Properties

Label 9800.2.a.cx.1.6
Level $9800$
Weight $2$
Character 9800.1
Self dual yes
Analytic conductor $78.253$
Analytic rank $0$
Dimension $6$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [9800,2,Mod(1,9800)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(9800, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("9800.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 9800 = 2^{3} \cdot 5^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 9800.a (trivial)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(78.2533939809\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: 6.6.239575536.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - x^{5} - 12x^{4} + 8x^{3} + 35x^{2} - 15x - 8 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 280)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.6
Root \(3.04302\) of defining polynomial
Character \(\chi\) \(=\) 9800.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+3.04302 q^{3} +6.25996 q^{9} +O(q^{10})\) \(q+3.04302 q^{3} +6.25996 q^{9} +3.20055 q^{11} +6.25996 q^{13} -1.78144 q^{17} -3.67654 q^{19} -3.21694 q^{23} +9.92011 q^{27} +2.88463 q^{29} +2.81854 q^{31} +9.73933 q^{33} -5.08265 q^{37} +19.0492 q^{39} -0.899340 q^{41} +7.26767 q^{43} +2.52063 q^{47} -5.42097 q^{51} +9.04054 q^{53} -11.1878 q^{57} -5.58427 q^{59} +10.9028 q^{61} -1.40019 q^{67} -9.78921 q^{69} -15.3970 q^{71} +7.01571 q^{73} +0.186696 q^{79} +11.4072 q^{81} +0.134286 q^{83} +8.77797 q^{87} +18.4329 q^{89} +8.57688 q^{93} -9.75419 q^{97} +20.0353 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + q^{3} + 7 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q + q^{3} + 7 q^{9} + q^{11} + 7 q^{13} + 4 q^{17} - 5 q^{19} - 6 q^{23} + 7 q^{27} - 3 q^{29} - 2 q^{31} + 16 q^{33} - 9 q^{37} + 10 q^{39} + 6 q^{41} + 3 q^{43} + 27 q^{47} + 5 q^{53} + 26 q^{57} - 24 q^{59} + 9 q^{61} - 17 q^{67} + 15 q^{69} + 4 q^{71} + 18 q^{73} - 22 q^{79} - 6 q^{81} + 9 q^{83} + 39 q^{87} + 15 q^{89} - 10 q^{93} + 12 q^{97} + 11 q^{99}+O(q^{100}) \) 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
\(3\) 3.04302 1.75689 0.878444 0.477846i \(-0.158582\pi\)
0.878444 + 0.477846i \(0.158582\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) 0 0
\(8\) 0 0
\(9\) 6.25996 2.08665
\(10\) 0 0
\(11\) 3.20055 0.965002 0.482501 0.875895i \(-0.339728\pi\)
0.482501 + 0.875895i \(0.339728\pi\)
\(12\) 0 0
\(13\) 6.25996 1.73620 0.868100 0.496389i \(-0.165341\pi\)
0.868100 + 0.496389i \(0.165341\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −1.78144 −0.432064 −0.216032 0.976386i \(-0.569311\pi\)
−0.216032 + 0.976386i \(0.569311\pi\)
\(18\) 0 0
\(19\) −3.67654 −0.843457 −0.421728 0.906722i \(-0.638576\pi\)
−0.421728 + 0.906722i \(0.638576\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −3.21694 −0.670778 −0.335389 0.942080i \(-0.608868\pi\)
−0.335389 + 0.942080i \(0.608868\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 9.92011 1.90913
\(28\) 0 0
\(29\) 2.88463 0.535662 0.267831 0.963466i \(-0.413693\pi\)
0.267831 + 0.963466i \(0.413693\pi\)
\(30\) 0 0
\(31\) 2.81854 0.506226 0.253113 0.967437i \(-0.418546\pi\)
0.253113 + 0.967437i \(0.418546\pi\)
\(32\) 0 0
\(33\) 9.73933 1.69540
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −5.08265 −0.835583 −0.417792 0.908543i \(-0.637196\pi\)
−0.417792 + 0.908543i \(0.637196\pi\)
\(38\) 0 0
\(39\) 19.0492 3.05031
\(40\) 0 0
\(41\) −0.899340 −0.140453 −0.0702266 0.997531i \(-0.522372\pi\)
−0.0702266 + 0.997531i \(0.522372\pi\)
\(42\) 0 0
\(43\) 7.26767 1.10831 0.554155 0.832414i \(-0.313041\pi\)
0.554155 + 0.832414i \(0.313041\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 2.52063 0.367671 0.183836 0.982957i \(-0.441149\pi\)
0.183836 + 0.982957i \(0.441149\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 0 0
\(51\) −5.42097 −0.759087
\(52\) 0 0
\(53\) 9.04054 1.24181 0.620907 0.783884i \(-0.286764\pi\)
0.620907 + 0.783884i \(0.286764\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) −11.1878 −1.48186
\(58\) 0 0
\(59\) −5.58427 −0.727011 −0.363505 0.931592i \(-0.618420\pi\)
−0.363505 + 0.931592i \(0.618420\pi\)
\(60\) 0 0
\(61\) 10.9028 1.39596 0.697979 0.716118i \(-0.254083\pi\)
0.697979 + 0.716118i \(0.254083\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) −1.40019 −0.171061 −0.0855305 0.996336i \(-0.527259\pi\)
−0.0855305 + 0.996336i \(0.527259\pi\)
\(68\) 0 0
\(69\) −9.78921 −1.17848
\(70\) 0 0
\(71\) −15.3970 −1.82729 −0.913644 0.406514i \(-0.866744\pi\)
−0.913644 + 0.406514i \(0.866744\pi\)
\(72\) 0 0
\(73\) 7.01571 0.821127 0.410564 0.911832i \(-0.365332\pi\)
0.410564 + 0.911832i \(0.365332\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 0.186696 0.0210050 0.0105025 0.999945i \(-0.496657\pi\)
0.0105025 + 0.999945i \(0.496657\pi\)
\(80\) 0 0
\(81\) 11.4072 1.26747
\(82\) 0 0
\(83\) 0.134286 0.0147398 0.00736988 0.999973i \(-0.497654\pi\)
0.00736988 + 0.999973i \(0.497654\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 8.77797 0.941097
\(88\) 0 0
\(89\) 18.4329 1.95388 0.976941 0.213509i \(-0.0684891\pi\)
0.976941 + 0.213509i \(0.0684891\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 8.57688 0.889381
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −9.75419 −0.990388 −0.495194 0.868783i \(-0.664903\pi\)
−0.495194 + 0.868783i \(0.664903\pi\)
\(98\) 0 0
\(99\) 20.0353 2.01362
\(100\) 0 0
\(101\) 2.01385 0.200386 0.100193 0.994968i \(-0.468054\pi\)
0.100193 + 0.994968i \(0.468054\pi\)
\(102\) 0 0
\(103\) −11.0070 −1.08455 −0.542276 0.840200i \(-0.682437\pi\)
−0.542276 + 0.840200i \(0.682437\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 1.20223 0.116224 0.0581118 0.998310i \(-0.481492\pi\)
0.0581118 + 0.998310i \(0.481492\pi\)
\(108\) 0 0
\(109\) 8.55494 0.819414 0.409707 0.912217i \(-0.365631\pi\)
0.409707 + 0.912217i \(0.365631\pi\)
\(110\) 0 0
\(111\) −15.4666 −1.46803
\(112\) 0 0
\(113\) −6.56879 −0.617940 −0.308970 0.951072i \(-0.599984\pi\)
−0.308970 + 0.951072i \(0.599984\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 39.1871 3.62285
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) −0.756478 −0.0687707
\(122\) 0 0
\(123\) −2.73671 −0.246761
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) 13.0999 1.16243 0.581214 0.813751i \(-0.302578\pi\)
0.581214 + 0.813751i \(0.302578\pi\)
\(128\) 0 0
\(129\) 22.1157 1.94718
\(130\) 0 0
\(131\) −7.54839 −0.659506 −0.329753 0.944067i \(-0.606966\pi\)
−0.329753 + 0.944067i \(0.606966\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −5.54631 −0.473854 −0.236927 0.971528i \(-0.576140\pi\)
−0.236927 + 0.971528i \(0.576140\pi\)
\(138\) 0 0
\(139\) −15.2856 −1.29651 −0.648256 0.761423i \(-0.724501\pi\)
−0.648256 + 0.761423i \(0.724501\pi\)
\(140\) 0 0
\(141\) 7.67031 0.645957
\(142\) 0 0
\(143\) 20.0353 1.67544
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) −8.74119 −0.716106 −0.358053 0.933701i \(-0.616559\pi\)
−0.358053 + 0.933701i \(0.616559\pi\)
\(150\) 0 0
\(151\) 14.1042 1.14778 0.573891 0.818931i \(-0.305433\pi\)
0.573891 + 0.818931i \(0.305433\pi\)
\(152\) 0 0
\(153\) −11.1518 −0.901567
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 0.910583 0.0726724 0.0363362 0.999340i \(-0.488431\pi\)
0.0363362 + 0.999340i \(0.488431\pi\)
\(158\) 0 0
\(159\) 27.5105 2.18173
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) 10.3923 0.813991 0.406996 0.913430i \(-0.366576\pi\)
0.406996 + 0.913430i \(0.366576\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 7.35994 0.569529 0.284765 0.958597i \(-0.408085\pi\)
0.284765 + 0.958597i \(0.408085\pi\)
\(168\) 0 0
\(169\) 26.1871 2.01439
\(170\) 0 0
\(171\) −23.0150 −1.76000
\(172\) 0 0
\(173\) −4.38925 −0.333708 −0.166854 0.985982i \(-0.553361\pi\)
−0.166854 + 0.985982i \(0.553361\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) −16.9930 −1.27728
\(178\) 0 0
\(179\) 6.98085 0.521774 0.260887 0.965369i \(-0.415985\pi\)
0.260887 + 0.965369i \(0.415985\pi\)
\(180\) 0 0
\(181\) 10.5217 0.782073 0.391036 0.920375i \(-0.372117\pi\)
0.391036 + 0.920375i \(0.372117\pi\)
\(182\) 0 0
\(183\) 33.1774 2.45254
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) −5.70160 −0.416942
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 1.29282 0.0935450 0.0467725 0.998906i \(-0.485106\pi\)
0.0467725 + 0.998906i \(0.485106\pi\)
\(192\) 0 0
\(193\) −12.5051 −0.900134 −0.450067 0.892995i \(-0.648600\pi\)
−0.450067 + 0.892995i \(0.648600\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 17.2608 1.22978 0.614891 0.788612i \(-0.289200\pi\)
0.614891 + 0.788612i \(0.289200\pi\)
\(198\) 0 0
\(199\) −19.9502 −1.41423 −0.707117 0.707097i \(-0.750004\pi\)
−0.707117 + 0.707097i \(0.750004\pi\)
\(200\) 0 0
\(201\) −4.26082 −0.300535
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) −20.1379 −1.39968
\(208\) 0 0
\(209\) −11.7670 −0.813938
\(210\) 0 0
\(211\) 7.53377 0.518646 0.259323 0.965791i \(-0.416501\pi\)
0.259323 + 0.965791i \(0.416501\pi\)
\(212\) 0 0
\(213\) −46.8534 −3.21034
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) 21.3489 1.44263
\(220\) 0 0
\(221\) −11.1518 −0.750149
\(222\) 0 0
\(223\) −22.2471 −1.48977 −0.744887 0.667191i \(-0.767496\pi\)
−0.744887 + 0.667191i \(0.767496\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) −12.4944 −0.829281 −0.414640 0.909985i \(-0.636093\pi\)
−0.414640 + 0.909985i \(0.636093\pi\)
\(228\) 0 0
\(229\) 2.75196 0.181854 0.0909272 0.995858i \(-0.471017\pi\)
0.0909272 + 0.995858i \(0.471017\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −24.0742 −1.57715 −0.788577 0.614936i \(-0.789182\pi\)
−0.788577 + 0.614936i \(0.789182\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 0.568120 0.0369033
\(238\) 0 0
\(239\) −12.0011 −0.776290 −0.388145 0.921598i \(-0.626884\pi\)
−0.388145 + 0.921598i \(0.626884\pi\)
\(240\) 0 0
\(241\) −11.4468 −0.737355 −0.368677 0.929557i \(-0.620189\pi\)
−0.368677 + 0.929557i \(0.620189\pi\)
\(242\) 0 0
\(243\) 4.95199 0.317670
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) −23.0150 −1.46441
\(248\) 0 0
\(249\) 0.408633 0.0258961
\(250\) 0 0
\(251\) 4.13791 0.261183 0.130591 0.991436i \(-0.458312\pi\)
0.130591 + 0.991436i \(0.458312\pi\)
\(252\) 0 0
\(253\) −10.2960 −0.647303
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −20.2697 −1.26439 −0.632196 0.774809i \(-0.717846\pi\)
−0.632196 + 0.774809i \(0.717846\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) 18.0576 1.11774
\(262\) 0 0
\(263\) −15.9590 −0.984074 −0.492037 0.870574i \(-0.663747\pi\)
−0.492037 + 0.870574i \(0.663747\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 56.0916 3.43275
\(268\) 0 0
\(269\) −10.6587 −0.649872 −0.324936 0.945736i \(-0.605343\pi\)
−0.324936 + 0.945736i \(0.605343\pi\)
\(270\) 0 0
\(271\) 32.3420 1.96464 0.982318 0.187221i \(-0.0599481\pi\)
0.982318 + 0.187221i \(0.0599481\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) −11.3636 −0.682770 −0.341385 0.939924i \(-0.610896\pi\)
−0.341385 + 0.939924i \(0.610896\pi\)
\(278\) 0 0
\(279\) 17.6440 1.05632
\(280\) 0 0
\(281\) 2.49337 0.148742 0.0743709 0.997231i \(-0.476305\pi\)
0.0743709 + 0.997231i \(0.476305\pi\)
\(282\) 0 0
\(283\) 17.7882 1.05740 0.528700 0.848809i \(-0.322680\pi\)
0.528700 + 0.848809i \(0.322680\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) −13.8265 −0.813321
\(290\) 0 0
\(291\) −29.6822 −1.74000
\(292\) 0 0
\(293\) 14.0630 0.821570 0.410785 0.911732i \(-0.365255\pi\)
0.410785 + 0.911732i \(0.365255\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 31.7498 1.84231
\(298\) 0 0
\(299\) −20.1379 −1.16461
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) 6.12819 0.352056
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) −10.1448 −0.578992 −0.289496 0.957179i \(-0.593488\pi\)
−0.289496 + 0.957179i \(0.593488\pi\)
\(308\) 0 0
\(309\) −33.4945 −1.90544
\(310\) 0 0
\(311\) 5.13868 0.291388 0.145694 0.989330i \(-0.453459\pi\)
0.145694 + 0.989330i \(0.453459\pi\)
\(312\) 0 0
\(313\) 3.22113 0.182069 0.0910347 0.995848i \(-0.470983\pi\)
0.0910347 + 0.995848i \(0.470983\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 11.7919 0.662301 0.331150 0.943578i \(-0.392563\pi\)
0.331150 + 0.943578i \(0.392563\pi\)
\(318\) 0 0
\(319\) 9.23239 0.516915
\(320\) 0 0
\(321\) 3.65840 0.204192
\(322\) 0 0
\(323\) 6.54956 0.364427
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) 26.0328 1.43962
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) −20.2514 −1.11312 −0.556560 0.830808i \(-0.687879\pi\)
−0.556560 + 0.830808i \(0.687879\pi\)
\(332\) 0 0
\(333\) −31.8172 −1.74357
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) −23.6086 −1.28604 −0.643022 0.765848i \(-0.722320\pi\)
−0.643022 + 0.765848i \(0.722320\pi\)
\(338\) 0 0
\(339\) −19.9889 −1.08565
\(340\) 0 0
\(341\) 9.02089 0.488509
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 34.8844 1.87269 0.936346 0.351078i \(-0.114185\pi\)
0.936346 + 0.351078i \(0.114185\pi\)
\(348\) 0 0
\(349\) 33.4410 1.79006 0.895028 0.446010i \(-0.147155\pi\)
0.895028 + 0.446010i \(0.147155\pi\)
\(350\) 0 0
\(351\) 62.0995 3.31462
\(352\) 0 0
\(353\) 13.5633 0.721904 0.360952 0.932584i \(-0.382452\pi\)
0.360952 + 0.932584i \(0.382452\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −20.9079 −1.10348 −0.551739 0.834017i \(-0.686036\pi\)
−0.551739 + 0.834017i \(0.686036\pi\)
\(360\) 0 0
\(361\) −5.48303 −0.288581
\(362\) 0 0
\(363\) −2.30198 −0.120822
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 19.7555 1.03123 0.515615 0.856820i \(-0.327564\pi\)
0.515615 + 0.856820i \(0.327564\pi\)
\(368\) 0 0
\(369\) −5.62983 −0.293077
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) −8.06795 −0.417743 −0.208871 0.977943i \(-0.566979\pi\)
−0.208871 + 0.977943i \(0.566979\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 18.0576 0.930016
\(378\) 0 0
\(379\) 28.1761 1.44731 0.723655 0.690162i \(-0.242461\pi\)
0.723655 + 0.690162i \(0.242461\pi\)
\(380\) 0 0
\(381\) 39.8632 2.04225
\(382\) 0 0
\(383\) 29.3412 1.49926 0.749632 0.661855i \(-0.230231\pi\)
0.749632 + 0.661855i \(0.230231\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 45.4953 2.31266
\(388\) 0 0
\(389\) −4.52293 −0.229322 −0.114661 0.993405i \(-0.536578\pi\)
−0.114661 + 0.993405i \(0.536578\pi\)
\(390\) 0 0
\(391\) 5.73080 0.289819
\(392\) 0 0
\(393\) −22.9699 −1.15868
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) 25.0352 1.25648 0.628239 0.778020i \(-0.283776\pi\)
0.628239 + 0.778020i \(0.283776\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 27.5942 1.37799 0.688995 0.724766i \(-0.258052\pi\)
0.688995 + 0.724766i \(0.258052\pi\)
\(402\) 0 0
\(403\) 17.6440 0.878909
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) −16.2673 −0.806340
\(408\) 0 0
\(409\) −31.5428 −1.55969 −0.779846 0.625971i \(-0.784703\pi\)
−0.779846 + 0.625971i \(0.784703\pi\)
\(410\) 0 0
\(411\) −16.8775 −0.832507
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) −46.5145 −2.27782
\(418\) 0 0
\(419\) −33.8997 −1.65611 −0.828055 0.560647i \(-0.810552\pi\)
−0.828055 + 0.560647i \(0.810552\pi\)
\(420\) 0 0
\(421\) −8.93350 −0.435392 −0.217696 0.976017i \(-0.569854\pi\)
−0.217696 + 0.976017i \(0.569854\pi\)
\(422\) 0 0
\(423\) 15.7790 0.767202
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) 60.9678 2.94355
\(430\) 0 0
\(431\) −9.10828 −0.438731 −0.219365 0.975643i \(-0.570399\pi\)
−0.219365 + 0.975643i \(0.570399\pi\)
\(432\) 0 0
\(433\) 38.2953 1.84035 0.920177 0.391504i \(-0.128045\pi\)
0.920177 + 0.391504i \(0.128045\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 11.8272 0.565773
\(438\) 0 0
\(439\) −9.14647 −0.436537 −0.218269 0.975889i \(-0.570041\pi\)
−0.218269 + 0.975889i \(0.570041\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −3.81570 −0.181289 −0.0906447 0.995883i \(-0.528893\pi\)
−0.0906447 + 0.995883i \(0.528893\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) −26.5996 −1.25812
\(448\) 0 0
\(449\) −37.1990 −1.75553 −0.877764 0.479092i \(-0.840966\pi\)
−0.877764 + 0.479092i \(0.840966\pi\)
\(450\) 0 0
\(451\) −2.87838 −0.135538
\(452\) 0 0
\(453\) 42.9193 2.01652
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 4.77887 0.223546 0.111773 0.993734i \(-0.464347\pi\)
0.111773 + 0.993734i \(0.464347\pi\)
\(458\) 0 0
\(459\) −17.6721 −0.824864
\(460\) 0 0
\(461\) 32.6903 1.52254 0.761269 0.648436i \(-0.224577\pi\)
0.761269 + 0.648436i \(0.224577\pi\)
\(462\) 0 0
\(463\) 6.36482 0.295798 0.147899 0.989002i \(-0.452749\pi\)
0.147899 + 0.989002i \(0.452749\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 26.0339 1.20470 0.602352 0.798231i \(-0.294231\pi\)
0.602352 + 0.798231i \(0.294231\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 2.77092 0.127677
\(472\) 0 0
\(473\) 23.2606 1.06952
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 56.5934 2.59123
\(478\) 0 0
\(479\) −1.66850 −0.0762357 −0.0381178 0.999273i \(-0.512136\pi\)
−0.0381178 + 0.999273i \(0.512136\pi\)
\(480\) 0 0
\(481\) −31.8172 −1.45074
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) −1.58671 −0.0719006 −0.0359503 0.999354i \(-0.511446\pi\)
−0.0359503 + 0.999354i \(0.511446\pi\)
\(488\) 0 0
\(489\) 31.6241 1.43009
\(490\) 0 0
\(491\) −26.8237 −1.21054 −0.605269 0.796021i \(-0.706934\pi\)
−0.605269 + 0.796021i \(0.706934\pi\)
\(492\) 0 0
\(493\) −5.13880 −0.231440
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) −22.0985 −0.989263 −0.494631 0.869103i \(-0.664697\pi\)
−0.494631 + 0.869103i \(0.664697\pi\)
\(500\) 0 0
\(501\) 22.3964 1.00060
\(502\) 0 0
\(503\) 0.937839 0.0418162 0.0209081 0.999781i \(-0.493344\pi\)
0.0209081 + 0.999781i \(0.493344\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 79.6877 3.53906
\(508\) 0 0
\(509\) 10.7200 0.475157 0.237578 0.971368i \(-0.423646\pi\)
0.237578 + 0.971368i \(0.423646\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) −36.4717 −1.61027
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 8.06739 0.354803
\(518\) 0 0
\(519\) −13.3566 −0.586288
\(520\) 0 0
\(521\) −43.4900 −1.90533 −0.952666 0.304018i \(-0.901672\pi\)
−0.952666 + 0.304018i \(0.901672\pi\)
\(522\) 0 0
\(523\) 1.58671 0.0693819 0.0346909 0.999398i \(-0.488955\pi\)
0.0346909 + 0.999398i \(0.488955\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −5.02108 −0.218722
\(528\) 0 0
\(529\) −12.6513 −0.550056
\(530\) 0 0
\(531\) −34.9573 −1.51702
\(532\) 0 0
\(533\) −5.62983 −0.243855
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 21.2429 0.916697
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) 6.70954 0.288466 0.144233 0.989544i \(-0.453929\pi\)
0.144233 + 0.989544i \(0.453929\pi\)
\(542\) 0 0
\(543\) 32.0178 1.37401
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 6.34916 0.271470 0.135735 0.990745i \(-0.456660\pi\)
0.135735 + 0.990745i \(0.456660\pi\)
\(548\) 0 0
\(549\) 68.2510 2.91288
\(550\) 0 0
\(551\) −10.6055 −0.451807
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 40.8845 1.73233 0.866166 0.499756i \(-0.166577\pi\)
0.866166 + 0.499756i \(0.166577\pi\)
\(558\) 0 0
\(559\) 45.4953 1.92425
\(560\) 0 0
\(561\) −17.3501 −0.732521
\(562\) 0 0
\(563\) −37.3938 −1.57596 −0.787981 0.615700i \(-0.788873\pi\)
−0.787981 + 0.615700i \(0.788873\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −0.893881 −0.0374734 −0.0187367 0.999824i \(-0.505964\pi\)
−0.0187367 + 0.999824i \(0.505964\pi\)
\(570\) 0 0
\(571\) 23.1786 0.969994 0.484997 0.874516i \(-0.338821\pi\)
0.484997 + 0.874516i \(0.338821\pi\)
\(572\) 0 0
\(573\) 3.93407 0.164348
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) −4.25249 −0.177033 −0.0885166 0.996075i \(-0.528213\pi\)
−0.0885166 + 0.996075i \(0.528213\pi\)
\(578\) 0 0
\(579\) −38.0531 −1.58143
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 28.9347 1.19835
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 23.3127 0.962218 0.481109 0.876661i \(-0.340234\pi\)
0.481109 + 0.876661i \(0.340234\pi\)
\(588\) 0 0
\(589\) −10.3625 −0.426979
\(590\) 0 0
\(591\) 52.5250 2.16059
\(592\) 0 0
\(593\) −0.347733 −0.0142797 −0.00713984 0.999975i \(-0.502273\pi\)
−0.00713984 + 0.999975i \(0.502273\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) −60.7089 −2.48465
\(598\) 0 0
\(599\) −31.1256 −1.27176 −0.635880 0.771788i \(-0.719363\pi\)
−0.635880 + 0.771788i \(0.719363\pi\)
\(600\) 0 0
\(601\) −28.9573 −1.18119 −0.590597 0.806967i \(-0.701108\pi\)
−0.590597 + 0.806967i \(0.701108\pi\)
\(602\) 0 0
\(603\) −8.76516 −0.356945
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) 22.0209 0.893799 0.446900 0.894584i \(-0.352528\pi\)
0.446900 + 0.894584i \(0.352528\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 15.7790 0.638351
\(612\) 0 0
\(613\) −3.18218 −0.128527 −0.0642636 0.997933i \(-0.520470\pi\)
−0.0642636 + 0.997933i \(0.520470\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 40.7426 1.64024 0.820119 0.572194i \(-0.193907\pi\)
0.820119 + 0.572194i \(0.193907\pi\)
\(618\) 0 0
\(619\) 15.5019 0.623075 0.311537 0.950234i \(-0.399156\pi\)
0.311537 + 0.950234i \(0.399156\pi\)
\(620\) 0 0
\(621\) −31.9124 −1.28060
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) −35.8071 −1.43000
\(628\) 0 0
\(629\) 9.05446 0.361025
\(630\) 0 0
\(631\) 9.15519 0.364462 0.182231 0.983256i \(-0.441668\pi\)
0.182231 + 0.983256i \(0.441668\pi\)
\(632\) 0 0
\(633\) 22.9254 0.911203
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) −96.3846 −3.81292
\(640\) 0 0
\(641\) 3.41967 0.135069 0.0675343 0.997717i \(-0.478487\pi\)
0.0675343 + 0.997717i \(0.478487\pi\)
\(642\) 0 0
\(643\) 43.9753 1.73421 0.867107 0.498121i \(-0.165977\pi\)
0.867107 + 0.498121i \(0.165977\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −14.5869 −0.573470 −0.286735 0.958010i \(-0.592570\pi\)
−0.286735 + 0.958010i \(0.592570\pi\)
\(648\) 0 0
\(649\) −17.8728 −0.701567
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −23.6812 −0.926715 −0.463358 0.886171i \(-0.653355\pi\)
−0.463358 + 0.886171i \(0.653355\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 43.9181 1.71341
\(658\) 0 0
\(659\) −22.0130 −0.857503 −0.428751 0.903422i \(-0.641046\pi\)
−0.428751 + 0.903422i \(0.641046\pi\)
\(660\) 0 0
\(661\) −20.1884 −0.785239 −0.392619 0.919701i \(-0.628431\pi\)
−0.392619 + 0.919701i \(0.628431\pi\)
\(662\) 0 0
\(663\) −33.9350 −1.31793
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) −9.27967 −0.359310
\(668\) 0 0
\(669\) −67.6982 −2.61736
\(670\) 0 0
\(671\) 34.8949 1.34710
\(672\) 0 0
\(673\) −32.7957 −1.26418 −0.632091 0.774894i \(-0.717803\pi\)
−0.632091 + 0.774894i \(0.717803\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 7.98825 0.307013 0.153507 0.988148i \(-0.450943\pi\)
0.153507 + 0.988148i \(0.450943\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) −38.0206 −1.45695
\(682\) 0 0
\(683\) −26.0090 −0.995207 −0.497604 0.867405i \(-0.665787\pi\)
−0.497604 + 0.867405i \(0.665787\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) 8.37425 0.319498
\(688\) 0 0
\(689\) 56.5934 2.15604
\(690\) 0 0
\(691\) 9.63696 0.366607 0.183304 0.983056i \(-0.441321\pi\)
0.183304 + 0.983056i \(0.441321\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 1.60212 0.0606847
\(698\) 0 0
\(699\) −73.2583 −2.77088
\(700\) 0 0
\(701\) −37.3814 −1.41188 −0.705938 0.708274i \(-0.749474\pi\)
−0.705938 + 0.708274i \(0.749474\pi\)
\(702\) 0 0
\(703\) 18.6866 0.704778
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) −7.21700 −0.271040 −0.135520 0.990775i \(-0.543270\pi\)
−0.135520 + 0.990775i \(0.543270\pi\)
\(710\) 0 0
\(711\) 1.16871 0.0438301
\(712\) 0 0
\(713\) −9.06709 −0.339565
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) −36.5197 −1.36385
\(718\) 0 0
\(719\) 45.3761 1.69224 0.846121 0.532991i \(-0.178932\pi\)
0.846121 + 0.532991i \(0.178932\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) −34.8329 −1.29545
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) 16.3119 0.604976 0.302488 0.953153i \(-0.402183\pi\)
0.302488 + 0.953153i \(0.402183\pi\)
\(728\) 0 0
\(729\) −19.1526 −0.709356
\(730\) 0 0
\(731\) −12.9470 −0.478860
\(732\) 0 0
\(733\) 29.1208 1.07560 0.537801 0.843072i \(-0.319255\pi\)
0.537801 + 0.843072i \(0.319255\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −4.48139 −0.165074
\(738\) 0 0
\(739\) −39.5275 −1.45404 −0.727021 0.686616i \(-0.759096\pi\)
−0.727021 + 0.686616i \(0.759096\pi\)
\(740\) 0 0
\(741\) −70.0351 −2.57280
\(742\) 0 0
\(743\) −15.7213 −0.576759 −0.288379 0.957516i \(-0.593116\pi\)
−0.288379 + 0.957516i \(0.593116\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 0.840622 0.0307568
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) −33.2601 −1.21368 −0.606840 0.794824i \(-0.707563\pi\)
−0.606840 + 0.794824i \(0.707563\pi\)
\(752\) 0 0
\(753\) 12.5917 0.458868
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 35.5431 1.29183 0.645917 0.763407i \(-0.276475\pi\)
0.645917 + 0.763407i \(0.276475\pi\)
\(758\) 0 0
\(759\) −31.3308 −1.13724
\(760\) 0 0
\(761\) 12.5071 0.453384 0.226692 0.973967i \(-0.427209\pi\)
0.226692 + 0.973967i \(0.427209\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) −34.9573 −1.26224
\(768\) 0 0
\(769\) 19.2730 0.695003 0.347501 0.937679i \(-0.387030\pi\)
0.347501 + 0.937679i \(0.387030\pi\)
\(770\) 0 0
\(771\) −61.6812 −2.22139
\(772\) 0 0
\(773\) −11.9871 −0.431146 −0.215573 0.976488i \(-0.569162\pi\)
−0.215573 + 0.976488i \(0.569162\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 3.30646 0.118466
\(780\) 0 0
\(781\) −49.2789 −1.76334
\(782\) 0 0
\(783\) 28.6158 1.02265
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) 28.0423 0.999599 0.499800 0.866141i \(-0.333407\pi\)
0.499800 + 0.866141i \(0.333407\pi\)
\(788\) 0 0
\(789\) −48.5635 −1.72891
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 68.2510 2.42366
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −14.4252 −0.510967 −0.255484 0.966813i \(-0.582235\pi\)
−0.255484 + 0.966813i \(0.582235\pi\)
\(798\) 0 0
\(799\) −4.49035 −0.158857
\(800\) 0 0
\(801\) 115.389 4.07707
\(802\) 0 0
\(803\) 22.4541 0.792390
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) −32.4346 −1.14175
\(808\) 0 0
\(809\) −17.8536 −0.627698 −0.313849 0.949473i \(-0.601619\pi\)
−0.313849 + 0.949473i \(0.601619\pi\)
\(810\) 0 0
\(811\) 11.0944 0.389576 0.194788 0.980845i \(-0.437598\pi\)
0.194788 + 0.980845i \(0.437598\pi\)
\(812\) 0 0
\(813\) 98.4172 3.45164
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) −26.7199 −0.934812
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −40.7820 −1.42330 −0.711650 0.702534i \(-0.752052\pi\)
−0.711650 + 0.702534i \(0.752052\pi\)
\(822\) 0 0
\(823\) −6.57749 −0.229277 −0.114638 0.993407i \(-0.536571\pi\)
−0.114638 + 0.993407i \(0.536571\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −28.3109 −0.984468 −0.492234 0.870463i \(-0.663820\pi\)
−0.492234 + 0.870463i \(0.663820\pi\)
\(828\) 0 0
\(829\) 7.95880 0.276421 0.138210 0.990403i \(-0.455865\pi\)
0.138210 + 0.990403i \(0.455865\pi\)
\(830\) 0 0
\(831\) −34.5795 −1.19955
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 27.9603 0.966449
\(838\) 0 0
\(839\) −39.8178 −1.37466 −0.687331 0.726344i \(-0.741218\pi\)
−0.687331 + 0.726344i \(0.741218\pi\)
\(840\) 0 0
\(841\) −20.6789 −0.713067
\(842\) 0 0
\(843\) 7.58736 0.261323
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) 54.1298 1.85773
\(850\) 0 0
\(851\) 16.3506 0.560491
\(852\) 0 0
\(853\) −21.5544 −0.738007 −0.369004 0.929428i \(-0.620301\pi\)
−0.369004 + 0.929428i \(0.620301\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 1.01622 0.0347133 0.0173566 0.999849i \(-0.494475\pi\)
0.0173566 + 0.999849i \(0.494475\pi\)
\(858\) 0 0
\(859\) −31.9605 −1.09048 −0.545240 0.838280i \(-0.683561\pi\)
−0.545240 + 0.838280i \(0.683561\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) −24.9315 −0.848679 −0.424339 0.905503i \(-0.639494\pi\)
−0.424339 + 0.905503i \(0.639494\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) −42.0742 −1.42891
\(868\) 0 0
\(869\) 0.597531 0.0202698
\(870\) 0 0
\(871\) −8.76516 −0.296996
\(872\) 0 0
\(873\) −61.0608 −2.06659
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 45.9702 1.55230 0.776151 0.630547i \(-0.217169\pi\)
0.776151 + 0.630547i \(0.217169\pi\)
\(878\) 0 0
\(879\) 42.7940 1.44341
\(880\) 0 0
\(881\) 20.7324 0.698493 0.349247 0.937031i \(-0.386438\pi\)
0.349247 + 0.937031i \(0.386438\pi\)
\(882\) 0 0
\(883\) −2.32252 −0.0781589 −0.0390795 0.999236i \(-0.512443\pi\)
−0.0390795 + 0.999236i \(0.512443\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 14.2867 0.479702 0.239851 0.970810i \(-0.422901\pi\)
0.239851 + 0.970810i \(0.422901\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) 36.5093 1.22311
\(892\) 0 0
\(893\) −9.26719 −0.310115
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) −61.2800 −2.04608
\(898\) 0 0
\(899\) 8.13045 0.271166
\(900\) 0 0
\(901\) −16.1052 −0.536543
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) −37.0737 −1.23101 −0.615506 0.788132i \(-0.711049\pi\)
−0.615506 + 0.788132i \(0.711049\pi\)
\(908\) 0 0
\(909\) 12.6066 0.418136
\(910\) 0 0
\(911\) 16.9023 0.559999 0.280000 0.960000i \(-0.409666\pi\)
0.280000 + 0.960000i \(0.409666\pi\)
\(912\) 0 0
\(913\) 0.429788 0.0142239
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) −1.01558 −0.0335010 −0.0167505 0.999860i \(-0.505332\pi\)
−0.0167505 + 0.999860i \(0.505332\pi\)
\(920\) 0 0
\(921\) −30.8707 −1.01722
\(922\) 0 0
\(923\) −96.3846 −3.17254
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) −68.9034 −2.26308
\(928\) 0 0
\(929\) −48.1114 −1.57848 −0.789242 0.614083i \(-0.789526\pi\)
−0.789242 + 0.614083i \(0.789526\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 0 0
\(933\) 15.6371 0.511936
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 3.00982 0.0983267 0.0491633 0.998791i \(-0.484345\pi\)
0.0491633 + 0.998791i \(0.484345\pi\)
\(938\) 0 0
\(939\) 9.80197 0.319875
\(940\) 0 0
\(941\) 19.5682 0.637904 0.318952 0.947771i \(-0.396669\pi\)
0.318952 + 0.947771i \(0.396669\pi\)
\(942\) 0 0
\(943\) 2.89312 0.0942130
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −10.5651 −0.343321 −0.171661 0.985156i \(-0.554913\pi\)
−0.171661 + 0.985156i \(0.554913\pi\)
\(948\) 0 0
\(949\) 43.9181 1.42564
\(950\) 0 0
\(951\) 35.8830 1.16359
\(952\) 0 0
\(953\) 48.3529 1.56630 0.783152 0.621830i \(-0.213611\pi\)
0.783152 + 0.621830i \(0.213611\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 28.0943 0.908161
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) −23.0558 −0.743736
\(962\) 0 0
\(963\) 7.52589 0.242518
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) −38.4047 −1.23501 −0.617506 0.786566i \(-0.711857\pi\)
−0.617506 + 0.786566i \(0.711857\pi\)
\(968\) 0 0
\(969\) 19.9304 0.640257
\(970\) 0 0
\(971\) 9.71416 0.311742 0.155871 0.987777i \(-0.450182\pi\)
0.155871 + 0.987777i \(0.450182\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −18.3191 −0.586080 −0.293040 0.956100i \(-0.594667\pi\)
−0.293040 + 0.956100i \(0.594667\pi\)
\(978\) 0 0
\(979\) 58.9954 1.88550
\(980\) 0 0
\(981\) 53.5535 1.70983
\(982\) 0 0
\(983\) −12.9575 −0.413280 −0.206640 0.978417i \(-0.566253\pi\)
−0.206640 + 0.978417i \(0.566253\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) −23.3797 −0.743430
\(990\) 0 0
\(991\) −25.4708 −0.809107 −0.404553 0.914514i \(-0.632573\pi\)
−0.404553 + 0.914514i \(0.632573\pi\)
\(992\) 0 0
\(993\) −61.6255 −1.95563
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) −49.7938 −1.57699 −0.788493 0.615043i \(-0.789138\pi\)
−0.788493 + 0.615043i \(0.789138\pi\)
\(998\) 0 0
\(999\) −50.4205 −1.59523
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 9800.2.a.cx.1.6 6
5.2 odd 4 1960.2.g.f.1569.1 12
5.3 odd 4 1960.2.g.f.1569.12 12
5.4 even 2 9800.2.a.cv.1.1 6
7.2 even 3 1400.2.q.n.1201.1 12
7.4 even 3 1400.2.q.n.401.1 12
7.6 odd 2 9800.2.a.cw.1.1 6
35.2 odd 12 280.2.bg.a.249.12 yes 24
35.4 even 6 1400.2.q.o.401.6 12
35.9 even 6 1400.2.q.o.1201.6 12
35.13 even 4 1960.2.g.e.1569.1 12
35.18 odd 12 280.2.bg.a.9.12 yes 24
35.23 odd 12 280.2.bg.a.249.1 yes 24
35.27 even 4 1960.2.g.e.1569.12 12
35.32 odd 12 280.2.bg.a.9.1 24
35.34 odd 2 9800.2.a.cy.1.6 6
140.23 even 12 560.2.bw.f.529.12 24
140.67 even 12 560.2.bw.f.289.12 24
140.107 even 12 560.2.bw.f.529.1 24
140.123 even 12 560.2.bw.f.289.1 24
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
280.2.bg.a.9.1 24 35.32 odd 12
280.2.bg.a.9.12 yes 24 35.18 odd 12
280.2.bg.a.249.1 yes 24 35.23 odd 12
280.2.bg.a.249.12 yes 24 35.2 odd 12
560.2.bw.f.289.1 24 140.123 even 12
560.2.bw.f.289.12 24 140.67 even 12
560.2.bw.f.529.1 24 140.107 even 12
560.2.bw.f.529.12 24 140.23 even 12
1400.2.q.n.401.1 12 7.4 even 3
1400.2.q.n.1201.1 12 7.2 even 3
1400.2.q.o.401.6 12 35.4 even 6
1400.2.q.o.1201.6 12 35.9 even 6
1960.2.g.e.1569.1 12 35.13 even 4
1960.2.g.e.1569.12 12 35.27 even 4
1960.2.g.f.1569.1 12 5.2 odd 4
1960.2.g.f.1569.12 12 5.3 odd 4
9800.2.a.cv.1.1 6 5.4 even 2
9800.2.a.cw.1.1 6 7.6 odd 2
9800.2.a.cx.1.6 6 1.1 even 1 trivial
9800.2.a.cy.1.6 6 35.34 odd 2