pi\)
\(80\) −0.601731 −0.0672755
\(81\) 0 0
\(82\) 7.32326i 0.808718i
\(83\) −5.58632 −0.613178 −0.306589 0.951842i \(-0.599188\pi\)
−0.306589 + 0.951842i \(0.599188\pi\)
\(84\) 0 0
\(85\) 0.339406 0.0368138
\(86\) − 9.36414i − 1.00976i
\(87\) 0 0
\(88\) −1.20041 −0.127964
\(89\) −7.62781 −0.808547 −0.404273 0.914638i \(-0.632476\pi\)
−0.404273 + 0.914638i \(0.632476\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 3.95367i 0.412198i
\(93\) 0 0
\(94\) 9.12845i 0.941528i
\(95\) − 0.149978i − 0.0153875i
\(96\) 0 0
\(97\) 13.2012i 1.34038i 0.742191 + 0.670189i \(0.233787\pi\)
−0.742191 + 0.670189i \(0.766213\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 4.63792 0.463792
\(101\) 5.20842 0.518257 0.259129 0.965843i \(-0.416565\pi\)
0.259129 + 0.965843i \(0.416565\pi\)
\(102\) 0 0
\(103\) 6.31086i 0.621828i 0.950438 + 0.310914i \(0.100635\pi\)
−0.950438 + 0.310914i \(0.899365\pi\)
\(104\) 0.649659 0.0637043
\(105\) 0 0
\(106\) 10.3178 1.00215
\(107\) 14.7551i 1.42643i 0.700948 + 0.713213i \(0.252761\pi\)
−0.700948 + 0.713213i \(0.747239\pi\)
\(108\) 0 0
\(109\) −1.94812 −0.186596 −0.0932982 0.995638i \(-0.529741\pi\)
−0.0932982 + 0.995638i \(0.529741\pi\)
\(110\) −0.722325 −0.0688710
\(111\) 0 0
\(112\) 0 0
\(113\) − 8.35305i − 0.785789i −0.919584 0.392894i \(-0.871474\pi\)
0.919584 0.392894i \(-0.128526\pi\)
\(114\) 0 0
\(115\) 2.37904i 0.221847i
\(116\) − 1.37199i − 0.127386i
\(117\) 0 0
\(118\) 7.26508i 0.668805i
\(119\) 0 0
\(120\) 0 0
\(121\) 9.55901 0.869001
\(122\) −0.865467 −0.0783557
\(123\) 0 0
\(124\) − 7.11169i − 0.638649i
\(125\) 5.79943 0.518717
\(126\) 0 0
\(127\) 1.09410 0.0970861 0.0485431 0.998821i \(-0.484542\pi\)
0.0485431 + 0.998821i \(0.484542\pi\)
\(128\) 1.00000i 0.0883883i
\(129\) 0 0
\(130\) 0.390920 0.0342859
\(131\) 6.60896 0.577427 0.288714 0.957415i \(-0.406772\pi\)
0.288714 + 0.957415i \(0.406772\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 9.37779i 0.810117i
\(135\) 0 0
\(136\) − 0.564050i − 0.0483669i
\(137\) 9.61488i 0.821455i 0.911758 + 0.410727i \(0.134725\pi\)
−0.911758 + 0.410727i \(0.865275\pi\)
\(138\) 0 0
\(139\) 1.95858i 0.166125i 0.996544 + 0.0830623i \(0.0264701\pi\)
−0.996544 + 0.0830623i \(0.973530\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 4.28897 0.359923
\(143\) 0.779858 0.0652150
\(144\) 0 0
\(145\) − 0.825566i − 0.0685595i
\(146\) 15.7896 1.30675
\(147\) 0 0
\(148\) −5.17738 −0.425578
\(149\) − 16.4159i − 1.34484i −0.740168 0.672422i \(-0.765254\pi\)
0.740168 0.672422i \(-0.234746\pi\)
\(150\) 0 0
\(151\) −7.35039 −0.598166 −0.299083 0.954227i \(-0.596681\pi\)
−0.299083 + 0.954227i \(0.596681\pi\)
\(152\) −0.249245 −0.0202164
\(153\) 0 0
\(154\) 0 0
\(155\) − 4.27932i − 0.343724i
\(156\) 0 0
\(157\) 13.3202i 1.06307i 0.847036 + 0.531535i \(0.178385\pi\)
−0.847036 + 0.531535i \(0.821615\pi\)
\(158\) − 12.1462i − 0.966303i
\(159\) 0 0
\(160\) 0.601731i 0.0475710i
\(161\) 0 0
\(162\) 0 0
\(163\) 11.5302 0.903113 0.451556 0.892243i \(-0.350869\pi\)
0.451556 + 0.892243i \(0.350869\pi\)
\(164\) 7.32326 0.571850
\(165\) 0 0
\(166\) 5.58632i 0.433582i
\(167\) −25.7937 −1.99598 −0.997990 0.0633754i \(-0.979813\pi\)
−0.997990 + 0.0633754i \(0.979813\pi\)
\(168\) 0 0
\(169\) 12.5779 0.967534
\(170\) − 0.339406i − 0.0260313i
\(171\) 0 0
\(172\) −9.36414 −0.714009
\(173\) −4.04800 −0.307764 −0.153882 0.988089i \(-0.549178\pi\)
−0.153882 + 0.988089i \(0.549178\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 1.20041i 0.0904845i
\(177\) 0 0
\(178\) 7.62781i 0.571729i
\(179\) 8.94172i 0.668336i 0.942514 + 0.334168i \(0.108455\pi\)
−0.942514 + 0.334168i \(0.891545\pi\)
\(180\) 0 0
\(181\) 1.77776i 0.132140i 0.997815 + 0.0660699i \(0.0210460\pi\)
−0.997815 + 0.0660699i \(0.978954\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 3.95367 0.291468
\(185\) −3.11539 −0.229048
\(186\) 0 0
\(187\) − 0.677092i − 0.0495139i
\(188\) 9.12845 0.665760
\(189\) 0 0
\(190\) −0.149978 −0.0108806
\(191\) 18.7745i 1.35848i 0.733918 + 0.679238i \(0.237690\pi\)
−0.733918 + 0.679238i \(0.762310\pi\)
\(192\) 0 0
\(193\) −7.34315 −0.528571 −0.264286 0.964444i \(-0.585136\pi\)
−0.264286 + 0.964444i \(0.585136\pi\)
\(194\) 13.2012 0.947790
\(195\) 0 0
\(196\) 0 0
\(197\) − 14.8593i − 1.05868i −0.848409 0.529342i \(-0.822439\pi\)
0.848409 0.529342i \(-0.177561\pi\)
\(198\) 0 0
\(199\) 14.9366i 1.05883i 0.848364 + 0.529413i \(0.177588\pi\)
−0.848364 + 0.529413i \(0.822412\pi\)
\(200\) − 4.63792i − 0.327950i
\(201\) 0 0
\(202\) − 5.20842i − 0.366463i
\(203\) 0 0
\(204\) 0 0
\(205\) 4.40663 0.307772
\(206\) 6.31086 0.439699
\(207\) 0 0
\(208\) − 0.649659i − 0.0450457i
\(209\) −0.299197 −0.0206959
\(210\) 0 0
\(211\) −6.38537 −0.439587 −0.219794 0.975546i \(-0.570538\pi\)
−0.219794 + 0.975546i \(0.570538\pi\)
\(212\) − 10.3178i − 0.708630i
\(213\) 0 0
\(214\) 14.7551 1.00864
\(215\) −5.63469 −0.384283
\(216\) 0 0
\(217\) 0 0
\(218\) 1.94812i 0.131944i
\(219\) 0 0
\(220\) 0.722325i 0.0486991i
\(221\) 0.366440i 0.0246494i
\(222\) 0 0
\(223\) 13.1867i 0.883047i 0.897250 + 0.441523i \(0.145562\pi\)
−0.897250 + 0.441523i \(0.854438\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) −8.35305 −0.555637
\(227\) −6.70481 −0.445014 −0.222507 0.974931i \(-0.571424\pi\)
−0.222507 + 0.974931i \(0.571424\pi\)
\(228\) 0 0
\(229\) 14.7600i 0.975368i 0.873020 + 0.487684i \(0.162158\pi\)
−0.873020 + 0.487684i \(0.837842\pi\)
\(230\) 2.37904 0.156870
\(231\) 0 0
\(232\) −1.37199 −0.0900753
\(233\) 20.9327i 1.37135i 0.727910 + 0.685673i \(0.240492\pi\)
−0.727910 + 0.685673i \(0.759508\pi\)
\(234\) 0 0
\(235\) 5.49287 0.358315
\(236\) 7.26508 0.472916
\(237\) 0 0
\(238\) 0 0
\(239\) − 15.3681i − 0.994082i −0.867727 0.497041i \(-0.834420\pi\)
0.867727 0.497041i \(-0.165580\pi\)
\(240\) 0 0
\(241\) − 13.4328i − 0.865283i −0.901566 0.432641i \(-0.857582\pi\)
0.901566 0.432641i \(-0.142418\pi\)
\(242\) − 9.55901i − 0.614476i
\(243\) 0 0
\(244\) 0.865467i 0.0554059i
\(245\) 0 0
\(246\) 0 0
\(247\) 0.161924 0.0103030
\(248\) −7.11169 −0.451593
\(249\) 0 0
\(250\) − 5.79943i − 0.366788i
\(251\) 3.61325 0.228066 0.114033 0.993477i \(-0.463623\pi\)
0.114033 + 0.993477i \(0.463623\pi\)
\(252\) 0 0
\(253\) 4.74603 0.298381
\(254\) − 1.09410i − 0.0686502i
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) 31.3932 1.95825 0.979126 0.203254i \(-0.0651518\pi\)
0.979126 + 0.203254i \(0.0651518\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) − 0.390920i − 0.0242438i
\(261\) 0 0
\(262\) − 6.60896i − 0.408303i
\(263\) − 3.96187i − 0.244300i −0.992512 0.122150i \(-0.961021\pi\)
0.992512 0.122150i \(-0.0389788\pi\)
\(264\) 0 0
\(265\) − 6.20854i − 0.381388i
\(266\) 0 0
\(267\) 0 0
\(268\) 9.37779 0.572840
\(269\) 21.5509 1.31398 0.656990 0.753899i \(-0.271829\pi\)
0.656990 + 0.753899i \(0.271829\pi\)
\(270\) 0 0
\(271\) − 6.18603i − 0.375774i −0.982191 0.187887i \(-0.939836\pi\)
0.982191 0.187887i \(-0.0601639\pi\)
\(272\) −0.564050 −0.0342005
\(273\) 0 0
\(274\) 9.61488 0.580856
\(275\) − 5.56742i − 0.335728i
\(276\) 0 0
\(277\) −11.1419 −0.669451 −0.334725 0.942316i \(-0.608644\pi\)
−0.334725 + 0.942316i \(0.608644\pi\)
\(278\) 1.95858 0.117468
\(279\) 0 0
\(280\) 0 0
\(281\) − 15.4122i − 0.919413i −0.888071 0.459707i \(-0.847955\pi\)
0.888071 0.459707i \(-0.152045\pi\)
\(282\) 0 0
\(283\) − 0.871470i − 0.0518035i −0.999664 0.0259018i \(-0.991754\pi\)
0.999664 0.0259018i \(-0.00824571\pi\)
\(284\) − 4.28897i − 0.254504i
\(285\) 0 0
\(286\) − 0.779858i − 0.0461140i
\(287\) 0 0
\(288\) 0 0
\(289\) −16.6818 −0.981285
\(290\) −0.825566 −0.0484789
\(291\) 0 0
\(292\) − 15.7896i − 0.924015i
\(293\) −26.3170 −1.53745 −0.768727 0.639577i \(-0.779109\pi\)
−0.768727 + 0.639577i \(0.779109\pi\)
\(294\) 0 0
\(295\) 4.37162 0.254526
\(296\) 5.17738i 0.300929i
\(297\) 0 0
\(298\) −16.4159 −0.950948
\(299\) −2.56853 −0.148542
\(300\) 0 0
\(301\) 0 0
\(302\) 7.35039i 0.422967i
\(303\) 0 0
\(304\) 0.249245i 0.0142952i
\(305\) 0.520778i 0.0298197i
\(306\) 0 0
\(307\) − 25.0183i − 1.42787i −0.700211 0.713936i \(-0.746911\pi\)
0.700211 0.713936i \(-0.253089\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) −4.27932 −0.243049
\(311\) −27.2433 −1.54483 −0.772413 0.635120i \(-0.780950\pi\)
−0.772413 + 0.635120i \(0.780950\pi\)
\(312\) 0 0
\(313\) 0.936585i 0.0529389i 0.999650 + 0.0264694i \(0.00842647\pi\)
−0.999650 + 0.0264694i \(0.991574\pi\)
\(314\) 13.3202 0.751704
\(315\) 0 0
\(316\) −12.1462 −0.683279
\(317\) 8.93218i 0.501681i 0.968028 + 0.250841i \(0.0807070\pi\)
−0.968028 + 0.250841i \(0.919293\pi\)
\(318\) 0 0
\(319\) −1.64695 −0.0922114
\(320\) 0.601731 0.0336378
\(321\) 0 0
\(322\) 0 0
\(323\) − 0.140587i − 0.00782245i
\(324\) 0 0
\(325\) 3.01306i 0.167135i
\(326\) − 11.5302i − 0.638597i
\(327\) 0 0
\(328\) − 7.32326i − 0.404359i
\(329\) 0 0
\(330\) 0 0
\(331\) −13.6805 −0.751949 −0.375974 0.926630i \(-0.622692\pi\)
−0.375974 + 0.926630i \(0.622692\pi\)
\(332\) 5.58632 0.306589
\(333\) 0 0
\(334\) 25.7937i 1.41137i
\(335\) 5.64290 0.308305
\(336\) 0 0
\(337\) −11.3662 −0.619157 −0.309578 0.950874i \(-0.600188\pi\)
−0.309578 + 0.950874i \(0.600188\pi\)
\(338\) − 12.5779i − 0.684150i
\(339\) 0 0
\(340\) −0.339406 −0.0184069
\(341\) −8.53696 −0.462303
\(342\) 0 0
\(343\) 0 0
\(344\) 9.36414i 0.504881i
\(345\) 0 0
\(346\) 4.04800i 0.217622i
\(347\) − 15.1885i − 0.815360i −0.913125 0.407680i \(-0.866338\pi\)
0.913125 0.407680i \(-0.133662\pi\)
\(348\) 0 0
\(349\) 1.64664i 0.0881425i 0.999028 + 0.0440713i \(0.0140329\pi\)
−0.999028 + 0.0440713i \(0.985967\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 1.20041 0.0639822
\(353\) 18.2235 0.969937 0.484968 0.874532i \(-0.338831\pi\)
0.484968 + 0.874532i \(0.338831\pi\)
\(354\) 0 0
\(355\) − 2.58081i − 0.136975i
\(356\) 7.62781 0.404273
\(357\) 0 0
\(358\) 8.94172 0.472585
\(359\) 27.7530i 1.46475i 0.680902 + 0.732374i \(0.261588\pi\)
−0.680902 + 0.732374i \(0.738412\pi\)
\(360\) 0 0
\(361\) 18.9379 0.996730
\(362\) 1.77776 0.0934369
\(363\) 0 0
\(364\) 0 0
\(365\) − 9.50107i − 0.497309i
\(366\) 0 0
\(367\) − 30.5271i − 1.59350i −0.604309 0.796750i \(-0.706551\pi\)
0.604309 0.796750i \(-0.293449\pi\)
\(368\) − 3.95367i − 0.206099i
\(369\) 0 0
\(370\) 3.11539i 0.161961i
\(371\) 0 0
\(372\) 0 0
\(373\) −12.2467 −0.634112 −0.317056 0.948407i \(-0.602694\pi\)
−0.317056 + 0.948407i \(0.602694\pi\)
\(374\) −0.677092 −0.0350116
\(375\) 0 0
\(376\) − 9.12845i − 0.470764i
\(377\) 0.891322 0.0459054
\(378\) 0 0
\(379\) 3.76316 0.193301 0.0966503 0.995318i \(-0.469187\pi\)
0.0966503 + 0.995318i \(0.469187\pi\)
\(380\) 0.149978i 0.00769373i
\(381\) 0 0
\(382\) 18.7745 0.960588
\(383\) 25.3157 1.29357 0.646787 0.762671i \(-0.276112\pi\)
0.646787 + 0.762671i \(0.276112\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 7.34315i 0.373756i
\(387\) 0 0
\(388\) − 13.2012i − 0.670189i
\(389\) 30.7667i 1.55993i 0.625822 + 0.779966i \(0.284764\pi\)
−0.625822 + 0.779966i \(0.715236\pi\)
\(390\) 0 0
\(391\) 2.23007i 0.112779i
\(392\) 0 0
\(393\) 0 0
\(394\) −14.8593 −0.748602
\(395\) −7.30876 −0.367744
\(396\) 0 0
\(397\) − 34.8793i − 1.75054i −0.483631 0.875272i \(-0.660682\pi\)
0.483631 0.875272i \(-0.339318\pi\)
\(398\) 14.9366 0.748703
\(399\) 0 0
\(400\) −4.63792 −0.231896
\(401\) − 31.8975i − 1.59288i −0.604716 0.796441i \(-0.706713\pi\)
0.604716 0.796441i \(-0.293287\pi\)
\(402\) 0 0
\(403\) 4.62017 0.230147
\(404\) −5.20842 −0.259129
\(405\) 0 0
\(406\) 0 0
\(407\) 6.21499i 0.308065i
\(408\) 0 0
\(409\) − 26.3923i − 1.30502i −0.757782 0.652508i \(-0.773717\pi\)
0.757782 0.652508i \(-0.226283\pi\)
\(410\) − 4.40663i − 0.217628i
\(411\) 0 0
\(412\) − 6.31086i − 0.310914i
\(413\) 0 0
\(414\) 0 0
\(415\) 3.36146 0.165008
\(416\) −0.649659 −0.0318521
\(417\) 0 0
\(418\) 0.299197i 0.0146342i
\(419\) −23.9692 −1.17097 −0.585485 0.810683i \(-0.699096\pi\)
−0.585485 + 0.810683i \(0.699096\pi\)
\(420\) 0 0
\(421\) −15.5279 −0.756782 −0.378391 0.925646i \(-0.623523\pi\)
−0.378391 + 0.925646i \(0.623523\pi\)
\(422\) 6.38537i 0.310835i
\(423\) 0 0
\(424\) −10.3178 −0.501077
\(425\) 2.61602 0.126896
\(426\) 0 0
\(427\) 0 0
\(428\) − 14.7551i − 0.713213i
\(429\) 0 0
\(430\) 5.63469i 0.271729i
\(431\) − 15.3414i − 0.738967i −0.929237 0.369484i \(-0.879535\pi\)
0.929237 0.369484i \(-0.120465\pi\)
\(432\) 0 0
\(433\) − 13.1662i − 0.632727i −0.948638 0.316364i \(-0.897538\pi\)
0.948638 0.316364i \(-0.102462\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 1.94812 0.0932982
\(437\) 0.985432 0.0471396
\(438\) 0 0
\(439\) 19.7473i 0.942486i 0.882003 + 0.471243i \(0.156195\pi\)
−0.882003 + 0.471243i \(0.843805\pi\)
\(440\) 0.722325 0.0344355
\(441\) 0 0
\(442\) 0.366440 0.0174298
\(443\) 24.2237i 1.15090i 0.817836 + 0.575452i \(0.195174\pi\)
−0.817836 + 0.575452i \(0.804826\pi\)
\(444\) 0 0
\(445\) 4.58989 0.217582
\(446\) 13.1867 0.624408
\(447\) 0 0
\(448\) 0 0
\(449\) − 16.9561i − 0.800208i −0.916470 0.400104i \(-0.868974\pi\)
0.916470 0.400104i \(-0.131026\pi\)
\(450\) 0 0
\(451\) − 8.79093i − 0.413949i
\(452\) 8.35305i 0.392894i
\(453\) 0 0
\(454\) 6.70481i 0.314672i
\(455\) 0 0
\(456\) 0 0
\(457\) −5.24264 −0.245240 −0.122620 0.992454i \(-0.539130\pi\)
−0.122620 + 0.992454i \(0.539130\pi\)
\(458\) 14.7600 0.689689
\(459\) 0 0
\(460\) − 2.37904i − 0.110923i
\(461\) 18.7628 0.873869 0.436935 0.899493i \(-0.356064\pi\)
0.436935 + 0.899493i \(0.356064\pi\)
\(462\) 0 0
\(463\) −3.27122 −0.152027 −0.0760133 0.997107i \(-0.524219\pi\)
−0.0760133 + 0.997107i \(0.524219\pi\)
\(464\) 1.37199i 0.0636928i
\(465\) 0 0
\(466\) 20.9327 0.969688
\(467\) −35.0751 −1.62308 −0.811541 0.584295i \(-0.801371\pi\)
−0.811541 + 0.584295i \(0.801371\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) − 5.49287i − 0.253367i
\(471\) 0 0
\(472\) − 7.26508i − 0.334402i
\(473\) 11.2408i 0.516854i
\(474\) 0 0
\(475\) − 1.15598i − 0.0530399i
\(476\) 0 0
\(477\) 0 0
\(478\) −15.3681 −0.702922
\(479\) −7.79415 −0.356124 −0.178062 0.984019i \(-0.556983\pi\)
−0.178062 + 0.984019i \(0.556983\pi\)
\(480\) 0 0
\(481\) − 3.36353i − 0.153364i
\(482\) −13.4328 −0.611847
\(483\) 0 0
\(484\) −9.55901 −0.434500
\(485\) − 7.94356i − 0.360699i
\(486\) 0 0
\(487\) 29.1383 1.32038 0.660191 0.751098i \(-0.270475\pi\)
0.660191 + 0.751098i \(0.270475\pi\)
\(488\) 0.865467 0.0391779
\(489\) 0 0
\(490\) 0 0
\(491\) 19.7492i 0.891268i 0.895215 + 0.445634i \(0.147022\pi\)
−0.895215 + 0.445634i \(0.852978\pi\)
\(492\) 0 0
\(493\) − 0.773868i − 0.0348533i
\(494\) − 0.161924i − 0.00728531i
\(495\) 0 0
\(496\) 7.11169i 0.319324i
\(497\) 0 0
\(498\) 0 0
\(499\) −31.6108 −1.41509 −0.707546 0.706667i \(-0.750198\pi\)
−0.707546 + 0.706667i \(0.750198\pi\)
\(500\) −5.79943 −0.259358
\(501\) 0 0
\(502\) − 3.61325i − 0.161267i
\(503\) 13.8811 0.618930 0.309465 0.950911i \(-0.399850\pi\)
0.309465 + 0.950911i \(0.399850\pi\)
\(504\) 0 0
\(505\) −3.13407 −0.139464
\(506\) − 4.74603i − 0.210987i
\(507\) 0 0
\(508\) −1.09410 −0.0485431
\(509\) −14.6657 −0.650047 −0.325023 0.945706i \(-0.605372\pi\)
−0.325023 + 0.945706i \(0.605372\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) − 1.00000i − 0.0441942i
\(513\) 0 0
\(514\) − 31.3932i − 1.38469i
\(515\) − 3.79744i − 0.167335i
\(516\) 0 0
\(517\) − 10.9579i − 0.481928i
\(518\) 0 0
\(519\) 0 0
\(520\) −0.390920 −0.0171430
\(521\) 39.5873 1.73435 0.867175 0.498004i \(-0.165934\pi\)
0.867175 + 0.498004i \(0.165934\pi\)
\(522\) 0 0
\(523\) 1.68035i 0.0734767i 0.999325 + 0.0367384i \(0.0116968\pi\)
−0.999325 + 0.0367384i \(0.988303\pi\)
\(524\) −6.60896 −0.288714
\(525\) 0 0
\(526\) −3.96187 −0.172746
\(527\) − 4.01135i − 0.174737i
\(528\) 0 0
\(529\) 7.36850 0.320370
\(530\) −6.20854 −0.269682
\(531\) 0 0
\(532\) 0 0
\(533\) 4.75762i 0.206075i
\(534\) 0 0
\(535\) − 8.87857i − 0.383854i
\(536\) − 9.37779i − 0.405059i
\(537\) 0 0
\(538\) − 21.5509i − 0.929125i
\(539\) 0 0
\(540\) 0 0
\(541\) −17.2062 −0.739753 −0.369876 0.929081i \(-0.620600\pi\)
−0.369876 + 0.929081i \(0.620600\pi\)
\(542\) −6.18603 −0.265713
\(543\) 0 0
\(544\) 0.564050i 0.0241834i
\(545\) 1.17225 0.0502135
\(546\) 0 0
\(547\) −43.6062 −1.86447 −0.932233 0.361859i \(-0.882142\pi\)
−0.932233 + 0.361859i \(0.882142\pi\)
\(548\) − 9.61488i − 0.410727i
\(549\) 0 0
\(550\) −5.56742 −0.237395
\(551\) −0.341961 −0.0145680
\(552\) 0 0
\(553\) 0 0
\(554\) 11.1419i 0.473373i
\(555\) 0 0
\(556\) − 1.95858i − 0.0830623i
\(557\) 37.4256i 1.58577i 0.609369 + 0.792886i \(0.291423\pi\)
−0.609369 + 0.792886i \(0.708577\pi\)
\(558\) 0 0
\(559\) − 6.08350i − 0.257304i
\(560\) 0 0
\(561\) 0 0
\(562\) −15.4122 −0.650123
\(563\) −25.6620 −1.08152 −0.540762 0.841176i \(-0.681864\pi\)
−0.540762 + 0.841176i \(0.681864\pi\)
\(564\) 0 0
\(565\) 5.02629i 0.211457i
\(566\) −0.871470 −0.0366306
\(567\) 0 0
\(568\) −4.28897 −0.179961
\(569\) − 13.2717i − 0.556380i −0.960526 0.278190i \(-0.910265\pi\)
0.960526 0.278190i \(-0.0897345\pi\)
\(570\) 0 0
\(571\) 20.7867 0.869897 0.434948 0.900455i \(-0.356767\pi\)
0.434948 + 0.900455i \(0.356767\pi\)
\(572\) −0.779858 −0.0326075
\(573\) 0 0
\(574\) 0 0
\(575\) 18.3368i 0.764697i
\(576\) 0 0
\(577\) − 16.1318i − 0.671576i −0.941938 0.335788i \(-0.890997\pi\)
0.941938 0.335788i \(-0.109003\pi\)
\(578\) 16.6818i 0.693873i
\(579\) 0 0
\(580\) 0.825566i 0.0342798i
\(581\) 0 0
\(582\) 0 0
\(583\) −12.3856 −0.512960
\(584\) −15.7896 −0.653377
\(585\) 0 0
\(586\) 26.3170i 1.08714i
\(587\) −18.0600 −0.745417 −0.372708 0.927949i \(-0.621571\pi\)
−0.372708 + 0.927949i \(0.621571\pi\)
\(588\) 0 0
\(589\) −1.77255 −0.0730368
\(590\) − 4.37162i − 0.179977i
\(591\) 0 0
\(592\) 5.17738 0.212789
\(593\) −6.69679 −0.275004 −0.137502 0.990501i \(-0.543907\pi\)
−0.137502 + 0.990501i \(0.543907\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 16.4159i 0.672422i
\(597\) 0 0
\(598\) 2.56853i 0.105035i
\(599\) 3.28379i 0.134172i 0.997747 + 0.0670860i \(0.0213702\pi\)
−0.997747 + 0.0670860i \(0.978630\pi\)
\(600\) 0 0
\(601\) − 4.59868i − 0.187584i −0.995592 0.0937920i \(-0.970101\pi\)
0.995592 0.0937920i \(-0.0298989\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 7.35039 0.299083
\(605\) −5.75195 −0.233850
\(606\) 0 0
\(607\) − 27.1633i − 1.10252i −0.834333 0.551261i \(-0.814147\pi\)
0.834333 0.551261i \(-0.185853\pi\)
\(608\) 0.249245 0.0101082
\(609\) 0 0
\(610\) 0.520778 0.0210857
\(611\) 5.93037i 0.239917i
\(612\) 0 0
\(613\) 32.9907 1.33248 0.666241 0.745736i \(-0.267902\pi\)
0.666241 + 0.745736i \(0.267902\pi\)
\(614\) −25.0183 −1.00966
\(615\) 0 0
\(616\) 0 0
\(617\) − 20.1479i − 0.811125i −0.914067 0.405563i \(-0.867076\pi\)
0.914067 0.405563i \(-0.132924\pi\)
\(618\) 0 0
\(619\) − 16.0966i − 0.646975i −0.946232 0.323488i \(-0.895145\pi\)
0.946232 0.323488i \(-0.104855\pi\)
\(620\) 4.27932i 0.171862i
\(621\) 0 0
\(622\) 27.2433i 1.09236i
\(623\) 0 0
\(624\) 0 0
\(625\) 19.6999 0.787996
\(626\) 0.936585 0.0374334
\(627\) 0 0
\(628\) − 13.3202i − 0.531535i
\(629\) −2.92030 −0.116440
\(630\) 0 0
\(631\) −2.81684 −0.112137 −0.0560684 0.998427i \(-0.517856\pi\)
−0.0560684 + 0.998427i \(0.517856\pi\)
\(632\) 12.1462i 0.483151i
\(633\) 0 0
\(634\) 8.93218 0.354742
\(635\) −0.658356 −0.0261261
\(636\) 0 0
\(637\) 0 0
\(638\) 1.64695i 0.0652033i
\(639\) 0 0
\(640\) − 0.601731i − 0.0237855i
\(641\) − 17.2433i − 0.681068i −0.940232 0.340534i \(-0.889392\pi\)
0.940232 0.340534i \(-0.110608\pi\)
\(642\) 0 0
\(643\) 43.9469i 1.73310i 0.499093 + 0.866549i \(0.333667\pi\)
−0.499093 + 0.866549i \(0.666333\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) −0.140587 −0.00553131
\(647\) −15.4124 −0.605922 −0.302961 0.953003i \(-0.597975\pi\)
−0.302961 + 0.953003i \(0.597975\pi\)
\(648\) 0 0
\(649\) − 8.72109i − 0.342333i
\(650\) 3.01306 0.118182
\(651\) 0 0
\(652\) −11.5302 −0.451556
\(653\) − 34.4687i − 1.34886i −0.738337 0.674432i \(-0.764389\pi\)
0.738337 0.674432i \(-0.235611\pi\)
\(654\) 0 0
\(655\) −3.97681 −0.155387
\(656\) −7.32326 −0.285925
\(657\) 0 0
\(658\) 0 0
\(659\) 16.2348i 0.632418i 0.948690 + 0.316209i \(0.102410\pi\)
−0.948690 + 0.316209i \(0.897590\pi\)
\(660\) 0 0
\(661\) − 47.2554i − 1.83802i −0.394230 0.919012i \(-0.628989\pi\)
0.394230 0.919012i \(-0.371011\pi\)
\(662\) 13.6805i 0.531708i
\(663\) 0 0
\(664\) − 5.58632i − 0.216791i
\(665\) 0 0
\(666\) 0 0
\(667\) 5.42438 0.210033
\(668\) 25.7937 0.997990
\(669\) 0 0
\(670\) − 5.64290i − 0.218004i
\(671\) 1.03892 0.0401070
\(672\) 0 0
\(673\) −16.2598 −0.626769 −0.313384 0.949626i \(-0.601463\pi\)
−0.313384 + 0.949626i \(0.601463\pi\)
\(674\) 11.3662i 0.437810i
\(675\) 0 0
\(676\) −12.5779 −0.483767
\(677\) −12.9879 −0.499165 −0.249582 0.968354i \(-0.580293\pi\)
−0.249582 + 0.968354i \(0.580293\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0.339406i 0.0130156i
\(681\) 0 0
\(682\) 8.53696i 0.326897i
\(683\) − 30.7325i − 1.17595i −0.808880 0.587974i \(-0.799926\pi\)
0.808880 0.587974i \(-0.200074\pi\)
\(684\) 0 0
\(685\) − 5.78557i − 0.221055i
\(686\) 0 0
\(687\) 0 0
\(688\) 9.36414 0.357005
\(689\) 6.70306 0.255366
\(690\) 0 0
\(691\) 16.0781i 0.611638i 0.952090 + 0.305819i \(0.0989302\pi\)
−0.952090 + 0.305819i \(0.901070\pi\)
\(692\) 4.04800 0.153882
\(693\) 0 0
\(694\) −15.1885 −0.576546
\(695\) − 1.17854i − 0.0447045i
\(696\) 0 0
\(697\) 4.13068 0.156461
\(698\) 1.64664 0.0623262
\(699\) 0 0
\(700\) 0 0
\(701\) 37.8263i 1.42868i 0.699799 + 0.714339i \(0.253273\pi\)
−0.699799 + 0.714339i \(0.746727\pi\)
\(702\) 0 0
\(703\) 1.29043i 0.0486697i
\(704\) − 1.20041i − 0.0452423i
\(705\) 0 0
\(706\) − 18.2235i − 0.685849i
\(707\) 0 0
\(708\) 0 0
\(709\) −41.7113 −1.56650 −0.783251 0.621706i \(-0.786440\pi\)
−0.783251 + 0.621706i \(0.786440\pi\)
\(710\) −2.58081 −0.0968559
\(711\) 0 0
\(712\) − 7.62781i − 0.285864i
\(713\) 28.1173 1.05300
\(714\) 0 0
\(715\) −0.469265 −0.0175495
\(716\) − 8.94172i − 0.334168i
\(717\) 0 0
\(718\) 27.7530 1.03573
\(719\) 2.06421 0.0769820 0.0384910 0.999259i \(-0.487745\pi\)
0.0384910 + 0.999259i \(0.487745\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) − 18.9379i − 0.704795i
\(723\) 0 0
\(724\) − 1.77776i − 0.0660699i
\(725\) − 6.36316i − 0.236322i
\(726\) 0 0
\(727\) − 31.5303i − 1.16939i −0.811251 0.584697i \(-0.801213\pi\)
0.811251 0.584697i \(-0.198787\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) −9.50107 −0.351650
\(731\) −5.28184 −0.195356
\(732\) 0 0
\(733\) 40.4015i 1.49226i 0.665799 + 0.746131i \(0.268091\pi\)
−0.665799 + 0.746131i \(0.731909\pi\)
\(734\) −30.5271 −1.12677
\(735\) 0 0
\(736\) −3.95367 −0.145734
\(737\) − 11.2572i − 0.414665i
\(738\) 0 0
\(739\) 40.5607 1.49205 0.746025 0.665918i \(-0.231960\pi\)
0.746025 + 0.665918i \(0.231960\pi\)
\(740\) 3.11539 0.114524
\(741\) 0 0
\(742\) 0 0
\(743\) 29.9301i 1.09803i 0.835812 + 0.549015i \(0.184997\pi\)
−0.835812 + 0.549015i \(0.815003\pi\)
\(744\) 0 0
\(745\) 9.87796i 0.361900i
\(746\) 12.2467i 0.448385i
\(747\) 0 0
\(748\) 0.677092i 0.0247570i
\(749\) 0 0
\(750\) 0 0
\(751\) −39.8892 −1.45558 −0.727790 0.685800i \(-0.759452\pi\)
−0.727790 + 0.685800i \(0.759452\pi\)
\(752\) −9.12845 −0.332880
\(753\) 0 0
\(754\) − 0.891322i − 0.0324600i
\(755\) 4.42296 0.160968
\(756\) 0 0
\(757\) −17.9258 −0.651525 −0.325762 0.945452i \(-0.605621\pi\)
−0.325762 + 0.945452i \(0.605621\pi\)
\(758\) − 3.76316i − 0.136684i
\(759\) 0 0
\(760\) 0.149978 0.00544029
\(761\) 15.6453 0.567142 0.283571 0.958951i \(-0.408481\pi\)
0.283571 + 0.958951i \(0.408481\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) − 18.7745i − 0.679238i
\(765\) 0 0
\(766\) − 25.3157i − 0.914695i
\(767\) 4.71982i 0.170423i
\(768\) 0 0
\(769\) − 2.25067i − 0.0811613i −0.999176 0.0405806i \(-0.987079\pi\)
0.999176 0.0405806i \(-0.0129208\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 7.34315 0.264286
\(773\) −21.9049 −0.787866 −0.393933 0.919139i \(-0.628886\pi\)
−0.393933 + 0.919139i \(0.628886\pi\)
\(774\) 0 0
\(775\) − 32.9835i − 1.18480i
\(776\) −13.2012 −0.473895
\(777\) 0 0
\(778\) 30.7667 1.10304
\(779\) − 1.82528i − 0.0653976i
\(780\) 0 0
\(781\) −5.14854 −0.184229
\(782\) 2.23007 0.0797470
\(783\) 0 0
\(784\) 0 0
\(785\) − 8.01519i − 0.286075i
\(786\) 0 0
\(787\) − 16.8406i − 0.600301i −0.953892 0.300151i \(-0.902963\pi\)
0.953892 0.300151i \(-0.0970369\pi\)
\(788\) 14.8593i 0.529342i
\(789\) 0 0
\(790\) 7.30876i 0.260034i
\(791\) 0 0
\(792\) 0 0
\(793\) −0.562258 −0.0199664
\(794\) −34.8793 −1.23782
\(795\) 0 0
\(796\) − 14.9366i − 0.529413i
\(797\) 13.8756 0.491500 0.245750 0.969333i \(-0.420966\pi\)
0.245750 + 0.969333i \(0.420966\pi\)
\(798\) 0 0
\(799\) 5.14890 0.182155
\(800\) 4.63792i 0.163975i
\(801\) 0 0
\(802\) −31.8975 −1.12634
\(803\) −18.9540 −0.668872
\(804\) 0 0
\(805\) 0 0
\(806\) − 4.62017i − 0.162739i
\(807\) 0 0
\(808\) 5.20842i 0.183232i
\(809\) 16.0990i 0.566010i 0.959118 + 0.283005i \(0.0913313\pi\)
−0.959118 + 0.283005i \(0.908669\pi\)
\(810\) 0 0
\(811\) − 8.57333i − 0.301050i −0.988606 0.150525i \(-0.951904\pi\)
0.988606 0.150525i \(-0.0480965\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 6.21499 0.217835
\(815\) −6.93806 −0.243030
\(816\) 0 0
\(817\) 2.33397i 0.0816551i
\(818\) −26.3923 −0.922786
\(819\) 0 0
\(820\) −4.40663 −0.153886
\(821\) − 41.2485i − 1.43958i −0.694191 0.719791i \(-0.744238\pi\)
0.694191 0.719791i \(-0.255762\pi\)
\(822\) 0 0
\(823\) −25.1482 −0.876611 −0.438305 0.898826i \(-0.644421\pi\)
−0.438305 + 0.898826i \(0.644421\pi\)
\(824\) −6.31086 −0.219849
\(825\) 0 0
\(826\) 0 0
\(827\) 12.8906i 0.448249i 0.974561 + 0.224125i \(0.0719523\pi\)
−0.974561 + 0.224125i \(0.928048\pi\)
\(828\) 0 0
\(829\) 24.5664i 0.853228i 0.904434 + 0.426614i \(0.140294\pi\)
−0.904434 + 0.426614i \(0.859706\pi\)
\(830\) − 3.36146i − 0.116678i
\(831\) 0 0
\(832\) 0.649659i 0.0225229i
\(833\) 0 0
\(834\) 0 0
\(835\) 15.5209 0.537122
\(836\) 0.299197 0.0103479
\(837\) 0 0
\(838\) 23.9692i 0.828001i
\(839\) −20.4459 −0.705872 −0.352936 0.935647i \(-0.614817\pi\)
−0.352936 + 0.935647i \(0.614817\pi\)
\(840\) 0 0
\(841\) 27.1177 0.935092
\(842\) 15.5279i 0.535126i
\(843\) 0 0
\(844\) 6.38537 0.219794
\(845\) −7.56853 −0.260366
\(846\) 0 0
\(847\) 0 0
\(848\) 10.3178i 0.354315i
\(849\) 0 0
\(850\) − 2.61602i − 0.0897287i
\(851\) − 20.4696i − 0.701690i
\(852\) 0 0
\(853\) 3.15935i 0.108174i 0.998536 + 0.0540870i \(0.0172248\pi\)
−0.998536 + 0.0540870i \(0.982775\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) −14.7551 −0.504318
\(857\) 38.2349 1.30608 0.653040 0.757323i \(-0.273493\pi\)
0.653040 + 0.757323i \(0.273493\pi\)
\(858\) 0 0
\(859\) − 21.9370i − 0.748480i −0.927332 0.374240i \(-0.877904\pi\)
0.927332 0.374240i \(-0.122096\pi\)
\(860\) 5.63469 0.192141
\(861\) 0 0
\(862\) −15.3414 −0.522529
\(863\) 1.54212i 0.0524942i 0.999655 + 0.0262471i \(0.00835568\pi\)
−0.999655 + 0.0262471i \(0.991644\pi\)
\(864\) 0 0
\(865\) 2.43581 0.0828200
\(866\) −13.1662 −0.447406
\(867\) 0 0
\(868\) 0 0
\(869\) 14.5805i 0.494610i
\(870\) 0 0
\(871\) 6.09236i 0.206432i
\(872\) − 1.94812i − 0.0659718i
\(873\) 0 0
\(874\) − 0.985432i − 0.0333327i
\(875\) 0 0
\(876\) 0 0
\(877\) 45.3364 1.53090 0.765451 0.643494i \(-0.222516\pi\)
0.765451 + 0.643494i \(0.222516\pi\)
\(878\) 19.7473 0.666439
\(879\) 0 0
\(880\) − 0.722325i − 0.0243496i
\(881\) 31.3187 1.05516 0.527578 0.849507i \(-0.323100\pi\)
0.527578 + 0.849507i \(0.323100\pi\)
\(882\) 0 0
\(883\) 31.9125 1.07394 0.536971 0.843601i \(-0.319568\pi\)
0.536971 + 0.843601i \(0.319568\pi\)
\(884\) − 0.366440i − 0.0123247i
\(885\) 0 0
\(886\) 24.2237 0.813811
\(887\) −43.0139 −1.44426 −0.722132 0.691755i \(-0.756838\pi\)
−0.722132 + 0.691755i \(0.756838\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) − 4.58989i − 0.153853i
\(891\) 0 0
\(892\) − 13.1867i − 0.441523i
\(893\) − 2.27522i − 0.0761373i
\(894\) 0 0
\(895\) − 5.38051i − 0.179851i
\(896\) 0 0
\(897\) 0 0
\(898\) −16.9561 −0.565832
\(899\) −9.75714 −0.325419
\(900\) 0 0
\(901\) − 5.81976i − 0.193884i
\(902\) −8.79093 −0.292706
\(903\) 0 0
\(904\) 8.35305 0.277818
\(905\) − 1.06973i − 0.0355591i
\(906\) 0 0
\(907\) −40.6686 −1.35038 −0.675189 0.737645i \(-0.735938\pi\)
−0.675189 + 0.737645i \(0.735938\pi\)
\(908\) 6.70481 0.222507
\(909\) 0 0
\(910\) 0 0
\(911\) 16.4174i 0.543932i 0.962307 + 0.271966i \(0.0876739\pi\)
−0.962307 + 0.271966i \(0.912326\pi\)
\(912\) 0 0
\(913\) − 6.70589i − 0.221932i
\(914\) 5.24264i 0.173411i
\(915\) 0 0
\(916\) − 14.7600i − 0.487684i
\(917\) 0 0
\(918\) 0 0
\(919\) 6.36995 0.210125 0.105063 0.994466i \(-0.466496\pi\)
0.105063 + 0.994466i \(0.466496\pi\)
\(920\) −2.37904 −0.0784348
\(921\) 0 0
\(922\) − 18.7628i − 0.617919i
\(923\) 2.78637 0.0917144
\(924\) 0 0
\(925\) −24.0123 −0.789518
\(926\) 3.27122i 0.107499i
\(927\) 0 0
\(928\) 1.37199 0.0450376
\(929\) 43.2708 1.41967 0.709834 0.704369i \(-0.248770\pi\)
0.709834 + 0.704369i \(0.248770\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) − 20.9327i − 0.685673i
\(933\) 0 0
\(934\) 35.0751i 1.14769i
\(935\) 0.407427i 0.0133243i
\(936\) 0 0
\(937\) 41.8956i 1.36867i 0.729168 + 0.684335i \(0.239907\pi\)
−0.729168 + 0.684335i \(0.760093\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) −5.49287 −0.179158
\(941\) 39.7919 1.29718 0.648590 0.761138i \(-0.275359\pi\)
0.648590 + 0.761138i \(0.275359\pi\)
\(942\) 0 0
\(943\) 28.9537i 0.942863i
\(944\) −7.26508 −0.236458
\(945\) 0 0
\(946\) 11.2408 0.365471
\(947\) 20.0515i 0.651587i 0.945441 + 0.325794i \(0.105631\pi\)
−0.945441 + 0.325794i \(0.894369\pi\)
\(948\) 0 0
\(949\) 10.2578 0.332983
\(950\) −1.15598 −0.0375049
\(951\) 0 0
\(952\) 0 0
\(953\) − 26.4300i − 0.856153i −0.903743 0.428076i \(-0.859191\pi\)
0.903743 0.428076i \(-0.140809\pi\)
\(954\) 0 0
\(955\) − 11.2972i − 0.365569i
\(956\) 15.3681i 0.497041i
\(957\) 0 0
\(958\) 7.79415i 0.251817i
\(959\) 0 0
\(960\) 0 0
\(961\) −19.5762 −0.631489
\(962\) −3.36353 −0.108444
\(963\) 0 0
\(964\) 13.4328i 0.432641i
\(965\) 4.41860 0.142240
\(966\) 0 0
\(967\) −17.7780 −0.571702 −0.285851 0.958274i \(-0.592276\pi\)
−0.285851 + 0.958274i \(0.592276\pi\)
\(968\) 9.55901i 0.307238i
\(969\) 0 0
\(970\) −7.94356 −0.255052
\(971\) 46.5597 1.49417 0.747086 0.664728i \(-0.231452\pi\)
0.747086 + 0.664728i \(0.231452\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) − 29.1383i − 0.933651i
\(975\) 0 0
\(976\) − 0.865467i − 0.0277029i
\(977\) 16.9303i 0.541647i 0.962629 + 0.270824i \(0.0872960\pi\)
−0.962629 + 0.270824i \(0.912704\pi\)
\(978\) 0 0
\(979\) − 9.15652i − 0.292644i
\(980\) 0 0
\(981\) 0 0
\(982\) 19.7492 0.630222
\(983\) −41.6087 −1.32711 −0.663556 0.748126i \(-0.730954\pi\)
−0.663556 + 0.748126i \(0.730954\pi\)
\(984\) 0 0
\(985\) 8.94131i 0.284894i
\(986\) −0.773868 −0.0246450
\(987\) 0 0
\(988\) −0.161924 −0.00515149
\(989\) − 37.0227i − 1.17725i
\(990\) 0 0
\(991\) −7.67254 −0.243726 −0.121863 0.992547i \(-0.538887\pi\)
−0.121863 + 0.992547i \(0.538887\pi\)
\(992\) 7.11169 0.225796
\(993\) 0 0
\(994\) 0 0
\(995\) − 8.98780i − 0.284932i
\(996\) 0 0
\(997\) − 2.76497i − 0.0875673i −0.999041 0.0437837i \(-0.986059\pi\)
0.999041 0.0437837i \(-0.0139412\pi\)
\(998\) 31.6108i 1.00062i
\(999\) 0 0
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 2646.2.d.e.2645.4 16
3.2 odd 2 inner 2646.2.d.e.2645.13 yes 16
7.6 odd 2 inner 2646.2.d.e.2645.5 yes 16
21.20 even 2 inner 2646.2.d.e.2645.12 yes 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2646.2.d.e.2645.4 16 1.1 even 1 trivial
2646.2.d.e.2645.5 yes 16 7.6 odd 2 inner
2646.2.d.e.2645.12 yes 16 21.20 even 2 inner
2646.2.d.e.2645.13 yes 16 3.2 odd 2 inner