Properties

Label 4200.2.a.bp.1.1
Level $4200$
Weight $2$
Character 4200.1
Self dual yes
Analytic conductor $33.537$
Analytic rank $1$
Dimension $3$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 4200 = 2^{3} \cdot 3 \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4200.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(33.5371688489\)
Analytic rank: \(1\)
Dimension: \(3\)
Coefficient field: 3.3.148.1
Defining polynomial: \(x^{3} - x^{2} - 3 x + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 840)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(0.311108\) of defining polynomial
Character \(\chi\) \(=\) 4200.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{3} +1.00000 q^{7} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{3} +1.00000 q^{7} +1.00000 q^{9} -5.05086 q^{11} -3.37778 q^{13} -7.18421 q^{17} +8.23506 q^{19} +1.00000 q^{21} +6.23506 q^{23} +1.00000 q^{27} -2.00000 q^{29} -4.62222 q^{31} -5.05086 q^{33} +4.85728 q^{37} -3.37778 q^{39} -3.37778 q^{41} -1.24443 q^{43} +1.00000 q^{49} -7.18421 q^{51} -4.62222 q^{53} +8.23506 q^{57} -11.6128 q^{59} +0.488863 q^{61} +1.00000 q^{63} -3.61285 q^{67} +6.23506 q^{69} -10.2953 q^{71} -16.2351 q^{73} -5.05086 q^{77} +1.24443 q^{79} +1.00000 q^{81} -11.6128 q^{83} -2.00000 q^{87} +6.99063 q^{89} -3.37778 q^{91} -4.62222 q^{93} +8.23506 q^{97} -5.05086 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 3 q^{3} + 3 q^{7} + 3 q^{9} + O(q^{10}) \) \( 3 q + 3 q^{3} + 3 q^{7} + 3 q^{9} - 2 q^{11} - 10 q^{13} - 8 q^{17} - 2 q^{19} + 3 q^{21} - 8 q^{23} + 3 q^{27} - 6 q^{29} - 14 q^{31} - 2 q^{33} - 12 q^{37} - 10 q^{39} - 10 q^{41} - 4 q^{43} + 3 q^{49} - 8 q^{51} - 14 q^{53} - 2 q^{57} - 8 q^{59} + 2 q^{61} + 3 q^{63} + 16 q^{67} - 8 q^{69} - 18 q^{71} - 22 q^{73} - 2 q^{77} + 4 q^{79} + 3 q^{81} - 8 q^{83} - 6 q^{87} - 6 q^{89} - 10 q^{91} - 14 q^{93} - 2 q^{97} - 2 q^{99} + O(q^{100}) \)

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
\(3\) 1.00000 0.577350
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) 1.00000 0.377964
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) −5.05086 −1.52289 −0.761445 0.648229i \(-0.775510\pi\)
−0.761445 + 0.648229i \(0.775510\pi\)
\(12\) 0 0
\(13\) −3.37778 −0.936829 −0.468414 0.883509i \(-0.655175\pi\)
−0.468414 + 0.883509i \(0.655175\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −7.18421 −1.74243 −0.871213 0.490905i \(-0.836666\pi\)
−0.871213 + 0.490905i \(0.836666\pi\)
\(18\) 0 0
\(19\) 8.23506 1.88925 0.944627 0.328147i \(-0.106424\pi\)
0.944627 + 0.328147i \(0.106424\pi\)
\(20\) 0 0
\(21\) 1.00000 0.218218
\(22\) 0 0
\(23\) 6.23506 1.30010 0.650050 0.759891i \(-0.274748\pi\)
0.650050 + 0.759891i \(0.274748\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 1.00000 0.192450
\(28\) 0 0
\(29\) −2.00000 −0.371391 −0.185695 0.982607i \(-0.559454\pi\)
−0.185695 + 0.982607i \(0.559454\pi\)
\(30\) 0 0
\(31\) −4.62222 −0.830174 −0.415087 0.909782i \(-0.636249\pi\)
−0.415087 + 0.909782i \(0.636249\pi\)
\(32\) 0 0
\(33\) −5.05086 −0.879241
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 4.85728 0.798532 0.399266 0.916835i \(-0.369265\pi\)
0.399266 + 0.916835i \(0.369265\pi\)
\(38\) 0 0
\(39\) −3.37778 −0.540878
\(40\) 0 0
\(41\) −3.37778 −0.527521 −0.263761 0.964588i \(-0.584963\pi\)
−0.263761 + 0.964588i \(0.584963\pi\)
\(42\) 0 0
\(43\) −1.24443 −0.189774 −0.0948870 0.995488i \(-0.530249\pi\)
−0.0948870 + 0.995488i \(0.530249\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) −7.18421 −1.00599
\(52\) 0 0
\(53\) −4.62222 −0.634910 −0.317455 0.948273i \(-0.602828\pi\)
−0.317455 + 0.948273i \(0.602828\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 8.23506 1.09076
\(58\) 0 0
\(59\) −11.6128 −1.51186 −0.755932 0.654650i \(-0.772816\pi\)
−0.755932 + 0.654650i \(0.772816\pi\)
\(60\) 0 0
\(61\) 0.488863 0.0625924 0.0312962 0.999510i \(-0.490036\pi\)
0.0312962 + 0.999510i \(0.490036\pi\)
\(62\) 0 0
\(63\) 1.00000 0.125988
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) −3.61285 −0.441380 −0.220690 0.975344i \(-0.570831\pi\)
−0.220690 + 0.975344i \(0.570831\pi\)
\(68\) 0 0
\(69\) 6.23506 0.750613
\(70\) 0 0
\(71\) −10.2953 −1.22183 −0.610913 0.791698i \(-0.709197\pi\)
−0.610913 + 0.791698i \(0.709197\pi\)
\(72\) 0 0
\(73\) −16.2351 −1.90017 −0.950085 0.311991i \(-0.899004\pi\)
−0.950085 + 0.311991i \(0.899004\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −5.05086 −0.575598
\(78\) 0 0
\(79\) 1.24443 0.140009 0.0700047 0.997547i \(-0.477699\pi\)
0.0700047 + 0.997547i \(0.477699\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) −11.6128 −1.27468 −0.637338 0.770585i \(-0.719964\pi\)
−0.637338 + 0.770585i \(0.719964\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) −2.00000 −0.214423
\(88\) 0 0
\(89\) 6.99063 0.741006 0.370503 0.928831i \(-0.379185\pi\)
0.370503 + 0.928831i \(0.379185\pi\)
\(90\) 0 0
\(91\) −3.37778 −0.354088
\(92\) 0 0
\(93\) −4.62222 −0.479301
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 8.23506 0.836144 0.418072 0.908414i \(-0.362706\pi\)
0.418072 + 0.908414i \(0.362706\pi\)
\(98\) 0 0
\(99\) −5.05086 −0.507630
\(100\) 0 0
\(101\) −9.47949 −0.943245 −0.471622 0.881801i \(-0.656331\pi\)
−0.471622 + 0.881801i \(0.656331\pi\)
\(102\) 0 0
\(103\) −16.8573 −1.66100 −0.830499 0.557021i \(-0.811944\pi\)
−0.830499 + 0.557021i \(0.811944\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −15.4795 −1.49646 −0.748230 0.663440i \(-0.769096\pi\)
−0.748230 + 0.663440i \(0.769096\pi\)
\(108\) 0 0
\(109\) −1.61285 −0.154483 −0.0772414 0.997012i \(-0.524611\pi\)
−0.0772414 + 0.997012i \(0.524611\pi\)
\(110\) 0 0
\(111\) 4.85728 0.461033
\(112\) 0 0
\(113\) −1.86665 −0.175599 −0.0877997 0.996138i \(-0.527984\pi\)
−0.0877997 + 0.996138i \(0.527984\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) −3.37778 −0.312276
\(118\) 0 0
\(119\) −7.18421 −0.658575
\(120\) 0 0
\(121\) 14.5111 1.31919
\(122\) 0 0
\(123\) −3.37778 −0.304565
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) −12.8573 −1.14090 −0.570450 0.821333i \(-0.693231\pi\)
−0.570450 + 0.821333i \(0.693231\pi\)
\(128\) 0 0
\(129\) −1.24443 −0.109566
\(130\) 0 0
\(131\) 21.7146 1.89721 0.948605 0.316463i \(-0.102495\pi\)
0.948605 + 0.316463i \(0.102495\pi\)
\(132\) 0 0
\(133\) 8.23506 0.714071
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −9.47949 −0.809888 −0.404944 0.914342i \(-0.632709\pi\)
−0.404944 + 0.914342i \(0.632709\pi\)
\(138\) 0 0
\(139\) 10.1334 0.859500 0.429750 0.902948i \(-0.358602\pi\)
0.429750 + 0.902948i \(0.358602\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 17.0607 1.42669
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 1.00000 0.0824786
\(148\) 0 0
\(149\) 7.24443 0.593487 0.296743 0.954957i \(-0.404099\pi\)
0.296743 + 0.954957i \(0.404099\pi\)
\(150\) 0 0
\(151\) −8.85728 −0.720795 −0.360398 0.932799i \(-0.617359\pi\)
−0.360398 + 0.932799i \(0.617359\pi\)
\(152\) 0 0
\(153\) −7.18421 −0.580809
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) −13.4795 −1.07578 −0.537890 0.843015i \(-0.680779\pi\)
−0.537890 + 0.843015i \(0.680779\pi\)
\(158\) 0 0
\(159\) −4.62222 −0.366566
\(160\) 0 0
\(161\) 6.23506 0.491392
\(162\) 0 0
\(163\) −4.00000 −0.313304 −0.156652 0.987654i \(-0.550070\pi\)
−0.156652 + 0.987654i \(0.550070\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −2.10171 −0.162635 −0.0813176 0.996688i \(-0.525913\pi\)
−0.0813176 + 0.996688i \(0.525913\pi\)
\(168\) 0 0
\(169\) −1.59057 −0.122352
\(170\) 0 0
\(171\) 8.23506 0.629751
\(172\) 0 0
\(173\) −11.1842 −0.850320 −0.425160 0.905118i \(-0.639782\pi\)
−0.425160 + 0.905118i \(0.639782\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) −11.6128 −0.872875
\(178\) 0 0
\(179\) 20.6637 1.54448 0.772239 0.635332i \(-0.219137\pi\)
0.772239 + 0.635332i \(0.219137\pi\)
\(180\) 0 0
\(181\) 24.9590 1.85519 0.927594 0.373591i \(-0.121874\pi\)
0.927594 + 0.373591i \(0.121874\pi\)
\(182\) 0 0
\(183\) 0.488863 0.0361378
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 36.2864 2.65352
\(188\) 0 0
\(189\) 1.00000 0.0727393
\(190\) 0 0
\(191\) 9.52098 0.688914 0.344457 0.938802i \(-0.388063\pi\)
0.344457 + 0.938802i \(0.388063\pi\)
\(192\) 0 0
\(193\) 22.9590 1.65262 0.826312 0.563212i \(-0.190435\pi\)
0.826312 + 0.563212i \(0.190435\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 19.8479 1.41411 0.707053 0.707161i \(-0.250024\pi\)
0.707053 + 0.707161i \(0.250024\pi\)
\(198\) 0 0
\(199\) −8.23506 −0.583768 −0.291884 0.956454i \(-0.594282\pi\)
−0.291884 + 0.956454i \(0.594282\pi\)
\(200\) 0 0
\(201\) −3.61285 −0.254831
\(202\) 0 0
\(203\) −2.00000 −0.140372
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 6.23506 0.433367
\(208\) 0 0
\(209\) −41.5941 −2.87712
\(210\) 0 0
\(211\) −11.6128 −0.799461 −0.399731 0.916633i \(-0.630896\pi\)
−0.399731 + 0.916633i \(0.630896\pi\)
\(212\) 0 0
\(213\) −10.2953 −0.705421
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) −4.62222 −0.313776
\(218\) 0 0
\(219\) −16.2351 −1.09706
\(220\) 0 0
\(221\) 24.2667 1.63236
\(222\) 0 0
\(223\) −2.48886 −0.166667 −0.0833333 0.996522i \(-0.526557\pi\)
−0.0833333 + 0.996522i \(0.526557\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 0.857279 0.0568996 0.0284498 0.999595i \(-0.490943\pi\)
0.0284498 + 0.999595i \(0.490943\pi\)
\(228\) 0 0
\(229\) −14.4701 −0.956213 −0.478106 0.878302i \(-0.658677\pi\)
−0.478106 + 0.878302i \(0.658677\pi\)
\(230\) 0 0
\(231\) −5.05086 −0.332322
\(232\) 0 0
\(233\) 3.96836 0.259976 0.129988 0.991516i \(-0.458506\pi\)
0.129988 + 0.991516i \(0.458506\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 1.24443 0.0808345
\(238\) 0 0
\(239\) 1.90813 0.123427 0.0617135 0.998094i \(-0.480344\pi\)
0.0617135 + 0.998094i \(0.480344\pi\)
\(240\) 0 0
\(241\) 0.755569 0.0486705 0.0243352 0.999704i \(-0.492253\pi\)
0.0243352 + 0.999704i \(0.492253\pi\)
\(242\) 0 0
\(243\) 1.00000 0.0641500
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) −27.8163 −1.76991
\(248\) 0 0
\(249\) −11.6128 −0.735934
\(250\) 0 0
\(251\) −9.12399 −0.575901 −0.287950 0.957645i \(-0.592974\pi\)
−0.287950 + 0.957645i \(0.592974\pi\)
\(252\) 0 0
\(253\) −31.4924 −1.97991
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 1.28592 0.0802134 0.0401067 0.999195i \(-0.487230\pi\)
0.0401067 + 0.999195i \(0.487230\pi\)
\(258\) 0 0
\(259\) 4.85728 0.301817
\(260\) 0 0
\(261\) −2.00000 −0.123797
\(262\) 0 0
\(263\) −7.00937 −0.432216 −0.216108 0.976369i \(-0.569336\pi\)
−0.216108 + 0.976369i \(0.569336\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 6.99063 0.427820
\(268\) 0 0
\(269\) 23.1941 1.41417 0.707083 0.707130i \(-0.250011\pi\)
0.707083 + 0.707130i \(0.250011\pi\)
\(270\) 0 0
\(271\) 7.11108 0.431967 0.215984 0.976397i \(-0.430704\pi\)
0.215984 + 0.976397i \(0.430704\pi\)
\(272\) 0 0
\(273\) −3.37778 −0.204433
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) −7.34614 −0.441387 −0.220693 0.975343i \(-0.570832\pi\)
−0.220693 + 0.975343i \(0.570832\pi\)
\(278\) 0 0
\(279\) −4.62222 −0.276725
\(280\) 0 0
\(281\) −19.9813 −1.19198 −0.595991 0.802991i \(-0.703241\pi\)
−0.595991 + 0.802991i \(0.703241\pi\)
\(282\) 0 0
\(283\) −4.85728 −0.288735 −0.144368 0.989524i \(-0.546115\pi\)
−0.144368 + 0.989524i \(0.546115\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) −3.37778 −0.199384
\(288\) 0 0
\(289\) 34.6128 2.03605
\(290\) 0 0
\(291\) 8.23506 0.482748
\(292\) 0 0
\(293\) 11.1842 0.653388 0.326694 0.945130i \(-0.394065\pi\)
0.326694 + 0.945130i \(0.394065\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) −5.05086 −0.293080
\(298\) 0 0
\(299\) −21.0607 −1.21797
\(300\) 0 0
\(301\) −1.24443 −0.0717278
\(302\) 0 0
\(303\) −9.47949 −0.544583
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) 23.3461 1.33243 0.666217 0.745758i \(-0.267912\pi\)
0.666217 + 0.745758i \(0.267912\pi\)
\(308\) 0 0
\(309\) −16.8573 −0.958977
\(310\) 0 0
\(311\) −32.8573 −1.86317 −0.931583 0.363530i \(-0.881572\pi\)
−0.931583 + 0.363530i \(0.881572\pi\)
\(312\) 0 0
\(313\) −28.8256 −1.62932 −0.814661 0.579938i \(-0.803077\pi\)
−0.814661 + 0.579938i \(0.803077\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −25.3590 −1.42431 −0.712153 0.702024i \(-0.752280\pi\)
−0.712153 + 0.702024i \(0.752280\pi\)
\(318\) 0 0
\(319\) 10.1017 0.565587
\(320\) 0 0
\(321\) −15.4795 −0.863981
\(322\) 0 0
\(323\) −59.1624 −3.29188
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) −1.61285 −0.0891907
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) 1.89829 0.104339 0.0521697 0.998638i \(-0.483386\pi\)
0.0521697 + 0.998638i \(0.483386\pi\)
\(332\) 0 0
\(333\) 4.85728 0.266177
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) −23.2257 −1.26518 −0.632592 0.774485i \(-0.718009\pi\)
−0.632592 + 0.774485i \(0.718009\pi\)
\(338\) 0 0
\(339\) −1.86665 −0.101382
\(340\) 0 0
\(341\) 23.3461 1.26426
\(342\) 0 0
\(343\) 1.00000 0.0539949
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −7.47949 −0.401520 −0.200760 0.979640i \(-0.564341\pi\)
−0.200760 + 0.979640i \(0.564341\pi\)
\(348\) 0 0
\(349\) 29.2257 1.56442 0.782208 0.623018i \(-0.214094\pi\)
0.782208 + 0.623018i \(0.214094\pi\)
\(350\) 0 0
\(351\) −3.37778 −0.180293
\(352\) 0 0
\(353\) 30.4099 1.61856 0.809278 0.587426i \(-0.199859\pi\)
0.809278 + 0.587426i \(0.199859\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) −7.18421 −0.380229
\(358\) 0 0
\(359\) −16.1936 −0.854664 −0.427332 0.904095i \(-0.640546\pi\)
−0.427332 + 0.904095i \(0.640546\pi\)
\(360\) 0 0
\(361\) 48.8163 2.56928
\(362\) 0 0
\(363\) 14.5111 0.761637
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 5.51114 0.287679 0.143840 0.989601i \(-0.454055\pi\)
0.143840 + 0.989601i \(0.454055\pi\)
\(368\) 0 0
\(369\) −3.37778 −0.175840
\(370\) 0 0
\(371\) −4.62222 −0.239973
\(372\) 0 0
\(373\) −1.63158 −0.0844802 −0.0422401 0.999107i \(-0.513449\pi\)
−0.0422401 + 0.999107i \(0.513449\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 6.75557 0.347929
\(378\) 0 0
\(379\) 20.8573 1.07137 0.535683 0.844419i \(-0.320054\pi\)
0.535683 + 0.844419i \(0.320054\pi\)
\(380\) 0 0
\(381\) −12.8573 −0.658698
\(382\) 0 0
\(383\) −20.0830 −1.02619 −0.513096 0.858331i \(-0.671502\pi\)
−0.513096 + 0.858331i \(0.671502\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) −1.24443 −0.0632580
\(388\) 0 0
\(389\) 33.2257 1.68461 0.842305 0.539001i \(-0.181198\pi\)
0.842305 + 0.539001i \(0.181198\pi\)
\(390\) 0 0
\(391\) −44.7940 −2.26533
\(392\) 0 0
\(393\) 21.7146 1.09535
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) −22.7239 −1.14048 −0.570241 0.821478i \(-0.693150\pi\)
−0.570241 + 0.821478i \(0.693150\pi\)
\(398\) 0 0
\(399\) 8.23506 0.412269
\(400\) 0 0
\(401\) 12.7556 0.636983 0.318491 0.947926i \(-0.396824\pi\)
0.318491 + 0.947926i \(0.396824\pi\)
\(402\) 0 0
\(403\) 15.6128 0.777731
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) −24.5334 −1.21608
\(408\) 0 0
\(409\) 27.4479 1.35721 0.678604 0.734504i \(-0.262585\pi\)
0.678604 + 0.734504i \(0.262585\pi\)
\(410\) 0 0
\(411\) −9.47949 −0.467589
\(412\) 0 0
\(413\) −11.6128 −0.571431
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 10.1334 0.496232
\(418\) 0 0
\(419\) 2.36842 0.115705 0.0578524 0.998325i \(-0.481575\pi\)
0.0578524 + 0.998325i \(0.481575\pi\)
\(420\) 0 0
\(421\) 39.3274 1.91670 0.958350 0.285596i \(-0.0921914\pi\)
0.958350 + 0.285596i \(0.0921914\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 0.488863 0.0236577
\(428\) 0 0
\(429\) 17.0607 0.823698
\(430\) 0 0
\(431\) 8.19358 0.394671 0.197335 0.980336i \(-0.436771\pi\)
0.197335 + 0.980336i \(0.436771\pi\)
\(432\) 0 0
\(433\) 14.6035 0.701798 0.350899 0.936413i \(-0.385876\pi\)
0.350899 + 0.936413i \(0.385876\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 51.3461 2.45622
\(438\) 0 0
\(439\) 5.27607 0.251813 0.125907 0.992042i \(-0.459816\pi\)
0.125907 + 0.992042i \(0.459816\pi\)
\(440\) 0 0
\(441\) 1.00000 0.0476190
\(442\) 0 0
\(443\) −18.5018 −0.879046 −0.439523 0.898231i \(-0.644852\pi\)
−0.439523 + 0.898231i \(0.644852\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 7.24443 0.342650
\(448\) 0 0
\(449\) 11.7146 0.552844 0.276422 0.961036i \(-0.410851\pi\)
0.276422 + 0.961036i \(0.410851\pi\)
\(450\) 0 0
\(451\) 17.0607 0.803357
\(452\) 0 0
\(453\) −8.85728 −0.416151
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 18.9590 0.886864 0.443432 0.896308i \(-0.353761\pi\)
0.443432 + 0.896308i \(0.353761\pi\)
\(458\) 0 0
\(459\) −7.18421 −0.335330
\(460\) 0 0
\(461\) −15.1111 −0.703793 −0.351897 0.936039i \(-0.614463\pi\)
−0.351897 + 0.936039i \(0.614463\pi\)
\(462\) 0 0
\(463\) 22.5718 1.04900 0.524501 0.851410i \(-0.324252\pi\)
0.524501 + 0.851410i \(0.324252\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 22.3684 1.03509 0.517543 0.855657i \(-0.326847\pi\)
0.517543 + 0.855657i \(0.326847\pi\)
\(468\) 0 0
\(469\) −3.61285 −0.166826
\(470\) 0 0
\(471\) −13.4795 −0.621102
\(472\) 0 0
\(473\) 6.28544 0.289005
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) −4.62222 −0.211637
\(478\) 0 0
\(479\) −32.8573 −1.50129 −0.750644 0.660707i \(-0.770256\pi\)
−0.750644 + 0.660707i \(0.770256\pi\)
\(480\) 0 0
\(481\) −16.4068 −0.748088
\(482\) 0 0
\(483\) 6.23506 0.283705
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) −3.40943 −0.154496 −0.0772479 0.997012i \(-0.524613\pi\)
−0.0772479 + 0.997012i \(0.524613\pi\)
\(488\) 0 0
\(489\) −4.00000 −0.180886
\(490\) 0 0
\(491\) −33.7877 −1.52482 −0.762409 0.647096i \(-0.775983\pi\)
−0.762409 + 0.647096i \(0.775983\pi\)
\(492\) 0 0
\(493\) 14.3684 0.647121
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −10.2953 −0.461807
\(498\) 0 0
\(499\) 13.6316 0.610233 0.305117 0.952315i \(-0.401305\pi\)
0.305117 + 0.952315i \(0.401305\pi\)
\(500\) 0 0
\(501\) −2.10171 −0.0938975
\(502\) 0 0
\(503\) −7.34614 −0.327548 −0.163774 0.986498i \(-0.552367\pi\)
−0.163774 + 0.986498i \(0.552367\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) −1.59057 −0.0706398
\(508\) 0 0
\(509\) −15.9684 −0.707785 −0.353892 0.935286i \(-0.615142\pi\)
−0.353892 + 0.935286i \(0.615142\pi\)
\(510\) 0 0
\(511\) −16.2351 −0.718197
\(512\) 0 0
\(513\) 8.23506 0.363587
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) −11.1842 −0.490932
\(520\) 0 0
\(521\) −1.09234 −0.0478564 −0.0239282 0.999714i \(-0.507617\pi\)
−0.0239282 + 0.999714i \(0.507617\pi\)
\(522\) 0 0
\(523\) −17.5111 −0.765709 −0.382854 0.923809i \(-0.625059\pi\)
−0.382854 + 0.923809i \(0.625059\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 33.2070 1.44652
\(528\) 0 0
\(529\) 15.8760 0.690262
\(530\) 0 0
\(531\) −11.6128 −0.503955
\(532\) 0 0
\(533\) 11.4094 0.494197
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 20.6637 0.891705
\(538\) 0 0
\(539\) −5.05086 −0.217556
\(540\) 0 0
\(541\) −15.3274 −0.658977 −0.329488 0.944160i \(-0.606876\pi\)
−0.329488 + 0.944160i \(0.606876\pi\)
\(542\) 0 0
\(543\) 24.9590 1.07109
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 5.32741 0.227783 0.113892 0.993493i \(-0.463668\pi\)
0.113892 + 0.993493i \(0.463668\pi\)
\(548\) 0 0
\(549\) 0.488863 0.0208641
\(550\) 0 0
\(551\) −16.4701 −0.701651
\(552\) 0 0
\(553\) 1.24443 0.0529186
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 0.355509 0.0150634 0.00753171 0.999972i \(-0.497603\pi\)
0.00753171 + 0.999972i \(0.497603\pi\)
\(558\) 0 0
\(559\) 4.20342 0.177786
\(560\) 0 0
\(561\) 36.2864 1.53201
\(562\) 0 0
\(563\) −16.4701 −0.694133 −0.347067 0.937841i \(-0.612822\pi\)
−0.347067 + 0.937841i \(0.612822\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 1.00000 0.0419961
\(568\) 0 0
\(569\) −20.9590 −0.878647 −0.439323 0.898329i \(-0.644782\pi\)
−0.439323 + 0.898329i \(0.644782\pi\)
\(570\) 0 0
\(571\) 17.5111 0.732818 0.366409 0.930454i \(-0.380587\pi\)
0.366409 + 0.930454i \(0.380587\pi\)
\(572\) 0 0
\(573\) 9.52098 0.397745
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) 0.152089 0.00633155 0.00316577 0.999995i \(-0.498992\pi\)
0.00316577 + 0.999995i \(0.498992\pi\)
\(578\) 0 0
\(579\) 22.9590 0.954143
\(580\) 0 0
\(581\) −11.6128 −0.481782
\(582\) 0 0
\(583\) 23.3461 0.966898
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 34.1847 1.41095 0.705476 0.708733i \(-0.250733\pi\)
0.705476 + 0.708733i \(0.250733\pi\)
\(588\) 0 0
\(589\) −38.0642 −1.56841
\(590\) 0 0
\(591\) 19.8479 0.816434
\(592\) 0 0
\(593\) 19.3047 0.792747 0.396374 0.918089i \(-0.370269\pi\)
0.396374 + 0.918089i \(0.370269\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) −8.23506 −0.337039
\(598\) 0 0
\(599\) −25.9081 −1.05858 −0.529289 0.848442i \(-0.677541\pi\)
−0.529289 + 0.848442i \(0.677541\pi\)
\(600\) 0 0
\(601\) −22.7368 −0.927455 −0.463727 0.885978i \(-0.653488\pi\)
−0.463727 + 0.885978i \(0.653488\pi\)
\(602\) 0 0
\(603\) −3.61285 −0.147127
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) −32.9403 −1.33700 −0.668502 0.743711i \(-0.733064\pi\)
−0.668502 + 0.743711i \(0.733064\pi\)
\(608\) 0 0
\(609\) −2.00000 −0.0810441
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) −13.1240 −0.530073 −0.265036 0.964238i \(-0.585384\pi\)
−0.265036 + 0.964238i \(0.585384\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 18.2538 0.734870 0.367435 0.930049i \(-0.380236\pi\)
0.367435 + 0.930049i \(0.380236\pi\)
\(618\) 0 0
\(619\) 7.64449 0.307258 0.153629 0.988129i \(-0.450904\pi\)
0.153629 + 0.988129i \(0.450904\pi\)
\(620\) 0 0
\(621\) 6.23506 0.250204
\(622\) 0 0
\(623\) 6.99063 0.280074
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) −41.5941 −1.66111
\(628\) 0 0
\(629\) −34.8957 −1.39138
\(630\) 0 0
\(631\) 28.6735 1.14148 0.570738 0.821132i \(-0.306657\pi\)
0.570738 + 0.821132i \(0.306657\pi\)
\(632\) 0 0
\(633\) −11.6128 −0.461569
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) −3.37778 −0.133833
\(638\) 0 0
\(639\) −10.2953 −0.407275
\(640\) 0 0
\(641\) 15.4479 0.610153 0.305077 0.952328i \(-0.401318\pi\)
0.305077 + 0.952328i \(0.401318\pi\)
\(642\) 0 0
\(643\) 25.8350 1.01883 0.509417 0.860520i \(-0.329861\pi\)
0.509417 + 0.860520i \(0.329861\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −23.6128 −0.928317 −0.464158 0.885752i \(-0.653643\pi\)
−0.464158 + 0.885752i \(0.653643\pi\)
\(648\) 0 0
\(649\) 58.6548 2.30240
\(650\) 0 0
\(651\) −4.62222 −0.181159
\(652\) 0 0
\(653\) 36.7052 1.43639 0.718193 0.695844i \(-0.244970\pi\)
0.718193 + 0.695844i \(0.244970\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) −16.2351 −0.633390
\(658\) 0 0
\(659\) 37.9911 1.47992 0.739962 0.672649i \(-0.234844\pi\)
0.739962 + 0.672649i \(0.234844\pi\)
\(660\) 0 0
\(661\) −49.2257 −1.91466 −0.957329 0.289001i \(-0.906677\pi\)
−0.957329 + 0.289001i \(0.906677\pi\)
\(662\) 0 0
\(663\) 24.2667 0.942441
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) −12.4701 −0.482845
\(668\) 0 0
\(669\) −2.48886 −0.0962250
\(670\) 0 0
\(671\) −2.46917 −0.0953214
\(672\) 0 0
\(673\) −11.2257 −0.432719 −0.216359 0.976314i \(-0.569418\pi\)
−0.216359 + 0.976314i \(0.569418\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 9.55262 0.367137 0.183569 0.983007i \(-0.441235\pi\)
0.183569 + 0.983007i \(0.441235\pi\)
\(678\) 0 0
\(679\) 8.23506 0.316033
\(680\) 0 0
\(681\) 0.857279 0.0328510
\(682\) 0 0
\(683\) −17.9684 −0.687540 −0.343770 0.939054i \(-0.611704\pi\)
−0.343770 + 0.939054i \(0.611704\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) −14.4701 −0.552070
\(688\) 0 0
\(689\) 15.6128 0.594802
\(690\) 0 0
\(691\) 0.355509 0.0135242 0.00676211 0.999977i \(-0.497848\pi\)
0.00676211 + 0.999977i \(0.497848\pi\)
\(692\) 0 0
\(693\) −5.05086 −0.191866
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 24.2667 0.919167
\(698\) 0 0
\(699\) 3.96836 0.150097
\(700\) 0 0
\(701\) 13.2257 0.499528 0.249764 0.968307i \(-0.419647\pi\)
0.249764 + 0.968307i \(0.419647\pi\)
\(702\) 0 0
\(703\) 40.0000 1.50863
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) −9.47949 −0.356513
\(708\) 0 0
\(709\) −10.9777 −0.412277 −0.206139 0.978523i \(-0.566090\pi\)
−0.206139 + 0.978523i \(0.566090\pi\)
\(710\) 0 0
\(711\) 1.24443 0.0466698
\(712\) 0 0
\(713\) −28.8198 −1.07931
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 1.90813 0.0712606
\(718\) 0 0
\(719\) −21.9813 −0.819763 −0.409881 0.912139i \(-0.634430\pi\)
−0.409881 + 0.912139i \(0.634430\pi\)
\(720\) 0 0
\(721\) −16.8573 −0.627798
\(722\) 0 0
\(723\) 0.755569 0.0280999
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) −13.0607 −0.484395 −0.242197 0.970227i \(-0.577868\pi\)
−0.242197 + 0.970227i \(0.577868\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) 8.94025 0.330667
\(732\) 0 0
\(733\) 13.5625 0.500941 0.250471 0.968124i \(-0.419415\pi\)
0.250471 + 0.968124i \(0.419415\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 18.2480 0.672173
\(738\) 0 0
\(739\) 10.2854 0.378356 0.189178 0.981943i \(-0.439418\pi\)
0.189178 + 0.981943i \(0.439418\pi\)
\(740\) 0 0
\(741\) −27.8163 −1.02186
\(742\) 0 0
\(743\) 21.4608 0.787319 0.393659 0.919256i \(-0.371209\pi\)
0.393659 + 0.919256i \(0.371209\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) −11.6128 −0.424892
\(748\) 0 0
\(749\) −15.4795 −0.565608
\(750\) 0 0
\(751\) 46.0642 1.68091 0.840454 0.541883i \(-0.182288\pi\)
0.840454 + 0.541883i \(0.182288\pi\)
\(752\) 0 0
\(753\) −9.12399 −0.332497
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 27.1052 0.985157 0.492579 0.870268i \(-0.336054\pi\)
0.492579 + 0.870268i \(0.336054\pi\)
\(758\) 0 0
\(759\) −31.4924 −1.14310
\(760\) 0 0
\(761\) 39.4608 1.43045 0.715226 0.698894i \(-0.246324\pi\)
0.715226 + 0.698894i \(0.246324\pi\)
\(762\) 0 0
\(763\) −1.61285 −0.0583890
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 39.2257 1.41636
\(768\) 0 0
\(769\) −7.51114 −0.270859 −0.135429 0.990787i \(-0.543241\pi\)
−0.135429 + 0.990787i \(0.543241\pi\)
\(770\) 0 0
\(771\) 1.28592 0.0463112
\(772\) 0 0
\(773\) 1.46965 0.0528596 0.0264298 0.999651i \(-0.491586\pi\)
0.0264298 + 0.999651i \(0.491586\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 4.85728 0.174254
\(778\) 0 0
\(779\) −27.8163 −0.996621
\(780\) 0 0
\(781\) 52.0000 1.86071
\(782\) 0 0
\(783\) −2.00000 −0.0714742
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) −16.2034 −0.577590 −0.288795 0.957391i \(-0.593255\pi\)
−0.288795 + 0.957391i \(0.593255\pi\)
\(788\) 0 0
\(789\) −7.00937 −0.249540
\(790\) 0 0
\(791\) −1.86665 −0.0663703
\(792\) 0 0
\(793\) −1.65127 −0.0586384
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −0.161933 −0.00573597 −0.00286799 0.999996i \(-0.500913\pi\)
−0.00286799 + 0.999996i \(0.500913\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 0 0
\(801\) 6.99063 0.247002
\(802\) 0 0
\(803\) 82.0010 2.89375
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 23.1941 0.816469
\(808\) 0 0
\(809\) −42.9403 −1.50970 −0.754849 0.655898i \(-0.772290\pi\)
−0.754849 + 0.655898i \(0.772290\pi\)
\(810\) 0 0
\(811\) −41.2958 −1.45009 −0.725045 0.688701i \(-0.758181\pi\)
−0.725045 + 0.688701i \(0.758181\pi\)
\(812\) 0 0
\(813\) 7.11108 0.249396
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) −10.2480 −0.358531
\(818\) 0 0
\(819\) −3.37778 −0.118029
\(820\) 0 0
\(821\) −43.2070 −1.50793 −0.753967 0.656913i \(-0.771862\pi\)
−0.753967 + 0.656913i \(0.771862\pi\)
\(822\) 0 0
\(823\) 11.1427 0.388411 0.194205 0.980961i \(-0.437787\pi\)
0.194205 + 0.980961i \(0.437787\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −36.9906 −1.28629 −0.643145 0.765744i \(-0.722371\pi\)
−0.643145 + 0.765744i \(0.722371\pi\)
\(828\) 0 0
\(829\) −22.6735 −0.787485 −0.393742 0.919221i \(-0.628820\pi\)
−0.393742 + 0.919221i \(0.628820\pi\)
\(830\) 0 0
\(831\) −7.34614 −0.254835
\(832\) 0 0
\(833\) −7.18421 −0.248918
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) −4.62222 −0.159767
\(838\) 0 0
\(839\) −21.8983 −0.756013 −0.378006 0.925803i \(-0.623390\pi\)
−0.378006 + 0.925803i \(0.623390\pi\)
\(840\) 0 0
\(841\) −25.0000 −0.862069
\(842\) 0 0
\(843\) −19.9813 −0.688191
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) 14.5111 0.498609
\(848\) 0 0
\(849\) −4.85728 −0.166701
\(850\) 0 0
\(851\) 30.2854 1.03817
\(852\) 0 0
\(853\) −52.4010 −1.79418 −0.897088 0.441851i \(-0.854322\pi\)
−0.897088 + 0.441851i \(0.854322\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 24.5116 0.837301 0.418650 0.908147i \(-0.362503\pi\)
0.418650 + 0.908147i \(0.362503\pi\)
\(858\) 0 0
\(859\) 57.4795 1.96118 0.980588 0.196082i \(-0.0628218\pi\)
0.980588 + 0.196082i \(0.0628218\pi\)
\(860\) 0 0
\(861\) −3.37778 −0.115115
\(862\) 0 0
\(863\) −21.1941 −0.721454 −0.360727 0.932671i \(-0.617471\pi\)
−0.360727 + 0.932671i \(0.617471\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) 34.6128 1.17551
\(868\) 0 0
\(869\) −6.28544 −0.213219
\(870\) 0 0
\(871\) 12.2034 0.413497
\(872\) 0 0
\(873\) 8.23506 0.278715
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) −8.38715 −0.283214 −0.141607 0.989923i \(-0.545227\pi\)
−0.141607 + 0.989923i \(0.545227\pi\)
\(878\) 0 0
\(879\) 11.1842 0.377234
\(880\) 0 0
\(881\) 29.1753 0.982941 0.491471 0.870894i \(-0.336459\pi\)
0.491471 + 0.870894i \(0.336459\pi\)
\(882\) 0 0
\(883\) 19.8796 0.669000 0.334500 0.942396i \(-0.391433\pi\)
0.334500 + 0.942396i \(0.391433\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) −14.6923 −0.493319 −0.246659 0.969102i \(-0.579333\pi\)
−0.246659 + 0.969102i \(0.579333\pi\)
\(888\) 0 0
\(889\) −12.8573 −0.431219
\(890\) 0 0
\(891\) −5.05086 −0.169210
\(892\) 0 0
\(893\) 0 0
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) −21.0607 −0.703196
\(898\) 0 0
\(899\) 9.24443 0.308319
\(900\) 0 0
\(901\) 33.2070 1.10628
\(902\) 0 0
\(903\) −1.24443 −0.0414121
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) 19.8796 0.660090 0.330045 0.943965i \(-0.392936\pi\)
0.330045 + 0.943965i \(0.392936\pi\)
\(908\) 0 0
\(909\) −9.47949 −0.314415
\(910\) 0 0
\(911\) −53.3372 −1.76714 −0.883571 0.468297i \(-0.844868\pi\)
−0.883571 + 0.468297i \(0.844868\pi\)
\(912\) 0 0
\(913\) 58.6548 1.94119
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 21.7146 0.717078
\(918\) 0 0
\(919\) −21.1240 −0.696816 −0.348408 0.937343i \(-0.613278\pi\)
−0.348408 + 0.937343i \(0.613278\pi\)
\(920\) 0 0
\(921\) 23.3461 0.769282
\(922\) 0 0
\(923\) 34.7753 1.14464
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) −16.8573 −0.553666
\(928\) 0 0
\(929\) 2.72393 0.0893691 0.0446846 0.999001i \(-0.485772\pi\)
0.0446846 + 0.999001i \(0.485772\pi\)
\(930\) 0 0
\(931\) 8.23506 0.269893
\(932\) 0 0
\(933\) −32.8573 −1.07570
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 15.3145 0.500303 0.250151 0.968207i \(-0.419520\pi\)
0.250151 + 0.968207i \(0.419520\pi\)
\(938\) 0 0
\(939\) −28.8256 −0.940689
\(940\) 0 0
\(941\) −30.6035 −0.997645 −0.498822 0.866704i \(-0.666234\pi\)
−0.498822 + 0.866704i \(0.666234\pi\)
\(942\) 0 0
\(943\) −21.0607 −0.685831
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 2.74266 0.0891245 0.0445623 0.999007i \(-0.485811\pi\)
0.0445623 + 0.999007i \(0.485811\pi\)
\(948\) 0 0
\(949\) 54.8385 1.78013
\(950\) 0 0
\(951\) −25.3590 −0.822323
\(952\) 0 0
\(953\) 47.8479 1.54995 0.774973 0.631994i \(-0.217763\pi\)
0.774973 + 0.631994i \(0.217763\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 10.1017 0.326542
\(958\) 0 0
\(959\) −9.47949 −0.306109
\(960\) 0 0
\(961\) −9.63512 −0.310810
\(962\) 0 0
\(963\) −15.4795 −0.498820
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) 38.7181 1.24509 0.622545 0.782584i \(-0.286099\pi\)
0.622545 + 0.782584i \(0.286099\pi\)
\(968\) 0 0
\(969\) −59.1624 −1.90057
\(970\) 0 0
\(971\) 13.7778 0.442152 0.221076 0.975257i \(-0.429043\pi\)
0.221076 + 0.975257i \(0.429043\pi\)
\(972\) 0 0
\(973\) 10.1334 0.324860
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −4.03164 −0.128984 −0.0644918 0.997918i \(-0.520543\pi\)
−0.0644918 + 0.997918i \(0.520543\pi\)
\(978\) 0 0
\(979\) −35.3087 −1.12847
\(980\) 0 0
\(981\) −1.61285 −0.0514943
\(982\) 0 0
\(983\) −47.4924 −1.51477 −0.757386 0.652967i \(-0.773524\pi\)
−0.757386 + 0.652967i \(0.773524\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) −7.75911 −0.246725
\(990\) 0 0
\(991\) 1.16146 0.0368949 0.0184474 0.999830i \(-0.494128\pi\)
0.0184474 + 0.999830i \(0.494128\pi\)
\(992\) 0 0
\(993\) 1.89829 0.0602404
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) 18.4572 0.584546 0.292273 0.956335i \(-0.405588\pi\)
0.292273 + 0.956335i \(0.405588\pi\)
\(998\) 0 0
\(999\) 4.85728 0.153678
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 4200.2.a.bp.1.1 3
4.3 odd 2 8400.2.a.di.1.3 3
5.2 odd 4 840.2.t.d.169.2 6
5.3 odd 4 840.2.t.d.169.5 yes 6
5.4 even 2 4200.2.a.bn.1.1 3
15.2 even 4 2520.2.t.k.1009.3 6
15.8 even 4 2520.2.t.k.1009.4 6
20.3 even 4 1680.2.t.j.1009.2 6
20.7 even 4 1680.2.t.j.1009.5 6
20.19 odd 2 8400.2.a.dl.1.3 3
60.23 odd 4 5040.2.t.z.1009.4 6
60.47 odd 4 5040.2.t.z.1009.3 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
840.2.t.d.169.2 6 5.2 odd 4
840.2.t.d.169.5 yes 6 5.3 odd 4
1680.2.t.j.1009.2 6 20.3 even 4
1680.2.t.j.1009.5 6 20.7 even 4
2520.2.t.k.1009.3 6 15.2 even 4
2520.2.t.k.1009.4 6 15.8 even 4
4200.2.a.bn.1.1 3 5.4 even 2
4200.2.a.bp.1.1 3 1.1 even 1 trivial
5040.2.t.z.1009.3 6 60.47 odd 4
5040.2.t.z.1009.4 6 60.23 odd 4
8400.2.a.di.1.3 3 4.3 odd 2
8400.2.a.dl.1.3 3 20.19 odd 2