Properties

Label 7800.2.a.y.1.2
Level $7800$
Weight $2$
Character 7800.1
Self dual yes
Analytic conductor $62.283$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

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

Newform invariants

Self dual: yes
Analytic conductor: \(62.2833135766\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{12})^+\)
Defining polynomial: \( x^{2} - 3 \) Copy content Toggle raw display
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 \(1.73205\) of defining polynomial
Character \(\chi\) \(=\) 7800.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{3} +1.73205 q^{7} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{3} +1.73205 q^{7} +1.00000 q^{9} +1.73205 q^{11} +1.00000 q^{13} -2.46410 q^{17} -3.46410 q^{19} -1.73205 q^{21} +2.00000 q^{23} -1.00000 q^{27} +3.92820 q^{29} -9.19615 q^{31} -1.73205 q^{33} -0.535898 q^{37} -1.00000 q^{39} -2.53590 q^{41} +8.92820 q^{43} -6.66025 q^{47} -4.00000 q^{49} +2.46410 q^{51} -11.9282 q^{53} +3.46410 q^{57} +1.73205 q^{59} -12.4641 q^{61} +1.73205 q^{63} +4.26795 q^{67} -2.00000 q^{69} -14.3923 q^{71} -3.46410 q^{73} +3.00000 q^{77} +14.0000 q^{79} +1.00000 q^{81} +13.7321 q^{83} -3.92820 q^{87} -2.92820 q^{89} +1.73205 q^{91} +9.19615 q^{93} +4.92820 q^{97} +1.73205 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{3} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{3} + 2 q^{9} + 2 q^{13} + 2 q^{17} + 4 q^{23} - 2 q^{27} - 6 q^{29} - 8 q^{31} - 8 q^{37} - 2 q^{39} - 12 q^{41} + 4 q^{43} + 4 q^{47} - 8 q^{49} - 2 q^{51} - 10 q^{53} - 18 q^{61} + 12 q^{67} - 4 q^{69} - 8 q^{71} + 6 q^{77} + 28 q^{79} + 2 q^{81} + 24 q^{83} + 6 q^{87} + 8 q^{89} + 8 q^{93} - 4 q^{97}+O(q^{100}) \) Copy content Toggle raw display

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
\(7\) 1.73205 0.654654 0.327327 0.944911i \(-0.393852\pi\)
0.327327 + 0.944911i \(0.393852\pi\)
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 1.73205 0.522233 0.261116 0.965307i \(-0.415909\pi\)
0.261116 + 0.965307i \(0.415909\pi\)
\(12\) 0 0
\(13\) 1.00000 0.277350
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −2.46410 −0.597632 −0.298816 0.954311i \(-0.596592\pi\)
−0.298816 + 0.954311i \(0.596592\pi\)
\(18\) 0 0
\(19\) −3.46410 −0.794719 −0.397360 0.917663i \(-0.630073\pi\)
−0.397360 + 0.917663i \(0.630073\pi\)
\(20\) 0 0
\(21\) −1.73205 −0.377964
\(22\) 0 0
\(23\) 2.00000 0.417029 0.208514 0.978019i \(-0.433137\pi\)
0.208514 + 0.978019i \(0.433137\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) −1.00000 −0.192450
\(28\) 0 0
\(29\) 3.92820 0.729449 0.364725 0.931115i \(-0.381163\pi\)
0.364725 + 0.931115i \(0.381163\pi\)
\(30\) 0 0
\(31\) −9.19615 −1.65168 −0.825839 0.563906i \(-0.809298\pi\)
−0.825839 + 0.563906i \(0.809298\pi\)
\(32\) 0 0
\(33\) −1.73205 −0.301511
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −0.535898 −0.0881012 −0.0440506 0.999029i \(-0.514026\pi\)
−0.0440506 + 0.999029i \(0.514026\pi\)
\(38\) 0 0
\(39\) −1.00000 −0.160128
\(40\) 0 0
\(41\) −2.53590 −0.396041 −0.198020 0.980198i \(-0.563451\pi\)
−0.198020 + 0.980198i \(0.563451\pi\)
\(42\) 0 0
\(43\) 8.92820 1.36154 0.680769 0.732498i \(-0.261646\pi\)
0.680769 + 0.732498i \(0.261646\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −6.66025 −0.971498 −0.485749 0.874098i \(-0.661453\pi\)
−0.485749 + 0.874098i \(0.661453\pi\)
\(48\) 0 0
\(49\) −4.00000 −0.571429
\(50\) 0 0
\(51\) 2.46410 0.345043
\(52\) 0 0
\(53\) −11.9282 −1.63846 −0.819232 0.573462i \(-0.805600\pi\)
−0.819232 + 0.573462i \(0.805600\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 3.46410 0.458831
\(58\) 0 0
\(59\) 1.73205 0.225494 0.112747 0.993624i \(-0.464035\pi\)
0.112747 + 0.993624i \(0.464035\pi\)
\(60\) 0 0
\(61\) −12.4641 −1.59586 −0.797932 0.602747i \(-0.794073\pi\)
−0.797932 + 0.602747i \(0.794073\pi\)
\(62\) 0 0
\(63\) 1.73205 0.218218
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 4.26795 0.521413 0.260706 0.965418i \(-0.416045\pi\)
0.260706 + 0.965418i \(0.416045\pi\)
\(68\) 0 0
\(69\) −2.00000 −0.240772
\(70\) 0 0
\(71\) −14.3923 −1.70805 −0.854026 0.520230i \(-0.825846\pi\)
−0.854026 + 0.520230i \(0.825846\pi\)
\(72\) 0 0
\(73\) −3.46410 −0.405442 −0.202721 0.979236i \(-0.564979\pi\)
−0.202721 + 0.979236i \(0.564979\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 3.00000 0.341882
\(78\) 0 0
\(79\) 14.0000 1.57512 0.787562 0.616236i \(-0.211343\pi\)
0.787562 + 0.616236i \(0.211343\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) 13.7321 1.50729 0.753644 0.657283i \(-0.228294\pi\)
0.753644 + 0.657283i \(0.228294\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) −3.92820 −0.421148
\(88\) 0 0
\(89\) −2.92820 −0.310389 −0.155194 0.987884i \(-0.549600\pi\)
−0.155194 + 0.987884i \(0.549600\pi\)
\(90\) 0 0
\(91\) 1.73205 0.181568
\(92\) 0 0
\(93\) 9.19615 0.953597
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 4.92820 0.500383 0.250192 0.968196i \(-0.419506\pi\)
0.250192 + 0.968196i \(0.419506\pi\)
\(98\) 0 0
\(99\) 1.73205 0.174078
\(100\) 0 0
\(101\) 3.00000 0.298511 0.149256 0.988799i \(-0.452312\pi\)
0.149256 + 0.988799i \(0.452312\pi\)
\(102\) 0 0
\(103\) −15.3205 −1.50957 −0.754787 0.655970i \(-0.772260\pi\)
−0.754787 + 0.655970i \(0.772260\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −7.46410 −0.721582 −0.360791 0.932647i \(-0.617493\pi\)
−0.360791 + 0.932647i \(0.617493\pi\)
\(108\) 0 0
\(109\) −2.53590 −0.242895 −0.121448 0.992598i \(-0.538754\pi\)
−0.121448 + 0.992598i \(0.538754\pi\)
\(110\) 0 0
\(111\) 0.535898 0.0508652
\(112\) 0 0
\(113\) 10.0000 0.940721 0.470360 0.882474i \(-0.344124\pi\)
0.470360 + 0.882474i \(0.344124\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 1.00000 0.0924500
\(118\) 0 0
\(119\) −4.26795 −0.391242
\(120\) 0 0
\(121\) −8.00000 −0.727273
\(122\) 0 0
\(123\) 2.53590 0.228654
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) 0.928203 0.0823647 0.0411824 0.999152i \(-0.486888\pi\)
0.0411824 + 0.999152i \(0.486888\pi\)
\(128\) 0 0
\(129\) −8.92820 −0.786084
\(130\) 0 0
\(131\) −21.4641 −1.87533 −0.937664 0.347544i \(-0.887016\pi\)
−0.937664 + 0.347544i \(0.887016\pi\)
\(132\) 0 0
\(133\) −6.00000 −0.520266
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 2.92820 0.250173 0.125087 0.992146i \(-0.460079\pi\)
0.125087 + 0.992146i \(0.460079\pi\)
\(138\) 0 0
\(139\) −14.3923 −1.22074 −0.610370 0.792117i \(-0.708979\pi\)
−0.610370 + 0.792117i \(0.708979\pi\)
\(140\) 0 0
\(141\) 6.66025 0.560895
\(142\) 0 0
\(143\) 1.73205 0.144841
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 4.00000 0.329914
\(148\) 0 0
\(149\) 11.8564 0.971315 0.485657 0.874149i \(-0.338580\pi\)
0.485657 + 0.874149i \(0.338580\pi\)
\(150\) 0 0
\(151\) 8.12436 0.661151 0.330575 0.943780i \(-0.392757\pi\)
0.330575 + 0.943780i \(0.392757\pi\)
\(152\) 0 0
\(153\) −2.46410 −0.199211
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) −9.39230 −0.749588 −0.374794 0.927108i \(-0.622286\pi\)
−0.374794 + 0.927108i \(0.622286\pi\)
\(158\) 0 0
\(159\) 11.9282 0.945968
\(160\) 0 0
\(161\) 3.46410 0.273009
\(162\) 0 0
\(163\) −6.39230 −0.500684 −0.250342 0.968157i \(-0.580543\pi\)
−0.250342 + 0.968157i \(0.580543\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 17.3205 1.34030 0.670151 0.742225i \(-0.266230\pi\)
0.670151 + 0.742225i \(0.266230\pi\)
\(168\) 0 0
\(169\) 1.00000 0.0769231
\(170\) 0 0
\(171\) −3.46410 −0.264906
\(172\) 0 0
\(173\) 3.92820 0.298656 0.149328 0.988788i \(-0.452289\pi\)
0.149328 + 0.988788i \(0.452289\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) −1.73205 −0.130189
\(178\) 0 0
\(179\) −0.392305 −0.0293222 −0.0146611 0.999893i \(-0.504667\pi\)
−0.0146611 + 0.999893i \(0.504667\pi\)
\(180\) 0 0
\(181\) −3.53590 −0.262821 −0.131411 0.991328i \(-0.541951\pi\)
−0.131411 + 0.991328i \(0.541951\pi\)
\(182\) 0 0
\(183\) 12.4641 0.921373
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) −4.26795 −0.312103
\(188\) 0 0
\(189\) −1.73205 −0.125988
\(190\) 0 0
\(191\) 14.3923 1.04139 0.520695 0.853743i \(-0.325673\pi\)
0.520695 + 0.853743i \(0.325673\pi\)
\(192\) 0 0
\(193\) 1.46410 0.105388 0.0526942 0.998611i \(-0.483219\pi\)
0.0526942 + 0.998611i \(0.483219\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 4.39230 0.312939 0.156469 0.987683i \(-0.449989\pi\)
0.156469 + 0.987683i \(0.449989\pi\)
\(198\) 0 0
\(199\) 21.3205 1.51137 0.755685 0.654935i \(-0.227304\pi\)
0.755685 + 0.654935i \(0.227304\pi\)
\(200\) 0 0
\(201\) −4.26795 −0.301038
\(202\) 0 0
\(203\) 6.80385 0.477536
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 2.00000 0.139010
\(208\) 0 0
\(209\) −6.00000 −0.415029
\(210\) 0 0
\(211\) −26.0000 −1.78991 −0.894957 0.446153i \(-0.852794\pi\)
−0.894957 + 0.446153i \(0.852794\pi\)
\(212\) 0 0
\(213\) 14.3923 0.986144
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) −15.9282 −1.08128
\(218\) 0 0
\(219\) 3.46410 0.234082
\(220\) 0 0
\(221\) −2.46410 −0.165753
\(222\) 0 0
\(223\) 28.2487 1.89167 0.945837 0.324642i \(-0.105244\pi\)
0.945837 + 0.324642i \(0.105244\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) −2.26795 −0.150529 −0.0752645 0.997164i \(-0.523980\pi\)
−0.0752645 + 0.997164i \(0.523980\pi\)
\(228\) 0 0
\(229\) −18.7846 −1.24132 −0.620661 0.784079i \(-0.713136\pi\)
−0.620661 + 0.784079i \(0.713136\pi\)
\(230\) 0 0
\(231\) −3.00000 −0.197386
\(232\) 0 0
\(233\) 19.8564 1.30084 0.650418 0.759576i \(-0.274594\pi\)
0.650418 + 0.759576i \(0.274594\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) −14.0000 −0.909398
\(238\) 0 0
\(239\) 3.19615 0.206742 0.103371 0.994643i \(-0.467037\pi\)
0.103371 + 0.994643i \(0.467037\pi\)
\(240\) 0 0
\(241\) −20.3923 −1.31358 −0.656792 0.754072i \(-0.728087\pi\)
−0.656792 + 0.754072i \(0.728087\pi\)
\(242\) 0 0
\(243\) −1.00000 −0.0641500
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) −3.46410 −0.220416
\(248\) 0 0
\(249\) −13.7321 −0.870233
\(250\) 0 0
\(251\) 4.39230 0.277240 0.138620 0.990346i \(-0.455733\pi\)
0.138620 + 0.990346i \(0.455733\pi\)
\(252\) 0 0
\(253\) 3.46410 0.217786
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −19.2487 −1.20070 −0.600351 0.799737i \(-0.704972\pi\)
−0.600351 + 0.799737i \(0.704972\pi\)
\(258\) 0 0
\(259\) −0.928203 −0.0576757
\(260\) 0 0
\(261\) 3.92820 0.243150
\(262\) 0 0
\(263\) 23.3205 1.43800 0.719002 0.695008i \(-0.244599\pi\)
0.719002 + 0.695008i \(0.244599\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 2.92820 0.179203
\(268\) 0 0
\(269\) −19.0000 −1.15845 −0.579225 0.815168i \(-0.696645\pi\)
−0.579225 + 0.815168i \(0.696645\pi\)
\(270\) 0 0
\(271\) −21.7321 −1.32013 −0.660064 0.751209i \(-0.729471\pi\)
−0.660064 + 0.751209i \(0.729471\pi\)
\(272\) 0 0
\(273\) −1.73205 −0.104828
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) −20.9282 −1.25745 −0.628727 0.777626i \(-0.716424\pi\)
−0.628727 + 0.777626i \(0.716424\pi\)
\(278\) 0 0
\(279\) −9.19615 −0.550559
\(280\) 0 0
\(281\) 12.9282 0.771232 0.385616 0.922659i \(-0.373989\pi\)
0.385616 + 0.922659i \(0.373989\pi\)
\(282\) 0 0
\(283\) 9.32051 0.554047 0.277023 0.960863i \(-0.410652\pi\)
0.277023 + 0.960863i \(0.410652\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) −4.39230 −0.259270
\(288\) 0 0
\(289\) −10.9282 −0.642835
\(290\) 0 0
\(291\) −4.92820 −0.288896
\(292\) 0 0
\(293\) −15.8564 −0.926341 −0.463171 0.886269i \(-0.653288\pi\)
−0.463171 + 0.886269i \(0.653288\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) −1.73205 −0.100504
\(298\) 0 0
\(299\) 2.00000 0.115663
\(300\) 0 0
\(301\) 15.4641 0.891336
\(302\) 0 0
\(303\) −3.00000 −0.172345
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) 4.53590 0.258877 0.129439 0.991587i \(-0.458682\pi\)
0.129439 + 0.991587i \(0.458682\pi\)
\(308\) 0 0
\(309\) 15.3205 0.871553
\(310\) 0 0
\(311\) −7.46410 −0.423250 −0.211625 0.977351i \(-0.567876\pi\)
−0.211625 + 0.977351i \(0.567876\pi\)
\(312\) 0 0
\(313\) −1.92820 −0.108988 −0.0544942 0.998514i \(-0.517355\pi\)
−0.0544942 + 0.998514i \(0.517355\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −12.5359 −0.704086 −0.352043 0.935984i \(-0.614513\pi\)
−0.352043 + 0.935984i \(0.614513\pi\)
\(318\) 0 0
\(319\) 6.80385 0.380942
\(320\) 0 0
\(321\) 7.46410 0.416606
\(322\) 0 0
\(323\) 8.53590 0.474950
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) 2.53590 0.140236
\(328\) 0 0
\(329\) −11.5359 −0.635995
\(330\) 0 0
\(331\) −4.53590 −0.249316 −0.124658 0.992200i \(-0.539783\pi\)
−0.124658 + 0.992200i \(0.539783\pi\)
\(332\) 0 0
\(333\) −0.535898 −0.0293671
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) −3.14359 −0.171242 −0.0856212 0.996328i \(-0.527287\pi\)
−0.0856212 + 0.996328i \(0.527287\pi\)
\(338\) 0 0
\(339\) −10.0000 −0.543125
\(340\) 0 0
\(341\) −15.9282 −0.862561
\(342\) 0 0
\(343\) −19.0526 −1.02874
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 16.2487 0.872277 0.436138 0.899880i \(-0.356346\pi\)
0.436138 + 0.899880i \(0.356346\pi\)
\(348\) 0 0
\(349\) −14.3923 −0.770402 −0.385201 0.922833i \(-0.625868\pi\)
−0.385201 + 0.922833i \(0.625868\pi\)
\(350\) 0 0
\(351\) −1.00000 −0.0533761
\(352\) 0 0
\(353\) −7.32051 −0.389631 −0.194816 0.980840i \(-0.562411\pi\)
−0.194816 + 0.980840i \(0.562411\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 4.26795 0.225884
\(358\) 0 0
\(359\) 3.73205 0.196970 0.0984851 0.995139i \(-0.468600\pi\)
0.0984851 + 0.995139i \(0.468600\pi\)
\(360\) 0 0
\(361\) −7.00000 −0.368421
\(362\) 0 0
\(363\) 8.00000 0.419891
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 6.92820 0.361649 0.180825 0.983515i \(-0.442123\pi\)
0.180825 + 0.983515i \(0.442123\pi\)
\(368\) 0 0
\(369\) −2.53590 −0.132014
\(370\) 0 0
\(371\) −20.6603 −1.07263
\(372\) 0 0
\(373\) −21.3923 −1.10765 −0.553826 0.832633i \(-0.686833\pi\)
−0.553826 + 0.832633i \(0.686833\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 3.92820 0.202313
\(378\) 0 0
\(379\) 4.80385 0.246757 0.123379 0.992360i \(-0.460627\pi\)
0.123379 + 0.992360i \(0.460627\pi\)
\(380\) 0 0
\(381\) −0.928203 −0.0475533
\(382\) 0 0
\(383\) 8.53590 0.436164 0.218082 0.975930i \(-0.430020\pi\)
0.218082 + 0.975930i \(0.430020\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 8.92820 0.453846
\(388\) 0 0
\(389\) −7.85641 −0.398336 −0.199168 0.979965i \(-0.563824\pi\)
−0.199168 + 0.979965i \(0.563824\pi\)
\(390\) 0 0
\(391\) −4.92820 −0.249230
\(392\) 0 0
\(393\) 21.4641 1.08272
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) −22.0000 −1.10415 −0.552074 0.833795i \(-0.686163\pi\)
−0.552074 + 0.833795i \(0.686163\pi\)
\(398\) 0 0
\(399\) 6.00000 0.300376
\(400\) 0 0
\(401\) −38.1051 −1.90288 −0.951439 0.307836i \(-0.900395\pi\)
−0.951439 + 0.307836i \(0.900395\pi\)
\(402\) 0 0
\(403\) −9.19615 −0.458093
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) −0.928203 −0.0460093
\(408\) 0 0
\(409\) 11.3205 0.559763 0.279882 0.960035i \(-0.409705\pi\)
0.279882 + 0.960035i \(0.409705\pi\)
\(410\) 0 0
\(411\) −2.92820 −0.144438
\(412\) 0 0
\(413\) 3.00000 0.147620
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 14.3923 0.704794
\(418\) 0 0
\(419\) −23.3205 −1.13928 −0.569641 0.821894i \(-0.692918\pi\)
−0.569641 + 0.821894i \(0.692918\pi\)
\(420\) 0 0
\(421\) −5.07180 −0.247184 −0.123592 0.992333i \(-0.539441\pi\)
−0.123592 + 0.992333i \(0.539441\pi\)
\(422\) 0 0
\(423\) −6.66025 −0.323833
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) −21.5885 −1.04474
\(428\) 0 0
\(429\) −1.73205 −0.0836242
\(430\) 0 0
\(431\) 2.39230 0.115233 0.0576166 0.998339i \(-0.481650\pi\)
0.0576166 + 0.998339i \(0.481650\pi\)
\(432\) 0 0
\(433\) 0.928203 0.0446066 0.0223033 0.999751i \(-0.492900\pi\)
0.0223033 + 0.999751i \(0.492900\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −6.92820 −0.331421
\(438\) 0 0
\(439\) 10.7846 0.514721 0.257361 0.966315i \(-0.417147\pi\)
0.257361 + 0.966315i \(0.417147\pi\)
\(440\) 0 0
\(441\) −4.00000 −0.190476
\(442\) 0 0
\(443\) 8.92820 0.424192 0.212096 0.977249i \(-0.431971\pi\)
0.212096 + 0.977249i \(0.431971\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) −11.8564 −0.560789
\(448\) 0 0
\(449\) −34.9282 −1.64836 −0.824182 0.566325i \(-0.808365\pi\)
−0.824182 + 0.566325i \(0.808365\pi\)
\(450\) 0 0
\(451\) −4.39230 −0.206826
\(452\) 0 0
\(453\) −8.12436 −0.381716
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 11.3205 0.529551 0.264776 0.964310i \(-0.414702\pi\)
0.264776 + 0.964310i \(0.414702\pi\)
\(458\) 0 0
\(459\) 2.46410 0.115014
\(460\) 0 0
\(461\) −7.07180 −0.329366 −0.164683 0.986347i \(-0.552660\pi\)
−0.164683 + 0.986347i \(0.552660\pi\)
\(462\) 0 0
\(463\) −6.26795 −0.291296 −0.145648 0.989336i \(-0.546527\pi\)
−0.145648 + 0.989336i \(0.546527\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −25.8564 −1.19649 −0.598246 0.801313i \(-0.704135\pi\)
−0.598246 + 0.801313i \(0.704135\pi\)
\(468\) 0 0
\(469\) 7.39230 0.341345
\(470\) 0 0
\(471\) 9.39230 0.432775
\(472\) 0 0
\(473\) 15.4641 0.711040
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) −11.9282 −0.546155
\(478\) 0 0
\(479\) −25.0526 −1.14468 −0.572340 0.820016i \(-0.693964\pi\)
−0.572340 + 0.820016i \(0.693964\pi\)
\(480\) 0 0
\(481\) −0.535898 −0.0244349
\(482\) 0 0
\(483\) −3.46410 −0.157622
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) −6.80385 −0.308312 −0.154156 0.988047i \(-0.549266\pi\)
−0.154156 + 0.988047i \(0.549266\pi\)
\(488\) 0 0
\(489\) 6.39230 0.289070
\(490\) 0 0
\(491\) −2.67949 −0.120924 −0.0604619 0.998171i \(-0.519257\pi\)
−0.0604619 + 0.998171i \(0.519257\pi\)
\(492\) 0 0
\(493\) −9.67949 −0.435942
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −24.9282 −1.11818
\(498\) 0 0
\(499\) 9.58846 0.429238 0.214619 0.976698i \(-0.431149\pi\)
0.214619 + 0.976698i \(0.431149\pi\)
\(500\) 0 0
\(501\) −17.3205 −0.773823
\(502\) 0 0
\(503\) 14.5359 0.648124 0.324062 0.946036i \(-0.394951\pi\)
0.324062 + 0.946036i \(0.394951\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) −1.00000 −0.0444116
\(508\) 0 0
\(509\) −14.2487 −0.631563 −0.315782 0.948832i \(-0.602267\pi\)
−0.315782 + 0.948832i \(0.602267\pi\)
\(510\) 0 0
\(511\) −6.00000 −0.265424
\(512\) 0 0
\(513\) 3.46410 0.152944
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) −11.5359 −0.507348
\(518\) 0 0
\(519\) −3.92820 −0.172429
\(520\) 0 0
\(521\) 30.0000 1.31432 0.657162 0.753749i \(-0.271757\pi\)
0.657162 + 0.753749i \(0.271757\pi\)
\(522\) 0 0
\(523\) −21.3205 −0.932281 −0.466140 0.884711i \(-0.654356\pi\)
−0.466140 + 0.884711i \(0.654356\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 22.6603 0.987096
\(528\) 0 0
\(529\) −19.0000 −0.826087
\(530\) 0 0
\(531\) 1.73205 0.0751646
\(532\) 0 0
\(533\) −2.53590 −0.109842
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 0.392305 0.0169292
\(538\) 0 0
\(539\) −6.92820 −0.298419
\(540\) 0 0
\(541\) −18.3923 −0.790747 −0.395373 0.918520i \(-0.629385\pi\)
−0.395373 + 0.918520i \(0.629385\pi\)
\(542\) 0 0
\(543\) 3.53590 0.151740
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) −9.46410 −0.404656 −0.202328 0.979318i \(-0.564851\pi\)
−0.202328 + 0.979318i \(0.564851\pi\)
\(548\) 0 0
\(549\) −12.4641 −0.531955
\(550\) 0 0
\(551\) −13.6077 −0.579707
\(552\) 0 0
\(553\) 24.2487 1.03116
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −32.3923 −1.37251 −0.686253 0.727363i \(-0.740746\pi\)
−0.686253 + 0.727363i \(0.740746\pi\)
\(558\) 0 0
\(559\) 8.92820 0.377623
\(560\) 0 0
\(561\) 4.26795 0.180193
\(562\) 0 0
\(563\) 18.0000 0.758610 0.379305 0.925272i \(-0.376163\pi\)
0.379305 + 0.925272i \(0.376163\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 1.73205 0.0727393
\(568\) 0 0
\(569\) 19.2487 0.806948 0.403474 0.914991i \(-0.367803\pi\)
0.403474 + 0.914991i \(0.367803\pi\)
\(570\) 0 0
\(571\) −27.8564 −1.16575 −0.582877 0.812560i \(-0.698073\pi\)
−0.582877 + 0.812560i \(0.698073\pi\)
\(572\) 0 0
\(573\) −14.3923 −0.601247
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) −32.6410 −1.35886 −0.679432 0.733739i \(-0.737774\pi\)
−0.679432 + 0.733739i \(0.737774\pi\)
\(578\) 0 0
\(579\) −1.46410 −0.0608460
\(580\) 0 0
\(581\) 23.7846 0.986752
\(582\) 0 0
\(583\) −20.6603 −0.855660
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 23.5885 0.973600 0.486800 0.873514i \(-0.338164\pi\)
0.486800 + 0.873514i \(0.338164\pi\)
\(588\) 0 0
\(589\) 31.8564 1.31262
\(590\) 0 0
\(591\) −4.39230 −0.180675
\(592\) 0 0
\(593\) −12.9282 −0.530898 −0.265449 0.964125i \(-0.585520\pi\)
−0.265449 + 0.964125i \(0.585520\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) −21.3205 −0.872590
\(598\) 0 0
\(599\) 29.1769 1.19214 0.596068 0.802934i \(-0.296729\pi\)
0.596068 + 0.802934i \(0.296729\pi\)
\(600\) 0 0
\(601\) 22.7128 0.926475 0.463237 0.886234i \(-0.346688\pi\)
0.463237 + 0.886234i \(0.346688\pi\)
\(602\) 0 0
\(603\) 4.26795 0.173804
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) 10.6795 0.433467 0.216734 0.976231i \(-0.430460\pi\)
0.216734 + 0.976231i \(0.430460\pi\)
\(608\) 0 0
\(609\) −6.80385 −0.275706
\(610\) 0 0
\(611\) −6.66025 −0.269445
\(612\) 0 0
\(613\) 11.0718 0.447186 0.223593 0.974683i \(-0.428221\pi\)
0.223593 + 0.974683i \(0.428221\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −7.85641 −0.316287 −0.158144 0.987416i \(-0.550551\pi\)
−0.158144 + 0.987416i \(0.550551\pi\)
\(618\) 0 0
\(619\) 25.3205 1.01772 0.508859 0.860850i \(-0.330068\pi\)
0.508859 + 0.860850i \(0.330068\pi\)
\(620\) 0 0
\(621\) −2.00000 −0.0802572
\(622\) 0 0
\(623\) −5.07180 −0.203197
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) 6.00000 0.239617
\(628\) 0 0
\(629\) 1.32051 0.0526521
\(630\) 0 0
\(631\) 40.2487 1.60228 0.801138 0.598480i \(-0.204228\pi\)
0.801138 + 0.598480i \(0.204228\pi\)
\(632\) 0 0
\(633\) 26.0000 1.03341
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) −4.00000 −0.158486
\(638\) 0 0
\(639\) −14.3923 −0.569351
\(640\) 0 0
\(641\) 49.3923 1.95088 0.975439 0.220268i \(-0.0706932\pi\)
0.975439 + 0.220268i \(0.0706932\pi\)
\(642\) 0 0
\(643\) −21.3205 −0.840799 −0.420399 0.907339i \(-0.638110\pi\)
−0.420399 + 0.907339i \(0.638110\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −47.7128 −1.87578 −0.937892 0.346927i \(-0.887225\pi\)
−0.937892 + 0.346927i \(0.887225\pi\)
\(648\) 0 0
\(649\) 3.00000 0.117760
\(650\) 0 0
\(651\) 15.9282 0.624276
\(652\) 0 0
\(653\) −35.9282 −1.40598 −0.702990 0.711200i \(-0.748152\pi\)
−0.702990 + 0.711200i \(0.748152\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) −3.46410 −0.135147
\(658\) 0 0
\(659\) 29.7128 1.15745 0.578723 0.815524i \(-0.303551\pi\)
0.578723 + 0.815524i \(0.303551\pi\)
\(660\) 0 0
\(661\) −36.3923 −1.41550 −0.707748 0.706465i \(-0.750289\pi\)
−0.707748 + 0.706465i \(0.750289\pi\)
\(662\) 0 0
\(663\) 2.46410 0.0956978
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 7.85641 0.304201
\(668\) 0 0
\(669\) −28.2487 −1.09216
\(670\) 0 0
\(671\) −21.5885 −0.833413
\(672\) 0 0
\(673\) −38.8564 −1.49780 −0.748902 0.662681i \(-0.769419\pi\)
−0.748902 + 0.662681i \(0.769419\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 40.6410 1.56196 0.780981 0.624555i \(-0.214720\pi\)
0.780981 + 0.624555i \(0.214720\pi\)
\(678\) 0 0
\(679\) 8.53590 0.327578
\(680\) 0 0
\(681\) 2.26795 0.0869080
\(682\) 0 0
\(683\) 43.0526 1.64736 0.823680 0.567055i \(-0.191917\pi\)
0.823680 + 0.567055i \(0.191917\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) 18.7846 0.716678
\(688\) 0 0
\(689\) −11.9282 −0.454428
\(690\) 0 0
\(691\) −38.1244 −1.45032 −0.725159 0.688581i \(-0.758234\pi\)
−0.725159 + 0.688581i \(0.758234\pi\)
\(692\) 0 0
\(693\) 3.00000 0.113961
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 6.24871 0.236687
\(698\) 0 0
\(699\) −19.8564 −0.751038
\(700\) 0 0
\(701\) 1.14359 0.0431929 0.0215965 0.999767i \(-0.493125\pi\)
0.0215965 + 0.999767i \(0.493125\pi\)
\(702\) 0 0
\(703\) 1.85641 0.0700157
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 5.19615 0.195421
\(708\) 0 0
\(709\) −17.4641 −0.655878 −0.327939 0.944699i \(-0.606354\pi\)
−0.327939 + 0.944699i \(0.606354\pi\)
\(710\) 0 0
\(711\) 14.0000 0.525041
\(712\) 0 0
\(713\) −18.3923 −0.688797
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) −3.19615 −0.119362
\(718\) 0 0
\(719\) −13.4641 −0.502126 −0.251063 0.967971i \(-0.580780\pi\)
−0.251063 + 0.967971i \(0.580780\pi\)
\(720\) 0 0
\(721\) −26.5359 −0.988248
\(722\) 0 0
\(723\) 20.3923 0.758398
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) 22.2487 0.825159 0.412580 0.910922i \(-0.364628\pi\)
0.412580 + 0.910922i \(0.364628\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) −22.0000 −0.813699
\(732\) 0 0
\(733\) 16.7846 0.619954 0.309977 0.950744i \(-0.399679\pi\)
0.309977 + 0.950744i \(0.399679\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 7.39230 0.272299
\(738\) 0 0
\(739\) 36.2679 1.33414 0.667069 0.744996i \(-0.267549\pi\)
0.667069 + 0.744996i \(0.267549\pi\)
\(740\) 0 0
\(741\) 3.46410 0.127257
\(742\) 0 0
\(743\) 32.8038 1.20346 0.601728 0.798701i \(-0.294479\pi\)
0.601728 + 0.798701i \(0.294479\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 13.7321 0.502429
\(748\) 0 0
\(749\) −12.9282 −0.472386
\(750\) 0 0
\(751\) −15.8564 −0.578608 −0.289304 0.957237i \(-0.593424\pi\)
−0.289304 + 0.957237i \(0.593424\pi\)
\(752\) 0 0
\(753\) −4.39230 −0.160064
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) −34.4641 −1.25262 −0.626310 0.779574i \(-0.715435\pi\)
−0.626310 + 0.779574i \(0.715435\pi\)
\(758\) 0 0
\(759\) −3.46410 −0.125739
\(760\) 0 0
\(761\) 38.3923 1.39172 0.695860 0.718177i \(-0.255023\pi\)
0.695860 + 0.718177i \(0.255023\pi\)
\(762\) 0 0
\(763\) −4.39230 −0.159012
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 1.73205 0.0625407
\(768\) 0 0
\(769\) 20.7846 0.749512 0.374756 0.927123i \(-0.377726\pi\)
0.374756 + 0.927123i \(0.377726\pi\)
\(770\) 0 0
\(771\) 19.2487 0.693225
\(772\) 0 0
\(773\) 11.8564 0.426445 0.213223 0.977004i \(-0.431604\pi\)
0.213223 + 0.977004i \(0.431604\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 0.928203 0.0332991
\(778\) 0 0
\(779\) 8.78461 0.314741
\(780\) 0 0
\(781\) −24.9282 −0.892001
\(782\) 0 0
\(783\) −3.92820 −0.140383
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) 12.8038 0.456408 0.228204 0.973613i \(-0.426715\pi\)
0.228204 + 0.973613i \(0.426715\pi\)
\(788\) 0 0
\(789\) −23.3205 −0.830232
\(790\) 0 0
\(791\) 17.3205 0.615846
\(792\) 0 0
\(793\) −12.4641 −0.442613
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 22.8564 0.809615 0.404808 0.914402i \(-0.367338\pi\)
0.404808 + 0.914402i \(0.367338\pi\)
\(798\) 0 0
\(799\) 16.4115 0.580599
\(800\) 0 0
\(801\) −2.92820 −0.103463
\(802\) 0 0
\(803\) −6.00000 −0.211735
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 19.0000 0.668832
\(808\) 0 0
\(809\) 11.0718 0.389264 0.194632 0.980876i \(-0.437649\pi\)
0.194632 + 0.980876i \(0.437649\pi\)
\(810\) 0 0
\(811\) 1.58846 0.0557783 0.0278891 0.999611i \(-0.491121\pi\)
0.0278891 + 0.999611i \(0.491121\pi\)
\(812\) 0 0
\(813\) 21.7321 0.762176
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) −30.9282 −1.08204
\(818\) 0 0
\(819\) 1.73205 0.0605228
\(820\) 0 0
\(821\) 27.4641 0.958504 0.479252 0.877677i \(-0.340908\pi\)
0.479252 + 0.877677i \(0.340908\pi\)
\(822\) 0 0
\(823\) −10.1436 −0.353583 −0.176792 0.984248i \(-0.556572\pi\)
−0.176792 + 0.984248i \(0.556572\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 20.1244 0.699792 0.349896 0.936788i \(-0.386217\pi\)
0.349896 + 0.936788i \(0.386217\pi\)
\(828\) 0 0
\(829\) 14.6077 0.507346 0.253673 0.967290i \(-0.418361\pi\)
0.253673 + 0.967290i \(0.418361\pi\)
\(830\) 0 0
\(831\) 20.9282 0.725991
\(832\) 0 0
\(833\) 9.85641 0.341504
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 9.19615 0.317866
\(838\) 0 0
\(839\) −29.3205 −1.01226 −0.506128 0.862458i \(-0.668924\pi\)
−0.506128 + 0.862458i \(0.668924\pi\)
\(840\) 0 0
\(841\) −13.5692 −0.467904
\(842\) 0 0
\(843\) −12.9282 −0.445271
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) −13.8564 −0.476112
\(848\) 0 0
\(849\) −9.32051 −0.319879
\(850\) 0 0
\(851\) −1.07180 −0.0367407
\(852\) 0 0
\(853\) −48.7846 −1.67035 −0.835177 0.549982i \(-0.814635\pi\)
−0.835177 + 0.549982i \(0.814635\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 21.7128 0.741696 0.370848 0.928694i \(-0.379067\pi\)
0.370848 + 0.928694i \(0.379067\pi\)
\(858\) 0 0
\(859\) −9.71281 −0.331397 −0.165698 0.986176i \(-0.552988\pi\)
−0.165698 + 0.986176i \(0.552988\pi\)
\(860\) 0 0
\(861\) 4.39230 0.149689
\(862\) 0 0
\(863\) 10.9474 0.372655 0.186328 0.982488i \(-0.440341\pi\)
0.186328 + 0.982488i \(0.440341\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) 10.9282 0.371141
\(868\) 0 0
\(869\) 24.2487 0.822581
\(870\) 0 0
\(871\) 4.26795 0.144614
\(872\) 0 0
\(873\) 4.92820 0.166794
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) −46.9282 −1.58465 −0.792326 0.610097i \(-0.791130\pi\)
−0.792326 + 0.610097i \(0.791130\pi\)
\(878\) 0 0
\(879\) 15.8564 0.534823
\(880\) 0 0
\(881\) −0.464102 −0.0156360 −0.00781799 0.999969i \(-0.502489\pi\)
−0.00781799 + 0.999969i \(0.502489\pi\)
\(882\) 0 0
\(883\) −21.0718 −0.709122 −0.354561 0.935033i \(-0.615370\pi\)
−0.354561 + 0.935033i \(0.615370\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) −2.78461 −0.0934980 −0.0467490 0.998907i \(-0.514886\pi\)
−0.0467490 + 0.998907i \(0.514886\pi\)
\(888\) 0 0
\(889\) 1.60770 0.0539204
\(890\) 0 0
\(891\) 1.73205 0.0580259
\(892\) 0 0
\(893\) 23.0718 0.772068
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) −2.00000 −0.0667781
\(898\) 0 0
\(899\) −36.1244 −1.20481
\(900\) 0 0
\(901\) 29.3923 0.979200
\(902\) 0 0
\(903\) −15.4641 −0.514613
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(908\) 0 0
\(909\) 3.00000 0.0995037
\(910\) 0 0
\(911\) −0.928203 −0.0307527 −0.0153764 0.999882i \(-0.504895\pi\)
−0.0153764 + 0.999882i \(0.504895\pi\)
\(912\) 0 0
\(913\) 23.7846 0.787156
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) −37.1769 −1.22769
\(918\) 0 0
\(919\) −57.0333 −1.88136 −0.940678 0.339301i \(-0.889809\pi\)
−0.940678 + 0.339301i \(0.889809\pi\)
\(920\) 0 0
\(921\) −4.53590 −0.149463
\(922\) 0 0
\(923\) −14.3923 −0.473728
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) −15.3205 −0.503192
\(928\) 0 0
\(929\) −52.6410 −1.72710 −0.863548 0.504267i \(-0.831763\pi\)
−0.863548 + 0.504267i \(0.831763\pi\)
\(930\) 0 0
\(931\) 13.8564 0.454125
\(932\) 0 0
\(933\) 7.46410 0.244364
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) −45.6410 −1.49103 −0.745514 0.666491i \(-0.767796\pi\)
−0.745514 + 0.666491i \(0.767796\pi\)
\(938\) 0 0
\(939\) 1.92820 0.0629245
\(940\) 0 0
\(941\) 3.75129 0.122289 0.0611443 0.998129i \(-0.480525\pi\)
0.0611443 + 0.998129i \(0.480525\pi\)
\(942\) 0 0
\(943\) −5.07180 −0.165160
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 7.33975 0.238510 0.119255 0.992864i \(-0.461949\pi\)
0.119255 + 0.992864i \(0.461949\pi\)
\(948\) 0 0
\(949\) −3.46410 −0.112449
\(950\) 0 0
\(951\) 12.5359 0.406504
\(952\) 0 0
\(953\) 29.2487 0.947459 0.473729 0.880670i \(-0.342907\pi\)
0.473729 + 0.880670i \(0.342907\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) −6.80385 −0.219937
\(958\) 0 0
\(959\) 5.07180 0.163777
\(960\) 0 0
\(961\) 53.5692 1.72804
\(962\) 0 0
\(963\) −7.46410 −0.240527
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) −33.4449 −1.07551 −0.537757 0.843100i \(-0.680728\pi\)
−0.537757 + 0.843100i \(0.680728\pi\)
\(968\) 0 0
\(969\) −8.53590 −0.274213
\(970\) 0 0
\(971\) −27.1769 −0.872149 −0.436074 0.899911i \(-0.643632\pi\)
−0.436074 + 0.899911i \(0.643632\pi\)
\(972\) 0 0
\(973\) −24.9282 −0.799162
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −41.5692 −1.32992 −0.664959 0.746880i \(-0.731551\pi\)
−0.664959 + 0.746880i \(0.731551\pi\)
\(978\) 0 0
\(979\) −5.07180 −0.162095
\(980\) 0 0
\(981\) −2.53590 −0.0809650
\(982\) 0 0
\(983\) −15.7321 −0.501774 −0.250887 0.968016i \(-0.580722\pi\)
−0.250887 + 0.968016i \(0.580722\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 11.5359 0.367192
\(988\) 0 0
\(989\) 17.8564 0.567801
\(990\) 0 0
\(991\) −49.7128 −1.57918 −0.789590 0.613635i \(-0.789707\pi\)
−0.789590 + 0.613635i \(0.789707\pi\)
\(992\) 0 0
\(993\) 4.53590 0.143942
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) 46.0333 1.45789 0.728945 0.684572i \(-0.240011\pi\)
0.728945 + 0.684572i \(0.240011\pi\)
\(998\) 0 0
\(999\) 0.535898 0.0169551
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.y.1.2 2
5.4 even 2 7800.2.a.bc.1.1 yes 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
7800.2.a.y.1.2 2 1.1 even 1 trivial
7800.2.a.bc.1.1 yes 2 5.4 even 2