Properties

Label 2475.2.a.w.1.2
Level $2475$
Weight $2$
Character 2475.1
Self dual yes
Analytic conductor $19.763$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 2475 = 3^{2} \cdot 5^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2475.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(19.7629745003\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{8})^+\)
Defining polynomial: \(x^{2} - 2\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 825)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(1.41421\) of defining polynomial
Character \(\chi\) \(=\) 2475.1

$q$-expansion

\(f(q)\) \(=\) \(q+2.41421 q^{2} +3.82843 q^{4} -0.414214 q^{7} +4.41421 q^{8} +O(q^{10})\) \(q+2.41421 q^{2} +3.82843 q^{4} -0.414214 q^{7} +4.41421 q^{8} +1.00000 q^{11} +2.82843 q^{13} -1.00000 q^{14} +3.00000 q^{16} +2.41421 q^{17} +6.41421 q^{19} +2.41421 q^{22} +1.00000 q^{23} +6.82843 q^{26} -1.58579 q^{28} -1.17157 q^{29} -8.48528 q^{31} -1.58579 q^{32} +5.82843 q^{34} +0.171573 q^{37} +15.4853 q^{38} +10.8995 q^{41} +11.6569 q^{43} +3.82843 q^{44} +2.41421 q^{46} -7.48528 q^{47} -6.82843 q^{49} +10.8284 q^{52} +7.65685 q^{53} -1.82843 q^{56} -2.82843 q^{58} -11.0000 q^{59} +8.82843 q^{61} -20.4853 q^{62} -9.82843 q^{64} +0.343146 q^{67} +9.24264 q^{68} -7.82843 q^{71} +8.82843 q^{73} +0.414214 q^{74} +24.5563 q^{76} -0.414214 q^{77} +13.2426 q^{79} +26.3137 q^{82} -4.48528 q^{83} +28.1421 q^{86} +4.41421 q^{88} -3.65685 q^{89} -1.17157 q^{91} +3.82843 q^{92} -18.0711 q^{94} -5.82843 q^{97} -16.4853 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{2} + 2 q^{4} + 2 q^{7} + 6 q^{8} + O(q^{10}) \) \( 2 q + 2 q^{2} + 2 q^{4} + 2 q^{7} + 6 q^{8} + 2 q^{11} - 2 q^{14} + 6 q^{16} + 2 q^{17} + 10 q^{19} + 2 q^{22} + 2 q^{23} + 8 q^{26} - 6 q^{28} - 8 q^{29} - 6 q^{32} + 6 q^{34} + 6 q^{37} + 14 q^{38} + 2 q^{41} + 12 q^{43} + 2 q^{44} + 2 q^{46} + 2 q^{47} - 8 q^{49} + 16 q^{52} + 4 q^{53} + 2 q^{56} - 22 q^{59} + 12 q^{61} - 24 q^{62} - 14 q^{64} + 12 q^{67} + 10 q^{68} - 10 q^{71} + 12 q^{73} - 2 q^{74} + 18 q^{76} + 2 q^{77} + 18 q^{79} + 30 q^{82} + 8 q^{83} + 28 q^{86} + 6 q^{88} + 4 q^{89} - 8 q^{91} + 2 q^{92} - 22 q^{94} - 6 q^{97} - 16 q^{98} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.41421 1.70711 0.853553 0.521005i \(-0.174443\pi\)
0.853553 + 0.521005i \(0.174443\pi\)
\(3\) 0 0
\(4\) 3.82843 1.91421
\(5\) 0 0
\(6\) 0 0
\(7\) −0.414214 −0.156558 −0.0782790 0.996931i \(-0.524942\pi\)
−0.0782790 + 0.996931i \(0.524942\pi\)
\(8\) 4.41421 1.56066
\(9\) 0 0
\(10\) 0 0
\(11\) 1.00000 0.301511
\(12\) 0 0
\(13\) 2.82843 0.784465 0.392232 0.919866i \(-0.371703\pi\)
0.392232 + 0.919866i \(0.371703\pi\)
\(14\) −1.00000 −0.267261
\(15\) 0 0
\(16\) 3.00000 0.750000
\(17\) 2.41421 0.585533 0.292766 0.956184i \(-0.405424\pi\)
0.292766 + 0.956184i \(0.405424\pi\)
\(18\) 0 0
\(19\) 6.41421 1.47152 0.735761 0.677242i \(-0.236825\pi\)
0.735761 + 0.677242i \(0.236825\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 2.41421 0.514712
\(23\) 1.00000 0.208514 0.104257 0.994550i \(-0.466753\pi\)
0.104257 + 0.994550i \(0.466753\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 6.82843 1.33916
\(27\) 0 0
\(28\) −1.58579 −0.299685
\(29\) −1.17157 −0.217556 −0.108778 0.994066i \(-0.534694\pi\)
−0.108778 + 0.994066i \(0.534694\pi\)
\(30\) 0 0
\(31\) −8.48528 −1.52400 −0.762001 0.647576i \(-0.775783\pi\)
−0.762001 + 0.647576i \(0.775783\pi\)
\(32\) −1.58579 −0.280330
\(33\) 0 0
\(34\) 5.82843 0.999567
\(35\) 0 0
\(36\) 0 0
\(37\) 0.171573 0.0282064 0.0141032 0.999901i \(-0.495511\pi\)
0.0141032 + 0.999901i \(0.495511\pi\)
\(38\) 15.4853 2.51204
\(39\) 0 0
\(40\) 0 0
\(41\) 10.8995 1.70222 0.851108 0.524991i \(-0.175931\pi\)
0.851108 + 0.524991i \(0.175931\pi\)
\(42\) 0 0
\(43\) 11.6569 1.77765 0.888827 0.458243i \(-0.151521\pi\)
0.888827 + 0.458243i \(0.151521\pi\)
\(44\) 3.82843 0.577157
\(45\) 0 0
\(46\) 2.41421 0.355956
\(47\) −7.48528 −1.09184 −0.545920 0.837837i \(-0.683820\pi\)
−0.545920 + 0.837837i \(0.683820\pi\)
\(48\) 0 0
\(49\) −6.82843 −0.975490
\(50\) 0 0
\(51\) 0 0
\(52\) 10.8284 1.50163
\(53\) 7.65685 1.05175 0.525875 0.850562i \(-0.323738\pi\)
0.525875 + 0.850562i \(0.323738\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) −1.82843 −0.244334
\(57\) 0 0
\(58\) −2.82843 −0.371391
\(59\) −11.0000 −1.43208 −0.716039 0.698060i \(-0.754047\pi\)
−0.716039 + 0.698060i \(0.754047\pi\)
\(60\) 0 0
\(61\) 8.82843 1.13036 0.565182 0.824966i \(-0.308806\pi\)
0.565182 + 0.824966i \(0.308806\pi\)
\(62\) −20.4853 −2.60163
\(63\) 0 0
\(64\) −9.82843 −1.22855
\(65\) 0 0
\(66\) 0 0
\(67\) 0.343146 0.0419219 0.0209610 0.999780i \(-0.493327\pi\)
0.0209610 + 0.999780i \(0.493327\pi\)
\(68\) 9.24264 1.12083
\(69\) 0 0
\(70\) 0 0
\(71\) −7.82843 −0.929063 −0.464532 0.885556i \(-0.653777\pi\)
−0.464532 + 0.885556i \(0.653777\pi\)
\(72\) 0 0
\(73\) 8.82843 1.03329 0.516645 0.856200i \(-0.327181\pi\)
0.516645 + 0.856200i \(0.327181\pi\)
\(74\) 0.414214 0.0481513
\(75\) 0 0
\(76\) 24.5563 2.81681
\(77\) −0.414214 −0.0472040
\(78\) 0 0
\(79\) 13.2426 1.48991 0.744957 0.667113i \(-0.232470\pi\)
0.744957 + 0.667113i \(0.232470\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 26.3137 2.90586
\(83\) −4.48528 −0.492324 −0.246162 0.969229i \(-0.579169\pi\)
−0.246162 + 0.969229i \(0.579169\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 28.1421 3.03464
\(87\) 0 0
\(88\) 4.41421 0.470557
\(89\) −3.65685 −0.387626 −0.193813 0.981039i \(-0.562085\pi\)
−0.193813 + 0.981039i \(0.562085\pi\)
\(90\) 0 0
\(91\) −1.17157 −0.122814
\(92\) 3.82843 0.399141
\(93\) 0 0
\(94\) −18.0711 −1.86389
\(95\) 0 0
\(96\) 0 0
\(97\) −5.82843 −0.591787 −0.295894 0.955221i \(-0.595617\pi\)
−0.295894 + 0.955221i \(0.595617\pi\)
\(98\) −16.4853 −1.66526
\(99\) 0 0
\(100\) 0 0
\(101\) −14.8995 −1.48256 −0.741278 0.671199i \(-0.765780\pi\)
−0.741278 + 0.671199i \(0.765780\pi\)
\(102\) 0 0
\(103\) 13.6569 1.34565 0.672825 0.739802i \(-0.265081\pi\)
0.672825 + 0.739802i \(0.265081\pi\)
\(104\) 12.4853 1.22428
\(105\) 0 0
\(106\) 18.4853 1.79545
\(107\) −5.31371 −0.513696 −0.256848 0.966452i \(-0.582684\pi\)
−0.256848 + 0.966452i \(0.582684\pi\)
\(108\) 0 0
\(109\) 5.31371 0.508961 0.254480 0.967078i \(-0.418096\pi\)
0.254480 + 0.967078i \(0.418096\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) −1.24264 −0.117419
\(113\) −10.0000 −0.940721 −0.470360 0.882474i \(-0.655876\pi\)
−0.470360 + 0.882474i \(0.655876\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) −4.48528 −0.416448
\(117\) 0 0
\(118\) −26.5563 −2.44471
\(119\) −1.00000 −0.0916698
\(120\) 0 0
\(121\) 1.00000 0.0909091
\(122\) 21.3137 1.92965
\(123\) 0 0
\(124\) −32.4853 −2.91726
\(125\) 0 0
\(126\) 0 0
\(127\) −7.24264 −0.642680 −0.321340 0.946964i \(-0.604133\pi\)
−0.321340 + 0.946964i \(0.604133\pi\)
\(128\) −20.5563 −1.81694
\(129\) 0 0
\(130\) 0 0
\(131\) −6.82843 −0.596602 −0.298301 0.954472i \(-0.596420\pi\)
−0.298301 + 0.954472i \(0.596420\pi\)
\(132\) 0 0
\(133\) −2.65685 −0.230378
\(134\) 0.828427 0.0715652
\(135\) 0 0
\(136\) 10.6569 0.913818
\(137\) −12.1421 −1.03737 −0.518686 0.854965i \(-0.673579\pi\)
−0.518686 + 0.854965i \(0.673579\pi\)
\(138\) 0 0
\(139\) −18.9706 −1.60906 −0.804531 0.593911i \(-0.797583\pi\)
−0.804531 + 0.593911i \(0.797583\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) −18.8995 −1.58601
\(143\) 2.82843 0.236525
\(144\) 0 0
\(145\) 0 0
\(146\) 21.3137 1.76394
\(147\) 0 0
\(148\) 0.656854 0.0539931
\(149\) −7.72792 −0.633096 −0.316548 0.948576i \(-0.602524\pi\)
−0.316548 + 0.948576i \(0.602524\pi\)
\(150\) 0 0
\(151\) 14.0000 1.13930 0.569652 0.821886i \(-0.307078\pi\)
0.569652 + 0.821886i \(0.307078\pi\)
\(152\) 28.3137 2.29655
\(153\) 0 0
\(154\) −1.00000 −0.0805823
\(155\) 0 0
\(156\) 0 0
\(157\) −6.00000 −0.478852 −0.239426 0.970915i \(-0.576959\pi\)
−0.239426 + 0.970915i \(0.576959\pi\)
\(158\) 31.9706 2.54344
\(159\) 0 0
\(160\) 0 0
\(161\) −0.414214 −0.0326446
\(162\) 0 0
\(163\) −15.7990 −1.23747 −0.618736 0.785599i \(-0.712355\pi\)
−0.618736 + 0.785599i \(0.712355\pi\)
\(164\) 41.7279 3.25840
\(165\) 0 0
\(166\) −10.8284 −0.840449
\(167\) 21.7990 1.68686 0.843428 0.537242i \(-0.180534\pi\)
0.843428 + 0.537242i \(0.180534\pi\)
\(168\) 0 0
\(169\) −5.00000 −0.384615
\(170\) 0 0
\(171\) 0 0
\(172\) 44.6274 3.40281
\(173\) −12.5563 −0.954642 −0.477321 0.878729i \(-0.658392\pi\)
−0.477321 + 0.878729i \(0.658392\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 3.00000 0.226134
\(177\) 0 0
\(178\) −8.82843 −0.661719
\(179\) −16.7990 −1.25562 −0.627808 0.778368i \(-0.716048\pi\)
−0.627808 + 0.778368i \(0.716048\pi\)
\(180\) 0 0
\(181\) −21.9706 −1.63306 −0.816530 0.577304i \(-0.804105\pi\)
−0.816530 + 0.577304i \(0.804105\pi\)
\(182\) −2.82843 −0.209657
\(183\) 0 0
\(184\) 4.41421 0.325420
\(185\) 0 0
\(186\) 0 0
\(187\) 2.41421 0.176545
\(188\) −28.6569 −2.09002
\(189\) 0 0
\(190\) 0 0
\(191\) −6.17157 −0.446559 −0.223280 0.974754i \(-0.571676\pi\)
−0.223280 + 0.974754i \(0.571676\pi\)
\(192\) 0 0
\(193\) −3.31371 −0.238526 −0.119263 0.992863i \(-0.538053\pi\)
−0.119263 + 0.992863i \(0.538053\pi\)
\(194\) −14.0711 −1.01024
\(195\) 0 0
\(196\) −26.1421 −1.86730
\(197\) 4.75736 0.338948 0.169474 0.985535i \(-0.445793\pi\)
0.169474 + 0.985535i \(0.445793\pi\)
\(198\) 0 0
\(199\) 10.8284 0.767607 0.383803 0.923415i \(-0.374614\pi\)
0.383803 + 0.923415i \(0.374614\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) −35.9706 −2.53088
\(203\) 0.485281 0.0340601
\(204\) 0 0
\(205\) 0 0
\(206\) 32.9706 2.29717
\(207\) 0 0
\(208\) 8.48528 0.588348
\(209\) 6.41421 0.443680
\(210\) 0 0
\(211\) −13.3137 −0.916553 −0.458277 0.888810i \(-0.651533\pi\)
−0.458277 + 0.888810i \(0.651533\pi\)
\(212\) 29.3137 2.01327
\(213\) 0 0
\(214\) −12.8284 −0.876933
\(215\) 0 0
\(216\) 0 0
\(217\) 3.51472 0.238595
\(218\) 12.8284 0.868851
\(219\) 0 0
\(220\) 0 0
\(221\) 6.82843 0.459330
\(222\) 0 0
\(223\) 21.1716 1.41775 0.708877 0.705332i \(-0.249202\pi\)
0.708877 + 0.705332i \(0.249202\pi\)
\(224\) 0.656854 0.0438879
\(225\) 0 0
\(226\) −24.1421 −1.60591
\(227\) −18.4853 −1.22691 −0.613456 0.789729i \(-0.710221\pi\)
−0.613456 + 0.789729i \(0.710221\pi\)
\(228\) 0 0
\(229\) −2.51472 −0.166177 −0.0830886 0.996542i \(-0.526478\pi\)
−0.0830886 + 0.996542i \(0.526478\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) −5.17157 −0.339530
\(233\) 16.5563 1.08464 0.542321 0.840171i \(-0.317546\pi\)
0.542321 + 0.840171i \(0.317546\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) −42.1127 −2.74130
\(237\) 0 0
\(238\) −2.41421 −0.156490
\(239\) −23.6569 −1.53023 −0.765117 0.643891i \(-0.777319\pi\)
−0.765117 + 0.643891i \(0.777319\pi\)
\(240\) 0 0
\(241\) 14.1421 0.910975 0.455488 0.890242i \(-0.349465\pi\)
0.455488 + 0.890242i \(0.349465\pi\)
\(242\) 2.41421 0.155192
\(243\) 0 0
\(244\) 33.7990 2.16376
\(245\) 0 0
\(246\) 0 0
\(247\) 18.1421 1.15436
\(248\) −37.4558 −2.37845
\(249\) 0 0
\(250\) 0 0
\(251\) −8.97056 −0.566217 −0.283108 0.959088i \(-0.591366\pi\)
−0.283108 + 0.959088i \(0.591366\pi\)
\(252\) 0 0
\(253\) 1.00000 0.0628695
\(254\) −17.4853 −1.09712
\(255\) 0 0
\(256\) −29.9706 −1.87316
\(257\) 9.31371 0.580973 0.290487 0.956879i \(-0.406183\pi\)
0.290487 + 0.956879i \(0.406183\pi\)
\(258\) 0 0
\(259\) −0.0710678 −0.00441594
\(260\) 0 0
\(261\) 0 0
\(262\) −16.4853 −1.01846
\(263\) 22.9706 1.41643 0.708213 0.705999i \(-0.249502\pi\)
0.708213 + 0.705999i \(0.249502\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) −6.41421 −0.393281
\(267\) 0 0
\(268\) 1.31371 0.0802475
\(269\) 15.7990 0.963281 0.481641 0.876369i \(-0.340041\pi\)
0.481641 + 0.876369i \(0.340041\pi\)
\(270\) 0 0
\(271\) 8.89949 0.540606 0.270303 0.962775i \(-0.412876\pi\)
0.270303 + 0.962775i \(0.412876\pi\)
\(272\) 7.24264 0.439150
\(273\) 0 0
\(274\) −29.3137 −1.77091
\(275\) 0 0
\(276\) 0 0
\(277\) −4.82843 −0.290112 −0.145056 0.989423i \(-0.546336\pi\)
−0.145056 + 0.989423i \(0.546336\pi\)
\(278\) −45.7990 −2.74684
\(279\) 0 0
\(280\) 0 0
\(281\) −32.0711 −1.91320 −0.956600 0.291405i \(-0.905877\pi\)
−0.956600 + 0.291405i \(0.905877\pi\)
\(282\) 0 0
\(283\) 0.899495 0.0534694 0.0267347 0.999643i \(-0.491489\pi\)
0.0267347 + 0.999643i \(0.491489\pi\)
\(284\) −29.9706 −1.77843
\(285\) 0 0
\(286\) 6.82843 0.403773
\(287\) −4.51472 −0.266495
\(288\) 0 0
\(289\) −11.1716 −0.657151
\(290\) 0 0
\(291\) 0 0
\(292\) 33.7990 1.97794
\(293\) 17.5858 1.02737 0.513686 0.857978i \(-0.328280\pi\)
0.513686 + 0.857978i \(0.328280\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0.757359 0.0440206
\(297\) 0 0
\(298\) −18.6569 −1.08076
\(299\) 2.82843 0.163572
\(300\) 0 0
\(301\) −4.82843 −0.278306
\(302\) 33.7990 1.94491
\(303\) 0 0
\(304\) 19.2426 1.10364
\(305\) 0 0
\(306\) 0 0
\(307\) −6.68629 −0.381607 −0.190803 0.981628i \(-0.561109\pi\)
−0.190803 + 0.981628i \(0.561109\pi\)
\(308\) −1.58579 −0.0903586
\(309\) 0 0
\(310\) 0 0
\(311\) −13.6569 −0.774409 −0.387205 0.921994i \(-0.626559\pi\)
−0.387205 + 0.921994i \(0.626559\pi\)
\(312\) 0 0
\(313\) 27.1421 1.53416 0.767082 0.641549i \(-0.221708\pi\)
0.767082 + 0.641549i \(0.221708\pi\)
\(314\) −14.4853 −0.817452
\(315\) 0 0
\(316\) 50.6985 2.85201
\(317\) 30.8284 1.73150 0.865748 0.500479i \(-0.166843\pi\)
0.865748 + 0.500479i \(0.166843\pi\)
\(318\) 0 0
\(319\) −1.17157 −0.0655955
\(320\) 0 0
\(321\) 0 0
\(322\) −1.00000 −0.0557278
\(323\) 15.4853 0.861624
\(324\) 0 0
\(325\) 0 0
\(326\) −38.1421 −2.11250
\(327\) 0 0
\(328\) 48.1127 2.65658
\(329\) 3.10051 0.170936
\(330\) 0 0
\(331\) −32.1421 −1.76669 −0.883346 0.468722i \(-0.844715\pi\)
−0.883346 + 0.468722i \(0.844715\pi\)
\(332\) −17.1716 −0.942412
\(333\) 0 0
\(334\) 52.6274 2.87964
\(335\) 0 0
\(336\) 0 0
\(337\) −4.14214 −0.225637 −0.112818 0.993616i \(-0.535988\pi\)
−0.112818 + 0.993616i \(0.535988\pi\)
\(338\) −12.0711 −0.656580
\(339\) 0 0
\(340\) 0 0
\(341\) −8.48528 −0.459504
\(342\) 0 0
\(343\) 5.72792 0.309279
\(344\) 51.4558 2.77431
\(345\) 0 0
\(346\) −30.3137 −1.62968
\(347\) −21.1716 −1.13655 −0.568275 0.822839i \(-0.692389\pi\)
−0.568275 + 0.822839i \(0.692389\pi\)
\(348\) 0 0
\(349\) −2.48528 −0.133034 −0.0665170 0.997785i \(-0.521189\pi\)
−0.0665170 + 0.997785i \(0.521189\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) −1.58579 −0.0845227
\(353\) −4.48528 −0.238727 −0.119364 0.992851i \(-0.538085\pi\)
−0.119364 + 0.992851i \(0.538085\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) −14.0000 −0.741999
\(357\) 0 0
\(358\) −40.5563 −2.14347
\(359\) −15.5147 −0.818836 −0.409418 0.912347i \(-0.634268\pi\)
−0.409418 + 0.912347i \(0.634268\pi\)
\(360\) 0 0
\(361\) 22.1421 1.16538
\(362\) −53.0416 −2.78781
\(363\) 0 0
\(364\) −4.48528 −0.235093
\(365\) 0 0
\(366\) 0 0
\(367\) −1.31371 −0.0685750 −0.0342875 0.999412i \(-0.510916\pi\)
−0.0342875 + 0.999412i \(0.510916\pi\)
\(368\) 3.00000 0.156386
\(369\) 0 0
\(370\) 0 0
\(371\) −3.17157 −0.164660
\(372\) 0 0
\(373\) 23.6569 1.22491 0.612453 0.790507i \(-0.290183\pi\)
0.612453 + 0.790507i \(0.290183\pi\)
\(374\) 5.82843 0.301381
\(375\) 0 0
\(376\) −33.0416 −1.70399
\(377\) −3.31371 −0.170665
\(378\) 0 0
\(379\) 9.17157 0.471112 0.235556 0.971861i \(-0.424309\pi\)
0.235556 + 0.971861i \(0.424309\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) −14.8995 −0.762324
\(383\) 20.0000 1.02195 0.510976 0.859595i \(-0.329284\pi\)
0.510976 + 0.859595i \(0.329284\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) −8.00000 −0.407189
\(387\) 0 0
\(388\) −22.3137 −1.13281
\(389\) 17.6569 0.895238 0.447619 0.894224i \(-0.352272\pi\)
0.447619 + 0.894224i \(0.352272\pi\)
\(390\) 0 0
\(391\) 2.41421 0.122092
\(392\) −30.1421 −1.52241
\(393\) 0 0
\(394\) 11.4853 0.578620
\(395\) 0 0
\(396\) 0 0
\(397\) 35.9411 1.80383 0.901917 0.431910i \(-0.142160\pi\)
0.901917 + 0.431910i \(0.142160\pi\)
\(398\) 26.1421 1.31039
\(399\) 0 0
\(400\) 0 0
\(401\) −31.7990 −1.58797 −0.793983 0.607940i \(-0.791996\pi\)
−0.793983 + 0.607940i \(0.791996\pi\)
\(402\) 0 0
\(403\) −24.0000 −1.19553
\(404\) −57.0416 −2.83793
\(405\) 0 0
\(406\) 1.17157 0.0581442
\(407\) 0.171573 0.00850455
\(408\) 0 0
\(409\) −4.14214 −0.204815 −0.102408 0.994743i \(-0.532655\pi\)
−0.102408 + 0.994743i \(0.532655\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 52.2843 2.57586
\(413\) 4.55635 0.224203
\(414\) 0 0
\(415\) 0 0
\(416\) −4.48528 −0.219909
\(417\) 0 0
\(418\) 15.4853 0.757410
\(419\) 25.4853 1.24504 0.622519 0.782605i \(-0.286109\pi\)
0.622519 + 0.782605i \(0.286109\pi\)
\(420\) 0 0
\(421\) −27.0000 −1.31590 −0.657950 0.753062i \(-0.728576\pi\)
−0.657950 + 0.753062i \(0.728576\pi\)
\(422\) −32.1421 −1.56465
\(423\) 0 0
\(424\) 33.7990 1.64142
\(425\) 0 0
\(426\) 0 0
\(427\) −3.65685 −0.176968
\(428\) −20.3431 −0.983323
\(429\) 0 0
\(430\) 0 0
\(431\) −1.17157 −0.0564327 −0.0282163 0.999602i \(-0.508983\pi\)
−0.0282163 + 0.999602i \(0.508983\pi\)
\(432\) 0 0
\(433\) 13.3137 0.639816 0.319908 0.947449i \(-0.396348\pi\)
0.319908 + 0.947449i \(0.396348\pi\)
\(434\) 8.48528 0.407307
\(435\) 0 0
\(436\) 20.3431 0.974260
\(437\) 6.41421 0.306833
\(438\) 0 0
\(439\) 2.27208 0.108440 0.0542202 0.998529i \(-0.482733\pi\)
0.0542202 + 0.998529i \(0.482733\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 16.4853 0.784125
\(443\) −7.97056 −0.378693 −0.189346 0.981910i \(-0.560637\pi\)
−0.189346 + 0.981910i \(0.560637\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 51.1127 2.42026
\(447\) 0 0
\(448\) 4.07107 0.192340
\(449\) 6.48528 0.306059 0.153030 0.988222i \(-0.451097\pi\)
0.153030 + 0.988222i \(0.451097\pi\)
\(450\) 0 0
\(451\) 10.8995 0.513237
\(452\) −38.2843 −1.80074
\(453\) 0 0
\(454\) −44.6274 −2.09447
\(455\) 0 0
\(456\) 0 0
\(457\) −3.85786 −0.180463 −0.0902316 0.995921i \(-0.528761\pi\)
−0.0902316 + 0.995921i \(0.528761\pi\)
\(458\) −6.07107 −0.283682
\(459\) 0 0
\(460\) 0 0
\(461\) 32.7696 1.52623 0.763115 0.646263i \(-0.223669\pi\)
0.763115 + 0.646263i \(0.223669\pi\)
\(462\) 0 0
\(463\) −34.9706 −1.62522 −0.812610 0.582808i \(-0.801954\pi\)
−0.812610 + 0.582808i \(0.801954\pi\)
\(464\) −3.51472 −0.163167
\(465\) 0 0
\(466\) 39.9706 1.85160
\(467\) 10.6274 0.491778 0.245889 0.969298i \(-0.420920\pi\)
0.245889 + 0.969298i \(0.420920\pi\)
\(468\) 0 0
\(469\) −0.142136 −0.00656321
\(470\) 0 0
\(471\) 0 0
\(472\) −48.5563 −2.23499
\(473\) 11.6569 0.535983
\(474\) 0 0
\(475\) 0 0
\(476\) −3.82843 −0.175476
\(477\) 0 0
\(478\) −57.1127 −2.61227
\(479\) 24.4853 1.11876 0.559381 0.828911i \(-0.311039\pi\)
0.559381 + 0.828911i \(0.311039\pi\)
\(480\) 0 0
\(481\) 0.485281 0.0221269
\(482\) 34.1421 1.55513
\(483\) 0 0
\(484\) 3.82843 0.174019
\(485\) 0 0
\(486\) 0 0
\(487\) 6.48528 0.293876 0.146938 0.989146i \(-0.453058\pi\)
0.146938 + 0.989146i \(0.453058\pi\)
\(488\) 38.9706 1.76411
\(489\) 0 0
\(490\) 0 0
\(491\) 24.1421 1.08952 0.544760 0.838592i \(-0.316621\pi\)
0.544760 + 0.838592i \(0.316621\pi\)
\(492\) 0 0
\(493\) −2.82843 −0.127386
\(494\) 43.7990 1.97061
\(495\) 0 0
\(496\) −25.4558 −1.14300
\(497\) 3.24264 0.145452
\(498\) 0 0
\(499\) 35.1716 1.57450 0.787248 0.616637i \(-0.211505\pi\)
0.787248 + 0.616637i \(0.211505\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) −21.6569 −0.966593
\(503\) −34.2843 −1.52866 −0.764330 0.644825i \(-0.776930\pi\)
−0.764330 + 0.644825i \(0.776930\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 2.41421 0.107325
\(507\) 0 0
\(508\) −27.7279 −1.23023
\(509\) −4.62742 −0.205107 −0.102553 0.994728i \(-0.532701\pi\)
−0.102553 + 0.994728i \(0.532701\pi\)
\(510\) 0 0
\(511\) −3.65685 −0.161770
\(512\) −31.2426 −1.38074
\(513\) 0 0
\(514\) 22.4853 0.991783
\(515\) 0 0
\(516\) 0 0
\(517\) −7.48528 −0.329202
\(518\) −0.171573 −0.00753848
\(519\) 0 0
\(520\) 0 0
\(521\) 36.1421 1.58342 0.791708 0.610900i \(-0.209192\pi\)
0.791708 + 0.610900i \(0.209192\pi\)
\(522\) 0 0
\(523\) −42.2132 −1.84585 −0.922927 0.384974i \(-0.874210\pi\)
−0.922927 + 0.384974i \(0.874210\pi\)
\(524\) −26.1421 −1.14202
\(525\) 0 0
\(526\) 55.4558 2.41799
\(527\) −20.4853 −0.892353
\(528\) 0 0
\(529\) −22.0000 −0.956522
\(530\) 0 0
\(531\) 0 0
\(532\) −10.1716 −0.440994
\(533\) 30.8284 1.33533
\(534\) 0 0
\(535\) 0 0
\(536\) 1.51472 0.0654259
\(537\) 0 0
\(538\) 38.1421 1.64442
\(539\) −6.82843 −0.294121
\(540\) 0 0
\(541\) 11.3137 0.486414 0.243207 0.969974i \(-0.421801\pi\)
0.243207 + 0.969974i \(0.421801\pi\)
\(542\) 21.4853 0.922872
\(543\) 0 0
\(544\) −3.82843 −0.164142
\(545\) 0 0
\(546\) 0 0
\(547\) −35.8701 −1.53369 −0.766846 0.641831i \(-0.778175\pi\)
−0.766846 + 0.641831i \(0.778175\pi\)
\(548\) −46.4853 −1.98575
\(549\) 0 0
\(550\) 0 0
\(551\) −7.51472 −0.320138
\(552\) 0 0
\(553\) −5.48528 −0.233258
\(554\) −11.6569 −0.495252
\(555\) 0 0
\(556\) −72.6274 −3.08009
\(557\) −5.17157 −0.219127 −0.109563 0.993980i \(-0.534945\pi\)
−0.109563 + 0.993980i \(0.534945\pi\)
\(558\) 0 0
\(559\) 32.9706 1.39451
\(560\) 0 0
\(561\) 0 0
\(562\) −77.4264 −3.26604
\(563\) 15.3137 0.645396 0.322698 0.946502i \(-0.395410\pi\)
0.322698 + 0.946502i \(0.395410\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 2.17157 0.0912780
\(567\) 0 0
\(568\) −34.5563 −1.44995
\(569\) −15.2426 −0.639005 −0.319502 0.947585i \(-0.603516\pi\)
−0.319502 + 0.947585i \(0.603516\pi\)
\(570\) 0 0
\(571\) −9.02944 −0.377870 −0.188935 0.981990i \(-0.560504\pi\)
−0.188935 + 0.981990i \(0.560504\pi\)
\(572\) 10.8284 0.452759
\(573\) 0 0
\(574\) −10.8995 −0.454936
\(575\) 0 0
\(576\) 0 0
\(577\) −23.9706 −0.997908 −0.498954 0.866629i \(-0.666282\pi\)
−0.498954 + 0.866629i \(0.666282\pi\)
\(578\) −26.9706 −1.12183
\(579\) 0 0
\(580\) 0 0
\(581\) 1.85786 0.0770772
\(582\) 0 0
\(583\) 7.65685 0.317115
\(584\) 38.9706 1.61261
\(585\) 0 0
\(586\) 42.4558 1.75383
\(587\) −36.6569 −1.51299 −0.756495 0.653999i \(-0.773090\pi\)
−0.756495 + 0.653999i \(0.773090\pi\)
\(588\) 0 0
\(589\) −54.4264 −2.24260
\(590\) 0 0
\(591\) 0 0
\(592\) 0.514719 0.0211548
\(593\) 3.79899 0.156006 0.0780029 0.996953i \(-0.475146\pi\)
0.0780029 + 0.996953i \(0.475146\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) −29.5858 −1.21188
\(597\) 0 0
\(598\) 6.82843 0.279235
\(599\) −36.3137 −1.48374 −0.741869 0.670545i \(-0.766060\pi\)
−0.741869 + 0.670545i \(0.766060\pi\)
\(600\) 0 0
\(601\) −14.8284 −0.604864 −0.302432 0.953171i \(-0.597799\pi\)
−0.302432 + 0.953171i \(0.597799\pi\)
\(602\) −11.6569 −0.475098
\(603\) 0 0
\(604\) 53.5980 2.18087
\(605\) 0 0
\(606\) 0 0
\(607\) 2.97056 0.120571 0.0602857 0.998181i \(-0.480799\pi\)
0.0602857 + 0.998181i \(0.480799\pi\)
\(608\) −10.1716 −0.412512
\(609\) 0 0
\(610\) 0 0
\(611\) −21.1716 −0.856510
\(612\) 0 0
\(613\) 42.0000 1.69636 0.848182 0.529705i \(-0.177697\pi\)
0.848182 + 0.529705i \(0.177697\pi\)
\(614\) −16.1421 −0.651444
\(615\) 0 0
\(616\) −1.82843 −0.0736694
\(617\) −9.85786 −0.396863 −0.198431 0.980115i \(-0.563585\pi\)
−0.198431 + 0.980115i \(0.563585\pi\)
\(618\) 0 0
\(619\) 38.6274 1.55257 0.776283 0.630384i \(-0.217103\pi\)
0.776283 + 0.630384i \(0.217103\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) −32.9706 −1.32200
\(623\) 1.51472 0.0606859
\(624\) 0 0
\(625\) 0 0
\(626\) 65.5269 2.61898
\(627\) 0 0
\(628\) −22.9706 −0.916625
\(629\) 0.414214 0.0165158
\(630\) 0 0
\(631\) 26.6274 1.06002 0.530010 0.847991i \(-0.322188\pi\)
0.530010 + 0.847991i \(0.322188\pi\)
\(632\) 58.4558 2.32525
\(633\) 0 0
\(634\) 74.4264 2.95585
\(635\) 0 0
\(636\) 0 0
\(637\) −19.3137 −0.765237
\(638\) −2.82843 −0.111979
\(639\) 0 0
\(640\) 0 0
\(641\) 42.4853 1.67807 0.839034 0.544079i \(-0.183121\pi\)
0.839034 + 0.544079i \(0.183121\pi\)
\(642\) 0 0
\(643\) 32.9706 1.30023 0.650116 0.759835i \(-0.274720\pi\)
0.650116 + 0.759835i \(0.274720\pi\)
\(644\) −1.58579 −0.0624887
\(645\) 0 0
\(646\) 37.3848 1.47088
\(647\) 17.3431 0.681829 0.340915 0.940094i \(-0.389263\pi\)
0.340915 + 0.940094i \(0.389263\pi\)
\(648\) 0 0
\(649\) −11.0000 −0.431788
\(650\) 0 0
\(651\) 0 0
\(652\) −60.4853 −2.36879
\(653\) −40.4853 −1.58431 −0.792156 0.610319i \(-0.791041\pi\)
−0.792156 + 0.610319i \(0.791041\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 32.6985 1.27666
\(657\) 0 0
\(658\) 7.48528 0.291807
\(659\) −15.1127 −0.588707 −0.294354 0.955697i \(-0.595104\pi\)
−0.294354 + 0.955697i \(0.595104\pi\)
\(660\) 0 0
\(661\) −34.6569 −1.34800 −0.673998 0.738733i \(-0.735424\pi\)
−0.673998 + 0.738733i \(0.735424\pi\)
\(662\) −77.5980 −3.01593
\(663\) 0 0
\(664\) −19.7990 −0.768350
\(665\) 0 0
\(666\) 0 0
\(667\) −1.17157 −0.0453635
\(668\) 83.4558 3.22900
\(669\) 0 0
\(670\) 0 0
\(671\) 8.82843 0.340818
\(672\) 0 0
\(673\) −11.6569 −0.449339 −0.224669 0.974435i \(-0.572130\pi\)
−0.224669 + 0.974435i \(0.572130\pi\)
\(674\) −10.0000 −0.385186
\(675\) 0 0
\(676\) −19.1421 −0.736236
\(677\) 40.4853 1.55598 0.777988 0.628279i \(-0.216240\pi\)
0.777988 + 0.628279i \(0.216240\pi\)
\(678\) 0 0
\(679\) 2.41421 0.0926490
\(680\) 0 0
\(681\) 0 0
\(682\) −20.4853 −0.784422
\(683\) 29.4853 1.12822 0.564111 0.825699i \(-0.309219\pi\)
0.564111 + 0.825699i \(0.309219\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 13.8284 0.527972
\(687\) 0 0
\(688\) 34.9706 1.33324
\(689\) 21.6569 0.825060
\(690\) 0 0
\(691\) 15.4558 0.587968 0.293984 0.955810i \(-0.405019\pi\)
0.293984 + 0.955810i \(0.405019\pi\)
\(692\) −48.0711 −1.82739
\(693\) 0 0
\(694\) −51.1127 −1.94021
\(695\) 0 0
\(696\) 0 0
\(697\) 26.3137 0.996703
\(698\) −6.00000 −0.227103
\(699\) 0 0
\(700\) 0 0
\(701\) 4.61522 0.174315 0.0871573 0.996195i \(-0.472222\pi\)
0.0871573 + 0.996195i \(0.472222\pi\)
\(702\) 0 0
\(703\) 1.10051 0.0415063
\(704\) −9.82843 −0.370423
\(705\) 0 0
\(706\) −10.8284 −0.407533
\(707\) 6.17157 0.232106
\(708\) 0 0
\(709\) −0.857864 −0.0322178 −0.0161089 0.999870i \(-0.505128\pi\)
−0.0161089 + 0.999870i \(0.505128\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) −16.1421 −0.604952
\(713\) −8.48528 −0.317776
\(714\) 0 0
\(715\) 0 0
\(716\) −64.3137 −2.40352
\(717\) 0 0
\(718\) −37.4558 −1.39784
\(719\) 1.65685 0.0617902 0.0308951 0.999523i \(-0.490164\pi\)
0.0308951 + 0.999523i \(0.490164\pi\)
\(720\) 0 0
\(721\) −5.65685 −0.210672
\(722\) 53.4558 1.98942
\(723\) 0 0
\(724\) −84.1127 −3.12602
\(725\) 0 0
\(726\) 0 0
\(727\) 16.9706 0.629403 0.314702 0.949191i \(-0.398096\pi\)
0.314702 + 0.949191i \(0.398096\pi\)
\(728\) −5.17157 −0.191671
\(729\) 0 0
\(730\) 0 0
\(731\) 28.1421 1.04087
\(732\) 0 0
\(733\) −3.85786 −0.142493 −0.0712467 0.997459i \(-0.522698\pi\)
−0.0712467 + 0.997459i \(0.522698\pi\)
\(734\) −3.17157 −0.117065
\(735\) 0 0
\(736\) −1.58579 −0.0584529
\(737\) 0.343146 0.0126399
\(738\) 0 0
\(739\) 17.5858 0.646904 0.323452 0.946245i \(-0.395157\pi\)
0.323452 + 0.946245i \(0.395157\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) −7.65685 −0.281092
\(743\) 31.1127 1.14141 0.570707 0.821154i \(-0.306669\pi\)
0.570707 + 0.821154i \(0.306669\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 57.1127 2.09104
\(747\) 0 0
\(748\) 9.24264 0.337944
\(749\) 2.20101 0.0804232
\(750\) 0 0
\(751\) −47.5980 −1.73687 −0.868437 0.495799i \(-0.834875\pi\)
−0.868437 + 0.495799i \(0.834875\pi\)
\(752\) −22.4558 −0.818880
\(753\) 0 0
\(754\) −8.00000 −0.291343
\(755\) 0 0
\(756\) 0 0
\(757\) 33.3137 1.21081 0.605404 0.795919i \(-0.293012\pi\)
0.605404 + 0.795919i \(0.293012\pi\)
\(758\) 22.1421 0.804239
\(759\) 0 0
\(760\) 0 0
\(761\) 10.8284 0.392530 0.196265 0.980551i \(-0.437119\pi\)
0.196265 + 0.980551i \(0.437119\pi\)
\(762\) 0 0
\(763\) −2.20101 −0.0796819
\(764\) −23.6274 −0.854810
\(765\) 0 0
\(766\) 48.2843 1.74458
\(767\) −31.1127 −1.12341
\(768\) 0 0
\(769\) 3.65685 0.131870 0.0659348 0.997824i \(-0.478997\pi\)
0.0659348 + 0.997824i \(0.478997\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) −12.6863 −0.456590
\(773\) 4.82843 0.173666 0.0868332 0.996223i \(-0.472325\pi\)
0.0868332 + 0.996223i \(0.472325\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) −25.7279 −0.923579
\(777\) 0 0
\(778\) 42.6274 1.52827
\(779\) 69.9117 2.50485
\(780\) 0 0
\(781\) −7.82843 −0.280123
\(782\) 5.82843 0.208424
\(783\) 0 0
\(784\) −20.4853 −0.731617
\(785\) 0 0
\(786\) 0 0
\(787\) 22.0711 0.786749 0.393374 0.919378i \(-0.371308\pi\)
0.393374 + 0.919378i \(0.371308\pi\)
\(788\) 18.2132 0.648819
\(789\) 0 0
\(790\) 0 0
\(791\) 4.14214 0.147277
\(792\) 0 0
\(793\) 24.9706 0.886731
\(794\) 86.7696 3.07934
\(795\) 0 0
\(796\) 41.4558 1.46936
\(797\) 7.02944 0.248995 0.124498 0.992220i \(-0.460268\pi\)
0.124498 + 0.992220i \(0.460268\pi\)
\(798\) 0 0
\(799\) −18.0711 −0.639308
\(800\) 0 0
\(801\) 0 0
\(802\) −76.7696 −2.71083
\(803\) 8.82843 0.311548
\(804\) 0 0
\(805\) 0 0
\(806\) −57.9411 −2.04089
\(807\) 0 0
\(808\) −65.7696 −2.31376
\(809\) 17.7279 0.623281 0.311640 0.950200i \(-0.399122\pi\)
0.311640 + 0.950200i \(0.399122\pi\)
\(810\) 0 0
\(811\) −32.2132 −1.13116 −0.565579 0.824694i \(-0.691347\pi\)
−0.565579 + 0.824694i \(0.691347\pi\)
\(812\) 1.85786 0.0651983
\(813\) 0 0
\(814\) 0.414214 0.0145182
\(815\) 0 0
\(816\) 0 0
\(817\) 74.7696 2.61586
\(818\) −10.0000 −0.349642
\(819\) 0 0
\(820\) 0 0
\(821\) 23.7990 0.830590 0.415295 0.909687i \(-0.363678\pi\)
0.415295 + 0.909687i \(0.363678\pi\)
\(822\) 0 0
\(823\) −18.9706 −0.661272 −0.330636 0.943758i \(-0.607263\pi\)
−0.330636 + 0.943758i \(0.607263\pi\)
\(824\) 60.2843 2.10010
\(825\) 0 0
\(826\) 11.0000 0.382739
\(827\) −35.3137 −1.22798 −0.613989 0.789315i \(-0.710436\pi\)
−0.613989 + 0.789315i \(0.710436\pi\)
\(828\) 0 0
\(829\) 19.9411 0.692584 0.346292 0.938127i \(-0.387441\pi\)
0.346292 + 0.938127i \(0.387441\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) −27.7990 −0.963757
\(833\) −16.4853 −0.571181
\(834\) 0 0
\(835\) 0 0
\(836\) 24.5563 0.849299
\(837\) 0 0
\(838\) 61.5269 2.12541
\(839\) 24.6863 0.852265 0.426133 0.904661i \(-0.359876\pi\)
0.426133 + 0.904661i \(0.359876\pi\)
\(840\) 0 0
\(841\) −27.6274 −0.952670
\(842\) −65.1838 −2.24638
\(843\) 0 0
\(844\) −50.9706 −1.75448
\(845\) 0 0
\(846\) 0 0
\(847\) −0.414214 −0.0142325
\(848\) 22.9706 0.788812
\(849\) 0 0
\(850\) 0 0
\(851\) 0.171573 0.00588144
\(852\) 0 0
\(853\) 24.8284 0.850109 0.425055 0.905168i \(-0.360255\pi\)
0.425055 + 0.905168i \(0.360255\pi\)
\(854\) −8.82843 −0.302103
\(855\) 0 0
\(856\) −23.4558 −0.801704
\(857\) 22.6985 0.775365 0.387683 0.921793i \(-0.373276\pi\)
0.387683 + 0.921793i \(0.373276\pi\)
\(858\) 0 0
\(859\) −3.51472 −0.119921 −0.0599603 0.998201i \(-0.519097\pi\)
−0.0599603 + 0.998201i \(0.519097\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) −2.82843 −0.0963366
\(863\) −51.3137 −1.74674 −0.873369 0.487058i \(-0.838070\pi\)
−0.873369 + 0.487058i \(0.838070\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 32.1421 1.09223
\(867\) 0 0
\(868\) 13.4558 0.456721
\(869\) 13.2426 0.449226
\(870\) 0 0
\(871\) 0.970563 0.0328863
\(872\) 23.4558 0.794315
\(873\) 0 0
\(874\) 15.4853 0.523797
\(875\) 0 0
\(876\) 0 0
\(877\) 47.1127 1.59088 0.795441 0.606031i \(-0.207239\pi\)
0.795441 + 0.606031i \(0.207239\pi\)
\(878\) 5.48528 0.185119
\(879\) 0 0
\(880\) 0 0
\(881\) 24.0000 0.808581 0.404290 0.914631i \(-0.367519\pi\)
0.404290 + 0.914631i \(0.367519\pi\)
\(882\) 0 0
\(883\) 6.62742 0.223030 0.111515 0.993763i \(-0.464430\pi\)
0.111515 + 0.993763i \(0.464430\pi\)
\(884\) 26.1421 0.879255
\(885\) 0 0
\(886\) −19.2426 −0.646469
\(887\) 6.14214 0.206233 0.103116 0.994669i \(-0.467119\pi\)
0.103116 + 0.994669i \(0.467119\pi\)
\(888\) 0 0
\(889\) 3.00000 0.100617
\(890\) 0 0
\(891\) 0 0
\(892\) 81.0538 2.71388
\(893\) −48.0122 −1.60667
\(894\) 0 0
\(895\) 0 0
\(896\) 8.51472 0.284457
\(897\) 0 0
\(898\) 15.6569 0.522476
\(899\) 9.94113 0.331555
\(900\) 0 0
\(901\) 18.4853 0.615834
\(902\) 26.3137 0.876151
\(903\) 0 0
\(904\) −44.1421 −1.46815
\(905\) 0 0
\(906\) 0 0
\(907\) −14.4853 −0.480976 −0.240488 0.970652i \(-0.577307\pi\)
−0.240488 + 0.970652i \(0.577307\pi\)
\(908\) −70.7696 −2.34857
\(909\) 0 0
\(910\) 0 0
\(911\) −45.4853 −1.50699 −0.753497 0.657451i \(-0.771635\pi\)
−0.753497 + 0.657451i \(0.771635\pi\)
\(912\) 0 0
\(913\) −4.48528 −0.148441
\(914\) −9.31371 −0.308070
\(915\) 0 0
\(916\) −9.62742 −0.318099
\(917\) 2.82843 0.0934029
\(918\) 0 0
\(919\) 44.2132 1.45846 0.729230 0.684269i \(-0.239879\pi\)
0.729230 + 0.684269i \(0.239879\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 79.1127 2.60544
\(923\) −22.1421 −0.728817
\(924\) 0 0
\(925\) 0 0
\(926\) −84.4264 −2.77442
\(927\) 0 0
\(928\) 1.85786 0.0609874
\(929\) −25.7990 −0.846437 −0.423219 0.906028i \(-0.639100\pi\)
−0.423219 + 0.906028i \(0.639100\pi\)
\(930\) 0 0
\(931\) −43.7990 −1.43545
\(932\) 63.3848 2.07624
\(933\) 0 0
\(934\) 25.6569 0.839518
\(935\) 0 0
\(936\) 0 0
\(937\) 16.0000 0.522697 0.261349 0.965244i \(-0.415833\pi\)
0.261349 + 0.965244i \(0.415833\pi\)
\(938\) −0.343146 −0.0112041
\(939\) 0 0
\(940\) 0 0
\(941\) 3.10051 0.101074 0.0505368 0.998722i \(-0.483907\pi\)
0.0505368 + 0.998722i \(0.483907\pi\)
\(942\) 0 0
\(943\) 10.8995 0.354936
\(944\) −33.0000 −1.07406
\(945\) 0 0
\(946\) 28.1421 0.914980
\(947\) 2.79899 0.0909549 0.0454775 0.998965i \(-0.485519\pi\)
0.0454775 + 0.998965i \(0.485519\pi\)
\(948\) 0 0
\(949\) 24.9706 0.810579
\(950\) 0 0
\(951\) 0 0
\(952\) −4.41421 −0.143065
\(953\) 19.0416 0.616819 0.308409 0.951254i \(-0.400203\pi\)
0.308409 + 0.951254i \(0.400203\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) −90.5685 −2.92920
\(957\) 0 0
\(958\) 59.1127 1.90984
\(959\) 5.02944 0.162409
\(960\) 0 0
\(961\) 41.0000 1.32258
\(962\) 1.17157 0.0377730
\(963\) 0 0
\(964\) 54.1421 1.74380
\(965\) 0 0
\(966\) 0 0
\(967\) −38.0000 −1.22200 −0.610999 0.791632i \(-0.709232\pi\)
−0.610999 + 0.791632i \(0.709232\pi\)
\(968\) 4.41421 0.141878
\(969\) 0 0
\(970\) 0 0
\(971\) 27.6863 0.888495 0.444248 0.895904i \(-0.353471\pi\)
0.444248 + 0.895904i \(0.353471\pi\)
\(972\) 0 0
\(973\) 7.85786 0.251912
\(974\) 15.6569 0.501678
\(975\) 0 0
\(976\) 26.4853 0.847773
\(977\) −52.5685 −1.68182 −0.840908 0.541178i \(-0.817979\pi\)
−0.840908 + 0.541178i \(0.817979\pi\)
\(978\) 0 0
\(979\) −3.65685 −0.116874
\(980\) 0 0
\(981\) 0 0
\(982\) 58.2843 1.85993
\(983\) −5.28427 −0.168542 −0.0842710 0.996443i \(-0.526856\pi\)
−0.0842710 + 0.996443i \(0.526856\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) −6.82843 −0.217461
\(987\) 0 0
\(988\) 69.4558 2.20968
\(989\) 11.6569 0.370666
\(990\) 0 0
\(991\) 14.2843 0.453755 0.226877 0.973923i \(-0.427148\pi\)
0.226877 + 0.973923i \(0.427148\pi\)
\(992\) 13.4558 0.427223
\(993\) 0 0
\(994\) 7.82843 0.248303
\(995\) 0 0
\(996\) 0 0
\(997\) −43.2548 −1.36989 −0.684947 0.728593i \(-0.740175\pi\)
−0.684947 + 0.728593i \(0.740175\pi\)
\(998\) 84.9117 2.68783
\(999\) 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 2475.2.a.w.1.2 2
3.2 odd 2 825.2.a.d.1.1 2
5.2 odd 4 2475.2.c.o.199.4 4
5.3 odd 4 2475.2.c.o.199.1 4
5.4 even 2 2475.2.a.l.1.1 2
15.2 even 4 825.2.c.d.199.1 4
15.8 even 4 825.2.c.d.199.4 4
15.14 odd 2 825.2.a.f.1.2 yes 2
33.32 even 2 9075.2.a.ca.1.2 2
165.164 even 2 9075.2.a.w.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
825.2.a.d.1.1 2 3.2 odd 2
825.2.a.f.1.2 yes 2 15.14 odd 2
825.2.c.d.199.1 4 15.2 even 4
825.2.c.d.199.4 4 15.8 even 4
2475.2.a.l.1.1 2 5.4 even 2
2475.2.a.w.1.2 2 1.1 even 1 trivial
2475.2.c.o.199.1 4 5.3 odd 4
2475.2.c.o.199.4 4 5.2 odd 4
9075.2.a.w.1.1 2 165.164 even 2
9075.2.a.ca.1.2 2 33.32 even 2