Properties

Label 6840.2.a.bh.1.2
Level $6840$
Weight $2$
Character 6840.1
Self dual yes
Analytic conductor $54.618$
Analytic rank $0$
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: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.1772.1
Defining polynomial: \(x^{3} - x^{2} - 12 x + 8\)
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 \(3.67370\) of defining polynomial
Character \(\chi\) \(=\) 6840.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{5} +0.911179 q^{7} +O(q^{10})\) \(q-1.00000 q^{5} +0.911179 q^{7} -2.91118 q^{11} -6.25857 q^{13} -4.00000 q^{17} +1.00000 q^{19} -5.34740 q^{23} +1.00000 q^{25} +0.911179 q^{29} -2.00000 q^{31} -0.911179 q^{35} +4.43622 q^{37} -6.43622 q^{41} +8.25857 q^{43} +5.34740 q^{47} -6.16975 q^{49} -3.34740 q^{53} +2.91118 q^{55} -5.52504 q^{59} +13.1698 q^{61} +6.25857 q^{65} +4.00000 q^{67} +16.5171 q^{71} +6.00000 q^{73} -2.65260 q^{77} -10.9921 q^{79} +3.82236 q^{83} +4.00000 q^{85} +2.43622 q^{89} -5.70268 q^{91} -1.00000 q^{95} +16.4362 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3q - 3q^{5} + O(q^{10}) \) \( 3q - 3q^{5} - 6q^{11} + 4q^{13} - 12q^{17} + 3q^{19} + 4q^{23} + 3q^{25} - 6q^{31} - 4q^{37} - 2q^{41} + 2q^{43} - 4q^{47} + 7q^{49} + 10q^{53} + 6q^{55} - 2q^{59} + 14q^{61} - 4q^{65} + 12q^{67} + 4q^{71} + 18q^{73} - 28q^{77} - 2q^{79} + 6q^{83} + 12q^{85} - 10q^{89} - 8q^{91} - 3q^{95} + 32q^{97} + O(q^{100}) \)

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
\(7\) 0.911179 0.344393 0.172197 0.985063i \(-0.444914\pi\)
0.172197 + 0.985063i \(0.444914\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) −2.91118 −0.877753 −0.438877 0.898547i \(-0.644624\pi\)
−0.438877 + 0.898547i \(0.644624\pi\)
\(12\) 0 0
\(13\) −6.25857 −1.73582 −0.867908 0.496725i \(-0.834536\pi\)
−0.867908 + 0.496725i \(0.834536\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −4.00000 −0.970143 −0.485071 0.874475i \(-0.661206\pi\)
−0.485071 + 0.874475i \(0.661206\pi\)
\(18\) 0 0
\(19\) 1.00000 0.229416
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −5.34740 −1.11501 −0.557505 0.830174i \(-0.688241\pi\)
−0.557505 + 0.830174i \(0.688241\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 0.911179 0.169202 0.0846008 0.996415i \(-0.473038\pi\)
0.0846008 + 0.996415i \(0.473038\pi\)
\(30\) 0 0
\(31\) −2.00000 −0.359211 −0.179605 0.983739i \(-0.557482\pi\)
−0.179605 + 0.983739i \(0.557482\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −0.911179 −0.154017
\(36\) 0 0
\(37\) 4.43622 0.729310 0.364655 0.931143i \(-0.381187\pi\)
0.364655 + 0.931143i \(0.381187\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −6.43622 −1.00517 −0.502584 0.864528i \(-0.667617\pi\)
−0.502584 + 0.864528i \(0.667617\pi\)
\(42\) 0 0
\(43\) 8.25857 1.25942 0.629710 0.776830i \(-0.283174\pi\)
0.629710 + 0.776830i \(0.283174\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 5.34740 0.779998 0.389999 0.920815i \(-0.372475\pi\)
0.389999 + 0.920815i \(0.372475\pi\)
\(48\) 0 0
\(49\) −6.16975 −0.881393
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −3.34740 −0.459800 −0.229900 0.973214i \(-0.573840\pi\)
−0.229900 + 0.973214i \(0.573840\pi\)
\(54\) 0 0
\(55\) 2.91118 0.392543
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −5.52504 −0.719299 −0.359649 0.933088i \(-0.617104\pi\)
−0.359649 + 0.933088i \(0.617104\pi\)
\(60\) 0 0
\(61\) 13.1698 1.68621 0.843107 0.537746i \(-0.180724\pi\)
0.843107 + 0.537746i \(0.180724\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 6.25857 0.776281
\(66\) 0 0
\(67\) 4.00000 0.488678 0.244339 0.969690i \(-0.421429\pi\)
0.244339 + 0.969690i \(0.421429\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 16.5171 1.96022 0.980112 0.198443i \(-0.0635884\pi\)
0.980112 + 0.198443i \(0.0635884\pi\)
\(72\) 0 0
\(73\) 6.00000 0.702247 0.351123 0.936329i \(-0.385800\pi\)
0.351123 + 0.936329i \(0.385800\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −2.65260 −0.302292
\(78\) 0 0
\(79\) −10.9921 −1.23671 −0.618355 0.785899i \(-0.712200\pi\)
−0.618355 + 0.785899i \(0.712200\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 3.82236 0.419558 0.209779 0.977749i \(-0.432726\pi\)
0.209779 + 0.977749i \(0.432726\pi\)
\(84\) 0 0
\(85\) 4.00000 0.433861
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 2.43622 0.258238 0.129119 0.991629i \(-0.458785\pi\)
0.129119 + 0.991629i \(0.458785\pi\)
\(90\) 0 0
\(91\) −5.70268 −0.597803
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −1.00000 −0.102598
\(96\) 0 0
\(97\) 16.4362 1.66884 0.834422 0.551125i \(-0.185801\pi\)
0.834422 + 0.551125i \(0.185801\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 0.177642 0.0176761 0.00883804 0.999961i \(-0.497187\pi\)
0.00883804 + 0.999961i \(0.497187\pi\)
\(102\) 0 0
\(103\) −3.64472 −0.359124 −0.179562 0.983747i \(-0.557468\pi\)
−0.179562 + 0.983747i \(0.557468\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 3.64472 0.352348 0.176174 0.984359i \(-0.443628\pi\)
0.176174 + 0.984359i \(0.443628\pi\)
\(108\) 0 0
\(109\) −3.52504 −0.337637 −0.168819 0.985647i \(-0.553995\pi\)
−0.168819 + 0.985647i \(0.553995\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 13.5250 1.27233 0.636164 0.771554i \(-0.280520\pi\)
0.636164 + 0.771554i \(0.280520\pi\)
\(114\) 0 0
\(115\) 5.34740 0.498647
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) −3.64472 −0.334110
\(120\) 0 0
\(121\) −2.52504 −0.229549
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) −1.00000 −0.0894427
\(126\) 0 0
\(127\) 22.3395 1.98231 0.991155 0.132707i \(-0.0423669\pi\)
0.991155 + 0.132707i \(0.0423669\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 8.73354 0.763053 0.381526 0.924358i \(-0.375399\pi\)
0.381526 + 0.924358i \(0.375399\pi\)
\(132\) 0 0
\(133\) 0.911179 0.0790092
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −13.3474 −1.14034 −0.570172 0.821525i \(-0.693124\pi\)
−0.570172 + 0.821525i \(0.693124\pi\)
\(138\) 0 0
\(139\) 1.82236 0.154570 0.0772852 0.997009i \(-0.475375\pi\)
0.0772852 + 0.997009i \(0.475375\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 18.2198 1.52362
\(144\) 0 0
\(145\) −0.911179 −0.0756693
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) −4.69479 −0.384612 −0.192306 0.981335i \(-0.561597\pi\)
−0.192306 + 0.981335i \(0.561597\pi\)
\(150\) 0 0
\(151\) 4.69479 0.382057 0.191028 0.981585i \(-0.438818\pi\)
0.191028 + 0.981585i \(0.438818\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 2.00000 0.160644
\(156\) 0 0
\(157\) −3.16975 −0.252974 −0.126487 0.991968i \(-0.540370\pi\)
−0.126487 + 0.991968i \(0.540370\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) −4.87243 −0.384002
\(162\) 0 0
\(163\) −6.43622 −0.504123 −0.252062 0.967711i \(-0.581109\pi\)
−0.252062 + 0.967711i \(0.581109\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −3.16975 −0.245283 −0.122641 0.992451i \(-0.539137\pi\)
−0.122641 + 0.992451i \(0.539137\pi\)
\(168\) 0 0
\(169\) 26.1698 2.01306
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 9.52504 0.724175 0.362088 0.932144i \(-0.382064\pi\)
0.362088 + 0.932144i \(0.382064\pi\)
\(174\) 0 0
\(175\) 0.911179 0.0688786
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) −5.16975 −0.386405 −0.193203 0.981159i \(-0.561888\pi\)
−0.193203 + 0.981159i \(0.561888\pi\)
\(180\) 0 0
\(181\) 18.8145 1.39847 0.699234 0.714893i \(-0.253524\pi\)
0.699234 + 0.714893i \(0.253524\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) −4.43622 −0.326157
\(186\) 0 0
\(187\) 11.6447 0.851546
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −7.78361 −0.563202 −0.281601 0.959532i \(-0.590866\pi\)
−0.281601 + 0.959532i \(0.590866\pi\)
\(192\) 0 0
\(193\) −15.7256 −1.13196 −0.565978 0.824420i \(-0.691501\pi\)
−0.565978 + 0.824420i \(0.691501\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 7.70268 0.548793 0.274397 0.961617i \(-0.411522\pi\)
0.274397 + 0.961617i \(0.411522\pi\)
\(198\) 0 0
\(199\) −20.5171 −1.45442 −0.727211 0.686414i \(-0.759184\pi\)
−0.727211 + 0.686414i \(0.759184\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 0.830247 0.0582719
\(204\) 0 0
\(205\) 6.43622 0.449525
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) −2.91118 −0.201370
\(210\) 0 0
\(211\) −22.3395 −1.53792 −0.768958 0.639300i \(-0.779224\pi\)
−0.768958 + 0.639300i \(0.779224\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) −8.25857 −0.563230
\(216\) 0 0
\(217\) −1.82236 −0.123710
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 25.0343 1.68399
\(222\) 0 0
\(223\) −18.3395 −1.22810 −0.614052 0.789266i \(-0.710462\pi\)
−0.614052 + 0.789266i \(0.710462\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 17.8645 1.18571 0.592856 0.805309i \(-0.298000\pi\)
0.592856 + 0.805309i \(0.298000\pi\)
\(228\) 0 0
\(229\) 25.1698 1.66326 0.831632 0.555327i \(-0.187407\pi\)
0.831632 + 0.555327i \(0.187407\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −15.6869 −1.02768 −0.513842 0.857885i \(-0.671778\pi\)
−0.513842 + 0.857885i \(0.671778\pi\)
\(234\) 0 0
\(235\) −5.34740 −0.348826
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) −3.42833 −0.221760 −0.110880 0.993834i \(-0.535367\pi\)
−0.110880 + 0.993834i \(0.535367\pi\)
\(240\) 0 0
\(241\) −23.0343 −1.48377 −0.741885 0.670527i \(-0.766068\pi\)
−0.741885 + 0.670527i \(0.766068\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 6.16975 0.394171
\(246\) 0 0
\(247\) −6.25857 −0.398224
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) −19.0730 −1.20388 −0.601940 0.798541i \(-0.705605\pi\)
−0.601940 + 0.798541i \(0.705605\pi\)
\(252\) 0 0
\(253\) 15.5672 0.978703
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 15.8645 0.989603 0.494802 0.869006i \(-0.335241\pi\)
0.494802 + 0.869006i \(0.335241\pi\)
\(258\) 0 0
\(259\) 4.04219 0.251169
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 3.16975 0.195455 0.0977277 0.995213i \(-0.468843\pi\)
0.0977277 + 0.995213i \(0.468843\pi\)
\(264\) 0 0
\(265\) 3.34740 0.205629
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 13.7836 0.840402 0.420201 0.907431i \(-0.361960\pi\)
0.420201 + 0.907431i \(0.361960\pi\)
\(270\) 0 0
\(271\) −17.8224 −1.08263 −0.541316 0.840820i \(-0.682074\pi\)
−0.541316 + 0.840820i \(0.682074\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −2.91118 −0.175551
\(276\) 0 0
\(277\) 21.3474 1.28264 0.641320 0.767273i \(-0.278387\pi\)
0.641320 + 0.767273i \(0.278387\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) −14.4362 −0.861192 −0.430596 0.902545i \(-0.641697\pi\)
−0.430596 + 0.902545i \(0.641697\pi\)
\(282\) 0 0
\(283\) −6.08093 −0.361474 −0.180737 0.983531i \(-0.557848\pi\)
−0.180737 + 0.983531i \(0.557848\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) −5.86454 −0.346173
\(288\) 0 0
\(289\) −1.00000 −0.0588235
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 25.1698 1.47043 0.735216 0.677833i \(-0.237081\pi\)
0.735216 + 0.677833i \(0.237081\pi\)
\(294\) 0 0
\(295\) 5.52504 0.321680
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 33.4671 1.93545
\(300\) 0 0
\(301\) 7.52504 0.433736
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) −13.1698 −0.754098
\(306\) 0 0
\(307\) 22.1776 1.26574 0.632872 0.774256i \(-0.281876\pi\)
0.632872 + 0.774256i \(0.281876\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 23.7836 1.34864 0.674322 0.738437i \(-0.264436\pi\)
0.674322 + 0.738437i \(0.264436\pi\)
\(312\) 0 0
\(313\) 15.4671 0.874251 0.437125 0.899401i \(-0.355997\pi\)
0.437125 + 0.899401i \(0.355997\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −32.3817 −1.81874 −0.909369 0.415991i \(-0.863435\pi\)
−0.909369 + 0.415991i \(0.863435\pi\)
\(318\) 0 0
\(319\) −2.65260 −0.148517
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) −4.00000 −0.222566
\(324\) 0 0
\(325\) −6.25857 −0.347163
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 4.87243 0.268626
\(330\) 0 0
\(331\) −20.0422 −1.10162 −0.550809 0.834631i \(-0.685681\pi\)
−0.550809 + 0.834631i \(0.685681\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) −4.00000 −0.218543
\(336\) 0 0
\(337\) 13.9033 0.757360 0.378680 0.925528i \(-0.376378\pi\)
0.378680 + 0.925528i \(0.376378\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 5.82236 0.315298
\(342\) 0 0
\(343\) −12.0000 −0.647939
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −7.46707 −0.400853 −0.200427 0.979709i \(-0.564233\pi\)
−0.200427 + 0.979709i \(0.564233\pi\)
\(348\) 0 0
\(349\) 16.6948 0.893652 0.446826 0.894621i \(-0.352554\pi\)
0.446826 + 0.894621i \(0.352554\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 9.70268 0.516422 0.258211 0.966089i \(-0.416867\pi\)
0.258211 + 0.966089i \(0.416867\pi\)
\(354\) 0 0
\(355\) −16.5171 −0.876639
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 29.6060 1.56254 0.781272 0.624191i \(-0.214571\pi\)
0.781272 + 0.624191i \(0.214571\pi\)
\(360\) 0 0
\(361\) 1.00000 0.0526316
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) −6.00000 −0.314054
\(366\) 0 0
\(367\) −13.7836 −0.719499 −0.359749 0.933049i \(-0.617138\pi\)
−0.359749 + 0.933049i \(0.617138\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) −3.05008 −0.158352
\(372\) 0 0
\(373\) −2.25857 −0.116945 −0.0584723 0.998289i \(-0.518623\pi\)
−0.0584723 + 0.998289i \(0.518623\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −5.70268 −0.293703
\(378\) 0 0
\(379\) −17.7027 −0.909326 −0.454663 0.890664i \(-0.650240\pi\)
−0.454663 + 0.890664i \(0.650240\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) −15.0501 −0.769023 −0.384511 0.923120i \(-0.625630\pi\)
−0.384511 + 0.923120i \(0.625630\pi\)
\(384\) 0 0
\(385\) 2.65260 0.135189
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 14.0000 0.709828 0.354914 0.934899i \(-0.384510\pi\)
0.354914 + 0.934899i \(0.384510\pi\)
\(390\) 0 0
\(391\) 21.3896 1.08172
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 10.9921 0.553073
\(396\) 0 0
\(397\) 0.313098 0.0157139 0.00785697 0.999969i \(-0.497499\pi\)
0.00785697 + 0.999969i \(0.497499\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 7.03086 0.351104 0.175552 0.984470i \(-0.443829\pi\)
0.175552 + 0.984470i \(0.443829\pi\)
\(402\) 0 0
\(403\) 12.5171 0.623524
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) −12.9146 −0.640154
\(408\) 0 0
\(409\) 20.1776 0.997720 0.498860 0.866683i \(-0.333752\pi\)
0.498860 + 0.866683i \(0.333752\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) −5.03430 −0.247722
\(414\) 0 0
\(415\) −3.82236 −0.187632
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 15.4283 0.753723 0.376862 0.926270i \(-0.377003\pi\)
0.376862 + 0.926270i \(0.377003\pi\)
\(420\) 0 0
\(421\) 27.3316 1.33206 0.666031 0.745924i \(-0.267992\pi\)
0.666031 + 0.745924i \(0.267992\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) −4.00000 −0.194029
\(426\) 0 0
\(427\) 12.0000 0.580721
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 1.22772 0.0591371 0.0295686 0.999563i \(-0.490587\pi\)
0.0295686 + 0.999563i \(0.490587\pi\)
\(432\) 0 0
\(433\) −24.4362 −1.17433 −0.587165 0.809467i \(-0.699756\pi\)
−0.587165 + 0.809467i \(0.699756\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −5.34740 −0.255801
\(438\) 0 0
\(439\) −34.9921 −1.67008 −0.835041 0.550187i \(-0.814556\pi\)
−0.835041 + 0.550187i \(0.814556\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 31.3896 1.49136 0.745682 0.666302i \(-0.232124\pi\)
0.745682 + 0.666302i \(0.232124\pi\)
\(444\) 0 0
\(445\) −2.43622 −0.115488
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 12.2586 0.578518 0.289259 0.957251i \(-0.406591\pi\)
0.289259 + 0.957251i \(0.406591\pi\)
\(450\) 0 0
\(451\) 18.7370 0.882290
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 5.70268 0.267346
\(456\) 0 0
\(457\) −17.2119 −0.805141 −0.402570 0.915389i \(-0.631883\pi\)
−0.402570 + 0.915389i \(0.631883\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 28.6948 1.33645 0.668225 0.743959i \(-0.267054\pi\)
0.668225 + 0.743959i \(0.267054\pi\)
\(462\) 0 0
\(463\) −7.08882 −0.329445 −0.164723 0.986340i \(-0.552673\pi\)
−0.164723 + 0.986340i \(0.552673\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 26.6790 1.23456 0.617279 0.786745i \(-0.288235\pi\)
0.617279 + 0.786745i \(0.288235\pi\)
\(468\) 0 0
\(469\) 3.64472 0.168297
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) −24.0422 −1.10546
\(474\) 0 0
\(475\) 1.00000 0.0458831
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 29.0888 1.32910 0.664551 0.747243i \(-0.268623\pi\)
0.664551 + 0.747243i \(0.268623\pi\)
\(480\) 0 0
\(481\) −27.7644 −1.26595
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) −16.4362 −0.746330
\(486\) 0 0
\(487\) 23.8066 1.07878 0.539390 0.842056i \(-0.318655\pi\)
0.539390 + 0.842056i \(0.318655\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 13.6060 0.614029 0.307014 0.951705i \(-0.400670\pi\)
0.307014 + 0.951705i \(0.400670\pi\)
\(492\) 0 0
\(493\) −3.64472 −0.164150
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 15.0501 0.675088
\(498\) 0 0
\(499\) −6.17764 −0.276549 −0.138275 0.990394i \(-0.544156\pi\)
−0.138275 + 0.990394i \(0.544156\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) −2.81447 −0.125491 −0.0627455 0.998030i \(-0.519986\pi\)
−0.0627455 + 0.998030i \(0.519986\pi\)
\(504\) 0 0
\(505\) −0.177642 −0.00790498
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 39.7678 1.76268 0.881339 0.472484i \(-0.156643\pi\)
0.881339 + 0.472484i \(0.156643\pi\)
\(510\) 0 0
\(511\) 5.46707 0.241849
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 3.64472 0.160605
\(516\) 0 0
\(517\) −15.5672 −0.684646
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 38.5981 1.69101 0.845506 0.533965i \(-0.179299\pi\)
0.845506 + 0.533965i \(0.179299\pi\)
\(522\) 0 0
\(523\) 38.5014 1.68355 0.841774 0.539831i \(-0.181512\pi\)
0.841774 + 0.539831i \(0.181512\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 8.00000 0.348485
\(528\) 0 0
\(529\) 5.59464 0.243245
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 40.2815 1.74479
\(534\) 0 0
\(535\) −3.64472 −0.157575
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 17.9613 0.773646
\(540\) 0 0
\(541\) −10.0000 −0.429934 −0.214967 0.976621i \(-0.568964\pi\)
−0.214967 + 0.976621i \(0.568964\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 3.52504 0.150996
\(546\) 0 0
\(547\) 14.1776 0.606192 0.303096 0.952960i \(-0.401980\pi\)
0.303096 + 0.952960i \(0.401980\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 0.911179 0.0388175
\(552\) 0 0
\(553\) −10.0158 −0.425914
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −6.00000 −0.254228 −0.127114 0.991888i \(-0.540571\pi\)
−0.127114 + 0.991888i \(0.540571\pi\)
\(558\) 0 0
\(559\) −51.6869 −2.18612
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) −28.8302 −1.21505 −0.607525 0.794301i \(-0.707838\pi\)
−0.607525 + 0.794301i \(0.707838\pi\)
\(564\) 0 0
\(565\) −13.5250 −0.569003
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 28.2586 1.18466 0.592331 0.805695i \(-0.298208\pi\)
0.592331 + 0.805695i \(0.298208\pi\)
\(570\) 0 0
\(571\) 0.872434 0.0365102 0.0182551 0.999833i \(-0.494189\pi\)
0.0182551 + 0.999833i \(0.494189\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) −5.34740 −0.223002
\(576\) 0 0
\(577\) 25.8066 1.07434 0.537171 0.843473i \(-0.319493\pi\)
0.537171 + 0.843473i \(0.319493\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 3.48285 0.144493
\(582\) 0 0
\(583\) 9.74487 0.403591
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −16.6948 −0.689068 −0.344534 0.938774i \(-0.611963\pi\)
−0.344534 + 0.938774i \(0.611963\pi\)
\(588\) 0 0
\(589\) −2.00000 −0.0824086
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 20.6368 0.847453 0.423726 0.905790i \(-0.360722\pi\)
0.423726 + 0.905790i \(0.360722\pi\)
\(594\) 0 0
\(595\) 3.64472 0.149419
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 14.9341 0.610193 0.305096 0.952321i \(-0.401311\pi\)
0.305096 + 0.952321i \(0.401311\pi\)
\(600\) 0 0
\(601\) −6.51715 −0.265840 −0.132920 0.991127i \(-0.542435\pi\)
−0.132920 + 0.991127i \(0.542435\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 2.52504 0.102657
\(606\) 0 0
\(607\) −34.5014 −1.40037 −0.700184 0.713963i \(-0.746899\pi\)
−0.700184 + 0.713963i \(0.746899\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) −33.4671 −1.35393
\(612\) 0 0
\(613\) −28.5593 −1.15350 −0.576750 0.816920i \(-0.695679\pi\)
−0.576750 + 0.816920i \(0.695679\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −25.9842 −1.04609 −0.523043 0.852306i \(-0.675203\pi\)
−0.523043 + 0.852306i \(0.675203\pi\)
\(618\) 0 0
\(619\) 5.22772 0.210120 0.105060 0.994466i \(-0.466497\pi\)
0.105060 + 0.994466i \(0.466497\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 2.21983 0.0889356
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) −17.7449 −0.707534
\(630\) 0 0
\(631\) −45.3896 −1.80693 −0.903465 0.428661i \(-0.858985\pi\)
−0.903465 + 0.428661i \(0.858985\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) −22.3395 −0.886516
\(636\) 0 0
\(637\) 38.6139 1.52994
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) −11.5480 −0.456119 −0.228059 0.973647i \(-0.573238\pi\)
−0.228059 + 0.973647i \(0.573238\pi\)
\(642\) 0 0
\(643\) 15.7414 0.620781 0.310391 0.950609i \(-0.399540\pi\)
0.310391 + 0.950609i \(0.399540\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 17.8645 0.702328 0.351164 0.936314i \(-0.385786\pi\)
0.351164 + 0.936314i \(0.385786\pi\)
\(648\) 0 0
\(649\) 16.0844 0.631367
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 42.9921 1.68241 0.841206 0.540715i \(-0.181846\pi\)
0.841206 + 0.540715i \(0.181846\pi\)
\(654\) 0 0
\(655\) −8.73354 −0.341248
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) −48.3817 −1.88468 −0.942342 0.334653i \(-0.891381\pi\)
−0.942342 + 0.334653i \(0.891381\pi\)
\(660\) 0 0
\(661\) −25.8645 −1.00601 −0.503007 0.864282i \(-0.667773\pi\)
−0.503007 + 0.864282i \(0.667773\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) −0.911179 −0.0353340
\(666\) 0 0
\(667\) −4.87243 −0.188661
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) −38.3395 −1.48008
\(672\) 0 0
\(673\) 42.9376 1.65512 0.827561 0.561376i \(-0.189728\pi\)
0.827561 + 0.561376i \(0.189728\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) −10.0422 −0.385953 −0.192976 0.981203i \(-0.561814\pi\)
−0.192976 + 0.981203i \(0.561814\pi\)
\(678\) 0 0
\(679\) 14.9763 0.574739
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) −30.6948 −1.17450 −0.587252 0.809404i \(-0.699790\pi\)
−0.587252 + 0.809404i \(0.699790\pi\)
\(684\) 0 0
\(685\) 13.3474 0.509978
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 20.9499 0.798129
\(690\) 0 0
\(691\) −3.48285 −0.132494 −0.0662470 0.997803i \(-0.521103\pi\)
−0.0662470 + 0.997803i \(0.521103\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −1.82236 −0.0691260
\(696\) 0 0
\(697\) 25.7449 0.975156
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 23.3896 0.883412 0.441706 0.897160i \(-0.354373\pi\)
0.441706 + 0.897160i \(0.354373\pi\)
\(702\) 0 0
\(703\) 4.43622 0.167315
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 0.161864 0.00608752
\(708\) 0 0
\(709\) −20.4592 −0.768361 −0.384180 0.923258i \(-0.625516\pi\)
−0.384180 + 0.923258i \(0.625516\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 10.6948 0.400523
\(714\) 0 0
\(715\) −18.2198 −0.681383
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) −12.3783 −0.461631 −0.230815 0.972998i \(-0.574139\pi\)
−0.230815 + 0.972998i \(0.574139\pi\)
\(720\) 0 0
\(721\) −3.32099 −0.123680
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 0.911179 0.0338403
\(726\) 0 0
\(727\) 2.49418 0.0925041 0.0462520 0.998930i \(-0.485272\pi\)
0.0462520 + 0.998930i \(0.485272\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) −33.0343 −1.22182
\(732\) 0 0
\(733\) 14.6526 0.541206 0.270603 0.962691i \(-0.412777\pi\)
0.270603 + 0.962691i \(0.412777\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −11.6447 −0.428939
\(738\) 0 0
\(739\) 21.0343 0.773759 0.386880 0.922130i \(-0.373553\pi\)
0.386880 + 0.922130i \(0.373553\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 0.949924 0.0348493 0.0174247 0.999848i \(-0.494453\pi\)
0.0174247 + 0.999848i \(0.494453\pi\)
\(744\) 0 0
\(745\) 4.69479 0.172004
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 3.32099 0.121346
\(750\) 0 0
\(751\) −20.6948 −0.755164 −0.377582 0.925976i \(-0.623244\pi\)
−0.377582 + 0.925976i \(0.623244\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) −4.69479 −0.170861
\(756\) 0 0
\(757\) 23.8803 0.867945 0.433973 0.900926i \(-0.357111\pi\)
0.433973 + 0.900926i \(0.357111\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −53.3738 −1.93480 −0.967399 0.253255i \(-0.918499\pi\)
−0.967399 + 0.253255i \(0.918499\pi\)
\(762\) 0 0
\(763\) −3.21194 −0.116280
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 34.5789 1.24857
\(768\) 0 0
\(769\) 29.0501 1.04757 0.523786 0.851850i \(-0.324519\pi\)
0.523786 + 0.851850i \(0.324519\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 31.3474 1.12749 0.563744 0.825950i \(-0.309361\pi\)
0.563744 + 0.825950i \(0.309361\pi\)
\(774\) 0 0
\(775\) −2.00000 −0.0718421
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) −6.43622 −0.230601
\(780\) 0 0
\(781\) −48.0844 −1.72059
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 3.16975 0.113133
\(786\) 0 0
\(787\) 23.2119 0.827416 0.413708 0.910410i \(-0.364233\pi\)
0.413708 + 0.910410i \(0.364233\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 12.3237 0.438181
\(792\) 0 0
\(793\) −82.4239 −2.92696
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 35.8645 1.27039 0.635194 0.772353i \(-0.280920\pi\)
0.635194 + 0.772353i \(0.280920\pi\)
\(798\) 0 0
\(799\) −21.3896 −0.756709
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) −17.4671 −0.616400
\(804\) 0 0
\(805\) 4.87243 0.171731
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) −2.35528 −0.0828074 −0.0414037 0.999142i \(-0.513183\pi\)
−0.0414037 + 0.999142i \(0.513183\pi\)
\(810\) 0 0
\(811\) −43.0185 −1.51058 −0.755292 0.655388i \(-0.772505\pi\)
−0.755292 + 0.655388i \(0.772505\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 6.43622 0.225451
\(816\) 0 0
\(817\) 8.25857 0.288931
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −47.9067 −1.67196 −0.835978 0.548763i \(-0.815099\pi\)
−0.835978 + 0.548763i \(0.815099\pi\)
\(822\) 0 0
\(823\) −12.3940 −0.432029 −0.216014 0.976390i \(-0.569306\pi\)
−0.216014 + 0.976390i \(0.569306\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −11.1698 −0.388410 −0.194205 0.980961i \(-0.562213\pi\)
−0.194205 + 0.980961i \(0.562213\pi\)
\(828\) 0 0
\(829\) −0.830247 −0.0288357 −0.0144178 0.999896i \(-0.504589\pi\)
−0.0144178 + 0.999896i \(0.504589\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 24.6790 0.855077
\(834\) 0 0
\(835\) 3.16975 0.109694
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 12.8724 0.444406 0.222203 0.975000i \(-0.428675\pi\)
0.222203 + 0.975000i \(0.428675\pi\)
\(840\) 0 0
\(841\) −28.1698 −0.971371
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) −26.1698 −0.900267
\(846\) 0 0
\(847\) −2.30076 −0.0790551
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) −23.7222 −0.813187
\(852\) 0 0
\(853\) 12.8302 0.439299 0.219650 0.975579i \(-0.429509\pi\)
0.219650 + 0.975579i \(0.429509\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 35.0765 1.19819 0.599095 0.800678i \(-0.295527\pi\)
0.599095 + 0.800678i \(0.295527\pi\)
\(858\) 0 0
\(859\) −22.6948 −0.774336 −0.387168 0.922009i \(-0.626547\pi\)
−0.387168 + 0.922009i \(0.626547\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 18.6948 0.636378 0.318189 0.948027i \(-0.396925\pi\)
0.318189 + 0.948027i \(0.396925\pi\)
\(864\) 0 0
\(865\) −9.52504 −0.323861
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 32.0000 1.08553
\(870\) 0 0
\(871\) −25.0343 −0.848255
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) −0.911179 −0.0308035
\(876\) 0 0
\(877\) −10.7757 −0.363870 −0.181935 0.983311i \(-0.558236\pi\)
−0.181935 + 0.983311i \(0.558236\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) −1.56722 −0.0528011 −0.0264006 0.999651i \(-0.508405\pi\)
−0.0264006 + 0.999651i \(0.508405\pi\)
\(882\) 0 0
\(883\) −5.13101 −0.172672 −0.0863361 0.996266i \(-0.527516\pi\)
−0.0863361 + 0.996266i \(0.527516\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 9.74487 0.327201 0.163600 0.986527i \(-0.447689\pi\)
0.163600 + 0.986527i \(0.447689\pi\)
\(888\) 0 0
\(889\) 20.3553 0.682694
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 5.34740 0.178944
\(894\) 0 0
\(895\) 5.16975 0.172806
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) −1.82236 −0.0607790
\(900\) 0 0
\(901\) 13.3896 0.446072
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) −18.8145 −0.625414
\(906\) 0 0
\(907\) −4.51715 −0.149989 −0.0749947 0.997184i \(-0.523894\pi\)
−0.0749947 + 0.997184i \(0.523894\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) −21.0343 −0.696897 −0.348449 0.937328i \(-0.613291\pi\)
−0.348449 + 0.937328i \(0.613291\pi\)
\(912\) 0 0
\(913\) −11.1276 −0.368269
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 7.95781 0.262790
\(918\) 0 0
\(919\) −0.949924 −0.0313351 −0.0156676 0.999877i \(-0.504987\pi\)
−0.0156676 + 0.999877i \(0.504987\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) −103.374 −3.40259
\(924\) 0 0
\(925\) 4.43622 0.145862
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) −29.4513 −0.966266 −0.483133 0.875547i \(-0.660501\pi\)
−0.483133 + 0.875547i \(0.660501\pi\)
\(930\) 0 0
\(931\) −6.16975 −0.202205
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) −11.6447 −0.380823
\(936\) 0 0
\(937\) −54.6015 −1.78375 −0.891877 0.452278i \(-0.850611\pi\)
−0.891877 + 0.452278i \(0.850611\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) −32.2006 −1.04971 −0.524855 0.851192i \(-0.675880\pi\)
−0.524855 + 0.851192i \(0.675880\pi\)
\(942\) 0 0
\(943\) 34.4170 1.12077
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −42.5171 −1.38162 −0.690811 0.723036i \(-0.742746\pi\)
−0.690811 + 0.723036i \(0.742746\pi\)
\(948\) 0 0
\(949\) −37.5514 −1.21897
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) −28.1039 −0.910375 −0.455187 0.890396i \(-0.650428\pi\)
−0.455187 + 0.890396i \(0.650428\pi\)
\(954\) 0 0
\(955\) 7.78361 0.251872
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) −12.1619 −0.392727
\(960\) 0 0
\(961\) −27.0000 −0.870968
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 15.7256 0.506226
\(966\) 0 0
\(967\) 42.4626 1.36551 0.682753 0.730649i \(-0.260783\pi\)
0.682753 + 0.730649i \(0.260783\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 22.0422 0.707367 0.353684 0.935365i \(-0.384929\pi\)
0.353684 + 0.935365i \(0.384929\pi\)
\(972\) 0 0
\(973\) 1.66049 0.0532330
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 15.3474 0.491007 0.245503 0.969396i \(-0.421047\pi\)
0.245503 + 0.969396i \(0.421047\pi\)
\(978\) 0 0
\(979\) −7.09226 −0.226670
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) 5.78017 0.184359 0.0921794 0.995742i \(-0.470617\pi\)
0.0921794 + 0.995742i \(0.470617\pi\)
\(984\) 0 0
\(985\) −7.70268 −0.245428
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) −44.1619 −1.40427
\(990\) 0 0
\(991\) 60.7370 1.92937 0.964687 0.263399i \(-0.0848437\pi\)
0.964687 + 0.263399i \(0.0848437\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 20.5171 0.650437
\(996\) 0 0
\(997\) 41.5093 1.31461 0.657306 0.753624i \(-0.271696\pi\)
0.657306 + 0.753624i \(0.271696\pi\)
\(998\) 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.bh.1.2 3
3.2 odd 2 2280.2.a.u.1.2 3
12.11 even 2 4560.2.a.bs.1.2 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2280.2.a.u.1.2 3 3.2 odd 2
4560.2.a.bs.1.2 3 12.11 even 2
6840.2.a.bh.1.2 3 1.1 even 1 trivial