Properties

Label 2312.1.p.e.179.2
Level $2312$
Weight $1$
Character 2312.179
Analytic conductor $1.154$
Analytic rank $0$
Dimension $8$
Projective image $D_{4}$
CM discriminant -8
Inner twists $16$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2312,1,Mod(155,2312)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2312, base_ring=CyclotomicField(8))
 
chi = DirichletCharacter(H, H._module([4, 4, 7]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2312.155");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2312 = 2^{3} \cdot 17^{2} \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 2312.p (of order \(8\), degree \(4\), not 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: \(1.15383830921\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(2\) over \(\Q(\zeta_{8})\)
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: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 136)
Projective image: \(D_{4}\)
Projective field: Galois closure of 4.0.314432.1

Embedding invariants

Embedding label 179.2
Root \(0.382683 - 0.923880i\) of defining polynomial
Character \(\chi\) \(=\) 2312.179
Dual form 2312.1.p.e.155.2

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(0.707107 - 0.707107i) q^{2} +(1.30656 + 0.541196i) q^{3} -1.00000i q^{4} +(1.30656 - 0.541196i) q^{6} +(-0.707107 - 0.707107i) q^{8} +(0.707107 + 0.707107i) q^{9} +O(q^{10})\) \(q+(0.707107 - 0.707107i) q^{2} +(1.30656 + 0.541196i) q^{3} -1.00000i q^{4} +(1.30656 - 0.541196i) q^{6} +(-0.707107 - 0.707107i) q^{8} +(0.707107 + 0.707107i) q^{9} +(1.30656 - 0.541196i) q^{11} +(0.541196 - 1.30656i) q^{12} -1.00000 q^{16} +1.00000 q^{18} +(-1.41421 + 1.41421i) q^{19} +(0.541196 - 1.30656i) q^{22} +(-0.541196 - 1.30656i) q^{24} +(-0.707107 - 0.707107i) q^{25} +(-0.707107 + 0.707107i) q^{32} +2.00000 q^{33} +(0.707107 - 0.707107i) q^{36} +2.00000i q^{38} +(0.541196 + 1.30656i) q^{41} +(-0.541196 - 1.30656i) q^{44} +(-1.30656 - 0.541196i) q^{48} +(-0.707107 + 0.707107i) q^{49} -1.00000 q^{50} +(-2.61313 + 1.08239i) q^{57} +1.00000i q^{64} +(1.41421 - 1.41421i) q^{66} -1.00000i q^{72} +(0.541196 - 1.30656i) q^{73} +(-0.541196 - 1.30656i) q^{75} +(1.41421 + 1.41421i) q^{76} -1.00000i q^{81} +(1.30656 + 0.541196i) q^{82} +(-1.30656 - 0.541196i) q^{88} +(-1.30656 + 0.541196i) q^{96} +(0.541196 - 1.30656i) q^{97} +1.00000i q^{98} +(1.30656 + 0.541196i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 8 q^{16} + 8 q^{18} + 16 q^{33} - 8 q^{50}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2312\mathbb{Z}\right)^\times\).

\(n\) \(1157\) \(1735\) \(1737\)
\(\chi(n)\) \(-1\) \(-1\) \(e\left(\frac{1}{8}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.707107 0.707107i 0.707107 0.707107i
\(3\) 1.30656 + 0.541196i 1.30656 + 0.541196i 0.923880 0.382683i \(-0.125000\pi\)
0.382683 + 0.923880i \(0.375000\pi\)
\(4\) 1.00000i 1.00000i
\(5\) 0 0 0.382683 0.923880i \(-0.375000\pi\)
−0.382683 + 0.923880i \(0.625000\pi\)
\(6\) 1.30656 0.541196i 1.30656 0.541196i
\(7\) 0 0 −0.382683 0.923880i \(-0.625000\pi\)
0.382683 + 0.923880i \(0.375000\pi\)
\(8\) −0.707107 0.707107i −0.707107 0.707107i
\(9\) 0.707107 + 0.707107i 0.707107 + 0.707107i
\(10\) 0 0
\(11\) 1.30656 0.541196i 1.30656 0.541196i 0.382683 0.923880i \(-0.375000\pi\)
0.923880 + 0.382683i \(0.125000\pi\)
\(12\) 0.541196 1.30656i 0.541196 1.30656i
\(13\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) −1.00000 −1.00000
\(17\) 0 0
\(18\) 1.00000 1.00000
\(19\) −1.41421 + 1.41421i −1.41421 + 1.41421i −0.707107 + 0.707107i \(0.750000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0.541196 1.30656i 0.541196 1.30656i
\(23\) 0 0 0.923880 0.382683i \(-0.125000\pi\)
−0.923880 + 0.382683i \(0.875000\pi\)
\(24\) −0.541196 1.30656i −0.541196 1.30656i
\(25\) −0.707107 0.707107i −0.707107 0.707107i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 0 0 0.382683 0.923880i \(-0.375000\pi\)
−0.382683 + 0.923880i \(0.625000\pi\)
\(30\) 0 0
\(31\) 0 0 −0.923880 0.382683i \(-0.875000\pi\)
0.923880 + 0.382683i \(0.125000\pi\)
\(32\) −0.707107 + 0.707107i −0.707107 + 0.707107i
\(33\) 2.00000 2.00000
\(34\) 0 0
\(35\) 0 0
\(36\) 0.707107 0.707107i 0.707107 0.707107i
\(37\) 0 0 −0.923880 0.382683i \(-0.875000\pi\)
0.923880 + 0.382683i \(0.125000\pi\)
\(38\) 2.00000i 2.00000i
\(39\) 0 0
\(40\) 0 0
\(41\) 0.541196 + 1.30656i 0.541196 + 1.30656i 0.923880 + 0.382683i \(0.125000\pi\)
−0.382683 + 0.923880i \(0.625000\pi\)
\(42\) 0 0
\(43\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(44\) −0.541196 1.30656i −0.541196 1.30656i
\(45\) 0 0
\(46\) 0 0
\(47\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(48\) −1.30656 0.541196i −1.30656 0.541196i
\(49\) −0.707107 + 0.707107i −0.707107 + 0.707107i
\(50\) −1.00000 −1.00000
\(51\) 0 0
\(52\) 0 0
\(53\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) −2.61313 + 1.08239i −2.61313 + 1.08239i
\(58\) 0 0
\(59\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(60\) 0 0
\(61\) 0 0 −0.382683 0.923880i \(-0.625000\pi\)
0.382683 + 0.923880i \(0.375000\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 1.00000i 1.00000i
\(65\) 0 0
\(66\) 1.41421 1.41421i 1.41421 1.41421i
\(67\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 −0.923880 0.382683i \(-0.875000\pi\)
0.923880 + 0.382683i \(0.125000\pi\)
\(72\) 1.00000i 1.00000i
\(73\) 0.541196 1.30656i 0.541196 1.30656i −0.382683 0.923880i \(-0.625000\pi\)
0.923880 0.382683i \(-0.125000\pi\)
\(74\) 0 0
\(75\) −0.541196 1.30656i −0.541196 1.30656i
\(76\) 1.41421 + 1.41421i 1.41421 + 1.41421i
\(77\) 0 0
\(78\) 0 0
\(79\) 0 0 0.923880 0.382683i \(-0.125000\pi\)
−0.923880 + 0.382683i \(0.875000\pi\)
\(80\) 0 0
\(81\) 1.00000i 1.00000i
\(82\) 1.30656 + 0.541196i 1.30656 + 0.541196i
\(83\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) −1.30656 0.541196i −1.30656 0.541196i
\(89\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) −1.30656 + 0.541196i −1.30656 + 0.541196i
\(97\) 0.541196 1.30656i 0.541196 1.30656i −0.382683 0.923880i \(-0.625000\pi\)
0.923880 0.382683i \(-0.125000\pi\)
\(98\) 1.00000i 1.00000i
\(99\) 1.30656 + 0.541196i 1.30656 + 0.541196i
\(100\) −0.707107 + 0.707107i −0.707107 + 0.707107i
\(101\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(102\) 0 0
\(103\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −0.541196 + 1.30656i −0.541196 + 1.30656i 0.382683 + 0.923880i \(0.375000\pi\)
−0.923880 + 0.382683i \(0.875000\pi\)
\(108\) 0 0
\(109\) 0 0 −0.382683 0.923880i \(-0.625000\pi\)
0.382683 + 0.923880i \(0.375000\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) −1.30656 + 0.541196i −1.30656 + 0.541196i −0.923880 0.382683i \(-0.875000\pi\)
−0.382683 + 0.923880i \(0.625000\pi\)
\(114\) −1.08239 + 2.61313i −1.08239 + 2.61313i
\(115\) 0 0
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) 0.707107 0.707107i 0.707107 0.707107i
\(122\) 0 0
\(123\) 2.00000i 2.00000i
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(128\) 0.707107 + 0.707107i 0.707107 + 0.707107i
\(129\) 0 0
\(130\) 0 0
\(131\) −0.541196 + 1.30656i −0.541196 + 1.30656i 0.382683 + 0.923880i \(0.375000\pi\)
−0.923880 + 0.382683i \(0.875000\pi\)
\(132\) 2.00000i 2.00000i
\(133\) 0 0
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −2.00000 −2.00000 −1.00000 \(\pi\)
−1.00000 \(\pi\)
\(138\) 0 0
\(139\) 1.30656 + 0.541196i 1.30656 + 0.541196i 0.923880 0.382683i \(-0.125000\pi\)
0.382683 + 0.923880i \(0.375000\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 0 0
\(144\) −0.707107 0.707107i −0.707107 0.707107i
\(145\) 0 0
\(146\) −0.541196 1.30656i −0.541196 1.30656i
\(147\) −1.30656 + 0.541196i −1.30656 + 0.541196i
\(148\) 0 0
\(149\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(150\) −1.30656 0.541196i −1.30656 0.541196i
\(151\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(152\) 2.00000 2.00000
\(153\) 0 0
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 0 0
\(162\) −0.707107 0.707107i −0.707107 0.707107i
\(163\) 0.541196 + 1.30656i 0.541196 + 1.30656i 0.923880 + 0.382683i \(0.125000\pi\)
−0.382683 + 0.923880i \(0.625000\pi\)
\(164\) 1.30656 0.541196i 1.30656 0.541196i
\(165\) 0 0
\(166\) 0 0
\(167\) 0 0 −0.923880 0.382683i \(-0.875000\pi\)
0.923880 + 0.382683i \(0.125000\pi\)
\(168\) 0 0
\(169\) −1.00000 −1.00000
\(170\) 0 0
\(171\) −2.00000 −2.00000
\(172\) 0 0
\(173\) 0 0 −0.923880 0.382683i \(-0.875000\pi\)
0.923880 + 0.382683i \(0.125000\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) −1.30656 + 0.541196i −1.30656 + 0.541196i
\(177\) 0 0
\(178\) 0 0
\(179\) −1.41421 1.41421i −1.41421 1.41421i −0.707107 0.707107i \(-0.750000\pi\)
−0.707107 0.707107i \(-0.750000\pi\)
\(180\) 0 0
\(181\) 0 0 0.923880 0.382683i \(-0.125000\pi\)
−0.923880 + 0.382683i \(0.875000\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(192\) −0.541196 + 1.30656i −0.541196 + 1.30656i
\(193\) 1.30656 0.541196i 1.30656 0.541196i 0.382683 0.923880i \(-0.375000\pi\)
0.923880 + 0.382683i \(0.125000\pi\)
\(194\) −0.541196 1.30656i −0.541196 1.30656i
\(195\) 0 0
\(196\) 0.707107 + 0.707107i 0.707107 + 0.707107i
\(197\) 0 0 −0.382683 0.923880i \(-0.625000\pi\)
0.382683 + 0.923880i \(0.375000\pi\)
\(198\) 1.30656 0.541196i 1.30656 0.541196i
\(199\) 0 0 0.382683 0.923880i \(-0.375000\pi\)
−0.382683 + 0.923880i \(0.625000\pi\)
\(200\) 1.00000i 1.00000i
\(201\) 0 0
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) −1.08239 + 2.61313i −1.08239 + 2.61313i
\(210\) 0 0
\(211\) 0.541196 + 1.30656i 0.541196 + 1.30656i 0.923880 + 0.382683i \(0.125000\pi\)
−0.382683 + 0.923880i \(0.625000\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0.541196 + 1.30656i 0.541196 + 1.30656i
\(215\) 0 0
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) 1.41421 1.41421i 1.41421 1.41421i
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(224\) 0 0
\(225\) 1.00000i 1.00000i
\(226\) −0.541196 + 1.30656i −0.541196 + 1.30656i
\(227\) 1.30656 0.541196i 1.30656 0.541196i 0.382683 0.923880i \(-0.375000\pi\)
0.923880 + 0.382683i \(0.125000\pi\)
\(228\) 1.08239 + 2.61313i 1.08239 + 2.61313i
\(229\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 0.541196 1.30656i 0.541196 1.30656i −0.382683 0.923880i \(-0.625000\pi\)
0.923880 0.382683i \(-0.125000\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(240\) 0 0
\(241\) −1.30656 0.541196i −1.30656 0.541196i −0.382683 0.923880i \(-0.625000\pi\)
−0.923880 + 0.382683i \(0.875000\pi\)
\(242\) 1.00000i 1.00000i
\(243\) 0.541196 1.30656i 0.541196 1.30656i
\(244\) 0 0
\(245\) 0 0
\(246\) 1.41421 + 1.41421i 1.41421 + 1.41421i
\(247\) 0 0
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 0 0
\(255\) 0 0
\(256\) 1.00000 1.00000
\(257\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) 0 0
\(262\) 0.541196 + 1.30656i 0.541196 + 1.30656i
\(263\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(264\) −1.41421 1.41421i −1.41421 1.41421i
\(265\) 0 0
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 0 0 −0.923880 0.382683i \(-0.875000\pi\)
0.923880 + 0.382683i \(0.125000\pi\)
\(270\) 0 0
\(271\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) −1.41421 + 1.41421i −1.41421 + 1.41421i
\(275\) −1.30656 0.541196i −1.30656 0.541196i
\(276\) 0 0
\(277\) 0 0 0.382683 0.923880i \(-0.375000\pi\)
−0.382683 + 0.923880i \(0.625000\pi\)
\(278\) 1.30656 0.541196i 1.30656 0.541196i
\(279\) 0 0
\(280\) 0 0
\(281\) −1.41421 1.41421i −1.41421 1.41421i −0.707107 0.707107i \(-0.750000\pi\)
−0.707107 0.707107i \(-0.750000\pi\)
\(282\) 0 0
\(283\) 1.30656 0.541196i 1.30656 0.541196i 0.382683 0.923880i \(-0.375000\pi\)
0.923880 + 0.382683i \(0.125000\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) −1.00000 −1.00000
\(289\) 0 0
\(290\) 0 0
\(291\) 1.41421 1.41421i 1.41421 1.41421i
\(292\) −1.30656 0.541196i −1.30656 0.541196i
\(293\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(294\) −0.541196 + 1.30656i −0.541196 + 1.30656i
\(295\) 0 0
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 0 0
\(300\) −1.30656 + 0.541196i −1.30656 + 0.541196i
\(301\) 0 0
\(302\) 0 0
\(303\) 0 0
\(304\) 1.41421 1.41421i 1.41421 1.41421i
\(305\) 0 0
\(306\) 0 0
\(307\) 2.00000 2.00000 1.00000 \(0\)
1.00000 \(0\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 0 0 0.382683 0.923880i \(-0.375000\pi\)
−0.382683 + 0.923880i \(0.625000\pi\)
\(312\) 0 0
\(313\) −0.541196 1.30656i −0.541196 1.30656i −0.923880 0.382683i \(-0.875000\pi\)
0.382683 0.923880i \(-0.375000\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 0 0 0.923880 0.382683i \(-0.125000\pi\)
−0.923880 + 0.382683i \(0.875000\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0 0
\(321\) −1.41421 + 1.41421i −1.41421 + 1.41421i
\(322\) 0 0
\(323\) 0 0
\(324\) −1.00000 −1.00000
\(325\) 0 0
\(326\) 1.30656 + 0.541196i 1.30656 + 0.541196i
\(327\) 0 0
\(328\) 0.541196 1.30656i 0.541196 1.30656i
\(329\) 0 0
\(330\) 0 0
\(331\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) −1.30656 0.541196i −1.30656 0.541196i −0.382683 0.923880i \(-0.625000\pi\)
−0.923880 + 0.382683i \(0.875000\pi\)
\(338\) −0.707107 + 0.707107i −0.707107 + 0.707107i
\(339\) −2.00000 −2.00000
\(340\) 0 0
\(341\) 0 0
\(342\) −1.41421 + 1.41421i −1.41421 + 1.41421i
\(343\) 0 0
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 0.541196 + 1.30656i 0.541196 + 1.30656i 0.923880 + 0.382683i \(0.125000\pi\)
−0.382683 + 0.923880i \(0.625000\pi\)
\(348\) 0 0
\(349\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) −0.541196 + 1.30656i −0.541196 + 1.30656i
\(353\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 0 0
\(358\) −2.00000 −2.00000
\(359\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(360\) 0 0
\(361\) 3.00000i 3.00000i
\(362\) 0 0
\(363\) 1.30656 0.541196i 1.30656 0.541196i
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 0 0 −0.382683 0.923880i \(-0.625000\pi\)
0.382683 + 0.923880i \(0.375000\pi\)
\(368\) 0 0
\(369\) −0.541196 + 1.30656i −0.541196 + 1.30656i
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) 0.541196 1.30656i 0.541196 1.30656i −0.382683 0.923880i \(-0.625000\pi\)
0.923880 0.382683i \(-0.125000\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(384\) 0.541196 + 1.30656i 0.541196 + 1.30656i
\(385\) 0 0
\(386\) 0.541196 1.30656i 0.541196 1.30656i
\(387\) 0 0
\(388\) −1.30656 0.541196i −1.30656 0.541196i
\(389\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 1.00000 1.00000
\(393\) −1.41421 + 1.41421i −1.41421 + 1.41421i
\(394\) 0 0
\(395\) 0 0
\(396\) 0.541196 1.30656i 0.541196 1.30656i
\(397\) 0 0 0.923880 0.382683i \(-0.125000\pi\)
−0.923880 + 0.382683i \(0.875000\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0.707107 + 0.707107i 0.707107 + 0.707107i
\(401\) 0.541196 + 1.30656i 0.541196 + 1.30656i 0.923880 + 0.382683i \(0.125000\pi\)
−0.382683 + 0.923880i \(0.625000\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(410\) 0 0
\(411\) −2.61313 1.08239i −2.61313 1.08239i
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 1.41421 + 1.41421i 1.41421 + 1.41421i
\(418\) 1.08239 + 2.61313i 1.08239 + 2.61313i
\(419\) −1.30656 + 0.541196i −1.30656 + 0.541196i −0.923880 0.382683i \(-0.875000\pi\)
−0.382683 + 0.923880i \(0.625000\pi\)
\(420\) 0 0
\(421\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(422\) 1.30656 + 0.541196i 1.30656 + 0.541196i
\(423\) 0 0
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 0 0
\(428\) 1.30656 + 0.541196i 1.30656 + 0.541196i
\(429\) 0 0
\(430\) 0 0
\(431\) 0 0 0.923880 0.382683i \(-0.125000\pi\)
−0.923880 + 0.382683i \(0.875000\pi\)
\(432\) 0 0
\(433\) 1.41421 + 1.41421i 1.41421 + 1.41421i 0.707107 + 0.707107i \(0.250000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 0 0
\(438\) 2.00000i 2.00000i
\(439\) 0 0 −0.923880 0.382683i \(-0.875000\pi\)
0.923880 + 0.382683i \(0.125000\pi\)
\(440\) 0 0
\(441\) −1.00000 −1.00000
\(442\) 0 0
\(443\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) −0.541196 1.30656i −0.541196 1.30656i −0.923880 0.382683i \(-0.875000\pi\)
0.382683 0.923880i \(-0.375000\pi\)
\(450\) −0.707107 0.707107i −0.707107 0.707107i
\(451\) 1.41421 + 1.41421i 1.41421 + 1.41421i
\(452\) 0.541196 + 1.30656i 0.541196 + 1.30656i
\(453\) 0 0
\(454\) 0.541196 1.30656i 0.541196 1.30656i
\(455\) 0 0
\(456\) 2.61313 + 1.08239i 2.61313 + 1.08239i
\(457\) 1.41421 1.41421i 1.41421 1.41421i 0.707107 0.707107i \(-0.250000\pi\)
0.707107 0.707107i \(-0.250000\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(462\) 0 0
\(463\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) −0.541196 1.30656i −0.541196 1.30656i
\(467\) 1.41421 + 1.41421i 1.41421 + 1.41421i 0.707107 + 0.707107i \(0.250000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) 2.00000 2.00000
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 0 0 −0.923880 0.382683i \(-0.875000\pi\)
0.923880 + 0.382683i \(0.125000\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) −1.30656 + 0.541196i −1.30656 + 0.541196i
\(483\) 0 0
\(484\) −0.707107 0.707107i −0.707107 0.707107i
\(485\) 0 0
\(486\) −0.541196 1.30656i −0.541196 1.30656i
\(487\) 0 0 0.923880 0.382683i \(-0.125000\pi\)
−0.923880 + 0.382683i \(0.875000\pi\)
\(488\) 0 0
\(489\) 2.00000i 2.00000i
\(490\) 0 0
\(491\) 1.41421 1.41421i 1.41421 1.41421i 0.707107 0.707107i \(-0.250000\pi\)
0.707107 0.707107i \(-0.250000\pi\)
\(492\) 2.00000 2.00000
\(493\) 0 0
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) −1.30656 + 0.541196i −1.30656 + 0.541196i −0.923880 0.382683i \(-0.875000\pi\)
−0.382683 + 0.923880i \(0.625000\pi\)
\(500\) 0 0
\(501\) 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
\(505\) 0 0
\(506\) 0 0
\(507\) −1.30656 0.541196i −1.30656 0.541196i
\(508\) 0 0
\(509\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0.707107 0.707107i 0.707107 0.707107i
\(513\) 0 0
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 1.30656 0.541196i 1.30656 0.541196i 0.382683 0.923880i \(-0.375000\pi\)
0.923880 + 0.382683i \(0.125000\pi\)
\(522\) 0 0
\(523\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(524\) 1.30656 + 0.541196i 1.30656 + 0.541196i
\(525\) 0 0
\(526\) 0 0
\(527\) 0 0
\(528\) −2.00000 −2.00000
\(529\) 0.707107 0.707107i 0.707107 0.707107i
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 0 0
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) −1.08239 2.61313i −1.08239 2.61313i
\(538\) 0 0
\(539\) −0.541196 + 1.30656i −0.541196 + 1.30656i
\(540\) 0 0
\(541\) 0 0 −0.923880 0.382683i \(-0.875000\pi\)
0.923880 + 0.382683i \(0.125000\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 1.30656 + 0.541196i 1.30656 + 0.541196i 0.923880 0.382683i \(-0.125000\pi\)
0.382683 + 0.923880i \(0.375000\pi\)
\(548\) 2.00000i 2.00000i
\(549\) 0 0
\(550\) −1.30656 + 0.541196i −1.30656 + 0.541196i
\(551\) 0 0
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) 0 0
\(556\) 0.541196 1.30656i 0.541196 1.30656i
\(557\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 0 0
\(561\) 0 0
\(562\) −2.00000 −2.00000
\(563\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0.541196 1.30656i 0.541196 1.30656i
\(567\) 0 0
\(568\) 0 0
\(569\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(570\) 0 0
\(571\) −0.541196 1.30656i −0.541196 1.30656i −0.923880 0.382683i \(-0.875000\pi\)
0.382683 0.923880i \(-0.375000\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 0 0
\(576\) −0.707107 + 0.707107i −0.707107 + 0.707107i
\(577\) −2.00000 −2.00000 −1.00000 \(\pi\)
−1.00000 \(\pi\)
\(578\) 0 0
\(579\) 2.00000 2.00000
\(580\) 0 0
\(581\) 0 0
\(582\) 2.00000i 2.00000i
\(583\) 0 0
\(584\) −1.30656 + 0.541196i −1.30656 + 0.541196i
\(585\) 0 0
\(586\) 0 0
\(587\) 1.41421 + 1.41421i 1.41421 + 1.41421i 0.707107 + 0.707107i \(0.250000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(588\) 0.541196 + 1.30656i 0.541196 + 1.30656i
\(589\) 0 0
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(600\) −0.541196 + 1.30656i −0.541196 + 1.30656i
\(601\) 1.30656 0.541196i 1.30656 0.541196i 0.382683 0.923880i \(-0.375000\pi\)
0.923880 + 0.382683i \(0.125000\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) 0 0 0.382683 0.923880i \(-0.375000\pi\)
−0.382683 + 0.923880i \(0.625000\pi\)
\(608\) 2.00000i 2.00000i
\(609\) 0 0
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(614\) 1.41421 1.41421i 1.41421 1.41421i
\(615\) 0 0
\(616\) 0 0
\(617\) 0.541196 1.30656i 0.541196 1.30656i −0.382683 0.923880i \(-0.625000\pi\)
0.923880 0.382683i \(-0.125000\pi\)
\(618\) 0 0
\(619\) −0.541196 1.30656i −0.541196 1.30656i −0.923880 0.382683i \(-0.875000\pi\)
0.382683 0.923880i \(-0.375000\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) 1.00000i 1.00000i
\(626\) −1.30656 0.541196i −1.30656 0.541196i
\(627\) −2.82843 + 2.82843i −2.82843 + 2.82843i
\(628\) 0 0
\(629\) 0 0
\(630\) 0 0
\(631\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(632\) 0 0
\(633\) 2.00000i 2.00000i
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) −0.541196 + 1.30656i −0.541196 + 1.30656i 0.382683 + 0.923880i \(0.375000\pi\)
−0.923880 + 0.382683i \(0.875000\pi\)
\(642\) 2.00000i 2.00000i
\(643\) 1.30656 + 0.541196i 1.30656 + 0.541196i 0.923880 0.382683i \(-0.125000\pi\)
0.382683 + 0.923880i \(0.375000\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(648\) −0.707107 + 0.707107i −0.707107 + 0.707107i
\(649\) 0 0
\(650\) 0 0
\(651\) 0 0
\(652\) 1.30656 0.541196i 1.30656 0.541196i
\(653\) 0 0 −0.382683 0.923880i \(-0.625000\pi\)
0.382683 + 0.923880i \(0.375000\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) −0.541196 1.30656i −0.541196 1.30656i
\(657\) 1.30656 0.541196i 1.30656 0.541196i
\(658\) 0 0
\(659\) 2.00000i 2.00000i 1.00000i \(0.5\pi\)
1.00000i \(0.5\pi\)
\(660\) 0 0
\(661\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(662\) 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
\(673\) −0.541196 1.30656i −0.541196 1.30656i −0.923880 0.382683i \(-0.875000\pi\)
0.382683 0.923880i \(-0.375000\pi\)
\(674\) −1.30656 + 0.541196i −1.30656 + 0.541196i
\(675\) 0 0
\(676\) 1.00000i 1.00000i
\(677\) 0 0 −0.923880 0.382683i \(-0.875000\pi\)
0.923880 + 0.382683i \(0.125000\pi\)
\(678\) −1.41421 + 1.41421i −1.41421 + 1.41421i
\(679\) 0 0
\(680\) 0 0
\(681\) 2.00000 2.00000
\(682\) 0 0
\(683\) −1.30656 0.541196i −1.30656 0.541196i −0.382683 0.923880i \(-0.625000\pi\)
−0.923880 + 0.382683i \(0.875000\pi\)
\(684\) 2.00000i 2.00000i
\(685\) 0 0
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 0 0
\(690\) 0 0
\(691\) 1.30656 0.541196i 1.30656 0.541196i 0.382683 0.923880i \(-0.375000\pi\)
0.923880 + 0.382683i \(0.125000\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 1.30656 + 0.541196i 1.30656 + 0.541196i
\(695\) 0 0
\(696\) 0 0
\(697\) 0 0
\(698\) 0 0
\(699\) 1.41421 1.41421i 1.41421 1.41421i
\(700\) 0 0
\(701\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) 0.541196 + 1.30656i 0.541196 + 1.30656i
\(705\) 0 0
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) 0 0 0.382683 0.923880i \(-0.375000\pi\)
−0.382683 + 0.923880i \(0.625000\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 0 0
\(714\) 0 0
\(715\) 0 0
\(716\) −1.41421 + 1.41421i −1.41421 + 1.41421i
\(717\) 0 0
\(718\) 0 0
\(719\) 0 0 0.382683 0.923880i \(-0.375000\pi\)
−0.382683 + 0.923880i \(0.625000\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) −2.12132 2.12132i −2.12132 2.12132i
\(723\) −1.41421 1.41421i −1.41421 1.41421i
\(724\) 0 0
\(725\) 0 0
\(726\) 0.541196 1.30656i 0.541196 1.30656i
\(727\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(728\) 0 0
\(729\) 0.707107 0.707107i 0.707107 0.707107i
\(730\) 0 0
\(731\) 0 0
\(732\) 0 0
\(733\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 0 0
\(738\) 0.541196 + 1.30656i 0.541196 + 1.30656i
\(739\) −1.41421 1.41421i −1.41421 1.41421i −0.707107 0.707107i \(-0.750000\pi\)
−0.707107 0.707107i \(-0.750000\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 0 0 0.382683 0.923880i \(-0.375000\pi\)
−0.382683 + 0.923880i \(0.625000\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) 0 0 −0.923880 0.382683i \(-0.875000\pi\)
0.923880 + 0.382683i \(0.125000\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(758\) −0.541196 1.30656i −0.541196 1.30656i
\(759\) 0 0
\(760\) 0 0
\(761\) 2.00000i 2.00000i 1.00000i \(-0.5\pi\)
1.00000i \(-0.5\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 0 0
\(768\) 1.30656 + 0.541196i 1.30656 + 0.541196i
\(769\) 2.00000i 2.00000i 1.00000i \(0.5\pi\)
1.00000i \(0.5\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) −0.541196 1.30656i −0.541196 1.30656i
\(773\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) −1.30656 + 0.541196i −1.30656 + 0.541196i
\(777\) 0 0
\(778\) 0 0
\(779\) −2.61313 1.08239i −2.61313 1.08239i
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 0 0
\(784\) 0.707107 0.707107i 0.707107 0.707107i
\(785\) 0 0
\(786\) 2.00000i 2.00000i
\(787\) −0.541196 + 1.30656i −0.541196 + 1.30656i 0.382683 + 0.923880i \(0.375000\pi\)
−0.923880 + 0.382683i \(0.875000\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) −0.541196 1.30656i −0.541196 1.30656i
\(793\) 0 0
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 1.00000 1.00000
\(801\) 0 0
\(802\) 1.30656 + 0.541196i 1.30656 + 0.541196i
\(803\) 2.00000i 2.00000i
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 0.541196 + 1.30656i 0.541196 + 1.30656i 0.923880 + 0.382683i \(0.125000\pi\)
−0.382683 + 0.923880i \(0.625000\pi\)
\(810\) 0 0
\(811\) 0.541196 1.30656i 0.541196 1.30656i −0.382683 0.923880i \(-0.625000\pi\)
0.923880 0.382683i \(-0.125000\pi\)
\(812\) 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
\(821\) 0 0 0.382683 0.923880i \(-0.375000\pi\)
−0.382683 + 0.923880i \(0.625000\pi\)
\(822\) −2.61313 + 1.08239i −2.61313 + 1.08239i
\(823\) 0 0 −0.382683 0.923880i \(-0.625000\pi\)
0.382683 + 0.923880i \(0.375000\pi\)
\(824\) 0 0
\(825\) −1.41421 1.41421i −1.41421 1.41421i
\(826\) 0 0
\(827\) −1.30656 + 0.541196i −1.30656 + 0.541196i −0.923880 0.382683i \(-0.875000\pi\)
−0.382683 + 0.923880i \(0.625000\pi\)
\(828\) 0 0
\(829\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 0 0
\(834\) 2.00000 2.00000
\(835\) 0 0
\(836\) 2.61313 + 1.08239i 2.61313 + 1.08239i
\(837\) 0 0
\(838\) −0.541196 + 1.30656i −0.541196 + 1.30656i
\(839\) 0 0 0.923880 0.382683i \(-0.125000\pi\)
−0.923880 + 0.382683i \(0.875000\pi\)
\(840\) 0 0
\(841\) −0.707107 0.707107i −0.707107 0.707107i
\(842\) 0 0
\(843\) −1.08239 2.61313i −1.08239 2.61313i
\(844\) 1.30656 0.541196i 1.30656 0.541196i
\(845\) 0 0
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) 2.00000 2.00000
\(850\) 0 0
\(851\) 0 0
\(852\) 0 0
\(853\) 0 0 −0.923880 0.382683i \(-0.875000\pi\)
0.923880 + 0.382683i \(0.125000\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 1.30656 0.541196i 1.30656 0.541196i
\(857\) 0.541196 + 1.30656i 0.541196 + 1.30656i 0.923880 + 0.382683i \(0.125000\pi\)
−0.382683 + 0.923880i \(0.625000\pi\)
\(858\) 0 0
\(859\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 2.00000 2.00000
\(867\) 0 0
\(868\) 0 0
\(869\) 0 0
\(870\) 0 0
\(871\) 0 0
\(872\) 0 0
\(873\) 1.30656 0.541196i 1.30656 0.541196i
\(874\) 0 0
\(875\) 0 0
\(876\) −1.41421 1.41421i −1.41421 1.41421i
\(877\) 0 0 −0.382683 0.923880i \(-0.625000\pi\)
0.382683 + 0.923880i \(0.375000\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 1.30656 + 0.541196i 1.30656 + 0.541196i 0.923880 0.382683i \(-0.125000\pi\)
0.382683 + 0.923880i \(0.375000\pi\)
\(882\) −0.707107 + 0.707107i −0.707107 + 0.707107i
\(883\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 0 0 −0.923880 0.382683i \(-0.875000\pi\)
0.923880 + 0.382683i \(0.125000\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) −0.541196 1.30656i −0.541196 1.30656i
\(892\) 0 0
\(893\) 0 0
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 0 0
\(898\) −1.30656 0.541196i −1.30656 0.541196i
\(899\) 0 0
\(900\) −1.00000 −1.00000
\(901\) 0 0
\(902\) 2.00000 2.00000
\(903\) 0 0
\(904\) 1.30656 + 0.541196i 1.30656 + 0.541196i
\(905\) 0 0
\(906\) 0 0
\(907\) −1.30656 + 0.541196i −1.30656 + 0.541196i −0.923880 0.382683i \(-0.875000\pi\)
−0.382683 + 0.923880i \(0.625000\pi\)
\(908\) −0.541196 1.30656i −0.541196 1.30656i
\(909\) 0 0
\(910\) 0 0
\(911\) 0 0 −0.382683 0.923880i \(-0.625000\pi\)
0.382683 + 0.923880i \(0.375000\pi\)
\(912\) 2.61313 1.08239i 2.61313 1.08239i
\(913\) 0 0
\(914\) 2.00000i 2.00000i
\(915\) 0 0
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(920\) 0 0
\(921\) 2.61313 + 1.08239i 2.61313 + 1.08239i
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) −1.30656 + 0.541196i −1.30656 + 0.541196i −0.923880 0.382683i \(-0.875000\pi\)
−0.382683 + 0.923880i \(0.625000\pi\)
\(930\) 0 0
\(931\) 2.00000i 2.00000i
\(932\) −1.30656 0.541196i −1.30656 0.541196i
\(933\) 0 0
\(934\) 2.00000 2.00000
\(935\) 0 0
\(936\) 0 0
\(937\) 1.41421 1.41421i 1.41421 1.41421i 0.707107 0.707107i \(-0.250000\pi\)
0.707107 0.707107i \(-0.250000\pi\)
\(938\) 0 0
\(939\) 2.00000i 2.00000i
\(940\) 0 0
\(941\) 0 0 0.923880 0.382683i \(-0.125000\pi\)
−0.923880 + 0.382683i \(0.875000\pi\)
\(942\) 0 0
\(943\) 0 0
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 0.541196 1.30656i 0.541196 1.30656i −0.382683 0.923880i \(-0.625000\pi\)
0.923880 0.382683i \(-0.125000\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) 1.41421 1.41421i 1.41421 1.41421i
\(951\) 0 0
\(952\) 0 0
\(953\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) 0.707107 + 0.707107i 0.707107 + 0.707107i
\(962\) 0 0
\(963\) −1.30656 + 0.541196i −1.30656 + 0.541196i
\(964\) −0.541196 + 1.30656i −0.541196 + 1.30656i
\(965\) 0 0
\(966\) 0 0
\(967\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(968\) −1.00000 −1.00000
\(969\) 0 0
\(970\) 0 0
\(971\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(972\) −1.30656 0.541196i −1.30656 0.541196i
\(973\) 0 0
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(978\) 1.41421 + 1.41421i 1.41421 + 1.41421i
\(979\) 0 0
\(980\) 0 0
\(981\) 0 0
\(982\) 2.00000i 2.00000i
\(983\) 0 0 −0.923880 0.382683i \(-0.875000\pi\)
0.923880 + 0.382683i \(0.125000\pi\)
\(984\) 1.41421 1.41421i 1.41421 1.41421i
\(985\) 0 0
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 0 0
\(990\) 0 0
\(991\) 0 0 0.382683 0.923880i \(-0.375000\pi\)
−0.382683 + 0.923880i \(0.625000\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) 0 0 0.923880 0.382683i \(-0.125000\pi\)
−0.923880 + 0.382683i \(0.875000\pi\)
\(998\) −0.541196 + 1.30656i −0.541196 + 1.30656i
\(999\) 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 2312.1.p.e.179.2 8
8.3 odd 2 CM 2312.1.p.e.179.2 8
17.2 even 8 inner 2312.1.p.e.155.2 8
17.3 odd 16 2312.1.j.b.251.1 2
17.4 even 4 inner 2312.1.p.e.1555.2 8
17.5 odd 16 136.1.j.a.123.1 yes 2
17.6 odd 16 2312.1.f.b.579.2 2
17.7 odd 16 2312.1.e.a.1155.1 2
17.8 even 8 inner 2312.1.p.e.1579.1 8
17.9 even 8 inner 2312.1.p.e.1579.2 8
17.10 odd 16 2312.1.e.a.1155.2 2
17.11 odd 16 2312.1.f.b.579.1 2
17.12 odd 16 2312.1.j.b.1483.1 2
17.13 even 4 inner 2312.1.p.e.1555.1 8
17.14 odd 16 136.1.j.a.115.1 2
17.15 even 8 inner 2312.1.p.e.155.1 8
17.16 even 2 inner 2312.1.p.e.179.1 8
51.5 even 16 1224.1.s.a.667.1 2
51.14 even 16 1224.1.s.a.523.1 2
68.31 even 16 544.1.n.a.47.1 2
68.39 even 16 544.1.n.a.463.1 2
85.14 odd 16 3400.1.y.a.251.1 2
85.22 even 16 3400.1.bc.a.2299.1 2
85.39 odd 16 3400.1.y.a.3251.1 2
85.48 even 16 3400.1.bc.a.2699.1 2
85.73 even 16 3400.1.bc.b.2299.1 2
85.82 even 16 3400.1.bc.b.2699.1 2
136.3 even 16 2312.1.j.b.251.1 2
136.5 odd 16 544.1.n.a.463.1 2
136.11 even 16 2312.1.f.b.579.1 2
136.19 odd 8 inner 2312.1.p.e.155.2 8
136.27 even 16 2312.1.e.a.1155.2 2
136.43 odd 8 inner 2312.1.p.e.1579.2 8
136.59 odd 8 inner 2312.1.p.e.1579.1 8
136.67 odd 2 inner 2312.1.p.e.179.1 8
136.75 even 16 2312.1.e.a.1155.1 2
136.83 odd 8 inner 2312.1.p.e.155.1 8
136.91 even 16 2312.1.f.b.579.2 2
136.99 even 16 136.1.j.a.115.1 2
136.107 even 16 136.1.j.a.123.1 yes 2
136.115 odd 4 inner 2312.1.p.e.1555.1 8
136.123 odd 4 inner 2312.1.p.e.1555.2 8
136.131 even 16 2312.1.j.b.1483.1 2
136.133 odd 16 544.1.n.a.47.1 2
408.107 odd 16 1224.1.s.a.667.1 2
408.371 odd 16 1224.1.s.a.523.1 2
680.99 even 16 3400.1.y.a.251.1 2
680.107 odd 16 3400.1.bc.a.2299.1 2
680.243 odd 16 3400.1.bc.b.2299.1 2
680.379 even 16 3400.1.y.a.3251.1 2
680.507 odd 16 3400.1.bc.b.2699.1 2
680.643 odd 16 3400.1.bc.a.2699.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
136.1.j.a.115.1 2 17.14 odd 16
136.1.j.a.115.1 2 136.99 even 16
136.1.j.a.123.1 yes 2 17.5 odd 16
136.1.j.a.123.1 yes 2 136.107 even 16
544.1.n.a.47.1 2 68.31 even 16
544.1.n.a.47.1 2 136.133 odd 16
544.1.n.a.463.1 2 68.39 even 16
544.1.n.a.463.1 2 136.5 odd 16
1224.1.s.a.523.1 2 51.14 even 16
1224.1.s.a.523.1 2 408.371 odd 16
1224.1.s.a.667.1 2 51.5 even 16
1224.1.s.a.667.1 2 408.107 odd 16
2312.1.e.a.1155.1 2 17.7 odd 16
2312.1.e.a.1155.1 2 136.75 even 16
2312.1.e.a.1155.2 2 17.10 odd 16
2312.1.e.a.1155.2 2 136.27 even 16
2312.1.f.b.579.1 2 17.11 odd 16
2312.1.f.b.579.1 2 136.11 even 16
2312.1.f.b.579.2 2 17.6 odd 16
2312.1.f.b.579.2 2 136.91 even 16
2312.1.j.b.251.1 2 17.3 odd 16
2312.1.j.b.251.1 2 136.3 even 16
2312.1.j.b.1483.1 2 17.12 odd 16
2312.1.j.b.1483.1 2 136.131 even 16
2312.1.p.e.155.1 8 17.15 even 8 inner
2312.1.p.e.155.1 8 136.83 odd 8 inner
2312.1.p.e.155.2 8 17.2 even 8 inner
2312.1.p.e.155.2 8 136.19 odd 8 inner
2312.1.p.e.179.1 8 17.16 even 2 inner
2312.1.p.e.179.1 8 136.67 odd 2 inner
2312.1.p.e.179.2 8 1.1 even 1 trivial
2312.1.p.e.179.2 8 8.3 odd 2 CM
2312.1.p.e.1555.1 8 17.13 even 4 inner
2312.1.p.e.1555.1 8 136.115 odd 4 inner
2312.1.p.e.1555.2 8 17.4 even 4 inner
2312.1.p.e.1555.2 8 136.123 odd 4 inner
2312.1.p.e.1579.1 8 17.8 even 8 inner
2312.1.p.e.1579.1 8 136.59 odd 8 inner
2312.1.p.e.1579.2 8 17.9 even 8 inner
2312.1.p.e.1579.2 8 136.43 odd 8 inner
3400.1.y.a.251.1 2 85.14 odd 16
3400.1.y.a.251.1 2 680.99 even 16
3400.1.y.a.3251.1 2 85.39 odd 16
3400.1.y.a.3251.1 2 680.379 even 16
3400.1.bc.a.2299.1 2 85.22 even 16
3400.1.bc.a.2299.1 2 680.107 odd 16
3400.1.bc.a.2699.1 2 85.48 even 16
3400.1.bc.a.2699.1 2 680.643 odd 16
3400.1.bc.b.2299.1 2 85.73 even 16
3400.1.bc.b.2299.1 2 680.243 odd 16
3400.1.bc.b.2699.1 2 85.82 even 16
3400.1.bc.b.2699.1 2 680.507 odd 16