Properties

Label 7800.2.a.bu.1.2
Level $7800$
Weight $2$
Character 7800.1
Self dual yes
Analytic conductor $62.283$
Analytic rank $0$
Dimension $4$
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: \(0\)
Dimension: \(4\)
Coefficient field: 4.4.461844.1
Defining polynomial: \(x^{4} - x^{3} - 14 x^{2} - 12 x + 6\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-1.60110\) of defining polynomial
Character \(\chi\) \(=\) 7800.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{3} -2.60110 q^{7} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{3} -2.60110 q^{7} +1.00000 q^{9} -5.61488 q^{11} -1.00000 q^{13} -5.77950 q^{17} -0.234280 q^{19} +2.60110 q^{21} -7.01378 q^{23} -1.00000 q^{27} +4.20220 q^{29} +4.41269 q^{31} +5.61488 q^{33} +2.96792 q^{37} +1.00000 q^{39} -10.9955 q^{41} -0.188415 q^{43} -7.84916 q^{47} -0.234280 q^{49} +5.77950 q^{51} -11.9496 q^{53} +0.234280 q^{57} -12.1601 q^{59} -2.98170 q^{61} -2.60110 q^{63} -3.56902 q^{67} +7.01378 q^{69} +3.76572 q^{71} +14.5728 q^{73} +14.6049 q^{77} -4.96792 q^{79} +1.00000 q^{81} -3.39890 q^{83} -4.20220 q^{87} +17.2298 q^{89} +2.60110 q^{91} -4.41269 q^{93} +5.20220 q^{97} -5.61488 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{3} - 3 q^{7} + 4 q^{9} + O(q^{10}) \) \( 4 q - 4 q^{3} - 3 q^{7} + 4 q^{9} + 5 q^{11} - 4 q^{13} - 7 q^{17} + 3 q^{19} + 3 q^{21} - 8 q^{23} - 4 q^{27} + 2 q^{29} + 5 q^{31} - 5 q^{33} + q^{37} + 4 q^{39} + 7 q^{41} - 6 q^{43} + 3 q^{49} + 7 q^{51} - 6 q^{53} - 3 q^{57} - 9 q^{59} + 19 q^{61} - 3 q^{63} + 4 q^{67} + 8 q^{69} + 19 q^{71} + 6 q^{73} + 17 q^{77} - 9 q^{79} + 4 q^{81} - 21 q^{83} - 2 q^{87} + 14 q^{89} + 3 q^{91} - 5 q^{93} + 6 q^{97} + 5 q^{99} + O(q^{100}) \)

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
\(7\) −2.60110 −0.983123 −0.491562 0.870843i \(-0.663574\pi\)
−0.491562 + 0.870843i \(0.663574\pi\)
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) −5.61488 −1.69295 −0.846476 0.532427i \(-0.821280\pi\)
−0.846476 + 0.532427i \(0.821280\pi\)
\(12\) 0 0
\(13\) −1.00000 −0.277350
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −5.77950 −1.40174 −0.700868 0.713291i \(-0.747204\pi\)
−0.700868 + 0.713291i \(0.747204\pi\)
\(18\) 0 0
\(19\) −0.234280 −0.0537475 −0.0268738 0.999639i \(-0.508555\pi\)
−0.0268738 + 0.999639i \(0.508555\pi\)
\(20\) 0 0
\(21\) 2.60110 0.567607
\(22\) 0 0
\(23\) −7.01378 −1.46248 −0.731238 0.682123i \(-0.761057\pi\)
−0.731238 + 0.682123i \(0.761057\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) −1.00000 −0.192450
\(28\) 0 0
\(29\) 4.20220 0.780329 0.390164 0.920745i \(-0.372418\pi\)
0.390164 + 0.920745i \(0.372418\pi\)
\(30\) 0 0
\(31\) 4.41269 0.792542 0.396271 0.918134i \(-0.370304\pi\)
0.396271 + 0.918134i \(0.370304\pi\)
\(32\) 0 0
\(33\) 5.61488 0.977426
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 2.96792 0.487923 0.243961 0.969785i \(-0.421553\pi\)
0.243961 + 0.969785i \(0.421553\pi\)
\(38\) 0 0
\(39\) 1.00000 0.160128
\(40\) 0 0
\(41\) −10.9955 −1.71721 −0.858603 0.512640i \(-0.828667\pi\)
−0.858603 + 0.512640i \(0.828667\pi\)
\(42\) 0 0
\(43\) −0.188415 −0.0287330 −0.0143665 0.999897i \(-0.504573\pi\)
−0.0143665 + 0.999897i \(0.504573\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −7.84916 −1.14492 −0.572459 0.819933i \(-0.694010\pi\)
−0.572459 + 0.819933i \(0.694010\pi\)
\(48\) 0 0
\(49\) −0.234280 −0.0334686
\(50\) 0 0
\(51\) 5.77950 0.809293
\(52\) 0 0
\(53\) −11.9496 −1.64141 −0.820704 0.571354i \(-0.806418\pi\)
−0.820704 + 0.571354i \(0.806418\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 0.234280 0.0310312
\(58\) 0 0
\(59\) −12.1601 −1.58311 −0.791556 0.611097i \(-0.790729\pi\)
−0.791556 + 0.611097i \(0.790729\pi\)
\(60\) 0 0
\(61\) −2.98170 −0.381768 −0.190884 0.981613i \(-0.561135\pi\)
−0.190884 + 0.981613i \(0.561135\pi\)
\(62\) 0 0
\(63\) −2.60110 −0.327708
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) −3.56902 −0.436025 −0.218013 0.975946i \(-0.569957\pi\)
−0.218013 + 0.975946i \(0.569957\pi\)
\(68\) 0 0
\(69\) 7.01378 0.844360
\(70\) 0 0
\(71\) 3.76572 0.446909 0.223454 0.974714i \(-0.428267\pi\)
0.223454 + 0.974714i \(0.428267\pi\)
\(72\) 0 0
\(73\) 14.5728 1.70562 0.852808 0.522224i \(-0.174898\pi\)
0.852808 + 0.522224i \(0.174898\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 14.6049 1.66438
\(78\) 0 0
\(79\) −4.96792 −0.558935 −0.279467 0.960155i \(-0.590158\pi\)
−0.279467 + 0.960155i \(0.590158\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) −3.39890 −0.373078 −0.186539 0.982448i \(-0.559727\pi\)
−0.186539 + 0.982448i \(0.559727\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) −4.20220 −0.450523
\(88\) 0 0
\(89\) 17.2298 1.82635 0.913176 0.407566i \(-0.133622\pi\)
0.913176 + 0.407566i \(0.133622\pi\)
\(90\) 0 0
\(91\) 2.60110 0.272669
\(92\) 0 0
\(93\) −4.41269 −0.457574
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 5.20220 0.528203 0.264102 0.964495i \(-0.414925\pi\)
0.264102 + 0.964495i \(0.414925\pi\)
\(98\) 0 0
\(99\) −5.61488 −0.564317
\(100\) 0 0
\(101\) 5.96792 0.593830 0.296915 0.954904i \(-0.404042\pi\)
0.296915 + 0.954904i \(0.404042\pi\)
\(102\) 0 0
\(103\) −15.8392 −1.56068 −0.780339 0.625357i \(-0.784954\pi\)
−0.780339 + 0.625357i \(0.784954\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −4.04587 −0.391129 −0.195564 0.980691i \(-0.562654\pi\)
−0.195564 + 0.980691i \(0.562654\pi\)
\(108\) 0 0
\(109\) −17.7291 −1.69814 −0.849071 0.528278i \(-0.822838\pi\)
−0.849071 + 0.528278i \(0.822838\pi\)
\(110\) 0 0
\(111\) −2.96792 −0.281702
\(112\) 0 0
\(113\) 0.733639 0.0690150 0.0345075 0.999404i \(-0.489014\pi\)
0.0345075 + 0.999404i \(0.489014\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) −1.00000 −0.0924500
\(118\) 0 0
\(119\) 15.0331 1.37808
\(120\) 0 0
\(121\) 20.5269 1.86608
\(122\) 0 0
\(123\) 10.9955 0.991430
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) 4.17012 0.370038 0.185019 0.982735i \(-0.440765\pi\)
0.185019 + 0.982735i \(0.440765\pi\)
\(128\) 0 0
\(129\) 0.188415 0.0165890
\(130\) 0 0
\(131\) 18.2618 1.59555 0.797773 0.602958i \(-0.206012\pi\)
0.797773 + 0.602958i \(0.206012\pi\)
\(132\) 0 0
\(133\) 0.609386 0.0528405
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 18.0093 1.53864 0.769318 0.638866i \(-0.220596\pi\)
0.769318 + 0.638866i \(0.220596\pi\)
\(138\) 0 0
\(139\) 3.55901 0.301871 0.150936 0.988544i \(-0.451771\pi\)
0.150936 + 0.988544i \(0.451771\pi\)
\(140\) 0 0
\(141\) 7.84916 0.661019
\(142\) 0 0
\(143\) 5.61488 0.469540
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 0.234280 0.0193231
\(148\) 0 0
\(149\) 9.20220 0.753874 0.376937 0.926239i \(-0.376977\pi\)
0.376937 + 0.926239i \(0.376977\pi\)
\(150\) 0 0
\(151\) 3.86746 0.314729 0.157365 0.987541i \(-0.449700\pi\)
0.157365 + 0.987541i \(0.449700\pi\)
\(152\) 0 0
\(153\) −5.77950 −0.467245
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 11.2343 0.896593 0.448297 0.893885i \(-0.352031\pi\)
0.448297 + 0.893885i \(0.352031\pi\)
\(158\) 0 0
\(159\) 11.9496 0.947667
\(160\) 0 0
\(161\) 18.2436 1.43779
\(162\) 0 0
\(163\) 1.48234 0.116106 0.0580531 0.998313i \(-0.481511\pi\)
0.0580531 + 0.998313i \(0.481511\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −9.37232 −0.725252 −0.362626 0.931935i \(-0.618120\pi\)
−0.362626 + 0.931935i \(0.618120\pi\)
\(168\) 0 0
\(169\) 1.00000 0.0769231
\(170\) 0 0
\(171\) −0.234280 −0.0179158
\(172\) 0 0
\(173\) −21.7121 −1.65074 −0.825371 0.564591i \(-0.809034\pi\)
−0.825371 + 0.564591i \(0.809034\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 12.1601 0.914010
\(178\) 0 0
\(179\) −18.6204 −1.39175 −0.695876 0.718162i \(-0.744984\pi\)
−0.695876 + 0.718162i \(0.744984\pi\)
\(180\) 0 0
\(181\) 14.4365 1.07306 0.536528 0.843883i \(-0.319736\pi\)
0.536528 + 0.843883i \(0.319736\pi\)
\(182\) 0 0
\(183\) 2.98170 0.220414
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 32.4513 2.37307
\(188\) 0 0
\(189\) 2.60110 0.189202
\(190\) 0 0
\(191\) 20.4044 1.47641 0.738205 0.674576i \(-0.235674\pi\)
0.738205 + 0.674576i \(0.235674\pi\)
\(192\) 0 0
\(193\) −26.1318 −1.88101 −0.940504 0.339782i \(-0.889647\pi\)
−0.940504 + 0.339782i \(0.889647\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −21.8026 −1.55337 −0.776684 0.629890i \(-0.783100\pi\)
−0.776684 + 0.629890i \(0.783100\pi\)
\(198\) 0 0
\(199\) −5.22050 −0.370071 −0.185036 0.982732i \(-0.559240\pi\)
−0.185036 + 0.982732i \(0.559240\pi\)
\(200\) 0 0
\(201\) 3.56902 0.251739
\(202\) 0 0
\(203\) −10.9303 −0.767159
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) −7.01378 −0.487492
\(208\) 0 0
\(209\) 1.31546 0.0909920
\(210\) 0 0
\(211\) 11.0138 0.758220 0.379110 0.925352i \(-0.376230\pi\)
0.379110 + 0.925352i \(0.376230\pi\)
\(212\) 0 0
\(213\) −3.76572 −0.258023
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) −11.4778 −0.779166
\(218\) 0 0
\(219\) −14.5728 −0.984738
\(220\) 0 0
\(221\) 5.77950 0.388772
\(222\) 0 0
\(223\) −27.4733 −1.83975 −0.919875 0.392212i \(-0.871710\pi\)
−0.919875 + 0.392212i \(0.871710\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) −1.47556 −0.0979367 −0.0489683 0.998800i \(-0.515593\pi\)
−0.0489683 + 0.998800i \(0.515593\pi\)
\(228\) 0 0
\(229\) −9.65246 −0.637853 −0.318926 0.947779i \(-0.603322\pi\)
−0.318926 + 0.947779i \(0.603322\pi\)
\(230\) 0 0
\(231\) −14.6049 −0.960930
\(232\) 0 0
\(233\) −19.2298 −1.25978 −0.629892 0.776683i \(-0.716901\pi\)
−0.629892 + 0.776683i \(0.716901\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 4.96792 0.322701
\(238\) 0 0
\(239\) −0.244290 −0.0158018 −0.00790089 0.999969i \(-0.502515\pi\)
−0.00790089 + 0.999969i \(0.502515\pi\)
\(240\) 0 0
\(241\) 0.986215 0.0635277 0.0317639 0.999495i \(-0.489888\pi\)
0.0317639 + 0.999495i \(0.489888\pi\)
\(242\) 0 0
\(243\) −1.00000 −0.0641500
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 0.234280 0.0149069
\(248\) 0 0
\(249\) 3.39890 0.215397
\(250\) 0 0
\(251\) −0.995489 −0.0628347 −0.0314174 0.999506i \(-0.510002\pi\)
−0.0314174 + 0.999506i \(0.510002\pi\)
\(252\) 0 0
\(253\) 39.3816 2.47590
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −20.3523 −1.26954 −0.634771 0.772701i \(-0.718905\pi\)
−0.634771 + 0.772701i \(0.718905\pi\)
\(258\) 0 0
\(259\) −7.71985 −0.479688
\(260\) 0 0
\(261\) 4.20220 0.260110
\(262\) 0 0
\(263\) −0.252576 −0.0155745 −0.00778724 0.999970i \(-0.502479\pi\)
−0.00778724 + 0.999970i \(0.502479\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) −17.2298 −1.05444
\(268\) 0 0
\(269\) 14.9358 0.910654 0.455327 0.890324i \(-0.349522\pi\)
0.455327 + 0.890324i \(0.349522\pi\)
\(270\) 0 0
\(271\) −3.09723 −0.188143 −0.0940716 0.995565i \(-0.529988\pi\)
−0.0940716 + 0.995565i \(0.529988\pi\)
\(272\) 0 0
\(273\) −2.60110 −0.157426
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) 21.2939 1.27943 0.639714 0.768613i \(-0.279053\pi\)
0.639714 + 0.768613i \(0.279053\pi\)
\(278\) 0 0
\(279\) 4.41269 0.264181
\(280\) 0 0
\(281\) 13.1839 0.786486 0.393243 0.919435i \(-0.371353\pi\)
0.393243 + 0.919435i \(0.371353\pi\)
\(282\) 0 0
\(283\) 7.18218 0.426936 0.213468 0.976950i \(-0.431524\pi\)
0.213468 + 0.976950i \(0.431524\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 28.6004 1.68823
\(288\) 0 0
\(289\) 16.4027 0.964863
\(290\) 0 0
\(291\) −5.20220 −0.304958
\(292\) 0 0
\(293\) 18.9005 1.10418 0.552090 0.833784i \(-0.313830\pi\)
0.552090 + 0.833784i \(0.313830\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 5.61488 0.325809
\(298\) 0 0
\(299\) 7.01378 0.405618
\(300\) 0 0
\(301\) 0.490085 0.0282481
\(302\) 0 0
\(303\) −5.96792 −0.342848
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) 4.56352 0.260454 0.130227 0.991484i \(-0.458429\pi\)
0.130227 + 0.991484i \(0.458429\pi\)
\(308\) 0 0
\(309\) 15.8392 0.901058
\(310\) 0 0
\(311\) −25.5500 −1.44881 −0.724403 0.689376i \(-0.757885\pi\)
−0.724403 + 0.689376i \(0.757885\pi\)
\(312\) 0 0
\(313\) 30.4092 1.71883 0.859414 0.511281i \(-0.170829\pi\)
0.859414 + 0.511281i \(0.170829\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 32.0401 1.79955 0.899775 0.436354i \(-0.143731\pi\)
0.899775 + 0.436354i \(0.143731\pi\)
\(318\) 0 0
\(319\) −23.5949 −1.32106
\(320\) 0 0
\(321\) 4.04587 0.225818
\(322\) 0 0
\(323\) 1.35402 0.0753398
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) 17.7291 0.980423
\(328\) 0 0
\(329\) 20.4165 1.12560
\(330\) 0 0
\(331\) 7.45478 0.409752 0.204876 0.978788i \(-0.434321\pi\)
0.204876 + 0.978788i \(0.434321\pi\)
\(332\) 0 0
\(333\) 2.96792 0.162641
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) −12.4916 −0.680462 −0.340231 0.940342i \(-0.610505\pi\)
−0.340231 + 0.940342i \(0.610505\pi\)
\(338\) 0 0
\(339\) −0.733639 −0.0398458
\(340\) 0 0
\(341\) −24.7767 −1.34173
\(342\) 0 0
\(343\) 18.8171 1.01603
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 23.2115 1.24606 0.623029 0.782199i \(-0.285902\pi\)
0.623029 + 0.782199i \(0.285902\pi\)
\(348\) 0 0
\(349\) −17.3982 −0.931302 −0.465651 0.884968i \(-0.654180\pi\)
−0.465651 + 0.884968i \(0.654180\pi\)
\(350\) 0 0
\(351\) 1.00000 0.0533761
\(352\) 0 0
\(353\) 6.65525 0.354223 0.177112 0.984191i \(-0.443325\pi\)
0.177112 + 0.984191i \(0.443325\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) −15.0331 −0.795634
\(358\) 0 0
\(359\) 8.55523 0.451528 0.225764 0.974182i \(-0.427512\pi\)
0.225764 + 0.974182i \(0.427512\pi\)
\(360\) 0 0
\(361\) −18.9451 −0.997111
\(362\) 0 0
\(363\) −20.5269 −1.07738
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) −14.7153 −0.768135 −0.384067 0.923305i \(-0.625477\pi\)
−0.384067 + 0.923305i \(0.625477\pi\)
\(368\) 0 0
\(369\) −10.9955 −0.572402
\(370\) 0 0
\(371\) 31.0822 1.61371
\(372\) 0 0
\(373\) 8.87747 0.459658 0.229829 0.973231i \(-0.426183\pi\)
0.229829 + 0.973231i \(0.426183\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −4.20220 −0.216424
\(378\) 0 0
\(379\) −1.11875 −0.0574666 −0.0287333 0.999587i \(-0.509147\pi\)
−0.0287333 + 0.999587i \(0.509147\pi\)
\(380\) 0 0
\(381\) −4.17012 −0.213642
\(382\) 0 0
\(383\) −17.3247 −0.885252 −0.442626 0.896706i \(-0.645953\pi\)
−0.442626 + 0.896706i \(0.645953\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) −0.188415 −0.00957766
\(388\) 0 0
\(389\) 0.459286 0.0232867 0.0116434 0.999932i \(-0.496294\pi\)
0.0116434 + 0.999932i \(0.496294\pi\)
\(390\) 0 0
\(391\) 40.5362 2.05000
\(392\) 0 0
\(393\) −18.2618 −0.921188
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) 20.5144 1.02959 0.514795 0.857313i \(-0.327868\pi\)
0.514795 + 0.857313i \(0.327868\pi\)
\(398\) 0 0
\(399\) −0.609386 −0.0305074
\(400\) 0 0
\(401\) −7.31546 −0.365316 −0.182658 0.983176i \(-0.558470\pi\)
−0.182658 + 0.983176i \(0.558470\pi\)
\(402\) 0 0
\(403\) −4.41269 −0.219812
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) −16.6645 −0.826030
\(408\) 0 0
\(409\) 29.0889 1.43836 0.719178 0.694826i \(-0.244519\pi\)
0.719178 + 0.694826i \(0.244519\pi\)
\(410\) 0 0
\(411\) −18.0093 −0.888332
\(412\) 0 0
\(413\) 31.6297 1.55639
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) −3.55901 −0.174285
\(418\) 0 0
\(419\) −3.90504 −0.190774 −0.0953868 0.995440i \(-0.530409\pi\)
−0.0953868 + 0.995440i \(0.530409\pi\)
\(420\) 0 0
\(421\) 2.06416 0.100601 0.0503005 0.998734i \(-0.483982\pi\)
0.0503005 + 0.998734i \(0.483982\pi\)
\(422\) 0 0
\(423\) −7.84916 −0.381639
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 7.75571 0.375325
\(428\) 0 0
\(429\) −5.61488 −0.271089
\(430\) 0 0
\(431\) 33.7567 1.62600 0.813001 0.582262i \(-0.197832\pi\)
0.813001 + 0.582262i \(0.197832\pi\)
\(432\) 0 0
\(433\) −22.4137 −1.07713 −0.538566 0.842583i \(-0.681034\pi\)
−0.538566 + 0.842583i \(0.681034\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 1.64319 0.0786044
\(438\) 0 0
\(439\) 10.6662 0.509072 0.254536 0.967063i \(-0.418077\pi\)
0.254536 + 0.967063i \(0.418077\pi\)
\(440\) 0 0
\(441\) −0.234280 −0.0111562
\(442\) 0 0
\(443\) 14.5821 0.692815 0.346407 0.938084i \(-0.387401\pi\)
0.346407 + 0.938084i \(0.387401\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) −9.20220 −0.435249
\(448\) 0 0
\(449\) −22.3019 −1.05249 −0.526246 0.850332i \(-0.676401\pi\)
−0.526246 + 0.850332i \(0.676401\pi\)
\(450\) 0 0
\(451\) 61.7384 2.90715
\(452\) 0 0
\(453\) −3.86746 −0.181709
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 21.3706 0.999674 0.499837 0.866119i \(-0.333393\pi\)
0.499837 + 0.866119i \(0.333393\pi\)
\(458\) 0 0
\(459\) 5.77950 0.269764
\(460\) 0 0
\(461\) −18.4686 −0.860167 −0.430083 0.902789i \(-0.641516\pi\)
−0.430083 + 0.902789i \(0.641516\pi\)
\(462\) 0 0
\(463\) −1.52315 −0.0707870 −0.0353935 0.999373i \(-0.511268\pi\)
−0.0353935 + 0.999373i \(0.511268\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 15.0446 0.696180 0.348090 0.937461i \(-0.386830\pi\)
0.348090 + 0.937461i \(0.386830\pi\)
\(468\) 0 0
\(469\) 9.28337 0.428666
\(470\) 0 0
\(471\) −11.2343 −0.517648
\(472\) 0 0
\(473\) 1.05793 0.0486435
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) −11.9496 −0.547136
\(478\) 0 0
\(479\) 32.4695 1.48357 0.741786 0.670637i \(-0.233979\pi\)
0.741786 + 0.670637i \(0.233979\pi\)
\(480\) 0 0
\(481\) −2.96792 −0.135325
\(482\) 0 0
\(483\) −18.2436 −0.830110
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) −28.0469 −1.27092 −0.635462 0.772132i \(-0.719190\pi\)
−0.635462 + 0.772132i \(0.719190\pi\)
\(488\) 0 0
\(489\) −1.48234 −0.0670340
\(490\) 0 0
\(491\) 29.2097 1.31822 0.659109 0.752048i \(-0.270934\pi\)
0.659109 + 0.752048i \(0.270934\pi\)
\(492\) 0 0
\(493\) −24.2866 −1.09381
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −9.79501 −0.439366
\(498\) 0 0
\(499\) −34.7497 −1.55561 −0.777805 0.628506i \(-0.783667\pi\)
−0.777805 + 0.628506i \(0.783667\pi\)
\(500\) 0 0
\(501\) 9.37232 0.418724
\(502\) 0 0
\(503\) 22.5728 1.00647 0.503236 0.864149i \(-0.332143\pi\)
0.503236 + 0.864149i \(0.332143\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) −1.00000 −0.0444116
\(508\) 0 0
\(509\) 25.8502 1.14579 0.572894 0.819630i \(-0.305821\pi\)
0.572894 + 0.819630i \(0.305821\pi\)
\(510\) 0 0
\(511\) −37.9053 −1.67683
\(512\) 0 0
\(513\) 0.234280 0.0103437
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 44.0722 1.93829
\(518\) 0 0
\(519\) 21.7121 0.953056
\(520\) 0 0
\(521\) −25.9634 −1.13748 −0.568739 0.822518i \(-0.692568\pi\)
−0.568739 + 0.822518i \(0.692568\pi\)
\(522\) 0 0
\(523\) −38.1579 −1.66853 −0.834264 0.551366i \(-0.814107\pi\)
−0.834264 + 0.551366i \(0.814107\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −25.5031 −1.11093
\(528\) 0 0
\(529\) 26.1932 1.13883
\(530\) 0 0
\(531\) −12.1601 −0.527704
\(532\) 0 0
\(533\) 10.9955 0.476268
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 18.6204 0.803529
\(538\) 0 0
\(539\) 1.31546 0.0566607
\(540\) 0 0
\(541\) 27.0414 1.16260 0.581299 0.813690i \(-0.302545\pi\)
0.581299 + 0.813690i \(0.302545\pi\)
\(542\) 0 0
\(543\) −14.4365 −0.619529
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 14.5287 0.621200 0.310600 0.950541i \(-0.399470\pi\)
0.310600 + 0.950541i \(0.399470\pi\)
\(548\) 0 0
\(549\) −2.98170 −0.127256
\(550\) 0 0
\(551\) −0.984492 −0.0419408
\(552\) 0 0
\(553\) 12.9221 0.549502
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −14.1684 −0.600334 −0.300167 0.953887i \(-0.597042\pi\)
−0.300167 + 0.953887i \(0.597042\pi\)
\(558\) 0 0
\(559\) 0.188415 0.00796909
\(560\) 0 0
\(561\) −32.4513 −1.37009
\(562\) 0 0
\(563\) −28.1701 −1.18723 −0.593614 0.804750i \(-0.702300\pi\)
−0.593614 + 0.804750i \(0.702300\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) −2.60110 −0.109236
\(568\) 0 0
\(569\) −36.0140 −1.50979 −0.754893 0.655847i \(-0.772311\pi\)
−0.754893 + 0.655847i \(0.772311\pi\)
\(570\) 0 0
\(571\) 13.9233 0.582673 0.291337 0.956621i \(-0.405900\pi\)
0.291337 + 0.956621i \(0.405900\pi\)
\(572\) 0 0
\(573\) −20.4044 −0.852406
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) −4.32022 −0.179853 −0.0899265 0.995948i \(-0.528663\pi\)
−0.0899265 + 0.995948i \(0.528663\pi\)
\(578\) 0 0
\(579\) 26.1318 1.08600
\(580\) 0 0
\(581\) 8.84088 0.366781
\(582\) 0 0
\(583\) 67.0958 2.77882
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 42.4878 1.75366 0.876830 0.480800i \(-0.159654\pi\)
0.876830 + 0.480800i \(0.159654\pi\)
\(588\) 0 0
\(589\) −1.03380 −0.0425972
\(590\) 0 0
\(591\) 21.8026 0.896838
\(592\) 0 0
\(593\) −18.8712 −0.774949 −0.387474 0.921880i \(-0.626652\pi\)
−0.387474 + 0.921880i \(0.626652\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 5.22050 0.213661
\(598\) 0 0
\(599\) −43.4367 −1.77478 −0.887388 0.461023i \(-0.847483\pi\)
−0.887388 + 0.461023i \(0.847483\pi\)
\(600\) 0 0
\(601\) −27.6645 −1.12846 −0.564230 0.825618i \(-0.690827\pi\)
−0.564230 + 0.825618i \(0.690827\pi\)
\(602\) 0 0
\(603\) −3.56902 −0.145342
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) −10.9188 −0.443181 −0.221591 0.975140i \(-0.571125\pi\)
−0.221591 + 0.975140i \(0.571125\pi\)
\(608\) 0 0
\(609\) 10.9303 0.442920
\(610\) 0 0
\(611\) 7.84916 0.317543
\(612\) 0 0
\(613\) −0.459538 −0.0185606 −0.00928029 0.999957i \(-0.502954\pi\)
−0.00928029 + 0.999957i \(0.502954\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 13.2280 0.532541 0.266271 0.963898i \(-0.414209\pi\)
0.266271 + 0.963898i \(0.414209\pi\)
\(618\) 0 0
\(619\) 36.6480 1.47301 0.736503 0.676434i \(-0.236476\pi\)
0.736503 + 0.676434i \(0.236476\pi\)
\(620\) 0 0
\(621\) 7.01378 0.281453
\(622\) 0 0
\(623\) −44.8163 −1.79553
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) −1.31546 −0.0525342
\(628\) 0 0
\(629\) −17.1531 −0.683939
\(630\) 0 0
\(631\) 9.15310 0.364379 0.182190 0.983263i \(-0.441682\pi\)
0.182190 + 0.983263i \(0.441682\pi\)
\(632\) 0 0
\(633\) −11.0138 −0.437759
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 0.234280 0.00928251
\(638\) 0 0
\(639\) 3.76572 0.148970
\(640\) 0 0
\(641\) −32.5883 −1.28716 −0.643580 0.765379i \(-0.722552\pi\)
−0.643580 + 0.765379i \(0.722552\pi\)
\(642\) 0 0
\(643\) −8.28942 −0.326903 −0.163451 0.986551i \(-0.552263\pi\)
−0.163451 + 0.986551i \(0.552263\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 7.72590 0.303736 0.151868 0.988401i \(-0.451471\pi\)
0.151868 + 0.988401i \(0.451471\pi\)
\(648\) 0 0
\(649\) 68.2776 2.68013
\(650\) 0 0
\(651\) 11.4778 0.449852
\(652\) 0 0
\(653\) 15.1210 0.591731 0.295866 0.955230i \(-0.404392\pi\)
0.295866 + 0.955230i \(0.404392\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 14.5728 0.568539
\(658\) 0 0
\(659\) −7.65397 −0.298156 −0.149078 0.988825i \(-0.547631\pi\)
−0.149078 + 0.988825i \(0.547631\pi\)
\(660\) 0 0
\(661\) 33.3633 1.29768 0.648841 0.760924i \(-0.275254\pi\)
0.648841 + 0.760924i \(0.275254\pi\)
\(662\) 0 0
\(663\) −5.77950 −0.224457
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) −29.4733 −1.14121
\(668\) 0 0
\(669\) 27.4733 1.06218
\(670\) 0 0
\(671\) 16.7419 0.646315
\(672\) 0 0
\(673\) −23.9020 −0.921356 −0.460678 0.887567i \(-0.652394\pi\)
−0.460678 + 0.887567i \(0.652394\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 36.0276 1.38465 0.692326 0.721585i \(-0.256586\pi\)
0.692326 + 0.721585i \(0.256586\pi\)
\(678\) 0 0
\(679\) −13.5314 −0.519289
\(680\) 0 0
\(681\) 1.47556 0.0565438
\(682\) 0 0
\(683\) −43.2015 −1.65306 −0.826529 0.562894i \(-0.809688\pi\)
−0.826529 + 0.562894i \(0.809688\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) 9.65246 0.368264
\(688\) 0 0
\(689\) 11.9496 0.455244
\(690\) 0 0
\(691\) −11.7008 −0.445120 −0.222560 0.974919i \(-0.571441\pi\)
−0.222560 + 0.974919i \(0.571441\pi\)
\(692\) 0 0
\(693\) 14.6049 0.554793
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 63.5485 2.40707
\(698\) 0 0
\(699\) 19.2298 0.727337
\(700\) 0 0
\(701\) −52.7552 −1.99254 −0.996268 0.0863132i \(-0.972491\pi\)
−0.996268 + 0.0863132i \(0.972491\pi\)
\(702\) 0 0
\(703\) −0.695324 −0.0262247
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) −15.5232 −0.583808
\(708\) 0 0
\(709\) 30.1428 1.13204 0.566018 0.824393i \(-0.308483\pi\)
0.566018 + 0.824393i \(0.308483\pi\)
\(710\) 0 0
\(711\) −4.96792 −0.186312
\(712\) 0 0
\(713\) −30.9496 −1.15907
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 0.244290 0.00912317
\(718\) 0 0
\(719\) −2.06009 −0.0768285 −0.0384142 0.999262i \(-0.512231\pi\)
−0.0384142 + 0.999262i \(0.512231\pi\)
\(720\) 0 0
\(721\) 41.1992 1.53434
\(722\) 0 0
\(723\) −0.986215 −0.0366777
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) −5.25129 −0.194760 −0.0973799 0.995247i \(-0.531046\pi\)
−0.0973799 + 0.995247i \(0.531046\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) 1.08894 0.0402760
\(732\) 0 0
\(733\) 24.7795 0.915252 0.457626 0.889145i \(-0.348700\pi\)
0.457626 + 0.889145i \(0.348700\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 20.0396 0.738169
\(738\) 0 0
\(739\) −19.7635 −0.727011 −0.363506 0.931592i \(-0.618420\pi\)
−0.363506 + 0.931592i \(0.618420\pi\)
\(740\) 0 0
\(741\) −0.234280 −0.00860649
\(742\) 0 0
\(743\) −12.3944 −0.454706 −0.227353 0.973812i \(-0.573007\pi\)
−0.227353 + 0.973812i \(0.573007\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) −3.39890 −0.124359
\(748\) 0 0
\(749\) 10.5237 0.384528
\(750\) 0 0
\(751\) 23.3799 0.853144 0.426572 0.904454i \(-0.359721\pi\)
0.426572 + 0.904454i \(0.359721\pi\)
\(752\) 0 0
\(753\) 0.995489 0.0362776
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) −14.3398 −0.521189 −0.260594 0.965448i \(-0.583919\pi\)
−0.260594 + 0.965448i \(0.583919\pi\)
\(758\) 0 0
\(759\) −39.3816 −1.42946
\(760\) 0 0
\(761\) 11.0706 0.401311 0.200655 0.979662i \(-0.435693\pi\)
0.200655 + 0.979662i \(0.435693\pi\)
\(762\) 0 0
\(763\) 46.1152 1.66948
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 12.1601 0.439076
\(768\) 0 0
\(769\) −49.6251 −1.78953 −0.894764 0.446539i \(-0.852656\pi\)
−0.894764 + 0.446539i \(0.852656\pi\)
\(770\) 0 0
\(771\) 20.3523 0.732970
\(772\) 0 0
\(773\) −7.66174 −0.275574 −0.137787 0.990462i \(-0.543999\pi\)
−0.137787 + 0.990462i \(0.543999\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 7.71985 0.276948
\(778\) 0 0
\(779\) 2.57602 0.0922956
\(780\) 0 0
\(781\) −21.1441 −0.756595
\(782\) 0 0
\(783\) −4.20220 −0.150174
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) 29.8234 1.06309 0.531544 0.847031i \(-0.321612\pi\)
0.531544 + 0.847031i \(0.321612\pi\)
\(788\) 0 0
\(789\) 0.252576 0.00899194
\(790\) 0 0
\(791\) −1.90827 −0.0678502
\(792\) 0 0
\(793\) 2.98170 0.105883
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 52.9085 1.87412 0.937058 0.349174i \(-0.113538\pi\)
0.937058 + 0.349174i \(0.113538\pi\)
\(798\) 0 0
\(799\) 45.3643 1.60487
\(800\) 0 0
\(801\) 17.2298 0.608784
\(802\) 0 0
\(803\) −81.8246 −2.88753
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) −14.9358 −0.525766
\(808\) 0 0
\(809\) 16.6354 0.584871 0.292436 0.956285i \(-0.405534\pi\)
0.292436 + 0.956285i \(0.405534\pi\)
\(810\) 0 0
\(811\) −19.0972 −0.670594 −0.335297 0.942112i \(-0.608837\pi\)
−0.335297 + 0.942112i \(0.608837\pi\)
\(812\) 0 0
\(813\) 3.09723 0.108625
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 0.0441418 0.00154433
\(818\) 0 0
\(819\) 2.60110 0.0908898
\(820\) 0 0
\(821\) 16.6845 0.582295 0.291147 0.956678i \(-0.405963\pi\)
0.291147 + 0.956678i \(0.405963\pi\)
\(822\) 0 0
\(823\) −1.66904 −0.0581789 −0.0290895 0.999577i \(-0.509261\pi\)
−0.0290895 + 0.999577i \(0.509261\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −41.1007 −1.42921 −0.714606 0.699527i \(-0.753394\pi\)
−0.714606 + 0.699527i \(0.753394\pi\)
\(828\) 0 0
\(829\) −20.7001 −0.718943 −0.359471 0.933156i \(-0.617043\pi\)
−0.359471 + 0.933156i \(0.617043\pi\)
\(830\) 0 0
\(831\) −21.2939 −0.738678
\(832\) 0 0
\(833\) 1.35402 0.0469141
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) −4.41269 −0.152525
\(838\) 0 0
\(839\) 56.4720 1.94963 0.974816 0.223012i \(-0.0715889\pi\)
0.974816 + 0.223012i \(0.0715889\pi\)
\(840\) 0 0
\(841\) −11.3415 −0.391087
\(842\) 0 0
\(843\) −13.1839 −0.454078
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) −53.3926 −1.83459
\(848\) 0 0
\(849\) −7.18218 −0.246492
\(850\) 0 0
\(851\) −20.8163 −0.713575
\(852\) 0 0
\(853\) 45.1197 1.54487 0.772435 0.635093i \(-0.219038\pi\)
0.772435 + 0.635093i \(0.219038\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 10.8088 0.369222 0.184611 0.982812i \(-0.440898\pi\)
0.184611 + 0.982812i \(0.440898\pi\)
\(858\) 0 0
\(859\) 31.0248 1.05855 0.529276 0.848450i \(-0.322464\pi\)
0.529276 + 0.848450i \(0.322464\pi\)
\(860\) 0 0
\(861\) −28.6004 −0.974698
\(862\) 0 0
\(863\) 46.8264 1.59399 0.796994 0.603987i \(-0.206422\pi\)
0.796994 + 0.603987i \(0.206422\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) −16.4027 −0.557064
\(868\) 0 0
\(869\) 27.8943 0.946249
\(870\) 0 0
\(871\) 3.56902 0.120932
\(872\) 0 0
\(873\) 5.20220 0.176068
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 27.6049 0.932150 0.466075 0.884745i \(-0.345668\pi\)
0.466075 + 0.884745i \(0.345668\pi\)
\(878\) 0 0
\(879\) −18.9005 −0.637499
\(880\) 0 0
\(881\) −24.1348 −0.813122 −0.406561 0.913624i \(-0.633272\pi\)
−0.406561 + 0.913624i \(0.633272\pi\)
\(882\) 0 0
\(883\) 32.1579 1.08220 0.541099 0.840959i \(-0.318008\pi\)
0.541099 + 0.840959i \(0.318008\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) −22.3077 −0.749020 −0.374510 0.927223i \(-0.622189\pi\)
−0.374510 + 0.927223i \(0.622189\pi\)
\(888\) 0 0
\(889\) −10.8469 −0.363793
\(890\) 0 0
\(891\) −5.61488 −0.188106
\(892\) 0 0
\(893\) 1.83890 0.0615365
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) −7.01378 −0.234183
\(898\) 0 0
\(899\) 18.5430 0.618443
\(900\) 0 0
\(901\) 69.0629 2.30082
\(902\) 0 0
\(903\) −0.490085 −0.0163090
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) −35.9634 −1.19415 −0.597073 0.802187i \(-0.703670\pi\)
−0.597073 + 0.802187i \(0.703670\pi\)
\(908\) 0 0
\(909\) 5.96792 0.197943
\(910\) 0 0
\(911\) 24.2911 0.804801 0.402401 0.915464i \(-0.368176\pi\)
0.402401 + 0.915464i \(0.368176\pi\)
\(912\) 0 0
\(913\) 19.0844 0.631603
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) −47.5009 −1.56862
\(918\) 0 0
\(919\) −53.0997 −1.75160 −0.875799 0.482676i \(-0.839665\pi\)
−0.875799 + 0.482676i \(0.839665\pi\)
\(920\) 0 0
\(921\) −4.56352 −0.150373
\(922\) 0 0
\(923\) −3.76572 −0.123950
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) −15.8392 −0.520226
\(928\) 0 0
\(929\) −51.6986 −1.69618 −0.848088 0.529856i \(-0.822246\pi\)
−0.848088 + 0.529856i \(0.822246\pi\)
\(930\) 0 0
\(931\) 0.0548871 0.00179885
\(932\) 0 0
\(933\) 25.5500 0.836469
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 11.3310 0.370166 0.185083 0.982723i \(-0.440745\pi\)
0.185083 + 0.982723i \(0.440745\pi\)
\(938\) 0 0
\(939\) −30.4092 −0.992366
\(940\) 0 0
\(941\) −39.3816 −1.28380 −0.641902 0.766787i \(-0.721854\pi\)
−0.641902 + 0.766787i \(0.721854\pi\)
\(942\) 0 0
\(943\) 77.1200 2.51137
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −23.0481 −0.748964 −0.374482 0.927234i \(-0.622179\pi\)
−0.374482 + 0.927234i \(0.622179\pi\)
\(948\) 0 0
\(949\) −14.5728 −0.473053
\(950\) 0 0
\(951\) −32.0401 −1.03897
\(952\) 0 0
\(953\) −40.0231 −1.29647 −0.648237 0.761439i \(-0.724493\pi\)
−0.648237 + 0.761439i \(0.724493\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 23.5949 0.762714
\(958\) 0 0
\(959\) −46.8439 −1.51267
\(960\) 0 0
\(961\) −11.5282 −0.371878
\(962\) 0 0
\(963\) −4.04587 −0.130376
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) −37.8875 −1.21838 −0.609190 0.793024i \(-0.708505\pi\)
−0.609190 + 0.793024i \(0.708505\pi\)
\(968\) 0 0
\(969\) −1.35402 −0.0434975
\(970\) 0 0
\(971\) −23.5593 −0.756053 −0.378026 0.925795i \(-0.623397\pi\)
−0.378026 + 0.925795i \(0.623397\pi\)
\(972\) 0 0
\(973\) −9.25734 −0.296777
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −23.0564 −0.737640 −0.368820 0.929501i \(-0.620238\pi\)
−0.368820 + 0.929501i \(0.620238\pi\)
\(978\) 0 0
\(979\) −96.7432 −3.09192
\(980\) 0 0
\(981\) −17.7291 −0.566048
\(982\) 0 0
\(983\) 0.844654 0.0269403 0.0134701 0.999909i \(-0.495712\pi\)
0.0134701 + 0.999909i \(0.495712\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) −20.4165 −0.649863
\(988\) 0 0
\(989\) 1.32150 0.0420213
\(990\) 0 0
\(991\) −14.2709 −0.453329 −0.226665 0.973973i \(-0.572782\pi\)
−0.226665 + 0.973973i \(0.572782\pi\)
\(992\) 0 0
\(993\) −7.45478 −0.236570
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) 16.3874 0.518994 0.259497 0.965744i \(-0.416443\pi\)
0.259497 + 0.965744i \(0.416443\pi\)
\(998\) 0 0
\(999\) −2.96792 −0.0939008
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.bu.1.2 4
5.4 even 2 7800.2.a.bx.1.3 yes 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
7800.2.a.bu.1.2 4 1.1 even 1 trivial
7800.2.a.bx.1.3 yes 4 5.4 even 2