Properties

Label 3332.1.ce.d.619.1
Level $3332$
Weight $1$
Character 3332.619
Analytic conductor $1.663$
Analytic rank $0$
Dimension $16$
Projective image $D_{16}$
CM discriminant -4
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [3332,1,Mod(31,3332)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3332, base_ring=CyclotomicField(48))
 
chi = DirichletCharacter(H, H._module([24, 8, 27]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("3332.31");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3332 = 2^{2} \cdot 7^{2} \cdot 17 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 3332.ce (of order \(48\), degree \(16\), 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: \(1.66288462209\)
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: \(D_{16}\)
Projective field: Galois closure of \(\mathbb{Q}[x]/(x^{16} - \cdots)\)

Embedding invariants

Embedding label 619.1
Root \(-0.991445 + 0.130526i\) of defining polynomial
Character \(\chi\) \(=\) 3332.619
Dual form 3332.1.ce.d.1195.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(0.991445 - 0.130526i) q^{2} +(0.965926 - 0.258819i) q^{4} +(-0.732626 - 0.835400i) q^{5} +(0.923880 - 0.382683i) q^{8} +(0.608761 - 0.793353i) q^{9} +O(q^{10})\) \(q+(0.991445 - 0.130526i) q^{2} +(0.965926 - 0.258819i) q^{4} +(-0.732626 - 0.835400i) q^{5} +(0.923880 - 0.382683i) q^{8} +(0.608761 - 0.793353i) q^{9} +(-0.835400 - 0.732626i) q^{10} +(1.00000 + 1.00000i) q^{13} +(0.866025 - 0.500000i) q^{16} +(-0.608761 - 0.793353i) q^{17} +(0.500000 - 0.866025i) q^{18} +(-0.923880 - 0.617317i) q^{20} +(-0.0306258 + 0.232626i) q^{25} +(1.12197 + 0.860919i) q^{26} +(-1.63099 + 0.324423i) q^{29} +(0.793353 - 0.608761i) q^{32} +(-0.707107 - 0.707107i) q^{34} +(0.382683 - 0.923880i) q^{36} +(-1.10876 - 0.0726721i) q^{37} +(-0.996552 - 0.491445i) q^{40} +(0.382683 + 0.0761205i) q^{41} +(-1.10876 + 0.0726721i) q^{45} +0.234633i q^{50} +(1.22474 + 0.707107i) q^{52} +(1.12484 + 1.46593i) q^{53} +(-1.57469 + 0.534534i) q^{58} +(1.85747 - 0.630526i) q^{61} +(0.707107 - 0.707107i) q^{64} +(0.102774 - 1.56803i) q^{65} +(-0.793353 - 0.608761i) q^{68} +(0.258819 - 0.965926i) q^{72} +(0.534534 - 1.57469i) q^{73} +(-1.10876 + 0.0726721i) q^{74} +(-1.05217 - 0.357164i) q^{80} +(-0.258819 - 0.965926i) q^{81} +(0.389345 + 0.0255190i) q^{82} +(-0.216773 + 1.08979i) q^{85} +(-1.93185 - 0.517638i) q^{89} +(-1.08979 + 0.216773i) q^{90} +(0.324423 + 1.63099i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q+O(q^{10}) \) Copy content Toggle raw display \( 16 q + 16 q^{13} + 8 q^{18} - 8 q^{37} - 8 q^{45} - 8 q^{74}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(785\) \(885\) \(1667\)
\(\chi(n)\) \(e\left(\frac{11}{16}\right)\) \(e\left(\frac{1}{6}\right)\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.991445 0.130526i 0.991445 0.130526i
\(3\) 0 0 0.896873 0.442289i \(-0.145833\pi\)
−0.896873 + 0.442289i \(0.854167\pi\)
\(4\) 0.965926 0.258819i 0.965926 0.258819i
\(5\) −0.732626 0.835400i −0.732626 0.835400i 0.258819 0.965926i \(-0.416667\pi\)
−0.991445 + 0.130526i \(0.958333\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) 0.923880 0.382683i 0.923880 0.382683i
\(9\) 0.608761 0.793353i 0.608761 0.793353i
\(10\) −0.835400 0.732626i −0.835400 0.732626i
\(11\) 0 0 −0.0654031 0.997859i \(-0.520833\pi\)
0.0654031 + 0.997859i \(0.479167\pi\)
\(12\) 0 0
\(13\) 1.00000 + 1.00000i 1.00000 + 1.00000i 1.00000 \(0\)
1.00000i \(0.5\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0.866025 0.500000i 0.866025 0.500000i
\(17\) −0.608761 0.793353i −0.608761 0.793353i
\(18\) 0.500000 0.866025i 0.500000 0.866025i
\(19\) 0 0 −0.130526 0.991445i \(-0.541667\pi\)
0.130526 + 0.991445i \(0.458333\pi\)
\(20\) −0.923880 0.617317i −0.923880 0.617317i
\(21\) 0 0
\(22\) 0 0
\(23\) 0 0 0.442289 0.896873i \(-0.354167\pi\)
−0.442289 + 0.896873i \(0.645833\pi\)
\(24\) 0 0
\(25\) −0.0306258 + 0.232626i −0.0306258 + 0.232626i
\(26\) 1.12197 + 0.860919i 1.12197 + 0.860919i
\(27\) 0 0
\(28\) 0 0
\(29\) −1.63099 + 0.324423i −1.63099 + 0.324423i −0.923880 0.382683i \(-0.875000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(30\) 0 0
\(31\) 0 0 −0.442289 0.896873i \(-0.645833\pi\)
0.442289 + 0.896873i \(0.354167\pi\)
\(32\) 0.793353 0.608761i 0.793353 0.608761i
\(33\) 0 0
\(34\) −0.707107 0.707107i −0.707107 0.707107i
\(35\) 0 0
\(36\) 0.382683 0.923880i 0.382683 0.923880i
\(37\) −1.10876 0.0726721i −1.10876 0.0726721i −0.500000 0.866025i \(-0.666667\pi\)
−0.608761 + 0.793353i \(0.708333\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) −0.996552 0.491445i −0.996552 0.491445i
\(41\) 0.382683 + 0.0761205i 0.382683 + 0.0761205i 0.382683 0.923880i \(-0.375000\pi\)
1.00000i \(0.5\pi\)
\(42\) 0 0
\(43\) 0 0 −0.382683 0.923880i \(-0.625000\pi\)
0.382683 + 0.923880i \(0.375000\pi\)
\(44\) 0 0
\(45\) −1.10876 + 0.0726721i −1.10876 + 0.0726721i
\(46\) 0 0
\(47\) 0 0 0.258819 0.965926i \(-0.416667\pi\)
−0.258819 + 0.965926i \(0.583333\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 0.234633i 0.234633i
\(51\) 0 0
\(52\) 1.22474 + 0.707107i 1.22474 + 0.707107i
\(53\) 1.12484 + 1.46593i 1.12484 + 1.46593i 0.866025 + 0.500000i \(0.166667\pi\)
0.258819 + 0.965926i \(0.416667\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 0 0
\(58\) −1.57469 + 0.534534i −1.57469 + 0.534534i
\(59\) 0 0 −0.991445 0.130526i \(-0.958333\pi\)
0.991445 + 0.130526i \(0.0416667\pi\)
\(60\) 0 0
\(61\) 1.85747 0.630526i 1.85747 0.630526i 0.866025 0.500000i \(-0.166667\pi\)
0.991445 0.130526i \(-0.0416667\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0.707107 0.707107i 0.707107 0.707107i
\(65\) 0.102774 1.56803i 0.102774 1.56803i
\(66\) 0 0
\(67\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(68\) −0.793353 0.608761i −0.793353 0.608761i
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 0.555570 0.831470i \(-0.312500\pi\)
−0.555570 + 0.831470i \(0.687500\pi\)
\(72\) 0.258819 0.965926i 0.258819 0.965926i
\(73\) 0.534534 1.57469i 0.534534 1.57469i −0.258819 0.965926i \(-0.583333\pi\)
0.793353 0.608761i \(-0.208333\pi\)
\(74\) −1.10876 + 0.0726721i −1.10876 + 0.0726721i
\(75\) 0 0
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 0 0 −0.896873 0.442289i \(-0.854167\pi\)
0.896873 + 0.442289i \(0.145833\pi\)
\(80\) −1.05217 0.357164i −1.05217 0.357164i
\(81\) −0.258819 0.965926i −0.258819 0.965926i
\(82\) 0.389345 + 0.0255190i 0.389345 + 0.0255190i
\(83\) 0 0 0.382683 0.923880i \(-0.375000\pi\)
−0.382683 + 0.923880i \(0.625000\pi\)
\(84\) 0 0
\(85\) −0.216773 + 1.08979i −0.216773 + 1.08979i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −1.93185 0.517638i −1.93185 0.517638i −0.965926 0.258819i \(-0.916667\pi\)
−0.965926 0.258819i \(-0.916667\pi\)
\(90\) −1.08979 + 0.216773i −1.08979 + 0.216773i
\(91\) 0 0
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 0.324423 + 1.63099i 0.324423 + 1.63099i 0.707107 + 0.707107i \(0.250000\pi\)
−0.382683 + 0.923880i \(0.625000\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0.0306258 + 0.232626i 0.0306258 + 0.232626i
\(101\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(102\) 0 0
\(103\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(104\) 1.30656 + 0.541196i 1.30656 + 0.541196i
\(105\) 0 0
\(106\) 1.30656 + 1.30656i 1.30656 + 1.30656i
\(107\) 0 0 0.751840 0.659346i \(-0.229167\pi\)
−0.751840 + 0.659346i \(0.770833\pi\)
\(108\) 0 0
\(109\) 1.47479 + 1.29335i 1.47479 + 1.29335i 0.866025 + 0.500000i \(0.166667\pi\)
0.608761 + 0.793353i \(0.291667\pi\)
\(110\) 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
\(116\) −1.49144 + 0.735499i −1.49144 + 0.735499i
\(117\) 1.40211 0.184592i 1.40211 0.184592i
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) −0.991445 + 0.130526i −0.991445 + 0.130526i
\(122\) 1.75928 0.867580i 1.75928 0.867580i
\(123\) 0 0
\(124\) 0 0
\(125\) −0.707107 + 0.472474i −0.707107 + 0.472474i
\(126\) 0 0
\(127\) 0 0 0.923880 0.382683i \(-0.125000\pi\)
−0.923880 + 0.382683i \(0.875000\pi\)
\(128\) 0.608761 0.793353i 0.608761 0.793353i
\(129\) 0 0
\(130\) −0.102774 1.56803i −0.102774 1.56803i
\(131\) 0 0 0.751840 0.659346i \(-0.229167\pi\)
−0.751840 + 0.659346i \(0.770833\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) 0 0
\(136\) −0.866025 0.500000i −0.866025 0.500000i
\(137\) −0.923880 + 1.60021i −0.923880 + 1.60021i −0.130526 + 0.991445i \(0.541667\pi\)
−0.793353 + 0.608761i \(0.791667\pi\)
\(138\) 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
\(144\) 0.130526 0.991445i 0.130526 0.991445i
\(145\) 1.46593 + 1.12484i 1.46593 + 1.12484i
\(146\) 0.324423 1.63099i 0.324423 1.63099i
\(147\) 0 0
\(148\) −1.08979 + 0.216773i −1.08979 + 0.216773i
\(149\) −1.36603 0.366025i −1.36603 0.366025i −0.500000 0.866025i \(-0.666667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(150\) 0 0
\(151\) 0 0 0.793353 0.608761i \(-0.208333\pi\)
−0.793353 + 0.608761i \(0.791667\pi\)
\(152\) 0 0
\(153\) −1.00000 −1.00000
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 0.198092 + 0.739288i 0.198092 + 0.739288i 0.991445 + 0.130526i \(0.0416667\pi\)
−0.793353 + 0.608761i \(0.791667\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) −1.08979 0.216773i −1.08979 0.216773i
\(161\) 0 0
\(162\) −0.382683 0.923880i −0.382683 0.923880i
\(163\) 0 0 −0.321439 0.946930i \(-0.604167\pi\)
0.321439 + 0.946930i \(0.395833\pi\)
\(164\) 0.389345 0.0255190i 0.389345 0.0255190i
\(165\) 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
\(170\) −0.0726721 + 1.10876i −0.0726721 + 1.10876i
\(171\) 0 0
\(172\) 0 0
\(173\) −0.0255190 + 0.389345i −0.0255190 + 0.389345i 0.965926 + 0.258819i \(0.0833333\pi\)
−0.991445 + 0.130526i \(0.958333\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 0 0
\(178\) −1.98289 0.261052i −1.98289 0.261052i
\(179\) 0 0 −0.991445 0.130526i \(-0.958333\pi\)
0.991445 + 0.130526i \(0.0416667\pi\)
\(180\) −1.05217 + 0.357164i −1.05217 + 0.357164i
\(181\) 1.08979 + 1.63099i 1.08979 + 1.63099i 0.707107 + 0.707107i \(0.250000\pi\)
0.382683 + 0.923880i \(0.375000\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 0.751597 + 0.979500i 0.751597 + 0.979500i
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 0 0 0.258819 0.965926i \(-0.416667\pi\)
−0.258819 + 0.965926i \(0.583333\pi\)
\(192\) 0 0
\(193\) 1.10876 0.0726721i 1.10876 0.0726721i 0.500000 0.866025i \(-0.333333\pi\)
0.608761 + 0.793353i \(0.291667\pi\)
\(194\) 0.534534 + 1.57469i 0.534534 + 1.57469i
\(195\) 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 −0.946930 0.321439i \(-0.895833\pi\)
0.946930 + 0.321439i \(0.104167\pi\)
\(200\) 0.0607275 + 0.226638i 0.0607275 + 0.226638i
\(201\) 0 0
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) −0.216773 0.375461i −0.216773 0.375461i
\(206\) 0 0
\(207\) 0 0
\(208\) 1.36603 + 0.366025i 1.36603 + 0.366025i
\(209\) 0 0
\(210\) 0 0
\(211\) 0 0 0.195090 0.980785i \(-0.437500\pi\)
−0.195090 + 0.980785i \(0.562500\pi\)
\(212\) 1.46593 + 1.12484i 1.46593 + 1.12484i
\(213\) 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
\(221\) 0.184592 1.40211i 0.184592 1.40211i
\(222\) 0 0
\(223\) 0 0 −0.923880 0.382683i \(-0.875000\pi\)
0.923880 + 0.382683i \(0.125000\pi\)
\(224\) 0 0
\(225\) 0.165911 + 0.165911i 0.165911 + 0.165911i
\(226\) −0.293353 + 0.257264i −0.293353 + 0.257264i
\(227\) 0 0 −0.0654031 0.997859i \(-0.520833\pi\)
0.0654031 + 0.997859i \(0.479167\pi\)
\(228\) 0 0
\(229\) −1.12484 + 1.46593i −1.12484 + 1.46593i −0.258819 + 0.965926i \(0.583333\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) −1.38268 + 0.923880i −1.38268 + 0.923880i
\(233\) 1.29335 + 1.47479i 1.29335 + 1.47479i 0.793353 + 0.608761i \(0.208333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(234\) 1.36603 0.366025i 1.36603 0.366025i
\(235\) 0 0
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(240\) 0 0
\(241\) −1.49144 + 0.735499i −1.49144 + 0.735499i −0.991445 0.130526i \(-0.958333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(242\) −0.965926 + 0.258819i −0.965926 + 0.258819i
\(243\) 0 0
\(244\) 1.63099 1.08979i 1.63099 1.08979i
\(245\) 0 0
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) 0 0
\(250\) −0.639387 + 0.560728i −0.639387 + 0.560728i
\(251\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 0 0
\(255\) 0 0
\(256\) 0.500000 0.866025i 0.500000 0.866025i
\(257\) 0.241181 + 1.83195i 0.241181 + 1.83195i 0.500000 + 0.866025i \(0.333333\pi\)
−0.258819 + 0.965926i \(0.583333\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) −0.306563 1.54120i −0.306563 1.54120i
\(261\) −0.735499 + 1.49144i −0.735499 + 1.49144i
\(262\) 0 0
\(263\) 0 0 0.130526 0.991445i \(-0.458333\pi\)
−0.130526 + 0.991445i \(0.541667\pi\)
\(264\) 0 0
\(265\) 0.400544 2.01367i 0.400544 2.01367i
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 0.735499 + 1.49144i 0.735499 + 1.49144i 0.866025 + 0.500000i \(0.166667\pi\)
−0.130526 + 0.991445i \(0.541667\pi\)
\(270\) 0 0
\(271\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(272\) −0.923880 0.382683i −0.923880 0.382683i
\(273\) 0 0
\(274\) −0.707107 + 1.70711i −0.707107 + 1.70711i
\(275\) 0 0
\(276\) 0 0
\(277\) −1.85747 0.630526i −1.85747 0.630526i −0.991445 0.130526i \(-0.958333\pi\)
−0.866025 0.500000i \(-0.833333\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) −0.707107 1.70711i −0.707107 1.70711i −0.707107 0.707107i \(-0.750000\pi\)
1.00000i \(-0.5\pi\)
\(282\) 0 0
\(283\) 0 0 0.997859 0.0654031i \(-0.0208333\pi\)
−0.997859 + 0.0654031i \(0.979167\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 1.00000i 1.00000i
\(289\) −0.258819 + 0.965926i −0.258819 + 0.965926i
\(290\) 1.60021 + 0.923880i 1.60021 + 0.923880i
\(291\) 0 0
\(292\) 0.108761 1.65938i 0.108761 1.65938i
\(293\) 0.541196 0.541196i 0.541196 0.541196i −0.382683 0.923880i \(-0.625000\pi\)
0.923880 + 0.382683i \(0.125000\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) −1.05217 + 0.357164i −1.05217 + 0.357164i
\(297\) 0 0
\(298\) −1.40211 0.184592i −1.40211 0.184592i
\(299\) 0 0
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) −1.88757 1.08979i −1.88757 1.08979i
\(306\) −0.991445 + 0.130526i −0.991445 + 0.130526i
\(307\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 0 0 0.321439 0.946930i \(-0.395833\pi\)
−0.321439 + 0.946930i \(0.604167\pi\)
\(312\) 0 0
\(313\) −0.630526 1.85747i −0.630526 1.85747i −0.500000 0.866025i \(-0.666667\pi\)
−0.130526 0.991445i \(-0.541667\pi\)
\(314\) 0.292893 + 0.707107i 0.292893 + 0.707107i
\(315\) 0 0
\(316\) 0 0
\(317\) 1.49144 + 0.735499i 1.49144 + 0.735499i 0.991445 0.130526i \(-0.0416667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) −1.10876 0.0726721i −1.10876 0.0726721i
\(321\) 0 0
\(322\) 0 0
\(323\) 0 0
\(324\) −0.500000 0.866025i −0.500000 0.866025i
\(325\) −0.263252 + 0.202000i −0.263252 + 0.202000i
\(326\) 0 0
\(327\) 0 0
\(328\) 0.382683 0.0761205i 0.382683 0.0761205i
\(329\) 0 0
\(330\) 0 0
\(331\) 0 0 −0.793353 0.608761i \(-0.791667\pi\)
0.793353 + 0.608761i \(0.208333\pi\)
\(332\) 0 0
\(333\) −0.732626 + 0.835400i −0.732626 + 0.835400i
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 0.324423 + 0.216773i 0.324423 + 0.216773i 0.707107 0.707107i \(-0.250000\pi\)
−0.382683 + 0.923880i \(0.625000\pi\)
\(338\) 0.130526 + 0.991445i 0.130526 + 0.991445i
\(339\) 0 0
\(340\) 0.0726721 + 1.10876i 0.0726721 + 1.10876i
\(341\) 0 0
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) 0 0
\(346\) 0.0255190 + 0.389345i 0.0255190 + 0.389345i
\(347\) 0 0 −0.751840 0.659346i \(-0.770833\pi\)
0.751840 + 0.659346i \(0.229167\pi\)
\(348\) 0 0
\(349\) 1.84776 0.765367i 1.84776 0.765367i 0.923880 0.382683i \(-0.125000\pi\)
0.923880 0.382683i \(-0.125000\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 1.78480 0.478235i 1.78480 0.478235i 0.793353 0.608761i \(-0.208333\pi\)
0.991445 + 0.130526i \(0.0416667\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) −2.00000 −2.00000
\(357\) 0 0
\(358\) 0 0
\(359\) 0 0 0.991445 0.130526i \(-0.0416667\pi\)
−0.991445 + 0.130526i \(0.958333\pi\)
\(360\) −0.996552 + 0.491445i −0.996552 + 0.491445i
\(361\) −0.965926 + 0.258819i −0.965926 + 0.258819i
\(362\) 1.29335 + 1.47479i 1.29335 + 1.47479i
\(363\) 0 0
\(364\) 0 0
\(365\) −1.70711 + 0.707107i −1.70711 + 0.707107i
\(366\) 0 0
\(367\) 0 0 −0.751840 0.659346i \(-0.770833\pi\)
0.751840 + 0.659346i \(0.229167\pi\)
\(368\) 0 0
\(369\) 0.293353 0.257264i 0.293353 0.257264i
\(370\) 0.873017 + 0.873017i 0.873017 + 0.873017i
\(371\) 0 0
\(372\) 0 0
\(373\) −0.662827 + 0.382683i −0.662827 + 0.382683i −0.793353 0.608761i \(-0.791667\pi\)
0.130526 + 0.991445i \(0.458333\pi\)
\(374\) 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 −0.793353 0.608761i \(-0.791667\pi\)
0.793353 + 0.608761i \(0.208333\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 1.08979 0.216773i 1.08979 0.216773i
\(387\) 0 0
\(388\) 0.735499 + 1.49144i 0.735499 + 1.49144i
\(389\) 1.12197 0.860919i 1.12197 0.860919i 0.130526 0.991445i \(-0.458333\pi\)
0.991445 + 0.130526i \(0.0416667\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0 0
\(393\) 0 0
\(394\) −0.389345 0.0255190i −0.389345 0.0255190i
\(395\) 0 0
\(396\) 0 0
\(397\) −0.349942 0.172572i −0.349942 0.172572i 0.258819 0.965926i \(-0.416667\pi\)
−0.608761 + 0.793353i \(0.708333\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0.0897902 + 0.216773i 0.0897902 + 0.216773i
\(401\) −0.357164 1.05217i −0.357164 1.05217i −0.965926 0.258819i \(-0.916667\pi\)
0.608761 0.793353i \(-0.291667\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0 0
\(405\) −0.617317 + 0.923880i −0.617317 + 0.923880i
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) −0.662827 0.382683i −0.662827 0.382683i 0.130526 0.991445i \(-0.458333\pi\)
−0.793353 + 0.608761i \(0.791667\pi\)
\(410\) −0.263926 0.343955i −0.263926 0.343955i
\(411\) 0 0
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) 0 0
\(416\) 1.40211 + 0.184592i 1.40211 + 0.184592i
\(417\) 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
\(424\) 1.60021 + 0.923880i 1.60021 + 0.923880i
\(425\) 0.203198 0.117317i 0.203198 0.117317i
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 0 0 0.997859 0.0654031i \(-0.0208333\pi\)
−0.997859 + 0.0654031i \(0.979167\pi\)
\(432\) 0 0
\(433\) −0.292893 0.707107i −0.292893 0.707107i 0.707107 0.707107i \(-0.250000\pi\)
−1.00000 \(\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 1.75928 + 0.867580i 1.75928 + 0.867580i
\(437\) 0 0
\(438\) 0 0
\(439\) 0 0 −0.997859 0.0654031i \(-0.979167\pi\)
0.997859 + 0.0654031i \(0.0208333\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 1.41421i 1.41421i
\(443\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(444\) 0 0
\(445\) 0.982890 + 1.99310i 0.982890 + 1.99310i
\(446\) 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 \(-0.250000\pi\)
\(450\) 0.186147 + 0.142836i 0.186147 + 0.142836i
\(451\) 0 0
\(452\) −0.257264 + 0.293353i −0.257264 + 0.293353i
\(453\) 0 0
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 0.184592 + 1.40211i 0.184592 + 1.40211i 0.793353 + 0.608761i \(0.208333\pi\)
−0.608761 + 0.793353i \(0.708333\pi\)
\(458\) −0.923880 + 1.60021i −0.923880 + 1.60021i
\(459\) 0 0
\(460\) 0 0
\(461\) 0 0 0.382683 0.923880i \(-0.375000\pi\)
−0.382683 + 0.923880i \(0.625000\pi\)
\(462\) 0 0
\(463\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(464\) −1.25026 + 1.09645i −1.25026 + 1.09645i
\(465\) 0 0
\(466\) 1.47479 + 1.29335i 1.47479 + 1.29335i
\(467\) 0 0 0.608761 0.793353i \(-0.291667\pi\)
−0.608761 + 0.793353i \(0.708333\pi\)
\(468\) 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 0.896873 0.442289i \(-0.145833\pi\)
−0.896873 + 0.442289i \(0.854167\pi\)
\(480\) 0 0
\(481\) −1.03609 1.18143i −1.03609 1.18143i
\(482\) −1.38268 + 0.923880i −1.38268 + 0.923880i
\(483\) 0 0
\(484\) −0.923880 + 0.382683i −0.923880 + 0.382683i
\(485\) 1.12484 1.46593i 1.12484 1.46593i
\(486\) 0 0
\(487\) 0 0 −0.0654031 0.997859i \(-0.520833\pi\)
0.0654031 + 0.997859i \(0.479167\pi\)
\(488\) 1.47479 1.29335i 1.47479 1.29335i
\(489\) 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
\(493\) 1.25026 + 1.09645i 1.25026 + 1.09645i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) 0 0 0.442289 0.896873i \(-0.354167\pi\)
−0.442289 + 0.896873i \(0.645833\pi\)
\(500\) −0.560728 + 0.639387i −0.560728 + 0.639387i
\(501\) 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
\(505\) 0 0
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) −0.923880 1.60021i −0.923880 1.60021i −0.793353 0.608761i \(-0.791667\pi\)
−0.130526 0.991445i \(-0.541667\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0.382683 0.923880i 0.382683 0.923880i
\(513\) 0 0
\(514\) 0.478235 + 1.78480i 0.478235 + 1.78480i
\(515\) 0 0
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) 0 0
\(520\) −0.505107 1.48800i −0.505107 1.48800i
\(521\) −0.389345 + 0.0255190i −0.389345 + 0.0255190i −0.258819 0.965926i \(-0.583333\pi\)
−0.130526 + 0.991445i \(0.541667\pi\)
\(522\) −0.534534 + 1.57469i −0.534534 + 1.57469i
\(523\) 0 0 0.258819 0.965926i \(-0.416667\pi\)
−0.258819 + 0.965926i \(0.583333\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) −0.608761 0.793353i −0.608761 0.793353i
\(530\) 0.134280 2.04872i 0.134280 2.04872i
\(531\) 0 0
\(532\) 0 0
\(533\) 0.306563 + 0.458804i 0.306563 + 0.458804i
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 0 0
\(538\) 0.923880 + 1.38268i 0.923880 + 1.38268i
\(539\) 0 0
\(540\) 0 0
\(541\) 0.0255190 0.389345i 0.0255190 0.389345i −0.965926 0.258819i \(-0.916667\pi\)
0.991445 0.130526i \(-0.0416667\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) −0.965926 0.258819i −0.965926 0.258819i
\(545\) 2.17958i 2.17958i
\(546\) 0 0
\(547\) 0 0 0.555570 0.831470i \(-0.312500\pi\)
−0.555570 + 0.831470i \(0.687500\pi\)
\(548\) −0.478235 + 1.78480i −0.478235 + 1.78480i
\(549\) 0.630526 1.85747i 0.630526 1.85747i
\(550\) 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
\(557\) 0.198092 + 0.739288i 0.198092 + 0.739288i 0.991445 + 0.130526i \(0.0416667\pi\)
−0.793353 + 0.608761i \(0.791667\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 0 0
\(561\) 0 0
\(562\) −0.923880 1.60021i −0.923880 1.60021i
\(563\) 0 0 0.793353 0.608761i \(-0.208333\pi\)
−0.793353 + 0.608761i \(0.791667\pi\)
\(564\) 0 0
\(565\) 0.418773 + 0.112210i 0.418773 + 0.112210i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 0.607206 + 0.465926i 0.607206 + 0.465926i 0.866025 0.500000i \(-0.166667\pi\)
−0.258819 + 0.965926i \(0.583333\pi\)
\(570\) 0 0
\(571\) 0 0 0.659346 0.751840i \(-0.270833\pi\)
−0.659346 + 0.751840i \(0.729167\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 0 0
\(576\) −0.130526 0.991445i −0.130526 0.991445i
\(577\) 0.707107 1.22474i 0.707107 1.22474i −0.258819 0.965926i \(-0.583333\pi\)
0.965926 0.258819i \(-0.0833333\pi\)
\(578\) −0.130526 + 0.991445i −0.130526 + 0.991445i
\(579\) 0 0
\(580\) 1.70711 + 0.707107i 1.70711 + 0.707107i
\(581\) 0 0
\(582\) 0 0
\(583\) 0 0
\(584\) −0.108761 1.65938i −0.108761 1.65938i
\(585\) −1.18143 1.03609i −1.18143 1.03609i
\(586\) 0.465926 0.607206i 0.465926 0.607206i
\(587\) 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
\(592\) −0.996552 + 0.491445i −0.996552 + 0.491445i
\(593\) 0 0 −0.130526 0.991445i \(-0.541667\pi\)
0.130526 + 0.991445i \(0.458333\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) −1.41421 −1.41421
\(597\) 0 0
\(598\) 0 0
\(599\) 0 0 0.965926 0.258819i \(-0.0833333\pi\)
−0.965926 + 0.258819i \(0.916667\pi\)
\(600\) 0 0
\(601\) −1.63099 + 1.08979i −1.63099 + 1.08979i −0.707107 + 0.707107i \(0.750000\pi\)
−0.923880 + 0.382683i \(0.875000\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 0.835400 + 0.732626i 0.835400 + 0.732626i
\(606\) 0 0
\(607\) 0 0 0.751840 0.659346i \(-0.229167\pi\)
−0.751840 + 0.659346i \(0.770833\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) −2.01367 0.834089i −2.01367 0.834089i
\(611\) 0 0
\(612\) −0.965926 + 0.258819i −0.965926 + 0.258819i
\(613\) −0.707107 + 1.22474i −0.707107 + 1.22474i 0.258819 + 0.965926i \(0.416667\pi\)
−0.965926 + 0.258819i \(0.916667\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −0.382683 1.92388i −0.382683 1.92388i −0.382683 0.923880i \(-0.625000\pi\)
1.00000i \(-0.5\pi\)
\(618\) 0 0
\(619\) 0 0 0.659346 0.751840i \(-0.270833\pi\)
−0.659346 + 0.751840i \(0.729167\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) 1.13939 + 0.305298i 1.13939 + 0.305298i
\(626\) −0.867580 1.75928i −0.867580 1.75928i
\(627\) 0 0
\(628\) 0.382683 + 0.662827i 0.382683 + 0.662827i
\(629\) 0.617317 + 0.923880i 0.617317 + 0.923880i
\(630\) 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
\(634\) 1.57469 + 0.534534i 1.57469 + 0.534534i
\(635\) 0 0
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) 0 0
\(640\) −1.10876 + 0.0726721i −1.10876 + 0.0726721i
\(641\) 0.630526 1.85747i 0.630526 1.85747i 0.130526 0.991445i \(-0.458333\pi\)
0.500000 0.866025i \(-0.333333\pi\)
\(642\) 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 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(648\) −0.608761 0.793353i −0.608761 0.793353i
\(649\) 0 0
\(650\) −0.234633 + 0.234633i −0.234633 + 0.234633i
\(651\) 0 0
\(652\) 0 0
\(653\) 1.05217 0.357164i 1.05217 0.357164i 0.258819 0.965926i \(-0.416667\pi\)
0.793353 + 0.608761i \(0.208333\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0.369474 0.125419i 0.369474 0.125419i
\(657\) −0.923880 1.38268i −0.923880 1.38268i
\(658\) 0 0
\(659\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(660\) 0 0
\(661\) −0.860919 1.12197i −0.860919 1.12197i −0.991445 0.130526i \(-0.958333\pi\)
0.130526 0.991445i \(-0.458333\pi\)
\(662\) 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 0.382683 0.923880i \(-0.375000\pi\)
1.00000i \(0.5\pi\)
\(674\) 0.349942 + 0.172572i 0.349942 + 0.172572i
\(675\) 0 0
\(676\) 0.258819 + 0.965926i 0.258819 + 0.965926i
\(677\) −1.65938 0.108761i −1.65938 0.108761i −0.793353 0.608761i \(-0.791667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0.216773 + 1.08979i 0.216773 + 1.08979i
\(681\) 0 0
\(682\) 0 0
\(683\) 0 0 −0.442289 0.896873i \(-0.645833\pi\)
0.442289 + 0.896873i \(0.354167\pi\)
\(684\) 0 0
\(685\) 2.01367 0.400544i 2.01367 0.400544i
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) −0.341081 + 2.59077i −0.341081 + 2.59077i
\(690\) 0 0
\(691\) 0 0 0.442289 0.896873i \(-0.354167\pi\)
−0.442289 + 0.896873i \(0.645833\pi\)
\(692\) 0.0761205 + 0.382683i 0.0761205 + 0.382683i
\(693\) 0 0
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) −0.172572 0.349942i −0.172572 0.349942i
\(698\) 1.73205 1.00000i 1.73205 1.00000i
\(699\) 0 0
\(700\) 0 0
\(701\) 0.541196 + 0.541196i 0.541196 + 0.541196i 0.923880 0.382683i \(-0.125000\pi\)
−0.382683 + 0.923880i \(0.625000\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) 0 0
\(705\) 0 0
\(706\) 1.70711 0.707107i 1.70711 0.707107i
\(707\) 0 0
\(708\) 0 0
\(709\) −1.09645 1.25026i −1.09645 1.25026i −0.965926 0.258819i \(-0.916667\pi\)
−0.130526 0.991445i \(-0.541667\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) −1.98289 + 0.261052i −1.98289 + 0.261052i
\(713\) 0 0
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 0 0 −0.659346 0.751840i \(-0.729167\pi\)
0.659346 + 0.751840i \(0.270833\pi\)
\(720\) −0.923880 + 0.617317i −0.923880 + 0.617317i
\(721\) 0 0
\(722\) −0.923880 + 0.382683i −0.923880 + 0.382683i
\(723\) 0 0
\(724\) 1.47479 + 1.29335i 1.47479 + 1.29335i
\(725\) −0.0255190 0.389345i −0.0255190 0.389345i
\(726\) 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
\(730\) −1.60021 + 0.923880i −1.60021 + 0.923880i
\(731\) 0 0
\(732\) 0 0
\(733\) 0.261052 + 1.98289i 0.261052 + 1.98289i 0.130526 + 0.991445i \(0.458333\pi\)
0.130526 + 0.991445i \(0.458333\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 0 0
\(738\) 0.257264 0.293353i 0.257264 0.293353i
\(739\) 0 0 0.130526 0.991445i \(-0.458333\pi\)
−0.130526 + 0.991445i \(0.541667\pi\)
\(740\) 0.979500 + 0.751597i 0.979500 + 0.751597i
\(741\) 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
\(745\) 0.695008 + 1.40934i 0.695008 + 1.40934i
\(746\) −0.607206 + 0.465926i −0.607206 + 0.465926i
\(747\) 0 0
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) 0 0 −0.997859 0.0654031i \(-0.979167\pi\)
0.997859 + 0.0654031i \(0.0208333\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) −2.10922 1.04015i −2.10922 1.04015i
\(755\) 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
\(761\) −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
\(765\) 0.732626 + 0.835400i 0.732626 + 0.835400i
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 1.05217 0.357164i 1.05217 0.357164i
\(773\) −1.83195 0.241181i −1.83195 0.241181i −0.866025 0.500000i \(-0.833333\pi\)
−0.965926 + 0.258819i \(0.916667\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0.923880 + 1.38268i 0.923880 + 1.38268i
\(777\) 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
\(785\) 0.472474 0.707107i 0.472474 0.707107i
\(786\) 0 0
\(787\) 0 0 0.321439 0.946930i \(-0.395833\pi\)
−0.321439 + 0.946930i \(0.604167\pi\)
\(788\) −0.389345 + 0.0255190i −0.389345 + 0.0255190i
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 2.48800 + 1.22694i 2.48800 + 1.22694i
\(794\) −0.369474 0.125419i −0.369474 0.125419i
\(795\) 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
\(800\) 0.117317 + 0.203198i 0.117317 + 0.203198i
\(801\) −1.58671 + 1.21752i −1.58671 + 1.21752i
\(802\) −0.491445 0.996552i −0.491445 0.996552i
\(803\) 0 0
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 1.09645 1.25026i 1.09645 1.25026i 0.130526 0.991445i \(-0.458333\pi\)
0.965926 0.258819i \(-0.0833333\pi\)
\(810\) −0.491445 + 0.996552i −0.491445 + 0.996552i
\(811\) 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
\(818\) −0.707107 0.292893i −0.707107 0.292893i
\(819\) 0 0
\(820\) −0.306563 0.306563i −0.306563 0.306563i
\(821\) −1.47479 + 1.29335i −1.47479 + 1.29335i −0.608761 + 0.793353i \(0.708333\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(822\) 0 0
\(823\) 0 0 −0.751840 0.659346i \(-0.770833\pi\)
0.751840 + 0.659346i \(0.229167\pi\)
\(824\) 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
\(829\) 0.739288 0.198092i 0.739288 0.198092i 0.130526 0.991445i \(-0.458333\pi\)
0.608761 + 0.793353i \(0.291667\pi\)
\(830\) 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
\(842\) 0.465926 0.607206i 0.465926 0.607206i
\(843\) 0 0
\(844\) 0 0
\(845\) 0.835400 0.732626i 0.835400 0.732626i
\(846\) 0 0
\(847\) 0 0
\(848\) 1.70711 + 0.707107i 1.70711 + 0.707107i
\(849\) 0 0
\(850\) 0.186147 0.142836i 0.186147 0.142836i
\(851\) 0 0
\(852\) 0 0
\(853\) −1.38268 0.923880i −1.38268 0.923880i −0.382683 0.923880i \(-0.625000\pi\)
−1.00000 \(\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 1.09645 1.25026i 1.09645 1.25026i 0.130526 0.991445i \(-0.458333\pi\)
0.965926 0.258819i \(-0.0833333\pi\)
\(858\) 0 0
\(859\) 0 0 −0.793353 0.608761i \(-0.791667\pi\)
0.793353 + 0.608761i \(0.208333\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 0 0 −0.965926 0.258819i \(-0.916667\pi\)
0.965926 + 0.258819i \(0.0833333\pi\)
\(864\) 0 0
\(865\) 0.343955 0.263926i 0.343955 0.263926i
\(866\) −0.382683 0.662827i −0.382683 0.662827i
\(867\) 0 0
\(868\) 0 0
\(869\) 0 0
\(870\) 0 0
\(871\) 0 0
\(872\) 1.85747 + 0.630526i 1.85747 + 0.630526i
\(873\) 1.49144 + 0.735499i 1.49144 + 0.735499i
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 0.125419 + 0.369474i 0.125419 + 0.369474i 0.991445 0.130526i \(-0.0416667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 0.216773 0.324423i 0.216773 0.324423i −0.707107 0.707107i \(-0.750000\pi\)
0.923880 + 0.382683i \(0.125000\pi\)
\(882\) 0 0
\(883\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(884\) −0.184592 1.40211i −0.184592 1.40211i
\(885\) 0 0
\(886\) 0 0
\(887\) 0 0 0.0654031 0.997859i \(-0.479167\pi\)
−0.0654031 + 0.997859i \(0.520833\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 1.23463 + 1.84776i 1.23463 + 1.84776i
\(891\) 0 0
\(892\) 0 0
\(893\) 0 0
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 0 0
\(898\) 0.108761 1.65938i 0.108761 1.65938i
\(899\) 0 0
\(900\) 0.203198 + 0.117317i 0.203198 + 0.117317i
\(901\) 0.478235 1.78480i 0.478235 1.78480i
\(902\) 0 0
\(903\) 0 0
\(904\) −0.216773 + 0.324423i −0.216773 + 0.324423i
\(905\) 0.564117 2.10531i 0.564117 2.10531i
\(906\) 0 0
\(907\) 0 0 0.997859 0.0654031i \(-0.0208333\pi\)
−0.997859 + 0.0654031i \(0.979167\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 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
\(914\) 0.366025 + 1.36603i 0.366025 + 1.36603i
\(915\) 0 0
\(916\) −0.707107 + 1.70711i −0.707107 + 1.70711i
\(917\) 0 0
\(918\) 0 0
\(919\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) 0.0508621 0.255701i 0.0508621 0.255701i
\(926\) 0 0
\(927\) 0 0
\(928\) −1.09645 + 1.25026i −1.09645 + 1.25026i
\(929\) −0.867580 + 1.75928i −0.867580 + 1.75928i −0.258819 + 0.965926i \(0.583333\pi\)
−0.608761 + 0.793353i \(0.708333\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 1.63099 + 1.08979i 1.63099 + 1.08979i
\(933\) 0 0
\(934\) 0 0
\(935\) 0 0
\(936\) 1.22474 0.707107i 1.22474 0.707107i
\(937\) −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
\(941\) 0.0255190 + 0.389345i 0.0255190 + 0.389345i 0.991445 + 0.130526i \(0.0416667\pi\)
−0.965926 + 0.258819i \(0.916667\pi\)
\(942\) 0 0
\(943\) 0 0
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 0 0 −0.659346 0.751840i \(-0.729167\pi\)
0.659346 + 0.751840i \(0.270833\pi\)
\(948\) 0 0
\(949\) 2.10922 1.04015i 2.10922 1.04015i
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 0.765367 0.765367 0.382683 0.923880i \(-0.375000\pi\)
0.382683 + 0.923880i \(0.375000\pi\)
\(954\) 1.83195 0.241181i 1.83195 0.241181i
\(955\) 0 0
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) −0.608761 + 0.793353i −0.608761 + 0.793353i
\(962\) −1.18143 1.03609i −1.18143 1.03609i
\(963\) 0 0
\(964\) −1.25026 + 1.09645i −1.25026 + 1.09645i
\(965\) −0.873017 0.873017i −0.873017 0.873017i
\(966\) 0 0
\(967\) 0 0 −0.923880 0.382683i \(-0.875000\pi\)
0.923880 + 0.382683i \(0.125000\pi\)
\(968\) −0.866025 + 0.500000i −0.866025 + 0.500000i
\(969\) 0 0
\(970\) 0.923880 1.60021i 0.923880 1.60021i
\(971\) 0 0 −0.130526 0.991445i \(-0.541667\pi\)
0.130526 + 0.991445i \(0.458333\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) 0 0
\(976\) 1.29335 1.47479i 1.29335 1.47479i
\(977\) 0.241181 1.83195i 0.241181 1.83195i −0.258819 0.965926i \(-0.583333\pi\)
0.500000 0.866025i \(-0.333333\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 0 0
\(981\) 1.92388 0.382683i 1.92388 0.382683i
\(982\) 0 0
\(983\) 0 0 −0.442289 0.896873i \(-0.645833\pi\)
0.442289 + 0.896873i \(0.354167\pi\)
\(984\) 0 0
\(985\) 0.216773 + 0.375461i 0.216773 + 0.375461i
\(986\) 1.38268 + 0.923880i 1.38268 + 0.923880i
\(987\) 0 0
\(988\) 0 0
\(989\) 0 0
\(990\) 0 0
\(991\) 0 0 −0.946930 0.321439i \(-0.895833\pi\)
0.946930 + 0.321439i \(0.104167\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) −1.95737 + 0.128293i −1.95737 + 0.128293i −0.991445 0.130526i \(-0.958333\pi\)
−0.965926 + 0.258819i \(0.916667\pi\)
\(998\) 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.ce.d.619.1 16
4.3 odd 2 CM 3332.1.ce.d.619.1 16
7.2 even 3 inner 3332.1.ce.d.1979.1 16
7.3 odd 6 3332.1.bn.a.3135.1 yes 8
7.4 even 3 3332.1.bn.c.3135.1 yes 8
7.5 odd 6 3332.1.ce.b.1979.1 16
7.6 odd 2 3332.1.ce.b.619.1 16
17.5 odd 16 3332.1.ce.b.3167.1 16
28.3 even 6 3332.1.bn.a.3135.1 yes 8
28.11 odd 6 3332.1.bn.c.3135.1 yes 8
28.19 even 6 3332.1.ce.b.1979.1 16
28.23 odd 6 inner 3332.1.ce.d.1979.1 16
28.27 even 2 3332.1.ce.b.619.1 16
68.39 even 16 3332.1.ce.b.3167.1 16
119.5 even 48 inner 3332.1.ce.d.1195.1 16
119.39 odd 48 3332.1.bn.a.2351.1 8
119.73 even 48 3332.1.bn.c.2351.1 yes 8
119.90 even 16 inner 3332.1.ce.d.3167.1 16
119.107 odd 48 3332.1.ce.b.1195.1 16
476.39 even 48 3332.1.bn.a.2351.1 8
476.107 even 48 3332.1.ce.b.1195.1 16
476.243 odd 48 inner 3332.1.ce.d.1195.1 16
476.311 odd 48 3332.1.bn.c.2351.1 yes 8
476.447 odd 16 inner 3332.1.ce.d.3167.1 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
3332.1.bn.a.2351.1 8 119.39 odd 48
3332.1.bn.a.2351.1 8 476.39 even 48
3332.1.bn.a.3135.1 yes 8 7.3 odd 6
3332.1.bn.a.3135.1 yes 8 28.3 even 6
3332.1.bn.c.2351.1 yes 8 119.73 even 48
3332.1.bn.c.2351.1 yes 8 476.311 odd 48
3332.1.bn.c.3135.1 yes 8 7.4 even 3
3332.1.bn.c.3135.1 yes 8 28.11 odd 6
3332.1.ce.b.619.1 16 7.6 odd 2
3332.1.ce.b.619.1 16 28.27 even 2
3332.1.ce.b.1195.1 16 119.107 odd 48
3332.1.ce.b.1195.1 16 476.107 even 48
3332.1.ce.b.1979.1 16 7.5 odd 6
3332.1.ce.b.1979.1 16 28.19 even 6
3332.1.ce.b.3167.1 16 17.5 odd 16
3332.1.ce.b.3167.1 16 68.39 even 16
3332.1.ce.d.619.1 16 1.1 even 1 trivial
3332.1.ce.d.619.1 16 4.3 odd 2 CM
3332.1.ce.d.1195.1 16 119.5 even 48 inner
3332.1.ce.d.1195.1 16 476.243 odd 48 inner
3332.1.ce.d.1979.1 16 7.2 even 3 inner
3332.1.ce.d.1979.1 16 28.23 odd 6 inner
3332.1.ce.d.3167.1 16 119.90 even 16 inner
3332.1.ce.d.3167.1 16 476.447 odd 16 inner