Properties

Label 1224.1.cx.b.299.1
Level $1224$
Weight $1$
Character 1224.299
Analytic conductor $0.611$
Analytic rank $0$
Dimension $16$
Projective image $D_{48}$
CM discriminant -8
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1224,1,Mod(11,1224)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1224, base_ring=CyclotomicField(48))
 
chi = DirichletCharacter(H, H._module([24, 24, 8, 21]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1224.11");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1224 = 2^{3} \cdot 3^{2} \cdot 17 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 1224.cx (of order \(48\), 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: \(0.610855575463\)
Analytic rank: \(0\)
Dimension: \(16\)
Coefficient field: \(\Q(\zeta_{48})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - x^{8} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{48}\)
Projective field: Galois closure of \(\mathbb{Q}[x]/(x^{48} - \cdots)\)

Embedding invariants

Embedding label 299.1
Root \(-0.991445 + 0.130526i\) of defining polynomial
Character \(\chi\) \(=\) 1224.299
Dual form 1224.1.cx.b.131.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(0.793353 - 0.608761i) q^{2} +(0.608761 + 0.793353i) q^{3} +(0.258819 - 0.965926i) q^{4} +(0.965926 + 0.258819i) q^{6} +(-0.382683 - 0.923880i) q^{8} +(-0.258819 + 0.965926i) q^{9} +(0.0578541 + 0.882683i) q^{11} +(0.923880 - 0.382683i) q^{12} +(-0.866025 - 0.500000i) q^{16} +(0.991445 - 0.130526i) q^{17} +(0.382683 + 0.923880i) q^{18} +(0.198092 - 0.478235i) q^{19} +(0.583242 + 0.665060i) q^{22} +(0.500000 - 0.866025i) q^{24} +(0.130526 - 0.991445i) q^{25} +(-0.923880 + 0.382683i) q^{27} +(-0.991445 + 0.130526i) q^{32} +(-0.665060 + 0.583242i) q^{33} +(0.707107 - 0.707107i) q^{34} +(0.866025 + 0.500000i) q^{36} +(-0.133975 - 0.500000i) q^{38} +(-1.69855 + 0.576581i) q^{41} +(-1.96593 - 0.258819i) q^{43} +(0.867580 + 0.172572i) q^{44} +(-0.130526 - 0.991445i) q^{48} +(-0.130526 - 0.991445i) q^{49} +(-0.500000 - 0.866025i) q^{50} +(0.707107 + 0.707107i) q^{51} +(-0.500000 + 0.866025i) q^{54} +(0.500000 - 0.133975i) q^{57} +(-0.965926 + 1.25882i) q^{59} +(-0.707107 + 0.707107i) q^{64} +(-0.172572 + 0.867580i) q^{66} +(-0.226078 + 0.130526i) q^{67} +(0.130526 - 0.991445i) q^{68} +(0.991445 - 0.130526i) q^{72} +(-0.128293 + 0.0255190i) q^{73} +(0.866025 - 0.500000i) q^{75} +(-0.410670 - 0.315118i) q^{76} +(-0.866025 - 0.500000i) q^{81} +(-0.996552 + 1.49144i) q^{82} +(0.465926 + 0.607206i) q^{83} +(-1.71723 + 0.991445i) q^{86} +(0.793353 - 0.391239i) q^{88} +(1.30656 - 1.30656i) q^{89} +(-0.707107 - 0.707107i) q^{96} +(0.608761 + 0.206647i) q^{97} +(-0.707107 - 0.707107i) q^{98} +(-0.867580 - 0.172572i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 8 q^{24} - 16 q^{38} - 16 q^{43} - 8 q^{50} - 8 q^{54} + 8 q^{57} - 8 q^{83}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1224\mathbb{Z}\right)^\times\).

\(n\) \(137\) \(613\) \(649\) \(919\)
\(\chi(n)\) \(e\left(\frac{1}{6}\right)\) \(-1\) \(e\left(\frac{3}{16}\right)\) \(-1\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.793353 0.608761i 0.793353 0.608761i
\(3\) 0.608761 + 0.793353i 0.608761 + 0.793353i
\(4\) 0.258819 0.965926i 0.258819 0.965926i
\(5\) 0 0 0.751840 0.659346i \(-0.229167\pi\)
−0.751840 + 0.659346i \(0.770833\pi\)
\(6\) 0.965926 + 0.258819i 0.965926 + 0.258819i
\(7\) 0 0 0.659346 0.751840i \(-0.270833\pi\)
−0.659346 + 0.751840i \(0.729167\pi\)
\(8\) −0.382683 0.923880i −0.382683 0.923880i
\(9\) −0.258819 + 0.965926i −0.258819 + 0.965926i
\(10\) 0 0
\(11\) 0.0578541 + 0.882683i 0.0578541 + 0.882683i 0.923880 + 0.382683i \(0.125000\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(12\) 0.923880 0.382683i 0.923880 0.382683i
\(13\) 0 0 −0.965926 0.258819i \(-0.916667\pi\)
0.965926 + 0.258819i \(0.0833333\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) −0.866025 0.500000i −0.866025 0.500000i
\(17\) 0.991445 0.130526i 0.991445 0.130526i
\(18\) 0.382683 + 0.923880i 0.382683 + 0.923880i
\(19\) 0.198092 0.478235i 0.198092 0.478235i −0.793353 0.608761i \(-0.791667\pi\)
0.991445 + 0.130526i \(0.0416667\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0.583242 + 0.665060i 0.583242 + 0.665060i
\(23\) 0 0 0.442289 0.896873i \(-0.354167\pi\)
−0.442289 + 0.896873i \(0.645833\pi\)
\(24\) 0.500000 0.866025i 0.500000 0.866025i
\(25\) 0.130526 0.991445i 0.130526 0.991445i
\(26\) 0 0
\(27\) −0.923880 + 0.382683i −0.923880 + 0.382683i
\(28\) 0 0
\(29\) 0 0 0.321439 0.946930i \(-0.395833\pi\)
−0.321439 + 0.946930i \(0.604167\pi\)
\(30\) 0 0
\(31\) 0 0 −0.997859 0.0654031i \(-0.979167\pi\)
0.997859 + 0.0654031i \(0.0208333\pi\)
\(32\) −0.991445 + 0.130526i −0.991445 + 0.130526i
\(33\) −0.665060 + 0.583242i −0.665060 + 0.583242i
\(34\) 0.707107 0.707107i 0.707107 0.707107i
\(35\) 0 0
\(36\) 0.866025 + 0.500000i 0.866025 + 0.500000i
\(37\) 0 0 −0.831470 0.555570i \(-0.812500\pi\)
0.831470 + 0.555570i \(0.187500\pi\)
\(38\) −0.133975 0.500000i −0.133975 0.500000i
\(39\) 0 0
\(40\) 0 0
\(41\) −1.69855 + 0.576581i −1.69855 + 0.576581i −0.991445 0.130526i \(-0.958333\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(42\) 0 0
\(43\) −1.96593 0.258819i −1.96593 0.258819i −0.965926 0.258819i \(-0.916667\pi\)
−1.00000 \(\pi\)
\(44\) 0.867580 + 0.172572i 0.867580 + 0.172572i
\(45\) 0 0
\(46\) 0 0
\(47\) 0 0 0.965926 0.258819i \(-0.0833333\pi\)
−0.965926 + 0.258819i \(0.916667\pi\)
\(48\) −0.130526 0.991445i −0.130526 0.991445i
\(49\) −0.130526 0.991445i −0.130526 0.991445i
\(50\) −0.500000 0.866025i −0.500000 0.866025i
\(51\) 0.707107 + 0.707107i 0.707107 + 0.707107i
\(52\) 0 0
\(53\) 0 0 −0.923880 0.382683i \(-0.875000\pi\)
0.923880 + 0.382683i \(0.125000\pi\)
\(54\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(55\) 0 0
\(56\) 0 0
\(57\) 0.500000 0.133975i 0.500000 0.133975i
\(58\) 0 0
\(59\) −0.965926 + 1.25882i −0.965926 + 1.25882i 1.00000i \(0.5\pi\)
−0.965926 + 0.258819i \(0.916667\pi\)
\(60\) 0 0
\(61\) 0 0 −0.751840 0.659346i \(-0.770833\pi\)
0.751840 + 0.659346i \(0.229167\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) −0.707107 + 0.707107i −0.707107 + 0.707107i
\(65\) 0 0
\(66\) −0.172572 + 0.867580i −0.172572 + 0.867580i
\(67\) −0.226078 + 0.130526i −0.226078 + 0.130526i −0.608761 0.793353i \(-0.708333\pi\)
0.382683 + 0.923880i \(0.375000\pi\)
\(68\) 0.130526 0.991445i 0.130526 0.991445i
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 0.555570 0.831470i \(-0.312500\pi\)
−0.555570 + 0.831470i \(0.687500\pi\)
\(72\) 0.991445 0.130526i 0.991445 0.130526i
\(73\) −0.128293 + 0.0255190i −0.128293 + 0.0255190i −0.258819 0.965926i \(-0.583333\pi\)
0.130526 + 0.991445i \(0.458333\pi\)
\(74\) 0 0
\(75\) 0.866025 0.500000i 0.866025 0.500000i
\(76\) −0.410670 0.315118i −0.410670 0.315118i
\(77\) 0 0
\(78\) 0 0
\(79\) 0 0 0.997859 0.0654031i \(-0.0208333\pi\)
−0.997859 + 0.0654031i \(0.979167\pi\)
\(80\) 0 0
\(81\) −0.866025 0.500000i −0.866025 0.500000i
\(82\) −0.996552 + 1.49144i −0.996552 + 1.49144i
\(83\) 0.465926 + 0.607206i 0.465926 + 0.607206i 0.965926 0.258819i \(-0.0833333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) −1.71723 + 0.991445i −1.71723 + 0.991445i
\(87\) 0 0
\(88\) 0.793353 0.391239i 0.793353 0.391239i
\(89\) 1.30656 1.30656i 1.30656 1.30656i 0.382683 0.923880i \(-0.375000\pi\)
0.923880 0.382683i \(-0.125000\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) −0.707107 0.707107i −0.707107 0.707107i
\(97\) 0.608761 + 0.206647i 0.608761 + 0.206647i 0.608761 0.793353i \(-0.291667\pi\)
1.00000i \(0.5\pi\)
\(98\) −0.707107 0.707107i −0.707107 0.707107i
\(99\) −0.867580 0.172572i −0.867580 0.172572i
\(100\) −0.923880 0.382683i −0.923880 0.382683i
\(101\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(102\) 0.991445 + 0.130526i 0.991445 + 0.130526i
\(103\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 0.369474 + 1.85747i 0.369474 + 1.85747i 0.500000 + 0.866025i \(0.333333\pi\)
−0.130526 + 0.991445i \(0.541667\pi\)
\(108\) 0.130526 + 0.991445i 0.130526 + 0.991445i
\(109\) 0 0 −0.980785 0.195090i \(-0.937500\pi\)
0.980785 + 0.195090i \(0.0625000\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) −0.349942 0.172572i −0.349942 0.172572i 0.258819 0.965926i \(-0.416667\pi\)
−0.608761 + 0.793353i \(0.708333\pi\)
\(114\) 0.315118 0.410670i 0.315118 0.410670i
\(115\) 0 0
\(116\) 0 0
\(117\) 0 0
\(118\) 1.58671i 1.58671i
\(119\) 0 0
\(120\) 0 0
\(121\) 0.215662 0.0283924i 0.215662 0.0283924i
\(122\) 0 0
\(123\) −1.49144 0.996552i −1.49144 0.996552i
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) 0 0 0.923880 0.382683i \(-0.125000\pi\)
−0.923880 + 0.382683i \(0.875000\pi\)
\(128\) −0.130526 + 0.991445i −0.130526 + 0.991445i
\(129\) −0.991445 1.71723i −0.991445 1.71723i
\(130\) 0 0
\(131\) 1.09645 + 1.25026i 1.09645 + 1.25026i 0.965926 + 0.258819i \(0.0833333\pi\)
0.130526 + 0.991445i \(0.458333\pi\)
\(132\) 0.391239 + 0.793353i 0.391239 + 0.793353i
\(133\) 0 0
\(134\) −0.0999004 + 0.241181i −0.0999004 + 0.241181i
\(135\) 0 0
\(136\) −0.500000 0.866025i −0.500000 0.866025i
\(137\) −1.67303 0.965926i −1.67303 0.965926i −0.965926 0.258819i \(-0.916667\pi\)
−0.707107 0.707107i \(-0.750000\pi\)
\(138\) 0 0
\(139\) 0.0862466 1.31587i 0.0862466 1.31587i −0.707107 0.707107i \(-0.750000\pi\)
0.793353 0.608761i \(-0.208333\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 0 0
\(144\) 0.707107 0.707107i 0.707107 0.707107i
\(145\) 0 0
\(146\) −0.0862466 + 0.0983454i −0.0862466 + 0.0983454i
\(147\) 0.707107 0.707107i 0.707107 0.707107i
\(148\) 0 0
\(149\) 0 0 0.258819 0.965926i \(-0.416667\pi\)
−0.258819 + 0.965926i \(0.583333\pi\)
\(150\) 0.382683 0.923880i 0.382683 0.923880i
\(151\) 0 0 0.793353 0.608761i \(-0.208333\pi\)
−0.793353 + 0.608761i \(0.791667\pi\)
\(152\) −0.517638 −0.517638
\(153\) −0.130526 + 0.991445i −0.130526 + 0.991445i
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 0 0 0.258819 0.965926i \(-0.416667\pi\)
−0.258819 + 0.965926i \(0.583333\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 0 0
\(162\) −0.991445 + 0.130526i −0.991445 + 0.130526i
\(163\) 0.382683 1.92388i 0.382683 1.92388i 1.00000i \(-0.5\pi\)
0.382683 0.923880i \(-0.375000\pi\)
\(164\) 0.117317 + 1.78990i 0.117317 + 1.78990i
\(165\) 0 0
\(166\) 0.739288 + 0.198092i 0.739288 + 0.198092i
\(167\) 0 0 0.0654031 0.997859i \(-0.479167\pi\)
−0.0654031 + 0.997859i \(0.520833\pi\)
\(168\) 0 0
\(169\) 0.866025 + 0.500000i 0.866025 + 0.500000i
\(170\) 0 0
\(171\) 0.410670 + 0.315118i 0.410670 + 0.315118i
\(172\) −0.758819 + 1.83195i −0.758819 + 1.83195i
\(173\) 0 0 −0.442289 0.896873i \(-0.645833\pi\)
0.442289 + 0.896873i \(0.354167\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0.391239 0.793353i 0.391239 0.793353i
\(177\) −1.58671 −1.58671
\(178\) 0.241181 1.83195i 0.241181 1.83195i
\(179\) −1.30656 + 0.541196i −1.30656 + 0.541196i −0.923880 0.382683i \(-0.875000\pi\)
−0.382683 + 0.923880i \(0.625000\pi\)
\(180\) 0 0
\(181\) 0 0 −0.555570 0.831470i \(-0.687500\pi\)
0.555570 + 0.831470i \(0.312500\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 0.172572 + 0.867580i 0.172572 + 0.867580i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 0 0 −0.258819 0.965926i \(-0.583333\pi\)
0.258819 + 0.965926i \(0.416667\pi\)
\(192\) −0.991445 0.130526i −0.991445 0.130526i
\(193\) −1.34861 0.665060i −1.34861 0.665060i −0.382683 0.923880i \(-0.625000\pi\)
−0.965926 + 0.258819i \(0.916667\pi\)
\(194\) 0.608761 0.206647i 0.608761 0.206647i
\(195\) 0 0
\(196\) −0.991445 0.130526i −0.991445 0.130526i
\(197\) 0 0 −0.980785 0.195090i \(-0.937500\pi\)
0.980785 + 0.195090i \(0.0625000\pi\)
\(198\) −0.793353 + 0.391239i −0.793353 + 0.391239i
\(199\) 0 0 −0.195090 0.980785i \(-0.562500\pi\)
0.195090 + 0.980785i \(0.437500\pi\)
\(200\) −0.965926 + 0.258819i −0.965926 + 0.258819i
\(201\) −0.241181 0.0999004i −0.241181 0.0999004i
\(202\) 0 0
\(203\) 0 0
\(204\) 0.866025 0.500000i 0.866025 0.500000i
\(205\) 0 0
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 0.433591 + 0.147184i 0.433591 + 0.147184i
\(210\) 0 0
\(211\) 0.534534 + 1.57469i 0.534534 + 1.57469i 0.793353 + 0.608761i \(0.208333\pi\)
−0.258819 + 0.965926i \(0.583333\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 1.42388 + 1.24871i 1.42388 + 1.24871i
\(215\) 0 0
\(216\) 0.707107 + 0.707107i 0.707107 + 0.707107i
\(217\) 0 0
\(218\) 0 0
\(219\) −0.0983454 0.0862466i −0.0983454 0.0862466i
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) 0 0 −0.608761 0.793353i \(-0.708333\pi\)
0.608761 + 0.793353i \(0.291667\pi\)
\(224\) 0 0
\(225\) 0.923880 + 0.382683i 0.923880 + 0.382683i
\(226\) −0.382683 + 0.0761205i −0.382683 + 0.0761205i
\(227\) 1.50046 0.0983454i 1.50046 0.0983454i 0.707107 0.707107i \(-0.250000\pi\)
0.793353 + 0.608761i \(0.208333\pi\)
\(228\) 0.517638i 0.517638i
\(229\) 0 0 −0.793353 0.608761i \(-0.791667\pi\)
0.793353 + 0.608761i \(0.208333\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 1.29335 0.257264i 1.29335 0.257264i 0.500000 0.866025i \(-0.333333\pi\)
0.793353 + 0.608761i \(0.208333\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0.965926 + 1.25882i 0.965926 + 1.25882i
\(237\) 0 0
\(238\) 0 0
\(239\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(240\) 0 0
\(241\) 1.78990 0.882683i 1.78990 0.882683i 0.866025 0.500000i \(-0.166667\pi\)
0.923880 0.382683i \(-0.125000\pi\)
\(242\) 0.153812 0.153812i 0.153812 0.153812i
\(243\) −0.130526 0.991445i −0.130526 0.991445i
\(244\) 0 0
\(245\) 0 0
\(246\) −1.78990 + 0.117317i −1.78990 + 0.117317i
\(247\) 0 0
\(248\) 0 0
\(249\) −0.198092 + 0.739288i −0.198092 + 0.739288i
\(250\) 0 0
\(251\) −0.860919 0.860919i −0.860919 0.860919i 0.130526 0.991445i \(-0.458333\pi\)
−0.991445 + 0.130526i \(0.958333\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 0 0
\(255\) 0 0
\(256\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(257\) −0.158919 1.20711i −0.158919 1.20711i −0.866025 0.500000i \(-0.833333\pi\)
0.707107 0.707107i \(-0.250000\pi\)
\(258\) −1.83195 0.758819i −1.83195 0.758819i
\(259\) 0 0
\(260\) 0 0
\(261\) 0 0
\(262\) 1.63099 + 0.324423i 1.63099 + 0.324423i
\(263\) 0 0 −0.991445 0.130526i \(-0.958333\pi\)
0.991445 + 0.130526i \(0.0416667\pi\)
\(264\) 0.793353 + 0.391239i 0.793353 + 0.391239i
\(265\) 0 0
\(266\) 0 0
\(267\) 1.83195 + 0.241181i 1.83195 + 0.241181i
\(268\) 0.0675653 + 0.252157i 0.0675653 + 0.252157i
\(269\) 0 0 −0.831470 0.555570i \(-0.812500\pi\)
0.831470 + 0.555570i \(0.187500\pi\)
\(270\) 0 0
\(271\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(272\) −0.923880 0.382683i −0.923880 0.382683i
\(273\) 0 0
\(274\) −1.91532 + 0.252157i −1.91532 + 0.252157i
\(275\) 0.882683 + 0.0578541i 0.882683 + 0.0578541i
\(276\) 0 0
\(277\) 0 0 0.321439 0.946930i \(-0.395833\pi\)
−0.321439 + 0.946930i \(0.604167\pi\)
\(278\) −0.732626 1.09645i −0.732626 1.09645i
\(279\) 0 0
\(280\) 0 0
\(281\) −0.184592 + 1.40211i −0.184592 + 1.40211i 0.608761 + 0.793353i \(0.291667\pi\)
−0.793353 + 0.608761i \(0.791667\pi\)
\(282\) 0 0
\(283\) 0.491445 0.996552i 0.491445 0.996552i −0.500000 0.866025i \(-0.666667\pi\)
0.991445 0.130526i \(-0.0416667\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0.130526 0.991445i 0.130526 0.991445i
\(289\) 0.965926 0.258819i 0.965926 0.258819i
\(290\) 0 0
\(291\) 0.206647 + 0.608761i 0.206647 + 0.608761i
\(292\) −0.00855514 + 0.130526i −0.00855514 + 0.130526i
\(293\) 0 0 −0.965926 0.258819i \(-0.916667\pi\)
0.965926 + 0.258819i \(0.0833333\pi\)
\(294\) 0.130526 0.991445i 0.130526 0.991445i
\(295\) 0 0
\(296\) 0 0
\(297\) −0.391239 0.793353i −0.391239 0.793353i
\(298\) 0 0
\(299\) 0 0
\(300\) −0.258819 0.965926i −0.258819 0.965926i
\(301\) 0 0
\(302\) 0 0
\(303\) 0 0
\(304\) −0.410670 + 0.315118i −0.410670 + 0.315118i
\(305\) 0 0
\(306\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(307\) 1.73205 1.73205 0.866025 0.500000i \(-0.166667\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 0 0 0.751840 0.659346i \(-0.229167\pi\)
−0.751840 + 0.659346i \(0.770833\pi\)
\(312\) 0 0
\(313\) −1.24871 + 1.42388i −1.24871 + 1.42388i −0.382683 + 0.923880i \(0.625000\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 0 0 −0.0654031 0.997859i \(-0.520833\pi\)
0.0654031 + 0.997859i \(0.479167\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0 0
\(321\) −1.24871 + 1.42388i −1.24871 + 1.42388i
\(322\) 0 0
\(323\) 0.133975 0.500000i 0.133975 0.500000i
\(324\) −0.707107 + 0.707107i −0.707107 + 0.707107i
\(325\) 0 0
\(326\) −0.867580 1.75928i −0.867580 1.75928i
\(327\) 0 0
\(328\) 1.18270 + 1.34861i 1.18270 + 1.34861i
\(329\) 0 0
\(330\) 0 0
\(331\) −0.241181 + 1.83195i −0.241181 + 1.83195i 0.258819 + 0.965926i \(0.416667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(332\) 0.707107 0.292893i 0.707107 0.292893i
\(333\) 0 0
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 1.31587 + 0.0862466i 1.31587 + 0.0862466i 0.707107 0.707107i \(-0.250000\pi\)
0.608761 + 0.793353i \(0.291667\pi\)
\(338\) 0.991445 0.130526i 0.991445 0.130526i
\(339\) −0.0761205 0.382683i −0.0761205 0.382683i
\(340\) 0 0
\(341\) 0 0
\(342\) 0.517638 0.517638
\(343\) 0 0
\(344\) 0.513210 + 1.91532i 0.513210 + 1.91532i
\(345\) 0 0
\(346\) 0 0
\(347\) −0.123864 + 0.0420463i −0.123864 + 0.0420463i −0.382683 0.923880i \(-0.625000\pi\)
0.258819 + 0.965926i \(0.416667\pi\)
\(348\) 0 0
\(349\) 0 0 −0.991445 0.130526i \(-0.958333\pi\)
0.991445 + 0.130526i \(0.0416667\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) −0.172572 0.867580i −0.172572 0.867580i
\(353\) 0.252157 0.0675653i 0.252157 0.0675653i −0.130526 0.991445i \(-0.541667\pi\)
0.382683 + 0.923880i \(0.375000\pi\)
\(354\) −1.25882 + 0.965926i −1.25882 + 0.965926i
\(355\) 0 0
\(356\) −0.923880 1.60021i −0.923880 1.60021i
\(357\) 0 0
\(358\) −0.707107 + 1.22474i −0.707107 + 1.22474i
\(359\) 0 0 −0.923880 0.382683i \(-0.875000\pi\)
0.923880 + 0.382683i \(0.125000\pi\)
\(360\) 0 0
\(361\) 0.517638 + 0.517638i 0.517638 + 0.517638i
\(362\) 0 0
\(363\) 0.153812 + 0.153812i 0.153812 + 0.153812i
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 0 0 −0.751840 0.659346i \(-0.770833\pi\)
0.751840 + 0.659346i \(0.229167\pi\)
\(368\) 0 0
\(369\) −0.117317 1.78990i −0.117317 1.78990i
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(374\) 0.665060 + 0.583242i 0.665060 + 0.583242i
\(375\) 0 0
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) −1.95737 + 0.389345i −1.95737 + 0.389345i −0.965926 + 0.258819i \(0.916667\pi\)
−0.991445 + 0.130526i \(0.958333\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 0 0 −0.793353 0.608761i \(-0.791667\pi\)
0.793353 + 0.608761i \(0.208333\pi\)
\(384\) −0.866025 + 0.500000i −0.866025 + 0.500000i
\(385\) 0 0
\(386\) −1.47479 + 0.293353i −1.47479 + 0.293353i
\(387\) 0.758819 1.83195i 0.758819 1.83195i
\(388\) 0.357164 0.534534i 0.357164 0.534534i
\(389\) 0 0 −0.608761 0.793353i \(-0.708333\pi\)
0.608761 + 0.793353i \(0.291667\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) −0.866025 + 0.500000i −0.866025 + 0.500000i
\(393\) −0.324423 + 1.63099i −0.324423 + 1.63099i
\(394\) 0 0
\(395\) 0 0
\(396\) −0.391239 + 0.793353i −0.391239 + 0.793353i
\(397\) 0 0 0.831470 0.555570i \(-0.187500\pi\)
−0.831470 + 0.555570i \(0.812500\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) −0.608761 + 0.793353i −0.608761 + 0.793353i
\(401\) −0.206647 0.608761i −0.206647 0.608761i 0.793353 0.608761i \(-0.208333\pi\)
−1.00000 \(\pi\)
\(402\) −0.252157 + 0.0675653i −0.252157 + 0.0675653i
\(403\) 0 0
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 0 0
\(408\) 0.382683 0.923880i 0.382683 0.923880i
\(409\) 0.793353 + 1.37413i 0.793353 + 1.37413i 0.923880 + 0.382683i \(0.125000\pi\)
−0.130526 + 0.991445i \(0.541667\pi\)
\(410\) 0 0
\(411\) −0.252157 1.91532i −0.252157 1.91532i
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 1.09645 0.732626i 1.09645 0.732626i
\(418\) 0.433591 0.147184i 0.433591 0.147184i
\(419\) 0.349942 + 0.172572i 0.349942 + 0.172572i 0.608761 0.793353i \(-0.291667\pi\)
−0.258819 + 0.965926i \(0.583333\pi\)
\(420\) 0 0
\(421\) 0 0 −0.258819 0.965926i \(-0.583333\pi\)
0.258819 + 0.965926i \(0.416667\pi\)
\(422\) 1.38268 + 0.923880i 1.38268 + 0.923880i
\(423\) 0 0
\(424\) 0 0
\(425\) 1.00000i 1.00000i
\(426\) 0 0
\(427\) 0 0
\(428\) 1.88981 + 0.123864i 1.88981 + 0.123864i
\(429\) 0 0
\(430\) 0 0
\(431\) 0 0 −0.555570 0.831470i \(-0.687500\pi\)
0.555570 + 0.831470i \(0.312500\pi\)
\(432\) 0.991445 + 0.130526i 0.991445 + 0.130526i
\(433\) 1.78480 0.739288i 1.78480 0.739288i 0.793353 0.608761i \(-0.208333\pi\)
0.991445 0.130526i \(-0.0416667\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 0 0
\(438\) −0.130526 0.00855514i −0.130526 0.00855514i
\(439\) 0 0 −0.442289 0.896873i \(-0.645833\pi\)
0.442289 + 0.896873i \(0.354167\pi\)
\(440\) 0 0
\(441\) 0.991445 + 0.130526i 0.991445 + 0.130526i
\(442\) 0 0
\(443\) −0.226078 0.130526i −0.226078 0.130526i 0.382683 0.923880i \(-0.375000\pi\)
−0.608761 + 0.793353i \(0.708333\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) −0.369474 + 1.85747i −0.369474 + 1.85747i 0.130526 + 0.991445i \(0.458333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(450\) 0.965926 0.258819i 0.965926 0.258819i
\(451\) −0.607206 1.46593i −0.607206 1.46593i
\(452\) −0.257264 + 0.293353i −0.257264 + 0.293353i
\(453\) 0 0
\(454\) 1.13053 0.991445i 1.13053 0.991445i
\(455\) 0 0
\(456\) −0.315118 0.410670i −0.315118 0.410670i
\(457\) −1.53264 + 1.17604i −1.53264 + 1.17604i −0.608761 + 0.793353i \(0.708333\pi\)
−0.923880 + 0.382683i \(0.875000\pi\)
\(458\) 0 0
\(459\) −0.866025 + 0.500000i −0.866025 + 0.500000i
\(460\) 0 0
\(461\) 0 0 0.793353 0.608761i \(-0.208333\pi\)
−0.793353 + 0.608761i \(0.791667\pi\)
\(462\) 0 0
\(463\) 0 0 0.258819 0.965926i \(-0.416667\pi\)
−0.258819 + 0.965926i \(0.583333\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0.869474 0.991445i 0.869474 0.991445i
\(467\) −0.739288 1.78480i −0.739288 1.78480i −0.608761 0.793353i \(-0.708333\pi\)
−0.130526 0.991445i \(-0.541667\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 0 0
\(472\) 1.53264 + 0.410670i 1.53264 + 0.410670i
\(473\) 0.114718 1.75026i 0.114718 1.75026i
\(474\) 0 0
\(475\) −0.448288 0.258819i −0.448288 0.258819i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 0 0 −0.442289 0.896873i \(-0.645833\pi\)
0.442289 + 0.896873i \(0.354167\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 0.882683 1.78990i 0.882683 1.78990i
\(483\) 0 0
\(484\) 0.0283924 0.215662i 0.0283924 0.215662i
\(485\) 0 0
\(486\) −0.707107 0.707107i −0.707107 0.707107i
\(487\) 0 0 −0.555570 0.831470i \(-0.687500\pi\)
0.555570 + 0.831470i \(0.312500\pi\)
\(488\) 0 0
\(489\) 1.75928 0.867580i 1.75928 0.867580i
\(490\) 0 0
\(491\) 1.91532 0.252157i 1.91532 0.252157i 0.923880 0.382683i \(-0.125000\pi\)
0.991445 + 0.130526i \(0.0416667\pi\)
\(492\) −1.34861 + 1.18270i −1.34861 + 1.18270i
\(493\) 0 0
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 0 0
\(498\) 0.292893 + 0.707107i 0.292893 + 0.707107i
\(499\) −1.78990 0.882683i −1.78990 0.882683i −0.923880 0.382683i \(-0.875000\pi\)
−0.866025 0.500000i \(-0.833333\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) −1.20711 0.158919i −1.20711 0.158919i
\(503\) 0 0 −0.980785 0.195090i \(-0.937500\pi\)
0.980785 + 0.195090i \(0.0625000\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 0.130526 + 0.991445i 0.130526 + 0.991445i
\(508\) 0 0
\(509\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0.923880 + 0.382683i 0.923880 + 0.382683i
\(513\) 0.517638i 0.517638i
\(514\) −0.860919 0.860919i −0.860919 0.860919i
\(515\) 0 0
\(516\) −1.91532 + 0.513210i −1.91532 + 0.513210i
\(517\) 0 0
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 1.65938 1.10876i 1.65938 1.10876i 0.793353 0.608761i \(-0.208333\pi\)
0.866025 0.500000i \(-0.166667\pi\)
\(522\) 0 0
\(523\) 0.541196 0.541196i 0.541196 0.541196i −0.382683 0.923880i \(-0.625000\pi\)
0.923880 + 0.382683i \(0.125000\pi\)
\(524\) 1.49144 0.735499i 1.49144 0.735499i
\(525\) 0 0
\(526\) 0 0
\(527\) 0 0
\(528\) 0.867580 0.172572i 0.867580 0.172572i
\(529\) −0.608761 0.793353i −0.608761 0.793353i
\(530\) 0 0
\(531\) −0.965926 1.25882i −0.965926 1.25882i
\(532\) 0 0
\(533\) 0 0
\(534\) 1.60021 0.923880i 1.60021 0.923880i
\(535\) 0 0
\(536\) 0.207107 + 0.158919i 0.207107 + 0.158919i
\(537\) −1.22474 0.707107i −1.22474 0.707107i
\(538\) 0 0
\(539\) 0.867580 0.172572i 0.867580 0.172572i
\(540\) 0 0
\(541\) 0 0 0.555570 0.831470i \(-0.312500\pi\)
−0.555570 + 0.831470i \(0.687500\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) −0.965926 + 0.258819i −0.965926 + 0.258819i
\(545\) 0 0
\(546\) 0 0
\(547\) 0.576581 0.284338i 0.576581 0.284338i −0.130526 0.991445i \(-0.541667\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(548\) −1.36603 + 1.36603i −1.36603 + 1.36603i
\(549\) 0 0
\(550\) 0.735499 0.491445i 0.735499 0.491445i
\(551\) 0 0
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) 0 0
\(556\) −1.24871 0.423880i −1.24871 0.423880i
\(557\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 0 0
\(561\) −0.583242 + 0.665060i −0.583242 + 0.665060i
\(562\) 0.707107 + 1.22474i 0.707107 + 1.22474i
\(563\) −0.258819 1.96593i −0.258819 1.96593i −0.258819 0.965926i \(-0.583333\pi\)
1.00000i \(-0.5\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) −0.216773 1.08979i −0.216773 1.08979i
\(567\) 0 0
\(568\) 0 0
\(569\) 1.96593 + 0.258819i 1.96593 + 0.258819i 1.00000 \(0\)
0.965926 + 0.258819i \(0.0833333\pi\)
\(570\) 0 0
\(571\) −1.69855 + 0.576581i −1.69855 + 0.576581i −0.991445 0.130526i \(-0.958333\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 0 0
\(576\) −0.500000 0.866025i −0.500000 0.866025i
\(577\) 1.73205i 1.73205i 0.500000 + 0.866025i \(0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(578\) 0.608761 0.793353i 0.608761 0.793353i
\(579\) −0.293353 1.47479i −0.293353 1.47479i
\(580\) 0 0
\(581\) 0 0
\(582\) 0.534534 + 0.357164i 0.534534 + 0.357164i
\(583\) 0 0
\(584\) 0.0726721 + 0.108761i 0.0726721 + 0.108761i
\(585\) 0 0
\(586\) 0 0
\(587\) 0.226078 1.71723i 0.226078 1.71723i −0.382683 0.923880i \(-0.625000\pi\)
0.608761 0.793353i \(-0.291667\pi\)
\(588\) −0.500000 0.866025i −0.500000 0.866025i
\(589\) 0 0
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) −0.292893 + 0.707107i −0.292893 + 0.707107i 0.707107 + 0.707107i \(0.250000\pi\)
−1.00000 \(\pi\)
\(594\) −0.793353 0.391239i −0.793353 0.391239i
\(595\) 0 0
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 0 0 −0.965926 0.258819i \(-0.916667\pi\)
0.965926 + 0.258819i \(0.0833333\pi\)
\(600\) −0.793353 0.608761i −0.793353 0.608761i
\(601\) 0.123864 + 1.88981i 0.123864 + 1.88981i 0.382683 + 0.923880i \(0.375000\pi\)
−0.258819 + 0.965926i \(0.583333\pi\)
\(602\) 0 0
\(603\) −0.0675653 0.252157i −0.0675653 0.252157i
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) 0 0 0.751840 0.659346i \(-0.229167\pi\)
−0.751840 + 0.659346i \(0.770833\pi\)
\(608\) −0.133975 + 0.500000i −0.133975 + 0.500000i
\(609\) 0 0
\(610\) 0 0
\(611\) 0 0
\(612\) 0.923880 + 0.382683i 0.923880 + 0.382683i
\(613\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(614\) 1.37413 1.05441i 1.37413 1.05441i
\(615\) 0 0
\(616\) 0 0
\(617\) 0.0983454 0.0862466i 0.0983454 0.0862466i −0.608761 0.793353i \(-0.708333\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(618\) 0 0
\(619\) −0.423880 + 0.483342i −0.423880 + 0.483342i −0.923880 0.382683i \(-0.875000\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) −0.965926 0.258819i −0.965926 0.258819i
\(626\) −0.123864 + 1.88981i −0.123864 + 1.88981i
\(627\) 0.147184 + 0.433591i 0.147184 + 0.433591i
\(628\) 0 0
\(629\) 0 0
\(630\) 0 0
\(631\) 0 0 0.382683 0.923880i \(-0.375000\pi\)
−0.382683 + 0.923880i \(0.625000\pi\)
\(632\) 0 0
\(633\) −0.923880 + 1.38268i −0.923880 + 1.38268i
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) −0.641502 + 1.88981i −0.641502 + 1.88981i −0.258819 + 0.965926i \(0.583333\pi\)
−0.382683 + 0.923880i \(0.625000\pi\)
\(642\) −0.123864 + 1.88981i −0.123864 + 1.88981i
\(643\) 0.130526 + 0.00855514i 0.130526 + 0.00855514i 0.130526 0.991445i \(-0.458333\pi\)
1.00000i \(0.5\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) −0.198092 0.478235i −0.198092 0.478235i
\(647\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(648\) −0.130526 + 0.991445i −0.130526 + 0.991445i
\(649\) −1.16702 0.779779i −1.16702 0.779779i
\(650\) 0 0
\(651\) 0 0
\(652\) −1.75928 0.867580i −1.75928 0.867580i
\(653\) 0 0 0.946930 0.321439i \(-0.104167\pi\)
−0.946930 + 0.321439i \(0.895833\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 1.75928 + 0.349942i 1.75928 + 0.349942i
\(657\) 0.00855514 0.130526i 0.00855514 0.130526i
\(658\) 0 0
\(659\) 0 0 −0.258819 0.965926i \(-0.583333\pi\)
0.258819 + 0.965926i \(0.416667\pi\)
\(660\) 0 0
\(661\) 0 0 −0.130526 0.991445i \(-0.541667\pi\)
0.130526 + 0.991445i \(0.458333\pi\)
\(662\) 0.923880 + 1.60021i 0.923880 + 1.60021i
\(663\) 0 0
\(664\) 0.382683 0.662827i 0.382683 0.662827i
\(665\) 0 0
\(666\) 0 0
\(667\) 0 0
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) 0.835400 + 0.732626i 0.835400 + 0.732626i 0.965926 0.258819i \(-0.0833333\pi\)
−0.130526 + 0.991445i \(0.541667\pi\)
\(674\) 1.09645 0.732626i 1.09645 0.732626i
\(675\) 0.258819 + 0.965926i 0.258819 + 0.965926i
\(676\) 0.707107 0.707107i 0.707107 0.707107i
\(677\) 0 0 0.896873 0.442289i \(-0.145833\pi\)
−0.896873 + 0.442289i \(0.854167\pi\)
\(678\) −0.293353 0.257264i −0.293353 0.257264i
\(679\) 0 0
\(680\) 0 0
\(681\) 0.991445 + 1.13053i 0.991445 + 1.13053i
\(682\) 0 0
\(683\) 0.0726721 0.108761i 0.0726721 0.108761i −0.793353 0.608761i \(-0.791667\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(684\) 0.410670 0.315118i 0.410670 0.315118i
\(685\) 0 0
\(686\) 0 0
\(687\) 0 0
\(688\) 1.57313 + 1.20711i 1.57313 + 1.20711i
\(689\) 0 0
\(690\) 0 0
\(691\) 1.10876 0.0726721i 1.10876 0.0726721i 0.500000 0.866025i \(-0.333333\pi\)
0.608761 + 0.793353i \(0.291667\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) −0.0726721 + 0.108761i −0.0726721 + 0.108761i
\(695\) 0 0
\(696\) 0 0
\(697\) −1.60876 + 0.793353i −1.60876 + 0.793353i
\(698\) 0 0
\(699\) 0.991445 + 0.869474i 0.991445 + 0.869474i
\(700\) 0 0
\(701\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) −0.665060 0.583242i −0.665060 0.583242i
\(705\) 0 0
\(706\) 0.158919 0.207107i 0.158919 0.207107i
\(707\) 0 0
\(708\) −0.410670 + 1.53264i −0.410670 + 1.53264i
\(709\) 0 0 −0.946930 0.321439i \(-0.895833\pi\)
0.946930 + 0.321439i \(0.104167\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) −1.70711 0.707107i −1.70711 0.707107i
\(713\) 0 0
\(714\) 0 0
\(715\) 0 0
\(716\) 0.184592 + 1.40211i 0.184592 + 1.40211i
\(717\) 0 0
\(718\) 0 0
\(719\) 0 0 −0.195090 0.980785i \(-0.562500\pi\)
0.195090 + 0.980785i \(0.437500\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0.725788 + 0.0955518i 0.725788 + 0.0955518i
\(723\) 1.78990 + 0.882683i 1.78990 + 0.882683i
\(724\) 0 0
\(725\) 0 0
\(726\) 0.215662 + 0.0283924i 0.215662 + 0.0283924i
\(727\) 0 0 −0.258819 0.965926i \(-0.583333\pi\)
0.258819 + 0.965926i \(0.416667\pi\)
\(728\) 0 0
\(729\) 0.707107 0.707107i 0.707107 0.707107i
\(730\) 0 0
\(731\) −1.98289 −1.98289
\(732\) 0 0
\(733\) 0 0 0.991445 0.130526i \(-0.0416667\pi\)
−0.991445 + 0.130526i \(0.958333\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −0.128293 0.192004i −0.128293 0.192004i
\(738\) −1.18270 1.34861i −1.18270 1.34861i
\(739\) −1.60021 + 0.662827i −1.60021 + 0.662827i −0.991445 0.130526i \(-0.958333\pi\)
−0.608761 + 0.793353i \(0.708333\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 0 0 −0.659346 0.751840i \(-0.729167\pi\)
0.659346 + 0.751840i \(0.270833\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) −0.707107 + 0.292893i −0.707107 + 0.292893i
\(748\) 0.882683 + 0.0578541i 0.882683 + 0.0578541i
\(749\) 0 0
\(750\) 0 0
\(751\) 0 0 0.0654031 0.997859i \(-0.479167\pi\)
−0.0654031 + 0.997859i \(0.520833\pi\)
\(752\) 0 0
\(753\) 0.158919 1.20711i 0.158919 1.20711i
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 0 0 −0.382683 0.923880i \(-0.625000\pi\)
0.382683 + 0.923880i \(0.375000\pi\)
\(758\) −1.31587 + 1.50046i −1.31587 + 1.50046i
\(759\) 0 0
\(760\) 0 0
\(761\) −0.366025 + 1.36603i −0.366025 + 1.36603i 0.500000 + 0.866025i \(0.333333\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 0 0
\(768\) −0.382683 + 0.923880i −0.382683 + 0.923880i
\(769\) −0.366025 + 1.36603i −0.366025 + 1.36603i 0.500000 + 0.866025i \(0.333333\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(770\) 0 0
\(771\) 0.860919 0.860919i 0.860919 0.860919i
\(772\) −0.991445 + 1.13053i −0.991445 + 1.13053i
\(773\) 0 0 −0.382683 0.923880i \(-0.625000\pi\)
0.382683 + 0.923880i \(0.375000\pi\)
\(774\) −0.513210 1.91532i −0.513210 1.91532i
\(775\) 0 0
\(776\) −0.0420463 0.641502i −0.0420463 0.641502i
\(777\) 0 0
\(778\) 0 0
\(779\) −0.0607275 + 0.926523i −0.0607275 + 0.926523i
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 0 0
\(784\) −0.382683 + 0.923880i −0.382683 + 0.923880i
\(785\) 0 0
\(786\) 0.735499 + 1.49144i 0.735499 + 1.49144i
\(787\) −0.257264 0.293353i −0.257264 0.293353i 0.608761 0.793353i \(-0.291667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) 0.172572 + 0.867580i 0.172572 + 0.867580i
\(793\) 0 0
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 0 0 0.991445 0.130526i \(-0.0416667\pi\)
−0.991445 + 0.130526i \(0.958333\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 1.00000i 1.00000i
\(801\) 0.923880 + 1.60021i 0.923880 + 1.60021i
\(802\) −0.534534 0.357164i −0.534534 0.357164i
\(803\) −0.0299475 0.111766i −0.0299475 0.111766i
\(804\) −0.158919 + 0.207107i −0.158919 + 0.207107i
\(805\) 0 0
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) −1.29335 0.257264i −1.29335 0.257264i −0.500000 0.866025i \(-0.666667\pi\)
−0.793353 + 0.608761i \(0.791667\pi\)
\(810\) 0 0
\(811\) 0.369474 + 1.85747i 0.369474 + 1.85747i 0.500000 + 0.866025i \(0.333333\pi\)
−0.130526 + 0.991445i \(0.541667\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 0 0
\(816\) −0.258819 0.965926i −0.258819 0.965926i
\(817\) −0.513210 + 0.888905i −0.513210 + 0.888905i
\(818\) 1.46593 + 0.607206i 1.46593 + 0.607206i
\(819\) 0 0
\(820\) 0 0
\(821\) 0 0 −0.946930 0.321439i \(-0.895833\pi\)
0.946930 + 0.321439i \(0.104167\pi\)
\(822\) −1.36603 1.36603i −1.36603 1.36603i
\(823\) 0 0 −0.321439 0.946930i \(-0.604167\pi\)
0.321439 + 0.946930i \(0.395833\pi\)
\(824\) 0 0
\(825\) 0.491445 + 0.735499i 0.491445 + 0.735499i
\(826\) 0 0
\(827\) −1.63099 + 1.08979i −1.63099 + 1.08979i −0.707107 + 0.707107i \(0.750000\pi\)
−0.923880 + 0.382683i \(0.875000\pi\)
\(828\) 0 0
\(829\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) −0.258819 0.965926i −0.258819 0.965926i
\(834\) 0.423880 1.24871i 0.423880 1.24871i
\(835\) 0 0
\(836\) 0.254391 0.380722i 0.254391 0.380722i
\(837\) 0 0
\(838\) 0.382683 0.0761205i 0.382683 0.0761205i
\(839\) 0 0 0.997859 0.0654031i \(-0.0208333\pi\)
−0.997859 + 0.0654031i \(0.979167\pi\)
\(840\) 0 0
\(841\) −0.793353 0.608761i −0.793353 0.608761i
\(842\) 0 0
\(843\) −1.22474 + 0.707107i −1.22474 + 0.707107i
\(844\) 1.65938 0.108761i 1.65938 0.108761i
\(845\) 0 0
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) 1.08979 0.216773i 1.08979 0.216773i
\(850\) −0.608761 0.793353i −0.608761 0.793353i
\(851\) 0 0
\(852\) 0 0
\(853\) 0 0 0.896873 0.442289i \(-0.145833\pi\)
−0.896873 + 0.442289i \(0.854167\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 1.57469 1.05217i 1.57469 1.05217i
\(857\) −0.835400 0.732626i −0.835400 0.732626i 0.130526 0.991445i \(-0.458333\pi\)
−0.965926 + 0.258819i \(0.916667\pi\)
\(858\) 0 0
\(859\) 0.741181 0.965926i 0.741181 0.965926i −0.258819 0.965926i \(-0.583333\pi\)
1.00000 \(0\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(864\) 0.866025 0.500000i 0.866025 0.500000i
\(865\) 0 0
\(866\) 0.965926 1.67303i 0.965926 1.67303i
\(867\) 0.793353 + 0.608761i 0.793353 + 0.608761i
\(868\) 0 0
\(869\) 0 0
\(870\) 0 0
\(871\) 0 0
\(872\) 0 0
\(873\) −0.357164 + 0.534534i −0.357164 + 0.534534i
\(874\) 0 0
\(875\) 0 0
\(876\) −0.108761 + 0.0726721i −0.108761 + 0.0726721i
\(877\) 0 0 0.946930 0.321439i \(-0.104167\pi\)
−0.946930 + 0.321439i \(0.895833\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) −0.923880 0.617317i −0.923880 0.617317i 1.00000i \(-0.5\pi\)
−0.923880 + 0.382683i \(0.875000\pi\)
\(882\) 0.866025 0.500000i 0.866025 0.500000i
\(883\) 1.21752i 1.21752i 0.793353 + 0.608761i \(0.208333\pi\)
−0.793353 + 0.608761i \(0.791667\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) −0.258819 + 0.0340742i −0.258819 + 0.0340742i
\(887\) 0 0 −0.997859 0.0654031i \(-0.979167\pi\)
0.997859 + 0.0654031i \(0.0208333\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) 0.391239 0.793353i 0.391239 0.793353i
\(892\) 0 0
\(893\) 0 0
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 0 0
\(898\) 0.837633 + 1.69855i 0.837633 + 1.69855i
\(899\) 0 0
\(900\) 0.608761 0.793353i 0.608761 0.793353i
\(901\) 0 0
\(902\) −1.37413 0.793353i −1.37413 0.793353i
\(903\) 0 0
\(904\) −0.0255190 + 0.389345i −0.0255190 + 0.389345i
\(905\) 0 0
\(906\) 0 0
\(907\) −0.117317 1.78990i −0.117317 1.78990i −0.500000 0.866025i \(-0.666667\pi\)
0.382683 0.923880i \(-0.375000\pi\)
\(908\) 0.293353 1.47479i 0.293353 1.47479i
\(909\) 0 0
\(910\) 0 0
\(911\) 0 0 0.659346 0.751840i \(-0.270833\pi\)
−0.659346 + 0.751840i \(0.729167\pi\)
\(912\) −0.500000 0.133975i −0.500000 0.133975i
\(913\) −0.509015 + 0.446394i −0.509015 + 0.446394i
\(914\) −0.500000 + 1.86603i −0.500000 + 1.86603i
\(915\) 0 0
\(916\) 0 0
\(917\) 0 0
\(918\) −0.382683 + 0.923880i −0.382683 + 0.923880i
\(919\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(920\) 0 0
\(921\) 1.05441 + 1.37413i 1.05441 + 1.37413i
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 0.128293 + 1.95737i 0.128293 + 1.95737i 0.258819 + 0.965926i \(0.416667\pi\)
−0.130526 + 0.991445i \(0.541667\pi\)
\(930\) 0 0
\(931\) −0.500000 0.133975i −0.500000 0.133975i
\(932\) 0.0862466 1.31587i 0.0862466 1.31587i
\(933\) 0 0
\(934\) −1.67303 0.965926i −1.67303 0.965926i
\(935\) 0 0
\(936\) 0 0
\(937\) 0 0 −0.923880 0.382683i \(-0.875000\pi\)
0.923880 + 0.382683i \(0.125000\pi\)
\(938\) 0 0
\(939\) −1.88981 0.123864i −1.88981 0.123864i
\(940\) 0 0
\(941\) 0 0 0.442289 0.896873i \(-0.354167\pi\)
−0.442289 + 0.896873i \(0.645833\pi\)
\(942\) 0 0
\(943\) 0 0
\(944\) 1.46593 0.607206i 1.46593 0.607206i
\(945\) 0 0
\(946\) −0.974481 1.45841i −0.974481 1.45841i
\(947\) 0.206647 0.608761i 0.206647 0.608761i −0.793353 0.608761i \(-0.791667\pi\)
1.00000 \(0\)
\(948\) 0 0
\(949\) 0 0
\(950\) −0.513210 + 0.0675653i −0.513210 + 0.0675653i
\(951\) 0 0
\(952\) 0 0
\(953\) 1.58671i 1.58671i −0.608761 0.793353i \(-0.708333\pi\)
0.608761 0.793353i \(-0.291667\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) 0.991445 + 0.130526i 0.991445 + 0.130526i
\(962\) 0 0
\(963\) −1.88981 0.123864i −1.88981 0.123864i
\(964\) −0.389345 1.95737i −0.389345 1.95737i
\(965\) 0 0
\(966\) 0 0
\(967\) 0 0 −0.130526 0.991445i \(-0.541667\pi\)
0.130526 + 0.991445i \(0.458333\pi\)
\(968\) −0.108761 0.188380i −0.108761 0.188380i
\(969\) 0.478235 0.198092i 0.478235 0.198092i
\(970\) 0 0
\(971\) −1.70711 0.707107i −1.70711 0.707107i −0.707107 0.707107i \(-0.750000\pi\)
−1.00000 \(\pi\)
\(972\) −0.991445 0.130526i −0.991445 0.130526i
\(973\) 0 0
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 0.158919 0.207107i 0.158919 0.207107i −0.707107 0.707107i \(-0.750000\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(978\) 0.867580 1.75928i 0.867580 1.75928i
\(979\) 1.22887 + 1.07769i 1.22887 + 1.07769i
\(980\) 0 0
\(981\) 0 0
\(982\) 1.36603 1.36603i 1.36603 1.36603i
\(983\) 0 0 0.896873 0.442289i \(-0.145833\pi\)
−0.896873 + 0.442289i \(0.854167\pi\)
\(984\) −0.349942 + 1.75928i −0.349942 + 1.75928i
\(985\) 0 0
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 0 0
\(990\) 0 0
\(991\) 0 0 0.980785 0.195090i \(-0.0625000\pi\)
−0.980785 + 0.195090i \(0.937500\pi\)
\(992\) 0 0
\(993\) −1.60021 + 0.923880i −1.60021 + 0.923880i
\(994\) 0 0
\(995\) 0 0
\(996\) 0.662827 + 0.382683i 0.662827 + 0.382683i
\(997\) 0 0 0.997859 0.0654031i \(-0.0208333\pi\)
−0.997859 + 0.0654031i \(0.979167\pi\)
\(998\) −1.95737 + 0.389345i −1.95737 + 0.389345i
\(999\) 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 1224.1.cx.b.299.1 yes 16
3.2 odd 2 3672.1.dv.b.1115.1 16
8.3 odd 2 CM 1224.1.cx.b.299.1 yes 16
9.4 even 3 3672.1.dv.a.2339.1 16
9.5 odd 6 1224.1.cx.a.707.1 16
17.12 odd 16 1224.1.cx.a.947.1 yes 16
24.11 even 2 3672.1.dv.b.1115.1 16
51.29 even 16 3672.1.dv.a.1763.1 16
72.59 even 6 1224.1.cx.a.707.1 16
72.67 odd 6 3672.1.dv.a.2339.1 16
136.131 even 16 1224.1.cx.a.947.1 yes 16
153.131 even 48 inner 1224.1.cx.b.131.1 yes 16
153.148 odd 48 3672.1.dv.b.2987.1 16
408.131 odd 16 3672.1.dv.a.1763.1 16
1224.131 odd 48 inner 1224.1.cx.b.131.1 yes 16
1224.1219 even 48 3672.1.dv.b.2987.1 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1224.1.cx.a.707.1 16 9.5 odd 6
1224.1.cx.a.707.1 16 72.59 even 6
1224.1.cx.a.947.1 yes 16 17.12 odd 16
1224.1.cx.a.947.1 yes 16 136.131 even 16
1224.1.cx.b.131.1 yes 16 153.131 even 48 inner
1224.1.cx.b.131.1 yes 16 1224.131 odd 48 inner
1224.1.cx.b.299.1 yes 16 1.1 even 1 trivial
1224.1.cx.b.299.1 yes 16 8.3 odd 2 CM
3672.1.dv.a.1763.1 16 51.29 even 16
3672.1.dv.a.1763.1 16 408.131 odd 16
3672.1.dv.a.2339.1 16 9.4 even 3
3672.1.dv.a.2339.1 16 72.67 odd 6
3672.1.dv.b.1115.1 16 3.2 odd 2
3672.1.dv.b.1115.1 16 24.11 even 2
3672.1.dv.b.2987.1 16 153.148 odd 48
3672.1.dv.b.2987.1 16 1224.1219 even 48