Properties

Label 525.2.bf.c.107.2
Level $525$
Weight $2$
Character 525.107
Analytic conductor $4.192$
Analytic rank $0$
Dimension $8$
CM discriminant -3
Inner twists $16$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [525,2,Mod(32,525)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(525, base_ring=CyclotomicField(12))
 
chi = DirichletCharacter(H, H._module([6, 3, 8]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("525.32");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 525 = 3 \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 525.bf (of order \(12\), degree \(4\), 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: \(4.19214610612\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(2\) over \(\Q(\zeta_{12})\)
Coefficient field: \(\Q(\zeta_{24})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{12}]$

Embedding invariants

Embedding label 107.2
Root \(0.965926 - 0.258819i\) of defining polynomial
Character \(\chi\) \(=\) 525.107
Dual form 525.2.bf.c.368.2

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(1.67303 + 0.448288i) q^{3} +(1.73205 - 1.00000i) q^{4} +(2.63896 - 0.189469i) q^{7} +(2.59808 + 1.50000i) q^{9} +O(q^{10})\) \(q+(1.67303 + 0.448288i) q^{3} +(1.73205 - 1.00000i) q^{4} +(2.63896 - 0.189469i) q^{7} +(2.59808 + 1.50000i) q^{9} +(3.34607 - 0.896575i) q^{12} +(-4.89898 + 4.89898i) q^{13} +(2.00000 - 3.46410i) q^{16} +(-6.92820 - 4.00000i) q^{19} +(4.50000 + 0.866025i) q^{21} +(3.67423 + 3.67423i) q^{27} +(4.38134 - 2.96713i) q^{28} +(-3.50000 - 6.06218i) q^{31} +6.00000 q^{36} +(5.01910 - 1.34486i) q^{37} +(-10.3923 + 6.00000i) q^{39} +(-8.57321 + 8.57321i) q^{43} +(4.89898 - 4.89898i) q^{48} +(6.92820 - 1.00000i) q^{49} +(-3.58630 + 13.3843i) q^{52} +(-9.79796 - 9.79796i) q^{57} +(0.500000 - 0.866025i) q^{61} +(7.14042 + 3.46618i) q^{63} -8.00000i q^{64} +(0.896575 - 3.34607i) q^{67} +(-1.67303 - 0.448288i) q^{73} -16.0000 q^{76} +(14.7224 + 8.50000i) q^{79} +(4.50000 + 7.79423i) q^{81} +(8.66025 - 3.00000i) q^{84} +(-12.0000 + 13.8564i) q^{91} +(-3.13801 - 11.7112i) q^{93} +(-3.67423 - 3.67423i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 16 q^{16} + 36 q^{21} - 28 q^{31} + 48 q^{36} + 4 q^{61} - 128 q^{76} + 36 q^{81} - 96 q^{91}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/525\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(176\) \(451\)
\(\chi(n)\) \(e\left(\frac{1}{4}\right)\) \(-1\) \(e\left(\frac{1}{3}\right)\)

Coefficient data

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



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0 0.965926 0.258819i \(-0.0833333\pi\)
−0.965926 + 0.258819i \(0.916667\pi\)
\(3\) 1.67303 + 0.448288i 0.965926 + 0.258819i
\(4\) 1.73205 1.00000i 0.866025 0.500000i
\(5\) 0 0
\(6\) 0 0
\(7\) 2.63896 0.189469i 0.997433 0.0716124i
\(8\) 0 0
\(9\) 2.59808 + 1.50000i 0.866025 + 0.500000i
\(10\) 0 0
\(11\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(12\) 3.34607 0.896575i 0.965926 0.258819i
\(13\) −4.89898 + 4.89898i −1.35873 + 1.35873i −0.483250 + 0.875482i \(0.660544\pi\)
−0.875482 + 0.483250i \(0.839456\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 2.00000 3.46410i 0.500000 0.866025i
\(17\) 0 0 0.258819 0.965926i \(-0.416667\pi\)
−0.258819 + 0.965926i \(0.583333\pi\)
\(18\) 0 0
\(19\) −6.92820 4.00000i −1.58944 0.917663i −0.993399 0.114708i \(-0.963407\pi\)
−0.596040 0.802955i \(-0.703260\pi\)
\(20\) 0 0
\(21\) 4.50000 + 0.866025i 0.981981 + 0.188982i
\(22\) 0 0
\(23\) 0 0 −0.258819 0.965926i \(-0.583333\pi\)
0.258819 + 0.965926i \(0.416667\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 3.67423 + 3.67423i 0.707107 + 0.707107i
\(28\) 4.38134 2.96713i 0.827996 0.560734i
\(29\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(30\) 0 0
\(31\) −3.50000 6.06218i −0.628619 1.08880i −0.987829 0.155543i \(-0.950287\pi\)
0.359211 0.933257i \(-0.383046\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 6.00000 1.00000
\(37\) 5.01910 1.34486i 0.825135 0.221094i 0.178545 0.983932i \(-0.442861\pi\)
0.646590 + 0.762838i \(0.276194\pi\)
\(38\) 0 0
\(39\) −10.3923 + 6.00000i −1.66410 + 0.960769i
\(40\) 0 0
\(41\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(42\) 0 0
\(43\) −8.57321 + 8.57321i −1.30740 + 1.30740i −0.384120 + 0.923283i \(0.625495\pi\)
−0.923283 + 0.384120i \(0.874505\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 0 0 0.965926 0.258819i \(-0.0833333\pi\)
−0.965926 + 0.258819i \(0.916667\pi\)
\(48\) 4.89898 4.89898i 0.707107 0.707107i
\(49\) 6.92820 1.00000i 0.989743 0.142857i
\(50\) 0 0
\(51\) 0 0
\(52\) −3.58630 + 13.3843i −0.497331 + 1.85606i
\(53\) 0 0 −0.965926 0.258819i \(-0.916667\pi\)
0.965926 + 0.258819i \(0.0833333\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) −9.79796 9.79796i −1.29777 1.29777i
\(58\) 0 0
\(59\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(60\) 0 0
\(61\) 0.500000 0.866025i 0.0640184 0.110883i −0.832240 0.554416i \(-0.812942\pi\)
0.896258 + 0.443533i \(0.146275\pi\)
\(62\) 0 0
\(63\) 7.14042 + 3.46618i 0.899608 + 0.436698i
\(64\) 8.00000i 1.00000i
\(65\) 0 0
\(66\) 0 0
\(67\) 0.896575 3.34607i 0.109534 0.408787i −0.889286 0.457352i \(-0.848798\pi\)
0.998820 + 0.0485648i \(0.0154647\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(72\) 0 0
\(73\) −1.67303 0.448288i −0.195814 0.0524681i 0.159579 0.987185i \(-0.448986\pi\)
−0.355393 + 0.934717i \(0.615653\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) −16.0000 −1.83533
\(77\) 0 0
\(78\) 0 0
\(79\) 14.7224 + 8.50000i 1.65640 + 0.956325i 0.974355 + 0.225018i \(0.0722440\pi\)
0.682048 + 0.731307i \(0.261089\pi\)
\(80\) 0 0
\(81\) 4.50000 + 7.79423i 0.500000 + 0.866025i
\(82\) 0 0
\(83\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(84\) 8.66025 3.00000i 0.944911 0.327327i
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(90\) 0 0
\(91\) −12.0000 + 13.8564i −1.25794 + 1.45255i
\(92\) 0 0
\(93\) −3.13801 11.7112i −0.325397 1.21440i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −3.67423 3.67423i −0.373062 0.373062i 0.495529 0.868591i \(-0.334974\pi\)
−0.868591 + 0.495529i \(0.834974\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(102\) 0 0
\(103\) −4.03459 15.0573i −0.397540 1.48364i −0.817411 0.576055i \(-0.804591\pi\)
0.419871 0.907584i \(-0.362075\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 0 0 0.965926 0.258819i \(-0.0833333\pi\)
−0.965926 + 0.258819i \(0.916667\pi\)
\(108\) 10.0382 + 2.68973i 0.965926 + 0.258819i
\(109\) −1.73205 + 1.00000i −0.165900 + 0.0957826i −0.580651 0.814152i \(-0.697202\pi\)
0.414751 + 0.909935i \(0.363869\pi\)
\(110\) 0 0
\(111\) 9.00000 0.854242
\(112\) 4.62158 9.52056i 0.436698 0.899608i
\(113\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) −20.0764 + 5.37945i −1.85606 + 0.497331i
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) −5.50000 + 9.52628i −0.500000 + 0.866025i
\(122\) 0 0
\(123\) 0 0
\(124\) −12.1244 7.00000i −1.08880 0.628619i
\(125\) 0 0
\(126\) 0 0
\(127\) −15.9217 15.9217i −1.41282 1.41282i −0.737783 0.675038i \(-0.764127\pi\)
−0.675038 0.737783i \(-0.735873\pi\)
\(128\) 0 0
\(129\) −18.1865 + 10.5000i −1.60123 + 0.924473i
\(130\) 0 0
\(131\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(132\) 0 0
\(133\) −19.0411 9.24316i −1.65107 0.801483i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 0 0 0.258819 0.965926i \(-0.416667\pi\)
−0.258819 + 0.965926i \(0.583333\pi\)
\(138\) 0 0
\(139\) 23.0000i 1.95083i 0.220366 + 0.975417i \(0.429275\pi\)
−0.220366 + 0.975417i \(0.570725\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 0 0
\(144\) 10.3923 6.00000i 0.866025 0.500000i
\(145\) 0 0
\(146\) 0 0
\(147\) 12.0394 + 1.43280i 0.992993 + 0.118175i
\(148\) 7.34847 7.34847i 0.604040 0.604040i
\(149\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(150\) 0 0
\(151\) −9.50000 16.4545i −0.773099 1.33905i −0.935857 0.352381i \(-0.885372\pi\)
0.162758 0.986666i \(-0.447961\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 0 0
\(156\) −12.0000 + 20.7846i −0.960769 + 1.66410i
\(157\) −5.82774 + 21.7494i −0.465104 + 1.73579i 0.191439 + 0.981505i \(0.438685\pi\)
−0.656543 + 0.754288i \(0.727982\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) 4.93117 + 18.4034i 0.386239 + 1.44146i 0.836205 + 0.548417i \(0.184769\pi\)
−0.449966 + 0.893045i \(0.648564\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(168\) 0 0
\(169\) 35.0000i 2.69231i
\(170\) 0 0
\(171\) −12.0000 20.7846i −0.917663 1.58944i
\(172\) −6.27603 + 23.4225i −0.478543 + 1.78595i
\(173\) 0 0 −0.258819 0.965926i \(-0.583333\pi\)
0.258819 + 0.965926i \(0.416667\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(180\) 0 0
\(181\) 7.00000 0.520306 0.260153 0.965567i \(-0.416227\pi\)
0.260153 + 0.965567i \(0.416227\pi\)
\(182\) 0 0
\(183\) 1.22474 1.22474i 0.0905357 0.0905357i
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) 10.3923 + 9.00000i 0.755929 + 0.654654i
\(190\) 0 0
\(191\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(192\) 3.58630 13.3843i 0.258819 0.965926i
\(193\) 11.7112 + 3.13801i 0.842993 + 0.225879i 0.654374 0.756171i \(-0.272932\pi\)
0.188619 + 0.982050i \(0.439599\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 11.0000 8.66025i 0.785714 0.618590i
\(197\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(198\) 0 0
\(199\) −14.7224 + 8.50000i −1.04365 + 0.602549i −0.920864 0.389885i \(-0.872515\pi\)
−0.122782 + 0.992434i \(0.539182\pi\)
\(200\) 0 0
\(201\) 3.00000 5.19615i 0.211604 0.366508i
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 0 0
\(208\) 7.17260 + 26.7685i 0.497331 + 1.85606i
\(209\) 0 0
\(210\) 0 0
\(211\) 29.0000 1.99644 0.998221 0.0596196i \(-0.0189888\pi\)
0.998221 + 0.0596196i \(0.0189888\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) −10.3849 15.3347i −0.704976 1.04099i
\(218\) 0 0
\(219\) −2.59808 1.50000i −0.175562 0.101361i
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) 13.4722 13.4722i 0.902165 0.902165i −0.0934584 0.995623i \(-0.529792\pi\)
0.995623 + 0.0934584i \(0.0297922\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 0 0 0.258819 0.965926i \(-0.416667\pi\)
−0.258819 + 0.965926i \(0.583333\pi\)
\(228\) −26.7685 7.17260i −1.77279 0.475017i
\(229\) 19.0526 + 11.0000i 1.25903 + 0.726900i 0.972886 0.231287i \(-0.0742935\pi\)
0.286143 + 0.958187i \(0.407627\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 0 0 −0.258819 0.965926i \(-0.583333\pi\)
0.258819 + 0.965926i \(0.416667\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 20.8207 + 20.8207i 1.35245 + 1.35245i
\(238\) 0 0
\(239\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(240\) 0 0
\(241\) 15.5000 + 26.8468i 0.998443 + 1.72935i 0.547533 + 0.836784i \(0.315567\pi\)
0.450910 + 0.892570i \(0.351100\pi\)
\(242\) 0 0
\(243\) 4.03459 + 15.0573i 0.258819 + 0.965926i
\(244\) 2.00000i 0.128037i
\(245\) 0 0
\(246\) 0 0
\(247\) 53.5370 14.3452i 3.40648 0.912764i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(252\) 15.8338 1.13681i 0.997433 0.0716124i
\(253\) 0 0
\(254\) 0 0
\(255\) 0 0
\(256\) −8.00000 13.8564i −0.500000 0.866025i
\(257\) 0 0 0.965926 0.258819i \(-0.0833333\pi\)
−0.965926 + 0.258819i \(0.916667\pi\)
\(258\) 0 0
\(259\) 12.9904 4.50000i 0.807183 0.279616i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 0 0 −0.965926 0.258819i \(-0.916667\pi\)
0.965926 + 0.258819i \(0.0833333\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 0 0
\(268\) −1.79315 6.69213i −0.109534 0.408787i
\(269\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(270\) 0 0
\(271\) 14.0000 24.2487i 0.850439 1.47300i −0.0303728 0.999539i \(-0.509669\pi\)
0.880812 0.473466i \(-0.156997\pi\)
\(272\) 0 0
\(273\) −26.2880 + 17.8028i −1.59103 + 1.07747i
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) 5.37945 20.0764i 0.323220 1.20627i −0.592869 0.805299i \(-0.702005\pi\)
0.916089 0.400975i \(-0.131328\pi\)
\(278\) 0 0
\(279\) 21.0000i 1.25724i
\(280\) 0 0
\(281\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(282\) 0 0
\(283\) −10.0382 2.68973i −0.596709 0.159888i −0.0521913 0.998637i \(-0.516621\pi\)
−0.544518 + 0.838749i \(0.683287\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) −14.7224 8.50000i −0.866025 0.500000i
\(290\) 0 0
\(291\) −4.50000 7.79423i −0.263795 0.456906i
\(292\) −3.34607 + 0.896575i −0.195814 + 0.0524681i
\(293\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 0 0
\(300\) 0 0
\(301\) −21.0000 + 24.2487i −1.21042 + 1.39767i
\(302\) 0 0
\(303\) 0 0
\(304\) −27.7128 + 16.0000i −1.58944 + 0.917663i
\(305\) 0 0
\(306\) 0 0
\(307\) 1.22474 + 1.22474i 0.0698999 + 0.0698999i 0.741192 0.671293i \(-0.234261\pi\)
−0.671293 + 0.741192i \(0.734261\pi\)
\(308\) 0 0
\(309\) 27.0000i 1.53598i
\(310\) 0 0
\(311\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(312\) 0 0
\(313\) 1.34486 + 5.01910i 0.0760162 + 0.283696i 0.993462 0.114165i \(-0.0364192\pi\)
−0.917446 + 0.397861i \(0.869753\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 34.0000 1.91265
\(317\) 0 0 0.965926 0.258819i \(-0.0833333\pi\)
−0.965926 + 0.258819i \(0.916667\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 0 0
\(324\) 15.5885 + 9.00000i 0.866025 + 0.500000i
\(325\) 0 0
\(326\) 0 0
\(327\) −3.34607 + 0.896575i −0.185038 + 0.0495807i
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) 15.5000 26.8468i 0.851957 1.47563i −0.0274825 0.999622i \(-0.508749\pi\)
0.879440 0.476011i \(-0.157918\pi\)
\(332\) 0 0
\(333\) 15.0573 + 4.03459i 0.825135 + 0.221094i
\(334\) 0 0
\(335\) 0 0
\(336\) 12.0000 13.8564i 0.654654 0.755929i
\(337\) 25.7196 + 25.7196i 1.40104 + 1.40104i 0.796815 + 0.604223i \(0.206516\pi\)
0.604223 + 0.796815i \(0.293484\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) 18.0938 3.95164i 0.976972 0.213368i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 0 0 0.258819 0.965926i \(-0.416667\pi\)
−0.258819 + 0.965926i \(0.583333\pi\)
\(348\) 0 0
\(349\) 14.0000i 0.749403i 0.927146 + 0.374701i \(0.122255\pi\)
−0.927146 + 0.374701i \(0.877745\pi\)
\(350\) 0 0
\(351\) −36.0000 −1.92154
\(352\) 0 0
\(353\) 0 0 −0.965926 0.258819i \(-0.916667\pi\)
0.965926 + 0.258819i \(0.0833333\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(360\) 0 0
\(361\) 22.5000 + 38.9711i 1.18421 + 2.05111i
\(362\) 0 0
\(363\) −13.4722 + 13.4722i −0.707107 + 0.707107i
\(364\) −6.92820 + 36.0000i −0.363137 + 1.88691i
\(365\) 0 0
\(366\) 0 0
\(367\) −4.03459 + 15.0573i −0.210604 + 0.785984i 0.777064 + 0.629421i \(0.216708\pi\)
−0.987668 + 0.156563i \(0.949959\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 0 0
\(372\) −17.1464 17.1464i −0.889001 0.889001i
\(373\) −1.79315 6.69213i −0.0928458 0.346505i 0.903838 0.427874i \(-0.140737\pi\)
−0.996684 + 0.0813690i \(0.974071\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) 8.00000i 0.410932i 0.978664 + 0.205466i \(0.0658711\pi\)
−0.978664 + 0.205466i \(0.934129\pi\)
\(380\) 0 0
\(381\) −19.5000 33.7750i −0.999015 1.73035i
\(382\) 0 0
\(383\) 0 0 −0.258819 0.965926i \(-0.583333\pi\)
0.258819 + 0.965926i \(0.416667\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) −35.1337 + 9.41404i −1.78595 + 0.478543i
\(388\) −10.0382 2.68973i −0.509612 0.136550i
\(389\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) −18.4034 + 4.93117i −0.923638 + 0.247488i −0.689140 0.724628i \(-0.742011\pi\)
−0.234498 + 0.972117i \(0.575345\pi\)
\(398\) 0 0
\(399\) −27.7128 24.0000i −1.38738 1.20150i
\(400\) 0 0
\(401\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(402\) 0 0
\(403\) 46.8449 + 12.5521i 2.33351 + 0.625262i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) 32.9090 19.0000i 1.62724 0.939490i 0.642333 0.766426i \(-0.277967\pi\)
0.984911 0.173064i \(-0.0553667\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) −22.0454 22.0454i −1.08610 1.08610i
\(413\) 0 0
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) −10.3106 + 38.4797i −0.504913 + 1.88436i
\(418\) 0 0
\(419\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(420\) 0 0
\(421\) −41.0000 −1.99822 −0.999109 0.0422075i \(-0.986561\pi\)
−0.999109 + 0.0422075i \(0.986561\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 1.15539 2.38014i 0.0559135 0.115183i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(432\) 20.0764 5.37945i 0.965926 0.258819i
\(433\) 15.9217 15.9217i 0.765147 0.765147i −0.212101 0.977248i \(-0.568030\pi\)
0.977248 + 0.212101i \(0.0680304\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) −2.00000 + 3.46410i −0.0957826 + 0.165900i
\(437\) 0 0
\(438\) 0 0
\(439\) −11.2583 6.50000i −0.537331 0.310228i 0.206666 0.978412i \(-0.433739\pi\)
−0.743996 + 0.668184i \(0.767072\pi\)
\(440\) 0 0
\(441\) 19.5000 + 7.79423i 0.928571 + 0.371154i
\(442\) 0 0
\(443\) 0 0 −0.258819 0.965926i \(-0.583333\pi\)
0.258819 + 0.965926i \(0.416667\pi\)
\(444\) 15.5885 9.00000i 0.739795 0.427121i
\(445\) 0 0
\(446\) 0 0
\(447\) 0 0
\(448\) −1.51575 21.1117i −0.0716124 0.997433i
\(449\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 0 0
\(453\) −8.51747 31.7876i −0.400186 1.49351i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −28.4416 + 7.62089i −1.33044 + 0.356490i −0.852879 0.522108i \(-0.825146\pi\)
−0.477561 + 0.878599i \(0.658479\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(462\) 0 0
\(463\) 1.22474 1.22474i 0.0569187 0.0569187i −0.678074 0.734993i \(-0.737185\pi\)
0.734993 + 0.678074i \(0.237185\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 0 0 0.965926 0.258819i \(-0.0833333\pi\)
−0.965926 + 0.258819i \(0.916667\pi\)
\(468\) −29.3939 + 29.3939i −1.35873 + 1.35873i
\(469\) 1.73205 9.00000i 0.0799787 0.415581i
\(470\) 0 0
\(471\) −19.5000 + 33.7750i −0.898513 + 1.55627i
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(480\) 0 0
\(481\) −18.0000 + 31.1769i −0.820729 + 1.42154i
\(482\) 0 0
\(483\) 0 0
\(484\) 22.0000i 1.00000i
\(485\) 0 0
\(486\) 0 0
\(487\) 9.41404 35.1337i 0.426591 1.59206i −0.333833 0.942632i \(-0.608342\pi\)
0.760424 0.649427i \(-0.224991\pi\)
\(488\) 0 0
\(489\) 33.0000i 1.49231i
\(490\) 0 0
\(491\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) 0 0
\(496\) −28.0000 −1.25724
\(497\) 0 0
\(498\) 0 0
\(499\) 27.7128 + 16.0000i 1.24060 + 0.716258i 0.969216 0.246214i \(-0.0791865\pi\)
0.271380 + 0.962472i \(0.412520\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 15.6901 58.5561i 0.696821 2.60057i
\(508\) −43.4988 11.6555i −1.92995 0.517128i
\(509\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(510\) 0 0
\(511\) −4.50000 0.866025i −0.199068 0.0383107i
\(512\) 0 0
\(513\) −10.7589 40.1528i −0.475017 1.77279i
\(514\) 0 0
\(515\) 0 0
\(516\) −21.0000 + 36.3731i −0.924473 + 1.60123i
\(517\) 0 0
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(522\) 0 0
\(523\) −7.62089 28.4416i −0.333238 1.24366i −0.905766 0.423777i \(-0.860704\pi\)
0.572528 0.819885i \(-0.305963\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) −19.9186 + 11.5000i −0.866025 + 0.500000i
\(530\) 0 0
\(531\) 0 0
\(532\) −42.2233 + 3.03150i −1.83061 + 0.131432i
\(533\) 0 0
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) 23.0000 39.8372i 0.988847 1.71273i 0.365444 0.930834i \(-0.380917\pi\)
0.623404 0.781900i \(-0.285749\pi\)
\(542\) 0 0
\(543\) 11.7112 + 3.13801i 0.502577 + 0.134665i
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) −17.1464 17.1464i −0.733128 0.733128i 0.238110 0.971238i \(-0.423472\pi\)
−0.971238 + 0.238110i \(0.923472\pi\)
\(548\) 0 0
\(549\) 2.59808 1.50000i 0.110883 0.0640184i
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) 40.4624 + 19.6417i 1.72063 + 0.835250i
\(554\) 0 0
\(555\) 0 0
\(556\) 23.0000 + 39.8372i 0.975417 + 1.68947i
\(557\) 0 0 0.258819 0.965926i \(-0.416667\pi\)
−0.258819 + 0.965926i \(0.583333\pi\)
\(558\) 0 0
\(559\) 84.0000i 3.55282i
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 0 0 −0.965926 0.258819i \(-0.916667\pi\)
0.965926 + 0.258819i \(0.0833333\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 13.3521 + 19.7160i 0.560734 + 0.827996i
\(568\) 0 0
\(569\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(570\) 0 0
\(571\) 8.00000 + 13.8564i 0.334790 + 0.579873i 0.983444 0.181210i \(-0.0580014\pi\)
−0.648655 + 0.761083i \(0.724668\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 0 0
\(576\) 12.0000 20.7846i 0.500000 0.866025i
\(577\) −8.51747 + 31.7876i −0.354587 + 1.32334i 0.526417 + 0.850227i \(0.323535\pi\)
−0.881004 + 0.473109i \(0.843132\pi\)
\(578\) 0 0
\(579\) 18.1865 + 10.5000i 0.755807 + 0.436365i
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(588\) 22.2856 9.55772i 0.919044 0.394154i
\(589\) 56.0000i 2.30744i
\(590\) 0 0
\(591\) 0 0
\(592\) 5.37945 20.0764i 0.221094 0.825135i
\(593\) 0 0 −0.258819 0.965926i \(-0.583333\pi\)
0.258819 + 0.965926i \(0.416667\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) −28.4416 + 7.62089i −1.16404 + 0.311902i
\(598\) 0 0
\(599\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(600\) 0 0
\(601\) −26.0000 −1.06056 −0.530281 0.847822i \(-0.677914\pi\)
−0.530281 + 0.847822i \(0.677914\pi\)
\(602\) 0 0
\(603\) 7.34847 7.34847i 0.299253 0.299253i
\(604\) −32.9090 19.0000i −1.33905 0.773099i
\(605\) 0 0
\(606\) 0 0
\(607\) 38.4797 10.3106i 1.56184 0.418495i 0.628598 0.777730i \(-0.283629\pi\)
0.933247 + 0.359235i \(0.116962\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) −46.8449 12.5521i −1.89205 0.506973i −0.998292 0.0584195i \(-0.981394\pi\)
−0.893756 0.448553i \(-0.851939\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(618\) 0 0
\(619\) 42.4352 24.5000i 1.70562 0.984738i 0.765787 0.643094i \(-0.222350\pi\)
0.939829 0.341644i \(-0.110984\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) 48.0000i 1.92154i
\(625\) 0 0
\(626\) 0 0
\(627\) 0 0
\(628\) 11.6555 + 43.4988i 0.465104 + 1.73579i
\(629\) 0 0
\(630\) 0 0
\(631\) −1.00000 −0.0398094 −0.0199047 0.999802i \(-0.506336\pi\)
−0.0199047 + 0.999802i \(0.506336\pi\)
\(632\) 0 0
\(633\) 48.5179 + 13.0003i 1.92842 + 0.516717i
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) −29.0421 + 38.8401i −1.15069 + 1.53890i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(642\) 0 0
\(643\) 13.4722 13.4722i 0.531291 0.531291i −0.389665 0.920957i \(-0.627409\pi\)
0.920957 + 0.389665i \(0.127409\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 0 0 0.258819 0.965926i \(-0.416667\pi\)
−0.258819 + 0.965926i \(0.583333\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) −10.5000 30.3109i −0.411527 1.18798i
\(652\) 26.9444 + 26.9444i 1.05522 + 1.05522i
\(653\) 0 0 −0.258819 0.965926i \(-0.583333\pi\)
0.258819 + 0.965926i \(0.416667\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) −3.67423 3.67423i −0.143346 0.143346i
\(658\) 0 0
\(659\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(660\) 0 0
\(661\) 5.50000 + 9.52628i 0.213925 + 0.370529i 0.952940 0.303160i \(-0.0980418\pi\)
−0.739014 + 0.673690i \(0.764708\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 0 0
\(668\) 0 0
\(669\) 28.5788 16.5000i 1.10492 0.637927i
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) −35.5176 + 35.5176i −1.36910 + 1.36910i −0.507381 + 0.861722i \(0.669386\pi\)
−0.861722 + 0.507381i \(0.830614\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) −35.0000 60.6218i −1.34615 2.33161i
\(677\) 0 0 0.965926 0.258819i \(-0.0833333\pi\)
−0.965926 + 0.258819i \(0.916667\pi\)
\(678\) 0 0
\(679\) −10.3923 9.00000i −0.398820 0.345388i
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 0 0 −0.965926 0.258819i \(-0.916667\pi\)
0.965926 + 0.258819i \(0.0833333\pi\)
\(684\) −41.5692 24.0000i −1.58944 0.917663i
\(685\) 0 0
\(686\) 0 0
\(687\) 26.9444 + 26.9444i 1.02799 + 1.02799i
\(688\) 12.5521 + 46.8449i 0.478543 + 1.78595i
\(689\) 0 0
\(690\) 0 0
\(691\) 20.5000 35.5070i 0.779857 1.35075i −0.152167 0.988355i \(-0.548625\pi\)
0.932024 0.362397i \(-0.118041\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 0 0
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(702\) 0 0
\(703\) −40.1528 10.7589i −1.51439 0.405780i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) −45.8993 26.5000i −1.72379 0.995228i −0.910679 0.413114i \(-0.864441\pi\)
−0.813107 0.582115i \(-0.802225\pi\)
\(710\) 0 0
\(711\) 25.5000 + 44.1673i 0.956325 + 1.65640i
\(712\) 0 0
\(713\) 0 0
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(720\) 0 0
\(721\) −13.5000 38.9711i −0.502766 1.45136i
\(722\) 0 0
\(723\) 13.8969 + 51.8640i 0.516832 + 1.92884i
\(724\) 12.1244 7.00000i 0.450598 0.260153i
\(725\) 0 0
\(726\) 0 0
\(727\) −22.0454 22.0454i −0.817619 0.817619i 0.168144 0.985763i \(-0.446223\pi\)
−0.985763 + 0.168144i \(0.946223\pi\)
\(728\) 0 0
\(729\) 27.0000i 1.00000i
\(730\) 0 0
\(731\) 0 0
\(732\) 0.896575 3.34607i 0.0331384 0.123674i
\(733\) −8.51747 31.7876i −0.314600 1.17410i −0.924362 0.381518i \(-0.875402\pi\)
0.609762 0.792585i \(-0.291265\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) −32.0429 + 18.5000i −1.17872 + 0.680534i −0.955718 0.294285i \(-0.904919\pi\)
−0.223001 + 0.974818i \(0.571585\pi\)
\(740\) 0 0
\(741\) 96.0000 3.52665
\(742\) 0 0
\(743\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) 5.50000 9.52628i 0.200698 0.347619i −0.748056 0.663636i \(-0.769012\pi\)
0.948753 + 0.316017i \(0.102346\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 0 0
\(756\) 27.0000 + 5.19615i 0.981981 + 0.188982i
\(757\) 33.0681 + 33.0681i 1.20188 + 1.20188i 0.973594 + 0.228287i \(0.0733125\pi\)
0.228287 + 0.973594i \(0.426688\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(762\) 0 0
\(763\) −4.38134 + 2.96713i −0.158615 + 0.107417i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 0 0
\(768\) −7.17260 26.7685i −0.258819 0.965926i
\(769\) 49.0000i 1.76699i 0.468445 + 0.883493i \(0.344814\pi\)
−0.468445 + 0.883493i \(0.655186\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 23.4225 6.27603i 0.842993 0.225879i
\(773\) 0 0 −0.965926 0.258819i \(-0.916667\pi\)
0.965926 + 0.258819i \(0.0833333\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 23.7506 1.70522i 0.852049 0.0611744i
\(778\) 0 0
\(779\) 0 0
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 0 0
\(784\) 10.3923 26.0000i 0.371154 0.928571i
\(785\) 0 0
\(786\) 0 0
\(787\) 12.1038 45.1719i 0.431453 1.61020i −0.317962 0.948103i \(-0.602999\pi\)
0.749415 0.662100i \(-0.230335\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 1.79315 + 6.69213i 0.0636767 + 0.237645i
\(794\) 0 0
\(795\) 0 0
\(796\) −17.0000 + 29.4449i −0.602549 + 1.04365i
\(797\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 0 0
\(804\) 12.0000i 0.423207i
\(805\) 0 0
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(810\) 0 0
\(811\) 19.0000 0.667180 0.333590 0.942718i \(-0.391740\pi\)
0.333590 + 0.942718i \(0.391740\pi\)
\(812\) 0 0
\(813\) 34.2929 34.2929i 1.20270 1.20270i
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 93.6898 25.1041i 3.27779 0.878282i
\(818\) 0 0
\(819\) −51.9615 + 18.0000i −1.81568 + 0.628971i
\(820\) 0 0
\(821\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(822\) 0 0
\(823\) 31.7876 + 8.51747i 1.10805 + 0.296900i 0.766037 0.642796i \(-0.222226\pi\)
0.342010 + 0.939696i \(0.388892\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(828\) 0 0
\(829\) 45.8993 26.5000i 1.59415 0.920383i 0.601566 0.798823i \(-0.294544\pi\)
0.992584 0.121560i \(-0.0387897\pi\)
\(830\) 0 0
\(831\) 18.0000 31.1769i 0.624413 1.08152i
\(832\) 39.1918 + 39.1918i 1.35873 + 1.35873i
\(833\) 0 0
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 9.41404 35.1337i 0.325397 1.21440i
\(838\) 0 0
\(839\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(840\) 0 0
\(841\) −29.0000 −1.00000
\(842\) 0 0
\(843\) 0 0
\(844\) 50.2295 29.0000i 1.72897 0.998221i
\(845\) 0 0
\(846\) 0 0
\(847\) −12.7093 + 26.1815i −0.436698 + 0.899608i
\(848\) 0 0
\(849\) −15.5885 9.00000i −0.534994 0.308879i
\(850\) 0 0
\(851\) 0 0
\(852\) 0 0
\(853\) −33.0681 + 33.0681i −1.13223 + 1.13223i −0.142425 + 0.989806i \(0.545490\pi\)
−0.989806 + 0.142425i \(0.954510\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 0 0 0.258819 0.965926i \(-0.416667\pi\)
−0.258819 + 0.965926i \(0.583333\pi\)
\(858\) 0 0
\(859\) 48.4974 + 28.0000i 1.65471 + 0.955348i 0.975097 + 0.221777i \(0.0711857\pi\)
0.679613 + 0.733571i \(0.262148\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 0 0 −0.258819 0.965926i \(-0.583333\pi\)
0.258819 + 0.965926i \(0.416667\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) −20.8207 20.8207i −0.707107 0.707107i
\(868\) −33.3220 16.1755i −1.13102 0.549033i
\(869\) 0 0
\(870\) 0 0
\(871\) 12.0000 + 20.7846i 0.406604 + 0.704260i
\(872\) 0 0
\(873\) −4.03459 15.0573i −0.136550 0.509612i
\(874\) 0 0
\(875\) 0 0
\(876\) −6.00000 −0.202721
\(877\) 5.01910 1.34486i 0.169483 0.0454128i −0.173080 0.984908i \(-0.555372\pi\)
0.342563 + 0.939495i \(0.388705\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(882\) 0 0
\(883\) −41.6413 + 41.6413i −1.40134 + 1.40134i −0.605487 + 0.795855i \(0.707022\pi\)
−0.795855 + 0.605487i \(0.792978\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 0 0 0.965926 0.258819i \(-0.0833333\pi\)
−0.965926 + 0.258819i \(0.916667\pi\)
\(888\) 0 0
\(889\) −45.0333 39.0000i −1.51037 1.30802i
\(890\) 0 0
\(891\) 0 0
\(892\) 9.86233 36.8067i 0.330215 1.23238i
\(893\) 0 0
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 0 0
\(900\) 0 0
\(901\) 0 0
\(902\) 0 0
\(903\) −46.0041 + 31.1548i −1.53092 + 1.03677i
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) −14.7935 + 55.2101i −0.491210 + 1.83322i 0.0590889 + 0.998253i \(0.481180\pi\)
−0.550299 + 0.834968i \(0.685486\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(912\) −53.5370 + 14.3452i −1.77279 + 0.475017i
\(913\) 0 0
\(914\) 0 0
\(915\) 0 0
\(916\) 44.0000 1.45380
\(917\) 0 0
\(918\) 0 0
\(919\) 0.866025 + 0.500000i 0.0285675 + 0.0164935i 0.514216 0.857661i \(-0.328083\pi\)
−0.485648 + 0.874154i \(0.661416\pi\)
\(920\) 0 0
\(921\) 1.50000 + 2.59808i 0.0494267 + 0.0856095i
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) 12.1038 45.1719i 0.397540 1.48364i
\(928\) 0 0
\(929\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(930\) 0 0
\(931\) −52.0000 20.7846i −1.70423 0.681188i
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) −35.5176 35.5176i −1.16031 1.16031i −0.984408 0.175902i \(-0.943716\pi\)
−0.175902 0.984408i \(-0.556284\pi\)
\(938\) 0 0
\(939\) 9.00000i 0.293704i
\(940\) 0 0
\(941\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(942\) 0 0
\(943\) 0 0
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 0 0 0.965926 0.258819i \(-0.0833333\pi\)
−0.965926 + 0.258819i \(0.916667\pi\)
\(948\) 56.8831 + 15.2418i 1.84748 + 0.495030i
\(949\) 10.3923 6.00000i 0.337348 0.194768i
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) −9.00000 + 15.5885i −0.290323 + 0.502853i
\(962\) 0 0
\(963\) 0 0
\(964\) 53.6936 + 31.0000i 1.72935 + 0.998443i
\(965\) 0 0
\(966\) 0 0
\(967\) 33.0681 + 33.0681i 1.06340 + 1.06340i 0.997849 + 0.0655495i \(0.0208800\pi\)
0.0655495 + 0.997849i \(0.479120\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(972\) 22.0454 + 22.0454i 0.707107 + 0.707107i
\(973\) 4.35778 + 60.6960i 0.139704 + 1.94583i
\(974\) 0 0
\(975\) 0 0
\(976\) −2.00000 3.46410i −0.0640184 0.110883i
\(977\) 0 0 0.258819 0.965926i \(-0.416667\pi\)
−0.258819 + 0.965926i \(0.583333\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 0 0
\(981\) −6.00000 −0.191565
\(982\) 0 0
\(983\) 0 0 −0.965926 0.258819i \(-0.916667\pi\)
0.965926 + 0.258819i \(0.0833333\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 0 0
\(988\) 78.3837 78.3837i 2.49372 2.49372i
\(989\) 0 0
\(990\) 0 0
\(991\) −8.50000 14.7224i −0.270011 0.467673i 0.698853 0.715265i \(-0.253694\pi\)
−0.968864 + 0.247592i \(0.920361\pi\)
\(992\) 0 0
\(993\) 37.9671 37.9671i 1.20485 1.20485i
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) −5.82774 + 21.7494i −0.184566 + 0.688811i 0.810157 + 0.586214i \(0.199382\pi\)
−0.994723 + 0.102598i \(0.967285\pi\)
\(998\) 0 0
\(999\) 23.3827 + 13.5000i 0.739795 + 0.427121i
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 525.2.bf.c.107.2 yes 8
3.2 odd 2 CM 525.2.bf.c.107.2 yes 8
5.2 odd 4 inner 525.2.bf.c.443.2 yes 8
5.3 odd 4 inner 525.2.bf.c.443.1 yes 8
5.4 even 2 inner 525.2.bf.c.107.1 yes 8
7.4 even 3 inner 525.2.bf.c.32.1 8
15.2 even 4 inner 525.2.bf.c.443.2 yes 8
15.8 even 4 inner 525.2.bf.c.443.1 yes 8
15.14 odd 2 inner 525.2.bf.c.107.1 yes 8
21.11 odd 6 inner 525.2.bf.c.32.1 8
35.4 even 6 inner 525.2.bf.c.32.2 yes 8
35.18 odd 12 inner 525.2.bf.c.368.2 yes 8
35.32 odd 12 inner 525.2.bf.c.368.1 yes 8
105.32 even 12 inner 525.2.bf.c.368.1 yes 8
105.53 even 12 inner 525.2.bf.c.368.2 yes 8
105.74 odd 6 inner 525.2.bf.c.32.2 yes 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
525.2.bf.c.32.1 8 7.4 even 3 inner
525.2.bf.c.32.1 8 21.11 odd 6 inner
525.2.bf.c.32.2 yes 8 35.4 even 6 inner
525.2.bf.c.32.2 yes 8 105.74 odd 6 inner
525.2.bf.c.107.1 yes 8 5.4 even 2 inner
525.2.bf.c.107.1 yes 8 15.14 odd 2 inner
525.2.bf.c.107.2 yes 8 1.1 even 1 trivial
525.2.bf.c.107.2 yes 8 3.2 odd 2 CM
525.2.bf.c.368.1 yes 8 35.32 odd 12 inner
525.2.bf.c.368.1 yes 8 105.32 even 12 inner
525.2.bf.c.368.2 yes 8 35.18 odd 12 inner
525.2.bf.c.368.2 yes 8 105.53 even 12 inner
525.2.bf.c.443.1 yes 8 5.3 odd 4 inner
525.2.bf.c.443.1 yes 8 15.8 even 4 inner
525.2.bf.c.443.2 yes 8 5.2 odd 4 inner
525.2.bf.c.443.2 yes 8 15.2 even 4 inner