Properties

Label 1028.1.l.a.711.1
Level $1028$
Weight $1$
Character 1028.711
Analytic conductor $0.513$
Analytic rank $0$
Dimension $16$
Projective image $D_{32}$
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] = [1028,1,Mod(15,1028)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1028, base_ring=CyclotomicField(32))
 
chi = DirichletCharacter(H, H._module([16, 7]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1028.15");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1028 = 2^{2} \cdot 257 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 1028.l (of order \(32\), degree \(16\), 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: \(0.513038832987\)
Analytic rank: \(0\)
Dimension: \(16\)
Coefficient field: \(\Q(\zeta_{32})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{9}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{32}\)
Projective field: Galois closure of \(\mathbb{Q}[x]/(x^{32} - \cdots)\)

Embedding invariants

Embedding label 711.1
Root \(-0.555570 + 0.831470i\) of defining polynomial
Character \(\chi\) \(=\) 1028.711
Dual form 1028.1.l.a.227.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.00000i q^{2} -1.00000 q^{4} +(-1.36347 - 0.728789i) q^{5} -1.00000i q^{8} +(0.831470 + 0.555570i) q^{9} +O(q^{10})\) \(q+1.00000i q^{2} -1.00000 q^{4} +(-1.36347 - 0.728789i) q^{5} -1.00000i q^{8} +(0.831470 + 0.555570i) q^{9} +(0.728789 - 1.36347i) q^{10} +(1.08979 - 0.216773i) q^{13} +1.00000 q^{16} +(1.38704 + 1.38704i) q^{17} +(-0.555570 + 0.831470i) q^{18} +(1.36347 + 0.728789i) q^{20} +(0.772343 + 1.15589i) q^{25} +(0.216773 + 1.08979i) q^{26} +(-1.17588 + 0.785695i) q^{29} +1.00000i q^{32} +(-1.38704 + 1.38704i) q^{34} +(-0.831470 - 0.555570i) q^{36} +(0.151537 - 1.53858i) q^{37} +(-0.728789 + 1.36347i) q^{40} +(0.728789 - 0.598102i) q^{41} +(-0.728789 - 1.36347i) q^{45} +(0.980785 - 0.195090i) q^{49} +(-1.15589 + 0.772343i) q^{50} +(-1.08979 + 0.216773i) q^{52} +(0.598102 + 1.11897i) q^{53} +(-0.785695 - 1.17588i) q^{58} +(1.81225 - 0.360480i) q^{61} -1.00000 q^{64} +(-1.64388 - 0.498664i) q^{65} +(-1.38704 - 1.38704i) q^{68} +(0.555570 - 0.831470i) q^{72} +(-1.53636 - 0.636379i) q^{73} +(1.53858 + 0.151537i) q^{74} +(-1.36347 - 0.728789i) q^{80} +(0.382683 + 0.923880i) q^{81} +(0.598102 + 0.728789i) q^{82} +(-0.880326 - 2.90205i) q^{85} +(0.324423 - 1.63099i) q^{89} +(1.36347 - 0.728789i) q^{90} +(-0.273678 + 0.512016i) q^{97} +(0.195090 + 0.980785i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 16 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 16 q^{4} + 16 q^{16} - 16 q^{64} - 16 q^{65}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1028\mathbb{Z}\right)^\times\).

\(n\) \(515\) \(517\)
\(\chi(n)\) \(-1\) \(e\left(\frac{3}{32}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000i 1.00000i
\(3\) 0 0 −0.956940 0.290285i \(-0.906250\pi\)
0.956940 + 0.290285i \(0.0937500\pi\)
\(4\) −1.00000 −1.00000
\(5\) −1.36347 0.728789i −1.36347 0.728789i −0.382683 0.923880i \(-0.625000\pi\)
−0.980785 + 0.195090i \(0.937500\pi\)
\(6\) 0 0
\(7\) 0 0 0.995185 0.0980171i \(-0.0312500\pi\)
−0.995185 + 0.0980171i \(0.968750\pi\)
\(8\) 1.00000i 1.00000i
\(9\) 0.831470 + 0.555570i 0.831470 + 0.555570i
\(10\) 0.728789 1.36347i 0.728789 1.36347i
\(11\) 0 0 −0.382683 0.923880i \(-0.625000\pi\)
0.382683 + 0.923880i \(0.375000\pi\)
\(12\) 0 0
\(13\) 1.08979 0.216773i 1.08979 0.216773i 0.382683 0.923880i \(-0.375000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 1.00000 1.00000
\(17\) 1.38704 + 1.38704i 1.38704 + 1.38704i 0.831470 + 0.555570i \(0.187500\pi\)
0.555570 + 0.831470i \(0.312500\pi\)
\(18\) −0.555570 + 0.831470i −0.555570 + 0.831470i
\(19\) 0 0 0.634393 0.773010i \(-0.281250\pi\)
−0.634393 + 0.773010i \(0.718750\pi\)
\(20\) 1.36347 + 0.728789i 1.36347 + 0.728789i
\(21\) 0 0
\(22\) 0 0
\(23\) 0 0 0.382683 0.923880i \(-0.375000\pi\)
−0.382683 + 0.923880i \(0.625000\pi\)
\(24\) 0 0
\(25\) 0.772343 + 1.15589i 0.772343 + 1.15589i
\(26\) 0.216773 + 1.08979i 0.216773 + 1.08979i
\(27\) 0 0
\(28\) 0 0
\(29\) −1.17588 + 0.785695i −1.17588 + 0.785695i −0.980785 0.195090i \(-0.937500\pi\)
−0.195090 + 0.980785i \(0.562500\pi\)
\(30\) 0 0
\(31\) 0 0 0.555570 0.831470i \(-0.312500\pi\)
−0.555570 + 0.831470i \(0.687500\pi\)
\(32\) 1.00000i 1.00000i
\(33\) 0 0
\(34\) −1.38704 + 1.38704i −1.38704 + 1.38704i
\(35\) 0 0
\(36\) −0.831470 0.555570i −0.831470 0.555570i
\(37\) 0.151537 1.53858i 0.151537 1.53858i −0.555570 0.831470i \(-0.687500\pi\)
0.707107 0.707107i \(-0.250000\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) −0.728789 + 1.36347i −0.728789 + 1.36347i
\(41\) 0.728789 0.598102i 0.728789 0.598102i −0.195090 0.980785i \(-0.562500\pi\)
0.923880 + 0.382683i \(0.125000\pi\)
\(42\) 0 0
\(43\) 0 0 −0.290285 0.956940i \(-0.593750\pi\)
0.290285 + 0.956940i \(0.406250\pi\)
\(44\) 0 0
\(45\) −0.728789 1.36347i −0.728789 1.36347i
\(46\) 0 0
\(47\) 0 0 0.634393 0.773010i \(-0.281250\pi\)
−0.634393 + 0.773010i \(0.718750\pi\)
\(48\) 0 0
\(49\) 0.980785 0.195090i 0.980785 0.195090i
\(50\) −1.15589 + 0.772343i −1.15589 + 0.772343i
\(51\) 0 0
\(52\) −1.08979 + 0.216773i −1.08979 + 0.216773i
\(53\) 0.598102 + 1.11897i 0.598102 + 1.11897i 0.980785 + 0.195090i \(0.0625000\pi\)
−0.382683 + 0.923880i \(0.625000\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 0 0
\(58\) −0.785695 1.17588i −0.785695 1.17588i
\(59\) 0 0 −0.980785 0.195090i \(-0.937500\pi\)
0.980785 + 0.195090i \(0.0625000\pi\)
\(60\) 0 0
\(61\) 1.81225 0.360480i 1.81225 0.360480i 0.831470 0.555570i \(-0.187500\pi\)
0.980785 + 0.195090i \(0.0625000\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) −1.00000 −1.00000
\(65\) −1.64388 0.498664i −1.64388 0.498664i
\(66\) 0 0
\(67\) 0 0 −0.382683 0.923880i \(-0.625000\pi\)
0.382683 + 0.923880i \(0.375000\pi\)
\(68\) −1.38704 1.38704i −1.38704 1.38704i
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 −0.634393 0.773010i \(-0.718750\pi\)
0.634393 + 0.773010i \(0.281250\pi\)
\(72\) 0.555570 0.831470i 0.555570 0.831470i
\(73\) −1.53636 0.636379i −1.53636 0.636379i −0.555570 0.831470i \(-0.687500\pi\)
−0.980785 + 0.195090i \(0.937500\pi\)
\(74\) 1.53858 + 0.151537i 1.53858 + 0.151537i
\(75\) 0 0
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 0 0 −0.980785 0.195090i \(-0.937500\pi\)
0.980785 + 0.195090i \(0.0625000\pi\)
\(80\) −1.36347 0.728789i −1.36347 0.728789i
\(81\) 0.382683 + 0.923880i 0.382683 + 0.923880i
\(82\) 0.598102 + 0.728789i 0.598102 + 0.728789i
\(83\) 0 0 −0.290285 0.956940i \(-0.593750\pi\)
0.290285 + 0.956940i \(0.406250\pi\)
\(84\) 0 0
\(85\) −0.880326 2.90205i −0.880326 2.90205i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 0.324423 1.63099i 0.324423 1.63099i −0.382683 0.923880i \(-0.625000\pi\)
0.707107 0.707107i \(-0.250000\pi\)
\(90\) 1.36347 0.728789i 1.36347 0.728789i
\(91\) 0 0
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −0.273678 + 0.512016i −0.273678 + 0.512016i −0.980785 0.195090i \(-0.937500\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(98\) 0.195090 + 0.980785i 0.195090 + 0.980785i
\(99\) 0 0
\(100\) −0.772343 1.15589i −0.772343 1.15589i
\(101\) −1.75535 0.172887i −1.75535 0.172887i −0.831470 0.555570i \(-0.812500\pi\)
−0.923880 + 0.382683i \(0.875000\pi\)
\(102\) 0 0
\(103\) 0 0 0.881921 0.471397i \(-0.156250\pi\)
−0.881921 + 0.471397i \(0.843750\pi\)
\(104\) −0.216773 1.08979i −0.216773 1.08979i
\(105\) 0 0
\(106\) −1.11897 + 0.598102i −1.11897 + 0.598102i
\(107\) 0 0 0.956940 0.290285i \(-0.0937500\pi\)
−0.956940 + 0.290285i \(0.906250\pi\)
\(108\) 0 0
\(109\) −0.187593 + 1.90466i −0.187593 + 1.90466i 0.195090 + 0.980785i \(0.437500\pi\)
−0.382683 + 0.923880i \(0.625000\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) −0.923880 0.617317i −0.923880 0.617317i 1.00000i \(-0.5\pi\)
−0.923880 + 0.382683i \(0.875000\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 1.17588 0.785695i 1.17588 0.785695i
\(117\) 1.02656 + 0.425215i 1.02656 + 0.425215i
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) −0.707107 + 0.707107i −0.707107 + 0.707107i
\(122\) 0.360480 + 1.81225i 0.360480 + 1.81225i
\(123\) 0 0
\(124\) 0 0
\(125\) −0.0591266 0.600323i −0.0591266 0.600323i
\(126\) 0 0
\(127\) 0 0 0.290285 0.956940i \(-0.406250\pi\)
−0.290285 + 0.956940i \(0.593750\pi\)
\(128\) 1.00000i 1.00000i
\(129\) 0 0
\(130\) 0.498664 1.64388i 0.498664 1.64388i
\(131\) 0 0 0.471397 0.881921i \(-0.343750\pi\)
−0.471397 + 0.881921i \(0.656250\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) 0 0
\(136\) 1.38704 1.38704i 1.38704 1.38704i
\(137\) −1.38704 + 1.38704i −1.38704 + 1.38704i −0.555570 + 0.831470i \(0.687500\pi\)
−0.831470 + 0.555570i \(0.812500\pi\)
\(138\) 0 0
\(139\) 0 0 0.195090 0.980785i \(-0.437500\pi\)
−0.195090 + 0.980785i \(0.562500\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 0 0
\(144\) 0.831470 + 0.555570i 0.831470 + 0.555570i
\(145\) 2.17588 0.214305i 2.17588 0.214305i
\(146\) 0.636379 1.53636i 0.636379 1.53636i
\(147\) 0 0
\(148\) −0.151537 + 1.53858i −0.151537 + 1.53858i
\(149\) −0.368309 0.448786i −0.368309 0.448786i 0.555570 0.831470i \(-0.312500\pi\)
−0.923880 + 0.382683i \(0.875000\pi\)
\(150\) 0 0
\(151\) 0 0 0.881921 0.471397i \(-0.156250\pi\)
−0.881921 + 0.471397i \(0.843750\pi\)
\(152\) 0 0
\(153\) 0.382683 + 1.92388i 0.382683 + 1.92388i
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 0 0 0.831470 0.555570i \(-0.187500\pi\)
−0.831470 + 0.555570i \(0.812500\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0.728789 1.36347i 0.728789 1.36347i
\(161\) 0 0
\(162\) −0.923880 + 0.382683i −0.923880 + 0.382683i
\(163\) 0 0 −0.773010 0.634393i \(-0.781250\pi\)
0.773010 + 0.634393i \(0.218750\pi\)
\(164\) −0.728789 + 0.598102i −0.728789 + 0.598102i
\(165\) 0 0
\(166\) 0 0
\(167\) 0 0 0.881921 0.471397i \(-0.156250\pi\)
−0.881921 + 0.471397i \(0.843750\pi\)
\(168\) 0 0
\(169\) 0.216773 0.0897902i 0.216773 0.0897902i
\(170\) 2.90205 0.880326i 2.90205 0.880326i
\(171\) 0 0
\(172\) 0 0
\(173\) −0.750661 0.149316i −0.750661 0.149316i −0.195090 0.980785i \(-0.562500\pi\)
−0.555570 + 0.831470i \(0.687500\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 0 0
\(178\) 1.63099 + 0.324423i 1.63099 + 0.324423i
\(179\) 0 0 0.0980171 0.995185i \(-0.468750\pi\)
−0.0980171 + 0.995185i \(0.531250\pi\)
\(180\) 0.728789 + 1.36347i 0.728789 + 1.36347i
\(181\) 1.26268 1.53858i 1.26268 1.53858i 0.555570 0.831470i \(-0.312500\pi\)
0.707107 0.707107i \(-0.250000\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) −1.32791 + 1.98736i −1.32791 + 1.98736i
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 0 0 0.995185 0.0980171i \(-0.0312500\pi\)
−0.995185 + 0.0980171i \(0.968750\pi\)
\(192\) 0 0
\(193\) −0.765367 −0.765367 −0.382683 0.923880i \(-0.625000\pi\)
−0.382683 + 0.923880i \(0.625000\pi\)
\(194\) −0.512016 0.273678i −0.512016 0.273678i
\(195\) 0 0
\(196\) −0.980785 + 0.195090i −0.980785 + 0.195090i
\(197\) −1.00000 1.00000i −1.00000 1.00000i 1.00000i \(-0.5\pi\)
−1.00000 \(\pi\)
\(198\) 0 0
\(199\) 0 0 −0.555570 0.831470i \(-0.687500\pi\)
0.555570 + 0.831470i \(0.312500\pi\)
\(200\) 1.15589 0.772343i 1.15589 0.772343i
\(201\) 0 0
\(202\) 0.172887 1.75535i 0.172887 1.75535i
\(203\) 0 0
\(204\) 0 0
\(205\) −1.42957 + 0.284359i −1.42957 + 0.284359i
\(206\) 0 0
\(207\) 0 0
\(208\) 1.08979 0.216773i 1.08979 0.216773i
\(209\) 0 0
\(210\) 0 0
\(211\) 0 0 −0.923880 0.382683i \(-0.875000\pi\)
0.923880 + 0.382683i \(0.125000\pi\)
\(212\) −0.598102 1.11897i −0.598102 1.11897i
\(213\) 0 0
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 0 0
\(218\) −1.90466 0.187593i −1.90466 0.187593i
\(219\) 0 0
\(220\) 0 0
\(221\) 1.81225 + 1.21091i 1.81225 + 1.21091i
\(222\) 0 0
\(223\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(224\) 0 0
\(225\) 1.39018i 1.39018i
\(226\) 0.617317 0.923880i 0.617317 0.923880i
\(227\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(228\) 0 0
\(229\) −0.195090 + 0.0192147i −0.195090 + 0.0192147i −0.195090 0.980785i \(-0.562500\pi\)
1.00000i \(0.5\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0.785695 + 1.17588i 0.785695 + 1.17588i
\(233\) 0.577774 1.90466i 0.577774 1.90466i 0.195090 0.980785i \(-0.437500\pi\)
0.382683 0.923880i \(-0.375000\pi\)
\(234\) −0.425215 + 1.02656i −0.425215 + 1.02656i
\(235\) 0 0
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 0 0 0.555570 0.831470i \(-0.312500\pi\)
−0.555570 + 0.831470i \(0.687500\pi\)
\(240\) 0 0
\(241\) 1.84776 1.84776 0.923880 0.382683i \(-0.125000\pi\)
0.923880 + 0.382683i \(0.125000\pi\)
\(242\) −0.707107 0.707107i −0.707107 0.707107i
\(243\) 0 0
\(244\) −1.81225 + 0.360480i −1.81225 + 0.360480i
\(245\) −1.47945 0.448786i −1.47945 0.448786i
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) 0 0
\(250\) 0.600323 0.0591266i 0.600323 0.0591266i
\(251\) 0 0 0.290285 0.956940i \(-0.406250\pi\)
−0.290285 + 0.956940i \(0.593750\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 0 0
\(255\) 0 0
\(256\) 1.00000 1.00000
\(257\) −1.00000 −1.00000
\(258\) 0 0
\(259\) 0 0
\(260\) 1.64388 + 0.498664i 1.64388 + 0.498664i
\(261\) −1.41421 −1.41421
\(262\) 0 0
\(263\) 0 0 0.290285 0.956940i \(-0.406250\pi\)
−0.290285 + 0.956940i \(0.593750\pi\)
\(264\) 0 0
\(265\) 1.96157i 1.96157i
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 1.90466 + 0.577774i 1.90466 + 0.577774i 0.980785 + 0.195090i \(0.0625000\pi\)
0.923880 + 0.382683i \(0.125000\pi\)
\(270\) 0 0
\(271\) 0 0 −0.0980171 0.995185i \(-0.531250\pi\)
0.0980171 + 0.995185i \(0.468750\pi\)
\(272\) 1.38704 + 1.38704i 1.38704 + 1.38704i
\(273\) 0 0
\(274\) −1.38704 1.38704i −1.38704 1.38704i
\(275\) 0 0
\(276\) 0 0
\(277\) −1.75535 0.938254i −1.75535 0.938254i −0.923880 0.382683i \(-0.875000\pi\)
−0.831470 0.555570i \(-0.812500\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 0.168530 0.555570i 0.168530 0.555570i −0.831470 0.555570i \(-0.812500\pi\)
1.00000 \(0\)
\(282\) 0 0
\(283\) 0 0 −0.195090 0.980785i \(-0.562500\pi\)
0.195090 + 0.980785i \(0.437500\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) −0.555570 + 0.831470i −0.555570 + 0.831470i
\(289\) 2.84776i 2.84776i
\(290\) 0.214305 + 2.17588i 0.214305 + 2.17588i
\(291\) 0 0
\(292\) 1.53636 + 0.636379i 1.53636 + 0.636379i
\(293\) 0.923880 + 0.617317i 0.923880 + 0.617317i 0.923880 0.382683i \(-0.125000\pi\)
1.00000i \(0.5\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) −1.53858 0.151537i −1.53858 0.151537i
\(297\) 0 0
\(298\) 0.448786 0.368309i 0.448786 0.368309i
\(299\) 0 0
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) −2.73367 0.829249i −2.73367 0.829249i
\(306\) −1.92388 + 0.382683i −1.92388 + 0.382683i
\(307\) 0 0 0.831470 0.555570i \(-0.187500\pi\)
−0.831470 + 0.555570i \(0.812500\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 0 0 0.773010 0.634393i \(-0.218750\pi\)
−0.773010 + 0.634393i \(0.781250\pi\)
\(312\) 0 0
\(313\) −0.0569057 0.577774i −0.0569057 0.577774i −0.980785 0.195090i \(-0.937500\pi\)
0.923880 0.382683i \(-0.125000\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 1.36347 + 0.728789i 1.36347 + 0.728789i
\(321\) 0 0
\(322\) 0 0
\(323\) 0 0
\(324\) −0.382683 0.923880i −0.382683 0.923880i
\(325\) 1.09226 + 1.09226i 1.09226 + 1.09226i
\(326\) 0 0
\(327\) 0 0
\(328\) −0.598102 0.728789i −0.598102 0.728789i
\(329\) 0 0
\(330\) 0 0
\(331\) 0 0 −0.995185 0.0980171i \(-0.968750\pi\)
0.995185 + 0.0980171i \(0.0312500\pi\)
\(332\) 0 0
\(333\) 0.980785 1.19509i 0.980785 1.19509i
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) −1.55557 0.831470i −1.55557 0.831470i −0.555570 0.831470i \(-0.687500\pi\)
−1.00000 \(\pi\)
\(338\) 0.0897902 + 0.216773i 0.0897902 + 0.216773i
\(339\) 0 0
\(340\) 0.880326 + 2.90205i 0.880326 + 2.90205i
\(341\) 0 0
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) 0 0
\(346\) 0.149316 0.750661i 0.149316 0.750661i
\(347\) 0 0 0.881921 0.471397i \(-0.156250\pi\)
−0.881921 + 0.471397i \(0.843750\pi\)
\(348\) 0 0
\(349\) −0.541196 + 1.30656i −0.541196 + 1.30656i 0.382683 + 0.923880i \(0.375000\pi\)
−0.923880 + 0.382683i \(0.875000\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 0.273678 0.902197i 0.273678 0.902197i −0.707107 0.707107i \(-0.750000\pi\)
0.980785 0.195090i \(-0.0625000\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) −0.324423 + 1.63099i −0.324423 + 1.63099i
\(357\) 0 0
\(358\) 0 0
\(359\) 0 0 0.881921 0.471397i \(-0.156250\pi\)
−0.881921 + 0.471397i \(0.843750\pi\)
\(360\) −1.36347 + 0.728789i −1.36347 + 0.728789i
\(361\) −0.195090 0.980785i −0.195090 0.980785i
\(362\) 1.53858 + 1.26268i 1.53858 + 1.26268i
\(363\) 0 0
\(364\) 0 0
\(365\) 1.63099 + 1.98736i 1.63099 + 1.98736i
\(366\) 0 0
\(367\) 0 0 −0.995185 0.0980171i \(-0.968750\pi\)
0.995185 + 0.0980171i \(0.0312500\pi\)
\(368\) 0 0
\(369\) 0.938254 0.0924099i 0.938254 0.0924099i
\(370\) −1.98736 1.32791i −1.98736 1.32791i
\(371\) 0 0
\(372\) 0 0
\(373\) −0.636379 + 0.425215i −0.636379 + 0.425215i −0.831470 0.555570i \(-0.812500\pi\)
0.195090 + 0.980785i \(0.437500\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −1.11114 + 1.11114i −1.11114 + 1.11114i
\(378\) 0 0
\(379\) 0 0 −0.195090 0.980785i \(-0.562500\pi\)
0.195090 + 0.980785i \(0.437500\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 0 0 0.471397 0.881921i \(-0.343750\pi\)
−0.471397 + 0.881921i \(0.656250\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0.765367i 0.765367i
\(387\) 0 0
\(388\) 0.273678 0.512016i 0.273678 0.512016i
\(389\) 0.172887 + 1.75535i 0.172887 + 1.75535i 0.555570 + 0.831470i \(0.312500\pi\)
−0.382683 + 0.923880i \(0.625000\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) −0.195090 0.980785i −0.195090 0.980785i
\(393\) 0 0
\(394\) 1.00000 1.00000i 1.00000 1.00000i
\(395\) 0 0
\(396\) 0 0
\(397\) −0.360480 0.149316i −0.360480 0.149316i 0.195090 0.980785i \(-0.437500\pi\)
−0.555570 + 0.831470i \(0.687500\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0.772343 + 1.15589i 0.772343 + 1.15589i
\(401\) 1.38268 + 0.923880i 1.38268 + 0.923880i 1.00000 \(0\)
0.382683 + 0.923880i \(0.375000\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 1.75535 + 0.172887i 1.75535 + 0.172887i
\(405\) 0.151537 1.53858i 0.151537 1.53858i
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) 0.980785 + 0.804910i 0.980785 + 0.804910i 0.980785 0.195090i \(-0.0625000\pi\)
1.00000i \(0.5\pi\)
\(410\) −0.284359 1.42957i −0.284359 1.42957i
\(411\) 0 0
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) 0 0
\(416\) 0.216773 + 1.08979i 0.216773 + 1.08979i
\(417\) 0 0
\(418\) 0 0
\(419\) 0 0 0.923880 0.382683i \(-0.125000\pi\)
−0.923880 + 0.382683i \(0.875000\pi\)
\(420\) 0 0
\(421\) 0.448786 0.368309i 0.448786 0.368309i −0.382683 0.923880i \(-0.625000\pi\)
0.831470 + 0.555570i \(0.187500\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 1.11897 0.598102i 1.11897 0.598102i
\(425\) −0.531999 + 2.67454i −0.531999 + 2.67454i
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 0 0 −0.290285 0.956940i \(-0.593750\pi\)
0.290285 + 0.956940i \(0.406250\pi\)
\(432\) 0 0
\(433\) 0.636379 + 1.53636i 0.636379 + 1.53636i 0.831470 + 0.555570i \(0.187500\pi\)
−0.195090 + 0.980785i \(0.562500\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0.187593 1.90466i 0.187593 1.90466i
\(437\) 0 0
\(438\) 0 0
\(439\) 0 0 −0.290285 0.956940i \(-0.593750\pi\)
0.290285 + 0.956940i \(0.406250\pi\)
\(440\) 0 0
\(441\) 0.923880 + 0.382683i 0.923880 + 0.382683i
\(442\) −1.21091 + 1.81225i −1.21091 + 1.81225i
\(443\) 0 0 −0.634393 0.773010i \(-0.718750\pi\)
0.634393 + 0.773010i \(0.281250\pi\)
\(444\) 0 0
\(445\) −1.63099 + 1.98736i −1.63099 + 1.98736i
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) −1.68789 0.512016i −1.68789 0.512016i −0.707107 0.707107i \(-0.750000\pi\)
−0.980785 + 0.195090i \(0.937500\pi\)
\(450\) −1.39018 −1.39018
\(451\) 0 0
\(452\) 0.923880 + 0.617317i 0.923880 + 0.617317i
\(453\) 0 0
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −1.66294 + 1.11114i −1.66294 + 1.11114i −0.831470 + 0.555570i \(0.812500\pi\)
−0.831470 + 0.555570i \(0.812500\pi\)
\(458\) −0.0192147 0.195090i −0.0192147 0.195090i
\(459\) 0 0
\(460\) 0 0
\(461\) −0.598102 1.11897i −0.598102 1.11897i −0.980785 0.195090i \(-0.937500\pi\)
0.382683 0.923880i \(-0.375000\pi\)
\(462\) 0 0
\(463\) 0 0 −0.471397 0.881921i \(-0.656250\pi\)
0.471397 + 0.881921i \(0.343750\pi\)
\(464\) −1.17588 + 0.785695i −1.17588 + 0.785695i
\(465\) 0 0
\(466\) 1.90466 + 0.577774i 1.90466 + 0.577774i
\(467\) 0 0 0.634393 0.773010i \(-0.281250\pi\)
−0.634393 + 0.773010i \(0.718750\pi\)
\(468\) −1.02656 0.425215i −1.02656 0.425215i
\(469\) 0 0
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) −0.124363 + 1.26268i −0.124363 + 1.26268i
\(478\) 0 0
\(479\) 0 0 −0.923880 0.382683i \(-0.875000\pi\)
0.923880 + 0.382683i \(0.125000\pi\)
\(480\) 0 0
\(481\) −0.168378 1.70957i −0.168378 1.70957i
\(482\) 1.84776i 1.84776i
\(483\) 0 0
\(484\) 0.707107 0.707107i 0.707107 0.707107i
\(485\) 0.746304 0.498664i 0.746304 0.498664i
\(486\) 0 0
\(487\) 0 0 −0.634393 0.773010i \(-0.718750\pi\)
0.634393 + 0.773010i \(0.281250\pi\)
\(488\) −0.360480 1.81225i −0.360480 1.81225i
\(489\) 0 0
\(490\) 0.448786 1.47945i 0.448786 1.47945i
\(491\) 0 0 0.382683 0.923880i \(-0.375000\pi\)
−0.382683 + 0.923880i \(0.625000\pi\)
\(492\) 0 0
\(493\) −2.72078 0.541196i −2.72078 0.541196i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(500\) 0.0591266 + 0.600323i 0.0591266 + 0.600323i
\(501\) 0 0
\(502\) 0 0
\(503\) 0 0 −0.382683 0.923880i \(-0.625000\pi\)
0.382683 + 0.923880i \(0.375000\pi\)
\(504\) 0 0
\(505\) 2.26737 + 1.51501i 2.26737 + 1.51501i
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 0.172887 + 0.0924099i 0.172887 + 0.0924099i 0.555570 0.831470i \(-0.312500\pi\)
−0.382683 + 0.923880i \(0.625000\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 1.00000i 1.00000i
\(513\) 0 0
\(514\) 1.00000i 1.00000i
\(515\) 0 0
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) 0 0
\(520\) −0.498664 + 1.64388i −0.498664 + 1.64388i
\(521\) 1.90466 0.187593i 1.90466 0.187593i 0.923880 0.382683i \(-0.125000\pi\)
0.980785 + 0.195090i \(0.0625000\pi\)
\(522\) 1.41421i 1.41421i
\(523\) 0 0 −0.831470 0.555570i \(-0.812500\pi\)
0.831470 + 0.555570i \(0.187500\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) −0.707107 0.707107i −0.707107 0.707107i
\(530\) 1.96157 1.96157
\(531\) 0 0
\(532\) 0 0
\(533\) 0.664575 0.809787i 0.664575 0.809787i
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 0 0
\(538\) −0.577774 + 1.90466i −0.577774 + 1.90466i
\(539\) 0 0
\(540\) 0 0
\(541\) −1.11897 1.36347i −1.11897 1.36347i −0.923880 0.382683i \(-0.875000\pi\)
−0.195090 0.980785i \(-0.562500\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) −1.38704 + 1.38704i −1.38704 + 1.38704i
\(545\) 1.64388 2.46024i 1.64388 2.46024i
\(546\) 0 0
\(547\) 0 0 −0.0980171 0.995185i \(-0.531250\pi\)
0.0980171 + 0.995185i \(0.468750\pi\)
\(548\) 1.38704 1.38704i 1.38704 1.38704i
\(549\) 1.70711 + 0.707107i 1.70711 + 0.707107i
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) 0 0
\(554\) 0.938254 1.75535i 0.938254 1.75535i
\(555\) 0 0
\(556\) 0 0
\(557\) −0.0569057 0.187593i −0.0569057 0.187593i 0.923880 0.382683i \(-0.125000\pi\)
−0.980785 + 0.195090i \(0.937500\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 0 0
\(561\) 0 0
\(562\) 0.555570 + 0.168530i 0.555570 + 0.168530i
\(563\) 0 0 0.980785 0.195090i \(-0.0625000\pi\)
−0.980785 + 0.195090i \(0.937500\pi\)
\(564\) 0 0
\(565\) 0.809787 + 1.51501i 0.809787 + 1.51501i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 0.187593 1.90466i 0.187593 1.90466i −0.195090 0.980785i \(-0.562500\pi\)
0.382683 0.923880i \(-0.375000\pi\)
\(570\) 0 0
\(571\) 0 0 0.831470 0.555570i \(-0.187500\pi\)
−0.831470 + 0.555570i \(0.812500\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 0 0
\(576\) −0.831470 0.555570i −0.831470 0.555570i
\(577\) −1.55557 0.831470i −1.55557 0.831470i −0.555570 0.831470i \(-0.687500\pi\)
−1.00000 \(\pi\)
\(578\) −2.84776 −2.84776
\(579\) 0 0
\(580\) −2.17588 + 0.214305i −2.17588 + 0.214305i
\(581\) 0 0
\(582\) 0 0
\(583\) 0 0
\(584\) −0.636379 + 1.53636i −0.636379 + 1.53636i
\(585\) −1.08979 1.32791i −1.08979 1.32791i
\(586\) −0.617317 + 0.923880i −0.617317 + 0.923880i
\(587\) 0 0 −0.923880 0.382683i \(-0.875000\pi\)
0.923880 + 0.382683i \(0.125000\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) 0 0
\(591\) 0 0
\(592\) 0.151537 1.53858i 0.151537 1.53858i
\(593\) 0.750661 + 0.149316i 0.750661 + 0.149316i 0.555570 0.831470i \(-0.312500\pi\)
0.195090 + 0.980785i \(0.437500\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0.368309 + 0.448786i 0.368309 + 0.448786i
\(597\) 0 0
\(598\) 0 0
\(599\) 0 0 −0.290285 0.956940i \(-0.593750\pi\)
0.290285 + 0.956940i \(0.406250\pi\)
\(600\) 0 0
\(601\) −1.90466 + 0.577774i −1.90466 + 0.577774i −0.923880 + 0.382683i \(0.875000\pi\)
−0.980785 + 0.195090i \(0.937500\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 1.47945 0.448786i 1.47945 0.448786i
\(606\) 0 0
\(607\) 0 0 0.773010 0.634393i \(-0.218750\pi\)
−0.773010 + 0.634393i \(0.781250\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0.829249 2.73367i 0.829249 2.73367i
\(611\) 0 0
\(612\) −0.382683 1.92388i −0.382683 1.92388i
\(613\) −0.149316 + 0.750661i −0.149316 + 0.750661i 0.831470 + 0.555570i \(0.187500\pi\)
−0.980785 + 0.195090i \(0.937500\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 0.172887 0.0924099i 0.172887 0.0924099i −0.382683 0.923880i \(-0.625000\pi\)
0.555570 + 0.831470i \(0.312500\pi\)
\(618\) 0 0
\(619\) 0 0 −0.773010 0.634393i \(-0.781250\pi\)
0.773010 + 0.634393i \(0.218750\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) 0.175108 0.422747i 0.175108 0.422747i
\(626\) 0.577774 0.0569057i 0.577774 0.0569057i
\(627\) 0 0
\(628\) 0 0
\(629\) 2.34425 1.92388i 2.34425 1.92388i
\(630\) 0 0
\(631\) 0 0 −0.923880 0.382683i \(-0.875000\pi\)
0.923880 + 0.382683i \(0.125000\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 1.02656 0.425215i 1.02656 0.425215i
\(638\) 0 0
\(639\) 0 0
\(640\) −0.728789 + 1.36347i −0.728789 + 1.36347i
\(641\) 0.0569057 0.187593i 0.0569057 0.187593i −0.923880 0.382683i \(-0.875000\pi\)
0.980785 + 0.195090i \(0.0625000\pi\)
\(642\) 0 0
\(643\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 0 0 0.555570 0.831470i \(-0.312500\pi\)
−0.555570 + 0.831470i \(0.687500\pi\)
\(648\) 0.923880 0.382683i 0.923880 0.382683i
\(649\) 0 0
\(650\) −1.09226 + 1.09226i −1.09226 + 1.09226i
\(651\) 0 0
\(652\) 0 0
\(653\) −0.390181 + 1.96157i −0.390181 + 1.96157i −0.195090 + 0.980785i \(0.562500\pi\)
−0.195090 + 0.980785i \(0.562500\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0.728789 0.598102i 0.728789 0.598102i
\(657\) −0.923880 1.38268i −0.923880 1.38268i
\(658\) 0 0
\(659\) 0 0 0.995185 0.0980171i \(-0.0312500\pi\)
−0.995185 + 0.0980171i \(0.968750\pi\)
\(660\) 0 0
\(661\) −0.195090 0.0192147i −0.195090 0.0192147i 1.00000i \(-0.5\pi\)
−0.195090 + 0.980785i \(0.562500\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 1.19509 + 0.980785i 1.19509 + 0.980785i
\(667\) 0 0
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) −0.216773 1.08979i −0.216773 1.08979i −0.923880 0.382683i \(-0.875000\pi\)
0.707107 0.707107i \(-0.250000\pi\)
\(674\) 0.831470 1.55557i 0.831470 1.55557i
\(675\) 0 0
\(676\) −0.216773 + 0.0897902i −0.216773 + 0.0897902i
\(677\) −1.47945 1.21415i −1.47945 1.21415i −0.923880 0.382683i \(-0.875000\pi\)
−0.555570 0.831470i \(-0.687500\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) −2.90205 + 0.880326i −2.90205 + 0.880326i
\(681\) 0 0
\(682\) 0 0
\(683\) 0 0 0.923880 0.382683i \(-0.125000\pi\)
−0.923880 + 0.382683i \(0.875000\pi\)
\(684\) 0 0
\(685\) 2.90205 0.880326i 2.90205 0.880326i
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 0.894368 + 1.08979i 0.894368 + 1.08979i
\(690\) 0 0
\(691\) 0 0 −0.881921 0.471397i \(-0.843750\pi\)
0.881921 + 0.471397i \(0.156250\pi\)
\(692\) 0.750661 + 0.149316i 0.750661 + 0.149316i
\(693\) 0 0
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 1.84045 + 0.181269i 1.84045 + 0.181269i
\(698\) −1.30656 0.541196i −1.30656 0.541196i
\(699\) 0 0
\(700\) 0 0
\(701\) 0.750661 1.81225i 0.750661 1.81225i 0.195090 0.980785i \(-0.437500\pi\)
0.555570 0.831470i \(-0.312500\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) 0 0
\(705\) 0 0
\(706\) 0.902197 + 0.273678i 0.902197 + 0.273678i
\(707\) 0 0
\(708\) 0 0
\(709\) 1.53636 + 1.02656i 1.53636 + 1.02656i 0.980785 + 0.195090i \(0.0625000\pi\)
0.555570 + 0.831470i \(0.312500\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) −1.63099 0.324423i −1.63099 0.324423i
\(713\) 0 0
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 0 0 0.980785 0.195090i \(-0.0625000\pi\)
−0.980785 + 0.195090i \(0.937500\pi\)
\(720\) −0.728789 1.36347i −0.728789 1.36347i
\(721\) 0 0
\(722\) 0.980785 0.195090i 0.980785 0.195090i
\(723\) 0 0
\(724\) −1.26268 + 1.53858i −1.26268 + 1.53858i
\(725\) −1.81636 0.752360i −1.81636 0.752360i
\(726\) 0 0
\(727\) 0 0 −0.382683 0.923880i \(-0.625000\pi\)
0.382683 + 0.923880i \(0.375000\pi\)
\(728\) 0 0
\(729\) −0.195090 + 0.980785i −0.195090 + 0.980785i
\(730\) −1.98736 + 1.63099i −1.98736 + 1.63099i
\(731\) 0 0
\(732\) 0 0
\(733\) 1.47945 + 1.21415i 1.47945 + 1.21415i 0.923880 + 0.382683i \(0.125000\pi\)
0.555570 + 0.831470i \(0.312500\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 0 0
\(738\) 0.0924099 + 0.938254i 0.0924099 + 0.938254i
\(739\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(740\) 1.32791 1.98736i 1.32791 1.98736i
\(741\) 0 0
\(742\) 0 0
\(743\) 0 0 0.995185 0.0980171i \(-0.0312500\pi\)
−0.995185 + 0.0980171i \(0.968750\pi\)
\(744\) 0 0
\(745\) 0.175108 + 0.880326i 0.175108 + 0.880326i
\(746\) −0.425215 0.636379i −0.425215 0.636379i
\(747\) 0 0
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) 0 0 −0.881921 0.471397i \(-0.843750\pi\)
0.881921 + 0.471397i \(0.156250\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) −1.11114 1.11114i −1.11114 1.11114i
\(755\) 0 0
\(756\) 0 0
\(757\) −0.0924099 0.938254i −0.0924099 0.938254i −0.923880 0.382683i \(-0.875000\pi\)
0.831470 0.555570i \(-0.187500\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −0.902197 + 1.68789i −0.902197 + 1.68789i −0.195090 + 0.980785i \(0.562500\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) 0.880326 2.90205i 0.880326 2.90205i
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) 1.66294i 1.66294i −0.555570 0.831470i \(-0.687500\pi\)
0.555570 0.831470i \(-0.312500\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0.765367 0.765367
\(773\) 0.390181i 0.390181i −0.980785 0.195090i \(-0.937500\pi\)
0.980785 0.195090i \(-0.0625000\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0.512016 + 0.273678i 0.512016 + 0.273678i
\(777\) 0 0
\(778\) −1.75535 + 0.172887i −1.75535 + 0.172887i
\(779\) 0 0
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 0 0
\(784\) 0.980785 0.195090i 0.980785 0.195090i
\(785\) 0 0
\(786\) 0 0
\(787\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(788\) 1.00000 + 1.00000i 1.00000 + 1.00000i
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 1.89684 0.785695i 1.89684 0.785695i
\(794\) 0.149316 0.360480i 0.149316 0.360480i
\(795\) 0 0
\(796\) 0 0
\(797\) −0.149316 0.750661i −0.149316 0.750661i −0.980785 0.195090i \(-0.937500\pi\)
0.831470 0.555570i \(-0.187500\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) −1.15589 + 0.772343i −1.15589 + 0.772343i
\(801\) 1.17588 1.17588i 1.17588 1.17588i
\(802\) −0.923880 + 1.38268i −0.923880 + 1.38268i
\(803\) 0 0
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 0 0
\(808\) −0.172887 + 1.75535i −0.172887 + 1.75535i
\(809\) 0.728789 + 0.598102i 0.728789 + 0.598102i 0.923880 0.382683i \(-0.125000\pi\)
−0.195090 + 0.980785i \(0.562500\pi\)
\(810\) 1.53858 + 0.151537i 1.53858 + 0.151537i
\(811\) 0 0 0.471397 0.881921i \(-0.343750\pi\)
−0.471397 + 0.881921i \(0.656250\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 0 0
\(818\) −0.804910 + 0.980785i −0.804910 + 0.980785i
\(819\) 0 0
\(820\) 1.42957 0.284359i 1.42957 0.284359i
\(821\) −1.38268 + 0.923880i −1.38268 + 0.923880i −0.382683 + 0.923880i \(0.625000\pi\)
−1.00000 \(\pi\)
\(822\) 0 0
\(823\) 0 0 0.980785 0.195090i \(-0.0625000\pi\)
−0.980785 + 0.195090i \(0.937500\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 0 0 −0.0980171 0.995185i \(-0.531250\pi\)
0.0980171 + 0.995185i \(0.468750\pi\)
\(828\) 0 0
\(829\) −1.08979 1.63099i −1.08979 1.63099i −0.707107 0.707107i \(-0.750000\pi\)
−0.382683 0.923880i \(-0.625000\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) −1.08979 + 0.216773i −1.08979 + 0.216773i
\(833\) 1.63099 + 1.08979i 1.63099 + 1.08979i
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(840\) 0 0
\(841\) 0.382683 0.923880i 0.382683 0.923880i
\(842\) 0.368309 + 0.448786i 0.368309 + 0.448786i
\(843\) 0 0
\(844\) 0 0
\(845\) −0.361001 0.0355555i −0.361001 0.0355555i
\(846\) 0 0
\(847\) 0 0
\(848\) 0.598102 + 1.11897i 0.598102 + 1.11897i
\(849\) 0 0
\(850\) −2.67454 0.531999i −2.67454 0.531999i
\(851\) 0 0
\(852\) 0 0
\(853\) −0.804910 0.980785i −0.804910 0.980785i 0.195090 0.980785i \(-0.437500\pi\)
−1.00000 \(\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 0.902197 0.273678i 0.902197 0.273678i 0.195090 0.980785i \(-0.437500\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(858\) 0 0
\(859\) 0 0 0.923880 0.382683i \(-0.125000\pi\)
−0.923880 + 0.382683i \(0.875000\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 0 0 0.382683 0.923880i \(-0.375000\pi\)
−0.382683 + 0.923880i \(0.625000\pi\)
\(864\) 0 0
\(865\) 0.914683 + 0.750661i 0.914683 + 0.750661i
\(866\) −1.53636 + 0.636379i −1.53636 + 0.636379i
\(867\) 0 0
\(868\) 0 0
\(869\) 0 0
\(870\) 0 0
\(871\) 0 0
\(872\) 1.90466 + 0.187593i 1.90466 + 0.187593i
\(873\) −0.512016 + 0.273678i −0.512016 + 0.273678i
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) −1.68789 + 0.902197i −1.68789 + 0.902197i −0.707107 + 0.707107i \(0.750000\pi\)
−0.980785 + 0.195090i \(0.937500\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) −1.75535 0.172887i −1.75535 0.172887i −0.831470 0.555570i \(-0.812500\pi\)
−0.923880 + 0.382683i \(0.875000\pi\)
\(882\) −0.382683 + 0.923880i −0.382683 + 0.923880i
\(883\) 0 0 0.995185 0.0980171i \(-0.0312500\pi\)
−0.995185 + 0.0980171i \(0.968750\pi\)
\(884\) −1.81225 1.21091i −1.81225 1.21091i
\(885\) 0 0
\(886\) 0 0
\(887\) 0 0 0.831470 0.555570i \(-0.187500\pi\)
−0.831470 + 0.555570i \(0.812500\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) −1.98736 1.63099i −1.98736 1.63099i
\(891\) 0 0
\(892\) 0 0
\(893\) 0 0
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 0 0
\(898\) 0.512016 1.68789i 0.512016 1.68789i
\(899\) 0 0
\(900\) 1.39018i 1.39018i
\(901\) −0.722465 + 2.38165i −0.722465 + 2.38165i
\(902\) 0 0
\(903\) 0 0
\(904\) −0.617317 + 0.923880i −0.617317 + 0.923880i
\(905\) −2.84292 + 1.17758i −2.84292 + 1.17758i
\(906\) 0 0
\(907\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(908\) 0 0
\(909\) −1.36347 1.11897i −1.36347 1.11897i
\(910\) 0 0
\(911\) 0 0 −0.923880 0.382683i \(-0.875000\pi\)
0.923880 + 0.382683i \(0.125000\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) −1.11114 1.66294i −1.11114 1.66294i
\(915\) 0 0
\(916\) 0.195090 0.0192147i 0.195090 0.0192147i
\(917\) 0 0
\(918\) 0 0
\(919\) 0 0 0.0980171 0.995185i \(-0.468750\pi\)
−0.0980171 + 0.995185i \(0.531250\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 1.11897 0.598102i 1.11897 0.598102i
\(923\) 0 0
\(924\) 0 0
\(925\) 1.89547 1.01315i 1.89547 1.01315i
\(926\) 0 0
\(927\) 0 0
\(928\) −0.785695 1.17588i −0.785695 1.17588i
\(929\) 0.275899 1.38704i 0.275899 1.38704i −0.555570 0.831470i \(-0.687500\pi\)
0.831470 0.555570i \(-0.187500\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) −0.577774 + 1.90466i −0.577774 + 1.90466i
\(933\) 0 0
\(934\) 0 0
\(935\) 0 0
\(936\) 0.425215 1.02656i 0.425215 1.02656i
\(937\) −0.902197 + 0.273678i −0.902197 + 0.273678i −0.707107 0.707107i \(-0.750000\pi\)
−0.195090 + 0.980785i \(0.562500\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) −0.555570 + 0.168530i −0.555570 + 0.168530i −0.555570 0.831470i \(-0.687500\pi\)
1.00000i \(0.5\pi\)
\(942\) 0 0
\(943\) 0 0
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 0 0 −0.382683 0.923880i \(-0.625000\pi\)
0.382683 + 0.923880i \(0.375000\pi\)
\(948\) 0 0
\(949\) −1.81225 0.360480i −1.81225 0.360480i
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 0.368309 + 1.21415i 0.368309 + 1.21415i 0.923880 + 0.382683i \(0.125000\pi\)
−0.555570 + 0.831470i \(0.687500\pi\)
\(954\) −1.26268 0.124363i −1.26268 0.124363i
\(955\) 0 0
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) −0.382683 0.923880i −0.382683 0.923880i
\(962\) 1.70957 0.168378i 1.70957 0.168378i
\(963\) 0 0
\(964\) −1.84776 −1.84776
\(965\) 1.04355 + 0.557791i 1.04355 + 0.557791i
\(966\) 0 0
\(967\) 0 0 0.980785 0.195090i \(-0.0625000\pi\)
−0.980785 + 0.195090i \(0.937500\pi\)
\(968\) 0.707107 + 0.707107i 0.707107 + 0.707107i
\(969\) 0 0
\(970\) 0.498664 + 0.746304i 0.498664 + 0.746304i
\(971\) 0 0 0.831470 0.555570i \(-0.187500\pi\)
−0.831470 + 0.555570i \(0.812500\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) 0 0
\(976\) 1.81225 0.360480i 1.81225 0.360480i
\(977\) −0.273678 0.512016i −0.273678 0.512016i 0.707107 0.707107i \(-0.250000\pi\)
−0.980785 + 0.195090i \(0.937500\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 1.47945 + 0.448786i 1.47945 + 0.448786i
\(981\) −1.21415 + 1.47945i −1.21415 + 1.47945i
\(982\) 0 0
\(983\) 0 0 −0.471397 0.881921i \(-0.656250\pi\)
0.471397 + 0.881921i \(0.343750\pi\)
\(984\) 0 0
\(985\) 0.634680 + 2.09226i 0.634680 + 2.09226i
\(986\) 0.541196 2.72078i 0.541196 2.72078i
\(987\) 0 0
\(988\) 0 0
\(989\) 0 0
\(990\) 0 0
\(991\) 0 0 0.0980171 0.995185i \(-0.468750\pi\)
−0.0980171 + 0.995185i \(0.531250\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) 0 0 −0.831470 0.555570i \(-0.812500\pi\)
0.831470 + 0.555570i \(0.187500\pi\)
\(998\) 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 1028.1.l.a.711.1 yes 16
4.3 odd 2 CM 1028.1.l.a.711.1 yes 16
257.227 even 32 inner 1028.1.l.a.227.1 16
1028.227 odd 32 inner 1028.1.l.a.227.1 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1028.1.l.a.227.1 16 257.227 even 32 inner
1028.1.l.a.227.1 16 1028.227 odd 32 inner
1028.1.l.a.711.1 yes 16 1.1 even 1 trivial
1028.1.l.a.711.1 yes 16 4.3 odd 2 CM