Properties

Label 6840.2.a.bd.1.2
Level $6840$
Weight $2$
Character 6840.1
Self dual yes
Analytic conductor $54.618$
Analytic rank $0$
Dimension $2$
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: \(2\)
Coefficient field: \(\Q(\zeta_{10})^+\)
Defining polynomial: \(x^{2} - x - 1\)
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 \(1.61803\) of defining polynomial
Character \(\chi\) \(=\) 6840.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{5} +5.23607 q^{7} +O(q^{10})\) \(q+1.00000 q^{5} +5.23607 q^{7} +1.23607 q^{11} +0.763932 q^{13} -4.47214 q^{17} +1.00000 q^{19} -2.47214 q^{23} +1.00000 q^{25} -0.763932 q^{29} -8.94427 q^{31} +5.23607 q^{35} +3.23607 q^{37} +9.70820 q^{41} +5.23607 q^{43} -2.47214 q^{47} +20.4164 q^{49} +0.472136 q^{53} +1.23607 q^{55} +10.4721 q^{59} +4.47214 q^{61} +0.763932 q^{65} +1.52786 q^{67} +12.4721 q^{73} +6.47214 q^{77} -4.00000 q^{83} -4.47214 q^{85} +1.70820 q^{89} +4.00000 q^{91} +1.00000 q^{95} -4.76393 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{5} + 6 q^{7} + O(q^{10}) \) \( 2 q + 2 q^{5} + 6 q^{7} - 2 q^{11} + 6 q^{13} + 2 q^{19} + 4 q^{23} + 2 q^{25} - 6 q^{29} + 6 q^{35} + 2 q^{37} + 6 q^{41} + 6 q^{43} + 4 q^{47} + 14 q^{49} - 8 q^{53} - 2 q^{55} + 12 q^{59} + 6 q^{65} + 12 q^{67} + 16 q^{73} + 4 q^{77} - 8 q^{83} - 10 q^{89} + 8 q^{91} + 2 q^{95} - 14 q^{97} + O(q^{100}) \)

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
\(7\) 5.23607 1.97905 0.989524 0.144370i \(-0.0461154\pi\)
0.989524 + 0.144370i \(0.0461154\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 1.23607 0.372689 0.186344 0.982485i \(-0.440336\pi\)
0.186344 + 0.982485i \(0.440336\pi\)
\(12\) 0 0
\(13\) 0.763932 0.211877 0.105938 0.994373i \(-0.466215\pi\)
0.105938 + 0.994373i \(0.466215\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −4.47214 −1.08465 −0.542326 0.840168i \(-0.682456\pi\)
−0.542326 + 0.840168i \(0.682456\pi\)
\(18\) 0 0
\(19\) 1.00000 0.229416
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −2.47214 −0.515476 −0.257738 0.966215i \(-0.582977\pi\)
−0.257738 + 0.966215i \(0.582977\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −0.763932 −0.141859 −0.0709293 0.997481i \(-0.522596\pi\)
−0.0709293 + 0.997481i \(0.522596\pi\)
\(30\) 0 0
\(31\) −8.94427 −1.60644 −0.803219 0.595683i \(-0.796881\pi\)
−0.803219 + 0.595683i \(0.796881\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 5.23607 0.885057
\(36\) 0 0
\(37\) 3.23607 0.532006 0.266003 0.963972i \(-0.414297\pi\)
0.266003 + 0.963972i \(0.414297\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 9.70820 1.51617 0.758083 0.652158i \(-0.226136\pi\)
0.758083 + 0.652158i \(0.226136\pi\)
\(42\) 0 0
\(43\) 5.23607 0.798493 0.399246 0.916844i \(-0.369272\pi\)
0.399246 + 0.916844i \(0.369272\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −2.47214 −0.360598 −0.180299 0.983612i \(-0.557707\pi\)
−0.180299 + 0.983612i \(0.557707\pi\)
\(48\) 0 0
\(49\) 20.4164 2.91663
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 0.472136 0.0648529 0.0324264 0.999474i \(-0.489677\pi\)
0.0324264 + 0.999474i \(0.489677\pi\)
\(54\) 0 0
\(55\) 1.23607 0.166671
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 10.4721 1.36336 0.681678 0.731652i \(-0.261251\pi\)
0.681678 + 0.731652i \(0.261251\pi\)
\(60\) 0 0
\(61\) 4.47214 0.572598 0.286299 0.958140i \(-0.407575\pi\)
0.286299 + 0.958140i \(0.407575\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 0.763932 0.0947541
\(66\) 0 0
\(67\) 1.52786 0.186658 0.0933292 0.995635i \(-0.470249\pi\)
0.0933292 + 0.995635i \(0.470249\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(72\) 0 0
\(73\) 12.4721 1.45975 0.729877 0.683579i \(-0.239578\pi\)
0.729877 + 0.683579i \(0.239578\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 6.47214 0.737568
\(78\) 0 0
\(79\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) −4.00000 −0.439057 −0.219529 0.975606i \(-0.570452\pi\)
−0.219529 + 0.975606i \(0.570452\pi\)
\(84\) 0 0
\(85\) −4.47214 −0.485071
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 1.70820 0.181069 0.0905346 0.995893i \(-0.471142\pi\)
0.0905346 + 0.995893i \(0.471142\pi\)
\(90\) 0 0
\(91\) 4.00000 0.419314
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 1.00000 0.102598
\(96\) 0 0
\(97\) −4.76393 −0.483704 −0.241852 0.970313i \(-0.577755\pi\)
−0.241852 + 0.970313i \(0.577755\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 13.4164 1.33498 0.667491 0.744618i \(-0.267368\pi\)
0.667491 + 0.744618i \(0.267368\pi\)
\(102\) 0 0
\(103\) 9.52786 0.938808 0.469404 0.882983i \(-0.344469\pi\)
0.469404 + 0.882983i \(0.344469\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 8.94427 0.864675 0.432338 0.901712i \(-0.357689\pi\)
0.432338 + 0.901712i \(0.357689\pi\)
\(108\) 0 0
\(109\) −4.47214 −0.428353 −0.214176 0.976795i \(-0.568707\pi\)
−0.214176 + 0.976795i \(0.568707\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 2.00000 0.188144 0.0940721 0.995565i \(-0.470012\pi\)
0.0940721 + 0.995565i \(0.470012\pi\)
\(114\) 0 0
\(115\) −2.47214 −0.230528
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) −23.4164 −2.14658
\(120\) 0 0
\(121\) −9.47214 −0.861103
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 1.00000 0.0894427
\(126\) 0 0
\(127\) 20.0000 1.77471 0.887357 0.461084i \(-0.152539\pi\)
0.887357 + 0.461084i \(0.152539\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) −14.1803 −1.23894 −0.619471 0.785020i \(-0.712653\pi\)
−0.619471 + 0.785020i \(0.712653\pi\)
\(132\) 0 0
\(133\) 5.23607 0.454025
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −15.8885 −1.35745 −0.678725 0.734393i \(-0.737467\pi\)
−0.678725 + 0.734393i \(0.737467\pi\)
\(138\) 0 0
\(139\) −14.4721 −1.22751 −0.613755 0.789496i \(-0.710342\pi\)
−0.613755 + 0.789496i \(0.710342\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 0.944272 0.0789640
\(144\) 0 0
\(145\) −0.763932 −0.0634411
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) −17.4164 −1.42681 −0.713404 0.700753i \(-0.752847\pi\)
−0.713404 + 0.700753i \(0.752847\pi\)
\(150\) 0 0
\(151\) 12.0000 0.976546 0.488273 0.872691i \(-0.337627\pi\)
0.488273 + 0.872691i \(0.337627\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) −8.94427 −0.718421
\(156\) 0 0
\(157\) −21.4164 −1.70922 −0.854608 0.519274i \(-0.826202\pi\)
−0.854608 + 0.519274i \(0.826202\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) −12.9443 −1.02015
\(162\) 0 0
\(163\) 20.6525 1.61763 0.808813 0.588065i \(-0.200110\pi\)
0.808813 + 0.588065i \(0.200110\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −13.8885 −1.07473 −0.537364 0.843350i \(-0.680580\pi\)
−0.537364 + 0.843350i \(0.680580\pi\)
\(168\) 0 0
\(169\) −12.4164 −0.955108
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 22.0000 1.67263 0.836315 0.548250i \(-0.184706\pi\)
0.836315 + 0.548250i \(0.184706\pi\)
\(174\) 0 0
\(175\) 5.23607 0.395810
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 15.4164 1.15228 0.576138 0.817352i \(-0.304559\pi\)
0.576138 + 0.817352i \(0.304559\pi\)
\(180\) 0 0
\(181\) 8.47214 0.629729 0.314864 0.949137i \(-0.398041\pi\)
0.314864 + 0.949137i \(0.398041\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 3.23607 0.237920
\(186\) 0 0
\(187\) −5.52786 −0.404237
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 6.18034 0.447194 0.223597 0.974682i \(-0.428220\pi\)
0.223597 + 0.974682i \(0.428220\pi\)
\(192\) 0 0
\(193\) 5.70820 0.410886 0.205443 0.978669i \(-0.434137\pi\)
0.205443 + 0.978669i \(0.434137\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 12.4721 0.888603 0.444301 0.895877i \(-0.353452\pi\)
0.444301 + 0.895877i \(0.353452\pi\)
\(198\) 0 0
\(199\) 18.4721 1.30945 0.654727 0.755865i \(-0.272783\pi\)
0.654727 + 0.755865i \(0.272783\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) −4.00000 −0.280745
\(204\) 0 0
\(205\) 9.70820 0.678050
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 1.23607 0.0855006
\(210\) 0 0
\(211\) −12.0000 −0.826114 −0.413057 0.910705i \(-0.635539\pi\)
−0.413057 + 0.910705i \(0.635539\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 5.23607 0.357097
\(216\) 0 0
\(217\) −46.8328 −3.17922
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) −3.41641 −0.229812
\(222\) 0 0
\(223\) 22.4721 1.50485 0.752423 0.658680i \(-0.228885\pi\)
0.752423 + 0.658680i \(0.228885\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(228\) 0 0
\(229\) −21.4164 −1.41524 −0.707618 0.706595i \(-0.750230\pi\)
−0.707618 + 0.706595i \(0.750230\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −23.8885 −1.56499 −0.782495 0.622657i \(-0.786053\pi\)
−0.782495 + 0.622657i \(0.786053\pi\)
\(234\) 0 0
\(235\) −2.47214 −0.161264
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 11.7082 0.757341 0.378670 0.925532i \(-0.376381\pi\)
0.378670 + 0.925532i \(0.376381\pi\)
\(240\) 0 0
\(241\) 6.94427 0.447320 0.223660 0.974667i \(-0.428199\pi\)
0.223660 + 0.974667i \(0.428199\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 20.4164 1.30436
\(246\) 0 0
\(247\) 0.763932 0.0486078
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) −3.70820 −0.234060 −0.117030 0.993128i \(-0.537337\pi\)
−0.117030 + 0.993128i \(0.537337\pi\)
\(252\) 0 0
\(253\) −3.05573 −0.192112
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 22.9443 1.43122 0.715612 0.698498i \(-0.246148\pi\)
0.715612 + 0.698498i \(0.246148\pi\)
\(258\) 0 0
\(259\) 16.9443 1.05287
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) −3.05573 −0.188424 −0.0942121 0.995552i \(-0.530033\pi\)
−0.0942121 + 0.995552i \(0.530033\pi\)
\(264\) 0 0
\(265\) 0.472136 0.0290031
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) −5.70820 −0.348035 −0.174018 0.984743i \(-0.555675\pi\)
−0.174018 + 0.984743i \(0.555675\pi\)
\(270\) 0 0
\(271\) −2.47214 −0.150172 −0.0750858 0.997177i \(-0.523923\pi\)
−0.0750858 + 0.997177i \(0.523923\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 1.23607 0.0745377
\(276\) 0 0
\(277\) −16.4721 −0.989715 −0.494857 0.868974i \(-0.664780\pi\)
−0.494857 + 0.868974i \(0.664780\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 15.2361 0.908908 0.454454 0.890770i \(-0.349834\pi\)
0.454454 + 0.890770i \(0.349834\pi\)
\(282\) 0 0
\(283\) −28.0689 −1.66852 −0.834261 0.551370i \(-0.814105\pi\)
−0.834261 + 0.551370i \(0.814105\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 50.8328 3.00057
\(288\) 0 0
\(289\) 3.00000 0.176471
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) −3.88854 −0.227171 −0.113586 0.993528i \(-0.536234\pi\)
−0.113586 + 0.993528i \(0.536234\pi\)
\(294\) 0 0
\(295\) 10.4721 0.609711
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) −1.88854 −0.109217
\(300\) 0 0
\(301\) 27.4164 1.58026
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 4.47214 0.256074
\(306\) 0 0
\(307\) −17.5279 −1.00037 −0.500184 0.865919i \(-0.666734\pi\)
−0.500184 + 0.865919i \(0.666734\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) −19.7082 −1.11755 −0.558775 0.829319i \(-0.688728\pi\)
−0.558775 + 0.829319i \(0.688728\pi\)
\(312\) 0 0
\(313\) −23.8885 −1.35026 −0.675130 0.737699i \(-0.735913\pi\)
−0.675130 + 0.737699i \(0.735913\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −12.4721 −0.700505 −0.350252 0.936655i \(-0.613904\pi\)
−0.350252 + 0.936655i \(0.613904\pi\)
\(318\) 0 0
\(319\) −0.944272 −0.0528691
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) −4.47214 −0.248836
\(324\) 0 0
\(325\) 0.763932 0.0423753
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) −12.9443 −0.713641
\(330\) 0 0
\(331\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 1.52786 0.0834761
\(336\) 0 0
\(337\) −28.7639 −1.56687 −0.783436 0.621473i \(-0.786535\pi\)
−0.783436 + 0.621473i \(0.786535\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) −11.0557 −0.598701
\(342\) 0 0
\(343\) 70.2492 3.79310
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 20.0000 1.07366 0.536828 0.843692i \(-0.319622\pi\)
0.536828 + 0.843692i \(0.319622\pi\)
\(348\) 0 0
\(349\) 10.9443 0.585833 0.292917 0.956138i \(-0.405374\pi\)
0.292917 + 0.956138i \(0.405374\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 14.9443 0.795403 0.397702 0.917515i \(-0.369808\pi\)
0.397702 + 0.917515i \(0.369808\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 22.1803 1.17063 0.585317 0.810805i \(-0.300970\pi\)
0.585317 + 0.810805i \(0.300970\pi\)
\(360\) 0 0
\(361\) 1.00000 0.0526316
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 12.4721 0.652821
\(366\) 0 0
\(367\) −29.2361 −1.52611 −0.763055 0.646333i \(-0.776302\pi\)
−0.763055 + 0.646333i \(0.776302\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 2.47214 0.128347
\(372\) 0 0
\(373\) −7.23607 −0.374669 −0.187335 0.982296i \(-0.559985\pi\)
−0.187335 + 0.982296i \(0.559985\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −0.583592 −0.0300565
\(378\) 0 0
\(379\) 1.88854 0.0970080 0.0485040 0.998823i \(-0.484555\pi\)
0.0485040 + 0.998823i \(0.484555\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 9.88854 0.505281 0.252640 0.967560i \(-0.418701\pi\)
0.252640 + 0.967560i \(0.418701\pi\)
\(384\) 0 0
\(385\) 6.47214 0.329851
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) −32.8328 −1.66469 −0.832345 0.554258i \(-0.813002\pi\)
−0.832345 + 0.554258i \(0.813002\pi\)
\(390\) 0 0
\(391\) 11.0557 0.559112
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) −6.58359 −0.330421 −0.165211 0.986258i \(-0.552830\pi\)
−0.165211 + 0.986258i \(0.552830\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 23.2361 1.16035 0.580177 0.814490i \(-0.302983\pi\)
0.580177 + 0.814490i \(0.302983\pi\)
\(402\) 0 0
\(403\) −6.83282 −0.340367
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 4.00000 0.198273
\(408\) 0 0
\(409\) 4.47214 0.221133 0.110566 0.993869i \(-0.464734\pi\)
0.110566 + 0.993869i \(0.464734\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 54.8328 2.69815
\(414\) 0 0
\(415\) −4.00000 −0.196352
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 22.1803 1.08358 0.541790 0.840514i \(-0.317747\pi\)
0.541790 + 0.840514i \(0.317747\pi\)
\(420\) 0 0
\(421\) −24.8328 −1.21028 −0.605139 0.796120i \(-0.706882\pi\)
−0.605139 + 0.796120i \(0.706882\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) −4.47214 −0.216930
\(426\) 0 0
\(427\) 23.4164 1.13320
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) −25.3050 −1.21890 −0.609448 0.792826i \(-0.708609\pi\)
−0.609448 + 0.792826i \(0.708609\pi\)
\(432\) 0 0
\(433\) 0.763932 0.0367122 0.0183561 0.999832i \(-0.494157\pi\)
0.0183561 + 0.999832i \(0.494157\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −2.47214 −0.118258
\(438\) 0 0
\(439\) 8.00000 0.381819 0.190910 0.981608i \(-0.438856\pi\)
0.190910 + 0.981608i \(0.438856\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −19.4164 −0.922501 −0.461251 0.887270i \(-0.652599\pi\)
−0.461251 + 0.887270i \(0.652599\pi\)
\(444\) 0 0
\(445\) 1.70820 0.0809766
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 9.70820 0.458158 0.229079 0.973408i \(-0.426429\pi\)
0.229079 + 0.973408i \(0.426429\pi\)
\(450\) 0 0
\(451\) 12.0000 0.565058
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 4.00000 0.187523
\(456\) 0 0
\(457\) 16.8328 0.787406 0.393703 0.919238i \(-0.371194\pi\)
0.393703 + 0.919238i \(0.371194\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 8.47214 0.394587 0.197293 0.980344i \(-0.436785\pi\)
0.197293 + 0.980344i \(0.436785\pi\)
\(462\) 0 0
\(463\) 25.5967 1.18958 0.594791 0.803880i \(-0.297235\pi\)
0.594791 + 0.803880i \(0.297235\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 4.58359 0.212103 0.106052 0.994361i \(-0.466179\pi\)
0.106052 + 0.994361i \(0.466179\pi\)
\(468\) 0 0
\(469\) 8.00000 0.369406
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 6.47214 0.297589
\(474\) 0 0
\(475\) 1.00000 0.0458831
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 14.1803 0.647916 0.323958 0.946071i \(-0.394986\pi\)
0.323958 + 0.946071i \(0.394986\pi\)
\(480\) 0 0
\(481\) 2.47214 0.112720
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) −4.76393 −0.216319
\(486\) 0 0
\(487\) 4.58359 0.207702 0.103851 0.994593i \(-0.466883\pi\)
0.103851 + 0.994593i \(0.466883\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 32.6525 1.47359 0.736793 0.676119i \(-0.236339\pi\)
0.736793 + 0.676119i \(0.236339\pi\)
\(492\) 0 0
\(493\) 3.41641 0.153867
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) −11.4164 −0.511069 −0.255534 0.966800i \(-0.582251\pi\)
−0.255534 + 0.966800i \(0.582251\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 12.9443 0.577157 0.288578 0.957456i \(-0.406817\pi\)
0.288578 + 0.957456i \(0.406817\pi\)
\(504\) 0 0
\(505\) 13.4164 0.597022
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) −30.2918 −1.34266 −0.671330 0.741158i \(-0.734277\pi\)
−0.671330 + 0.741158i \(0.734277\pi\)
\(510\) 0 0
\(511\) 65.3050 2.88892
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 9.52786 0.419848
\(516\) 0 0
\(517\) −3.05573 −0.134391
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 26.2918 1.15186 0.575932 0.817497i \(-0.304639\pi\)
0.575932 + 0.817497i \(0.304639\pi\)
\(522\) 0 0
\(523\) −0.944272 −0.0412901 −0.0206451 0.999787i \(-0.506572\pi\)
−0.0206451 + 0.999787i \(0.506572\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 40.0000 1.74243
\(528\) 0 0
\(529\) −16.8885 −0.734285
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 7.41641 0.321240
\(534\) 0 0
\(535\) 8.94427 0.386695
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 25.2361 1.08699
\(540\) 0 0
\(541\) −29.7771 −1.28022 −0.640108 0.768285i \(-0.721111\pi\)
−0.640108 + 0.768285i \(0.721111\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) −4.47214 −0.191565
\(546\) 0 0
\(547\) −37.3050 −1.59504 −0.797522 0.603289i \(-0.793856\pi\)
−0.797522 + 0.603289i \(0.793856\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) −0.763932 −0.0325446
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −32.8328 −1.39117 −0.695586 0.718443i \(-0.744855\pi\)
−0.695586 + 0.718443i \(0.744855\pi\)
\(558\) 0 0
\(559\) 4.00000 0.169182
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 9.88854 0.416752 0.208376 0.978049i \(-0.433182\pi\)
0.208376 + 0.978049i \(0.433182\pi\)
\(564\) 0 0
\(565\) 2.00000 0.0841406
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 32.5410 1.36419 0.682095 0.731263i \(-0.261069\pi\)
0.682095 + 0.731263i \(0.261069\pi\)
\(570\) 0 0
\(571\) −21.3050 −0.891584 −0.445792 0.895136i \(-0.647078\pi\)
−0.445792 + 0.895136i \(0.647078\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) −2.47214 −0.103095
\(576\) 0 0
\(577\) −9.05573 −0.376995 −0.188497 0.982074i \(-0.560362\pi\)
−0.188497 + 0.982074i \(0.560362\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) −20.9443 −0.868915
\(582\) 0 0
\(583\) 0.583592 0.0241699
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 11.4164 0.471206 0.235603 0.971849i \(-0.424294\pi\)
0.235603 + 0.971849i \(0.424294\pi\)
\(588\) 0 0
\(589\) −8.94427 −0.368542
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 2.00000 0.0821302 0.0410651 0.999156i \(-0.486925\pi\)
0.0410651 + 0.999156i \(0.486925\pi\)
\(594\) 0 0
\(595\) −23.4164 −0.959979
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) −47.4164 −1.93738 −0.968691 0.248270i \(-0.920138\pi\)
−0.968691 + 0.248270i \(0.920138\pi\)
\(600\) 0 0
\(601\) −10.3607 −0.422621 −0.211310 0.977419i \(-0.567773\pi\)
−0.211310 + 0.977419i \(0.567773\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) −9.47214 −0.385097
\(606\) 0 0
\(607\) −33.5279 −1.36085 −0.680427 0.732816i \(-0.738206\pi\)
−0.680427 + 0.732816i \(0.738206\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) −1.88854 −0.0764023
\(612\) 0 0
\(613\) −35.5279 −1.43496 −0.717478 0.696581i \(-0.754704\pi\)
−0.717478 + 0.696581i \(0.754704\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 39.3050 1.58236 0.791179 0.611585i \(-0.209468\pi\)
0.791179 + 0.611585i \(0.209468\pi\)
\(618\) 0 0
\(619\) 32.3607 1.30069 0.650343 0.759641i \(-0.274625\pi\)
0.650343 + 0.759641i \(0.274625\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 8.94427 0.358345
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) −14.4721 −0.577042
\(630\) 0 0
\(631\) 16.0000 0.636950 0.318475 0.947931i \(-0.396829\pi\)
0.318475 + 0.947931i \(0.396829\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 20.0000 0.793676
\(636\) 0 0
\(637\) 15.5967 0.617966
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 44.7639 1.76807 0.884035 0.467422i \(-0.154817\pi\)
0.884035 + 0.467422i \(0.154817\pi\)
\(642\) 0 0
\(643\) 17.5967 0.693948 0.346974 0.937875i \(-0.387209\pi\)
0.346974 + 0.937875i \(0.387209\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −14.8328 −0.583138 −0.291569 0.956550i \(-0.594177\pi\)
−0.291569 + 0.956550i \(0.594177\pi\)
\(648\) 0 0
\(649\) 12.9443 0.508107
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 31.5279 1.23378 0.616890 0.787049i \(-0.288392\pi\)
0.616890 + 0.787049i \(0.288392\pi\)
\(654\) 0 0
\(655\) −14.1803 −0.554072
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) −30.8328 −1.20108 −0.600538 0.799596i \(-0.705047\pi\)
−0.600538 + 0.799596i \(0.705047\pi\)
\(660\) 0 0
\(661\) 23.3050 0.906458 0.453229 0.891394i \(-0.350272\pi\)
0.453229 + 0.891394i \(0.350272\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 5.23607 0.203046
\(666\) 0 0
\(667\) 1.88854 0.0731247
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 5.52786 0.213401
\(672\) 0 0
\(673\) −27.5967 −1.06378 −0.531888 0.846815i \(-0.678517\pi\)
−0.531888 + 0.846815i \(0.678517\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) −15.5279 −0.596784 −0.298392 0.954443i \(-0.596450\pi\)
−0.298392 + 0.954443i \(0.596450\pi\)
\(678\) 0 0
\(679\) −24.9443 −0.957273
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) −31.7771 −1.21592 −0.607958 0.793969i \(-0.708011\pi\)
−0.607958 + 0.793969i \(0.708011\pi\)
\(684\) 0 0
\(685\) −15.8885 −0.607070
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 0.360680 0.0137408
\(690\) 0 0
\(691\) −25.5279 −0.971126 −0.485563 0.874202i \(-0.661385\pi\)
−0.485563 + 0.874202i \(0.661385\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −14.4721 −0.548959
\(696\) 0 0
\(697\) −43.4164 −1.64451
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 14.5836 0.550815 0.275407 0.961328i \(-0.411187\pi\)
0.275407 + 0.961328i \(0.411187\pi\)
\(702\) 0 0
\(703\) 3.23607 0.122051
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 70.2492 2.64199
\(708\) 0 0
\(709\) −31.3050 −1.17568 −0.587841 0.808976i \(-0.700022\pi\)
−0.587841 + 0.808976i \(0.700022\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 22.1115 0.828081
\(714\) 0 0
\(715\) 0.944272 0.0353138
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) −1.23607 −0.0460976 −0.0230488 0.999734i \(-0.507337\pi\)
−0.0230488 + 0.999734i \(0.507337\pi\)
\(720\) 0 0
\(721\) 49.8885 1.85795
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) −0.763932 −0.0283717
\(726\) 0 0
\(727\) 20.6525 0.765958 0.382979 0.923757i \(-0.374898\pi\)
0.382979 + 0.923757i \(0.374898\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) −23.4164 −0.866087
\(732\) 0 0
\(733\) −36.8328 −1.36045 −0.680226 0.733003i \(-0.738118\pi\)
−0.680226 + 0.733003i \(0.738118\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 1.88854 0.0695654
\(738\) 0 0
\(739\) −8.94427 −0.329020 −0.164510 0.986375i \(-0.552604\pi\)
−0.164510 + 0.986375i \(0.552604\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 44.9443 1.64885 0.824423 0.565975i \(-0.191500\pi\)
0.824423 + 0.565975i \(0.191500\pi\)
\(744\) 0 0
\(745\) −17.4164 −0.638088
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 46.8328 1.71123
\(750\) 0 0
\(751\) 8.94427 0.326381 0.163191 0.986595i \(-0.447821\pi\)
0.163191 + 0.986595i \(0.447821\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 12.0000 0.436725
\(756\) 0 0
\(757\) −2.94427 −0.107011 −0.0535057 0.998568i \(-0.517040\pi\)
−0.0535057 + 0.998568i \(0.517040\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −9.41641 −0.341345 −0.170672 0.985328i \(-0.554594\pi\)
−0.170672 + 0.985328i \(0.554594\pi\)
\(762\) 0 0
\(763\) −23.4164 −0.847731
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 8.00000 0.288863
\(768\) 0 0
\(769\) 8.11146 0.292507 0.146253 0.989247i \(-0.453279\pi\)
0.146253 + 0.989247i \(0.453279\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) −39.5279 −1.42172 −0.710859 0.703334i \(-0.751694\pi\)
−0.710859 + 0.703334i \(0.751694\pi\)
\(774\) 0 0
\(775\) −8.94427 −0.321288
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 9.70820 0.347833
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) −21.4164 −0.764384
\(786\) 0 0
\(787\) 10.8328 0.386148 0.193074 0.981184i \(-0.438154\pi\)
0.193074 + 0.981184i \(0.438154\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 10.4721 0.372346
\(792\) 0 0
\(793\) 3.41641 0.121320
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −37.7771 −1.33813 −0.669067 0.743202i \(-0.733306\pi\)
−0.669067 + 0.743202i \(0.733306\pi\)
\(798\) 0 0
\(799\) 11.0557 0.391124
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 15.4164 0.544033
\(804\) 0 0
\(805\) −12.9443 −0.456226
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) −38.9443 −1.36921 −0.684604 0.728915i \(-0.740025\pi\)
−0.684604 + 0.728915i \(0.740025\pi\)
\(810\) 0 0
\(811\) 0.944272 0.0331579 0.0165789 0.999863i \(-0.494723\pi\)
0.0165789 + 0.999863i \(0.494723\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 20.6525 0.723425
\(816\) 0 0
\(817\) 5.23607 0.183187
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −15.5279 −0.541926 −0.270963 0.962590i \(-0.587342\pi\)
−0.270963 + 0.962590i \(0.587342\pi\)
\(822\) 0 0
\(823\) −2.18034 −0.0760019 −0.0380009 0.999278i \(-0.512099\pi\)
−0.0380009 + 0.999278i \(0.512099\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −8.00000 −0.278187 −0.139094 0.990279i \(-0.544419\pi\)
−0.139094 + 0.990279i \(0.544419\pi\)
\(828\) 0 0
\(829\) 47.3050 1.64297 0.821484 0.570231i \(-0.193146\pi\)
0.821484 + 0.570231i \(0.193146\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) −91.3050 −3.16353
\(834\) 0 0
\(835\) −13.8885 −0.480633
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) −40.0000 −1.38095 −0.690477 0.723355i \(-0.742599\pi\)
−0.690477 + 0.723355i \(0.742599\pi\)
\(840\) 0 0
\(841\) −28.4164 −0.979876
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) −12.4164 −0.427137
\(846\) 0 0
\(847\) −49.5967 −1.70416
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) −8.00000 −0.274236
\(852\) 0 0
\(853\) 42.7214 1.46275 0.731376 0.681975i \(-0.238879\pi\)
0.731376 + 0.681975i \(0.238879\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −24.4721 −0.835952 −0.417976 0.908458i \(-0.637260\pi\)
−0.417976 + 0.908458i \(0.637260\pi\)
\(858\) 0 0
\(859\) 10.8328 0.369611 0.184805 0.982775i \(-0.440834\pi\)
0.184805 + 0.982775i \(0.440834\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) −46.8328 −1.59421 −0.797104 0.603842i \(-0.793636\pi\)
−0.797104 + 0.603842i \(0.793636\pi\)
\(864\) 0 0
\(865\) 22.0000 0.748022
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 0 0
\(870\) 0 0
\(871\) 1.16718 0.0395485
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 5.23607 0.177011
\(876\) 0 0
\(877\) 37.7082 1.27332 0.636658 0.771146i \(-0.280316\pi\)
0.636658 + 0.771146i \(0.280316\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 55.3050 1.86327 0.931636 0.363394i \(-0.118382\pi\)
0.931636 + 0.363394i \(0.118382\pi\)
\(882\) 0 0
\(883\) 25.5967 0.861399 0.430700 0.902495i \(-0.358267\pi\)
0.430700 + 0.902495i \(0.358267\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 25.8885 0.869252 0.434626 0.900611i \(-0.356881\pi\)
0.434626 + 0.900611i \(0.356881\pi\)
\(888\) 0 0
\(889\) 104.721 3.51224
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) −2.47214 −0.0827269
\(894\) 0 0
\(895\) 15.4164 0.515314
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 6.83282 0.227887
\(900\) 0 0
\(901\) −2.11146 −0.0703428
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 8.47214 0.281623
\(906\) 0 0
\(907\) 10.1115 0.335745 0.167873 0.985809i \(-0.446310\pi\)
0.167873 + 0.985809i \(0.446310\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 43.7771 1.45040 0.725200 0.688538i \(-0.241747\pi\)
0.725200 + 0.688538i \(0.241747\pi\)
\(912\) 0 0
\(913\) −4.94427 −0.163632
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) −74.2492 −2.45193
\(918\) 0 0
\(919\) −24.0000 −0.791687 −0.395843 0.918318i \(-0.629548\pi\)
−0.395843 + 0.918318i \(0.629548\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) 3.23607 0.106401
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) −27.8885 −0.914993 −0.457497 0.889211i \(-0.651254\pi\)
−0.457497 + 0.889211i \(0.651254\pi\)
\(930\) 0 0
\(931\) 20.4164 0.669121
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) −5.52786 −0.180780
\(936\) 0 0
\(937\) −37.4164 −1.22234 −0.611170 0.791499i \(-0.709301\pi\)
−0.611170 + 0.791499i \(0.709301\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 17.1246 0.558246 0.279123 0.960255i \(-0.409956\pi\)
0.279123 + 0.960255i \(0.409956\pi\)
\(942\) 0 0
\(943\) −24.0000 −0.781548
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −4.00000 −0.129983 −0.0649913 0.997886i \(-0.520702\pi\)
−0.0649913 + 0.997886i \(0.520702\pi\)
\(948\) 0 0
\(949\) 9.52786 0.309288
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) −42.9443 −1.39110 −0.695551 0.718477i \(-0.744840\pi\)
−0.695551 + 0.718477i \(0.744840\pi\)
\(954\) 0 0
\(955\) 6.18034 0.199991
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) −83.1935 −2.68646
\(960\) 0 0
\(961\) 49.0000 1.58065
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 5.70820 0.183754
\(966\) 0 0
\(967\) −18.1803 −0.584640 −0.292320 0.956321i \(-0.594427\pi\)
−0.292320 + 0.956321i \(0.594427\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 38.8328 1.24620 0.623102 0.782140i \(-0.285872\pi\)
0.623102 + 0.782140i \(0.285872\pi\)
\(972\) 0 0
\(973\) −75.7771 −2.42930
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −30.5836 −0.978456 −0.489228 0.872156i \(-0.662721\pi\)
−0.489228 + 0.872156i \(0.662721\pi\)
\(978\) 0 0
\(979\) 2.11146 0.0674824
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) −4.00000 −0.127580 −0.0637901 0.997963i \(-0.520319\pi\)
−0.0637901 + 0.997963i \(0.520319\pi\)
\(984\) 0 0
\(985\) 12.4721 0.397395
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) −12.9443 −0.411604
\(990\) 0 0
\(991\) 28.9443 0.919445 0.459723 0.888063i \(-0.347949\pi\)
0.459723 + 0.888063i \(0.347949\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 18.4721 0.585606
\(996\) 0 0
\(997\) 1.41641 0.0448581 0.0224290 0.999748i \(-0.492860\pi\)
0.0224290 + 0.999748i \(0.492860\pi\)
\(998\) 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.bd.1.2 2
3.2 odd 2 2280.2.a.p.1.2 2
12.11 even 2 4560.2.a.be.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2280.2.a.p.1.2 2 3.2 odd 2
4560.2.a.be.1.1 2 12.11 even 2
6840.2.a.bd.1.2 2 1.1 even 1 trivial