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Properties

Label 512.2.e.d.129.1
Level $512$
Weight $2$
Character 512.129
Analytic conductor $4.088$
Analytic rank $0$
Dimension $2$
CM discriminant -4
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [512,2,Mod(129,512)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(512, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([0, 3]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("512.129");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 512 = 2^{9} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 512.e (of order \(4\), degree \(2\), 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: \(4.08834058349\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(i)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{4}]$

Embedding invariants

Embedding label 129.1
Root \(-1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 512.129
Dual form 512.2.e.d.385.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-1.00000 + 1.00000i) q^{5} -3.00000i q^{9} +O(q^{10})\) \(q+(-1.00000 + 1.00000i) q^{5} -3.00000i q^{9} +(5.00000 + 5.00000i) q^{13} +8.00000 q^{17} +3.00000i q^{25} +(3.00000 + 3.00000i) q^{29} +(-7.00000 + 7.00000i) q^{37} -8.00000i q^{41} +(3.00000 + 3.00000i) q^{45} +7.00000 q^{49} +(9.00000 - 9.00000i) q^{53} +(-11.0000 - 11.0000i) q^{61} -10.0000 q^{65} +6.00000i q^{73} -9.00000 q^{81} +(-8.00000 + 8.00000i) q^{85} +10.0000i q^{89} -8.00000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{5}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{5} + 10 q^{13} + 16 q^{17} + 6 q^{29} - 14 q^{37} + 6 q^{45} + 14 q^{49} + 18 q^{53} - 22 q^{61} - 20 q^{65} - 18 q^{81} - 16 q^{85} - 16 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/512\mathbb{Z}\right)^\times\).

\(n\) \(5\) \(511\)
\(\chi(n)\) \(e\left(\frac{3}{4}\right)\) \(1\)

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
\(3\) 0 0 0.707107 − 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(4\) 0 0
\(5\) −1.00000 + 1.00000i −0.447214 + 0.447214i −0.894427 − 0.447214i \(-0.852416\pi\)
0.447214 + 0.894427i \(0.352416\pi\)
\(6\) 0 0
\(7\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(8\) 0 0
\(9\) − 3.00000i − 1.00000i
\(10\) 0 0
\(11\) 0 0 −0.707107 − 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(12\) 0 0
\(13\) 5.00000 + 5.00000i 1.38675 + 1.38675i 0.832050 + 0.554700i \(0.187167\pi\)
0.554700 + 0.832050i \(0.312833\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 8.00000 1.94029 0.970143 − 0.242536i \(-0.0779791\pi\)
0.970143 + 0.242536i \(0.0779791\pi\)
\(18\) 0 0
\(19\) 0 0 0.707107 − 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(24\) 0 0
\(25\) 3.00000i 0.600000i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 3.00000 + 3.00000i 0.557086 + 0.557086i 0.928477 − 0.371391i \(-0.121119\pi\)
−0.371391 + 0.928477i \(0.621119\pi\)
\(30\) 0 0
\(31\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −7.00000 + 7.00000i −1.15079 + 1.15079i −0.164399 + 0.986394i \(0.552568\pi\)
−0.986394 + 0.164399i \(0.947432\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) − 8.00000i − 1.24939i −0.780869 − 0.624695i \(-0.785223\pi\)
0.780869 − 0.624695i \(-0.214777\pi\)
\(42\) 0 0
\(43\) 0 0 −0.707107 − 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(44\) 0 0
\(45\) 3.00000 + 3.00000i 0.447214 + 0.447214i
\(46\) 0 0
\(47\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(48\) 0 0
\(49\) 7.00000 1.00000
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 9.00000 − 9.00000i 1.23625 − 1.23625i 0.274721 − 0.961524i \(-0.411414\pi\)
0.961524 − 0.274721i \(-0.0885855\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 0 0 −0.707107 − 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(60\) 0 0
\(61\) −11.0000 − 11.0000i −1.40841 − 1.40841i −0.768221 − 0.640184i \(-0.778858\pi\)
−0.640184 − 0.768221i \(-0.721142\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −10.0000 −1.24035
\(66\) 0 0
\(67\) 0 0 0.707107 − 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(72\) 0 0
\(73\) 6.00000i 0.702247i 0.936329 + 0.351123i \(0.114200\pi\)
−0.936329 + 0.351123i \(0.885800\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(80\) 0 0
\(81\) −9.00000 −1.00000
\(82\) 0 0
\(83\) 0 0 0.707107 − 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(84\) 0 0
\(85\) −8.00000 + 8.00000i −0.867722 + 0.867722i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 10.0000i 1.06000i 0.847998 + 0.529999i \(0.177808\pi\)
−0.847998 + 0.529999i \(0.822192\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −8.00000 −0.812277 −0.406138 − 0.913812i \(-0.633125\pi\)
−0.406138 + 0.913812i \(0.633125\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 9.00000 − 9.00000i 0.895533 − 0.895533i −0.0995037 − 0.995037i \(-0.531726\pi\)
0.995037 + 0.0995037i \(0.0317255\pi\)
\(102\) 0 0
\(103\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 0 0 −0.707107 − 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(108\) 0 0
\(109\) −13.0000 − 13.0000i −1.24517 − 1.24517i −0.957826 − 0.287348i \(-0.907226\pi\)
−0.287348 − 0.957826i \(-0.592774\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) −14.0000 −1.31701 −0.658505 − 0.752577i \(-0.728811\pi\)
−0.658505 + 0.752577i \(0.728811\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 15.0000 − 15.0000i 1.38675 − 1.38675i
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) 11.0000i 1.00000i
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) −8.00000 − 8.00000i −0.715542 − 0.715542i
\(126\) 0 0
\(127\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 0 0 0.707107 − 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) − 8.00000i − 0.683486i −0.939793 − 0.341743i \(-0.888983\pi\)
0.939793 − 0.341743i \(-0.111017\pi\)
\(138\) 0 0
\(139\) 0 0 −0.707107 − 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) −6.00000 −0.498273
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) −17.0000 + 17.0000i −1.39269 + 1.39269i −0.573462 + 0.819232i \(0.694400\pi\)
−0.819232 + 0.573462i \(0.805600\pi\)
\(150\) 0 0
\(151\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(152\) 0 0
\(153\) − 24.0000i − 1.94029i
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 5.00000 + 5.00000i 0.399043 + 0.399043i 0.877896 − 0.478852i \(-0.158947\pi\)
−0.478852 + 0.877896i \(0.658947\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) 0 0 0.707107 − 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(168\) 0 0
\(169\) 37.0000i 2.84615i
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) −11.0000 − 11.0000i −0.836315 − 0.836315i 0.152057 − 0.988372i \(-0.451410\pi\)
−0.988372 + 0.152057i \(0.951410\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 0 0 0.707107 − 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(180\) 0 0
\(181\) −1.00000 + 1.00000i −0.0743294 + 0.0743294i −0.743294 − 0.668965i \(-0.766738\pi\)
0.668965 + 0.743294i \(0.266738\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) − 14.0000i − 1.02930i
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(192\) 0 0
\(193\) 24.0000 1.72756 0.863779 − 0.503871i \(-0.168091\pi\)
0.863779 + 0.503871i \(0.168091\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 15.0000 − 15.0000i 1.06871 − 1.06871i 0.0712470 − 0.997459i \(-0.477302\pi\)
0.997459 − 0.0712470i \(-0.0226979\pi\)
\(198\) 0 0
\(199\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 8.00000 + 8.00000i 0.558744 + 0.558744i
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 0 0
\(210\) 0 0
\(211\) 0 0 0.707107 − 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 40.0000 + 40.0000i 2.69069 + 2.69069i
\(222\) 0 0
\(223\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(224\) 0 0
\(225\) 9.00000 0.600000
\(226\) 0 0
\(227\) 0 0 0.707107 − 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(228\) 0 0
\(229\) −17.0000 + 17.0000i −1.12339 + 1.12339i −0.132164 + 0.991228i \(0.542192\pi\)
−0.991228 + 0.132164i \(0.957808\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) − 26.0000i − 1.70332i −0.524097 − 0.851658i \(-0.675597\pi\)
0.524097 − 0.851658i \(-0.324403\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(240\) 0 0
\(241\) 8.00000 0.515325 0.257663 − 0.966235i \(-0.417048\pi\)
0.257663 + 0.966235i \(0.417048\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) −7.00000 + 7.00000i −0.447214 + 0.447214i
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 0 0 −0.707107 − 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 2.00000 0.124757 0.0623783 − 0.998053i \(-0.480131\pi\)
0.0623783 + 0.998053i \(0.480131\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) 9.00000 − 9.00000i 0.557086 − 0.557086i
\(262\) 0 0
\(263\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(264\) 0 0
\(265\) 18.0000i 1.10573i
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 3.00000 + 3.00000i 0.182913 + 0.182913i 0.792624 − 0.609711i \(-0.208714\pi\)
−0.609711 + 0.792624i \(0.708714\pi\)
\(270\) 0 0
\(271\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) −23.0000 + 23.0000i −1.38194 + 1.38194i −0.540758 + 0.841178i \(0.681862\pi\)
−0.841178 + 0.540758i \(0.818138\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) − 10.0000i − 0.596550i −0.954480 − 0.298275i \(-0.903589\pi\)
0.954480 − 0.298275i \(-0.0964112\pi\)
\(282\) 0 0
\(283\) 0 0 −0.707107 − 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) 47.0000 2.76471
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 15.0000 − 15.0000i 0.876309 − 0.876309i −0.116841 − 0.993151i \(-0.537277\pi\)
0.993151 + 0.116841i \(0.0372769\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 0 0
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 22.0000 1.25972
\(306\) 0 0
\(307\) 0 0 0.707107 − 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(312\) 0 0
\(313\) − 24.0000i − 1.35656i −0.734803 − 0.678280i \(-0.762726\pi\)
0.734803 − 0.678280i \(-0.237274\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 3.00000 + 3.00000i 0.168497 + 0.168497i 0.786318 − 0.617822i \(-0.211985\pi\)
−0.617822 + 0.786318i \(0.711985\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 0 0
\(324\) 0 0
\(325\) −15.0000 + 15.0000i −0.832050 + 0.832050i
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) 0 0 −0.707107 − 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(332\) 0 0
\(333\) 21.0000 + 21.0000i 1.15079 + 1.15079i
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) −18.0000 −0.980522 −0.490261 − 0.871576i \(-0.663099\pi\)
−0.490261 + 0.871576i \(0.663099\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 0 0 −0.707107 − 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(348\) 0 0
\(349\) −13.0000 − 13.0000i −0.695874 − 0.695874i 0.267644 − 0.963518i \(-0.413755\pi\)
−0.963518 + 0.267644i \(0.913755\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) −34.0000 −1.80964 −0.904819 − 0.425797i \(-0.859994\pi\)
−0.904819 + 0.425797i \(0.859994\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(360\) 0 0
\(361\) − 19.0000i − 1.00000i
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) −6.00000 − 6.00000i −0.314054 − 0.314054i
\(366\) 0 0
\(367\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(368\) 0 0
\(369\) −24.0000 −1.24939
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) 25.0000 − 25.0000i 1.29445 − 1.29445i 0.362446 − 0.932005i \(-0.381942\pi\)
0.932005 − 0.362446i \(-0.118058\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 30.0000i 1.54508i
\(378\) 0 0
\(379\) 0 0 −0.707107 − 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) −7.00000 + 7.00000i −0.354914 + 0.354914i −0.861934 − 0.507020i \(-0.830747\pi\)
0.507020 + 0.861934i \(0.330747\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) −13.0000 − 13.0000i −0.652451 − 0.652451i 0.301131 − 0.953583i \(-0.402636\pi\)
−0.953583 + 0.301131i \(0.902636\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 40.0000 1.99750 0.998752 − 0.0499376i \(-0.0159023\pi\)
0.998752 + 0.0499376i \(0.0159023\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0 0
\(405\) 9.00000 − 9.00000i 0.447214 − 0.447214i
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) 40.0000i 1.97787i 0.148340 + 0.988936i \(0.452607\pi\)
−0.148340 + 0.988936i \(0.547393\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 0 0 0.707107 − 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(420\) 0 0
\(421\) −1.00000 + 1.00000i −0.0487370 + 0.0487370i −0.731055 − 0.682318i \(-0.760972\pi\)
0.682318 + 0.731055i \(0.260972\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 24.0000i 1.16417i
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(432\) 0 0
\(433\) −24.0000 −1.15337 −0.576683 − 0.816968i \(-0.695653\pi\)
−0.576683 + 0.816968i \(0.695653\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 0 0
\(438\) 0 0
\(439\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(440\) 0 0
\(441\) − 21.0000i − 1.00000i
\(442\) 0 0
\(443\) 0 0 −0.707107 − 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(444\) 0 0
\(445\) −10.0000 − 10.0000i −0.474045 − 0.474045i
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) −40.0000 −1.88772 −0.943858 − 0.330350i \(-0.892833\pi\)
−0.943858 + 0.330350i \(0.892833\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) − 8.00000i − 0.374224i −0.982339 − 0.187112i \(-0.940087\pi\)
0.982339 − 0.187112i \(-0.0599128\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) −29.0000 − 29.0000i −1.35066 − 1.35066i −0.884918 − 0.465746i \(-0.845786\pi\)
−0.465746 − 0.884918i \(-0.654214\pi\)
\(462\) 0 0
\(463\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 0 0 0.707107 − 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) −27.0000 − 27.0000i −1.23625 − 1.23625i
\(478\) 0 0
\(479\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(480\) 0 0
\(481\) −70.0000 −3.19173
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 8.00000 − 8.00000i 0.363261 − 0.363261i
\(486\) 0 0
\(487\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 0 0 −0.707107 − 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(492\) 0 0
\(493\) 24.0000 + 24.0000i 1.08091 + 1.08091i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) 0 0 0.707107 − 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(504\) 0 0
\(505\) 18.0000i 0.800989i
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) −27.0000 − 27.0000i −1.19675 − 1.19675i −0.975133 − 0.221621i \(-0.928865\pi\)
−0.221621 − 0.975133i \(-0.571135\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) − 40.0000i − 1.75243i −0.481919 − 0.876216i \(-0.660060\pi\)
0.481919 − 0.876216i \(-0.339940\pi\)
\(522\) 0 0
\(523\) 0 0 −0.707107 − 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) 23.0000 1.00000
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 40.0000 − 40.0000i 1.73259 − 1.73259i
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) −11.0000 − 11.0000i −0.472927 − 0.472927i 0.429934 − 0.902861i \(-0.358537\pi\)
−0.902861 + 0.429934i \(0.858537\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 26.0000 1.11372
\(546\) 0 0
\(547\) 0 0 0.707107 − 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(548\) 0 0
\(549\) −33.0000 + 33.0000i −1.40841 + 1.40841i
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 5.00000 + 5.00000i 0.211857 + 0.211857i 0.805056 − 0.593199i \(-0.202135\pi\)
−0.593199 + 0.805056i \(0.702135\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 0 0 0.707107 − 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(564\) 0 0
\(565\) 14.0000 − 14.0000i 0.588984 − 0.588984i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 40.0000i 1.67689i 0.544988 + 0.838444i \(0.316534\pi\)
−0.544988 + 0.838444i \(0.683466\pi\)
\(570\) 0 0
\(571\) 0 0 −0.707107 − 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) −2.00000 −0.0832611 −0.0416305 − 0.999133i \(-0.513255\pi\)
−0.0416305 + 0.999133i \(0.513255\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) 30.0000i 1.24035i
\(586\) 0 0
\(587\) 0 0 −0.707107 − 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) −46.0000 −1.88899 −0.944497 − 0.328521i \(-0.893450\pi\)
−0.944497 + 0.328521i \(0.893450\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(600\) 0 0
\(601\) 10.0000i 0.407909i 0.978980 + 0.203954i \(0.0653794\pi\)
−0.978980 + 0.203954i \(0.934621\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) −11.0000 − 11.0000i −0.447214 − 0.447214i
\(606\) 0 0
\(607\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) −1.00000 + 1.00000i −0.0403896 + 0.0403896i −0.727013 − 0.686624i \(-0.759092\pi\)
0.686624 + 0.727013i \(0.259092\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) − 38.0000i − 1.52982i −0.644136 − 0.764911i \(-0.722783\pi\)
0.644136 − 0.764911i \(-0.277217\pi\)
\(618\) 0 0
\(619\) 0 0 −0.707107 − 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) −56.0000 + 56.0000i −2.23287 + 2.23287i
\(630\) 0 0
\(631\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 35.0000 + 35.0000i 1.38675 + 1.38675i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) −8.00000 −0.315981 −0.157991 − 0.987441i \(-0.550502\pi\)
−0.157991 + 0.987441i \(0.550502\pi\)
\(642\) 0 0
\(643\) 0 0 0.707107 − 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 35.0000 + 35.0000i 1.36966 + 1.36966i 0.860927 + 0.508729i \(0.169885\pi\)
0.508729 + 0.860927i \(0.330115\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 18.0000 0.702247
\(658\) 0 0
\(659\) 0 0 0.707107 − 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(660\) 0 0
\(661\) 31.0000 − 31.0000i 1.20576 − 1.20576i 0.233373 − 0.972387i \(-0.425024\pi\)
0.972387 − 0.233373i \(-0.0749763\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 0 0
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) 24.0000 0.925132 0.462566 − 0.886585i \(-0.346929\pi\)
0.462566 + 0.886585i \(0.346929\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 25.0000 − 25.0000i 0.960828 − 0.960828i −0.0384331 − 0.999261i \(-0.512237\pi\)
0.999261 + 0.0384331i \(0.0122367\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 0 0 −0.707107 − 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(684\) 0 0
\(685\) 8.00000 + 8.00000i 0.305664 + 0.305664i
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 90.0000 3.42873
\(690\) 0 0
\(691\) 0 0 0.707107 − 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) − 64.0000i − 2.42417i
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 21.0000 + 21.0000i 0.793159 + 0.793159i 0.982006 − 0.188847i \(-0.0604752\pi\)
−0.188847 + 0.982006i \(0.560475\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) −7.00000 + 7.00000i −0.262891 + 0.262891i −0.826227 − 0.563337i \(-0.809517\pi\)
0.563337 + 0.826227i \(0.309517\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 0 0
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) −9.00000 + 9.00000i −0.334252 + 0.334252i
\(726\) 0 0
\(727\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(728\) 0 0
\(729\) 27.0000i 1.00000i
\(730\) 0 0
\(731\) 0 0
\(732\) 0 0
\(733\) −29.0000 − 29.0000i −1.07114 − 1.07114i −0.997268 − 0.0738717i \(-0.976464\pi\)
−0.0738717 − 0.997268i \(-0.523536\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) 0 0 0.707107 − 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(744\) 0 0
\(745\) − 34.0000i − 1.24566i
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) −17.0000 + 17.0000i −0.617876 + 0.617876i −0.944986 − 0.327111i \(-0.893925\pi\)
0.327111 + 0.944986i \(0.393925\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 40.0000i 1.45000i 0.688749 + 0.724999i \(0.258160\pi\)
−0.688749 + 0.724999i \(0.741840\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) 24.0000 + 24.0000i 0.867722 + 0.867722i
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) 24.0000 0.865462 0.432731 − 0.901523i \(-0.357550\pi\)
0.432731 + 0.901523i \(0.357550\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) −39.0000 + 39.0000i −1.40273 + 1.40273i −0.611448 + 0.791285i \(0.709412\pi\)
−0.791285 + 0.611448i \(0.790588\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 0 0
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) −10.0000 −0.356915
\(786\) 0 0
\(787\) 0 0 0.707107 − 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) − 110.000i − 3.90621i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 37.0000 + 37.0000i 1.31061 + 1.31061i 0.920967 + 0.389640i \(0.127401\pi\)
0.389640 + 0.920967i \(0.372599\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 0 0
\(801\) 30.0000 1.06000
\(802\) 0 0
\(803\) 0 0
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 56.0000i 1.96886i 0.175791 + 0.984428i \(0.443752\pi\)
−0.175791 + 0.984428i \(0.556248\pi\)
\(810\) 0 0
\(811\) 0 0 −0.707107 − 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 0 0
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −39.0000 + 39.0000i −1.36111 + 1.36111i −0.488603 + 0.872506i \(0.662493\pi\)
−0.872506 + 0.488603i \(0.837507\pi\)
\(822\) 0 0
\(823\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 0 0 −0.707107 − 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(828\) 0 0
\(829\) 37.0000 + 37.0000i 1.28506 + 1.28506i 0.937749 + 0.347314i \(0.112906\pi\)
0.347314 + 0.937749i \(0.387094\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 56.0000 1.94029
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(840\) 0 0
\(841\) − 11.0000i − 0.379310i
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) −37.0000 − 37.0000i −1.27284 − 1.27284i
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 0 0
\(852\) 0 0
\(853\) 41.0000 − 41.0000i 1.40381 − 1.40381i 0.616308 − 0.787505i \(-0.288628\pi\)
0.787505 − 0.616308i \(-0.211372\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 8.00000i 0.273275i 0.990621 + 0.136637i \(0.0436295\pi\)
−0.990621 + 0.136637i \(0.956370\pi\)
\(858\) 0 0
\(859\) 0 0 −0.707107 − 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(864\) 0 0
\(865\) 22.0000 0.748022
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 0 0
\(870\) 0 0
\(871\) 0 0
\(872\) 0 0
\(873\) 24.0000i 0.812277i
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 35.0000 + 35.0000i 1.18187 + 1.18187i 0.979260 + 0.202606i \(0.0649409\pi\)
0.202606 + 0.979260i \(0.435059\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) −50.0000 −1.68454 −0.842271 − 0.539054i \(-0.818782\pi\)
−0.842271 + 0.539054i \(0.818782\pi\)
\(882\) 0 0
\(883\) 0 0 0.707107 − 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 0 0
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 0 0
\(900\) 0 0
\(901\) 72.0000 − 72.0000i 2.39867 − 2.39867i
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) − 2.00000i − 0.0664822i
\(906\) 0 0
\(907\) 0 0 −0.707107 − 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(908\) 0 0
\(909\) −27.0000 − 27.0000i −0.895533 − 0.895533i
\(910\) 0 0
\(911\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) −21.0000 − 21.0000i −0.690476 − 0.690476i
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) −40.0000 −1.31236 −0.656179 − 0.754606i \(-0.727828\pi\)
−0.656179 + 0.754606i \(0.727828\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 38.0000i 1.24141i 0.784046 + 0.620703i \(0.213153\pi\)
−0.784046 + 0.620703i \(0.786847\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 19.0000 + 19.0000i 0.619382 + 0.619382i 0.945373 − 0.325991i \(-0.105698\pi\)
−0.325991 + 0.945373i \(0.605698\pi\)
\(942\) 0 0
\(943\) 0 0
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 0 0 0.707107 − 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(948\) 0 0
\(949\) −30.0000 + 30.0000i −0.973841 + 0.973841i
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) − 56.0000i − 1.81402i −0.421111 − 0.907009i \(-0.638360\pi\)
0.421111 − 0.907009i \(-0.361640\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) −31.0000 −1.00000
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) −24.0000 + 24.0000i −0.772587 + 0.772587i
\(966\) 0 0
\(967\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 0 0 −0.707107 − 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 8.00000 0.255943 0.127971 − 0.991778i \(-0.459153\pi\)
0.127971 + 0.991778i \(0.459153\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 0 0
\(981\) −39.0000 + 39.0000i −1.24517 + 1.24517i
\(982\) 0 0
\(983\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(984\) 0 0
\(985\) 30.0000i 0.955879i
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 0 0
\(990\) 0 0
\(991\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) 25.0000 − 25.0000i 0.791758 − 0.791758i −0.190022 − 0.981780i \(-0.560856\pi\)
0.981780 + 0.190022i \(0.0608559\pi\)
\(998\) 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 512.2.e.d.129.1 ✓ 2
3.2 odd 2 4608.2.k.r.1153.1 2
4.3 odd 2 CM 512.2.e.d.129.1 ✓ 2
8.3 odd 2 512.2.e.e.129.1 yes 2
8.5 even 2 512.2.e.e.129.1 yes 2
12.11 even 2 4608.2.k.r.1153.1 2
16.3 odd 4 512.2.e.e.385.1 yes 2
16.5 even 4 inner 512.2.e.d.385.1 yes 2
16.11 odd 4 inner 512.2.e.d.385.1 yes 2
16.13 even 4 512.2.e.e.385.1 yes 2
24.5 odd 2 4608.2.k.g.1153.1 2
24.11 even 2 4608.2.k.g.1153.1 2
32.3 odd 8 1024.2.b.c.513.1 2
32.5 even 8 1024.2.a.c.1.1 2
32.11 odd 8 1024.2.a.c.1.2 2
32.13 even 8 1024.2.b.c.513.2 2
32.19 odd 8 1024.2.b.c.513.2 2
32.21 even 8 1024.2.a.c.1.2 2
32.27 odd 8 1024.2.a.c.1.1 2
32.29 even 8 1024.2.b.c.513.1 2
48.5 odd 4 4608.2.k.r.3457.1 2
48.11 even 4 4608.2.k.r.3457.1 2
48.29 odd 4 4608.2.k.g.3457.1 2
48.35 even 4 4608.2.k.g.3457.1 2
96.5 odd 8 9216.2.a.p.1.2 2
96.11 even 8 9216.2.a.p.1.1 2
96.53 odd 8 9216.2.a.p.1.1 2
96.59 even 8 9216.2.a.p.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
512.2.e.d.129.1 ✓ 2 1.1 even 1 trivial
512.2.e.d.129.1 ✓ 2 4.3 odd 2 CM
512.2.e.d.385.1 yes 2 16.5 even 4 inner
512.2.e.d.385.1 yes 2 16.11 odd 4 inner
512.2.e.e.129.1 yes 2 8.3 odd 2
512.2.e.e.129.1 yes 2 8.5 even 2
512.2.e.e.385.1 yes 2 16.3 odd 4
512.2.e.e.385.1 yes 2 16.13 even 4
1024.2.a.c.1.1 2 32.5 even 8
1024.2.a.c.1.1 2 32.27 odd 8
1024.2.a.c.1.2 2 32.11 odd 8
1024.2.a.c.1.2 2 32.21 even 8
1024.2.b.c.513.1 2 32.3 odd 8
1024.2.b.c.513.1 2 32.29 even 8
1024.2.b.c.513.2 2 32.13 even 8
1024.2.b.c.513.2 2 32.19 odd 8
4608.2.k.g.1153.1 2 24.5 odd 2
4608.2.k.g.1153.1 2 24.11 even 2
4608.2.k.g.3457.1 2 48.29 odd 4
4608.2.k.g.3457.1 2 48.35 even 4
4608.2.k.r.1153.1 2 3.2 odd 2
4608.2.k.r.1153.1 2 12.11 even 2
4608.2.k.r.3457.1 2 48.5 odd 4
4608.2.k.r.3457.1 2 48.11 even 4
9216.2.a.p.1.1 2 96.11 even 8
9216.2.a.p.1.1 2 96.53 odd 8
9216.2.a.p.1.2 2 96.5 odd 8
9216.2.a.p.1.2 2 96.59 even 8