Properties

Label 3200.2.d.l.1601.4
Level $3200$
Weight $2$
Character 3200.1601
Analytic conductor $25.552$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [3200,2,Mod(1601,3200)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3200, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("3200.1601");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3200 = 2^{7} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3200.d (of order \(2\), degree \(1\), minimal)

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: no
Analytic conductor: \(25.5521286468\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(i, \sqrt{6})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 9 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{29}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 640)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 1601.4
Root \(1.22474 + 1.22474i\) of defining polynomial
Character \(\chi\) \(=\) 3200.1601
Dual form 3200.2.d.l.1601.2

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+2.44949i q^{3} +2.44949 q^{7} -3.00000 q^{9} +O(q^{10})\) \(q+2.44949i q^{3} +2.44949 q^{7} -3.00000 q^{9} +4.89898i q^{11} +4.00000 q^{17} +4.89898i q^{19} +6.00000i q^{21} +2.44949 q^{23} -8.00000i q^{29} +9.79796 q^{31} -12.0000 q^{33} +4.00000i q^{37} +8.00000 q^{41} -7.34847i q^{43} -12.2474 q^{47} -1.00000 q^{49} +9.79796i q^{51} +8.00000i q^{53} -12.0000 q^{57} +4.89898i q^{59} -6.00000i q^{61} -7.34847 q^{63} +2.44949i q^{67} +6.00000i q^{69} +9.79796 q^{71} -4.00000 q^{73} +12.0000i q^{77} +9.79796 q^{79} -9.00000 q^{81} -2.44949i q^{83} +19.5959 q^{87} -2.00000 q^{89} +24.0000i q^{93} -4.00000 q^{97} -14.6969i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 12 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 12 q^{9} + 16 q^{17} - 48 q^{33} + 32 q^{41} - 4 q^{49} - 48 q^{57} - 16 q^{73} - 36 q^{81} - 8 q^{89} - 16 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3200\mathbb{Z}\right)^\times\).

\(n\) \(901\) \(1151\) \(2177\)
\(\chi(n)\) \(-1\) \(1\) \(1\)

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
\(3\) 2.44949i 1.41421i 0.707107 + 0.707107i \(0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) 2.44949 0.925820 0.462910 0.886405i \(-0.346805\pi\)
0.462910 + 0.886405i \(0.346805\pi\)
\(8\) 0 0
\(9\) −3.00000 −1.00000
\(10\) 0 0
\(11\) 4.89898i 1.47710i 0.674200 + 0.738549i \(0.264489\pi\)
−0.674200 + 0.738549i \(0.735511\pi\)
\(12\) 0 0
\(13\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 4.00000 0.970143 0.485071 0.874475i \(-0.338794\pi\)
0.485071 + 0.874475i \(0.338794\pi\)
\(18\) 0 0
\(19\) 4.89898i 1.12390i 0.827170 + 0.561951i \(0.189949\pi\)
−0.827170 + 0.561951i \(0.810051\pi\)
\(20\) 0 0
\(21\) 6.00000i 1.30931i
\(22\) 0 0
\(23\) 2.44949 0.510754 0.255377 0.966842i \(-0.417800\pi\)
0.255377 + 0.966842i \(0.417800\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) − 8.00000i − 1.48556i −0.669534 0.742781i \(-0.733506\pi\)
0.669534 0.742781i \(-0.266494\pi\)
\(30\) 0 0
\(31\) 9.79796 1.75977 0.879883 0.475191i \(-0.157621\pi\)
0.879883 + 0.475191i \(0.157621\pi\)
\(32\) 0 0
\(33\) −12.0000 −2.08893
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 4.00000i 0.657596i 0.944400 + 0.328798i \(0.106644\pi\)
−0.944400 + 0.328798i \(0.893356\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 8.00000 1.24939 0.624695 0.780869i \(-0.285223\pi\)
0.624695 + 0.780869i \(0.285223\pi\)
\(42\) 0 0
\(43\) − 7.34847i − 1.12063i −0.828279 0.560316i \(-0.810680\pi\)
0.828279 0.560316i \(-0.189320\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −12.2474 −1.78647 −0.893237 0.449586i \(-0.851571\pi\)
−0.893237 + 0.449586i \(0.851571\pi\)
\(48\) 0 0
\(49\) −1.00000 −0.142857
\(50\) 0 0
\(51\) 9.79796i 1.37199i
\(52\) 0 0
\(53\) 8.00000i 1.09888i 0.835532 + 0.549442i \(0.185160\pi\)
−0.835532 + 0.549442i \(0.814840\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) −12.0000 −1.58944
\(58\) 0 0
\(59\) 4.89898i 0.637793i 0.947790 + 0.318896i \(0.103312\pi\)
−0.947790 + 0.318896i \(0.896688\pi\)
\(60\) 0 0
\(61\) − 6.00000i − 0.768221i −0.923287 0.384111i \(-0.874508\pi\)
0.923287 0.384111i \(-0.125492\pi\)
\(62\) 0 0
\(63\) −7.34847 −0.925820
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 2.44949i 0.299253i 0.988743 + 0.149626i \(0.0478071\pi\)
−0.988743 + 0.149626i \(0.952193\pi\)
\(68\) 0 0
\(69\) 6.00000i 0.722315i
\(70\) 0 0
\(71\) 9.79796 1.16280 0.581402 0.813617i \(-0.302504\pi\)
0.581402 + 0.813617i \(0.302504\pi\)
\(72\) 0 0
\(73\) −4.00000 −0.468165 −0.234082 0.972217i \(-0.575209\pi\)
−0.234082 + 0.972217i \(0.575209\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 12.0000i 1.36753i
\(78\) 0 0
\(79\) 9.79796 1.10236 0.551178 0.834388i \(-0.314178\pi\)
0.551178 + 0.834388i \(0.314178\pi\)
\(80\) 0 0
\(81\) −9.00000 −1.00000
\(82\) 0 0
\(83\) − 2.44949i − 0.268866i −0.990923 0.134433i \(-0.957079\pi\)
0.990923 0.134433i \(-0.0429214\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 19.5959 2.10090
\(88\) 0 0
\(89\) −2.00000 −0.212000 −0.106000 0.994366i \(-0.533804\pi\)
−0.106000 + 0.994366i \(0.533804\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 24.0000i 2.48868i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −4.00000 −0.406138 −0.203069 0.979164i \(-0.565092\pi\)
−0.203069 + 0.979164i \(0.565092\pi\)
\(98\) 0 0
\(99\) − 14.6969i − 1.47710i
\(100\) 0 0
\(101\) 8.00000i 0.796030i 0.917379 + 0.398015i \(0.130301\pi\)
−0.917379 + 0.398015i \(0.869699\pi\)
\(102\) 0 0
\(103\) 2.44949 0.241355 0.120678 0.992692i \(-0.461493\pi\)
0.120678 + 0.992692i \(0.461493\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) − 7.34847i − 0.710403i −0.934790 0.355202i \(-0.884412\pi\)
0.934790 0.355202i \(-0.115588\pi\)
\(108\) 0 0
\(109\) 10.0000i 0.957826i 0.877862 + 0.478913i \(0.158969\pi\)
−0.877862 + 0.478913i \(0.841031\pi\)
\(110\) 0 0
\(111\) −9.79796 −0.929981
\(112\) 0 0
\(113\) −16.0000 −1.50515 −0.752577 0.658505i \(-0.771189\pi\)
−0.752577 + 0.658505i \(0.771189\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 9.79796 0.898177
\(120\) 0 0
\(121\) −13.0000 −1.18182
\(122\) 0 0
\(123\) 19.5959i 1.76690i
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) −7.34847 −0.652071 −0.326036 0.945357i \(-0.605713\pi\)
−0.326036 + 0.945357i \(0.605713\pi\)
\(128\) 0 0
\(129\) 18.0000 1.58481
\(130\) 0 0
\(131\) − 4.89898i − 0.428026i −0.976831 0.214013i \(-0.931347\pi\)
0.976831 0.214013i \(-0.0686535\pi\)
\(132\) 0 0
\(133\) 12.0000i 1.04053i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −8.00000 −0.683486 −0.341743 0.939793i \(-0.611017\pi\)
−0.341743 + 0.939793i \(0.611017\pi\)
\(138\) 0 0
\(139\) − 4.89898i − 0.415526i −0.978179 0.207763i \(-0.933382\pi\)
0.978179 0.207763i \(-0.0666183\pi\)
\(140\) 0 0
\(141\) − 30.0000i − 2.52646i
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) − 2.44949i − 0.202031i
\(148\) 0 0
\(149\) 2.00000i 0.163846i 0.996639 + 0.0819232i \(0.0261062\pi\)
−0.996639 + 0.0819232i \(0.973894\pi\)
\(150\) 0 0
\(151\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(152\) 0 0
\(153\) −12.0000 −0.970143
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) − 20.0000i − 1.59617i −0.602542 0.798087i \(-0.705846\pi\)
0.602542 0.798087i \(-0.294154\pi\)
\(158\) 0 0
\(159\) −19.5959 −1.55406
\(160\) 0 0
\(161\) 6.00000 0.472866
\(162\) 0 0
\(163\) 2.44949i 0.191859i 0.995388 + 0.0959294i \(0.0305823\pi\)
−0.995388 + 0.0959294i \(0.969418\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −2.44949 −0.189547 −0.0947736 0.995499i \(-0.530213\pi\)
−0.0947736 + 0.995499i \(0.530213\pi\)
\(168\) 0 0
\(169\) 13.0000 1.00000
\(170\) 0 0
\(171\) − 14.6969i − 1.12390i
\(172\) 0 0
\(173\) − 4.00000i − 0.304114i −0.988372 0.152057i \(-0.951410\pi\)
0.988372 0.152057i \(-0.0485898\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) −12.0000 −0.901975
\(178\) 0 0
\(179\) 14.6969i 1.09850i 0.835658 + 0.549250i \(0.185087\pi\)
−0.835658 + 0.549250i \(0.814913\pi\)
\(180\) 0 0
\(181\) 8.00000i 0.594635i 0.954779 + 0.297318i \(0.0960920\pi\)
−0.954779 + 0.297318i \(0.903908\pi\)
\(182\) 0 0
\(183\) 14.6969 1.08643
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 19.5959i 1.43300i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −19.5959 −1.41791 −0.708955 0.705253i \(-0.750833\pi\)
−0.708955 + 0.705253i \(0.750833\pi\)
\(192\) 0 0
\(193\) 12.0000 0.863779 0.431889 0.901927i \(-0.357847\pi\)
0.431889 + 0.901927i \(0.357847\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 8.00000i 0.569976i 0.958531 + 0.284988i \(0.0919897\pi\)
−0.958531 + 0.284988i \(0.908010\pi\)
\(198\) 0 0
\(199\) −9.79796 −0.694559 −0.347279 0.937762i \(-0.612894\pi\)
−0.347279 + 0.937762i \(0.612894\pi\)
\(200\) 0 0
\(201\) −6.00000 −0.423207
\(202\) 0 0
\(203\) − 19.5959i − 1.37536i
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) −7.34847 −0.510754
\(208\) 0 0
\(209\) −24.0000 −1.66011
\(210\) 0 0
\(211\) − 24.4949i − 1.68630i −0.537680 0.843149i \(-0.680699\pi\)
0.537680 0.843149i \(-0.319301\pi\)
\(212\) 0 0
\(213\) 24.0000i 1.64445i
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 24.0000 1.62923
\(218\) 0 0
\(219\) − 9.79796i − 0.662085i
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) 7.34847 0.492090 0.246045 0.969258i \(-0.420869\pi\)
0.246045 + 0.969258i \(0.420869\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) − 2.44949i − 0.162578i −0.996691 0.0812892i \(-0.974096\pi\)
0.996691 0.0812892i \(-0.0259037\pi\)
\(228\) 0 0
\(229\) 8.00000i 0.528655i 0.964433 + 0.264327i \(0.0851500\pi\)
−0.964433 + 0.264327i \(0.914850\pi\)
\(230\) 0 0
\(231\) −29.3939 −1.93398
\(232\) 0 0
\(233\) −20.0000 −1.31024 −0.655122 0.755523i \(-0.727383\pi\)
−0.655122 + 0.755523i \(0.727383\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 24.0000i 1.55897i
\(238\) 0 0
\(239\) −9.79796 −0.633777 −0.316889 0.948463i \(-0.602638\pi\)
−0.316889 + 0.948463i \(0.602638\pi\)
\(240\) 0 0
\(241\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(242\) 0 0
\(243\) − 22.0454i − 1.41421i
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) 6.00000 0.380235
\(250\) 0 0
\(251\) 4.89898i 0.309221i 0.987976 + 0.154610i \(0.0494122\pi\)
−0.987976 + 0.154610i \(0.950588\pi\)
\(252\) 0 0
\(253\) 12.0000i 0.754434i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −8.00000 −0.499026 −0.249513 0.968371i \(-0.580271\pi\)
−0.249513 + 0.968371i \(0.580271\pi\)
\(258\) 0 0
\(259\) 9.79796i 0.608816i
\(260\) 0 0
\(261\) 24.0000i 1.48556i
\(262\) 0 0
\(263\) −22.0454 −1.35938 −0.679689 0.733500i \(-0.737885\pi\)
−0.679689 + 0.733500i \(0.737885\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) − 4.89898i − 0.299813i
\(268\) 0 0
\(269\) − 26.0000i − 1.58525i −0.609711 0.792624i \(-0.708714\pi\)
0.609711 0.792624i \(-0.291286\pi\)
\(270\) 0 0
\(271\) 9.79796 0.595184 0.297592 0.954693i \(-0.403817\pi\)
0.297592 + 0.954693i \(0.403817\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) − 12.0000i − 0.721010i −0.932757 0.360505i \(-0.882604\pi\)
0.932757 0.360505i \(-0.117396\pi\)
\(278\) 0 0
\(279\) −29.3939 −1.75977
\(280\) 0 0
\(281\) 16.0000 0.954480 0.477240 0.878773i \(-0.341637\pi\)
0.477240 + 0.878773i \(0.341637\pi\)
\(282\) 0 0
\(283\) 26.9444i 1.60168i 0.598880 + 0.800839i \(0.295613\pi\)
−0.598880 + 0.800839i \(0.704387\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 19.5959 1.15671
\(288\) 0 0
\(289\) −1.00000 −0.0588235
\(290\) 0 0
\(291\) − 9.79796i − 0.574367i
\(292\) 0 0
\(293\) − 20.0000i − 1.16841i −0.811605 0.584206i \(-0.801406\pi\)
0.811605 0.584206i \(-0.198594\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 0 0
\(300\) 0 0
\(301\) − 18.0000i − 1.03750i
\(302\) 0 0
\(303\) −19.5959 −1.12576
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) − 17.1464i − 0.978598i −0.872116 0.489299i \(-0.837253\pi\)
0.872116 0.489299i \(-0.162747\pi\)
\(308\) 0 0
\(309\) 6.00000i 0.341328i
\(310\) 0 0
\(311\) −19.5959 −1.11118 −0.555591 0.831456i \(-0.687508\pi\)
−0.555591 + 0.831456i \(0.687508\pi\)
\(312\) 0 0
\(313\) −24.0000 −1.35656 −0.678280 0.734803i \(-0.737274\pi\)
−0.678280 + 0.734803i \(0.737274\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 8.00000i 0.449325i 0.974437 + 0.224662i \(0.0721279\pi\)
−0.974437 + 0.224662i \(0.927872\pi\)
\(318\) 0 0
\(319\) 39.1918 2.19432
\(320\) 0 0
\(321\) 18.0000 1.00466
\(322\) 0 0
\(323\) 19.5959i 1.09035i
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) −24.4949 −1.35457
\(328\) 0 0
\(329\) −30.0000 −1.65395
\(330\) 0 0
\(331\) − 4.89898i − 0.269272i −0.990895 0.134636i \(-0.957013\pi\)
0.990895 0.134636i \(-0.0429866\pi\)
\(332\) 0 0
\(333\) − 12.0000i − 0.657596i
\(334\) 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
\(339\) − 39.1918i − 2.12861i
\(340\) 0 0
\(341\) 48.0000i 2.59935i
\(342\) 0 0
\(343\) −19.5959 −1.05808
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) − 12.2474i − 0.657477i −0.944421 0.328739i \(-0.893376\pi\)
0.944421 0.328739i \(-0.106624\pi\)
\(348\) 0 0
\(349\) − 24.0000i − 1.28469i −0.766415 0.642345i \(-0.777962\pi\)
0.766415 0.642345i \(-0.222038\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) −8.00000 −0.425797 −0.212899 0.977074i \(-0.568290\pi\)
−0.212899 + 0.977074i \(0.568290\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 24.0000i 1.27021i
\(358\) 0 0
\(359\) 19.5959 1.03423 0.517116 0.855915i \(-0.327005\pi\)
0.517116 + 0.855915i \(0.327005\pi\)
\(360\) 0 0
\(361\) −5.00000 −0.263158
\(362\) 0 0
\(363\) − 31.8434i − 1.67134i
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 31.8434 1.66221 0.831105 0.556115i \(-0.187709\pi\)
0.831105 + 0.556115i \(0.187709\pi\)
\(368\) 0 0
\(369\) −24.0000 −1.24939
\(370\) 0 0
\(371\) 19.5959i 1.01737i
\(372\) 0 0
\(373\) − 20.0000i − 1.03556i −0.855514 0.517780i \(-0.826758\pi\)
0.855514 0.517780i \(-0.173242\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) 14.6969i 0.754931i 0.926024 + 0.377466i \(0.123204\pi\)
−0.926024 + 0.377466i \(0.876796\pi\)
\(380\) 0 0
\(381\) − 18.0000i − 0.922168i
\(382\) 0 0
\(383\) 26.9444 1.37679 0.688397 0.725334i \(-0.258315\pi\)
0.688397 + 0.725334i \(0.258315\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 22.0454i 1.12063i
\(388\) 0 0
\(389\) 2.00000i 0.101404i 0.998714 + 0.0507020i \(0.0161459\pi\)
−0.998714 + 0.0507020i \(0.983854\pi\)
\(390\) 0 0
\(391\) 9.79796 0.495504
\(392\) 0 0
\(393\) 12.0000 0.605320
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) − 8.00000i − 0.401508i −0.979642 0.200754i \(-0.935661\pi\)
0.979642 0.200754i \(-0.0643393\pi\)
\(398\) 0 0
\(399\) −29.3939 −1.47153
\(400\) 0 0
\(401\) 10.0000 0.499376 0.249688 0.968326i \(-0.419672\pi\)
0.249688 + 0.968326i \(0.419672\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) −19.5959 −0.971334
\(408\) 0 0
\(409\) 24.0000 1.18672 0.593362 0.804936i \(-0.297800\pi\)
0.593362 + 0.804936i \(0.297800\pi\)
\(410\) 0 0
\(411\) − 19.5959i − 0.966595i
\(412\) 0 0
\(413\) 12.0000i 0.590481i
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 12.0000 0.587643
\(418\) 0 0
\(419\) 14.6969i 0.717992i 0.933339 + 0.358996i \(0.116881\pi\)
−0.933339 + 0.358996i \(0.883119\pi\)
\(420\) 0 0
\(421\) − 2.00000i − 0.0974740i −0.998812 0.0487370i \(-0.984480\pi\)
0.998812 0.0487370i \(-0.0155196\pi\)
\(422\) 0 0
\(423\) 36.7423 1.78647
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) − 14.6969i − 0.711235i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 29.3939 1.41585 0.707927 0.706286i \(-0.249631\pi\)
0.707927 + 0.706286i \(0.249631\pi\)
\(432\) 0 0
\(433\) 36.0000 1.73005 0.865025 0.501729i \(-0.167303\pi\)
0.865025 + 0.501729i \(0.167303\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 12.0000i 0.574038i
\(438\) 0 0
\(439\) 19.5959 0.935262 0.467631 0.883924i \(-0.345108\pi\)
0.467631 + 0.883924i \(0.345108\pi\)
\(440\) 0 0
\(441\) 3.00000 0.142857
\(442\) 0 0
\(443\) 26.9444i 1.28017i 0.768306 + 0.640083i \(0.221100\pi\)
−0.768306 + 0.640083i \(0.778900\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) −4.89898 −0.231714
\(448\) 0 0
\(449\) 32.0000 1.51017 0.755087 0.655625i \(-0.227595\pi\)
0.755087 + 0.655625i \(0.227595\pi\)
\(450\) 0 0
\(451\) 39.1918i 1.84547i
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 32.0000 1.49690 0.748448 0.663193i \(-0.230799\pi\)
0.748448 + 0.663193i \(0.230799\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) − 8.00000i − 0.372597i −0.982493 0.186299i \(-0.940351\pi\)
0.982493 0.186299i \(-0.0596492\pi\)
\(462\) 0 0
\(463\) 31.8434 1.47989 0.739943 0.672669i \(-0.234852\pi\)
0.739943 + 0.672669i \(0.234852\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 36.7423i 1.70023i 0.526595 + 0.850117i \(0.323469\pi\)
−0.526595 + 0.850117i \(0.676531\pi\)
\(468\) 0 0
\(469\) 6.00000i 0.277054i
\(470\) 0 0
\(471\) 48.9898 2.25733
\(472\) 0 0
\(473\) 36.0000 1.65528
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) − 24.0000i − 1.09888i
\(478\) 0 0
\(479\) 19.5959 0.895360 0.447680 0.894194i \(-0.352250\pi\)
0.447680 + 0.894194i \(0.352250\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 0 0
\(483\) 14.6969i 0.668734i
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) −17.1464 −0.776979 −0.388489 0.921453i \(-0.627003\pi\)
−0.388489 + 0.921453i \(0.627003\pi\)
\(488\) 0 0
\(489\) −6.00000 −0.271329
\(490\) 0 0
\(491\) 14.6969i 0.663264i 0.943409 + 0.331632i \(0.107599\pi\)
−0.943409 + 0.331632i \(0.892401\pi\)
\(492\) 0 0
\(493\) − 32.0000i − 1.44121i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 24.0000 1.07655
\(498\) 0 0
\(499\) − 4.89898i − 0.219308i −0.993970 0.109654i \(-0.965026\pi\)
0.993970 0.109654i \(-0.0349744\pi\)
\(500\) 0 0
\(501\) − 6.00000i − 0.268060i
\(502\) 0 0
\(503\) −17.1464 −0.764521 −0.382261 0.924055i \(-0.624854\pi\)
−0.382261 + 0.924055i \(0.624854\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 31.8434i 1.41421i
\(508\) 0 0
\(509\) − 8.00000i − 0.354594i −0.984157 0.177297i \(-0.943265\pi\)
0.984157 0.177297i \(-0.0567353\pi\)
\(510\) 0 0
\(511\) −9.79796 −0.433436
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) − 60.0000i − 2.63880i
\(518\) 0 0
\(519\) 9.79796 0.430083
\(520\) 0 0
\(521\) −22.0000 −0.963837 −0.481919 0.876216i \(-0.660060\pi\)
−0.481919 + 0.876216i \(0.660060\pi\)
\(522\) 0 0
\(523\) − 31.8434i − 1.39241i −0.717841 0.696207i \(-0.754870\pi\)
0.717841 0.696207i \(-0.245130\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 39.1918 1.70722
\(528\) 0 0
\(529\) −17.0000 −0.739130
\(530\) 0 0
\(531\) − 14.6969i − 0.637793i
\(532\) 0 0
\(533\) 0 0
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) −36.0000 −1.55351
\(538\) 0 0
\(539\) − 4.89898i − 0.211014i
\(540\) 0 0
\(541\) − 8.00000i − 0.343947i −0.985102 0.171973i \(-0.944986\pi\)
0.985102 0.171973i \(-0.0550143\pi\)
\(542\) 0 0
\(543\) −19.5959 −0.840941
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) − 22.0454i − 0.942594i −0.881975 0.471297i \(-0.843786\pi\)
0.881975 0.471297i \(-0.156214\pi\)
\(548\) 0 0
\(549\) 18.0000i 0.768221i
\(550\) 0 0
\(551\) 39.1918 1.66963
\(552\) 0 0
\(553\) 24.0000 1.02058
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) − 4.00000i − 0.169485i −0.996403 0.0847427i \(-0.972993\pi\)
0.996403 0.0847427i \(-0.0270068\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 0 0
\(561\) −48.0000 −2.02656
\(562\) 0 0
\(563\) 2.44949i 0.103234i 0.998667 + 0.0516168i \(0.0164375\pi\)
−0.998667 + 0.0516168i \(0.983563\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) −22.0454 −0.925820
\(568\) 0 0
\(569\) 8.00000 0.335377 0.167689 0.985840i \(-0.446370\pi\)
0.167689 + 0.985840i \(0.446370\pi\)
\(570\) 0 0
\(571\) 44.0908i 1.84514i 0.385826 + 0.922572i \(0.373917\pi\)
−0.385826 + 0.922572i \(0.626083\pi\)
\(572\) 0 0
\(573\) − 48.0000i − 2.00523i
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) 24.0000 0.999133 0.499567 0.866276i \(-0.333493\pi\)
0.499567 + 0.866276i \(0.333493\pi\)
\(578\) 0 0
\(579\) 29.3939i 1.22157i
\(580\) 0 0
\(581\) − 6.00000i − 0.248922i
\(582\) 0 0
\(583\) −39.1918 −1.62316
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) − 12.2474i − 0.505506i −0.967531 0.252753i \(-0.918664\pi\)
0.967531 0.252753i \(-0.0813361\pi\)
\(588\) 0 0
\(589\) 48.0000i 1.97781i
\(590\) 0 0
\(591\) −19.5959 −0.806068
\(592\) 0 0
\(593\) −8.00000 −0.328521 −0.164260 0.986417i \(-0.552524\pi\)
−0.164260 + 0.986417i \(0.552524\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) − 24.0000i − 0.982255i
\(598\) 0 0
\(599\) 9.79796 0.400334 0.200167 0.979762i \(-0.435852\pi\)
0.200167 + 0.979762i \(0.435852\pi\)
\(600\) 0 0
\(601\) 16.0000 0.652654 0.326327 0.945257i \(-0.394189\pi\)
0.326327 + 0.945257i \(0.394189\pi\)
\(602\) 0 0
\(603\) − 7.34847i − 0.299253i
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) 7.34847 0.298265 0.149133 0.988817i \(-0.452352\pi\)
0.149133 + 0.988817i \(0.452352\pi\)
\(608\) 0 0
\(609\) 48.0000 1.94506
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) − 16.0000i − 0.646234i −0.946359 0.323117i \(-0.895269\pi\)
0.946359 0.323117i \(-0.104731\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −28.0000 −1.12724 −0.563619 0.826035i \(-0.690591\pi\)
−0.563619 + 0.826035i \(0.690591\pi\)
\(618\) 0 0
\(619\) 24.4949i 0.984533i 0.870445 + 0.492267i \(0.163831\pi\)
−0.870445 + 0.492267i \(0.836169\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) −4.89898 −0.196273
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) − 58.7878i − 2.34776i
\(628\) 0 0
\(629\) 16.0000i 0.637962i
\(630\) 0 0
\(631\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(632\) 0 0
\(633\) 60.0000 2.38479
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) −29.3939 −1.16280
\(640\) 0 0
\(641\) −16.0000 −0.631962 −0.315981 0.948766i \(-0.602334\pi\)
−0.315981 + 0.948766i \(0.602334\pi\)
\(642\) 0 0
\(643\) − 2.44949i − 0.0965984i −0.998833 0.0482992i \(-0.984620\pi\)
0.998833 0.0482992i \(-0.0153801\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 22.0454 0.866694 0.433347 0.901227i \(-0.357332\pi\)
0.433347 + 0.901227i \(0.357332\pi\)
\(648\) 0 0
\(649\) −24.0000 −0.942082
\(650\) 0 0
\(651\) 58.7878i 2.30407i
\(652\) 0 0
\(653\) 16.0000i 0.626128i 0.949732 + 0.313064i \(0.101356\pi\)
−0.949732 + 0.313064i \(0.898644\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 12.0000 0.468165
\(658\) 0 0
\(659\) 14.6969i 0.572511i 0.958153 + 0.286256i \(0.0924107\pi\)
−0.958153 + 0.286256i \(0.907589\pi\)
\(660\) 0 0
\(661\) − 30.0000i − 1.16686i −0.812162 0.583432i \(-0.801709\pi\)
0.812162 0.583432i \(-0.198291\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) − 19.5959i − 0.758757i
\(668\) 0 0
\(669\) 18.0000i 0.695920i
\(670\) 0 0
\(671\) 29.3939 1.13474
\(672\) 0 0
\(673\) 36.0000 1.38770 0.693849 0.720121i \(-0.255914\pi\)
0.693849 + 0.720121i \(0.255914\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 32.0000i 1.22986i 0.788582 + 0.614930i \(0.210816\pi\)
−0.788582 + 0.614930i \(0.789184\pi\)
\(678\) 0 0
\(679\) −9.79796 −0.376011
\(680\) 0 0
\(681\) 6.00000 0.229920
\(682\) 0 0
\(683\) − 7.34847i − 0.281181i −0.990068 0.140591i \(-0.955100\pi\)
0.990068 0.140591i \(-0.0449002\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) −19.5959 −0.747631
\(688\) 0 0
\(689\) 0 0
\(690\) 0 0
\(691\) 14.6969i 0.559098i 0.960131 + 0.279549i \(0.0901849\pi\)
−0.960131 + 0.279549i \(0.909815\pi\)
\(692\) 0 0
\(693\) − 36.0000i − 1.36753i
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 32.0000 1.21209
\(698\) 0 0
\(699\) − 48.9898i − 1.85296i
\(700\) 0 0
\(701\) − 26.0000i − 0.982006i −0.871158 0.491003i \(-0.836630\pi\)
0.871158 0.491003i \(-0.163370\pi\)
\(702\) 0 0
\(703\) −19.5959 −0.739074
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 19.5959i 0.736980i
\(708\) 0 0
\(709\) − 40.0000i − 1.50223i −0.660171 0.751116i \(-0.729516\pi\)
0.660171 0.751116i \(-0.270484\pi\)
\(710\) 0 0
\(711\) −29.3939 −1.10236
\(712\) 0 0
\(713\) 24.0000 0.898807
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) − 24.0000i − 0.896296i
\(718\) 0 0
\(719\) 48.9898 1.82701 0.913506 0.406826i \(-0.133365\pi\)
0.913506 + 0.406826i \(0.133365\pi\)
\(720\) 0 0
\(721\) 6.00000 0.223452
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) −36.7423 −1.36270 −0.681349 0.731959i \(-0.738606\pi\)
−0.681349 + 0.731959i \(0.738606\pi\)
\(728\) 0 0
\(729\) 27.0000 1.00000
\(730\) 0 0
\(731\) − 29.3939i − 1.08717i
\(732\) 0 0
\(733\) − 12.0000i − 0.443230i −0.975134 0.221615i \(-0.928867\pi\)
0.975134 0.221615i \(-0.0711328\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −12.0000 −0.442026
\(738\) 0 0
\(739\) 14.6969i 0.540636i 0.962771 + 0.270318i \(0.0871288\pi\)
−0.962771 + 0.270318i \(0.912871\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 17.1464 0.629041 0.314521 0.949251i \(-0.398156\pi\)
0.314521 + 0.949251i \(0.398156\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 7.34847i 0.268866i
\(748\) 0 0
\(749\) − 18.0000i − 0.657706i
\(750\) 0 0
\(751\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(752\) 0 0
\(753\) −12.0000 −0.437304
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 12.0000i 0.436147i 0.975932 + 0.218074i \(0.0699773\pi\)
−0.975932 + 0.218074i \(0.930023\pi\)
\(758\) 0 0
\(759\) −29.3939 −1.06693
\(760\) 0 0
\(761\) 22.0000 0.797499 0.398750 0.917060i \(-0.369444\pi\)
0.398750 + 0.917060i \(0.369444\pi\)
\(762\) 0 0
\(763\) 24.4949i 0.886775i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) 26.0000 0.937584 0.468792 0.883309i \(-0.344689\pi\)
0.468792 + 0.883309i \(0.344689\pi\)
\(770\) 0 0
\(771\) − 19.5959i − 0.705730i
\(772\) 0 0
\(773\) 8.00000i 0.287740i 0.989597 + 0.143870i \(0.0459547\pi\)
−0.989597 + 0.143870i \(0.954045\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) −24.0000 −0.860995
\(778\) 0 0
\(779\) 39.1918i 1.40419i
\(780\) 0 0
\(781\) 48.0000i 1.71758i
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) 41.6413i 1.48435i 0.670205 + 0.742176i \(0.266206\pi\)
−0.670205 + 0.742176i \(0.733794\pi\)
\(788\) 0 0
\(789\) − 54.0000i − 1.92245i
\(790\) 0 0
\(791\) −39.1918 −1.39350
\(792\) 0 0
\(793\) 0 0
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 8.00000i 0.283375i 0.989911 + 0.141687i \(0.0452527\pi\)
−0.989911 + 0.141687i \(0.954747\pi\)
\(798\) 0 0
\(799\) −48.9898 −1.73313
\(800\) 0 0
\(801\) 6.00000 0.212000
\(802\) 0 0
\(803\) − 19.5959i − 0.691525i
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 63.6867 2.24188
\(808\) 0 0
\(809\) −10.0000 −0.351581 −0.175791 0.984428i \(-0.556248\pi\)
−0.175791 + 0.984428i \(0.556248\pi\)
\(810\) 0 0
\(811\) − 53.8888i − 1.89229i −0.323742 0.946145i \(-0.604941\pi\)
0.323742 0.946145i \(-0.395059\pi\)
\(812\) 0 0
\(813\) 24.0000i 0.841717i
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 36.0000 1.25948
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) − 14.0000i − 0.488603i −0.969699 0.244302i \(-0.921441\pi\)
0.969699 0.244302i \(-0.0785587\pi\)
\(822\) 0 0
\(823\) 22.0454 0.768455 0.384227 0.923239i \(-0.374468\pi\)
0.384227 + 0.923239i \(0.374468\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 46.5403i 1.61836i 0.587557 + 0.809182i \(0.300090\pi\)
−0.587557 + 0.809182i \(0.699910\pi\)
\(828\) 0 0
\(829\) 22.0000i 0.764092i 0.924143 + 0.382046i \(0.124780\pi\)
−0.924143 + 0.382046i \(0.875220\pi\)
\(830\) 0 0
\(831\) 29.3939 1.01966
\(832\) 0 0
\(833\) −4.00000 −0.138592
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) −39.1918 −1.35305 −0.676526 0.736419i \(-0.736515\pi\)
−0.676526 + 0.736419i \(0.736515\pi\)
\(840\) 0 0
\(841\) −35.0000 −1.20690
\(842\) 0 0
\(843\) 39.1918i 1.34984i
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) −31.8434 −1.09415
\(848\) 0 0
\(849\) −66.0000 −2.26511
\(850\) 0 0
\(851\) 9.79796i 0.335870i
\(852\) 0 0
\(853\) − 32.0000i − 1.09566i −0.836590 0.547830i \(-0.815454\pi\)
0.836590 0.547830i \(-0.184546\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −32.0000 −1.09310 −0.546550 0.837427i \(-0.684059\pi\)
−0.546550 + 0.837427i \(0.684059\pi\)
\(858\) 0 0
\(859\) 24.4949i 0.835755i 0.908503 + 0.417878i \(0.137226\pi\)
−0.908503 + 0.417878i \(0.862774\pi\)
\(860\) 0 0
\(861\) 48.0000i 1.63584i
\(862\) 0 0
\(863\) 51.4393 1.75101 0.875507 0.483206i \(-0.160528\pi\)
0.875507 + 0.483206i \(0.160528\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) − 2.44949i − 0.0831890i
\(868\) 0 0
\(869\) 48.0000i 1.62829i
\(870\) 0 0
\(871\) 0 0
\(872\) 0 0
\(873\) 12.0000 0.406138
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) − 28.0000i − 0.945493i −0.881199 0.472746i \(-0.843263\pi\)
0.881199 0.472746i \(-0.156737\pi\)
\(878\) 0 0
\(879\) 48.9898 1.65238
\(880\) 0 0
\(881\) −8.00000 −0.269527 −0.134763 0.990878i \(-0.543027\pi\)
−0.134763 + 0.990878i \(0.543027\pi\)
\(882\) 0 0
\(883\) − 36.7423i − 1.23648i −0.785990 0.618239i \(-0.787846\pi\)
0.785990 0.618239i \(-0.212154\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 2.44949 0.0822458 0.0411229 0.999154i \(-0.486906\pi\)
0.0411229 + 0.999154i \(0.486906\pi\)
\(888\) 0 0
\(889\) −18.0000 −0.603701
\(890\) 0 0
\(891\) − 44.0908i − 1.47710i
\(892\) 0 0
\(893\) − 60.0000i − 2.00782i
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) − 78.3837i − 2.61424i
\(900\) 0 0
\(901\) 32.0000i 1.06607i
\(902\) 0 0
\(903\) 44.0908 1.46725
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) 12.2474i 0.406670i 0.979109 + 0.203335i \(0.0651780\pi\)
−0.979109 + 0.203335i \(0.934822\pi\)
\(908\) 0 0
\(909\) − 24.0000i − 0.796030i
\(910\) 0 0
\(911\) −29.3939 −0.973863 −0.486931 0.873440i \(-0.661884\pi\)
−0.486931 + 0.873440i \(0.661884\pi\)
\(912\) 0 0
\(913\) 12.0000 0.397142
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) − 12.0000i − 0.396275i
\(918\) 0 0
\(919\) 48.9898 1.61602 0.808012 0.589166i \(-0.200544\pi\)
0.808012 + 0.589166i \(0.200544\pi\)
\(920\) 0 0
\(921\) 42.0000 1.38395
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) −7.34847 −0.241355
\(928\) 0 0
\(929\) 16.0000 0.524943 0.262471 0.964940i \(-0.415462\pi\)
0.262471 + 0.964940i \(0.415462\pi\)
\(930\) 0 0
\(931\) − 4.89898i − 0.160558i
\(932\) 0 0
\(933\) − 48.0000i − 1.57145i
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 60.0000 1.96011 0.980057 0.198715i \(-0.0636769\pi\)
0.980057 + 0.198715i \(0.0636769\pi\)
\(938\) 0 0
\(939\) − 58.7878i − 1.91847i
\(940\) 0 0
\(941\) − 8.00000i − 0.260793i −0.991462 0.130396i \(-0.958375\pi\)
0.991462 0.130396i \(-0.0416250\pi\)
\(942\) 0 0
\(943\) 19.5959 0.638131
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) − 17.1464i − 0.557184i −0.960410 0.278592i \(-0.910132\pi\)
0.960410 0.278592i \(-0.0898677\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) 0 0
\(951\) −19.5959 −0.635441
\(952\) 0 0
\(953\) −32.0000 −1.03658 −0.518291 0.855204i \(-0.673432\pi\)
−0.518291 + 0.855204i \(0.673432\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 96.0000i 3.10324i
\(958\) 0 0
\(959\) −19.5959 −0.632785
\(960\) 0 0
\(961\) 65.0000 2.09677
\(962\) 0 0
\(963\) 22.0454i 0.710403i
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) 17.1464 0.551392 0.275696 0.961245i \(-0.411092\pi\)
0.275696 + 0.961245i \(0.411092\pi\)
\(968\) 0 0
\(969\) −48.0000 −1.54198
\(970\) 0 0
\(971\) − 24.4949i − 0.786079i −0.919522 0.393039i \(-0.871424\pi\)
0.919522 0.393039i \(-0.128576\pi\)
\(972\) 0 0
\(973\) − 12.0000i − 0.384702i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −4.00000 −0.127971 −0.0639857 0.997951i \(-0.520381\pi\)
−0.0639857 + 0.997951i \(0.520381\pi\)
\(978\) 0 0
\(979\) − 9.79796i − 0.313144i
\(980\) 0 0
\(981\) − 30.0000i − 0.957826i
\(982\) 0 0
\(983\) −22.0454 −0.703139 −0.351570 0.936162i \(-0.614352\pi\)
−0.351570 + 0.936162i \(0.614352\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) − 73.4847i − 2.33904i
\(988\) 0 0
\(989\) − 18.0000i − 0.572367i
\(990\) 0 0
\(991\) −39.1918 −1.24497 −0.622485 0.782632i \(-0.713877\pi\)
−0.622485 + 0.782632i \(0.713877\pi\)
\(992\) 0 0
\(993\) 12.0000 0.380808
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) − 56.0000i − 1.77354i −0.462213 0.886769i \(-0.652944\pi\)
0.462213 0.886769i \(-0.347056\pi\)
\(998\) 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 3200.2.d.l.1601.4 4
4.3 odd 2 inner 3200.2.d.l.1601.1 4
5.2 odd 4 640.2.f.c.449.4 yes 4
5.3 odd 4 640.2.f.g.449.2 yes 4
5.4 even 2 3200.2.d.k.1601.1 4
8.3 odd 2 inner 3200.2.d.l.1601.3 4
8.5 even 2 inner 3200.2.d.l.1601.2 4
16.3 odd 4 6400.2.a.bx.1.2 2
16.5 even 4 6400.2.a.bw.1.2 2
16.11 odd 4 6400.2.a.bw.1.1 2
16.13 even 4 6400.2.a.bx.1.1 2
20.3 even 4 640.2.f.g.449.4 yes 4
20.7 even 4 640.2.f.c.449.2 yes 4
20.19 odd 2 3200.2.d.k.1601.4 4
40.3 even 4 640.2.f.c.449.1 4
40.13 odd 4 640.2.f.c.449.3 yes 4
40.19 odd 2 3200.2.d.k.1601.2 4
40.27 even 4 640.2.f.g.449.3 yes 4
40.29 even 2 3200.2.d.k.1601.3 4
40.37 odd 4 640.2.f.g.449.1 yes 4
80.3 even 4 1280.2.c.m.769.3 4
80.13 odd 4 1280.2.c.m.769.1 4
80.19 odd 4 6400.2.a.bv.1.1 2
80.27 even 4 1280.2.c.e.769.3 4
80.29 even 4 6400.2.a.bv.1.2 2
80.37 odd 4 1280.2.c.e.769.1 4
80.43 even 4 1280.2.c.e.769.2 4
80.53 odd 4 1280.2.c.e.769.4 4
80.59 odd 4 6400.2.a.bu.1.2 2
80.67 even 4 1280.2.c.m.769.2 4
80.69 even 4 6400.2.a.bu.1.1 2
80.77 odd 4 1280.2.c.m.769.4 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
640.2.f.c.449.1 4 40.3 even 4
640.2.f.c.449.2 yes 4 20.7 even 4
640.2.f.c.449.3 yes 4 40.13 odd 4
640.2.f.c.449.4 yes 4 5.2 odd 4
640.2.f.g.449.1 yes 4 40.37 odd 4
640.2.f.g.449.2 yes 4 5.3 odd 4
640.2.f.g.449.3 yes 4 40.27 even 4
640.2.f.g.449.4 yes 4 20.3 even 4
1280.2.c.e.769.1 4 80.37 odd 4
1280.2.c.e.769.2 4 80.43 even 4
1280.2.c.e.769.3 4 80.27 even 4
1280.2.c.e.769.4 4 80.53 odd 4
1280.2.c.m.769.1 4 80.13 odd 4
1280.2.c.m.769.2 4 80.67 even 4
1280.2.c.m.769.3 4 80.3 even 4
1280.2.c.m.769.4 4 80.77 odd 4
3200.2.d.k.1601.1 4 5.4 even 2
3200.2.d.k.1601.2 4 40.19 odd 2
3200.2.d.k.1601.3 4 40.29 even 2
3200.2.d.k.1601.4 4 20.19 odd 2
3200.2.d.l.1601.1 4 4.3 odd 2 inner
3200.2.d.l.1601.2 4 8.5 even 2 inner
3200.2.d.l.1601.3 4 8.3 odd 2 inner
3200.2.d.l.1601.4 4 1.1 even 1 trivial
6400.2.a.bu.1.1 2 80.69 even 4
6400.2.a.bu.1.2 2 80.59 odd 4
6400.2.a.bv.1.1 2 80.19 odd 4
6400.2.a.bv.1.2 2 80.29 even 4
6400.2.a.bw.1.1 2 16.11 odd 4
6400.2.a.bw.1.2 2 16.5 even 4
6400.2.a.bx.1.1 2 16.13 even 4
6400.2.a.bx.1.2 2 16.3 odd 4