Properties

Label 3332.1.ce.d.227.1
Level $3332$
Weight $1$
Character 3332.227
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 227.1
Root \(0.130526 + 0.991445i\) of defining polynomial
Character \(\chi\) \(=\) 3332.227
Dual form 3332.1.ce.d.411.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-0.130526 - 0.991445i) q^{2} +(-0.965926 + 0.258819i) q^{4} +(-0.128293 + 1.95737i) q^{5} +(0.382683 + 0.923880i) q^{8} +(0.793353 + 0.608761i) q^{9} +O(q^{10})\) \(q+(-0.130526 - 0.991445i) q^{2} +(-0.965926 + 0.258819i) q^{4} +(-0.128293 + 1.95737i) q^{5} +(0.382683 + 0.923880i) q^{8} +(0.793353 + 0.608761i) q^{9} +(1.95737 - 0.128293i) q^{10} +(1.00000 + 1.00000i) q^{13} +(0.866025 - 0.500000i) q^{16} +(-0.793353 + 0.608761i) q^{17} +(0.500000 - 0.866025i) q^{18} +(-0.382683 - 1.92388i) q^{20} +(-2.82340 - 0.371707i) q^{25} +(0.860919 - 1.12197i) q^{26} +(0.324423 + 0.216773i) q^{29} +(-0.608761 - 0.793353i) q^{32} +(0.707107 + 0.707107i) q^{34} +(-0.923880 - 0.382683i) q^{36} +(-1.29335 - 1.47479i) q^{37} +(-1.85747 + 0.630526i) q^{40} +(-0.923880 + 0.617317i) q^{41} +(-1.29335 + 1.47479i) q^{45} +2.84776i q^{50} +(-1.22474 - 0.707107i) q^{52} +(0.607206 - 0.465926i) q^{53} +(0.172572 - 0.349942i) q^{58} +(0.735499 - 1.49144i) q^{61} +(-0.707107 + 0.707107i) q^{64} +(-2.08566 + 1.82908i) q^{65} +(0.608761 - 0.793353i) q^{68} +(-0.258819 + 0.965926i) q^{72} +(-0.349942 + 0.172572i) q^{73} +(-1.29335 + 1.47479i) q^{74} +(0.867580 + 1.75928i) q^{80} +(0.258819 + 0.965926i) q^{81} +(0.732626 + 0.835400i) q^{82} +(-1.08979 - 1.63099i) q^{85} +(1.93185 + 0.517638i) q^{89} +(1.63099 + 1.08979i) q^{90} +(0.216773 - 0.324423i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q+O(q^{10}) \) 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 \(\left(\mathbb{Z}/3332\mathbb{Z}\right)^\times\).

\(n\) \(785\) \(885\) \(1667\)
\(\chi(n)\) \(e\left(\frac{15}{16}\right)\) \(e\left(\frac{1}{6}\right)\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.130526 0.991445i −0.130526 0.991445i
\(3\) 0 0 −0.946930 0.321439i \(-0.895833\pi\)
0.946930 + 0.321439i \(0.104167\pi\)
\(4\) −0.965926 + 0.258819i −0.965926 + 0.258819i
\(5\) −0.128293 + 1.95737i −0.128293 + 1.95737i 0.130526 + 0.991445i \(0.458333\pi\)
−0.258819 + 0.965926i \(0.583333\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) 0.382683 + 0.923880i 0.382683 + 0.923880i
\(9\) 0.793353 + 0.608761i 0.793353 + 0.608761i
\(10\) 1.95737 0.128293i 1.95737 0.128293i
\(11\) 0 0 −0.751840 0.659346i \(-0.770833\pi\)
0.751840 + 0.659346i \(0.229167\pi\)
\(12\) 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
\(17\) −0.793353 + 0.608761i −0.793353 + 0.608761i
\(18\) 0.500000 0.866025i 0.500000 0.866025i
\(19\) 0 0 0.991445 0.130526i \(-0.0416667\pi\)
−0.991445 + 0.130526i \(0.958333\pi\)
\(20\) −0.382683 1.92388i −0.382683 1.92388i
\(21\) 0 0
\(22\) 0 0
\(23\) 0 0 −0.321439 0.946930i \(-0.604167\pi\)
0.321439 + 0.946930i \(0.395833\pi\)
\(24\) 0 0
\(25\) −2.82340 0.371707i −2.82340 0.371707i
\(26\) 0.860919 1.12197i 0.860919 1.12197i
\(27\) 0 0
\(28\) 0 0
\(29\) 0.324423 + 0.216773i 0.324423 + 0.216773i 0.707107 0.707107i \(-0.250000\pi\)
−0.382683 + 0.923880i \(0.625000\pi\)
\(30\) 0 0
\(31\) 0 0 0.321439 0.946930i \(-0.395833\pi\)
−0.321439 + 0.946930i \(0.604167\pi\)
\(32\) −0.608761 0.793353i −0.608761 0.793353i
\(33\) 0 0
\(34\) 0.707107 + 0.707107i 0.707107 + 0.707107i
\(35\) 0 0
\(36\) −0.923880 0.382683i −0.923880 0.382683i
\(37\) −1.29335 1.47479i −1.29335 1.47479i −0.793353 0.608761i \(-0.791667\pi\)
−0.500000 0.866025i \(-0.666667\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) −1.85747 + 0.630526i −1.85747 + 0.630526i
\(41\) −0.923880 + 0.617317i −0.923880 + 0.617317i −0.923880 0.382683i \(-0.875000\pi\)
1.00000i \(0.5\pi\)
\(42\) 0 0
\(43\) 0 0 0.923880 0.382683i \(-0.125000\pi\)
−0.923880 + 0.382683i \(0.875000\pi\)
\(44\) 0 0
\(45\) −1.29335 + 1.47479i −1.29335 + 1.47479i
\(46\) 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
\(50\) 2.84776i 2.84776i
\(51\) 0 0
\(52\) −1.22474 0.707107i −1.22474 0.707107i
\(53\) 0.607206 0.465926i 0.607206 0.465926i −0.258819 0.965926i \(-0.583333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 0 0
\(58\) 0.172572 0.349942i 0.172572 0.349942i
\(59\) 0 0 0.130526 0.991445i \(-0.458333\pi\)
−0.130526 + 0.991445i \(0.541667\pi\)
\(60\) 0 0
\(61\) 0.735499 1.49144i 0.735499 1.49144i −0.130526 0.991445i \(-0.541667\pi\)
0.866025 0.500000i \(-0.166667\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) −0.707107 + 0.707107i −0.707107 + 0.707107i
\(65\) −2.08566 + 1.82908i −2.08566 + 1.82908i
\(66\) 0 0
\(67\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(68\) 0.608761 0.793353i 0.608761 0.793353i
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 0.980785 0.195090i \(-0.0625000\pi\)
−0.980785 + 0.195090i \(0.937500\pi\)
\(72\) −0.258819 + 0.965926i −0.258819 + 0.965926i
\(73\) −0.349942 + 0.172572i −0.349942 + 0.172572i −0.608761 0.793353i \(-0.708333\pi\)
0.258819 + 0.965926i \(0.416667\pi\)
\(74\) −1.29335 + 1.47479i −1.29335 + 1.47479i
\(75\) 0 0
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 0 0 0.946930 0.321439i \(-0.104167\pi\)
−0.946930 + 0.321439i \(0.895833\pi\)
\(80\) 0.867580 + 1.75928i 0.867580 + 1.75928i
\(81\) 0.258819 + 0.965926i 0.258819 + 0.965926i
\(82\) 0.732626 + 0.835400i 0.732626 + 0.835400i
\(83\) 0 0 −0.923880 0.382683i \(-0.875000\pi\)
0.923880 + 0.382683i \(0.125000\pi\)
\(84\) 0 0
\(85\) −1.08979 1.63099i −1.08979 1.63099i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 1.93185 + 0.517638i 1.93185 + 0.517638i 0.965926 + 0.258819i \(0.0833333\pi\)
0.965926 + 0.258819i \(0.0833333\pi\)
\(90\) 1.63099 + 1.08979i 1.63099 + 1.08979i
\(91\) 0 0
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 0.216773 0.324423i 0.216773 0.324423i −0.707107 0.707107i \(-0.750000\pi\)
0.923880 + 0.382683i \(0.125000\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 2.82340 0.371707i 2.82340 0.371707i
\(101\) 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 \(0.833333\pi\)
\(104\) −0.541196 + 1.30656i −0.541196 + 1.30656i
\(105\) 0 0
\(106\) −0.541196 0.541196i −0.541196 0.541196i
\(107\) 0 0 −0.997859 0.0654031i \(-0.979167\pi\)
0.997859 + 0.0654031i \(0.0208333\pi\)
\(108\) 0 0
\(109\) 1.65938 0.108761i 1.65938 0.108761i 0.793353 0.608761i \(-0.208333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) −0.216773 + 1.08979i −0.216773 + 1.08979i 0.707107 + 0.707107i \(0.250000\pi\)
−0.923880 + 0.382683i \(0.875000\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) −0.369474 0.125419i −0.369474 0.125419i
\(117\) 0.184592 + 1.40211i 0.184592 + 1.40211i
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) 0.130526 + 0.991445i 0.130526 + 0.991445i
\(122\) −1.57469 0.534534i −1.57469 0.534534i
\(123\) 0 0
\(124\) 0 0
\(125\) 0.707107 3.55487i 0.707107 3.55487i
\(126\) 0 0
\(127\) 0 0 −0.382683 0.923880i \(-0.625000\pi\)
0.382683 + 0.923880i \(0.375000\pi\)
\(128\) 0.793353 + 0.608761i 0.793353 + 0.608761i
\(129\) 0 0
\(130\) 2.08566 + 1.82908i 2.08566 + 1.82908i
\(131\) 0 0 −0.997859 0.0654031i \(-0.979167\pi\)
0.997859 + 0.0654031i \(0.0208333\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) 0 0
\(136\) −0.866025 0.500000i −0.866025 0.500000i
\(137\) −0.382683 + 0.662827i −0.382683 + 0.662827i −0.991445 0.130526i \(-0.958333\pi\)
0.608761 + 0.793353i \(0.291667\pi\)
\(138\) 0 0
\(139\) 0 0 −0.195090 0.980785i \(-0.562500\pi\)
0.195090 + 0.980785i \(0.437500\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 0 0
\(144\) 0.991445 + 0.130526i 0.991445 + 0.130526i
\(145\) −0.465926 + 0.607206i −0.465926 + 0.607206i
\(146\) 0.216773 + 0.324423i 0.216773 + 0.324423i
\(147\) 0 0
\(148\) 1.63099 + 1.08979i 1.63099 + 1.08979i
\(149\) −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 −0.608761 0.793353i \(-0.708333\pi\)
0.608761 + 0.793353i \(0.291667\pi\)
\(152\) 0 0
\(153\) −1.00000 −1.00000
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 0.478235 + 1.78480i 0.478235 + 1.78480i 0.608761 + 0.793353i \(0.291667\pi\)
−0.130526 + 0.991445i \(0.541667\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 1.63099 1.08979i 1.63099 1.08979i
\(161\) 0 0
\(162\) 0.923880 0.382683i 0.923880 0.382683i
\(163\) 0 0 −0.896873 0.442289i \(-0.854167\pi\)
0.896873 + 0.442289i \(0.145833\pi\)
\(164\) 0.732626 0.835400i 0.732626 0.835400i
\(165\) 0 0
\(166\) 0 0
\(167\) 0 0 0.980785 0.195090i \(-0.0625000\pi\)
−0.980785 + 0.195090i \(0.937500\pi\)
\(168\) 0 0
\(169\) 1.00000i 1.00000i
\(170\) −1.47479 + 1.29335i −1.47479 + 1.29335i
\(171\) 0 0
\(172\) 0 0
\(173\) −0.835400 + 0.732626i −0.835400 + 0.732626i −0.965926 0.258819i \(-0.916667\pi\)
0.130526 + 0.991445i \(0.458333\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 0 0
\(178\) 0.261052 1.98289i 0.261052 1.98289i
\(179\) 0 0 0.130526 0.991445i \(-0.458333\pi\)
−0.130526 + 0.991445i \(0.541667\pi\)
\(180\) 0.867580 1.75928i 0.867580 1.75928i
\(181\) −1.63099 0.324423i −1.63099 0.324423i −0.707107 0.707107i \(-0.750000\pi\)
−0.923880 + 0.382683i \(0.875000\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 3.05263 2.34237i 3.05263 2.34237i
\(186\) 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
\(193\) 1.29335 1.47479i 1.29335 1.47479i 0.500000 0.866025i \(-0.333333\pi\)
0.793353 0.608761i \(-0.208333\pi\)
\(194\) −0.349942 0.172572i −0.349942 0.172572i
\(195\) 0 0
\(196\) 0 0
\(197\) 0.923880 0.617317i 0.923880 0.617317i 1.00000i \(-0.5\pi\)
0.923880 + 0.382683i \(0.125000\pi\)
\(198\) 0 0
\(199\) 0 0 −0.442289 0.896873i \(-0.645833\pi\)
0.442289 + 0.896873i \(0.354167\pi\)
\(200\) −0.737054 2.75072i −0.737054 2.75072i
\(201\) 0 0
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) −1.08979 1.88757i −1.08979 1.88757i
\(206\) 0 0
\(207\) 0 0
\(208\) 1.36603 + 0.366025i 1.36603 + 0.366025i
\(209\) 0 0
\(210\) 0 0
\(211\) 0 0 −0.555570 0.831470i \(-0.687500\pi\)
0.555570 + 0.831470i \(0.312500\pi\)
\(212\) −0.465926 + 0.607206i −0.465926 + 0.607206i
\(213\) 0 0
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 0 0
\(218\) −0.324423 1.63099i −0.324423 1.63099i
\(219\) 0 0
\(220\) 0 0
\(221\) −1.40211 0.184592i −1.40211 0.184592i
\(222\) 0 0
\(223\) 0 0 0.382683 0.923880i \(-0.375000\pi\)
−0.382683 + 0.923880i \(0.625000\pi\)
\(224\) 0 0
\(225\) −2.01367 2.01367i −2.01367 2.01367i
\(226\) 1.10876 + 0.0726721i 1.10876 + 0.0726721i
\(227\) 0 0 −0.751840 0.659346i \(-0.770833\pi\)
0.751840 + 0.659346i \(0.229167\pi\)
\(228\) 0 0
\(229\) −0.607206 0.465926i −0.607206 0.465926i 0.258819 0.965926i \(-0.416667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) −0.0761205 + 0.382683i −0.0761205 + 0.382683i
\(233\) −0.108761 + 1.65938i −0.108761 + 1.65938i 0.500000 + 0.866025i \(0.333333\pi\)
−0.608761 + 0.793353i \(0.708333\pi\)
\(234\) 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
\(241\) −0.369474 0.125419i −0.369474 0.125419i 0.130526 0.991445i \(-0.458333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(242\) 0.965926 0.258819i 0.965926 0.258819i
\(243\) 0 0
\(244\) −0.324423 + 1.63099i −0.324423 + 1.63099i
\(245\) 0 0
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) 0 0
\(250\) −3.61675 0.237054i −3.61675 0.237054i
\(251\) 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
\(257\) 0.758819 0.0999004i 0.758819 0.0999004i 0.258819 0.965926i \(-0.416667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 1.54120 2.30656i 1.54120 2.30656i
\(261\) 0.125419 + 0.369474i 0.125419 + 0.369474i
\(262\) 0 0
\(263\) 0 0 −0.991445 0.130526i \(-0.958333\pi\)
0.991445 + 0.130526i \(0.0416667\pi\)
\(264\) 0 0
\(265\) 0.834089 + 1.24830i 0.834089 + 1.24830i
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) −0.125419 + 0.369474i −0.125419 + 0.369474i −0.991445 0.130526i \(-0.958333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(270\) 0 0
\(271\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(272\) −0.382683 + 0.923880i −0.382683 + 0.923880i
\(273\) 0 0
\(274\) 0.707107 + 0.292893i 0.707107 + 0.292893i
\(275\) 0 0
\(276\) 0 0
\(277\) −0.735499 1.49144i −0.735499 1.49144i −0.866025 0.500000i \(-0.833333\pi\)
0.130526 0.991445i \(-0.458333\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 0.707107 0.292893i 0.707107 0.292893i 1.00000i \(-0.5\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(282\) 0 0
\(283\) 0 0 0.659346 0.751840i \(-0.270833\pi\)
−0.659346 + 0.751840i \(0.729167\pi\)
\(284\) 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
\(290\) 0.662827 + 0.382683i 0.662827 + 0.382683i
\(291\) 0 0
\(292\) 0.293353 0.257264i 0.293353 0.257264i
\(293\) 1.30656 1.30656i 1.30656 1.30656i 0.382683 0.923880i \(-0.375000\pi\)
0.923880 0.382683i \(-0.125000\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0.867580 1.75928i 0.867580 1.75928i
\(297\) 0 0
\(298\) −0.184592 + 1.40211i −0.184592 + 1.40211i
\(299\) 0 0
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 2.82495 + 1.63099i 2.82495 + 1.63099i
\(306\) 0.130526 + 0.991445i 0.130526 + 0.991445i
\(307\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 0 0 0.896873 0.442289i \(-0.145833\pi\)
−0.896873 + 0.442289i \(0.854167\pi\)
\(312\) 0 0
\(313\) −1.49144 0.735499i −1.49144 0.735499i −0.500000 0.866025i \(-0.666667\pi\)
−0.991445 + 0.130526i \(0.958333\pi\)
\(314\) 1.70711 0.707107i 1.70711 0.707107i
\(315\) 0 0
\(316\) 0 0
\(317\) 0.369474 0.125419i 0.369474 0.125419i −0.130526 0.991445i \(-0.541667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) −1.29335 1.47479i −1.29335 1.47479i
\(321\) 0 0
\(322\) 0 0
\(323\) 0 0
\(324\) −0.500000 0.866025i −0.500000 0.866025i
\(325\) −2.45169 3.19510i −2.45169 3.19510i
\(326\) 0 0
\(327\) 0 0
\(328\) −0.923880 0.617317i −0.923880 0.617317i
\(329\) 0 0
\(330\) 0 0
\(331\) 0 0 0.608761 0.793353i \(-0.291667\pi\)
−0.608761 + 0.793353i \(0.708333\pi\)
\(332\) 0 0
\(333\) −0.128293 1.95737i −0.128293 1.95737i
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 0.216773 + 1.08979i 0.216773 + 1.08979i 0.923880 + 0.382683i \(0.125000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(338\) 0.991445 0.130526i 0.991445 0.130526i
\(339\) 0 0
\(340\) 1.47479 + 1.29335i 1.47479 + 1.29335i
\(341\) 0 0
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) 0 0
\(346\) 0.835400 + 0.732626i 0.835400 + 0.732626i
\(347\) 0 0 0.997859 0.0654031i \(-0.0208333\pi\)
−0.997859 + 0.0654031i \(0.979167\pi\)
\(348\) 0 0
\(349\) 0.765367 + 1.84776i 0.765367 + 1.84776i 0.382683 + 0.923880i \(0.375000\pi\)
0.382683 + 0.923880i \(0.375000\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) −0.739288 + 0.198092i −0.739288 + 0.198092i −0.608761 0.793353i \(-0.708333\pi\)
−0.130526 + 0.991445i \(0.541667\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) −2.00000 −2.00000
\(357\) 0 0
\(358\) 0 0
\(359\) 0 0 −0.130526 0.991445i \(-0.541667\pi\)
0.130526 + 0.991445i \(0.458333\pi\)
\(360\) −1.85747 0.630526i −1.85747 0.630526i
\(361\) 0.965926 0.258819i 0.965926 0.258819i
\(362\) −0.108761 + 1.65938i −0.108761 + 1.65938i
\(363\) 0 0
\(364\) 0 0
\(365\) −0.292893 0.707107i −0.292893 0.707107i
\(366\) 0 0
\(367\) 0 0 0.997859 0.0654031i \(-0.0208333\pi\)
−0.997859 + 0.0654031i \(0.979167\pi\)
\(368\) 0 0
\(369\) −1.10876 0.0726721i −1.10876 0.0726721i
\(370\) −2.72078 2.72078i −2.72078 2.72078i
\(371\) 0 0
\(372\) 0 0
\(373\) 1.60021 0.923880i 1.60021 0.923880i 0.608761 0.793353i \(-0.291667\pi\)
0.991445 0.130526i \(-0.0416667\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 0.107651 + 0.541196i 0.107651 + 0.541196i
\(378\) 0 0
\(379\) 0 0 0.555570 0.831470i \(-0.312500\pi\)
−0.555570 + 0.831470i \(0.687500\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 0 0 0.608761 0.793353i \(-0.291667\pi\)
−0.608761 + 0.793353i \(0.708333\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) −1.63099 1.08979i −1.63099 1.08979i
\(387\) 0 0
\(388\) −0.125419 + 0.369474i −0.125419 + 0.369474i
\(389\) 0.860919 + 1.12197i 0.860919 + 1.12197i 0.991445 + 0.130526i \(0.0416667\pi\)
−0.130526 + 0.991445i \(0.541667\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0 0
\(393\) 0 0
\(394\) −0.732626 0.835400i −0.732626 0.835400i
\(395\) 0 0
\(396\) 0 0
\(397\) −1.05217 + 0.357164i −1.05217 + 0.357164i −0.793353 0.608761i \(-0.791667\pi\)
−0.258819 + 0.965926i \(0.583333\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) −2.63099 + 1.08979i −2.63099 + 1.08979i
\(401\) 1.75928 + 0.867580i 1.75928 + 0.867580i 0.965926 + 0.258819i \(0.0833333\pi\)
0.793353 + 0.608761i \(0.208333\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0 0
\(405\) −1.92388 + 0.382683i −1.92388 + 0.382683i
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) 1.60021 + 0.923880i 1.60021 + 0.923880i 0.991445 + 0.130526i \(0.0416667\pi\)
0.608761 + 0.793353i \(0.291667\pi\)
\(410\) −1.72918 + 1.32684i −1.72918 + 1.32684i
\(411\) 0 0
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) 0 0
\(416\) 0.184592 1.40211i 0.184592 1.40211i
\(417\) 0 0
\(418\) 0 0
\(419\) 0 0 −0.980785 0.195090i \(-0.937500\pi\)
0.980785 + 0.195090i \(0.0625000\pi\)
\(420\) 0 0
\(421\) 1.30656 1.30656i 1.30656 1.30656i 0.382683 0.923880i \(-0.375000\pi\)
0.923880 0.382683i \(-0.125000\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0.662827 + 0.382683i 0.662827 + 0.382683i
\(425\) 2.46623 1.42388i 2.46623 1.42388i
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 0 0 0.659346 0.751840i \(-0.270833\pi\)
−0.659346 + 0.751840i \(0.729167\pi\)
\(432\) 0 0
\(433\) −1.70711 + 0.707107i −1.70711 + 0.707107i −0.707107 + 0.707107i \(0.750000\pi\)
−1.00000 \(\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) −1.57469 + 0.534534i −1.57469 + 0.534534i
\(437\) 0 0
\(438\) 0 0
\(439\) 0 0 −0.659346 0.751840i \(-0.729167\pi\)
0.659346 + 0.751840i \(0.270833\pi\)
\(440\) 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
\(445\) −1.26105 + 3.71494i −1.26105 + 3.71494i
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 0.216773 + 0.324423i 0.216773 + 0.324423i 0.923880 0.382683i \(-0.125000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(450\) −1.73361 + 2.25928i −1.73361 + 2.25928i
\(451\) 0 0
\(452\) −0.0726721 1.10876i −0.0726721 1.10876i
\(453\) 0 0
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −1.40211 + 0.184592i −1.40211 + 0.184592i −0.793353 0.608761i \(-0.791667\pi\)
−0.608761 + 0.793353i \(0.708333\pi\)
\(458\) −0.382683 + 0.662827i −0.382683 + 0.662827i
\(459\) 0 0
\(460\) 0 0
\(461\) 0 0 −0.923880 0.382683i \(-0.875000\pi\)
0.923880 + 0.382683i \(0.125000\pi\)
\(462\) 0 0
\(463\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(464\) 0.389345 + 0.0255190i 0.389345 + 0.0255190i
\(465\) 0 0
\(466\) 1.65938 0.108761i 1.65938 0.108761i
\(467\) 0 0 −0.793353 0.608761i \(-0.791667\pi\)
0.793353 + 0.608761i \(0.208333\pi\)
\(468\) −0.541196 1.30656i −0.541196 1.30656i
\(469\) 0 0
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 0.765367 0.765367
\(478\) 0 0
\(479\) 0 0 −0.946930 0.321439i \(-0.895833\pi\)
0.946930 + 0.321439i \(0.104167\pi\)
\(480\) 0 0
\(481\) 0.181433 2.76814i 0.181433 2.76814i
\(482\) −0.0761205 + 0.382683i −0.0761205 + 0.382683i
\(483\) 0 0
\(484\) −0.382683 0.923880i −0.382683 0.923880i
\(485\) 0.607206 + 0.465926i 0.607206 + 0.465926i
\(486\) 0 0
\(487\) 0 0 −0.751840 0.659346i \(-0.770833\pi\)
0.751840 + 0.659346i \(0.229167\pi\)
\(488\) 1.65938 + 0.108761i 1.65938 + 0.108761i
\(489\) 0 0
\(490\) 0 0
\(491\) 0 0 0.382683 0.923880i \(-0.375000\pi\)
−0.382683 + 0.923880i \(0.625000\pi\)
\(492\) 0 0
\(493\) −0.389345 + 0.0255190i −0.389345 + 0.0255190i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) 0 0 −0.321439 0.946930i \(-0.604167\pi\)
0.321439 + 0.946930i \(0.395833\pi\)
\(500\) 0.237054 + 3.61675i 0.237054 + 3.61675i
\(501\) 0 0
\(502\) 0 0
\(503\) 0 0 −0.555570 0.831470i \(-0.687500\pi\)
0.555570 + 0.831470i \(0.312500\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) −0.382683 0.662827i −0.382683 0.662827i 0.608761 0.793353i \(-0.291667\pi\)
−0.991445 + 0.130526i \(0.958333\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) −0.923880 0.382683i −0.923880 0.382683i
\(513\) 0 0
\(514\) −0.198092 0.739288i −0.198092 0.739288i
\(515\) 0 0
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) 0 0
\(520\) −2.48800 1.22694i −2.48800 1.22694i
\(521\) −0.732626 + 0.835400i −0.732626 + 0.835400i −0.991445 0.130526i \(-0.958333\pi\)
0.258819 + 0.965926i \(0.416667\pi\)
\(522\) 0.349942 0.172572i 0.349942 0.172572i
\(523\) 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
\(529\) −0.793353 + 0.608761i −0.793353 + 0.608761i
\(530\) 1.12875 0.989890i 1.12875 0.989890i
\(531\) 0 0
\(532\) 0 0
\(533\) −1.54120 0.306563i −1.54120 0.306563i
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 0 0
\(538\) 0.382683 + 0.0761205i 0.382683 + 0.0761205i
\(539\) 0 0
\(540\) 0 0
\(541\) 0.835400 0.732626i 0.835400 0.732626i −0.130526 0.991445i \(-0.541667\pi\)
0.965926 + 0.258819i \(0.0833333\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0.965926 + 0.258819i 0.965926 + 0.258819i
\(545\) 3.26197i 3.26197i
\(546\) 0 0
\(547\) 0 0 0.980785 0.195090i \(-0.0625000\pi\)
−0.980785 + 0.195090i \(0.937500\pi\)
\(548\) 0.198092 0.739288i 0.198092 0.739288i
\(549\) 1.49144 0.735499i 1.49144 0.735499i
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) 0 0
\(554\) −1.38268 + 0.923880i −1.38268 + 0.923880i
\(555\) 0 0
\(556\) 0 0
\(557\) 0.478235 + 1.78480i 0.478235 + 1.78480i 0.608761 + 0.793353i \(0.291667\pi\)
−0.130526 + 0.991445i \(0.541667\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 0 0
\(561\) 0 0
\(562\) −0.382683 0.662827i −0.382683 0.662827i
\(563\) 0 0 −0.608761 0.793353i \(-0.708333\pi\)
0.608761 + 0.793353i \(0.291667\pi\)
\(564\) 0 0
\(565\) −2.10531 0.564117i −2.10531 0.564117i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 1.12484 1.46593i 1.12484 1.46593i 0.258819 0.965926i \(-0.416667\pi\)
0.866025 0.500000i \(-0.166667\pi\)
\(570\) 0 0
\(571\) 0 0 −0.0654031 0.997859i \(-0.520833\pi\)
0.0654031 + 0.997859i \(0.479167\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 0 0
\(576\) −0.991445 + 0.130526i −0.991445 + 0.130526i
\(577\) −0.707107 + 1.22474i −0.707107 + 1.22474i 0.258819 + 0.965926i \(0.416667\pi\)
−0.965926 + 0.258819i \(0.916667\pi\)
\(578\) −0.991445 0.130526i −0.991445 0.130526i
\(579\) 0 0
\(580\) 0.292893 0.707107i 0.292893 0.707107i
\(581\) 0 0
\(582\) 0 0
\(583\) 0 0
\(584\) −0.293353 0.257264i −0.293353 0.257264i
\(585\) −2.76814 + 0.181433i −2.76814 + 0.181433i
\(586\) −1.46593 1.12484i −1.46593 1.12484i
\(587\) 0 0 −0.382683 0.923880i \(-0.625000\pi\)
0.382683 + 0.923880i \(0.375000\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) 0 0
\(591\) 0 0
\(592\) −1.85747 0.630526i −1.85747 0.630526i
\(593\) 0 0 0.991445 0.130526i \(-0.0416667\pi\)
−0.991445 + 0.130526i \(0.958333\pi\)
\(594\) 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
\(601\) 0.324423 1.63099i 0.324423 1.63099i −0.382683 0.923880i \(-0.625000\pi\)
0.707107 0.707107i \(-0.250000\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) −1.95737 + 0.128293i −1.95737 + 0.128293i
\(606\) 0 0
\(607\) 0 0 −0.997859 0.0654031i \(-0.979167\pi\)
0.997859 + 0.0654031i \(0.0208333\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 1.24830 3.01367i 1.24830 3.01367i
\(611\) 0 0
\(612\) 0.965926 0.258819i 0.965926 0.258819i
\(613\) 0.707107 1.22474i 0.707107 1.22474i −0.258819 0.965926i \(-0.583333\pi\)
0.965926 0.258819i \(-0.0833333\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 0.923880 1.38268i 0.923880 1.38268i 1.00000i \(-0.5\pi\)
0.923880 0.382683i \(-0.125000\pi\)
\(618\) 0 0
\(619\) 0 0 −0.0654031 0.997859i \(-0.520833\pi\)
0.0654031 + 0.997859i \(0.479167\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) 4.11675 + 1.10308i 4.11675 + 1.10308i
\(626\) −0.534534 + 1.57469i −0.534534 + 1.57469i
\(627\) 0 0
\(628\) −0.923880 1.60021i −0.923880 1.60021i
\(629\) 1.92388 + 0.382683i 1.92388 + 0.382683i
\(630\) 0 0
\(631\) 0 0 −0.923880 0.382683i \(-0.875000\pi\)
0.923880 + 0.382683i \(0.125000\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) −0.172572 0.349942i −0.172572 0.349942i
\(635\) 0 0
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) 0 0
\(640\) −1.29335 + 1.47479i −1.29335 + 1.47479i
\(641\) 1.49144 0.735499i 1.49144 0.735499i 0.500000 0.866025i \(-0.333333\pi\)
0.991445 + 0.130526i \(0.0416667\pi\)
\(642\) 0 0
\(643\) 0 0 0.980785 0.195090i \(-0.0625000\pi\)
−0.980785 + 0.195090i \(0.937500\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(648\) −0.793353 + 0.608761i −0.793353 + 0.608761i
\(649\) 0 0
\(650\) −2.84776 + 2.84776i −2.84776 + 2.84776i
\(651\) 0 0
\(652\) 0 0
\(653\) −0.867580 + 1.75928i −0.867580 + 1.75928i −0.258819 + 0.965926i \(0.583333\pi\)
−0.608761 + 0.793353i \(0.708333\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) −0.491445 + 0.996552i −0.491445 + 0.996552i
\(657\) −0.382683 0.0761205i −0.382683 0.0761205i
\(658\) 0 0
\(659\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(660\) 0 0
\(661\) 1.12197 0.860919i 1.12197 0.860919i 0.130526 0.991445i \(-0.458333\pi\)
0.991445 + 0.130526i \(0.0416667\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) −1.92388 + 0.382683i −1.92388 + 0.382683i
\(667\) 0 0
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) −0.923880 + 0.617317i −0.923880 + 0.617317i −0.923880 0.382683i \(-0.875000\pi\)
1.00000i \(0.5\pi\)
\(674\) 1.05217 0.357164i 1.05217 0.357164i
\(675\) 0 0
\(676\) −0.258819 0.965926i −0.258819 0.965926i
\(677\) −0.257264 0.293353i −0.257264 0.293353i 0.608761 0.793353i \(-0.291667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 1.08979 1.63099i 1.08979 1.63099i
\(681\) 0 0
\(682\) 0 0
\(683\) 0 0 0.321439 0.946930i \(-0.395833\pi\)
−0.321439 + 0.946930i \(0.604167\pi\)
\(684\) 0 0
\(685\) −1.24830 0.834089i −1.24830 0.834089i
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 1.07313 + 0.141281i 1.07313 + 0.141281i
\(690\) 0 0
\(691\) 0 0 −0.321439 0.946930i \(-0.604167\pi\)
0.321439 + 0.946930i \(0.395833\pi\)
\(692\) 0.617317 0.923880i 0.617317 0.923880i
\(693\) 0 0
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 0.357164 1.05217i 0.357164 1.05217i
\(698\) 1.73205 1.00000i 1.73205 1.00000i
\(699\) 0 0
\(700\) 0 0
\(701\) 1.30656 + 1.30656i 1.30656 + 1.30656i 0.923880 + 0.382683i \(0.125000\pi\)
0.382683 + 0.923880i \(0.375000\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) 0 0
\(705\) 0 0
\(706\) 0.292893 + 0.707107i 0.292893 + 0.707107i
\(707\) 0 0
\(708\) 0 0
\(709\) −0.0255190 + 0.389345i −0.0255190 + 0.389345i 0.965926 + 0.258819i \(0.0833333\pi\)
−0.991445 + 0.130526i \(0.958333\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0.261052 + 1.98289i 0.261052 + 1.98289i
\(713\) 0 0
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 0 0 0.0654031 0.997859i \(-0.479167\pi\)
−0.0654031 + 0.997859i \(0.520833\pi\)
\(720\) −0.382683 + 1.92388i −0.382683 + 1.92388i
\(721\) 0 0
\(722\) −0.382683 0.923880i −0.382683 0.923880i
\(723\) 0 0
\(724\) 1.65938 0.108761i 1.65938 0.108761i
\(725\) −0.835400 0.732626i −0.835400 0.732626i
\(726\) 0 0
\(727\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(728\) 0 0
\(729\) −0.382683 + 0.923880i −0.382683 + 0.923880i
\(730\) −0.662827 + 0.382683i −0.662827 + 0.382683i
\(731\) 0 0
\(732\) 0 0
\(733\) 1.98289 0.261052i 1.98289 0.261052i 0.991445 0.130526i \(-0.0416667\pi\)
0.991445 0.130526i \(-0.0416667\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 0 0
\(738\) 0.0726721 + 1.10876i 0.0726721 + 1.10876i
\(739\) 0 0 −0.991445 0.130526i \(-0.958333\pi\)
0.991445 + 0.130526i \(0.0416667\pi\)
\(740\) −2.34237 + 3.05263i −2.34237 + 3.05263i
\(741\) 0 0
\(742\) 0 0
\(743\) 0 0 −0.831470 0.555570i \(-0.812500\pi\)
0.831470 + 0.555570i \(0.187500\pi\)
\(744\) 0 0
\(745\) 0.891699 2.62686i 0.891699 2.62686i
\(746\) −1.12484 1.46593i −1.12484 1.46593i
\(747\) 0 0
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) 0 0 −0.659346 0.751840i \(-0.729167\pi\)
0.659346 + 0.751840i \(0.270833\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0.522515 0.177370i 0.522515 0.177370i
\(755\) 0 0
\(756\) 0 0
\(757\) 0 0 −0.382683 0.923880i \(-0.625000\pi\)
0.382683 + 0.923880i \(0.375000\pi\)
\(758\) 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
\(765\) 0.128293 1.95737i 0.128293 1.95737i
\(766\) 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
\(772\) −0.867580 + 1.75928i −0.867580 + 1.75928i
\(773\) 0.0999004 0.758819i 0.0999004 0.758819i −0.866025 0.500000i \(-0.833333\pi\)
0.965926 0.258819i \(-0.0833333\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0.382683 + 0.0761205i 0.382683 + 0.0761205i
\(777\) 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
\(785\) −3.55487 + 0.707107i −3.55487 + 0.707107i
\(786\) 0 0
\(787\) 0 0 0.896873 0.442289i \(-0.145833\pi\)
−0.896873 + 0.442289i \(0.854167\pi\)
\(788\) −0.732626 + 0.835400i −0.732626 + 0.835400i
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 2.22694 0.755946i 2.22694 0.755946i
\(794\) 0.491445 + 0.996552i 0.491445 + 0.996552i
\(795\) 0 0
\(796\) 0 0
\(797\) −1.30656 0.541196i −1.30656 0.541196i −0.382683 0.923880i \(-0.625000\pi\)
−0.923880 + 0.382683i \(0.875000\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 1.42388 + 2.46623i 1.42388 + 2.46623i
\(801\) 1.21752 + 1.58671i 1.21752 + 1.58671i
\(802\) 0.630526 1.85747i 0.630526 1.85747i
\(803\) 0 0
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 0.0255190 + 0.389345i 0.0255190 + 0.389345i 0.991445 + 0.130526i \(0.0416667\pi\)
−0.965926 + 0.258819i \(0.916667\pi\)
\(810\) 0.630526 + 1.85747i 0.630526 + 1.85747i
\(811\) 0 0 0.555570 0.831470i \(-0.312500\pi\)
−0.555570 + 0.831470i \(0.687500\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 0 0
\(818\) 0.707107 1.70711i 0.707107 1.70711i
\(819\) 0 0
\(820\) 1.54120 + 1.54120i 1.54120 + 1.54120i
\(821\) −1.65938 0.108761i −1.65938 0.108761i −0.793353 0.608761i \(-0.791667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(822\) 0 0
\(823\) 0 0 0.997859 0.0654031i \(-0.0208333\pi\)
−0.997859 + 0.0654031i \(0.979167\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 0 0 0.195090 0.980785i \(-0.437500\pi\)
−0.195090 + 0.980785i \(0.562500\pi\)
\(828\) 0 0
\(829\) 1.78480 0.478235i 1.78480 0.478235i 0.793353 0.608761i \(-0.208333\pi\)
0.991445 + 0.130526i \(0.0416667\pi\)
\(830\) 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 0.195090 0.980785i \(-0.437500\pi\)
−0.195090 + 0.980785i \(0.562500\pi\)
\(840\) 0 0
\(841\) −0.324423 0.783227i −0.324423 0.783227i
\(842\) −1.46593 1.12484i −1.46593 1.12484i
\(843\) 0 0
\(844\) 0 0
\(845\) −1.95737 0.128293i −1.95737 0.128293i
\(846\) 0 0
\(847\) 0 0
\(848\) 0.292893 0.707107i 0.292893 0.707107i
\(849\) 0 0
\(850\) −1.73361 2.25928i −1.73361 2.25928i
\(851\) 0 0
\(852\) 0 0
\(853\) −0.0761205 0.382683i −0.0761205 0.382683i 0.923880 0.382683i \(-0.125000\pi\)
−1.00000 \(\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 0.0255190 + 0.389345i 0.0255190 + 0.389345i 0.991445 + 0.130526i \(0.0416667\pi\)
−0.965926 + 0.258819i \(0.916667\pi\)
\(858\) 0 0
\(859\) 0 0 0.608761 0.793353i \(-0.291667\pi\)
−0.608761 + 0.793353i \(0.708333\pi\)
\(860\) 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
\(865\) −1.32684 1.72918i −1.32684 1.72918i
\(866\) 0.923880 + 1.60021i 0.923880 + 1.60021i
\(867\) 0 0
\(868\) 0 0
\(869\) 0 0
\(870\) 0 0
\(871\) 0 0
\(872\) 0.735499 + 1.49144i 0.735499 + 1.49144i
\(873\) 0.369474 0.125419i 0.369474 0.125419i
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) −0.996552 0.491445i −0.996552 0.491445i −0.130526 0.991445i \(-0.541667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 1.08979 0.216773i 1.08979 0.216773i 0.382683 0.923880i \(-0.375000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(882\) 0 0
\(883\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(884\) 1.40211 0.184592i 1.40211 0.184592i
\(885\) 0 0
\(886\) 0 0
\(887\) 0 0 0.751840 0.659346i \(-0.229167\pi\)
−0.751840 + 0.659346i \(0.770833\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 3.84776 + 0.765367i 3.84776 + 0.765367i
\(891\) 0 0
\(892\) 0 0
\(893\) 0 0
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 0 0
\(898\) 0.293353 0.257264i 0.293353 0.257264i
\(899\) 0 0
\(900\) 2.46623 + 1.42388i 2.46623 + 1.42388i
\(901\) −0.198092 + 0.739288i −0.198092 + 0.739288i
\(902\) 0 0
\(903\) 0 0
\(904\) −1.08979 + 0.216773i −1.08979 + 0.216773i
\(905\) 0.844261 3.15082i 0.844261 3.15082i
\(906\) 0 0
\(907\) 0 0 0.659346 0.751840i \(-0.270833\pi\)
−0.659346 + 0.751840i \(0.729167\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 0 0 0.831470 0.555570i \(-0.187500\pi\)
−0.831470 + 0.555570i \(0.812500\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) 0.366025 + 1.36603i 0.366025 + 1.36603i
\(915\) 0 0
\(916\) 0.707107 + 0.292893i 0.707107 + 0.292893i
\(917\) 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
\(925\) 3.10346 + 4.64466i 3.10346 + 4.64466i
\(926\) 0 0
\(927\) 0 0
\(928\) −0.0255190 0.389345i −0.0255190 0.389345i
\(929\) −0.534534 1.57469i −0.534534 1.57469i −0.793353 0.608761i \(-0.791667\pi\)
0.258819 0.965926i \(-0.416667\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) −0.324423 1.63099i −0.324423 1.63099i
\(933\) 0 0
\(934\) 0 0
\(935\) 0 0
\(936\) −1.22474 + 0.707107i −1.22474 + 0.707107i
\(937\) 0.541196 1.30656i 0.541196 1.30656i −0.382683 0.923880i \(-0.625000\pi\)
0.923880 0.382683i \(-0.125000\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 0.835400 + 0.732626i 0.835400 + 0.732626i 0.965926 0.258819i \(-0.0833333\pi\)
−0.130526 + 0.991445i \(0.541667\pi\)
\(942\) 0 0
\(943\) 0 0
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 0 0 0.0654031 0.997859i \(-0.479167\pi\)
−0.0654031 + 0.997859i \(0.520833\pi\)
\(948\) 0 0
\(949\) −0.522515 0.177370i −0.522515 0.177370i
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) −1.84776 −1.84776 −0.923880 0.382683i \(-0.875000\pi\)
−0.923880 + 0.382683i \(0.875000\pi\)
\(954\) −0.0999004 0.758819i −0.0999004 0.758819i
\(955\) 0 0
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) −0.793353 0.608761i −0.793353 0.608761i
\(962\) −2.76814 + 0.181433i −2.76814 + 0.181433i
\(963\) 0 0
\(964\) 0.389345 + 0.0255190i 0.389345 + 0.0255190i
\(965\) 2.72078 + 2.72078i 2.72078 + 2.72078i
\(966\) 0 0
\(967\) 0 0 0.382683 0.923880i \(-0.375000\pi\)
−0.382683 + 0.923880i \(0.625000\pi\)
\(968\) −0.866025 + 0.500000i −0.866025 + 0.500000i
\(969\) 0 0
\(970\) 0.382683 0.662827i 0.382683 0.662827i
\(971\) 0 0 0.991445 0.130526i \(-0.0416667\pi\)
−0.991445 + 0.130526i \(0.958333\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) 0 0
\(976\) −0.108761 1.65938i −0.108761 1.65938i
\(977\) 0.758819 + 0.0999004i 0.758819 + 0.0999004i 0.500000 0.866025i \(-0.333333\pi\)
0.258819 + 0.965926i \(0.416667\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 0 0
\(981\) 1.38268 + 0.923880i 1.38268 + 0.923880i
\(982\) 0 0
\(983\) 0 0 0.321439 0.946930i \(-0.395833\pi\)
−0.321439 + 0.946930i \(0.604167\pi\)
\(984\) 0 0
\(985\) 1.08979 + 1.88757i 1.08979 + 1.88757i
\(986\) 0.0761205 + 0.382683i 0.0761205 + 0.382683i
\(987\) 0 0
\(988\) 0 0
\(989\) 0 0
\(990\) 0 0
\(991\) 0 0 −0.442289 0.896873i \(-0.645833\pi\)
0.442289 + 0.896873i \(0.354167\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) 1.09645 1.25026i 1.09645 1.25026i 0.130526 0.991445i \(-0.458333\pi\)
0.965926 0.258819i \(-0.0833333\pi\)
\(998\) 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.227.1 16
4.3 odd 2 CM 3332.1.ce.d.227.1 16
7.2 even 3 inner 3332.1.ce.d.1587.1 16
7.3 odd 6 3332.1.bn.a.2743.1 yes 8
7.4 even 3 3332.1.bn.c.2743.1 yes 8
7.5 odd 6 3332.1.ce.b.1587.1 16
7.6 odd 2 3332.1.ce.b.227.1 16
17.3 odd 16 3332.1.ce.b.2383.1 16
28.3 even 6 3332.1.bn.a.2743.1 yes 8
28.11 odd 6 3332.1.bn.c.2743.1 yes 8
28.19 even 6 3332.1.ce.b.1587.1 16
28.23 odd 6 inner 3332.1.ce.d.1587.1 16
28.27 even 2 3332.1.ce.b.227.1 16
68.3 even 16 3332.1.ce.b.2383.1 16
119.3 even 48 3332.1.bn.c.1567.1 yes 8
119.20 even 16 inner 3332.1.ce.d.2383.1 16
119.37 odd 48 3332.1.ce.b.411.1 16
119.54 even 48 inner 3332.1.ce.d.411.1 16
119.88 odd 48 3332.1.bn.a.1567.1 8
476.3 odd 48 3332.1.bn.c.1567.1 yes 8
476.139 odd 16 inner 3332.1.ce.d.2383.1 16
476.207 even 48 3332.1.bn.a.1567.1 8
476.275 even 48 3332.1.ce.b.411.1 16
476.411 odd 48 inner 3332.1.ce.d.411.1 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
3332.1.bn.a.1567.1 8 119.88 odd 48
3332.1.bn.a.1567.1 8 476.207 even 48
3332.1.bn.a.2743.1 yes 8 7.3 odd 6
3332.1.bn.a.2743.1 yes 8 28.3 even 6
3332.1.bn.c.1567.1 yes 8 119.3 even 48
3332.1.bn.c.1567.1 yes 8 476.3 odd 48
3332.1.bn.c.2743.1 yes 8 7.4 even 3
3332.1.bn.c.2743.1 yes 8 28.11 odd 6
3332.1.ce.b.227.1 16 7.6 odd 2
3332.1.ce.b.227.1 16 28.27 even 2
3332.1.ce.b.411.1 16 119.37 odd 48
3332.1.ce.b.411.1 16 476.275 even 48
3332.1.ce.b.1587.1 16 7.5 odd 6
3332.1.ce.b.1587.1 16 28.19 even 6
3332.1.ce.b.2383.1 16 17.3 odd 16
3332.1.ce.b.2383.1 16 68.3 even 16
3332.1.ce.d.227.1 16 1.1 even 1 trivial
3332.1.ce.d.227.1 16 4.3 odd 2 CM
3332.1.ce.d.411.1 16 119.54 even 48 inner
3332.1.ce.d.411.1 16 476.411 odd 48 inner
3332.1.ce.d.1587.1 16 7.2 even 3 inner
3332.1.ce.d.1587.1 16 28.23 odd 6 inner
3332.1.ce.d.2383.1 16 119.20 even 16 inner
3332.1.ce.d.2383.1 16 476.139 odd 16 inner