Properties

Label 4200.2.a.bo.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: \( x^{3} - x^{2} - 3x + 1 \) Copy content Toggle raw display
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

\(f(q)\) \(=\) \(q-1.00000 q^{3} -1.00000 q^{7} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{3} -1.00000 q^{7} +1.00000 q^{9} -3.80642 q^{11} -0.622216 q^{13} +4.42864 q^{17} +0.622216 q^{19} +1.00000 q^{21} -2.62222 q^{23} -1.00000 q^{27} +9.61285 q^{29} -0.622216 q^{31} +3.80642 q^{33} +1.24443 q^{37} +0.622216 q^{39} +4.62222 q^{41} +4.85728 q^{43} -11.6128 q^{47} +1.00000 q^{49} -4.42864 q^{51} -13.4795 q^{53} -0.622216 q^{57} +11.6128 q^{59} -8.10171 q^{61} -1.00000 q^{63} +2.62222 q^{69} +2.56199 q^{71} -10.9906 q^{73} +3.80642 q^{77} -6.75557 q^{79} +1.00000 q^{81} -11.6128 q^{83} -9.61285 q^{87} -8.23506 q^{89} +0.622216 q^{91} +0.622216 q^{93} -4.23506 q^{97} -3.80642 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 3 q^{3} - 3 q^{7} + 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q - 3 q^{3} - 3 q^{7} + 3 q^{9} + 2 q^{11} - 2 q^{13} + 2 q^{19} + 3 q^{21} - 8 q^{23} - 3 q^{27} + 2 q^{29} - 2 q^{31} - 2 q^{33} + 4 q^{37} + 2 q^{39} + 14 q^{41} - 12 q^{43} - 8 q^{47} + 3 q^{49} - 14 q^{53} - 2 q^{57} + 8 q^{59} + 2 q^{61} - 3 q^{63} + 8 q^{69} - 6 q^{71} - 6 q^{73} - 2 q^{77} - 20 q^{79} + 3 q^{81} - 8 q^{83} - 2 q^{87} + 2 q^{89} + 2 q^{91} + 2 q^{93} + 14 q^{97} + 2 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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
\(11\) −3.80642 −1.14768 −0.573840 0.818967i \(-0.694547\pi\)
−0.573840 + 0.818967i \(0.694547\pi\)
\(12\) 0 0
\(13\) −0.622216 −0.172572 −0.0862858 0.996270i \(-0.527500\pi\)
−0.0862858 + 0.996270i \(0.527500\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 4.42864 1.07410 0.537051 0.843550i \(-0.319538\pi\)
0.537051 + 0.843550i \(0.319538\pi\)
\(18\) 0 0
\(19\) 0.622216 0.142746 0.0713730 0.997450i \(-0.477262\pi\)
0.0713730 + 0.997450i \(0.477262\pi\)
\(20\) 0 0
\(21\) 1.00000 0.218218
\(22\) 0 0
\(23\) −2.62222 −0.546770 −0.273385 0.961905i \(-0.588143\pi\)
−0.273385 + 0.961905i \(0.588143\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) −1.00000 −0.192450
\(28\) 0 0
\(29\) 9.61285 1.78506 0.892531 0.450987i \(-0.148928\pi\)
0.892531 + 0.450987i \(0.148928\pi\)
\(30\) 0 0
\(31\) −0.622216 −0.111753 −0.0558766 0.998438i \(-0.517795\pi\)
−0.0558766 + 0.998438i \(0.517795\pi\)
\(32\) 0 0
\(33\) 3.80642 0.662613
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 1.24443 0.204583 0.102292 0.994754i \(-0.467383\pi\)
0.102292 + 0.994754i \(0.467383\pi\)
\(38\) 0 0
\(39\) 0.622216 0.0996342
\(40\) 0 0
\(41\) 4.62222 0.721869 0.360934 0.932591i \(-0.382458\pi\)
0.360934 + 0.932591i \(0.382458\pi\)
\(42\) 0 0
\(43\) 4.85728 0.740728 0.370364 0.928887i \(-0.379233\pi\)
0.370364 + 0.928887i \(0.379233\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −11.6128 −1.69391 −0.846954 0.531666i \(-0.821566\pi\)
−0.846954 + 0.531666i \(0.821566\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) −4.42864 −0.620134
\(52\) 0 0
\(53\) −13.4795 −1.85155 −0.925775 0.378074i \(-0.876587\pi\)
−0.925775 + 0.378074i \(0.876587\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) −0.622216 −0.0824145
\(58\) 0 0
\(59\) 11.6128 1.51186 0.755932 0.654650i \(-0.227184\pi\)
0.755932 + 0.654650i \(0.227184\pi\)
\(60\) 0 0
\(61\) −8.10171 −1.03732 −0.518659 0.854981i \(-0.673569\pi\)
−0.518659 + 0.854981i \(0.673569\pi\)
\(62\) 0 0
\(63\) −1.00000 −0.125988
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(68\) 0 0
\(69\) 2.62222 0.315678
\(70\) 0 0
\(71\) 2.56199 0.304053 0.152026 0.988376i \(-0.451420\pi\)
0.152026 + 0.988376i \(0.451420\pi\)
\(72\) 0 0
\(73\) −10.9906 −1.28636 −0.643178 0.765717i \(-0.722385\pi\)
−0.643178 + 0.765717i \(0.722385\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 3.80642 0.433782
\(78\) 0 0
\(79\) −6.75557 −0.760061 −0.380030 0.924974i \(-0.624086\pi\)
−0.380030 + 0.924974i \(0.624086\pi\)
\(80\) 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
\(87\) −9.61285 −1.03061
\(88\) 0 0
\(89\) −8.23506 −0.872915 −0.436457 0.899725i \(-0.643767\pi\)
−0.436457 + 0.899725i \(0.643767\pi\)
\(90\) 0 0
\(91\) 0.622216 0.0652259
\(92\) 0 0
\(93\) 0.622216 0.0645208
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −4.23506 −0.430006 −0.215003 0.976613i \(-0.568976\pi\)
−0.215003 + 0.976613i \(0.568976\pi\)
\(98\) 0 0
\(99\) −3.80642 −0.382560
\(100\) 0 0
\(101\) 18.7239 1.86310 0.931550 0.363613i \(-0.118457\pi\)
0.931550 + 0.363613i \(0.118457\pi\)
\(102\) 0 0
\(103\) 0.857279 0.0844702 0.0422351 0.999108i \(-0.486552\pi\)
0.0422351 + 0.999108i \(0.486552\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 11.0923 1.07234 0.536169 0.844111i \(-0.319871\pi\)
0.536169 + 0.844111i \(0.319871\pi\)
\(108\) 0 0
\(109\) 5.61285 0.537613 0.268807 0.963194i \(-0.413371\pi\)
0.268807 + 0.963194i \(0.413371\pi\)
\(110\) 0 0
\(111\) −1.24443 −0.118116
\(112\) 0 0
\(113\) −16.2351 −1.52727 −0.763633 0.645650i \(-0.776586\pi\)
−0.763633 + 0.645650i \(0.776586\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) −0.622216 −0.0575239
\(118\) 0 0
\(119\) −4.42864 −0.405973
\(120\) 0 0
\(121\) 3.48886 0.317169
\(122\) 0 0
\(123\) −4.62222 −0.416771
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) −15.3461 −1.36175 −0.680875 0.732400i \(-0.738400\pi\)
−0.680875 + 0.732400i \(0.738400\pi\)
\(128\) 0 0
\(129\) −4.85728 −0.427660
\(130\) 0 0
\(131\) −4.00000 −0.349482 −0.174741 0.984614i \(-0.555909\pi\)
−0.174741 + 0.984614i \(0.555909\pi\)
\(132\) 0 0
\(133\) −0.622216 −0.0539529
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 17.0923 1.46030 0.730149 0.683288i \(-0.239451\pi\)
0.730149 + 0.683288i \(0.239451\pi\)
\(138\) 0 0
\(139\) −13.4795 −1.14332 −0.571658 0.820492i \(-0.693700\pi\)
−0.571658 + 0.820492i \(0.693700\pi\)
\(140\) 0 0
\(141\) 11.6128 0.977978
\(142\) 0 0
\(143\) 2.36842 0.198057
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) −1.00000 −0.0824786
\(148\) 0 0
\(149\) −9.34614 −0.765666 −0.382833 0.923818i \(-0.625051\pi\)
−0.382833 + 0.923818i \(0.625051\pi\)
\(150\) 0 0
\(151\) −7.14272 −0.581266 −0.290633 0.956835i \(-0.593866\pi\)
−0.290633 + 0.956835i \(0.593866\pi\)
\(152\) 0 0
\(153\) 4.42864 0.358034
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 6.99063 0.557913 0.278957 0.960304i \(-0.410011\pi\)
0.278957 + 0.960304i \(0.410011\pi\)
\(158\) 0 0
\(159\) 13.4795 1.06899
\(160\) 0 0
\(161\) 2.62222 0.206660
\(162\) 0 0
\(163\) −15.6128 −1.22289 −0.611446 0.791286i \(-0.709412\pi\)
−0.611446 + 0.791286i \(0.709412\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 1.51114 0.116935 0.0584677 0.998289i \(-0.481379\pi\)
0.0584677 + 0.998289i \(0.481379\pi\)
\(168\) 0 0
\(169\) −12.6128 −0.970219
\(170\) 0 0
\(171\) 0.622216 0.0475820
\(172\) 0 0
\(173\) 6.53035 0.496493 0.248247 0.968697i \(-0.420146\pi\)
0.248247 + 0.968697i \(0.420146\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) −11.6128 −0.872875
\(178\) 0 0
\(179\) −6.29529 −0.470532 −0.235266 0.971931i \(-0.575596\pi\)
−0.235266 + 0.971931i \(0.575596\pi\)
\(180\) 0 0
\(181\) −6.85728 −0.509698 −0.254849 0.966981i \(-0.582026\pi\)
−0.254849 + 0.966981i \(0.582026\pi\)
\(182\) 0 0
\(183\) 8.10171 0.598896
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) −16.8573 −1.23273
\(188\) 0 0
\(189\) 1.00000 0.0727393
\(190\) 0 0
\(191\) −10.5620 −0.764239 −0.382119 0.924113i \(-0.624806\pi\)
−0.382119 + 0.924113i \(0.624806\pi\)
\(192\) 0 0
\(193\) −5.24443 −0.377502 −0.188751 0.982025i \(-0.560444\pi\)
−0.188751 + 0.982025i \(0.560444\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −17.7462 −1.26436 −0.632182 0.774820i \(-0.717841\pi\)
−0.632182 + 0.774820i \(0.717841\pi\)
\(198\) 0 0
\(199\) −20.2351 −1.43443 −0.717213 0.696854i \(-0.754582\pi\)
−0.717213 + 0.696854i \(0.754582\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) −9.61285 −0.674690
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) −2.62222 −0.182257
\(208\) 0 0
\(209\) −2.36842 −0.163827
\(210\) 0 0
\(211\) 21.3274 1.46824 0.734120 0.679020i \(-0.237595\pi\)
0.734120 + 0.679020i \(0.237595\pi\)
\(212\) 0 0
\(213\) −2.56199 −0.175545
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 0.622216 0.0422387
\(218\) 0 0
\(219\) 10.9906 0.742678
\(220\) 0 0
\(221\) −2.75557 −0.185360
\(222\) 0 0
\(223\) 9.71456 0.650535 0.325267 0.945622i \(-0.394546\pi\)
0.325267 + 0.945622i \(0.394546\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 11.3461 0.753070 0.376535 0.926402i \(-0.377115\pi\)
0.376535 + 0.926402i \(0.377115\pi\)
\(228\) 0 0
\(229\) 1.34614 0.0889555 0.0444778 0.999010i \(-0.485838\pi\)
0.0444778 + 0.999010i \(0.485838\pi\)
\(230\) 0 0
\(231\) −3.80642 −0.250444
\(232\) 0 0
\(233\) −15.3778 −1.00743 −0.503716 0.863869i \(-0.668034\pi\)
−0.503716 + 0.863869i \(0.668034\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 6.75557 0.438821
\(238\) 0 0
\(239\) 7.53972 0.487704 0.243852 0.969812i \(-0.421589\pi\)
0.243852 + 0.969812i \(0.421589\pi\)
\(240\) 0 0
\(241\) 23.9813 1.54477 0.772385 0.635155i \(-0.219064\pi\)
0.772385 + 0.635155i \(0.219064\pi\)
\(242\) 0 0
\(243\) −1.00000 −0.0641500
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) −0.387152 −0.0246339
\(248\) 0 0
\(249\) 11.6128 0.735934
\(250\) 0 0
\(251\) 14.1017 0.890092 0.445046 0.895508i \(-0.353187\pi\)
0.445046 + 0.895508i \(0.353187\pi\)
\(252\) 0 0
\(253\) 9.98126 0.627517
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −17.0192 −1.06163 −0.530815 0.847488i \(-0.678114\pi\)
−0.530815 + 0.847488i \(0.678114\pi\)
\(258\) 0 0
\(259\) −1.24443 −0.0773252
\(260\) 0 0
\(261\) 9.61285 0.595020
\(262\) 0 0
\(263\) −12.6035 −0.777164 −0.388582 0.921414i \(-0.627035\pi\)
−0.388582 + 0.921414i \(0.627035\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 8.23506 0.503978
\(268\) 0 0
\(269\) 3.76494 0.229552 0.114776 0.993391i \(-0.463385\pi\)
0.114776 + 0.993391i \(0.463385\pi\)
\(270\) 0 0
\(271\) −17.8666 −1.08532 −0.542661 0.839952i \(-0.682583\pi\)
−0.542661 + 0.839952i \(0.682583\pi\)
\(272\) 0 0
\(273\) −0.622216 −0.0376582
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) 1.24443 0.0747706 0.0373853 0.999301i \(-0.488097\pi\)
0.0373853 + 0.999301i \(0.488097\pi\)
\(278\) 0 0
\(279\) −0.622216 −0.0372511
\(280\) 0 0
\(281\) −8.95899 −0.534448 −0.267224 0.963634i \(-0.586106\pi\)
−0.267224 + 0.963634i \(0.586106\pi\)
\(282\) 0 0
\(283\) −30.5718 −1.81731 −0.908654 0.417551i \(-0.862889\pi\)
−0.908654 + 0.417551i \(0.862889\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) −4.62222 −0.272841
\(288\) 0 0
\(289\) 2.61285 0.153697
\(290\) 0 0
\(291\) 4.23506 0.248264
\(292\) 0 0
\(293\) 5.67307 0.331424 0.165712 0.986174i \(-0.447008\pi\)
0.165712 + 0.986174i \(0.447008\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 3.80642 0.220871
\(298\) 0 0
\(299\) 1.63158 0.0943569
\(300\) 0 0
\(301\) −4.85728 −0.279969
\(302\) 0 0
\(303\) −18.7239 −1.07566
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) 4.85728 0.277220 0.138610 0.990347i \(-0.455737\pi\)
0.138610 + 0.990347i \(0.455737\pi\)
\(308\) 0 0
\(309\) −0.857279 −0.0487689
\(310\) 0 0
\(311\) −34.5718 −1.96039 −0.980195 0.198037i \(-0.936543\pi\)
−0.980195 + 0.198037i \(0.936543\pi\)
\(312\) 0 0
\(313\) 6.33677 0.358176 0.179088 0.983833i \(-0.442685\pi\)
0.179088 + 0.983833i \(0.442685\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −15.9684 −0.896872 −0.448436 0.893815i \(-0.648019\pi\)
−0.448436 + 0.893815i \(0.648019\pi\)
\(318\) 0 0
\(319\) −36.5906 −2.04868
\(320\) 0 0
\(321\) −11.0923 −0.619114
\(322\) 0 0
\(323\) 2.75557 0.153324
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) −5.61285 −0.310391
\(328\) 0 0
\(329\) 11.6128 0.640237
\(330\) 0 0
\(331\) −27.6128 −1.51774 −0.758870 0.651243i \(-0.774248\pi\)
−0.758870 + 0.651243i \(0.774248\pi\)
\(332\) 0 0
\(333\) 1.24443 0.0681944
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) −16.0000 −0.871576 −0.435788 0.900049i \(-0.643530\pi\)
−0.435788 + 0.900049i \(0.643530\pi\)
\(338\) 0 0
\(339\) 16.2351 0.881768
\(340\) 0 0
\(341\) 2.36842 0.128257
\(342\) 0 0
\(343\) −1.00000 −0.0539949
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 11.8666 0.637035 0.318517 0.947917i \(-0.396815\pi\)
0.318517 + 0.947917i \(0.396815\pi\)
\(348\) 0 0
\(349\) 21.8163 1.16780 0.583899 0.811826i \(-0.301526\pi\)
0.583899 + 0.811826i \(0.301526\pi\)
\(350\) 0 0
\(351\) 0.622216 0.0332114
\(352\) 0 0
\(353\) 2.79706 0.148872 0.0744361 0.997226i \(-0.476284\pi\)
0.0744361 + 0.997226i \(0.476284\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 4.42864 0.234388
\(358\) 0 0
\(359\) −13.0509 −0.688798 −0.344399 0.938823i \(-0.611917\pi\)
−0.344399 + 0.938823i \(0.611917\pi\)
\(360\) 0 0
\(361\) −18.6128 −0.979624
\(362\) 0 0
\(363\) −3.48886 −0.183118
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 10.4889 0.547514 0.273757 0.961799i \(-0.411734\pi\)
0.273757 + 0.961799i \(0.411734\pi\)
\(368\) 0 0
\(369\) 4.62222 0.240623
\(370\) 0 0
\(371\) 13.4795 0.699820
\(372\) 0 0
\(373\) −30.1847 −1.56290 −0.781452 0.623966i \(-0.785521\pi\)
−0.781452 + 0.623966i \(0.785521\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −5.98126 −0.308051
\(378\) 0 0
\(379\) −12.8573 −0.660434 −0.330217 0.943905i \(-0.607122\pi\)
−0.330217 + 0.943905i \(0.607122\pi\)
\(380\) 0 0
\(381\) 15.3461 0.786207
\(382\) 0 0
\(383\) 24.4701 1.25037 0.625183 0.780479i \(-0.285025\pi\)
0.625183 + 0.780479i \(0.285025\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 4.85728 0.246909
\(388\) 0 0
\(389\) −1.61285 −0.0817746 −0.0408873 0.999164i \(-0.513018\pi\)
−0.0408873 + 0.999164i \(0.513018\pi\)
\(390\) 0 0
\(391\) −11.6128 −0.587287
\(392\) 0 0
\(393\) 4.00000 0.201773
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) −22.2163 −1.11501 −0.557503 0.830175i \(-0.688240\pi\)
−0.557503 + 0.830175i \(0.688240\pi\)
\(398\) 0 0
\(399\) 0.622216 0.0311497
\(400\) 0 0
\(401\) 19.9813 0.997817 0.498908 0.866655i \(-0.333734\pi\)
0.498908 + 0.866655i \(0.333734\pi\)
\(402\) 0 0
\(403\) 0.387152 0.0192854
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) −4.73683 −0.234796
\(408\) 0 0
\(409\) −5.73329 −0.283493 −0.141747 0.989903i \(-0.545272\pi\)
−0.141747 + 0.989903i \(0.545272\pi\)
\(410\) 0 0
\(411\) −17.0923 −0.843103
\(412\) 0 0
\(413\) −11.6128 −0.571431
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 13.4795 0.660094
\(418\) 0 0
\(419\) 26.3684 1.28818 0.644091 0.764949i \(-0.277236\pi\)
0.644091 + 0.764949i \(0.277236\pi\)
\(420\) 0 0
\(421\) −19.3274 −0.941960 −0.470980 0.882144i \(-0.656100\pi\)
−0.470980 + 0.882144i \(0.656100\pi\)
\(422\) 0 0
\(423\) −11.6128 −0.564636
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 8.10171 0.392069
\(428\) 0 0
\(429\) −2.36842 −0.114348
\(430\) 0 0
\(431\) −26.9491 −1.29809 −0.649047 0.760748i \(-0.724832\pi\)
−0.649047 + 0.760748i \(0.724832\pi\)
\(432\) 0 0
\(433\) 2.13335 0.102522 0.0512612 0.998685i \(-0.483676\pi\)
0.0512612 + 0.998685i \(0.483676\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −1.63158 −0.0780492
\(438\) 0 0
\(439\) −10.5205 −0.502116 −0.251058 0.967972i \(-0.580779\pi\)
−0.251058 + 0.967972i \(0.580779\pi\)
\(440\) 0 0
\(441\) 1.00000 0.0476190
\(442\) 0 0
\(443\) 6.88892 0.327303 0.163651 0.986518i \(-0.447673\pi\)
0.163651 + 0.986518i \(0.447673\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 9.34614 0.442057
\(448\) 0 0
\(449\) 39.9180 1.88385 0.941923 0.335829i \(-0.109016\pi\)
0.941923 + 0.335829i \(0.109016\pi\)
\(450\) 0 0
\(451\) −17.5941 −0.828474
\(452\) 0 0
\(453\) 7.14272 0.335594
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 8.47013 0.396216 0.198108 0.980180i \(-0.436520\pi\)
0.198108 + 0.980180i \(0.436520\pi\)
\(458\) 0 0
\(459\) −4.42864 −0.206711
\(460\) 0 0
\(461\) 40.1146 1.86832 0.934162 0.356849i \(-0.116149\pi\)
0.934162 + 0.356849i \(0.116149\pi\)
\(462\) 0 0
\(463\) −33.5941 −1.56125 −0.780625 0.624999i \(-0.785099\pi\)
−0.780625 + 0.624999i \(0.785099\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −11.3461 −0.525037 −0.262518 0.964927i \(-0.584553\pi\)
−0.262518 + 0.964927i \(0.584553\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) −6.99063 −0.322111
\(472\) 0 0
\(473\) −18.4889 −0.850119
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) −13.4795 −0.617184
\(478\) 0 0
\(479\) 36.2864 1.65797 0.828984 0.559273i \(-0.188919\pi\)
0.828984 + 0.559273i \(0.188919\pi\)
\(480\) 0 0
\(481\) −0.774305 −0.0353053
\(482\) 0 0
\(483\) −2.62222 −0.119315
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) 38.8385 1.75994 0.879971 0.475027i \(-0.157562\pi\)
0.879971 + 0.475027i \(0.157562\pi\)
\(488\) 0 0
\(489\) 15.6128 0.706037
\(490\) 0 0
\(491\) −28.7467 −1.29732 −0.648660 0.761079i \(-0.724670\pi\)
−0.648660 + 0.761079i \(0.724670\pi\)
\(492\) 0 0
\(493\) 42.5718 1.91734
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −2.56199 −0.114921
\(498\) 0 0
\(499\) −5.63158 −0.252104 −0.126052 0.992024i \(-0.540231\pi\)
−0.126052 + 0.992024i \(0.540231\pi\)
\(500\) 0 0
\(501\) −1.51114 −0.0675126
\(502\) 0 0
\(503\) 34.9590 1.55874 0.779372 0.626561i \(-0.215538\pi\)
0.779372 + 0.626561i \(0.215538\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 12.6128 0.560156
\(508\) 0 0
\(509\) −10.9906 −0.487151 −0.243576 0.969882i \(-0.578320\pi\)
−0.243576 + 0.969882i \(0.578320\pi\)
\(510\) 0 0
\(511\) 10.9906 0.486197
\(512\) 0 0
\(513\) −0.622216 −0.0274715
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 44.2034 1.94406
\(518\) 0 0
\(519\) −6.53035 −0.286651
\(520\) 0 0
\(521\) 6.90766 0.302630 0.151315 0.988486i \(-0.451649\pi\)
0.151315 + 0.988486i \(0.451649\pi\)
\(522\) 0 0
\(523\) −37.7146 −1.64914 −0.824571 0.565758i \(-0.808584\pi\)
−0.824571 + 0.565758i \(0.808584\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −2.75557 −0.120034
\(528\) 0 0
\(529\) −16.1240 −0.701043
\(530\) 0 0
\(531\) 11.6128 0.503955
\(532\) 0 0
\(533\) −2.87601 −0.124574
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 6.29529 0.271662
\(538\) 0 0
\(539\) −3.80642 −0.163954
\(540\) 0 0
\(541\) −3.12399 −0.134311 −0.0671553 0.997743i \(-0.521392\pi\)
−0.0671553 + 0.997743i \(0.521392\pi\)
\(542\) 0 0
\(543\) 6.85728 0.294274
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 5.51114 0.235639 0.117820 0.993035i \(-0.462410\pi\)
0.117820 + 0.993035i \(0.462410\pi\)
\(548\) 0 0
\(549\) −8.10171 −0.345773
\(550\) 0 0
\(551\) 5.98126 0.254810
\(552\) 0 0
\(553\) 6.75557 0.287276
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −36.7052 −1.55525 −0.777624 0.628729i \(-0.783575\pi\)
−0.777624 + 0.628729i \(0.783575\pi\)
\(558\) 0 0
\(559\) −3.02227 −0.127829
\(560\) 0 0
\(561\) 16.8573 0.711715
\(562\) 0 0
\(563\) 27.4924 1.15867 0.579333 0.815091i \(-0.303313\pi\)
0.579333 + 0.815091i \(0.303313\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) −1.00000 −0.0419961
\(568\) 0 0
\(569\) 23.2444 0.974457 0.487229 0.873274i \(-0.338008\pi\)
0.487229 + 0.873274i \(0.338008\pi\)
\(570\) 0 0
\(571\) −25.5111 −1.06761 −0.533804 0.845608i \(-0.679238\pi\)
−0.533804 + 0.845608i \(0.679238\pi\)
\(572\) 0 0
\(573\) 10.5620 0.441234
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) −26.0701 −1.08531 −0.542656 0.839955i \(-0.682581\pi\)
−0.542656 + 0.839955i \(0.682581\pi\)
\(578\) 0 0
\(579\) 5.24443 0.217951
\(580\) 0 0
\(581\) 11.6128 0.481782
\(582\) 0 0
\(583\) 51.3087 2.12499
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 12.2667 0.506301 0.253151 0.967427i \(-0.418533\pi\)
0.253151 + 0.967427i \(0.418533\pi\)
\(588\) 0 0
\(589\) −0.387152 −0.0159523
\(590\) 0 0
\(591\) 17.7462 0.729981
\(592\) 0 0
\(593\) 14.9175 0.612588 0.306294 0.951937i \(-0.400911\pi\)
0.306294 + 0.951937i \(0.400911\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 20.2351 0.828166
\(598\) 0 0
\(599\) −26.5620 −1.08529 −0.542647 0.839961i \(-0.682578\pi\)
−0.542647 + 0.839961i \(0.682578\pi\)
\(600\) 0 0
\(601\) 39.7146 1.61999 0.809995 0.586436i \(-0.199470\pi\)
0.809995 + 0.586436i \(0.199470\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) −6.28544 −0.255118 −0.127559 0.991831i \(-0.540714\pi\)
−0.127559 + 0.991831i \(0.540714\pi\)
\(608\) 0 0
\(609\) 9.61285 0.389532
\(610\) 0 0
\(611\) 7.22570 0.292320
\(612\) 0 0
\(613\) 45.7146 1.84639 0.923197 0.384328i \(-0.125567\pi\)
0.923197 + 0.384328i \(0.125567\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −4.88892 −0.196821 −0.0984103 0.995146i \(-0.531376\pi\)
−0.0984103 + 0.995146i \(0.531376\pi\)
\(618\) 0 0
\(619\) 12.2351 0.491769 0.245884 0.969299i \(-0.420922\pi\)
0.245884 + 0.969299i \(0.420922\pi\)
\(620\) 0 0
\(621\) 2.62222 0.105226
\(622\) 0 0
\(623\) 8.23506 0.329931
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) 2.36842 0.0945854
\(628\) 0 0
\(629\) 5.51114 0.219743
\(630\) 0 0
\(631\) 1.24443 0.0495400 0.0247700 0.999693i \(-0.492115\pi\)
0.0247700 + 0.999693i \(0.492115\pi\)
\(632\) 0 0
\(633\) −21.3274 −0.847688
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) −0.622216 −0.0246531
\(638\) 0 0
\(639\) 2.56199 0.101351
\(640\) 0 0
\(641\) −48.1847 −1.90318 −0.951590 0.307369i \(-0.900551\pi\)
−0.951590 + 0.307369i \(0.900551\pi\)
\(642\) 0 0
\(643\) −4.85728 −0.191552 −0.0957762 0.995403i \(-0.530533\pi\)
−0.0957762 + 0.995403i \(0.530533\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −0.203420 −0.00799728 −0.00399864 0.999992i \(-0.501273\pi\)
−0.00399864 + 0.999992i \(0.501273\pi\)
\(648\) 0 0
\(649\) −44.2034 −1.73514
\(650\) 0 0
\(651\) −0.622216 −0.0243866
\(652\) 0 0
\(653\) 27.3145 1.06890 0.534449 0.845200i \(-0.320519\pi\)
0.534449 + 0.845200i \(0.320519\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) −10.9906 −0.428785
\(658\) 0 0
\(659\) 33.3176 1.29787 0.648934 0.760845i \(-0.275215\pi\)
0.648934 + 0.760845i \(0.275215\pi\)
\(660\) 0 0
\(661\) 14.5906 0.567508 0.283754 0.958897i \(-0.408420\pi\)
0.283754 + 0.958897i \(0.408420\pi\)
\(662\) 0 0
\(663\) 2.75557 0.107017
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) −25.2070 −0.976017
\(668\) 0 0
\(669\) −9.71456 −0.375587
\(670\) 0 0
\(671\) 30.8385 1.19051
\(672\) 0 0
\(673\) −4.53341 −0.174750 −0.0873751 0.996175i \(-0.527848\pi\)
−0.0873751 + 0.996175i \(0.527848\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 27.2672 1.04796 0.523981 0.851730i \(-0.324446\pi\)
0.523981 + 0.851730i \(0.324446\pi\)
\(678\) 0 0
\(679\) 4.23506 0.162527
\(680\) 0 0
\(681\) −11.3461 −0.434785
\(682\) 0 0
\(683\) 29.5812 1.13189 0.565947 0.824442i \(-0.308511\pi\)
0.565947 + 0.824442i \(0.308511\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) −1.34614 −0.0513585
\(688\) 0 0
\(689\) 8.38715 0.319525
\(690\) 0 0
\(691\) 2.99063 0.113769 0.0568845 0.998381i \(-0.481883\pi\)
0.0568845 + 0.998381i \(0.481883\pi\)
\(692\) 0 0
\(693\) 3.80642 0.144594
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 20.4701 0.775361
\(698\) 0 0
\(699\) 15.3778 0.581641
\(700\) 0 0
\(701\) −41.0420 −1.55013 −0.775067 0.631879i \(-0.782284\pi\)
−0.775067 + 0.631879i \(0.782284\pi\)
\(702\) 0 0
\(703\) 0.774305 0.0292035
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) −18.7239 −0.704186
\(708\) 0 0
\(709\) −41.4291 −1.55590 −0.777952 0.628324i \(-0.783741\pi\)
−0.777952 + 0.628324i \(0.783741\pi\)
\(710\) 0 0
\(711\) −6.75557 −0.253354
\(712\) 0 0
\(713\) 1.63158 0.0611033
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) −7.53972 −0.281576
\(718\) 0 0
\(719\) −18.9590 −0.707051 −0.353525 0.935425i \(-0.615017\pi\)
−0.353525 + 0.935425i \(0.615017\pi\)
\(720\) 0 0
\(721\) −0.857279 −0.0319267
\(722\) 0 0
\(723\) −23.9813 −0.891873
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) −41.7975 −1.55018 −0.775092 0.631848i \(-0.782297\pi\)
−0.775092 + 0.631848i \(0.782297\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) 21.5111 0.795618
\(732\) 0 0
\(733\) −15.3145 −0.565654 −0.282827 0.959171i \(-0.591272\pi\)
−0.282827 + 0.959171i \(0.591272\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) −32.7368 −1.20424 −0.602122 0.798404i \(-0.705678\pi\)
−0.602122 + 0.798404i \(0.705678\pi\)
\(740\) 0 0
\(741\) 0.387152 0.0142224
\(742\) 0 0
\(743\) 37.3778 1.37126 0.685629 0.727951i \(-0.259527\pi\)
0.685629 + 0.727951i \(0.259527\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) −11.6128 −0.424892
\(748\) 0 0
\(749\) −11.0923 −0.405305
\(750\) 0 0
\(751\) 20.3497 0.742570 0.371285 0.928519i \(-0.378917\pi\)
0.371285 + 0.928519i \(0.378917\pi\)
\(752\) 0 0
\(753\) −14.1017 −0.513895
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 6.95899 0.252929 0.126464 0.991971i \(-0.459637\pi\)
0.126464 + 0.991971i \(0.459637\pi\)
\(758\) 0 0
\(759\) −9.98126 −0.362297
\(760\) 0 0
\(761\) 48.6419 1.76327 0.881634 0.471934i \(-0.156444\pi\)
0.881634 + 0.471934i \(0.156444\pi\)
\(762\) 0 0
\(763\) −5.61285 −0.203199
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) −7.22570 −0.260905
\(768\) 0 0
\(769\) −24.6923 −0.890426 −0.445213 0.895425i \(-0.646872\pi\)
−0.445213 + 0.895425i \(0.646872\pi\)
\(770\) 0 0
\(771\) 17.0192 0.612932
\(772\) 0 0
\(773\) −36.0415 −1.29632 −0.648161 0.761503i \(-0.724462\pi\)
−0.648161 + 0.761503i \(0.724462\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 1.24443 0.0446437
\(778\) 0 0
\(779\) 2.87601 0.103044
\(780\) 0 0
\(781\) −9.75203 −0.348955
\(782\) 0 0
\(783\) −9.61285 −0.343535
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) 32.2034 1.14793 0.573964 0.818881i \(-0.305405\pi\)
0.573964 + 0.818881i \(0.305405\pi\)
\(788\) 0 0
\(789\) 12.6035 0.448696
\(790\) 0 0
\(791\) 16.2351 0.577252
\(792\) 0 0
\(793\) 5.04101 0.179012
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 17.5526 0.621746 0.310873 0.950451i \(-0.399379\pi\)
0.310873 + 0.950451i \(0.399379\pi\)
\(798\) 0 0
\(799\) −51.4291 −1.81943
\(800\) 0 0
\(801\) −8.23506 −0.290972
\(802\) 0 0
\(803\) 41.8350 1.47633
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) −3.76494 −0.132532
\(808\) 0 0
\(809\) 46.1659 1.62311 0.811554 0.584277i \(-0.198622\pi\)
0.811554 + 0.584277i \(0.198622\pi\)
\(810\) 0 0
\(811\) 18.5205 0.650343 0.325171 0.945655i \(-0.394578\pi\)
0.325171 + 0.945655i \(0.394578\pi\)
\(812\) 0 0
\(813\) 17.8666 0.626611
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 3.02227 0.105736
\(818\) 0 0
\(819\) 0.622216 0.0217420
\(820\) 0 0
\(821\) 43.0607 1.50283 0.751414 0.659831i \(-0.229372\pi\)
0.751414 + 0.659831i \(0.229372\pi\)
\(822\) 0 0
\(823\) −14.5718 −0.507942 −0.253971 0.967212i \(-0.581737\pi\)
−0.253971 + 0.967212i \(0.581737\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 55.8292 1.94137 0.970685 0.240354i \(-0.0772635\pi\)
0.970685 + 0.240354i \(0.0772635\pi\)
\(828\) 0 0
\(829\) −37.3087 −1.29578 −0.647892 0.761732i \(-0.724349\pi\)
−0.647892 + 0.761732i \(0.724349\pi\)
\(830\) 0 0
\(831\) −1.24443 −0.0431688
\(832\) 0 0
\(833\) 4.42864 0.153443
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 0.622216 0.0215069
\(838\) 0 0
\(839\) −51.0420 −1.76216 −0.881082 0.472963i \(-0.843184\pi\)
−0.881082 + 0.472963i \(0.843184\pi\)
\(840\) 0 0
\(841\) 63.4068 2.18644
\(842\) 0 0
\(843\) 8.95899 0.308564
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) −3.48886 −0.119879
\(848\) 0 0
\(849\) 30.5718 1.04922
\(850\) 0 0
\(851\) −3.26317 −0.111860
\(852\) 0 0
\(853\) −26.4197 −0.904595 −0.452297 0.891867i \(-0.649395\pi\)
−0.452297 + 0.891867i \(0.649395\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 0.161933 0.00553154 0.00276577 0.999996i \(-0.499120\pi\)
0.00276577 + 0.999996i \(0.499120\pi\)
\(858\) 0 0
\(859\) 51.3403 1.75171 0.875854 0.482575i \(-0.160299\pi\)
0.875854 + 0.482575i \(0.160299\pi\)
\(860\) 0 0
\(861\) 4.62222 0.157525
\(862\) 0 0
\(863\) −12.8702 −0.438106 −0.219053 0.975713i \(-0.570297\pi\)
−0.219053 + 0.975713i \(0.570297\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) −2.61285 −0.0887370
\(868\) 0 0
\(869\) 25.7146 0.872307
\(870\) 0 0
\(871\) 0 0
\(872\) 0 0
\(873\) −4.23506 −0.143335
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 43.2257 1.45963 0.729814 0.683646i \(-0.239607\pi\)
0.729814 + 0.683646i \(0.239607\pi\)
\(878\) 0 0
\(879\) −5.67307 −0.191348
\(880\) 0 0
\(881\) −16.5018 −0.555959 −0.277979 0.960587i \(-0.589665\pi\)
−0.277979 + 0.960587i \(0.589665\pi\)
\(882\) 0 0
\(883\) 46.1847 1.55424 0.777119 0.629353i \(-0.216680\pi\)
0.777119 + 0.629353i \(0.216680\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 37.3274 1.25333 0.626666 0.779288i \(-0.284419\pi\)
0.626666 + 0.779288i \(0.284419\pi\)
\(888\) 0 0
\(889\) 15.3461 0.514693
\(890\) 0 0
\(891\) −3.80642 −0.127520
\(892\) 0 0
\(893\) −7.22570 −0.241799
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) −1.63158 −0.0544770
\(898\) 0 0
\(899\) −5.98126 −0.199486
\(900\) 0 0
\(901\) −59.6958 −1.98876
\(902\) 0 0
\(903\) 4.85728 0.161640
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) −24.6735 −0.819272 −0.409636 0.912249i \(-0.634344\pi\)
−0.409636 + 0.912249i \(0.634344\pi\)
\(908\) 0 0
\(909\) 18.7239 0.621033
\(910\) 0 0
\(911\) 29.4380 0.975325 0.487662 0.873032i \(-0.337850\pi\)
0.487662 + 0.873032i \(0.337850\pi\)
\(912\) 0 0
\(913\) 44.2034 1.46292
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 4.00000 0.132092
\(918\) 0 0
\(919\) 40.0197 1.32013 0.660064 0.751209i \(-0.270529\pi\)
0.660064 + 0.751209i \(0.270529\pi\)
\(920\) 0 0
\(921\) −4.85728 −0.160053
\(922\) 0 0
\(923\) −1.59411 −0.0524708
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) 0.857279 0.0281567
\(928\) 0 0
\(929\) −22.4572 −0.736797 −0.368399 0.929668i \(-0.620094\pi\)
−0.368399 + 0.929668i \(0.620094\pi\)
\(930\) 0 0
\(931\) 0.622216 0.0203923
\(932\) 0 0
\(933\) 34.5718 1.13183
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 33.8292 1.10515 0.552575 0.833463i \(-0.313645\pi\)
0.552575 + 0.833463i \(0.313645\pi\)
\(938\) 0 0
\(939\) −6.33677 −0.206793
\(940\) 0 0
\(941\) 47.8479 1.55980 0.779899 0.625906i \(-0.215271\pi\)
0.779899 + 0.625906i \(0.215271\pi\)
\(942\) 0 0
\(943\) −12.1204 −0.394696
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −44.8069 −1.45603 −0.728014 0.685562i \(-0.759557\pi\)
−0.728014 + 0.685562i \(0.759557\pi\)
\(948\) 0 0
\(949\) 6.83854 0.221989
\(950\) 0 0
\(951\) 15.9684 0.517809
\(952\) 0 0
\(953\) −15.7017 −0.508626 −0.254313 0.967122i \(-0.581849\pi\)
−0.254313 + 0.967122i \(0.581849\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 36.5906 1.18281
\(958\) 0 0
\(959\) −17.0923 −0.551941
\(960\) 0 0
\(961\) −30.6128 −0.987511
\(962\) 0 0
\(963\) 11.0923 0.357446
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) 46.9590 1.51010 0.755050 0.655668i \(-0.227613\pi\)
0.755050 + 0.655668i \(0.227613\pi\)
\(968\) 0 0
\(969\) −2.75557 −0.0885216
\(970\) 0 0
\(971\) −18.7556 −0.601895 −0.300947 0.953641i \(-0.597303\pi\)
−0.300947 + 0.953641i \(0.597303\pi\)
\(972\) 0 0
\(973\) 13.4795 0.432133
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −37.8292 −1.21026 −0.605131 0.796126i \(-0.706879\pi\)
−0.605131 + 0.796126i \(0.706879\pi\)
\(978\) 0 0
\(979\) 31.3461 1.00183
\(980\) 0 0
\(981\) 5.61285 0.179204
\(982\) 0 0
\(983\) 17.6316 0.562360 0.281180 0.959655i \(-0.409274\pi\)
0.281180 + 0.959655i \(0.409274\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) −11.6128 −0.369641
\(988\) 0 0
\(989\) −12.7368 −0.405008
\(990\) 0 0
\(991\) −37.1240 −1.17928 −0.589641 0.807665i \(-0.700731\pi\)
−0.589641 + 0.807665i \(0.700731\pi\)
\(992\) 0 0
\(993\) 27.6128 0.876267
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) −42.4197 −1.34345 −0.671723 0.740802i \(-0.734446\pi\)
−0.671723 + 0.740802i \(0.734446\pi\)
\(998\) 0 0
\(999\) −1.24443 −0.0393721
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.bo.1.1 3
4.3 odd 2 8400.2.a.dk.1.3 3
5.2 odd 4 840.2.t.e.169.5 yes 6
5.3 odd 4 840.2.t.e.169.2 6
5.4 even 2 4200.2.a.bq.1.1 3
15.2 even 4 2520.2.t.j.1009.3 6
15.8 even 4 2520.2.t.j.1009.4 6
20.3 even 4 1680.2.t.i.1009.5 6
20.7 even 4 1680.2.t.i.1009.2 6
20.19 odd 2 8400.2.a.dh.1.3 3
60.23 odd 4 5040.2.t.ba.1009.4 6
60.47 odd 4 5040.2.t.ba.1009.3 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
840.2.t.e.169.2 6 5.3 odd 4
840.2.t.e.169.5 yes 6 5.2 odd 4
1680.2.t.i.1009.2 6 20.7 even 4
1680.2.t.i.1009.5 6 20.3 even 4
2520.2.t.j.1009.3 6 15.2 even 4
2520.2.t.j.1009.4 6 15.8 even 4
4200.2.a.bo.1.1 3 1.1 even 1 trivial
4200.2.a.bq.1.1 3 5.4 even 2
5040.2.t.ba.1009.3 6 60.47 odd 4
5040.2.t.ba.1009.4 6 60.23 odd 4
8400.2.a.dh.1.3 3 20.19 odd 2
8400.2.a.dk.1.3 3 4.3 odd 2