Properties

Label 2800.2.a.bq.1.1
Level $2800$
Weight $2$
Character 2800.1
Self dual yes
Analytic conductor $22.358$
Analytic rank $1$
Dimension $3$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 2800 = 2^{4} \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2800.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(22.3581125660\)
Analytic rank: \(1\)
Dimension: \(3\)
Coefficient field: 3.3.568.1
Defining polynomial: \(x^{3} - x^{2} - 6 x - 2\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 280)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(3.12489\) of defining polynomial
Character \(\chi\) \(=\) 2800.1

$q$-expansion

\(f(q)\) \(=\) \(q-3.12489 q^{3} +1.00000 q^{7} +6.76491 q^{9} +O(q^{10})\) \(q-3.12489 q^{3} +1.00000 q^{7} +6.76491 q^{9} -2.48486 q^{11} -4.15516 q^{13} +5.76491 q^{17} +1.60975 q^{19} -3.12489 q^{21} -7.28005 q^{23} -11.7649 q^{27} +1.45459 q^{29} +2.24977 q^{31} +7.76491 q^{33} -6.00000 q^{37} +12.9844 q^{39} +11.2800 q^{41} +5.28005 q^{43} -3.45459 q^{47} +1.00000 q^{49} -18.0147 q^{51} -9.21949 q^{53} -5.03028 q^{57} +5.92007 q^{59} +5.35998 q^{61} +6.76491 q^{63} +7.52982 q^{67} +22.7493 q^{69} +4.24977 q^{71} +7.28005 q^{73} -2.48486 q^{77} -16.9844 q^{79} +16.4693 q^{81} +10.1093 q^{83} -4.54541 q^{87} -11.4693 q^{89} -4.15516 q^{91} -7.03028 q^{93} -2.73463 q^{97} -16.8099 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3q - q^{3} + 3q^{7} + 4q^{9} + O(q^{10}) \) \( 3q - q^{3} + 3q^{7} + 4q^{9} - 7q^{11} - 5q^{13} + q^{17} - 4q^{19} - q^{21} - 6q^{23} - 19q^{27} + 3q^{29} - 10q^{31} + 7q^{33} - 18q^{37} + 5q^{39} + 18q^{41} - 9q^{47} + 3q^{49} - 21q^{51} - 10q^{53} - 16q^{57} - 6q^{59} + 24q^{61} + 4q^{63} - 10q^{67} + 18q^{69} - 4q^{71} + 6q^{73} - 7q^{77} - 17q^{79} + 15q^{81} - 12q^{83} - 15q^{87} - 5q^{91} - 22q^{93} + 9q^{97} - 2q^{99} + O(q^{100}) \)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) −3.12489 −1.80415 −0.902077 0.431576i \(-0.857958\pi\)
−0.902077 + 0.431576i \(0.857958\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) 1.00000 0.377964
\(8\) 0 0
\(9\) 6.76491 2.25497
\(10\) 0 0
\(11\) −2.48486 −0.749214 −0.374607 0.927184i \(-0.622222\pi\)
−0.374607 + 0.927184i \(0.622222\pi\)
\(12\) 0 0
\(13\) −4.15516 −1.15243 −0.576217 0.817297i \(-0.695472\pi\)
−0.576217 + 0.817297i \(0.695472\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 5.76491 1.39820 0.699098 0.715026i \(-0.253585\pi\)
0.699098 + 0.715026i \(0.253585\pi\)
\(18\) 0 0
\(19\) 1.60975 0.369301 0.184651 0.982804i \(-0.440885\pi\)
0.184651 + 0.982804i \(0.440885\pi\)
\(20\) 0 0
\(21\) −3.12489 −0.681906
\(22\) 0 0
\(23\) −7.28005 −1.51799 −0.758997 0.651094i \(-0.774310\pi\)
−0.758997 + 0.651094i \(0.774310\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) −11.7649 −2.26416
\(28\) 0 0
\(29\) 1.45459 0.270110 0.135055 0.990838i \(-0.456879\pi\)
0.135055 + 0.990838i \(0.456879\pi\)
\(30\) 0 0
\(31\) 2.24977 0.404071 0.202035 0.979378i \(-0.435244\pi\)
0.202035 + 0.979378i \(0.435244\pi\)
\(32\) 0 0
\(33\) 7.76491 1.35170
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −6.00000 −0.986394 −0.493197 0.869918i \(-0.664172\pi\)
−0.493197 + 0.869918i \(0.664172\pi\)
\(38\) 0 0
\(39\) 12.9844 2.07917
\(40\) 0 0
\(41\) 11.2800 1.76165 0.880824 0.473444i \(-0.156990\pi\)
0.880824 + 0.473444i \(0.156990\pi\)
\(42\) 0 0
\(43\) 5.28005 0.805200 0.402600 0.915376i \(-0.368107\pi\)
0.402600 + 0.915376i \(0.368107\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −3.45459 −0.503903 −0.251952 0.967740i \(-0.581072\pi\)
−0.251952 + 0.967740i \(0.581072\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) −18.0147 −2.52256
\(52\) 0 0
\(53\) −9.21949 −1.26639 −0.633197 0.773990i \(-0.718258\pi\)
−0.633197 + 0.773990i \(0.718258\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) −5.03028 −0.666276
\(58\) 0 0
\(59\) 5.92007 0.770728 0.385364 0.922765i \(-0.374076\pi\)
0.385364 + 0.922765i \(0.374076\pi\)
\(60\) 0 0
\(61\) 5.35998 0.686275 0.343137 0.939285i \(-0.388510\pi\)
0.343137 + 0.939285i \(0.388510\pi\)
\(62\) 0 0
\(63\) 6.76491 0.852298
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 7.52982 0.919914 0.459957 0.887941i \(-0.347865\pi\)
0.459957 + 0.887941i \(0.347865\pi\)
\(68\) 0 0
\(69\) 22.7493 2.73870
\(70\) 0 0
\(71\) 4.24977 0.504355 0.252178 0.967681i \(-0.418853\pi\)
0.252178 + 0.967681i \(0.418853\pi\)
\(72\) 0 0
\(73\) 7.28005 0.852065 0.426033 0.904708i \(-0.359911\pi\)
0.426033 + 0.904708i \(0.359911\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −2.48486 −0.283176
\(78\) 0 0
\(79\) −16.9844 −1.91089 −0.955447 0.295162i \(-0.904627\pi\)
−0.955447 + 0.295162i \(0.904627\pi\)
\(80\) 0 0
\(81\) 16.4693 1.82992
\(82\) 0 0
\(83\) 10.1093 1.10964 0.554819 0.831971i \(-0.312787\pi\)
0.554819 + 0.831971i \(0.312787\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) −4.54541 −0.487320
\(88\) 0 0
\(89\) −11.4693 −1.21574 −0.607870 0.794037i \(-0.707976\pi\)
−0.607870 + 0.794037i \(0.707976\pi\)
\(90\) 0 0
\(91\) −4.15516 −0.435579
\(92\) 0 0
\(93\) −7.03028 −0.729006
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −2.73463 −0.277660 −0.138830 0.990316i \(-0.544334\pi\)
−0.138830 + 0.990316i \(0.544334\pi\)
\(98\) 0 0
\(99\) −16.8099 −1.68945
\(100\) 0 0
\(101\) −4.57947 −0.455674 −0.227837 0.973699i \(-0.573165\pi\)
−0.227837 + 0.973699i \(0.573165\pi\)
\(102\) 0 0
\(103\) 2.48486 0.244841 0.122420 0.992478i \(-0.460934\pi\)
0.122420 + 0.992478i \(0.460934\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(108\) 0 0
\(109\) −9.95413 −0.953433 −0.476716 0.879057i \(-0.658173\pi\)
−0.476716 + 0.879057i \(0.658173\pi\)
\(110\) 0 0
\(111\) 18.7493 1.77961
\(112\) 0 0
\(113\) −18.4995 −1.74029 −0.870145 0.492795i \(-0.835975\pi\)
−0.870145 + 0.492795i \(0.835975\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) −28.1093 −2.59870
\(118\) 0 0
\(119\) 5.76491 0.528468
\(120\) 0 0
\(121\) −4.82546 −0.438678
\(122\) 0 0
\(123\) −35.2489 −3.17828
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) 7.15894 0.635253 0.317627 0.948216i \(-0.397114\pi\)
0.317627 + 0.948216i \(0.397114\pi\)
\(128\) 0 0
\(129\) −16.4995 −1.45270
\(130\) 0 0
\(131\) −7.85952 −0.686689 −0.343345 0.939209i \(-0.611560\pi\)
−0.343345 + 0.939209i \(0.611560\pi\)
\(132\) 0 0
\(133\) 1.60975 0.139583
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −10.5601 −0.902210 −0.451105 0.892471i \(-0.648970\pi\)
−0.451105 + 0.892471i \(0.648970\pi\)
\(138\) 0 0
\(139\) −9.79897 −0.831137 −0.415569 0.909562i \(-0.636417\pi\)
−0.415569 + 0.909562i \(0.636417\pi\)
\(140\) 0 0
\(141\) 10.7952 0.909119
\(142\) 0 0
\(143\) 10.3250 0.863420
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) −3.12489 −0.257736
\(148\) 0 0
\(149\) 2.49954 0.204770 0.102385 0.994745i \(-0.467353\pi\)
0.102385 + 0.994745i \(0.467353\pi\)
\(150\) 0 0
\(151\) 3.76491 0.306384 0.153192 0.988196i \(-0.451045\pi\)
0.153192 + 0.988196i \(0.451045\pi\)
\(152\) 0 0
\(153\) 38.9991 3.15289
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) −5.67030 −0.452539 −0.226270 0.974065i \(-0.572653\pi\)
−0.226270 + 0.974065i \(0.572653\pi\)
\(158\) 0 0
\(159\) 28.8099 2.28477
\(160\) 0 0
\(161\) −7.28005 −0.573748
\(162\) 0 0
\(163\) 10.2498 0.802824 0.401412 0.915898i \(-0.368520\pi\)
0.401412 + 0.915898i \(0.368520\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −2.95504 −0.228668 −0.114334 0.993442i \(-0.536473\pi\)
−0.114334 + 0.993442i \(0.536473\pi\)
\(168\) 0 0
\(169\) 4.26537 0.328105
\(170\) 0 0
\(171\) 10.8898 0.832763
\(172\) 0 0
\(173\) −22.4049 −1.70342 −0.851708 0.524017i \(-0.824433\pi\)
−0.851708 + 0.524017i \(0.824433\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) −18.4995 −1.39051
\(178\) 0 0
\(179\) −14.5601 −1.08827 −0.544136 0.838997i \(-0.683143\pi\)
−0.544136 + 0.838997i \(0.683143\pi\)
\(180\) 0 0
\(181\) −9.67030 −0.718788 −0.359394 0.933186i \(-0.617017\pi\)
−0.359394 + 0.933186i \(0.617017\pi\)
\(182\) 0 0
\(183\) −16.7493 −1.23814
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) −14.3250 −1.04755
\(188\) 0 0
\(189\) −11.7649 −0.855771
\(190\) 0 0
\(191\) −8.01468 −0.579922 −0.289961 0.957038i \(-0.593642\pi\)
−0.289961 + 0.957038i \(0.593642\pi\)
\(192\) 0 0
\(193\) −3.46927 −0.249723 −0.124862 0.992174i \(-0.539849\pi\)
−0.124862 + 0.992174i \(0.539849\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 16.2791 1.15984 0.579920 0.814673i \(-0.303084\pi\)
0.579920 + 0.814673i \(0.303084\pi\)
\(198\) 0 0
\(199\) −26.8704 −1.90479 −0.952397 0.304862i \(-0.901390\pi\)
−0.952397 + 0.304862i \(0.901390\pi\)
\(200\) 0 0
\(201\) −23.5298 −1.65967
\(202\) 0 0
\(203\) 1.45459 0.102092
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) −49.2489 −3.42303
\(208\) 0 0
\(209\) −4.00000 −0.276686
\(210\) 0 0
\(211\) −17.2342 −1.18645 −0.593225 0.805037i \(-0.702145\pi\)
−0.593225 + 0.805037i \(0.702145\pi\)
\(212\) 0 0
\(213\) −13.2800 −0.909934
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 2.24977 0.152724
\(218\) 0 0
\(219\) −22.7493 −1.53726
\(220\) 0 0
\(221\) −23.9541 −1.61133
\(222\) 0 0
\(223\) −0.235091 −0.0157429 −0.00787143 0.999969i \(-0.502506\pi\)
−0.00787143 + 0.999969i \(0.502506\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) −0.564792 −0.0374865 −0.0187433 0.999824i \(-0.505967\pi\)
−0.0187433 + 0.999824i \(0.505967\pi\)
\(228\) 0 0
\(229\) 7.11021 0.469856 0.234928 0.972013i \(-0.424515\pi\)
0.234928 + 0.972013i \(0.424515\pi\)
\(230\) 0 0
\(231\) 7.76491 0.510893
\(232\) 0 0
\(233\) −8.49954 −0.556823 −0.278412 0.960462i \(-0.589808\pi\)
−0.278412 + 0.960462i \(0.589808\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 53.0743 3.44755
\(238\) 0 0
\(239\) 7.26537 0.469958 0.234979 0.972001i \(-0.424498\pi\)
0.234979 + 0.972001i \(0.424498\pi\)
\(240\) 0 0
\(241\) 7.28005 0.468949 0.234475 0.972122i \(-0.424663\pi\)
0.234475 + 0.972122i \(0.424663\pi\)
\(242\) 0 0
\(243\) −16.1698 −1.03730
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) −6.68876 −0.425596
\(248\) 0 0
\(249\) −31.5904 −2.00196
\(250\) 0 0
\(251\) −22.9192 −1.44664 −0.723322 0.690511i \(-0.757386\pi\)
−0.723322 + 0.690511i \(0.757386\pi\)
\(252\) 0 0
\(253\) 18.0899 1.13730
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −0.719953 −0.0449094 −0.0224547 0.999748i \(-0.507148\pi\)
−0.0224547 + 0.999748i \(0.507148\pi\)
\(258\) 0 0
\(259\) −6.00000 −0.372822
\(260\) 0 0
\(261\) 9.84014 0.609089
\(262\) 0 0
\(263\) −18.5601 −1.14446 −0.572232 0.820092i \(-0.693922\pi\)
−0.572232 + 0.820092i \(0.693922\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 35.8401 2.19338
\(268\) 0 0
\(269\) −22.1698 −1.35172 −0.675860 0.737030i \(-0.736227\pi\)
−0.675860 + 0.737030i \(0.736227\pi\)
\(270\) 0 0
\(271\) 7.87890 0.478609 0.239304 0.970945i \(-0.423081\pi\)
0.239304 + 0.970945i \(0.423081\pi\)
\(272\) 0 0
\(273\) 12.9844 0.785852
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) 2.78051 0.167064 0.0835322 0.996505i \(-0.473380\pi\)
0.0835322 + 0.996505i \(0.473380\pi\)
\(278\) 0 0
\(279\) 15.2195 0.911167
\(280\) 0 0
\(281\) −23.7044 −1.41408 −0.707042 0.707172i \(-0.749971\pi\)
−0.707042 + 0.707172i \(0.749971\pi\)
\(282\) 0 0
\(283\) 26.8439 1.59571 0.797853 0.602852i \(-0.205969\pi\)
0.797853 + 0.602852i \(0.205969\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 11.2800 0.665840
\(288\) 0 0
\(289\) 16.2342 0.954951
\(290\) 0 0
\(291\) 8.54541 0.500941
\(292\) 0 0
\(293\) 9.93475 0.580394 0.290197 0.956967i \(-0.406279\pi\)
0.290197 + 0.956967i \(0.406279\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 29.2342 1.69634
\(298\) 0 0
\(299\) 30.2498 1.74939
\(300\) 0 0
\(301\) 5.28005 0.304337
\(302\) 0 0
\(303\) 14.3103 0.822107
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) −32.0946 −1.83174 −0.915868 0.401479i \(-0.868496\pi\)
−0.915868 + 0.401479i \(0.868496\pi\)
\(308\) 0 0
\(309\) −7.76491 −0.441730
\(310\) 0 0
\(311\) 10.0606 0.570482 0.285241 0.958456i \(-0.407926\pi\)
0.285241 + 0.958456i \(0.407926\pi\)
\(312\) 0 0
\(313\) −20.8245 −1.17707 −0.588536 0.808471i \(-0.700296\pi\)
−0.588536 + 0.808471i \(0.700296\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −15.7502 −0.884621 −0.442311 0.896862i \(-0.645841\pi\)
−0.442311 + 0.896862i \(0.645841\pi\)
\(318\) 0 0
\(319\) −3.61445 −0.202370
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 9.28005 0.516356
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) 31.1055 1.72014
\(328\) 0 0
\(329\) −3.45459 −0.190457
\(330\) 0 0
\(331\) 25.7796 1.41697 0.708487 0.705724i \(-0.249378\pi\)
0.708487 + 0.705724i \(0.249378\pi\)
\(332\) 0 0
\(333\) −40.5895 −2.22429
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) −0.0605522 −0.00329849 −0.00164924 0.999999i \(-0.500525\pi\)
−0.00164924 + 0.999999i \(0.500525\pi\)
\(338\) 0 0
\(339\) 57.8089 3.13975
\(340\) 0 0
\(341\) −5.59037 −0.302736
\(342\) 0 0
\(343\) 1.00000 0.0539949
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −20.9310 −1.12363 −0.561817 0.827262i \(-0.689897\pi\)
−0.561817 + 0.827262i \(0.689897\pi\)
\(348\) 0 0
\(349\) 5.48108 0.293396 0.146698 0.989181i \(-0.453136\pi\)
0.146698 + 0.989181i \(0.453136\pi\)
\(350\) 0 0
\(351\) 48.8851 2.60929
\(352\) 0 0
\(353\) 26.9239 1.43301 0.716506 0.697581i \(-0.245740\pi\)
0.716506 + 0.697581i \(0.245740\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) −18.0147 −0.953438
\(358\) 0 0
\(359\) −6.06055 −0.319864 −0.159932 0.987128i \(-0.551127\pi\)
−0.159932 + 0.987128i \(0.551127\pi\)
\(360\) 0 0
\(361\) −16.4087 −0.863616
\(362\) 0 0
\(363\) 15.0790 0.791443
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 30.7034 1.60271 0.801353 0.598191i \(-0.204114\pi\)
0.801353 + 0.598191i \(0.204114\pi\)
\(368\) 0 0
\(369\) 76.3085 3.97246
\(370\) 0 0
\(371\) −9.21949 −0.478652
\(372\) 0 0
\(373\) 10.8099 0.559714 0.279857 0.960042i \(-0.409713\pi\)
0.279857 + 0.960042i \(0.409713\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −6.04404 −0.311284
\(378\) 0 0
\(379\) 9.28005 0.476684 0.238342 0.971181i \(-0.423396\pi\)
0.238342 + 0.971181i \(0.423396\pi\)
\(380\) 0 0
\(381\) −22.3709 −1.14609
\(382\) 0 0
\(383\) −5.28005 −0.269798 −0.134899 0.990859i \(-0.543071\pi\)
−0.134899 + 0.990859i \(0.543071\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 35.7190 1.81570
\(388\) 0 0
\(389\) 5.48395 0.278047 0.139024 0.990289i \(-0.455604\pi\)
0.139024 + 0.990289i \(0.455604\pi\)
\(390\) 0 0
\(391\) −41.9688 −2.12245
\(392\) 0 0
\(393\) 24.5601 1.23889
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) −6.40493 −0.321454 −0.160727 0.986999i \(-0.551384\pi\)
−0.160727 + 0.986999i \(0.551384\pi\)
\(398\) 0 0
\(399\) −5.03028 −0.251829
\(400\) 0 0
\(401\) −1.20482 −0.0601656 −0.0300828 0.999547i \(-0.509577\pi\)
−0.0300828 + 0.999547i \(0.509577\pi\)
\(402\) 0 0
\(403\) −9.34816 −0.465665
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 14.9092 0.739020
\(408\) 0 0
\(409\) −2.31032 −0.114238 −0.0571191 0.998367i \(-0.518191\pi\)
−0.0571191 + 0.998367i \(0.518191\pi\)
\(410\) 0 0
\(411\) 32.9991 1.62772
\(412\) 0 0
\(413\) 5.92007 0.291308
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 30.6206 1.49950
\(418\) 0 0
\(419\) 19.7384 0.964285 0.482142 0.876093i \(-0.339859\pi\)
0.482142 + 0.876093i \(0.339859\pi\)
\(420\) 0 0
\(421\) 30.1433 1.46910 0.734548 0.678556i \(-0.237394\pi\)
0.734548 + 0.678556i \(0.237394\pi\)
\(422\) 0 0
\(423\) −23.3700 −1.13629
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 5.35998 0.259387
\(428\) 0 0
\(429\) −32.2645 −1.55774
\(430\) 0 0
\(431\) 4.61353 0.222226 0.111113 0.993808i \(-0.464558\pi\)
0.111113 + 0.993808i \(0.464558\pi\)
\(432\) 0 0
\(433\) 11.4399 0.549767 0.274883 0.961478i \(-0.411361\pi\)
0.274883 + 0.961478i \(0.411361\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −11.7190 −0.560598
\(438\) 0 0
\(439\) −12.0294 −0.574130 −0.287065 0.957911i \(-0.592680\pi\)
−0.287065 + 0.957911i \(0.592680\pi\)
\(440\) 0 0
\(441\) 6.76491 0.322139
\(442\) 0 0
\(443\) −14.4390 −0.686017 −0.343009 0.939332i \(-0.611446\pi\)
−0.343009 + 0.939332i \(0.611446\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) −7.81078 −0.369437
\(448\) 0 0
\(449\) 19.2342 0.907717 0.453858 0.891074i \(-0.350047\pi\)
0.453858 + 0.891074i \(0.350047\pi\)
\(450\) 0 0
\(451\) −28.0294 −1.31985
\(452\) 0 0
\(453\) −11.7649 −0.552764
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −34.8780 −1.63152 −0.815762 0.578388i \(-0.803682\pi\)
−0.815762 + 0.578388i \(0.803682\pi\)
\(458\) 0 0
\(459\) −67.8236 −3.16574
\(460\) 0 0
\(461\) −15.2607 −0.710760 −0.355380 0.934722i \(-0.615649\pi\)
−0.355380 + 0.934722i \(0.615649\pi\)
\(462\) 0 0
\(463\) 24.9991 1.16181 0.580903 0.813973i \(-0.302700\pi\)
0.580903 + 0.813973i \(0.302700\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 5.06433 0.234349 0.117175 0.993111i \(-0.462616\pi\)
0.117175 + 0.993111i \(0.462616\pi\)
\(468\) 0 0
\(469\) 7.52982 0.347695
\(470\) 0 0
\(471\) 17.7190 0.816450
\(472\) 0 0
\(473\) −13.1202 −0.603267
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) −62.3690 −2.85568
\(478\) 0 0
\(479\) −41.8089 −1.91030 −0.955150 0.296123i \(-0.904306\pi\)
−0.955150 + 0.296123i \(0.904306\pi\)
\(480\) 0 0
\(481\) 24.9310 1.13675
\(482\) 0 0
\(483\) 22.7493 1.03513
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) 21.3406 0.967035 0.483517 0.875335i \(-0.339359\pi\)
0.483517 + 0.875335i \(0.339359\pi\)
\(488\) 0 0
\(489\) −32.0294 −1.44842
\(490\) 0 0
\(491\) −8.35620 −0.377110 −0.188555 0.982063i \(-0.560380\pi\)
−0.188555 + 0.982063i \(0.560380\pi\)
\(492\) 0 0
\(493\) 8.38555 0.377666
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 4.24977 0.190628
\(498\) 0 0
\(499\) 38.8539 1.73934 0.869670 0.493634i \(-0.164332\pi\)
0.869670 + 0.493634i \(0.164332\pi\)
\(500\) 0 0
\(501\) 9.23417 0.412552
\(502\) 0 0
\(503\) −17.7044 −0.789398 −0.394699 0.918810i \(-0.629151\pi\)
−0.394699 + 0.918810i \(0.629151\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) −13.3288 −0.591952
\(508\) 0 0
\(509\) 7.11021 0.315154 0.157577 0.987507i \(-0.449632\pi\)
0.157577 + 0.987507i \(0.449632\pi\)
\(510\) 0 0
\(511\) 7.28005 0.322050
\(512\) 0 0
\(513\) −18.9385 −0.836157
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 8.58417 0.377531
\(518\) 0 0
\(519\) 70.0128 3.07322
\(520\) 0 0
\(521\) 18.1892 0.796884 0.398442 0.917194i \(-0.369551\pi\)
0.398442 + 0.917194i \(0.369551\pi\)
\(522\) 0 0
\(523\) −13.9882 −0.611661 −0.305830 0.952086i \(-0.598934\pi\)
−0.305830 + 0.952086i \(0.598934\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 12.9697 0.564970
\(528\) 0 0
\(529\) 29.9991 1.30431
\(530\) 0 0
\(531\) 40.0487 1.73797
\(532\) 0 0
\(533\) −46.8704 −2.03018
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 45.4986 1.96341
\(538\) 0 0
\(539\) −2.48486 −0.107031
\(540\) 0 0
\(541\) −20.8245 −0.895317 −0.447659 0.894205i \(-0.647742\pi\)
−0.447659 + 0.894205i \(0.647742\pi\)
\(542\) 0 0
\(543\) 30.2186 1.29680
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) −3.09839 −0.132478 −0.0662388 0.997804i \(-0.521100\pi\)
−0.0662388 + 0.997804i \(0.521100\pi\)
\(548\) 0 0
\(549\) 36.2598 1.54753
\(550\) 0 0
\(551\) 2.34152 0.0997519
\(552\) 0 0
\(553\) −16.9844 −0.722250
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −18.8099 −0.797000 −0.398500 0.917168i \(-0.630469\pi\)
−0.398500 + 0.917168i \(0.630469\pi\)
\(558\) 0 0
\(559\) −21.9394 −0.927940
\(560\) 0 0
\(561\) 44.7640 1.88994
\(562\) 0 0
\(563\) −28.9503 −1.22011 −0.610056 0.792358i \(-0.708853\pi\)
−0.610056 + 0.792358i \(0.708853\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 16.4693 0.691644
\(568\) 0 0
\(569\) −29.9688 −1.25636 −0.628179 0.778069i \(-0.716199\pi\)
−0.628179 + 0.778069i \(0.716199\pi\)
\(570\) 0 0
\(571\) −0.280964 −0.0117580 −0.00587898 0.999983i \(-0.501871\pi\)
−0.00587898 + 0.999983i \(0.501871\pi\)
\(572\) 0 0
\(573\) 25.0450 1.04627
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) 25.9541 1.08048 0.540242 0.841510i \(-0.318333\pi\)
0.540242 + 0.841510i \(0.318333\pi\)
\(578\) 0 0
\(579\) 10.8411 0.450539
\(580\) 0 0
\(581\) 10.1093 0.419404
\(582\) 0 0
\(583\) 22.9092 0.948801
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 22.2304 0.917547 0.458773 0.888553i \(-0.348289\pi\)
0.458773 + 0.888553i \(0.348289\pi\)
\(588\) 0 0
\(589\) 3.62156 0.149224
\(590\) 0 0
\(591\) −50.8704 −2.09253
\(592\) 0 0
\(593\) 35.0743 1.44033 0.720165 0.693803i \(-0.244066\pi\)
0.720165 + 0.693803i \(0.244066\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 83.9670 3.43654
\(598\) 0 0
\(599\) 21.7262 0.887707 0.443853 0.896099i \(-0.353611\pi\)
0.443853 + 0.896099i \(0.353611\pi\)
\(600\) 0 0
\(601\) −23.9688 −0.977708 −0.488854 0.872366i \(-0.662585\pi\)
−0.488854 + 0.872366i \(0.662585\pi\)
\(602\) 0 0
\(603\) 50.9385 2.07438
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) −43.3241 −1.75847 −0.879235 0.476388i \(-0.841946\pi\)
−0.879235 + 0.476388i \(0.841946\pi\)
\(608\) 0 0
\(609\) −4.54541 −0.184189
\(610\) 0 0
\(611\) 14.3544 0.580715
\(612\) 0 0
\(613\) −8.90917 −0.359838 −0.179919 0.983681i \(-0.557584\pi\)
−0.179919 + 0.983681i \(0.557584\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 27.2413 1.09669 0.548347 0.836251i \(-0.315257\pi\)
0.548347 + 0.836251i \(0.315257\pi\)
\(618\) 0 0
\(619\) −24.1405 −0.970288 −0.485144 0.874434i \(-0.661233\pi\)
−0.485144 + 0.874434i \(0.661233\pi\)
\(620\) 0 0
\(621\) 85.6491 3.43698
\(622\) 0 0
\(623\) −11.4693 −0.459506
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) 12.4995 0.499184
\(628\) 0 0
\(629\) −34.5895 −1.37917
\(630\) 0 0
\(631\) −8.01468 −0.319059 −0.159530 0.987193i \(-0.550998\pi\)
−0.159530 + 0.987193i \(0.550998\pi\)
\(632\) 0 0
\(633\) 53.8548 2.14054
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) −4.15516 −0.164633
\(638\) 0 0
\(639\) 28.7493 1.13731
\(640\) 0 0
\(641\) −49.1495 −1.94129 −0.970645 0.240516i \(-0.922683\pi\)
−0.970645 + 0.240516i \(0.922683\pi\)
\(642\) 0 0
\(643\) −7.24599 −0.285754 −0.142877 0.989740i \(-0.545635\pi\)
−0.142877 + 0.989740i \(0.545635\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −30.2791 −1.19040 −0.595198 0.803579i \(-0.702926\pi\)
−0.595198 + 0.803579i \(0.702926\pi\)
\(648\) 0 0
\(649\) −14.7106 −0.577440
\(650\) 0 0
\(651\) −7.03028 −0.275538
\(652\) 0 0
\(653\) 7.96881 0.311844 0.155922 0.987769i \(-0.450165\pi\)
0.155922 + 0.987769i \(0.450165\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 49.2489 1.92138
\(658\) 0 0
\(659\) −20.9239 −0.815078 −0.407539 0.913188i \(-0.633613\pi\)
−0.407539 + 0.913188i \(0.633613\pi\)
\(660\) 0 0
\(661\) 46.1992 1.79694 0.898470 0.439034i \(-0.144679\pi\)
0.898470 + 0.439034i \(0.144679\pi\)
\(662\) 0 0
\(663\) 74.8539 2.90708
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) −10.5895 −0.410025
\(668\) 0 0
\(669\) 0.734633 0.0284025
\(670\) 0 0
\(671\) −13.3188 −0.514167
\(672\) 0 0
\(673\) 40.3784 1.55647 0.778237 0.627970i \(-0.216114\pi\)
0.778237 + 0.627970i \(0.216114\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) −44.3737 −1.70542 −0.852711 0.522383i \(-0.825043\pi\)
−0.852711 + 0.522383i \(0.825043\pi\)
\(678\) 0 0
\(679\) −2.73463 −0.104946
\(680\) 0 0
\(681\) 1.76491 0.0676315
\(682\) 0 0
\(683\) −17.9394 −0.686434 −0.343217 0.939256i \(-0.611517\pi\)
−0.343217 + 0.939256i \(0.611517\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) −22.2186 −0.847692
\(688\) 0 0
\(689\) 38.3085 1.45944
\(690\) 0 0
\(691\) 12.7905 0.486573 0.243287 0.969954i \(-0.421774\pi\)
0.243287 + 0.969954i \(0.421774\pi\)
\(692\) 0 0
\(693\) −16.8099 −0.638554
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 65.0284 2.46313
\(698\) 0 0
\(699\) 26.5601 1.00460
\(700\) 0 0
\(701\) −19.4234 −0.733611 −0.366806 0.930298i \(-0.619549\pi\)
−0.366806 + 0.930298i \(0.619549\pi\)
\(702\) 0 0
\(703\) −9.65848 −0.364277
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) −4.57947 −0.172229
\(708\) 0 0
\(709\) −13.4839 −0.506400 −0.253200 0.967414i \(-0.581483\pi\)
−0.253200 + 0.967414i \(0.581483\pi\)
\(710\) 0 0
\(711\) −114.898 −4.30901
\(712\) 0 0
\(713\) −16.3784 −0.613377
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) −22.7034 −0.847876
\(718\) 0 0
\(719\) −15.3700 −0.573203 −0.286601 0.958050i \(-0.592526\pi\)
−0.286601 + 0.958050i \(0.592526\pi\)
\(720\) 0 0
\(721\) 2.48486 0.0925411
\(722\) 0 0
\(723\) −22.7493 −0.846056
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) 13.1589 0.488038 0.244019 0.969770i \(-0.421534\pi\)
0.244019 + 0.969770i \(0.421534\pi\)
\(728\) 0 0
\(729\) 1.12110 0.0415224
\(730\) 0 0
\(731\) 30.4390 1.12583
\(732\) 0 0
\(733\) 10.0265 0.370337 0.185169 0.982707i \(-0.440717\pi\)
0.185169 + 0.982707i \(0.440717\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −18.7106 −0.689212
\(738\) 0 0
\(739\) −19.2266 −0.707262 −0.353631 0.935385i \(-0.615053\pi\)
−0.353631 + 0.935385i \(0.615053\pi\)
\(740\) 0 0
\(741\) 20.9016 0.767840
\(742\) 0 0
\(743\) −5.21949 −0.191485 −0.0957423 0.995406i \(-0.530522\pi\)
−0.0957423 + 0.995406i \(0.530522\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 68.3884 2.50220
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) −29.8548 −1.08942 −0.544709 0.838625i \(-0.683360\pi\)
−0.544709 + 0.838625i \(0.683360\pi\)
\(752\) 0 0
\(753\) 71.6197 2.60997
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 18.0294 0.655288 0.327644 0.944801i \(-0.393745\pi\)
0.327644 + 0.944801i \(0.393745\pi\)
\(758\) 0 0
\(759\) −56.5289 −2.05187
\(760\) 0 0
\(761\) 8.22041 0.297990 0.148995 0.988838i \(-0.452396\pi\)
0.148995 + 0.988838i \(0.452396\pi\)
\(762\) 0 0
\(763\) −9.95413 −0.360364
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) −24.5988 −0.888213
\(768\) 0 0
\(769\) 50.6888 1.82788 0.913942 0.405845i \(-0.133023\pi\)
0.913942 + 0.405845i \(0.133023\pi\)
\(770\) 0 0
\(771\) 2.24977 0.0810235
\(772\) 0 0
\(773\) −6.99622 −0.251637 −0.125818 0.992053i \(-0.540156\pi\)
−0.125818 + 0.992053i \(0.540156\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 18.7493 0.672628
\(778\) 0 0
\(779\) 18.1580 0.650579
\(780\) 0 0
\(781\) −10.5601 −0.377870
\(782\) 0 0
\(783\) −17.1131 −0.611571
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) −14.3444 −0.511322 −0.255661 0.966767i \(-0.582293\pi\)
−0.255661 + 0.966767i \(0.582293\pi\)
\(788\) 0 0
\(789\) 57.9982 2.06479
\(790\) 0 0
\(791\) −18.4995 −0.657768
\(792\) 0 0
\(793\) −22.2716 −0.790887
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 13.9348 0.493594 0.246797 0.969067i \(-0.420622\pi\)
0.246797 + 0.969067i \(0.420622\pi\)
\(798\) 0 0
\(799\) −19.9154 −0.704555
\(800\) 0 0
\(801\) −77.5885 −2.74146
\(802\) 0 0
\(803\) −18.0899 −0.638379
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 69.2782 2.43871
\(808\) 0 0
\(809\) −5.58325 −0.196297 −0.0981483 0.995172i \(-0.531292\pi\)
−0.0981483 + 0.995172i \(0.531292\pi\)
\(810\) 0 0
\(811\) 6.57947 0.231036 0.115518 0.993305i \(-0.463147\pi\)
0.115518 + 0.993305i \(0.463147\pi\)
\(812\) 0 0
\(813\) −24.6206 −0.863484
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 8.49954 0.297361
\(818\) 0 0
\(819\) −28.1093 −0.982218
\(820\) 0 0
\(821\) 16.3250 0.569747 0.284873 0.958565i \(-0.408048\pi\)
0.284873 + 0.958565i \(0.408048\pi\)
\(822\) 0 0
\(823\) 12.3491 0.430462 0.215231 0.976563i \(-0.430950\pi\)
0.215231 + 0.976563i \(0.430950\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 6.40963 0.222885 0.111442 0.993771i \(-0.464453\pi\)
0.111442 + 0.993771i \(0.464453\pi\)
\(828\) 0 0
\(829\) −26.1698 −0.908916 −0.454458 0.890768i \(-0.650167\pi\)
−0.454458 + 0.890768i \(0.650167\pi\)
\(830\) 0 0
\(831\) −8.68876 −0.301410
\(832\) 0 0
\(833\) 5.76491 0.199742
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) −26.4683 −0.914880
\(838\) 0 0
\(839\) 1.12958 0.0389975 0.0194988 0.999810i \(-0.493793\pi\)
0.0194988 + 0.999810i \(0.493793\pi\)
\(840\) 0 0
\(841\) −26.8842 −0.927041
\(842\) 0 0
\(843\) 74.0734 2.55122
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) −4.82546 −0.165805
\(848\) 0 0
\(849\) −83.8842 −2.87890
\(850\) 0 0
\(851\) 43.6803 1.49734
\(852\) 0 0
\(853\) −6.29095 −0.215398 −0.107699 0.994184i \(-0.534348\pi\)
−0.107699 + 0.994184i \(0.534348\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 25.0596 0.856021 0.428010 0.903774i \(-0.359215\pi\)
0.428010 + 0.903774i \(0.359215\pi\)
\(858\) 0 0
\(859\) −53.7896 −1.83528 −0.917638 0.397417i \(-0.869907\pi\)
−0.917638 + 0.397417i \(0.869907\pi\)
\(860\) 0 0
\(861\) −35.2489 −1.20128
\(862\) 0 0
\(863\) −58.0899 −1.97740 −0.988702 0.149896i \(-0.952106\pi\)
−0.988702 + 0.149896i \(0.952106\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) −50.7299 −1.72288
\(868\) 0 0
\(869\) 42.2039 1.43167
\(870\) 0 0
\(871\) −31.2876 −1.06014
\(872\) 0 0
\(873\) −18.4995 −0.626115
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 24.0899 0.813459 0.406729 0.913549i \(-0.366669\pi\)
0.406729 + 0.913549i \(0.366669\pi\)
\(878\) 0 0
\(879\) −31.0450 −1.04712
\(880\) 0 0
\(881\) 40.9679 1.38024 0.690122 0.723693i \(-0.257557\pi\)
0.690122 + 0.723693i \(0.257557\pi\)
\(882\) 0 0
\(883\) −43.8014 −1.47403 −0.737017 0.675874i \(-0.763766\pi\)
−0.737017 + 0.675874i \(0.763766\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) −1.77959 −0.0597527 −0.0298764 0.999554i \(-0.509511\pi\)
−0.0298764 + 0.999554i \(0.509511\pi\)
\(888\) 0 0
\(889\) 7.15894 0.240103
\(890\) 0 0
\(891\) −40.9239 −1.37100
\(892\) 0 0
\(893\) −5.56101 −0.186092
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) −94.5271 −3.15617
\(898\) 0 0
\(899\) 3.27248 0.109143
\(900\) 0 0
\(901\) −53.1495 −1.77067
\(902\) 0 0
\(903\) −16.4995 −0.549070
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) 53.0284 1.76078 0.880390 0.474250i \(-0.157281\pi\)
0.880390 + 0.474250i \(0.157281\pi\)
\(908\) 0 0
\(909\) −30.9797 −1.02753
\(910\) 0 0
\(911\) −28.1798 −0.933639 −0.466820 0.884353i \(-0.654600\pi\)
−0.466820 + 0.884353i \(0.654600\pi\)
\(912\) 0 0
\(913\) −25.1202 −0.831357
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) −7.85952 −0.259544
\(918\) 0 0
\(919\) −38.4243 −1.26750 −0.633751 0.773538i \(-0.718485\pi\)
−0.633751 + 0.773538i \(0.718485\pi\)
\(920\) 0 0
\(921\) 100.292 3.30473
\(922\) 0 0
\(923\) −17.6585 −0.581236
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) 16.8099 0.552108
\(928\) 0 0
\(929\) −36.6576 −1.20270 −0.601348 0.798987i \(-0.705369\pi\)
−0.601348 + 0.798987i \(0.705369\pi\)
\(930\) 0 0
\(931\) 1.60975 0.0527573
\(932\) 0 0
\(933\) −31.4381 −1.02924
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) −5.83302 −0.190557 −0.0952783 0.995451i \(-0.530374\pi\)
−0.0952783 + 0.995451i \(0.530374\pi\)
\(938\) 0 0
\(939\) 65.0743 2.12362
\(940\) 0 0
\(941\) −2.23796 −0.0729553 −0.0364776 0.999334i \(-0.511614\pi\)
−0.0364776 + 0.999334i \(0.511614\pi\)
\(942\) 0 0
\(943\) −82.1193 −2.67417
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −17.7502 −0.576805 −0.288402 0.957509i \(-0.593124\pi\)
−0.288402 + 0.957509i \(0.593124\pi\)
\(948\) 0 0
\(949\) −30.2498 −0.981949
\(950\) 0 0
\(951\) 49.2177 1.59599
\(952\) 0 0
\(953\) 32.0294 1.03753 0.518766 0.854916i \(-0.326391\pi\)
0.518766 + 0.854916i \(0.326391\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 11.2947 0.365107
\(958\) 0 0
\(959\) −10.5601 −0.341003
\(960\) 0 0
\(961\) −25.9385 −0.836727
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) 43.4305 1.39663 0.698316 0.715790i \(-0.253933\pi\)
0.698316 + 0.715790i \(0.253933\pi\)
\(968\) 0 0
\(969\) −28.9991 −0.931585
\(970\) 0 0
\(971\) 1.79897 0.0577316 0.0288658 0.999583i \(-0.490810\pi\)
0.0288658 + 0.999583i \(0.490810\pi\)
\(972\) 0 0
\(973\) −9.79897 −0.314140
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 22.4390 0.717887 0.358943 0.933359i \(-0.383137\pi\)
0.358943 + 0.933359i \(0.383137\pi\)
\(978\) 0 0
\(979\) 28.4995 0.910849
\(980\) 0 0
\(981\) −67.3388 −2.14996
\(982\) 0 0
\(983\) 37.6950 1.20228 0.601141 0.799143i \(-0.294713\pi\)
0.601141 + 0.799143i \(0.294713\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 10.7952 0.343615
\(988\) 0 0
\(989\) −38.4390 −1.22229
\(990\) 0 0
\(991\) −13.5686 −0.431020 −0.215510 0.976502i \(-0.569141\pi\)
−0.215510 + 0.976502i \(0.569141\pi\)
\(992\) 0 0
\(993\) −80.5583 −2.55644
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) −14.2838 −0.452373 −0.226187 0.974084i \(-0.572626\pi\)
−0.226187 + 0.974084i \(0.572626\pi\)
\(998\) 0 0
\(999\) 70.5895 2.23335
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 2800.2.a.bq.1.1 3
4.3 odd 2 1400.2.a.t.1.3 3
5.2 odd 4 560.2.g.f.449.6 6
5.3 odd 4 560.2.g.f.449.1 6
5.4 even 2 2800.2.a.br.1.3 3
15.2 even 4 5040.2.t.y.1009.2 6
15.8 even 4 5040.2.t.y.1009.1 6
20.3 even 4 280.2.g.b.169.6 yes 6
20.7 even 4 280.2.g.b.169.1 6
20.19 odd 2 1400.2.a.s.1.1 3
28.27 even 2 9800.2.a.cd.1.1 3
40.3 even 4 2240.2.g.l.449.1 6
40.13 odd 4 2240.2.g.m.449.6 6
40.27 even 4 2240.2.g.l.449.6 6
40.37 odd 4 2240.2.g.m.449.1 6
60.23 odd 4 2520.2.t.g.1009.1 6
60.47 odd 4 2520.2.t.g.1009.2 6
140.27 odd 4 1960.2.g.c.1569.6 6
140.83 odd 4 1960.2.g.c.1569.1 6
140.139 even 2 9800.2.a.cg.1.3 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
280.2.g.b.169.1 6 20.7 even 4
280.2.g.b.169.6 yes 6 20.3 even 4
560.2.g.f.449.1 6 5.3 odd 4
560.2.g.f.449.6 6 5.2 odd 4
1400.2.a.s.1.1 3 20.19 odd 2
1400.2.a.t.1.3 3 4.3 odd 2
1960.2.g.c.1569.1 6 140.83 odd 4
1960.2.g.c.1569.6 6 140.27 odd 4
2240.2.g.l.449.1 6 40.3 even 4
2240.2.g.l.449.6 6 40.27 even 4
2240.2.g.m.449.1 6 40.37 odd 4
2240.2.g.m.449.6 6 40.13 odd 4
2520.2.t.g.1009.1 6 60.23 odd 4
2520.2.t.g.1009.2 6 60.47 odd 4
2800.2.a.bq.1.1 3 1.1 even 1 trivial
2800.2.a.br.1.3 3 5.4 even 2
5040.2.t.y.1009.1 6 15.8 even 4
5040.2.t.y.1009.2 6 15.2 even 4
9800.2.a.cd.1.1 3 28.27 even 2
9800.2.a.cg.1.3 3 140.139 even 2