Properties

Label 3332.1.bn.b.3135.1
Level $3332$
Weight $1$
Character 3332.3135
Analytic conductor $1.663$
Analytic rank $0$
Dimension $8$
Projective image $D_{16}$
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] = [3332,1,Mod(979,3332)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3332, base_ring=CyclotomicField(16))
 
chi = DirichletCharacter(H, H._module([8, 8, 3]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("3332.979");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3332 = 2^{2} \cdot 7^{2} \cdot 17 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 3332.bn (of order \(16\), degree \(8\), 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: \(1.66288462209\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\Q(\zeta_{16})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{16}\)
Projective field: Galois closure of \(\mathbb{Q}[x]/(x^{16} - \cdots)\)

Embedding invariants

Embedding label 3135.1
Root \(0.923880 + 0.382683i\) of defining polynomial
Character \(\chi\) \(=\) 3332.3135
Dual form 3332.1.bn.b.2351.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(0.382683 - 0.923880i) q^{2} +(-0.707107 - 0.707107i) q^{4} +(-1.63099 + 0.324423i) q^{5} +(-0.923880 + 0.382683i) q^{8} +(0.382683 + 0.923880i) q^{9} +O(q^{10})\) \(q+(0.382683 - 0.923880i) q^{2} +(-0.707107 - 0.707107i) q^{4} +(-1.63099 + 0.324423i) q^{5} +(-0.923880 + 0.382683i) q^{8} +(0.382683 + 0.923880i) q^{9} +(-0.324423 + 1.63099i) q^{10} +(-1.00000 - 1.00000i) q^{13} +1.00000i q^{16} +(0.923880 + 0.382683i) q^{17} +1.00000 q^{18} +(1.38268 + 0.923880i) q^{20} +(1.63099 - 0.675577i) q^{25} +(-1.30656 + 0.541196i) q^{26} +(1.63099 - 0.324423i) q^{29} +(0.923880 + 0.382683i) q^{32} +(0.707107 - 0.707107i) q^{34} +(0.382683 - 0.923880i) q^{36} +(-0.617317 + 0.923880i) q^{37} +(1.38268 - 0.923880i) q^{40} +(1.92388 + 0.382683i) q^{41} +(-0.923880 - 1.38268i) q^{45} -1.76537i q^{50} +1.41421i q^{52} +(0.707107 - 1.70711i) q^{53} +(0.324423 - 1.63099i) q^{58} +(-0.0761205 + 0.382683i) q^{61} +(0.707107 - 0.707107i) q^{64} +(1.95541 + 1.30656i) q^{65} +(-0.382683 - 0.923880i) q^{68} +(-0.707107 - 0.707107i) q^{72} +(1.08979 - 0.216773i) q^{73} +(0.617317 + 0.923880i) q^{74} +(-0.324423 - 1.63099i) q^{80} +(-0.707107 + 0.707107i) q^{81} +(1.08979 - 1.63099i) q^{82} +(-1.63099 - 0.324423i) q^{85} +(-1.63099 + 0.324423i) q^{90} +(0.216773 + 1.08979i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 8 q^{13} + 8 q^{18} + 8 q^{20} - 8 q^{37} + 8 q^{40} + 8 q^{41} - 8 q^{61} + 8 q^{74}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3332\mathbb{Z}\right)^\times\).

\(n\) \(785\) \(885\) \(1667\)
\(\chi(n)\) \(e\left(\frac{11}{16}\right)\) \(-1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.382683 0.923880i 0.382683 0.923880i
\(3\) 0 0 −0.831470 0.555570i \(-0.812500\pi\)
0.831470 + 0.555570i \(0.187500\pi\)
\(4\) −0.707107 0.707107i −0.707107 0.707107i
\(5\) −1.63099 + 0.324423i −1.63099 + 0.324423i −0.923880 0.382683i \(-0.875000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) −0.923880 + 0.382683i −0.923880 + 0.382683i
\(9\) 0.382683 + 0.923880i 0.382683 + 0.923880i
\(10\) −0.324423 + 1.63099i −0.324423 + 1.63099i
\(11\) 0 0 0.831470 0.555570i \(-0.187500\pi\)
−0.831470 + 0.555570i \(0.812500\pi\)
\(12\) 0 0
\(13\) −1.00000 1.00000i −1.00000 1.00000i 1.00000i \(-0.5\pi\)
−1.00000 \(\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 1.00000i 1.00000i
\(17\) 0.923880 + 0.382683i 0.923880 + 0.382683i
\(18\) 1.00000 1.00000
\(19\) 0 0 −0.923880 0.382683i \(-0.875000\pi\)
0.923880 + 0.382683i \(0.125000\pi\)
\(20\) 1.38268 + 0.923880i 1.38268 + 0.923880i
\(21\) 0 0
\(22\) 0 0
\(23\) 0 0 −0.555570 0.831470i \(-0.687500\pi\)
0.555570 + 0.831470i \(0.312500\pi\)
\(24\) 0 0
\(25\) 1.63099 0.675577i 1.63099 0.675577i
\(26\) −1.30656 + 0.541196i −1.30656 + 0.541196i
\(27\) 0 0
\(28\) 0 0
\(29\) 1.63099 0.324423i 1.63099 0.324423i 0.707107 0.707107i \(-0.250000\pi\)
0.923880 + 0.382683i \(0.125000\pi\)
\(30\) 0 0
\(31\) 0 0 0.555570 0.831470i \(-0.312500\pi\)
−0.555570 + 0.831470i \(0.687500\pi\)
\(32\) 0.923880 + 0.382683i 0.923880 + 0.382683i
\(33\) 0 0
\(34\) 0.707107 0.707107i 0.707107 0.707107i
\(35\) 0 0
\(36\) 0.382683 0.923880i 0.382683 0.923880i
\(37\) −0.617317 + 0.923880i −0.617317 + 0.923880i 0.382683 + 0.923880i \(0.375000\pi\)
−1.00000 \(\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 1.38268 0.923880i 1.38268 0.923880i
\(41\) 1.92388 + 0.382683i 1.92388 + 0.382683i 1.00000 \(0\)
0.923880 + 0.382683i \(0.125000\pi\)
\(42\) 0 0
\(43\) 0 0 −0.382683 0.923880i \(-0.625000\pi\)
0.382683 + 0.923880i \(0.375000\pi\)
\(44\) 0 0
\(45\) −0.923880 1.38268i −0.923880 1.38268i
\(46\) 0 0
\(47\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 1.76537i 1.76537i
\(51\) 0 0
\(52\) 1.41421i 1.41421i
\(53\) 0.707107 1.70711i 0.707107 1.70711i 1.00000i \(-0.5\pi\)
0.707107 0.707107i \(-0.250000\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 0 0
\(58\) 0.324423 1.63099i 0.324423 1.63099i
\(59\) 0 0 −0.382683 0.923880i \(-0.625000\pi\)
0.382683 + 0.923880i \(0.375000\pi\)
\(60\) 0 0
\(61\) −0.0761205 + 0.382683i −0.0761205 + 0.382683i 0.923880 + 0.382683i \(0.125000\pi\)
−1.00000 \(\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0.707107 0.707107i 0.707107 0.707107i
\(65\) 1.95541 + 1.30656i 1.95541 + 1.30656i
\(66\) 0 0
\(67\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(68\) −0.382683 0.923880i −0.382683 0.923880i
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 0.555570 0.831470i \(-0.312500\pi\)
−0.555570 + 0.831470i \(0.687500\pi\)
\(72\) −0.707107 0.707107i −0.707107 0.707107i
\(73\) 1.08979 0.216773i 1.08979 0.216773i 0.382683 0.923880i \(-0.375000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(74\) 0.617317 + 0.923880i 0.617317 + 0.923880i
\(75\) 0 0
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 0 0 0.831470 0.555570i \(-0.187500\pi\)
−0.831470 + 0.555570i \(0.812500\pi\)
\(80\) −0.324423 1.63099i −0.324423 1.63099i
\(81\) −0.707107 + 0.707107i −0.707107 + 0.707107i
\(82\) 1.08979 1.63099i 1.08979 1.63099i
\(83\) 0 0 0.382683 0.923880i \(-0.375000\pi\)
−0.382683 + 0.923880i \(0.625000\pi\)
\(84\) 0 0
\(85\) −1.63099 0.324423i −1.63099 0.324423i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(90\) −1.63099 + 0.324423i −1.63099 + 0.324423i
\(91\) 0 0
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 0.216773 + 1.08979i 0.216773 + 1.08979i 0.923880 + 0.382683i \(0.125000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) −1.63099 0.675577i −1.63099 0.675577i
\(101\) 2.00000 2.00000 1.00000 \(0\)
1.00000 \(0\)
\(102\) 0 0
\(103\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(104\) 1.30656 + 0.541196i 1.30656 + 0.541196i
\(105\) 0 0
\(106\) −1.30656 1.30656i −1.30656 1.30656i
\(107\) 0 0 −0.195090 0.980785i \(-0.562500\pi\)
0.195090 + 0.980785i \(0.437500\pi\)
\(108\) 0 0
\(109\) −0.382683 + 1.92388i −0.382683 + 1.92388i 1.00000i \(0.5\pi\)
−0.382683 + 0.923880i \(0.625000\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) −0.324423 + 0.216773i −0.324423 + 0.216773i −0.707107 0.707107i \(-0.750000\pi\)
0.382683 + 0.923880i \(0.375000\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) −1.38268 0.923880i −1.38268 0.923880i
\(117\) 0.541196 1.30656i 0.541196 1.30656i
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) 0.382683 0.923880i 0.382683 0.923880i
\(122\) 0.324423 + 0.216773i 0.324423 + 0.216773i
\(123\) 0 0
\(124\) 0 0
\(125\) −1.05826 + 0.707107i −1.05826 + 0.707107i
\(126\) 0 0
\(127\) 0 0 0.923880 0.382683i \(-0.125000\pi\)
−0.923880 + 0.382683i \(0.875000\pi\)
\(128\) −0.382683 0.923880i −0.382683 0.923880i
\(129\) 0 0
\(130\) 1.95541 1.30656i 1.95541 1.30656i
\(131\) 0 0 −0.195090 0.980785i \(-0.562500\pi\)
0.195090 + 0.980785i \(0.437500\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) 0 0
\(136\) −1.00000 −1.00000
\(137\) −1.84776 −1.84776 −0.923880 0.382683i \(-0.875000\pi\)
−0.923880 + 0.382683i \(0.875000\pi\)
\(138\) 0 0
\(139\) 0 0 −0.831470 0.555570i \(-0.812500\pi\)
0.831470 + 0.555570i \(0.187500\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 0 0
\(144\) −0.923880 + 0.382683i −0.923880 + 0.382683i
\(145\) −2.55487 + 1.05826i −2.55487 + 1.05826i
\(146\) 0.216773 1.08979i 0.216773 1.08979i
\(147\) 0 0
\(148\) 1.08979 0.216773i 1.08979 0.216773i
\(149\) 1.00000 1.00000i 1.00000 1.00000i 1.00000i \(-0.5\pi\)
1.00000 \(0\)
\(150\) 0 0
\(151\) 0 0 −0.923880 0.382683i \(-0.875000\pi\)
0.923880 + 0.382683i \(0.125000\pi\)
\(152\) 0 0
\(153\) 1.00000i 1.00000i
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) −1.30656 + 1.30656i −1.30656 + 1.30656i −0.382683 + 0.923880i \(0.625000\pi\)
−0.923880 + 0.382683i \(0.875000\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) −1.63099 0.324423i −1.63099 0.324423i
\(161\) 0 0
\(162\) 0.382683 + 0.923880i 0.382683 + 0.923880i
\(163\) 0 0 −0.980785 0.195090i \(-0.937500\pi\)
0.980785 + 0.195090i \(0.0625000\pi\)
\(164\) −1.08979 1.63099i −1.08979 1.63099i
\(165\) 0 0
\(166\) 0 0
\(167\) 0 0 0.555570 0.831470i \(-0.312500\pi\)
−0.555570 + 0.831470i \(0.687500\pi\)
\(168\) 0 0
\(169\) 1.00000i 1.00000i
\(170\) −0.923880 + 1.38268i −0.923880 + 1.38268i
\(171\) 0 0
\(172\) 0 0
\(173\) 1.63099 + 1.08979i 1.63099 + 1.08979i 0.923880 + 0.382683i \(0.125000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 0 0 −0.382683 0.923880i \(-0.625000\pi\)
0.382683 + 0.923880i \(0.375000\pi\)
\(180\) −0.324423 + 1.63099i −0.324423 + 1.63099i
\(181\) −0.216773 0.324423i −0.216773 0.324423i 0.707107 0.707107i \(-0.250000\pi\)
−0.923880 + 0.382683i \(0.875000\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 0.707107 1.70711i 0.707107 1.70711i
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(192\) 0 0
\(193\) −0.617317 0.923880i −0.617317 0.923880i 0.382683 0.923880i \(-0.375000\pi\)
−1.00000 \(\pi\)
\(194\) 1.08979 + 0.216773i 1.08979 + 0.216773i
\(195\) 0 0
\(196\) 0 0
\(197\) −0.382683 0.0761205i −0.382683 0.0761205i 1.00000i \(-0.5\pi\)
−0.382683 + 0.923880i \(0.625000\pi\)
\(198\) 0 0
\(199\) 0 0 −0.195090 0.980785i \(-0.562500\pi\)
0.195090 + 0.980785i \(0.437500\pi\)
\(200\) −1.24830 + 1.24830i −1.24830 + 1.24830i
\(201\) 0 0
\(202\) 0.765367 1.84776i 0.765367 1.84776i
\(203\) 0 0
\(204\) 0 0
\(205\) −3.26197 −3.26197
\(206\) 0 0
\(207\) 0 0
\(208\) 1.00000 1.00000i 1.00000 1.00000i
\(209\) 0 0
\(210\) 0 0
\(211\) 0 0 0.195090 0.980785i \(-0.437500\pi\)
−0.195090 + 0.980785i \(0.562500\pi\)
\(212\) −1.70711 + 0.707107i −1.70711 + 0.707107i
\(213\) 0 0
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 0 0
\(218\) 1.63099 + 1.08979i 1.63099 + 1.08979i
\(219\) 0 0
\(220\) 0 0
\(221\) −0.541196 1.30656i −0.541196 1.30656i
\(222\) 0 0
\(223\) 0 0 −0.923880 0.382683i \(-0.875000\pi\)
0.923880 + 0.382683i \(0.125000\pi\)
\(224\) 0 0
\(225\) 1.24830 + 1.24830i 1.24830 + 1.24830i
\(226\) 0.0761205 + 0.382683i 0.0761205 + 0.382683i
\(227\) 0 0 0.831470 0.555570i \(-0.187500\pi\)
−0.831470 + 0.555570i \(0.812500\pi\)
\(228\) 0 0
\(229\) −0.292893 0.707107i −0.292893 0.707107i 0.707107 0.707107i \(-0.250000\pi\)
−1.00000 \(\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) −1.38268 + 0.923880i −1.38268 + 0.923880i
\(233\) 1.92388 0.382683i 1.92388 0.382683i 0.923880 0.382683i \(-0.125000\pi\)
1.00000 \(0\)
\(234\) −1.00000 1.00000i −1.00000 1.00000i
\(235\) 0 0
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(240\) 0 0
\(241\) 0.923880 + 0.617317i 0.923880 + 0.617317i 0.923880 0.382683i \(-0.125000\pi\)
1.00000i \(0.5\pi\)
\(242\) −0.707107 0.707107i −0.707107 0.707107i
\(243\) 0 0
\(244\) 0.324423 0.216773i 0.324423 0.216773i
\(245\) 0 0
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) 0 0
\(250\) 0.248303 + 1.24830i 0.248303 + 1.24830i
\(251\) 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
\(256\) −1.00000 −1.00000
\(257\) −0.707107 0.292893i −0.707107 0.292893i 1.00000i \(-0.5\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) −0.458804 2.30656i −0.458804 2.30656i
\(261\) 0.923880 + 1.38268i 0.923880 + 1.38268i
\(262\) 0 0
\(263\) 0 0 0.923880 0.382683i \(-0.125000\pi\)
−0.923880 + 0.382683i \(0.875000\pi\)
\(264\) 0 0
\(265\) −0.599456 + 3.01367i −0.599456 + 3.01367i
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) −0.617317 + 0.923880i −0.617317 + 0.923880i 0.382683 + 0.923880i \(0.375000\pi\)
−1.00000 \(\pi\)
\(270\) 0 0
\(271\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(272\) −0.382683 + 0.923880i −0.382683 + 0.923880i
\(273\) 0 0
\(274\) −0.707107 + 1.70711i −0.707107 + 1.70711i
\(275\) 0 0
\(276\) 0 0
\(277\) 0.382683 + 1.92388i 0.382683 + 1.92388i 0.382683 + 0.923880i \(0.375000\pi\)
1.00000i \(0.5\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 0.707107 + 1.70711i 0.707107 + 1.70711i 0.707107 + 0.707107i \(0.250000\pi\)
1.00000i \(0.5\pi\)
\(282\) 0 0
\(283\) 0 0 −0.555570 0.831470i \(-0.687500\pi\)
0.555570 + 0.831470i \(0.312500\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 1.00000i 1.00000i
\(289\) 0.707107 + 0.707107i 0.707107 + 0.707107i
\(290\) 2.76537i 2.76537i
\(291\) 0 0
\(292\) −0.923880 0.617317i −0.923880 0.617317i
\(293\) 1.30656 1.30656i 1.30656 1.30656i 0.382683 0.923880i \(-0.375000\pi\)
0.923880 0.382683i \(-0.125000\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0.216773 1.08979i 0.216773 1.08979i
\(297\) 0 0
\(298\) −0.541196 1.30656i −0.541196 1.30656i
\(299\) 0 0
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 0.648847i 0.648847i
\(306\) 0.923880 + 0.382683i 0.923880 + 0.382683i
\(307\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 0 0 0.980785 0.195090i \(-0.0625000\pi\)
−0.980785 + 0.195090i \(0.937500\pi\)
\(312\) 0 0
\(313\) −0.382683 0.0761205i −0.382683 0.0761205i 1.00000i \(-0.5\pi\)
−0.382683 + 0.923880i \(0.625000\pi\)
\(314\) 0.707107 + 1.70711i 0.707107 + 1.70711i
\(315\) 0 0
\(316\) 0 0
\(317\) −1.38268 + 0.923880i −1.38268 + 0.923880i −0.382683 + 0.923880i \(0.625000\pi\)
−1.00000 \(\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) −0.923880 + 1.38268i −0.923880 + 1.38268i
\(321\) 0 0
\(322\) 0 0
\(323\) 0 0
\(324\) 1.00000 1.00000
\(325\) −2.30656 0.955410i −2.30656 0.955410i
\(326\) 0 0
\(327\) 0 0
\(328\) −1.92388 + 0.382683i −1.92388 + 0.382683i
\(329\) 0 0
\(330\) 0 0
\(331\) 0 0 0.923880 0.382683i \(-0.125000\pi\)
−0.923880 + 0.382683i \(0.875000\pi\)
\(332\) 0 0
\(333\) −1.08979 0.216773i −1.08979 0.216773i
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) −0.324423 0.216773i −0.324423 0.216773i 0.382683 0.923880i \(-0.375000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(338\) 0.923880 + 0.382683i 0.923880 + 0.382683i
\(339\) 0 0
\(340\) 0.923880 + 1.38268i 0.923880 + 1.38268i
\(341\) 0 0
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) 0 0
\(346\) 1.63099 1.08979i 1.63099 1.08979i
\(347\) 0 0 0.195090 0.980785i \(-0.437500\pi\)
−0.195090 + 0.980785i \(0.562500\pi\)
\(348\) 0 0
\(349\) 0 0 −0.382683 0.923880i \(-0.625000\pi\)
0.382683 + 0.923880i \(0.375000\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 0.541196 + 0.541196i 0.541196 + 0.541196i 0.923880 0.382683i \(-0.125000\pi\)
−0.382683 + 0.923880i \(0.625000\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 0 0 0.382683 0.923880i \(-0.375000\pi\)
−0.382683 + 0.923880i \(0.625000\pi\)
\(360\) 1.38268 + 0.923880i 1.38268 + 0.923880i
\(361\) 0.707107 + 0.707107i 0.707107 + 0.707107i
\(362\) −0.382683 + 0.0761205i −0.382683 + 0.0761205i
\(363\) 0 0
\(364\) 0 0
\(365\) −1.70711 + 0.707107i −1.70711 + 0.707107i
\(366\) 0 0
\(367\) 0 0 0.195090 0.980785i \(-0.437500\pi\)
−0.195090 + 0.980785i \(0.562500\pi\)
\(368\) 0 0
\(369\) 0.382683 + 1.92388i 0.382683 + 1.92388i
\(370\) −1.30656 1.30656i −1.30656 1.30656i
\(371\) 0 0
\(372\) 0 0
\(373\) 0.765367i 0.765367i −0.923880 0.382683i \(-0.875000\pi\)
0.923880 0.382683i \(-0.125000\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −1.95541 1.30656i −1.95541 1.30656i
\(378\) 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 0.923880 0.382683i \(-0.125000\pi\)
−0.923880 + 0.382683i \(0.875000\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) −1.08979 + 0.216773i −1.08979 + 0.216773i
\(387\) 0 0
\(388\) 0.617317 0.923880i 0.617317 0.923880i
\(389\) 1.30656 + 0.541196i 1.30656 + 0.541196i 0.923880 0.382683i \(-0.125000\pi\)
0.382683 + 0.923880i \(0.375000\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0 0
\(393\) 0 0
\(394\) −0.216773 + 0.324423i −0.216773 + 0.324423i
\(395\) 0 0
\(396\) 0 0
\(397\) 1.63099 1.08979i 1.63099 1.08979i 0.707107 0.707107i \(-0.250000\pi\)
0.923880 0.382683i \(-0.125000\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0.675577 + 1.63099i 0.675577 + 1.63099i
\(401\) −1.08979 0.216773i −1.08979 0.216773i −0.382683 0.923880i \(-0.625000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) −1.41421 1.41421i −1.41421 1.41421i
\(405\) 0.923880 1.38268i 0.923880 1.38268i
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) 1.84776i 1.84776i −0.382683 0.923880i \(-0.625000\pi\)
0.382683 0.923880i \(-0.375000\pi\)
\(410\) −1.24830 + 3.01367i −1.24830 + 3.01367i
\(411\) 0 0
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) 0 0
\(416\) −0.541196 1.30656i −0.541196 1.30656i
\(417\) 0 0
\(418\) 0 0
\(419\) 0 0 −0.555570 0.831470i \(-0.687500\pi\)
0.555570 + 0.831470i \(0.312500\pi\)
\(420\) 0 0
\(421\) 0.541196 0.541196i 0.541196 0.541196i −0.382683 0.923880i \(-0.625000\pi\)
0.923880 + 0.382683i \(0.125000\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 1.84776i 1.84776i
\(425\) 1.76537 1.76537
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 0 0 −0.555570 0.831470i \(-0.687500\pi\)
0.555570 + 0.831470i \(0.312500\pi\)
\(432\) 0 0
\(433\) −0.707107 1.70711i −0.707107 1.70711i −0.707107 0.707107i \(-0.750000\pi\)
1.00000i \(-0.5\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 1.63099 1.08979i 1.63099 1.08979i
\(437\) 0 0
\(438\) 0 0
\(439\) 0 0 0.555570 0.831470i \(-0.312500\pi\)
−0.555570 + 0.831470i \(0.687500\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) −1.41421 −1.41421
\(443\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 0.324423 1.63099i 0.324423 1.63099i −0.382683 0.923880i \(-0.625000\pi\)
0.707107 0.707107i \(-0.250000\pi\)
\(450\) 1.63099 0.675577i 1.63099 0.675577i
\(451\) 0 0
\(452\) 0.382683 + 0.0761205i 0.382683 + 0.0761205i
\(453\) 0 0
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −1.30656 0.541196i −1.30656 0.541196i −0.382683 0.923880i \(-0.625000\pi\)
−0.923880 + 0.382683i \(0.875000\pi\)
\(458\) −0.765367 −0.765367
\(459\) 0 0
\(460\) 0 0
\(461\) −1.84776 0.765367i −1.84776 0.765367i −0.923880 0.382683i \(-0.875000\pi\)
−0.923880 0.382683i \(-0.875000\pi\)
\(462\) 0 0
\(463\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(464\) 0.324423 + 1.63099i 0.324423 + 1.63099i
\(465\) 0 0
\(466\) 0.382683 1.92388i 0.382683 1.92388i
\(467\) 0 0 −0.382683 0.923880i \(-0.625000\pi\)
0.382683 + 0.923880i \(0.375000\pi\)
\(468\) −1.30656 + 0.541196i −1.30656 + 0.541196i
\(469\) 0 0
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 1.84776 1.84776
\(478\) 0 0
\(479\) 0 0 −0.831470 0.555570i \(-0.812500\pi\)
0.831470 + 0.555570i \(0.187500\pi\)
\(480\) 0 0
\(481\) 1.54120 0.306563i 1.54120 0.306563i
\(482\) 0.923880 0.617317i 0.923880 0.617317i
\(483\) 0 0
\(484\) −0.923880 + 0.382683i −0.923880 + 0.382683i
\(485\) −0.707107 1.70711i −0.707107 1.70711i
\(486\) 0 0
\(487\) 0 0 0.831470 0.555570i \(-0.187500\pi\)
−0.831470 + 0.555570i \(0.812500\pi\)
\(488\) −0.0761205 0.382683i −0.0761205 0.382683i
\(489\) 0 0
\(490\) 0 0
\(491\) 0 0 −0.923880 0.382683i \(-0.875000\pi\)
0.923880 + 0.382683i \(0.125000\pi\)
\(492\) 0 0
\(493\) 1.63099 + 0.324423i 1.63099 + 0.324423i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) 0 0 −0.555570 0.831470i \(-0.687500\pi\)
0.555570 + 0.831470i \(0.312500\pi\)
\(500\) 1.24830 + 0.248303i 1.24830 + 0.248303i
\(501\) 0 0
\(502\) 0 0
\(503\) 0 0 0.195090 0.980785i \(-0.437500\pi\)
−0.195090 + 0.980785i \(0.562500\pi\)
\(504\) 0 0
\(505\) −3.26197 + 0.648847i −3.26197 + 0.648847i
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 0.765367 0.765367 0.382683 0.923880i \(-0.375000\pi\)
0.382683 + 0.923880i \(0.375000\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) −0.382683 + 0.923880i −0.382683 + 0.923880i
\(513\) 0 0
\(514\) −0.541196 + 0.541196i −0.541196 + 0.541196i
\(515\) 0 0
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) 0 0
\(520\) −2.30656 0.458804i −2.30656 0.458804i
\(521\) −1.08979 1.63099i −1.08979 1.63099i −0.707107 0.707107i \(-0.750000\pi\)
−0.382683 0.923880i \(-0.625000\pi\)
\(522\) 1.63099 0.324423i 1.63099 0.324423i
\(523\) 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
\(529\) −0.382683 + 0.923880i −0.382683 + 0.923880i
\(530\) 2.55487 + 1.70711i 2.55487 + 1.70711i
\(531\) 0 0
\(532\) 0 0
\(533\) −1.54120 2.30656i −1.54120 2.30656i
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 0 0
\(538\) 0.617317 + 0.923880i 0.617317 + 0.923880i
\(539\) 0 0
\(540\) 0 0
\(541\) 0.324423 + 0.216773i 0.324423 + 0.216773i 0.707107 0.707107i \(-0.250000\pi\)
−0.382683 + 0.923880i \(0.625000\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0.707107 + 0.707107i 0.707107 + 0.707107i
\(545\) 3.26197i 3.26197i
\(546\) 0 0
\(547\) 0 0 0.555570 0.831470i \(-0.312500\pi\)
−0.555570 + 0.831470i \(0.687500\pi\)
\(548\) 1.30656 + 1.30656i 1.30656 + 1.30656i
\(549\) −0.382683 + 0.0761205i −0.382683 + 0.0761205i
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) 0 0
\(554\) 1.92388 + 0.382683i 1.92388 + 0.382683i
\(555\) 0 0
\(556\) 0 0
\(557\) 0.541196 0.541196i 0.541196 0.541196i −0.382683 0.923880i \(-0.625000\pi\)
0.923880 + 0.382683i \(0.125000\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 0 0
\(561\) 0 0
\(562\) 1.84776 1.84776
\(563\) 0 0 −0.923880 0.382683i \(-0.875000\pi\)
0.923880 + 0.382683i \(0.125000\pi\)
\(564\) 0 0
\(565\) 0.458804 0.458804i 0.458804 0.458804i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 0.707107 0.292893i 0.707107 0.292893i 1.00000i \(-0.5\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(570\) 0 0
\(571\) 0 0 −0.980785 0.195090i \(-0.937500\pi\)
0.980785 + 0.195090i \(0.0625000\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 0 0
\(576\) 0.923880 + 0.382683i 0.923880 + 0.382683i
\(577\) −1.41421 −1.41421 −0.707107 0.707107i \(-0.750000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(578\) 0.923880 0.382683i 0.923880 0.382683i
\(579\) 0 0
\(580\) 2.55487 + 1.05826i 2.55487 + 1.05826i
\(581\) 0 0
\(582\) 0 0
\(583\) 0 0
\(584\) −0.923880 + 0.617317i −0.923880 + 0.617317i
\(585\) −0.458804 + 2.30656i −0.458804 + 2.30656i
\(586\) −0.707107 1.70711i −0.707107 1.70711i
\(587\) 0 0 0.923880 0.382683i \(-0.125000\pi\)
−0.923880 + 0.382683i \(0.875000\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) 0 0
\(591\) 0 0
\(592\) −0.923880 0.617317i −0.923880 0.617317i
\(593\) −0.765367 + 1.84776i −0.765367 + 1.84776i −0.382683 + 0.923880i \(0.625000\pi\)
−0.382683 + 0.923880i \(0.625000\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) −1.41421 −1.41421
\(597\) 0 0
\(598\) 0 0
\(599\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(600\) 0 0
\(601\) −0.324423 + 0.216773i −0.324423 + 0.216773i −0.707107 0.707107i \(-0.750000\pi\)
0.382683 + 0.923880i \(0.375000\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) −0.324423 + 1.63099i −0.324423 + 1.63099i
\(606\) 0 0
\(607\) 0 0 −0.195090 0.980785i \(-0.562500\pi\)
0.195090 + 0.980785i \(0.437500\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) −0.599456 0.248303i −0.599456 0.248303i
\(611\) 0 0
\(612\) 0.707107 0.707107i 0.707107 0.707107i
\(613\) −1.41421 −1.41421 −0.707107 0.707107i \(-0.750000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 0.382683 + 1.92388i 0.382683 + 1.92388i 0.382683 + 0.923880i \(0.375000\pi\)
1.00000i \(0.5\pi\)
\(618\) 0 0
\(619\) 0 0 −0.980785 0.195090i \(-0.937500\pi\)
0.980785 + 0.195090i \(0.0625000\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) 0.248303 0.248303i 0.248303 0.248303i
\(626\) −0.216773 + 0.324423i −0.216773 + 0.324423i
\(627\) 0 0
\(628\) 1.84776 1.84776
\(629\) −0.923880 + 0.617317i −0.923880 + 0.617317i
\(630\) 0 0
\(631\) 0 0 0.382683 0.923880i \(-0.375000\pi\)
−0.382683 + 0.923880i \(0.625000\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0.324423 + 1.63099i 0.324423 + 1.63099i
\(635\) 0 0
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) 0 0
\(640\) 0.923880 + 1.38268i 0.923880 + 1.38268i
\(641\) 1.92388 0.382683i 1.92388 0.382683i 0.923880 0.382683i \(-0.125000\pi\)
1.00000 \(0\)
\(642\) 0 0
\(643\) 0 0 0.555570 0.831470i \(-0.312500\pi\)
−0.555570 + 0.831470i \(0.687500\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(648\) 0.382683 0.923880i 0.382683 0.923880i
\(649\) 0 0
\(650\) −1.76537 + 1.76537i −1.76537 + 1.76537i
\(651\) 0 0
\(652\) 0 0
\(653\) 0.216773 1.08979i 0.216773 1.08979i −0.707107 0.707107i \(-0.750000\pi\)
0.923880 0.382683i \(-0.125000\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) −0.382683 + 1.92388i −0.382683 + 1.92388i
\(657\) 0.617317 + 0.923880i 0.617317 + 0.923880i
\(658\) 0 0
\(659\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(660\) 0 0
\(661\) −0.541196 + 1.30656i −0.541196 + 1.30656i 0.382683 + 0.923880i \(0.375000\pi\)
−0.923880 + 0.382683i \(0.875000\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) −0.617317 + 0.923880i −0.617317 + 0.923880i
\(667\) 0 0
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) −0.382683 0.0761205i −0.382683 0.0761205i 1.00000i \(-0.5\pi\)
−0.382683 + 0.923880i \(0.625000\pi\)
\(674\) −0.324423 + 0.216773i −0.324423 + 0.216773i
\(675\) 0 0
\(676\) 0.707107 0.707107i 0.707107 0.707107i
\(677\) 0.617317 0.923880i 0.617317 0.923880i −0.382683 0.923880i \(-0.625000\pi\)
1.00000 \(0\)
\(678\) 0 0
\(679\) 0 0
\(680\) 1.63099 0.324423i 1.63099 0.324423i
\(681\) 0 0
\(682\) 0 0
\(683\) 0 0 0.555570 0.831470i \(-0.312500\pi\)
−0.555570 + 0.831470i \(0.687500\pi\)
\(684\) 0 0
\(685\) 3.01367 0.599456i 3.01367 0.599456i
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) −2.41421 + 1.00000i −2.41421 + 1.00000i
\(690\) 0 0
\(691\) 0 0 −0.555570 0.831470i \(-0.687500\pi\)
0.555570 + 0.831470i \(0.312500\pi\)
\(692\) −0.382683 1.92388i −0.382683 1.92388i
\(693\) 0 0
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 1.63099 + 1.08979i 1.63099 + 1.08979i
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) −0.541196 0.541196i −0.541196 0.541196i 0.382683 0.923880i \(-0.375000\pi\)
−0.923880 + 0.382683i \(0.875000\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) 0 0
\(705\) 0 0
\(706\) 0.707107 0.292893i 0.707107 0.292893i
\(707\) 0 0
\(708\) 0 0
\(709\) −1.63099 + 0.324423i −1.63099 + 0.324423i −0.923880 0.382683i \(-0.875000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(710\) 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 0.980785 0.195090i \(-0.0625000\pi\)
−0.980785 + 0.195090i \(0.937500\pi\)
\(720\) 1.38268 0.923880i 1.38268 0.923880i
\(721\) 0 0
\(722\) 0.923880 0.382683i 0.923880 0.382683i
\(723\) 0 0
\(724\) −0.0761205 + 0.382683i −0.0761205 + 0.382683i
\(725\) 2.44094 1.63099i 2.44094 1.63099i
\(726\) 0 0
\(727\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(728\) 0 0
\(729\) −0.923880 0.382683i −0.923880 0.382683i
\(730\) 1.84776i 1.84776i
\(731\) 0 0
\(732\) 0 0
\(733\) 0 0 0.382683 0.923880i \(-0.375000\pi\)
−0.382683 + 0.923880i \(0.625000\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 0 0
\(738\) 1.92388 + 0.382683i 1.92388 + 0.382683i
\(739\) 0 0 0.923880 0.382683i \(-0.125000\pi\)
−0.923880 + 0.382683i \(0.875000\pi\)
\(740\) −1.70711 + 0.707107i −1.70711 + 0.707107i
\(741\) 0 0
\(742\) 0 0
\(743\) 0 0 0.980785 0.195090i \(-0.0625000\pi\)
−0.980785 + 0.195090i \(0.937500\pi\)
\(744\) 0 0
\(745\) −1.30656 + 1.95541i −1.30656 + 1.95541i
\(746\) −0.707107 0.292893i −0.707107 0.292893i
\(747\) 0 0
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) 0 0 0.555570 0.831470i \(-0.312500\pi\)
−0.555570 + 0.831470i \(0.687500\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) −1.95541 + 1.30656i −1.95541 + 1.30656i
\(755\) 0 0
\(756\) 0 0
\(757\) 0 0 0.923880 0.382683i \(-0.125000\pi\)
−0.923880 + 0.382683i \(0.875000\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −1.00000 1.00000i −1.00000 1.00000i 1.00000i \(-0.5\pi\)
−1.00000 \(\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) −0.324423 1.63099i −0.324423 1.63099i
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) 1.41421 1.41421i 1.41421 1.41421i 0.707107 0.707107i \(-0.250000\pi\)
0.707107 0.707107i \(-0.250000\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) −0.216773 + 1.08979i −0.216773 + 1.08979i
\(773\) −0.292893 0.707107i −0.292893 0.707107i 0.707107 0.707107i \(-0.250000\pi\)
−1.00000 \(\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) −0.617317 0.923880i −0.617317 0.923880i
\(777\) 0 0
\(778\) 1.00000 1.00000i 1.00000 1.00000i
\(779\) 0 0
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 1.70711 2.55487i 1.70711 2.55487i
\(786\) 0 0
\(787\) 0 0 0.980785 0.195090i \(-0.0625000\pi\)
−0.980785 + 0.195090i \(0.937500\pi\)
\(788\) 0.216773 + 0.324423i 0.216773 + 0.324423i
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 0.458804 0.306563i 0.458804 0.306563i
\(794\) −0.382683 1.92388i −0.382683 1.92388i
\(795\) 0 0
\(796\) 0 0
\(797\) −0.541196 + 1.30656i −0.541196 + 1.30656i 0.382683 + 0.923880i \(0.375000\pi\)
−0.923880 + 0.382683i \(0.875000\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 1.76537 1.76537
\(801\) 0 0
\(802\) −0.617317 + 0.923880i −0.617317 + 0.923880i
\(803\) 0 0
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 0 0
\(808\) −1.84776 + 0.765367i −1.84776 + 0.765367i
\(809\) −1.63099 0.324423i −1.63099 0.324423i −0.707107 0.707107i \(-0.750000\pi\)
−0.923880 + 0.382683i \(0.875000\pi\)
\(810\) −0.923880 1.38268i −0.923880 1.38268i
\(811\) 0 0 −0.195090 0.980785i \(-0.562500\pi\)
0.195090 + 0.980785i \(0.437500\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 0 0
\(818\) −1.70711 0.707107i −1.70711 0.707107i
\(819\) 0 0
\(820\) 2.30656 + 2.30656i 2.30656 + 2.30656i
\(821\) −0.382683 1.92388i −0.382683 1.92388i −0.382683 0.923880i \(-0.625000\pi\)
1.00000i \(-0.5\pi\)
\(822\) 0 0
\(823\) 0 0 0.195090 0.980785i \(-0.437500\pi\)
−0.195090 + 0.980785i \(0.562500\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 0 0 0.831470 0.555570i \(-0.187500\pi\)
−0.831470 + 0.555570i \(0.812500\pi\)
\(828\) 0 0
\(829\) 1.30656 + 1.30656i 1.30656 + 1.30656i 0.923880 + 0.382683i \(0.125000\pi\)
0.382683 + 0.923880i \(0.375000\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) −1.41421 −1.41421
\(833\) 0 0
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 0 0 0.831470 0.555570i \(-0.187500\pi\)
−0.831470 + 0.555570i \(0.812500\pi\)
\(840\) 0 0
\(841\) 1.63099 0.675577i 1.63099 0.675577i
\(842\) −0.292893 0.707107i −0.292893 0.707107i
\(843\) 0 0
\(844\) 0 0
\(845\) −0.324423 1.63099i −0.324423 1.63099i
\(846\) 0 0
\(847\) 0 0
\(848\) 1.70711 + 0.707107i 1.70711 + 0.707107i
\(849\) 0 0
\(850\) 0.675577 1.63099i 0.675577 1.63099i
\(851\) 0 0
\(852\) 0 0
\(853\) 0.923880 + 0.617317i 0.923880 + 0.617317i 0.923880 0.382683i \(-0.125000\pi\)
1.00000i \(0.5\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 1.08979 + 0.216773i 1.08979 + 0.216773i 0.707107 0.707107i \(-0.250000\pi\)
0.382683 + 0.923880i \(0.375000\pi\)
\(858\) 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 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(864\) 0 0
\(865\) −3.01367 1.24830i −3.01367 1.24830i
\(866\) −1.84776 −1.84776
\(867\) 0 0
\(868\) 0 0
\(869\) 0 0
\(870\) 0 0
\(871\) 0 0
\(872\) −0.382683 1.92388i −0.382683 1.92388i
\(873\) −0.923880 + 0.617317i −0.923880 + 0.617317i
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 0.382683 + 0.0761205i 0.382683 + 0.0761205i 0.382683 0.923880i \(-0.375000\pi\)
1.00000i \(0.5\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) −1.08979 + 1.63099i −1.08979 + 1.63099i −0.382683 + 0.923880i \(0.625000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(882\) 0 0
\(883\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(884\) −0.541196 + 1.30656i −0.541196 + 1.30656i
\(885\) 0 0
\(886\) 0 0
\(887\) 0 0 −0.831470 0.555570i \(-0.812500\pi\)
0.831470 + 0.555570i \(0.187500\pi\)
\(888\) 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
\(898\) −1.38268 0.923880i −1.38268 0.923880i
\(899\) 0 0
\(900\) 1.76537i 1.76537i
\(901\) 1.30656 1.30656i 1.30656 1.30656i
\(902\) 0 0
\(903\) 0 0
\(904\) 0.216773 0.324423i 0.216773 0.324423i
\(905\) 0.458804 + 0.458804i 0.458804 + 0.458804i
\(906\) 0 0
\(907\) 0 0 −0.555570 0.831470i \(-0.687500\pi\)
0.555570 + 0.831470i \(0.312500\pi\)
\(908\) 0 0
\(909\) 0.765367 + 1.84776i 0.765367 + 1.84776i
\(910\) 0 0
\(911\) 0 0 −0.980785 0.195090i \(-0.937500\pi\)
0.980785 + 0.195090i \(0.0625000\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) −1.00000 + 1.00000i −1.00000 + 1.00000i
\(915\) 0 0
\(916\) −0.292893 + 0.707107i −0.292893 + 0.707107i
\(917\) 0 0
\(918\) 0 0
\(919\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) −1.41421 + 1.41421i −1.41421 + 1.41421i
\(923\) 0 0
\(924\) 0 0
\(925\) −0.382683 + 1.92388i −0.382683 + 1.92388i
\(926\) 0 0
\(927\) 0 0
\(928\) 1.63099 + 0.324423i 1.63099 + 0.324423i
\(929\) −0.216773 0.324423i −0.216773 0.324423i 0.707107 0.707107i \(-0.250000\pi\)
−0.923880 + 0.382683i \(0.875000\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) −1.63099 1.08979i −1.63099 1.08979i
\(933\) 0 0
\(934\) 0 0
\(935\) 0 0
\(936\) 1.41421i 1.41421i
\(937\) −1.30656 0.541196i −1.30656 0.541196i −0.382683 0.923880i \(-0.625000\pi\)
−0.923880 + 0.382683i \(0.875000\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) −1.63099 + 1.08979i −1.63099 + 1.08979i −0.707107 + 0.707107i \(0.750000\pi\)
−0.923880 + 0.382683i \(0.875000\pi\)
\(942\) 0 0
\(943\) 0 0
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 0 0 0.980785 0.195090i \(-0.0625000\pi\)
−0.980785 + 0.195090i \(0.937500\pi\)
\(948\) 0 0
\(949\) −1.30656 0.873017i −1.30656 0.873017i
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 0.765367 0.765367 0.382683 0.923880i \(-0.375000\pi\)
0.382683 + 0.923880i \(0.375000\pi\)
\(954\) 0.707107 1.70711i 0.707107 1.70711i
\(955\) 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
\(962\) 0.306563 1.54120i 0.306563 1.54120i
\(963\) 0 0
\(964\) −0.216773 1.08979i −0.216773 1.08979i
\(965\) 1.30656 + 1.30656i 1.30656 + 1.30656i
\(966\) 0 0
\(967\) 0 0 −0.923880 0.382683i \(-0.875000\pi\)
0.923880 + 0.382683i \(0.125000\pi\)
\(968\) 1.00000i 1.00000i
\(969\) 0 0
\(970\) −1.84776 −1.84776
\(971\) 0 0 −0.923880 0.382683i \(-0.875000\pi\)
0.923880 + 0.382683i \(0.125000\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) 0 0
\(976\) −0.382683 0.0761205i −0.382683 0.0761205i
\(977\) 1.70711 0.707107i 1.70711 0.707107i 0.707107 0.707107i \(-0.250000\pi\)
1.00000 \(0\)
\(978\) 0 0
\(979\) 0 0
\(980\) 0 0
\(981\) −1.92388 + 0.382683i −1.92388 + 0.382683i
\(982\) 0 0
\(983\) 0 0 0.555570 0.831470i \(-0.312500\pi\)
−0.555570 + 0.831470i \(0.687500\pi\)
\(984\) 0 0
\(985\) 0.648847 0.648847
\(986\) 0.923880 1.38268i 0.923880 1.38268i
\(987\) 0 0
\(988\) 0 0
\(989\) 0 0
\(990\) 0 0
\(991\) 0 0 −0.195090 0.980785i \(-0.562500\pi\)
0.195090 + 0.980785i \(0.437500\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) 0.216773 + 0.324423i 0.216773 + 0.324423i 0.923880 0.382683i \(-0.125000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(998\) 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 3332.1.bn.b.3135.1 yes 8
4.3 odd 2 CM 3332.1.bn.b.3135.1 yes 8
7.2 even 3 3332.1.ce.a.619.1 16
7.3 odd 6 3332.1.ce.c.1979.1 16
7.4 even 3 3332.1.ce.a.1979.1 16
7.5 odd 6 3332.1.ce.c.619.1 16
7.6 odd 2 3332.1.bn.d.3135.1 yes 8
17.5 odd 16 3332.1.bn.d.2351.1 yes 8
28.3 even 6 3332.1.ce.c.1979.1 16
28.11 odd 6 3332.1.ce.a.1979.1 16
28.19 even 6 3332.1.ce.c.619.1 16
28.23 odd 6 3332.1.ce.a.619.1 16
28.27 even 2 3332.1.bn.d.3135.1 yes 8
68.39 even 16 3332.1.bn.d.2351.1 yes 8
119.5 even 48 3332.1.ce.a.3167.1 16
119.39 odd 48 3332.1.ce.c.1195.1 16
119.73 even 48 3332.1.ce.a.1195.1 16
119.90 even 16 inner 3332.1.bn.b.2351.1 8
119.107 odd 48 3332.1.ce.c.3167.1 16
476.39 even 48 3332.1.ce.c.1195.1 16
476.107 even 48 3332.1.ce.c.3167.1 16
476.243 odd 48 3332.1.ce.a.3167.1 16
476.311 odd 48 3332.1.ce.a.1195.1 16
476.447 odd 16 inner 3332.1.bn.b.2351.1 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
3332.1.bn.b.2351.1 8 119.90 even 16 inner
3332.1.bn.b.2351.1 8 476.447 odd 16 inner
3332.1.bn.b.3135.1 yes 8 1.1 even 1 trivial
3332.1.bn.b.3135.1 yes 8 4.3 odd 2 CM
3332.1.bn.d.2351.1 yes 8 17.5 odd 16
3332.1.bn.d.2351.1 yes 8 68.39 even 16
3332.1.bn.d.3135.1 yes 8 7.6 odd 2
3332.1.bn.d.3135.1 yes 8 28.27 even 2
3332.1.ce.a.619.1 16 7.2 even 3
3332.1.ce.a.619.1 16 28.23 odd 6
3332.1.ce.a.1195.1 16 119.73 even 48
3332.1.ce.a.1195.1 16 476.311 odd 48
3332.1.ce.a.1979.1 16 7.4 even 3
3332.1.ce.a.1979.1 16 28.11 odd 6
3332.1.ce.a.3167.1 16 119.5 even 48
3332.1.ce.a.3167.1 16 476.243 odd 48
3332.1.ce.c.619.1 16 7.5 odd 6
3332.1.ce.c.619.1 16 28.19 even 6
3332.1.ce.c.1195.1 16 119.39 odd 48
3332.1.ce.c.1195.1 16 476.39 even 48
3332.1.ce.c.1979.1 16 7.3 odd 6
3332.1.ce.c.1979.1 16 28.3 even 6
3332.1.ce.c.3167.1 16 119.107 odd 48
3332.1.ce.c.3167.1 16 476.107 even 48