Properties

Label 6840.2.a.bk.1.2
Level $6840$
Weight $2$
Character 6840.1
Self dual yes
Analytic conductor $54.618$
Analytic rank $1$
Dimension $3$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 6840 = 2^{3} \cdot 3^{2} \cdot 5 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6840.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(54.6176749826\)
Analytic rank: \(1\)
Dimension: \(3\)
Coefficient field: 3.3.568.1
Defining polynomial: \(x^{3} - x^{2} - 6 x - 2\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 2280)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-0.363328\) of defining polynomial
Character \(\chi\) \(=\) 6840.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{5} -2.77801 q^{7} +O(q^{10})\) \(q+1.00000 q^{5} -2.77801 q^{7} +0.778008 q^{11} -5.50466 q^{13} +7.00933 q^{17} +1.00000 q^{19} +1.27334 q^{23} +1.00000 q^{25} -8.23132 q^{29} +5.00933 q^{31} -2.77801 q^{35} +8.51399 q^{37} -9.06068 q^{41} -10.5140 q^{43} +6.72666 q^{47} +0.717328 q^{49} -8.72666 q^{53} +0.778008 q^{55} +5.29200 q^{59} -7.73599 q^{61} -5.50466 q^{65} -1.45331 q^{67} +12.4626 q^{71} +14.5653 q^{73} -2.16131 q^{77} +2.28267 q^{79} -3.45331 q^{83} +7.00933 q^{85} +2.51399 q^{89} +15.2920 q^{91} +1.00000 q^{95} +8.51399 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 3 q^{5} - 2 q^{7} + O(q^{10}) \) \( 3 q + 3 q^{5} - 2 q^{7} - 4 q^{11} - 6 q^{13} + 3 q^{19} + 8 q^{23} + 3 q^{25} - 10 q^{29} - 6 q^{31} - 2 q^{35} - 6 q^{37} - 4 q^{41} + 16 q^{47} + 19 q^{49} - 22 q^{53} - 4 q^{55} - 22 q^{59} + 2 q^{61} - 6 q^{65} + 4 q^{67} + 8 q^{71} + 10 q^{73} - 36 q^{77} - 10 q^{79} - 2 q^{83} - 24 q^{89} + 8 q^{91} + 3 q^{95} - 6 q^{97} + O(q^{100}) \)

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
\(3\) 0 0
\(4\) 0 0
\(5\) 1.00000 0.447214
\(6\) 0 0
\(7\) −2.77801 −1.04999 −0.524994 0.851106i \(-0.675932\pi\)
−0.524994 + 0.851106i \(0.675932\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 0.778008 0.234578 0.117289 0.993098i \(-0.462580\pi\)
0.117289 + 0.993098i \(0.462580\pi\)
\(12\) 0 0
\(13\) −5.50466 −1.52672 −0.763360 0.645974i \(-0.776452\pi\)
−0.763360 + 0.645974i \(0.776452\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 7.00933 1.70001 0.850006 0.526773i \(-0.176598\pi\)
0.850006 + 0.526773i \(0.176598\pi\)
\(18\) 0 0
\(19\) 1.00000 0.229416
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 1.27334 0.265510 0.132755 0.991149i \(-0.457618\pi\)
0.132755 + 0.991149i \(0.457618\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −8.23132 −1.52852 −0.764259 0.644909i \(-0.776895\pi\)
−0.764259 + 0.644909i \(0.776895\pi\)
\(30\) 0 0
\(31\) 5.00933 0.899702 0.449851 0.893104i \(-0.351477\pi\)
0.449851 + 0.893104i \(0.351477\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −2.77801 −0.469569
\(36\) 0 0
\(37\) 8.51399 1.39969 0.699846 0.714294i \(-0.253252\pi\)
0.699846 + 0.714294i \(0.253252\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −9.06068 −1.41504 −0.707520 0.706693i \(-0.750186\pi\)
−0.707520 + 0.706693i \(0.750186\pi\)
\(42\) 0 0
\(43\) −10.5140 −1.60337 −0.801684 0.597747i \(-0.796063\pi\)
−0.801684 + 0.597747i \(0.796063\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 6.72666 0.981184 0.490592 0.871389i \(-0.336781\pi\)
0.490592 + 0.871389i \(0.336781\pi\)
\(48\) 0 0
\(49\) 0.717328 0.102475
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −8.72666 −1.19870 −0.599349 0.800488i \(-0.704574\pi\)
−0.599349 + 0.800488i \(0.704574\pi\)
\(54\) 0 0
\(55\) 0.778008 0.104907
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 5.29200 0.688960 0.344480 0.938794i \(-0.388055\pi\)
0.344480 + 0.938794i \(0.388055\pi\)
\(60\) 0 0
\(61\) −7.73599 −0.990491 −0.495246 0.868753i \(-0.664922\pi\)
−0.495246 + 0.868753i \(0.664922\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −5.50466 −0.682770
\(66\) 0 0
\(67\) −1.45331 −0.177550 −0.0887752 0.996052i \(-0.528295\pi\)
−0.0887752 + 0.996052i \(0.528295\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 12.4626 1.47904 0.739522 0.673133i \(-0.235052\pi\)
0.739522 + 0.673133i \(0.235052\pi\)
\(72\) 0 0
\(73\) 14.5653 1.70474 0.852372 0.522935i \(-0.175163\pi\)
0.852372 + 0.522935i \(0.175163\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −2.16131 −0.246304
\(78\) 0 0
\(79\) 2.28267 0.256821 0.128410 0.991721i \(-0.459013\pi\)
0.128410 + 0.991721i \(0.459013\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) −3.45331 −0.379050 −0.189525 0.981876i \(-0.560695\pi\)
−0.189525 + 0.981876i \(0.560695\pi\)
\(84\) 0 0
\(85\) 7.00933 0.760268
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 2.51399 0.266483 0.133241 0.991084i \(-0.457461\pi\)
0.133241 + 0.991084i \(0.457461\pi\)
\(90\) 0 0
\(91\) 15.2920 1.60304
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 1.00000 0.102598
\(96\) 0 0
\(97\) 8.51399 0.864465 0.432233 0.901762i \(-0.357726\pi\)
0.432233 + 0.901762i \(0.357726\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) −15.5560 −1.54788 −0.773941 0.633258i \(-0.781717\pi\)
−0.773941 + 0.633258i \(0.781717\pi\)
\(102\) 0 0
\(103\) −1.45331 −0.143199 −0.0715996 0.997433i \(-0.522810\pi\)
−0.0715996 + 0.997433i \(0.522810\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −14.0187 −1.35523 −0.677617 0.735415i \(-0.736987\pi\)
−0.677617 + 0.735415i \(0.736987\pi\)
\(108\) 0 0
\(109\) −7.29200 −0.698447 −0.349224 0.937039i \(-0.613555\pi\)
−0.349224 + 0.937039i \(0.613555\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) −19.7360 −1.85661 −0.928303 0.371825i \(-0.878732\pi\)
−0.928303 + 0.371825i \(0.878732\pi\)
\(114\) 0 0
\(115\) 1.27334 0.118740
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) −19.4720 −1.78499
\(120\) 0 0
\(121\) −10.3947 −0.944973
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 1.00000 0.0894427
\(126\) 0 0
\(127\) 1.45331 0.128961 0.0644803 0.997919i \(-0.479461\pi\)
0.0644803 + 0.997919i \(0.479461\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) −15.7873 −1.37935 −0.689673 0.724121i \(-0.742246\pi\)
−0.689673 + 0.724121i \(0.742246\pi\)
\(132\) 0 0
\(133\) −2.77801 −0.240884
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 0.282672 0.0241503 0.0120752 0.999927i \(-0.496156\pi\)
0.0120752 + 0.999927i \(0.496156\pi\)
\(138\) 0 0
\(139\) 16.4626 1.39634 0.698172 0.715931i \(-0.253997\pi\)
0.698172 + 0.715931i \(0.253997\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) −4.28267 −0.358135
\(144\) 0 0
\(145\) −8.23132 −0.683574
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) −14.0000 −1.14692 −0.573462 0.819232i \(-0.694400\pi\)
−0.573462 + 0.819232i \(0.694400\pi\)
\(150\) 0 0
\(151\) −21.5747 −1.75572 −0.877861 0.478916i \(-0.841030\pi\)
−0.877861 + 0.478916i \(0.841030\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 5.00933 0.402359
\(156\) 0 0
\(157\) −13.8387 −1.10445 −0.552224 0.833696i \(-0.686221\pi\)
−0.552224 + 0.833696i \(0.686221\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) −3.53736 −0.278783
\(162\) 0 0
\(163\) 18.6167 1.45817 0.729086 0.684422i \(-0.239945\pi\)
0.729086 + 0.684422i \(0.239945\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 9.73599 0.753393 0.376697 0.926337i \(-0.377060\pi\)
0.376697 + 0.926337i \(0.377060\pi\)
\(168\) 0 0
\(169\) 17.3013 1.33087
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) −4.82936 −0.367169 −0.183585 0.983004i \(-0.558770\pi\)
−0.183585 + 0.983004i \(0.558770\pi\)
\(174\) 0 0
\(175\) −2.77801 −0.209998
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) −23.8387 −1.78179 −0.890894 0.454212i \(-0.849921\pi\)
−0.890894 + 0.454212i \(0.849921\pi\)
\(180\) 0 0
\(181\) −14.7267 −1.09462 −0.547312 0.836929i \(-0.684349\pi\)
−0.547312 + 0.836929i \(0.684349\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 8.51399 0.625961
\(186\) 0 0
\(187\) 5.45331 0.398786
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 3.22199 0.233135 0.116568 0.993183i \(-0.462811\pi\)
0.116568 + 0.993183i \(0.462811\pi\)
\(192\) 0 0
\(193\) 5.50466 0.396234 0.198117 0.980178i \(-0.436517\pi\)
0.198117 + 0.980178i \(0.436517\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −6.17997 −0.440305 −0.220152 0.975466i \(-0.570655\pi\)
−0.220152 + 0.975466i \(0.570655\pi\)
\(198\) 0 0
\(199\) −17.0280 −1.20708 −0.603541 0.797332i \(-0.706244\pi\)
−0.603541 + 0.797332i \(0.706244\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 22.8667 1.60493
\(204\) 0 0
\(205\) −9.06068 −0.632825
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 0.778008 0.0538159
\(210\) 0 0
\(211\) 8.56534 0.589663 0.294831 0.955549i \(-0.404737\pi\)
0.294831 + 0.955549i \(0.404737\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) −10.5140 −0.717048
\(216\) 0 0
\(217\) −13.9160 −0.944677
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) −38.5840 −2.59544
\(222\) 0 0
\(223\) −1.98134 −0.132681 −0.0663403 0.997797i \(-0.521132\pi\)
−0.0663403 + 0.997797i \(0.521132\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) −0.282672 −0.0187616 −0.00938081 0.999956i \(-0.502986\pi\)
−0.00938081 + 0.999956i \(0.502986\pi\)
\(228\) 0 0
\(229\) 4.26401 0.281774 0.140887 0.990026i \(-0.455005\pi\)
0.140887 + 0.990026i \(0.455005\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 6.62395 0.433950 0.216975 0.976177i \(-0.430381\pi\)
0.216975 + 0.976177i \(0.430381\pi\)
\(234\) 0 0
\(235\) 6.72666 0.438799
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) −4.21266 −0.272495 −0.136247 0.990675i \(-0.543504\pi\)
−0.136247 + 0.990675i \(0.543504\pi\)
\(240\) 0 0
\(241\) −25.1120 −1.61761 −0.808804 0.588078i \(-0.799885\pi\)
−0.808804 + 0.588078i \(0.799885\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 0.717328 0.0458284
\(246\) 0 0
\(247\) −5.50466 −0.350253
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 22.3340 1.40971 0.704856 0.709351i \(-0.251012\pi\)
0.704856 + 0.709351i \(0.251012\pi\)
\(252\) 0 0
\(253\) 0.990671 0.0622830
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 0.829359 0.0517340 0.0258670 0.999665i \(-0.491765\pi\)
0.0258670 + 0.999665i \(0.491765\pi\)
\(258\) 0 0
\(259\) −23.6519 −1.46966
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) −5.73599 −0.353696 −0.176848 0.984238i \(-0.556590\pi\)
−0.176848 + 0.984238i \(0.556590\pi\)
\(264\) 0 0
\(265\) −8.72666 −0.536074
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 15.1379 0.922977 0.461488 0.887146i \(-0.347316\pi\)
0.461488 + 0.887146i \(0.347316\pi\)
\(270\) 0 0
\(271\) −24.6680 −1.49848 −0.749239 0.662300i \(-0.769580\pi\)
−0.749239 + 0.662300i \(0.769580\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 0.778008 0.0469156
\(276\) 0 0
\(277\) −4.28267 −0.257321 −0.128660 0.991689i \(-0.541068\pi\)
−0.128660 + 0.991689i \(0.541068\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) −30.5140 −1.82031 −0.910156 0.414265i \(-0.864039\pi\)
−0.910156 + 0.414265i \(0.864039\pi\)
\(282\) 0 0
\(283\) 17.5233 1.04165 0.520827 0.853662i \(-0.325624\pi\)
0.520827 + 0.853662i \(0.325624\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 25.1706 1.48578
\(288\) 0 0
\(289\) 32.1307 1.89004
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) −5.71733 −0.334010 −0.167005 0.985956i \(-0.553410\pi\)
−0.167005 + 0.985956i \(0.553410\pi\)
\(294\) 0 0
\(295\) 5.29200 0.308112
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) −7.00933 −0.405360
\(300\) 0 0
\(301\) 29.2080 1.68352
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) −7.73599 −0.442961
\(306\) 0 0
\(307\) −8.46264 −0.482988 −0.241494 0.970402i \(-0.577637\pi\)
−0.241494 + 0.970402i \(0.577637\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 10.2313 0.580165 0.290082 0.957002i \(-0.406317\pi\)
0.290082 + 0.957002i \(0.406317\pi\)
\(312\) 0 0
\(313\) 30.1400 1.70361 0.851807 0.523855i \(-0.175507\pi\)
0.851807 + 0.523855i \(0.175507\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −7.83869 −0.440265 −0.220132 0.975470i \(-0.570649\pi\)
−0.220132 + 0.975470i \(0.570649\pi\)
\(318\) 0 0
\(319\) −6.40403 −0.358557
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 7.00933 0.390009
\(324\) 0 0
\(325\) −5.50466 −0.305344
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) −18.6867 −1.03023
\(330\) 0 0
\(331\) −7.29200 −0.400805 −0.200402 0.979714i \(-0.564225\pi\)
−0.200402 + 0.979714i \(0.564225\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) −1.45331 −0.0794030
\(336\) 0 0
\(337\) −10.4953 −0.571717 −0.285859 0.958272i \(-0.592279\pi\)
−0.285859 + 0.958272i \(0.592279\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 3.89730 0.211050
\(342\) 0 0
\(343\) 17.4533 0.942390
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 5.43466 0.291748 0.145874 0.989303i \(-0.453401\pi\)
0.145874 + 0.989303i \(0.453401\pi\)
\(348\) 0 0
\(349\) −1.43466 −0.0767954 −0.0383977 0.999263i \(-0.512225\pi\)
−0.0383977 + 0.999263i \(0.512225\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 5.73599 0.305296 0.152648 0.988281i \(-0.451220\pi\)
0.152648 + 0.988281i \(0.451220\pi\)
\(354\) 0 0
\(355\) 12.4626 0.661448
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −28.7780 −1.51885 −0.759423 0.650598i \(-0.774518\pi\)
−0.759423 + 0.650598i \(0.774518\pi\)
\(360\) 0 0
\(361\) 1.00000 0.0526316
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 14.5653 0.762385
\(366\) 0 0
\(367\) −31.9087 −1.66562 −0.832810 0.553559i \(-0.813269\pi\)
−0.832810 + 0.553559i \(0.813269\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 24.2427 1.25862
\(372\) 0 0
\(373\) −23.5233 −1.21799 −0.608996 0.793174i \(-0.708427\pi\)
−0.608996 + 0.793174i \(0.708427\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 45.3107 2.33362
\(378\) 0 0
\(379\) 27.2920 1.40190 0.700948 0.713212i \(-0.252761\pi\)
0.700948 + 0.713212i \(0.252761\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) −5.65872 −0.289147 −0.144574 0.989494i \(-0.546181\pi\)
−0.144574 + 0.989494i \(0.546181\pi\)
\(384\) 0 0
\(385\) −2.16131 −0.110151
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) −30.9253 −1.56797 −0.783987 0.620777i \(-0.786817\pi\)
−0.783987 + 0.620777i \(0.786817\pi\)
\(390\) 0 0
\(391\) 8.92528 0.451371
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 2.28267 0.114854
\(396\) 0 0
\(397\) −11.1893 −0.561575 −0.280787 0.959770i \(-0.590596\pi\)
−0.280787 + 0.959770i \(0.590596\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 7.60737 0.379894 0.189947 0.981794i \(-0.439168\pi\)
0.189947 + 0.981794i \(0.439168\pi\)
\(402\) 0 0
\(403\) −27.5747 −1.37359
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 6.62395 0.328337
\(408\) 0 0
\(409\) −18.6680 −0.923076 −0.461538 0.887121i \(-0.652702\pi\)
−0.461538 + 0.887121i \(0.652702\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) −14.7012 −0.723400
\(414\) 0 0
\(415\) −3.45331 −0.169516
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) −30.8994 −1.50953 −0.754766 0.655994i \(-0.772250\pi\)
−0.754766 + 0.655994i \(0.772250\pi\)
\(420\) 0 0
\(421\) −21.7360 −1.05935 −0.529674 0.848202i \(-0.677686\pi\)
−0.529674 + 0.848202i \(0.677686\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 7.00933 0.340002
\(426\) 0 0
\(427\) 21.4906 1.04000
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) −24.1400 −1.16278 −0.581392 0.813624i \(-0.697492\pi\)
−0.581392 + 0.813624i \(0.697492\pi\)
\(432\) 0 0
\(433\) −6.53265 −0.313939 −0.156970 0.987603i \(-0.550172\pi\)
−0.156970 + 0.987603i \(0.550172\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 1.27334 0.0609123
\(438\) 0 0
\(439\) 27.2080 1.29856 0.649282 0.760547i \(-0.275069\pi\)
0.649282 + 0.760547i \(0.275069\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 36.1587 1.71795 0.858975 0.512017i \(-0.171102\pi\)
0.858975 + 0.512017i \(0.171102\pi\)
\(444\) 0 0
\(445\) 2.51399 0.119175
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 38.6540 1.82420 0.912098 0.409973i \(-0.134462\pi\)
0.912098 + 0.409973i \(0.134462\pi\)
\(450\) 0 0
\(451\) −7.04928 −0.331938
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 15.2920 0.716900
\(456\) 0 0
\(457\) 40.1587 1.87854 0.939272 0.343174i \(-0.111502\pi\)
0.939272 + 0.343174i \(0.111502\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 22.9253 1.06774 0.533868 0.845568i \(-0.320738\pi\)
0.533868 + 0.845568i \(0.320738\pi\)
\(462\) 0 0
\(463\) −5.11929 −0.237914 −0.118957 0.992899i \(-0.537955\pi\)
−0.118957 + 0.992899i \(0.537955\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −35.0280 −1.62090 −0.810451 0.585807i \(-0.800778\pi\)
−0.810451 + 0.585807i \(0.800778\pi\)
\(468\) 0 0
\(469\) 4.03731 0.186426
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) −8.17997 −0.376115
\(474\) 0 0
\(475\) 1.00000 0.0458831
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 40.8153 1.86490 0.932450 0.361299i \(-0.117667\pi\)
0.932450 + 0.361299i \(0.117667\pi\)
\(480\) 0 0
\(481\) −46.8667 −2.13694
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 8.51399 0.386601
\(486\) 0 0
\(487\) −21.9160 −0.993107 −0.496553 0.868006i \(-0.665401\pi\)
−0.496553 + 0.868006i \(0.665401\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 15.9927 0.721742 0.360871 0.932616i \(-0.382479\pi\)
0.360871 + 0.932616i \(0.382479\pi\)
\(492\) 0 0
\(493\) −57.6960 −2.59850
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −34.6213 −1.55298
\(498\) 0 0
\(499\) −0.668047 −0.0299059 −0.0149530 0.999888i \(-0.504760\pi\)
−0.0149530 + 0.999888i \(0.504760\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) −12.4881 −0.556816 −0.278408 0.960463i \(-0.589807\pi\)
−0.278408 + 0.960463i \(0.589807\pi\)
\(504\) 0 0
\(505\) −15.5560 −0.692234
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 1.32469 0.0587161 0.0293580 0.999569i \(-0.490654\pi\)
0.0293580 + 0.999569i \(0.490654\pi\)
\(510\) 0 0
\(511\) −40.4626 −1.78996
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) −1.45331 −0.0640406
\(516\) 0 0
\(517\) 5.23339 0.230164
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) −32.3272 −1.41628 −0.708141 0.706071i \(-0.750466\pi\)
−0.708141 + 0.706071i \(0.750466\pi\)
\(522\) 0 0
\(523\) 33.2334 1.45319 0.726597 0.687063i \(-0.241101\pi\)
0.726597 + 0.687063i \(0.241101\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 35.1120 1.52950
\(528\) 0 0
\(529\) −21.3786 −0.929504
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 49.8760 2.16037
\(534\) 0 0
\(535\) −14.0187 −0.606079
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 0.558087 0.0240385
\(540\) 0 0
\(541\) 11.0934 0.476941 0.238471 0.971150i \(-0.423354\pi\)
0.238471 + 0.971150i \(0.423354\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) −7.29200 −0.312355
\(546\) 0 0
\(547\) −5.55602 −0.237558 −0.118779 0.992921i \(-0.537898\pi\)
−0.118779 + 0.992921i \(0.537898\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) −8.23132 −0.350666
\(552\) 0 0
\(553\) −6.34128 −0.269659
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −30.2427 −1.28143 −0.640713 0.767781i \(-0.721361\pi\)
−0.640713 + 0.767781i \(0.721361\pi\)
\(558\) 0 0
\(559\) 57.8760 2.44789
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) −32.8853 −1.38595 −0.692976 0.720961i \(-0.743701\pi\)
−0.692976 + 0.720961i \(0.743701\pi\)
\(564\) 0 0
\(565\) −19.7360 −0.830299
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 9.48601 0.397674 0.198837 0.980033i \(-0.436284\pi\)
0.198837 + 0.980033i \(0.436284\pi\)
\(570\) 0 0
\(571\) −45.0653 −1.88592 −0.942962 0.332900i \(-0.891973\pi\)
−0.942962 + 0.332900i \(0.891973\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 1.27334 0.0531021
\(576\) 0 0
\(577\) −34.6680 −1.44325 −0.721625 0.692284i \(-0.756604\pi\)
−0.721625 + 0.692284i \(0.756604\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 9.59333 0.397998
\(582\) 0 0
\(583\) −6.78941 −0.281189
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 35.3879 1.46062 0.730308 0.683118i \(-0.239377\pi\)
0.730308 + 0.683118i \(0.239377\pi\)
\(588\) 0 0
\(589\) 5.00933 0.206406
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) −25.2080 −1.03517 −0.517583 0.855633i \(-0.673168\pi\)
−0.517583 + 0.855633i \(0.673168\pi\)
\(594\) 0 0
\(595\) −19.4720 −0.798273
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) −32.6027 −1.33211 −0.666054 0.745903i \(-0.732018\pi\)
−0.666054 + 0.745903i \(0.732018\pi\)
\(600\) 0 0
\(601\) 5.89730 0.240556 0.120278 0.992740i \(-0.461621\pi\)
0.120278 + 0.992740i \(0.461621\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) −10.3947 −0.422605
\(606\) 0 0
\(607\) 8.46264 0.343488 0.171744 0.985142i \(-0.445060\pi\)
0.171744 + 0.985142i \(0.445060\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) −37.0280 −1.49799
\(612\) 0 0
\(613\) 46.4413 1.87575 0.937874 0.346976i \(-0.112791\pi\)
0.937874 + 0.346976i \(0.112791\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −4.42533 −0.178157 −0.0890785 0.996025i \(-0.528392\pi\)
−0.0890785 + 0.996025i \(0.528392\pi\)
\(618\) 0 0
\(619\) 28.9907 1.16523 0.582617 0.812747i \(-0.302029\pi\)
0.582617 + 0.812747i \(0.302029\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) −6.98389 −0.279804
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 59.6774 2.37949
\(630\) 0 0
\(631\) −1.98134 −0.0788760 −0.0394380 0.999222i \(-0.512557\pi\)
−0.0394380 + 0.999222i \(0.512557\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 1.45331 0.0576730
\(636\) 0 0
\(637\) −3.94865 −0.156451
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 4.17271 0.164812 0.0824061 0.996599i \(-0.473740\pi\)
0.0824061 + 0.996599i \(0.473740\pi\)
\(642\) 0 0
\(643\) −14.6167 −0.576426 −0.288213 0.957566i \(-0.593061\pi\)
−0.288213 + 0.957566i \(0.593061\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −1.17064 −0.0460226 −0.0230113 0.999735i \(-0.507325\pi\)
−0.0230113 + 0.999735i \(0.507325\pi\)
\(648\) 0 0
\(649\) 4.11722 0.161615
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 17.6519 0.690774 0.345387 0.938460i \(-0.387748\pi\)
0.345387 + 0.938460i \(0.387748\pi\)
\(654\) 0 0
\(655\) −15.7873 −0.616862
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) −28.5067 −1.11046 −0.555232 0.831695i \(-0.687371\pi\)
−0.555232 + 0.831695i \(0.687371\pi\)
\(660\) 0 0
\(661\) 29.5161 1.14804 0.574021 0.818841i \(-0.305383\pi\)
0.574021 + 0.818841i \(0.305383\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) −2.77801 −0.107727
\(666\) 0 0
\(667\) −10.4813 −0.405838
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) −6.01866 −0.232348
\(672\) 0 0
\(673\) 1.50466 0.0580005 0.0290003 0.999579i \(-0.490768\pi\)
0.0290003 + 0.999579i \(0.490768\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 9.85735 0.378849 0.189424 0.981895i \(-0.439338\pi\)
0.189424 + 0.981895i \(0.439338\pi\)
\(678\) 0 0
\(679\) −23.6519 −0.907678
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) −1.65872 −0.0634691 −0.0317346 0.999496i \(-0.510103\pi\)
−0.0317346 + 0.999496i \(0.510103\pi\)
\(684\) 0 0
\(685\) 0.282672 0.0108004
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 48.0373 1.83008
\(690\) 0 0
\(691\) 19.2147 0.730963 0.365481 0.930819i \(-0.380904\pi\)
0.365481 + 0.930819i \(0.380904\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 16.4626 0.624464
\(696\) 0 0
\(697\) −63.5093 −2.40559
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) −34.5653 −1.30552 −0.652758 0.757567i \(-0.726388\pi\)
−0.652758 + 0.757567i \(0.726388\pi\)
\(702\) 0 0
\(703\) 8.51399 0.321111
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 43.2147 1.62526
\(708\) 0 0
\(709\) 2.84802 0.106960 0.0534798 0.998569i \(-0.482969\pi\)
0.0534798 + 0.998569i \(0.482969\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 6.37860 0.238880
\(714\) 0 0
\(715\) −4.28267 −0.160163
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 6.12862 0.228559 0.114279 0.993449i \(-0.463544\pi\)
0.114279 + 0.993449i \(0.463544\pi\)
\(720\) 0 0
\(721\) 4.03731 0.150357
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) −8.23132 −0.305704
\(726\) 0 0
\(727\) −10.6753 −0.395925 −0.197963 0.980210i \(-0.563432\pi\)
−0.197963 + 0.980210i \(0.563432\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) −73.6960 −2.72575
\(732\) 0 0
\(733\) 40.6426 1.50117 0.750585 0.660774i \(-0.229772\pi\)
0.750585 + 0.660774i \(0.229772\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −1.13069 −0.0416495
\(738\) 0 0
\(739\) 26.2241 0.964668 0.482334 0.875987i \(-0.339789\pi\)
0.482334 + 0.875987i \(0.339789\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 8.56534 0.314232 0.157116 0.987580i \(-0.449780\pi\)
0.157116 + 0.987580i \(0.449780\pi\)
\(744\) 0 0
\(745\) −14.0000 −0.512920
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 38.9439 1.42298
\(750\) 0 0
\(751\) −41.3693 −1.50959 −0.754793 0.655963i \(-0.772263\pi\)
−0.754793 + 0.655963i \(0.772263\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) −21.5747 −0.785183
\(756\) 0 0
\(757\) 34.1986 1.24297 0.621485 0.783426i \(-0.286530\pi\)
0.621485 + 0.783426i \(0.286530\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −6.00000 −0.217500 −0.108750 0.994069i \(-0.534685\pi\)
−0.108750 + 0.994069i \(0.534685\pi\)
\(762\) 0 0
\(763\) 20.2572 0.733361
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) −29.1307 −1.05185
\(768\) 0 0
\(769\) −24.5840 −0.886522 −0.443261 0.896393i \(-0.646178\pi\)
−0.443261 + 0.896393i \(0.646178\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) −42.2173 −1.51845 −0.759225 0.650828i \(-0.774422\pi\)
−0.759225 + 0.650828i \(0.774422\pi\)
\(774\) 0 0
\(775\) 5.00933 0.179940
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) −9.06068 −0.324633
\(780\) 0 0
\(781\) 9.69603 0.346951
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) −13.8387 −0.493924
\(786\) 0 0
\(787\) 24.8226 0.884829 0.442415 0.896811i \(-0.354122\pi\)
0.442415 + 0.896811i \(0.354122\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 54.8267 1.94941
\(792\) 0 0
\(793\) 42.5840 1.51220
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −11.1706 −0.395684 −0.197842 0.980234i \(-0.563393\pi\)
−0.197842 + 0.980234i \(0.563393\pi\)
\(798\) 0 0
\(799\) 47.1493 1.66802
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 11.3320 0.399896
\(804\) 0 0
\(805\) −3.53736 −0.124675
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) −24.9066 −0.875670 −0.437835 0.899055i \(-0.644255\pi\)
−0.437835 + 0.899055i \(0.644255\pi\)
\(810\) 0 0
\(811\) 43.6774 1.53372 0.766860 0.641814i \(-0.221818\pi\)
0.766860 + 0.641814i \(0.221818\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 18.6167 0.652114
\(816\) 0 0
\(817\) −10.5140 −0.367838
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 16.8039 0.586461 0.293230 0.956042i \(-0.405270\pi\)
0.293230 + 0.956042i \(0.405270\pi\)
\(822\) 0 0
\(823\) 8.33402 0.290506 0.145253 0.989395i \(-0.453600\pi\)
0.145253 + 0.989395i \(0.453600\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 11.7173 0.407451 0.203726 0.979028i \(-0.434695\pi\)
0.203726 + 0.979028i \(0.434695\pi\)
\(828\) 0 0
\(829\) 16.2173 0.563250 0.281625 0.959525i \(-0.409127\pi\)
0.281625 + 0.959525i \(0.409127\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 5.02799 0.174209
\(834\) 0 0
\(835\) 9.73599 0.336928
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 35.6120 1.22946 0.614731 0.788737i \(-0.289264\pi\)
0.614731 + 0.788737i \(0.289264\pi\)
\(840\) 0 0
\(841\) 38.7546 1.33637
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 17.3013 0.595184
\(846\) 0 0
\(847\) 28.8766 0.992211
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 10.8412 0.371633
\(852\) 0 0
\(853\) −24.3854 −0.834939 −0.417470 0.908691i \(-0.637083\pi\)
−0.417470 + 0.908691i \(0.637083\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 39.2733 1.34155 0.670776 0.741660i \(-0.265961\pi\)
0.670776 + 0.741660i \(0.265961\pi\)
\(858\) 0 0
\(859\) 15.4720 0.527897 0.263948 0.964537i \(-0.414975\pi\)
0.263948 + 0.964537i \(0.414975\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) −5.65872 −0.192625 −0.0963125 0.995351i \(-0.530705\pi\)
−0.0963125 + 0.995351i \(0.530705\pi\)
\(864\) 0 0
\(865\) −4.82936 −0.164203
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 1.77594 0.0602445
\(870\) 0 0
\(871\) 8.00000 0.271070
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) −2.77801 −0.0939138
\(876\) 0 0
\(877\) −3.06068 −0.103352 −0.0516759 0.998664i \(-0.516456\pi\)
−0.0516759 + 0.998664i \(0.516456\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) −12.6867 −0.427426 −0.213713 0.976896i \(-0.568556\pi\)
−0.213713 + 0.976896i \(0.568556\pi\)
\(882\) 0 0
\(883\) −27.0793 −0.911292 −0.455646 0.890161i \(-0.650592\pi\)
−0.455646 + 0.890161i \(0.650592\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 12.8880 0.432736 0.216368 0.976312i \(-0.430579\pi\)
0.216368 + 0.976312i \(0.430579\pi\)
\(888\) 0 0
\(889\) −4.03731 −0.135407
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 6.72666 0.225099
\(894\) 0 0
\(895\) −23.8387 −0.796839
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) −41.2334 −1.37521
\(900\) 0 0
\(901\) −61.1680 −2.03780
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) −14.7267 −0.489531
\(906\) 0 0
\(907\) −13.5560 −0.450120 −0.225060 0.974345i \(-0.572258\pi\)
−0.225060 + 0.974345i \(0.572258\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 17.0934 0.566329 0.283164 0.959071i \(-0.408616\pi\)
0.283164 + 0.959071i \(0.408616\pi\)
\(912\) 0 0
\(913\) −2.68670 −0.0889169
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 43.8573 1.44830
\(918\) 0 0
\(919\) 24.7708 0.817112 0.408556 0.912733i \(-0.366033\pi\)
0.408556 + 0.912733i \(0.366033\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) −68.6027 −2.25808
\(924\) 0 0
\(925\) 8.51399 0.279938
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) −1.57467 −0.0516634 −0.0258317 0.999666i \(-0.508223\pi\)
−0.0258317 + 0.999666i \(0.508223\pi\)
\(930\) 0 0
\(931\) 0.717328 0.0235095
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 5.45331 0.178342
\(936\) 0 0
\(937\) −27.5933 −0.901435 −0.450717 0.892667i \(-0.648832\pi\)
−0.450717 + 0.892667i \(0.648832\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) −52.3713 −1.70726 −0.853628 0.520882i \(-0.825603\pi\)
−0.853628 + 0.520882i \(0.825603\pi\)
\(942\) 0 0
\(943\) −11.5374 −0.375708
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 46.2427 1.50269 0.751343 0.659912i \(-0.229406\pi\)
0.751343 + 0.659912i \(0.229406\pi\)
\(948\) 0 0
\(949\) −80.1773 −2.60267
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) −37.1893 −1.20468 −0.602340 0.798240i \(-0.705765\pi\)
−0.602340 + 0.798240i \(0.705765\pi\)
\(954\) 0 0
\(955\) 3.22199 0.104261
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) −0.785266 −0.0253576
\(960\) 0 0
\(961\) −5.90663 −0.190536
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 5.50466 0.177201
\(966\) 0 0
\(967\) 42.2500 1.35867 0.679334 0.733829i \(-0.262269\pi\)
0.679334 + 0.733829i \(0.262269\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 7.73599 0.248260 0.124130 0.992266i \(-0.460386\pi\)
0.124130 + 0.992266i \(0.460386\pi\)
\(972\) 0 0
\(973\) −45.7333 −1.46614
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 38.9880 1.24734 0.623669 0.781689i \(-0.285641\pi\)
0.623669 + 0.781689i \(0.285641\pi\)
\(978\) 0 0
\(979\) 1.95591 0.0625110
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) 7.39470 0.235854 0.117927 0.993022i \(-0.462375\pi\)
0.117927 + 0.993022i \(0.462375\pi\)
\(984\) 0 0
\(985\) −6.17997 −0.196910
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) −13.3879 −0.425711
\(990\) 0 0
\(991\) 44.6613 1.41871 0.709356 0.704850i \(-0.248986\pi\)
0.709356 + 0.704850i \(0.248986\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) −17.0280 −0.539823
\(996\) 0 0
\(997\) −8.74531 −0.276967 −0.138483 0.990365i \(-0.544223\pi\)
−0.138483 + 0.990365i \(0.544223\pi\)
\(998\) 0 0
\(999\) 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 6840.2.a.bk.1.2 3
3.2 odd 2 2280.2.a.r.1.2 3
12.11 even 2 4560.2.a.bt.1.2 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2280.2.a.r.1.2 3 3.2 odd 2
4560.2.a.bt.1.2 3 12.11 even 2
6840.2.a.bk.1.2 3 1.1 even 1 trivial