Properties

Label 6840.2.a.bg.1.1
Level $6840$
Weight $2$
Character 6840.1
Self dual yes
Analytic conductor $54.618$
Analytic rank $1$
Dimension $3$
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] = [6840,2,Mod(1,6840)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(6840, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 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("6840.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 6840 = 2^{3} \cdot 3^{2} \cdot 5 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6840.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: \(54.6176749826\)
Analytic rank: \(1\)
Dimension: \(3\)
Coefficient field: 3.3.229.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - 4x - 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 760)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-0.254102\) of defining polynomial
Character \(\chi\) \(=\) 6840.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.00000 q^{5} -3.18953 q^{7} +O(q^{10})\) \(q-1.00000 q^{5} -3.18953 q^{7} -0.681331 q^{13} +1.18953 q^{17} +1.00000 q^{19} +2.17313 q^{23} +1.00000 q^{25} -2.81047 q^{29} +6.37907 q^{31} +3.18953 q^{35} +7.87086 q^{37} -0.983593 q^{41} +1.36266 q^{43} -11.7417 q^{47} +3.17313 q^{49} +1.69774 q^{53} -11.5358 q^{59} +7.36266 q^{61} +0.681331 q^{65} +7.02759 q^{67} +12.7581 q^{71} -5.53579 q^{73} +5.36266 q^{79} -2.37907 q^{83} -1.18953 q^{85} -3.01641 q^{89} +2.17313 q^{91} -1.00000 q^{95} +4.88727 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 3 q^{5} - q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 3 q - 3 q^{5} - q^{7} + 5 q^{13} - 5 q^{17} + 3 q^{19} + q^{23} + 3 q^{25} - 17 q^{29} + 2 q^{31} + q^{35} + 8 q^{37} - 6 q^{41} - 10 q^{43} - 4 q^{47} + 4 q^{49} - 5 q^{53} - 15 q^{59} + 8 q^{61} - 5 q^{65} + 3 q^{67} + 4 q^{71} + 3 q^{73} + 2 q^{79} + 10 q^{83} + 5 q^{85} - 6 q^{89} + q^{91} - 3 q^{95} - 4 q^{97}+O(q^{100}) \) 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
\(3\) 0 0
\(4\) 0 0
\(5\) −1.00000 −0.447214
\(6\) 0 0
\(7\) −3.18953 −1.20553 −0.602765 0.797919i \(-0.705934\pi\)
−0.602765 + 0.797919i \(0.705934\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(12\) 0 0
\(13\) −0.681331 −0.188967 −0.0944836 0.995526i \(-0.530120\pi\)
−0.0944836 + 0.995526i \(0.530120\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 1.18953 0.288504 0.144252 0.989541i \(-0.453922\pi\)
0.144252 + 0.989541i \(0.453922\pi\)
\(18\) 0 0
\(19\) 1.00000 0.229416
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 2.17313 0.453128 0.226564 0.973996i \(-0.427251\pi\)
0.226564 + 0.973996i \(0.427251\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −2.81047 −0.521890 −0.260945 0.965354i \(-0.584034\pi\)
−0.260945 + 0.965354i \(0.584034\pi\)
\(30\) 0 0
\(31\) 6.37907 1.14571 0.572857 0.819655i \(-0.305835\pi\)
0.572857 + 0.819655i \(0.305835\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 3.18953 0.539130
\(36\) 0 0
\(37\) 7.87086 1.29396 0.646981 0.762506i \(-0.276031\pi\)
0.646981 + 0.762506i \(0.276031\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −0.983593 −0.153611 −0.0768057 0.997046i \(-0.524472\pi\)
−0.0768057 + 0.997046i \(0.524472\pi\)
\(42\) 0 0
\(43\) 1.36266 0.207804 0.103902 0.994588i \(-0.466867\pi\)
0.103902 + 0.994588i \(0.466867\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −11.7417 −1.71271 −0.856354 0.516390i \(-0.827276\pi\)
−0.856354 + 0.516390i \(0.827276\pi\)
\(48\) 0 0
\(49\) 3.17313 0.453304
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 1.69774 0.233202 0.116601 0.993179i \(-0.462800\pi\)
0.116601 + 0.993179i \(0.462800\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −11.5358 −1.50183 −0.750916 0.660398i \(-0.770388\pi\)
−0.750916 + 0.660398i \(0.770388\pi\)
\(60\) 0 0
\(61\) 7.36266 0.942692 0.471346 0.881948i \(-0.343768\pi\)
0.471346 + 0.881948i \(0.343768\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 0.681331 0.0845087
\(66\) 0 0
\(67\) 7.02759 0.858556 0.429278 0.903172i \(-0.358768\pi\)
0.429278 + 0.903172i \(0.358768\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 12.7581 1.51411 0.757056 0.653350i \(-0.226637\pi\)
0.757056 + 0.653350i \(0.226637\pi\)
\(72\) 0 0
\(73\) −5.53579 −0.647915 −0.323958 0.946072i \(-0.605013\pi\)
−0.323958 + 0.946072i \(0.605013\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 5.36266 0.603347 0.301673 0.953411i \(-0.402455\pi\)
0.301673 + 0.953411i \(0.402455\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) −2.37907 −0.261137 −0.130568 0.991439i \(-0.541680\pi\)
−0.130568 + 0.991439i \(0.541680\pi\)
\(84\) 0 0
\(85\) −1.18953 −0.129023
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −3.01641 −0.319738 −0.159869 0.987138i \(-0.551107\pi\)
−0.159869 + 0.987138i \(0.551107\pi\)
\(90\) 0 0
\(91\) 2.17313 0.227806
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −1.00000 −0.102598
\(96\) 0 0
\(97\) 4.88727 0.496227 0.248114 0.968731i \(-0.420189\pi\)
0.248114 + 0.968731i \(0.420189\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) −12.1208 −1.20606 −0.603032 0.797717i \(-0.706041\pi\)
−0.603032 + 0.797717i \(0.706041\pi\)
\(102\) 0 0
\(103\) −17.8709 −1.76087 −0.880434 0.474168i \(-0.842749\pi\)
−0.880434 + 0.474168i \(0.842749\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −14.4231 −1.39433 −0.697165 0.716911i \(-0.745555\pi\)
−0.697165 + 0.716911i \(0.745555\pi\)
\(108\) 0 0
\(109\) 10.2059 0.977552 0.488776 0.872409i \(-0.337444\pi\)
0.488776 + 0.872409i \(0.337444\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 1.49180 0.140336 0.0701682 0.997535i \(-0.477646\pi\)
0.0701682 + 0.997535i \(0.477646\pi\)
\(114\) 0 0
\(115\) −2.17313 −0.202645
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) −3.79406 −0.347801
\(120\) 0 0
\(121\) −11.0000 −1.00000
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) −1.00000 −0.0894427
\(126\) 0 0
\(127\) 8.24993 0.732063 0.366032 0.930602i \(-0.380716\pi\)
0.366032 + 0.930602i \(0.380716\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) −8.75814 −0.765202 −0.382601 0.923914i \(-0.624972\pi\)
−0.382601 + 0.923914i \(0.624972\pi\)
\(132\) 0 0
\(133\) −3.18953 −0.276568
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 11.9149 1.01795 0.508977 0.860780i \(-0.330024\pi\)
0.508977 + 0.860780i \(0.330024\pi\)
\(138\) 0 0
\(139\) −23.4835 −1.99184 −0.995920 0.0902352i \(-0.971238\pi\)
−0.995920 + 0.0902352i \(0.971238\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) 2.81047 0.233396
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 18.8461 1.54393 0.771967 0.635662i \(-0.219273\pi\)
0.771967 + 0.635662i \(0.219273\pi\)
\(150\) 0 0
\(151\) −16.0880 −1.30922 −0.654611 0.755966i \(-0.727167\pi\)
−0.654611 + 0.755966i \(0.727167\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) −6.37907 −0.512379
\(156\) 0 0
\(157\) −13.7417 −1.09671 −0.548355 0.836246i \(-0.684746\pi\)
−0.548355 + 0.836246i \(0.684746\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) −6.93126 −0.546260
\(162\) 0 0
\(163\) −14.6373 −1.14648 −0.573242 0.819386i \(-0.694315\pi\)
−0.573242 + 0.819386i \(0.694315\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −17.9588 −1.38970 −0.694849 0.719156i \(-0.744529\pi\)
−0.694849 + 0.719156i \(0.744529\pi\)
\(168\) 0 0
\(169\) −12.5358 −0.964291
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) −6.85446 −0.521135 −0.260567 0.965456i \(-0.583910\pi\)
−0.260567 + 0.965456i \(0.583910\pi\)
\(174\) 0 0
\(175\) −3.18953 −0.241106
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) −5.01641 −0.374944 −0.187472 0.982270i \(-0.560029\pi\)
−0.187472 + 0.982270i \(0.560029\pi\)
\(180\) 0 0
\(181\) 16.7253 1.24318 0.621592 0.783341i \(-0.286486\pi\)
0.621592 + 0.783341i \(0.286486\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) −7.87086 −0.578677
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 11.2775 0.816013 0.408006 0.912979i \(-0.366224\pi\)
0.408006 + 0.912979i \(0.366224\pi\)
\(192\) 0 0
\(193\) 13.5798 0.977494 0.488747 0.872426i \(-0.337454\pi\)
0.488747 + 0.872426i \(0.337454\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 2.00000 0.142494 0.0712470 0.997459i \(-0.477302\pi\)
0.0712470 + 0.997459i \(0.477302\pi\)
\(198\) 0 0
\(199\) 25.6566 1.81875 0.909374 0.415980i \(-0.136562\pi\)
0.909374 + 0.415980i \(0.136562\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 8.96408 0.629155
\(204\) 0 0
\(205\) 0.983593 0.0686971
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 0 0
\(210\) 0 0
\(211\) 16.2939 1.12172 0.560860 0.827911i \(-0.310471\pi\)
0.560860 + 0.827911i \(0.310471\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) −1.36266 −0.0929327
\(216\) 0 0
\(217\) −20.3463 −1.38119
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) −0.810466 −0.0545178
\(222\) 0 0
\(223\) −24.2499 −1.62390 −0.811948 0.583730i \(-0.801593\pi\)
−0.811948 + 0.583730i \(0.801593\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) −5.31867 −0.353012 −0.176506 0.984300i \(-0.556480\pi\)
−0.176506 + 0.984300i \(0.556480\pi\)
\(228\) 0 0
\(229\) 11.7089 0.773747 0.386873 0.922133i \(-0.373555\pi\)
0.386873 + 0.922133i \(0.373555\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −27.5163 −1.80265 −0.901325 0.433142i \(-0.857405\pi\)
−0.901325 + 0.433142i \(0.857405\pi\)
\(234\) 0 0
\(235\) 11.7417 0.765946
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) −14.9313 −0.965823 −0.482912 0.875669i \(-0.660421\pi\)
−0.482912 + 0.875669i \(0.660421\pi\)
\(240\) 0 0
\(241\) −9.32985 −0.600988 −0.300494 0.953784i \(-0.597152\pi\)
−0.300494 + 0.953784i \(0.597152\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) −3.17313 −0.202724
\(246\) 0 0
\(247\) −0.681331 −0.0433520
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) −1.27468 −0.0804569 −0.0402285 0.999191i \(-0.512809\pi\)
−0.0402285 + 0.999191i \(0.512809\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −31.2887 −1.95174 −0.975868 0.218363i \(-0.929928\pi\)
−0.975868 + 0.218363i \(0.929928\pi\)
\(258\) 0 0
\(259\) −25.1044 −1.55991
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) −25.5163 −1.57340 −0.786700 0.617335i \(-0.788212\pi\)
−0.786700 + 0.617335i \(0.788212\pi\)
\(264\) 0 0
\(265\) −1.69774 −0.104291
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) −26.7581 −1.63147 −0.815736 0.578424i \(-0.803668\pi\)
−0.815736 + 0.578424i \(0.803668\pi\)
\(270\) 0 0
\(271\) 14.1731 0.860956 0.430478 0.902601i \(-0.358345\pi\)
0.430478 + 0.902601i \(0.358345\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) 6.08798 0.365791 0.182896 0.983132i \(-0.441453\pi\)
0.182896 + 0.983132i \(0.441453\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 15.7089 0.937115 0.468558 0.883433i \(-0.344774\pi\)
0.468558 + 0.883433i \(0.344774\pi\)
\(282\) 0 0
\(283\) 0.346255 0.0205827 0.0102913 0.999947i \(-0.496724\pi\)
0.0102913 + 0.999947i \(0.496724\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 3.13720 0.185183
\(288\) 0 0
\(289\) −15.5850 −0.916765
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) −6.71414 −0.392244 −0.196122 0.980579i \(-0.562835\pi\)
−0.196122 + 0.980579i \(0.562835\pi\)
\(294\) 0 0
\(295\) 11.5358 0.671640
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) −1.48062 −0.0856264
\(300\) 0 0
\(301\) −4.34625 −0.250514
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) −7.36266 −0.421585
\(306\) 0 0
\(307\) −17.6126 −1.00520 −0.502602 0.864518i \(-0.667624\pi\)
−0.502602 + 0.864518i \(0.667624\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 6.58501 0.373402 0.186701 0.982417i \(-0.440221\pi\)
0.186701 + 0.982417i \(0.440221\pi\)
\(312\) 0 0
\(313\) −18.6178 −1.05234 −0.526171 0.850379i \(-0.676373\pi\)
−0.526171 + 0.850379i \(0.676373\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 17.7857 0.998946 0.499473 0.866330i \(-0.333527\pi\)
0.499473 + 0.866330i \(0.333527\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 1.18953 0.0661874
\(324\) 0 0
\(325\) −0.681331 −0.0377934
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 37.4506 2.06472
\(330\) 0 0
\(331\) −3.53579 −0.194345 −0.0971723 0.995268i \(-0.530980\pi\)
−0.0971723 + 0.995268i \(0.530980\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) −7.02759 −0.383958
\(336\) 0 0
\(337\) −7.87086 −0.428753 −0.214377 0.976751i \(-0.568772\pi\)
−0.214377 + 0.976751i \(0.568772\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) 12.2059 0.659059
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −15.6537 −0.840337 −0.420169 0.907446i \(-0.638029\pi\)
−0.420169 + 0.907446i \(0.638029\pi\)
\(348\) 0 0
\(349\) −21.4178 −1.14647 −0.573235 0.819391i \(-0.694312\pi\)
−0.573235 + 0.819391i \(0.694312\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) −3.15672 −0.168015 −0.0840076 0.996465i \(-0.526772\pi\)
−0.0840076 + 0.996465i \(0.526772\pi\)
\(354\) 0 0
\(355\) −12.7581 −0.677132
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −17.8269 −0.940866 −0.470433 0.882436i \(-0.655902\pi\)
−0.470433 + 0.882436i \(0.655902\pi\)
\(360\) 0 0
\(361\) 1.00000 0.0526316
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 5.53579 0.289756
\(366\) 0 0
\(367\) 19.2252 1.00355 0.501773 0.864999i \(-0.332681\pi\)
0.501773 + 0.864999i \(0.332681\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) −5.41499 −0.281132
\(372\) 0 0
\(373\) 9.95601 0.515503 0.257751 0.966211i \(-0.417018\pi\)
0.257751 + 0.966211i \(0.417018\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 1.91486 0.0986201
\(378\) 0 0
\(379\) −19.6238 −1.00801 −0.504003 0.863702i \(-0.668140\pi\)
−0.504003 + 0.863702i \(0.668140\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) −11.9037 −0.608250 −0.304125 0.952632i \(-0.598364\pi\)
−0.304125 + 0.952632i \(0.598364\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) −10.3463 −0.524576 −0.262288 0.964990i \(-0.584477\pi\)
−0.262288 + 0.964990i \(0.584477\pi\)
\(390\) 0 0
\(391\) 2.58501 0.130730
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) −5.36266 −0.269825
\(396\) 0 0
\(397\) 26.1536 1.31261 0.656306 0.754495i \(-0.272118\pi\)
0.656306 + 0.754495i \(0.272118\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 32.7253 1.63422 0.817112 0.576479i \(-0.195574\pi\)
0.817112 + 0.576479i \(0.195574\pi\)
\(402\) 0 0
\(403\) −4.34625 −0.216502
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) 4.03281 0.199410 0.0997049 0.995017i \(-0.468210\pi\)
0.0997049 + 0.995017i \(0.468210\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 36.7938 1.81050
\(414\) 0 0
\(415\) 2.37907 0.116784
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 15.8297 0.773332 0.386666 0.922220i \(-0.373627\pi\)
0.386666 + 0.922220i \(0.373627\pi\)
\(420\) 0 0
\(421\) 6.05233 0.294973 0.147486 0.989064i \(-0.452882\pi\)
0.147486 + 0.989064i \(0.452882\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 1.18953 0.0577009
\(426\) 0 0
\(427\) −23.4835 −1.13644
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) −25.7089 −1.23835 −0.619177 0.785251i \(-0.712534\pi\)
−0.619177 + 0.785251i \(0.712534\pi\)
\(432\) 0 0
\(433\) 2.57383 0.123690 0.0618452 0.998086i \(-0.480301\pi\)
0.0618452 + 0.998086i \(0.480301\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 2.17313 0.103955
\(438\) 0 0
\(439\) 26.8133 1.27973 0.639865 0.768488i \(-0.278990\pi\)
0.639865 + 0.768488i \(0.278990\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −10.7909 −0.512693 −0.256347 0.966585i \(-0.582519\pi\)
−0.256347 + 0.966585i \(0.582519\pi\)
\(444\) 0 0
\(445\) 3.01641 0.142991
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 29.4835 1.39141 0.695705 0.718327i \(-0.255092\pi\)
0.695705 + 0.718327i \(0.255092\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) −2.17313 −0.101878
\(456\) 0 0
\(457\) −11.8269 −0.553238 −0.276619 0.960980i \(-0.589214\pi\)
−0.276619 + 0.960980i \(0.589214\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 7.45065 0.347011 0.173506 0.984833i \(-0.444491\pi\)
0.173506 + 0.984833i \(0.444491\pi\)
\(462\) 0 0
\(463\) −37.4506 −1.74048 −0.870240 0.492629i \(-0.836036\pi\)
−0.870240 + 0.492629i \(0.836036\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 27.4835 1.27178 0.635891 0.771779i \(-0.280633\pi\)
0.635891 + 0.771779i \(0.280633\pi\)
\(468\) 0 0
\(469\) −22.4147 −1.03502
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) 1.00000 0.0458831
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) −10.6597 −0.487054 −0.243527 0.969894i \(-0.578304\pi\)
−0.243527 + 0.969894i \(0.578304\pi\)
\(480\) 0 0
\(481\) −5.36266 −0.244516
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) −4.88727 −0.221920
\(486\) 0 0
\(487\) 7.92604 0.359163 0.179581 0.983743i \(-0.442526\pi\)
0.179581 + 0.983743i \(0.442526\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 1.68656 0.0761133 0.0380567 0.999276i \(-0.487883\pi\)
0.0380567 + 0.999276i \(0.487883\pi\)
\(492\) 0 0
\(493\) −3.34314 −0.150568
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −40.6925 −1.82531
\(498\) 0 0
\(499\) −20.3463 −0.910823 −0.455412 0.890281i \(-0.650508\pi\)
−0.455412 + 0.890281i \(0.650508\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) −1.06874 −0.0476526 −0.0238263 0.999716i \(-0.507585\pi\)
−0.0238263 + 0.999716i \(0.507585\pi\)
\(504\) 0 0
\(505\) 12.1208 0.539368
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 19.0164 0.842887 0.421444 0.906855i \(-0.361524\pi\)
0.421444 + 0.906855i \(0.361524\pi\)
\(510\) 0 0
\(511\) 17.6566 0.781081
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 17.8709 0.787484
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) −9.07158 −0.397433 −0.198717 0.980057i \(-0.563677\pi\)
−0.198717 + 0.980057i \(0.563677\pi\)
\(522\) 0 0
\(523\) −13.4946 −0.590079 −0.295040 0.955485i \(-0.595333\pi\)
−0.295040 + 0.955485i \(0.595333\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 7.58812 0.330544
\(528\) 0 0
\(529\) −18.2775 −0.794675
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 0.670152 0.0290275
\(534\) 0 0
\(535\) 14.4231 0.623563
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) 4.63734 0.199375 0.0996874 0.995019i \(-0.468216\pi\)
0.0996874 + 0.995019i \(0.468216\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) −10.2059 −0.437174
\(546\) 0 0
\(547\) −18.1291 −0.775146 −0.387573 0.921839i \(-0.626686\pi\)
−0.387573 + 0.921839i \(0.626686\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) −2.81047 −0.119730
\(552\) 0 0
\(553\) −17.1044 −0.727353
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −32.2968 −1.36846 −0.684229 0.729267i \(-0.739861\pi\)
−0.684229 + 0.729267i \(0.739861\pi\)
\(558\) 0 0
\(559\) −0.928423 −0.0392681
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 31.3871 1.32281 0.661405 0.750029i \(-0.269960\pi\)
0.661405 + 0.750029i \(0.269960\pi\)
\(564\) 0 0
\(565\) −1.49180 −0.0627604
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 31.7969 1.33300 0.666498 0.745507i \(-0.267793\pi\)
0.666498 + 0.745507i \(0.267793\pi\)
\(570\) 0 0
\(571\) 18.3134 0.766394 0.383197 0.923667i \(-0.374823\pi\)
0.383197 + 0.923667i \(0.374823\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 2.17313 0.0906257
\(576\) 0 0
\(577\) 26.7282 1.11271 0.556354 0.830945i \(-0.312200\pi\)
0.556354 + 0.830945i \(0.312200\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 7.58812 0.314808
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −24.3463 −1.00488 −0.502439 0.864613i \(-0.667564\pi\)
−0.502439 + 0.864613i \(0.667564\pi\)
\(588\) 0 0
\(589\) 6.37907 0.262845
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) −12.7253 −0.522566 −0.261283 0.965262i \(-0.584146\pi\)
−0.261283 + 0.965262i \(0.584146\pi\)
\(594\) 0 0
\(595\) 3.79406 0.155541
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) −10.6373 −0.434630 −0.217315 0.976102i \(-0.569730\pi\)
−0.217315 + 0.976102i \(0.569730\pi\)
\(600\) 0 0
\(601\) 11.5329 0.470439 0.235219 0.971942i \(-0.424419\pi\)
0.235219 + 0.971942i \(0.424419\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 11.0000 0.447214
\(606\) 0 0
\(607\) −18.2827 −0.742074 −0.371037 0.928618i \(-0.620998\pi\)
−0.371037 + 0.928618i \(0.620998\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 8.00000 0.323645
\(612\) 0 0
\(613\) −6.84612 −0.276512 −0.138256 0.990397i \(-0.544150\pi\)
−0.138256 + 0.990397i \(0.544150\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −15.0820 −0.607180 −0.303590 0.952803i \(-0.598185\pi\)
−0.303590 + 0.952803i \(0.598185\pi\)
\(618\) 0 0
\(619\) −42.7253 −1.71728 −0.858638 0.512583i \(-0.828689\pi\)
−0.858638 + 0.512583i \(0.828689\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 9.62093 0.385454
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 9.36266 0.373314
\(630\) 0 0
\(631\) 26.7909 1.06653 0.533265 0.845948i \(-0.320965\pi\)
0.533265 + 0.845948i \(0.320965\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) −8.24993 −0.327389
\(636\) 0 0
\(637\) −2.16195 −0.0856595
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) −28.1208 −1.11070 −0.555352 0.831615i \(-0.687417\pi\)
−0.555352 + 0.831615i \(0.687417\pi\)
\(642\) 0 0
\(643\) 15.1372 0.596953 0.298477 0.954417i \(-0.403522\pi\)
0.298477 + 0.954417i \(0.403522\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −1.06874 −0.0420164 −0.0210082 0.999779i \(-0.506688\pi\)
−0.0210082 + 0.999779i \(0.506688\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 21.3298 0.834701 0.417351 0.908745i \(-0.362959\pi\)
0.417351 + 0.908745i \(0.362959\pi\)
\(654\) 0 0
\(655\) 8.75814 0.342209
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) −45.4639 −1.77102 −0.885512 0.464617i \(-0.846192\pi\)
−0.885512 + 0.464617i \(0.846192\pi\)
\(660\) 0 0
\(661\) 28.6074 1.11270 0.556349 0.830949i \(-0.312202\pi\)
0.556349 + 0.830949i \(0.312202\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 3.18953 0.123685
\(666\) 0 0
\(667\) −6.10750 −0.236483
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) 47.1840 1.81881 0.909405 0.415911i \(-0.136537\pi\)
0.909405 + 0.415911i \(0.136537\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) −20.8573 −0.801611 −0.400806 0.916163i \(-0.631270\pi\)
−0.400806 + 0.916163i \(0.631270\pi\)
\(678\) 0 0
\(679\) −15.5881 −0.598217
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 17.2887 0.661534 0.330767 0.943713i \(-0.392693\pi\)
0.330767 + 0.943713i \(0.392693\pi\)
\(684\) 0 0
\(685\) −11.9149 −0.455243
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) −1.15672 −0.0440675
\(690\) 0 0
\(691\) −35.8297 −1.36303 −0.681513 0.731806i \(-0.738678\pi\)
−0.681513 + 0.731806i \(0.738678\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 23.4835 0.890778
\(696\) 0 0
\(697\) −1.17002 −0.0443176
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 38.0880 1.43856 0.719282 0.694719i \(-0.244471\pi\)
0.719282 + 0.694719i \(0.244471\pi\)
\(702\) 0 0
\(703\) 7.87086 0.296855
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 38.6597 1.45395
\(708\) 0 0
\(709\) −15.2747 −0.573653 −0.286826 0.957983i \(-0.592600\pi\)
−0.286826 + 0.957983i \(0.592600\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 13.8625 0.519156
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 27.8597 1.03899 0.519495 0.854473i \(-0.326120\pi\)
0.519495 + 0.854473i \(0.326120\pi\)
\(720\) 0 0
\(721\) 56.9997 2.12278
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) −2.81047 −0.104378
\(726\) 0 0
\(727\) −39.4311 −1.46242 −0.731210 0.682153i \(-0.761044\pi\)
−0.731210 + 0.682153i \(0.761044\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 1.62093 0.0599523
\(732\) 0 0
\(733\) 33.1372 1.22395 0.611975 0.790877i \(-0.290375\pi\)
0.611975 + 0.790877i \(0.290375\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) −25.8625 −0.951368 −0.475684 0.879616i \(-0.657799\pi\)
−0.475684 + 0.879616i \(0.657799\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 28.6842 1.05232 0.526160 0.850386i \(-0.323631\pi\)
0.526160 + 0.850386i \(0.323631\pi\)
\(744\) 0 0
\(745\) −18.8461 −0.690468
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 46.0028 1.68091
\(750\) 0 0
\(751\) −34.2968 −1.25151 −0.625753 0.780021i \(-0.715208\pi\)
−0.625753 + 0.780021i \(0.715208\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 16.0880 0.585502
\(756\) 0 0
\(757\) 20.6597 0.750889 0.375445 0.926845i \(-0.377490\pi\)
0.375445 + 0.926845i \(0.377490\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −38.1236 −1.38198 −0.690990 0.722864i \(-0.742825\pi\)
−0.690990 + 0.722864i \(0.742825\pi\)
\(762\) 0 0
\(763\) −32.5522 −1.17847
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 7.85969 0.283797
\(768\) 0 0
\(769\) 20.3267 0.733001 0.366500 0.930418i \(-0.380556\pi\)
0.366500 + 0.930418i \(0.380556\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) −26.4559 −0.951552 −0.475776 0.879567i \(-0.657833\pi\)
−0.475776 + 0.879567i \(0.657833\pi\)
\(774\) 0 0
\(775\) 6.37907 0.229143
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) −0.983593 −0.0352409
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 13.7417 0.490463
\(786\) 0 0
\(787\) −27.3738 −0.975772 −0.487886 0.872907i \(-0.662232\pi\)
−0.487886 + 0.872907i \(0.662232\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) −4.75814 −0.169180
\(792\) 0 0
\(793\) −5.01641 −0.178138
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −36.4454 −1.29096 −0.645481 0.763776i \(-0.723343\pi\)
−0.645481 + 0.763776i \(0.723343\pi\)
\(798\) 0 0
\(799\) −13.9672 −0.494124
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 0 0
\(804\) 0 0
\(805\) 6.93126 0.244295
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) −52.2444 −1.83682 −0.918408 0.395634i \(-0.870525\pi\)
−0.918408 + 0.395634i \(0.870525\pi\)
\(810\) 0 0
\(811\) 32.8984 1.15522 0.577610 0.816313i \(-0.303985\pi\)
0.577610 + 0.816313i \(0.303985\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 14.6373 0.512724
\(816\) 0 0
\(817\) 1.36266 0.0476735
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 4.44470 0.155121 0.0775605 0.996988i \(-0.475287\pi\)
0.0775605 + 0.996988i \(0.475287\pi\)
\(822\) 0 0
\(823\) −27.7774 −0.968259 −0.484129 0.874996i \(-0.660864\pi\)
−0.484129 + 0.874996i \(0.660864\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −46.5110 −1.61735 −0.808674 0.588257i \(-0.799814\pi\)
−0.808674 + 0.588257i \(0.799814\pi\)
\(828\) 0 0
\(829\) 28.2388 0.980772 0.490386 0.871505i \(-0.336856\pi\)
0.490386 + 0.871505i \(0.336856\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 3.77454 0.130780
\(834\) 0 0
\(835\) 17.9588 0.621492
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) −9.01641 −0.311281 −0.155640 0.987814i \(-0.549744\pi\)
−0.155640 + 0.987814i \(0.549744\pi\)
\(840\) 0 0
\(841\) −21.1013 −0.727630
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 12.5358 0.431244
\(846\) 0 0
\(847\) 35.0849 1.20553
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 17.1044 0.586331
\(852\) 0 0
\(853\) 5.30749 0.181725 0.0908625 0.995863i \(-0.471038\pi\)
0.0908625 + 0.995863i \(0.471038\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 38.9258 1.32968 0.664839 0.746986i \(-0.268500\pi\)
0.664839 + 0.746986i \(0.268500\pi\)
\(858\) 0 0
\(859\) 20.1760 0.688395 0.344198 0.938897i \(-0.388151\pi\)
0.344198 + 0.938897i \(0.388151\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 30.3051 1.03160 0.515799 0.856710i \(-0.327495\pi\)
0.515799 + 0.856710i \(0.327495\pi\)
\(864\) 0 0
\(865\) 6.85446 0.233059
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 0 0
\(870\) 0 0
\(871\) −4.78811 −0.162239
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 3.18953 0.107826
\(876\) 0 0
\(877\) 9.43947 0.318748 0.159374 0.987218i \(-0.449052\pi\)
0.159374 + 0.987218i \(0.449052\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 24.4671 0.824316 0.412158 0.911112i \(-0.364775\pi\)
0.412158 + 0.911112i \(0.364775\pi\)
\(882\) 0 0
\(883\) 6.12080 0.205981 0.102991 0.994682i \(-0.467159\pi\)
0.102991 + 0.994682i \(0.467159\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) −45.4200 −1.52505 −0.762526 0.646957i \(-0.776041\pi\)
−0.762526 + 0.646957i \(0.776041\pi\)
\(888\) 0 0
\(889\) −26.3134 −0.882524
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) −11.7417 −0.392922
\(894\) 0 0
\(895\) 5.01641 0.167680
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) −17.9282 −0.597937
\(900\) 0 0
\(901\) 2.01952 0.0672798
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) −16.7253 −0.555969
\(906\) 0 0
\(907\) −10.2694 −0.340991 −0.170496 0.985358i \(-0.554537\pi\)
−0.170496 + 0.985358i \(0.554537\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 10.9180 0.361728 0.180864 0.983508i \(-0.442111\pi\)
0.180864 + 0.983508i \(0.442111\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 27.9344 0.922474
\(918\) 0 0
\(919\) −38.9313 −1.28422 −0.642112 0.766611i \(-0.721942\pi\)
−0.642112 + 0.766611i \(0.721942\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) −8.69251 −0.286117
\(924\) 0 0
\(925\) 7.87086 0.258792
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) −21.9700 −0.720813 −0.360407 0.932795i \(-0.617362\pi\)
−0.360407 + 0.932795i \(0.617362\pi\)
\(930\) 0 0
\(931\) 3.17313 0.103995
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 10.6402 0.347600 0.173800 0.984781i \(-0.444395\pi\)
0.173800 + 0.984781i \(0.444395\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 20.8656 0.680200 0.340100 0.940389i \(-0.389539\pi\)
0.340100 + 0.940389i \(0.389539\pi\)
\(942\) 0 0
\(943\) −2.13747 −0.0696057
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −12.6925 −0.412451 −0.206226 0.978504i \(-0.566118\pi\)
−0.206226 + 0.978504i \(0.566118\pi\)
\(948\) 0 0
\(949\) 3.77170 0.122435
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) −16.7337 −0.542056 −0.271028 0.962571i \(-0.587364\pi\)
−0.271028 + 0.962571i \(0.587364\pi\)
\(954\) 0 0
\(955\) −11.2775 −0.364932
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) −38.0028 −1.22718
\(960\) 0 0
\(961\) 9.69251 0.312662
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) −13.5798 −0.437149
\(966\) 0 0
\(967\) 4.62688 0.148790 0.0743952 0.997229i \(-0.476297\pi\)
0.0743952 + 0.997229i \(0.476297\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 49.4506 1.58695 0.793473 0.608605i \(-0.208271\pi\)
0.793473 + 0.608605i \(0.208271\pi\)
\(972\) 0 0
\(973\) 74.9013 2.40123
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −22.5962 −0.722916 −0.361458 0.932388i \(-0.617721\pi\)
−0.361458 + 0.932388i \(0.617721\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) −32.9424 −1.05070 −0.525350 0.850886i \(-0.676066\pi\)
−0.525350 + 0.850886i \(0.676066\pi\)
\(984\) 0 0
\(985\) −2.00000 −0.0637253
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 2.96124 0.0941618
\(990\) 0 0
\(991\) −49.7802 −1.58132 −0.790660 0.612255i \(-0.790263\pi\)
−0.790660 + 0.612255i \(0.790263\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) −25.6566 −0.813368
\(996\) 0 0
\(997\) −36.4014 −1.15284 −0.576422 0.817152i \(-0.695552\pi\)
−0.576422 + 0.817152i \(0.695552\pi\)
\(998\) 0 0
\(999\) 0 0
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 6840.2.a.bg.1.1 3
3.2 odd 2 760.2.a.j.1.1 3
12.11 even 2 1520.2.a.s.1.3 3
15.2 even 4 3800.2.d.l.3649.6 6
15.8 even 4 3800.2.d.l.3649.1 6
15.14 odd 2 3800.2.a.x.1.3 3
24.5 odd 2 6080.2.a.bv.1.3 3
24.11 even 2 6080.2.a.bq.1.1 3
60.59 even 2 7600.2.a.bq.1.1 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
760.2.a.j.1.1 3 3.2 odd 2
1520.2.a.s.1.3 3 12.11 even 2
3800.2.a.x.1.3 3 15.14 odd 2
3800.2.d.l.3649.1 6 15.8 even 4
3800.2.d.l.3649.6 6 15.2 even 4
6080.2.a.bq.1.1 3 24.11 even 2
6080.2.a.bv.1.3 3 24.5 odd 2
6840.2.a.bg.1.1 3 1.1 even 1 trivial
7600.2.a.bq.1.1 3 60.59 even 2