Properties

Label 3024.2.b.u.1567.8
Level $3024$
Weight $2$
Character 3024.1567
Analytic conductor $24.147$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [3024,2,Mod(1567,3024)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3024, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0, 0, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("3024.1567");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3024 = 2^{4} \cdot 3^{3} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3024.b (of order \(2\), degree \(1\), 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: \(24.1467615712\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.4161798144.13
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 6x^{6} + 29x^{4} + 42x^{2} + 49 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{6}\cdot 3^{3} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 1567.8
Root \(1.05050 - 1.81952i\) of defining polynomial
Character \(\chi\) \(=\) 3024.1567
Dual form 3024.2.b.u.1567.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+3.63904i q^{5} +(-1.00000 + 2.44949i) q^{7} +O(q^{10})\) \(q+3.63904i q^{5} +(-1.00000 + 2.44949i) q^{7} -1.50734i q^{11} -5.91359i q^{13} -5.14639i q^{17} +5.24264 q^{19} -8.78543i q^{23} -8.24264 q^{25} -8.91380 q^{29} +3.24264 q^{31} +(-8.91380 - 3.63904i) q^{35} +1.00000 q^{37} +6.65373i q^{41} -9.37769i q^{43} -3.69222 q^{47} +(-5.00000 - 4.89898i) q^{49} -12.6060 q^{53} +5.48528 q^{55} +8.91380 q^{59} +3.46410i q^{61} +21.5198 q^{65} -10.8126i q^{67} +3.63904i q^{71} -0.420266i q^{73} +(3.69222 + 1.50734i) q^{77} -4.47871i q^{79} +3.69222 q^{83} +18.7279 q^{85} -13.9318i q^{89} +(14.4853 + 5.91359i) q^{91} +19.0782i q^{95} -13.2621i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 8 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 8 q^{7} + 8 q^{19} - 32 q^{25} - 8 q^{31} + 8 q^{37} - 40 q^{49} - 24 q^{55} + 48 q^{85} + 48 q^{91}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3024\mathbb{Z}\right)^\times\).

\(n\) \(757\) \(785\) \(1135\) \(2593\)
\(\chi(n)\) \(1\) \(1\) \(-1\) \(-1\)

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 0
\(3\) 0 0
\(4\) 0 0
\(5\) 3.63904i 1.62743i 0.581264 + 0.813715i \(0.302558\pi\)
−0.581264 + 0.813715i \(0.697442\pi\)
\(6\) 0 0
\(7\) −1.00000 + 2.44949i −0.377964 + 0.925820i
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 1.50734i 0.454481i −0.973839 0.227240i \(-0.927030\pi\)
0.973839 0.227240i \(-0.0729703\pi\)
\(12\) 0 0
\(13\) 5.91359i 1.64014i −0.572267 0.820068i \(-0.693936\pi\)
0.572267 0.820068i \(-0.306064\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 5.14639i 1.24818i −0.781352 0.624091i \(-0.785469\pi\)
0.781352 0.624091i \(-0.214531\pi\)
\(18\) 0 0
\(19\) 5.24264 1.20274 0.601372 0.798969i \(-0.294621\pi\)
0.601372 + 0.798969i \(0.294621\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 8.78543i 1.83189i −0.401305 0.915944i \(-0.631443\pi\)
0.401305 0.915944i \(-0.368557\pi\)
\(24\) 0 0
\(25\) −8.24264 −1.64853
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −8.91380 −1.65525 −0.827626 0.561281i \(-0.810309\pi\)
−0.827626 + 0.561281i \(0.810309\pi\)
\(30\) 0 0
\(31\) 3.24264 0.582395 0.291198 0.956663i \(-0.405946\pi\)
0.291198 + 0.956663i \(0.405946\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −8.91380 3.63904i −1.50671 0.615111i
\(36\) 0 0
\(37\) 1.00000 0.164399 0.0821995 0.996616i \(-0.473806\pi\)
0.0821995 + 0.996616i \(0.473806\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 6.65373i 1.03914i 0.854429 + 0.519569i \(0.173907\pi\)
−0.854429 + 0.519569i \(0.826093\pi\)
\(42\) 0 0
\(43\) 9.37769i 1.43008i −0.699081 0.715042i \(-0.746407\pi\)
0.699081 0.715042i \(-0.253593\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −3.69222 −0.538565 −0.269283 0.963061i \(-0.586787\pi\)
−0.269283 + 0.963061i \(0.586787\pi\)
\(48\) 0 0
\(49\) −5.00000 4.89898i −0.714286 0.699854i
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −12.6060 −1.73157 −0.865785 0.500416i \(-0.833180\pi\)
−0.865785 + 0.500416i \(0.833180\pi\)
\(54\) 0 0
\(55\) 5.48528 0.739635
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 8.91380 1.16048 0.580239 0.814446i \(-0.302959\pi\)
0.580239 + 0.814446i \(0.302959\pi\)
\(60\) 0 0
\(61\) 3.46410i 0.443533i 0.975100 + 0.221766i \(0.0711822\pi\)
−0.975100 + 0.221766i \(0.928818\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 21.5198 2.66921
\(66\) 0 0
\(67\) 10.8126i 1.32097i −0.750841 0.660483i \(-0.770352\pi\)
0.750841 0.660483i \(-0.229648\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 3.63904i 0.431875i 0.976407 + 0.215938i \(0.0692808\pi\)
−0.976407 + 0.215938i \(0.930719\pi\)
\(72\) 0 0
\(73\) 0.420266i 0.0491884i −0.999698 0.0245942i \(-0.992171\pi\)
0.999698 0.0245942i \(-0.00782937\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 3.69222 + 1.50734i 0.420767 + 0.171777i
\(78\) 0 0
\(79\) 4.47871i 0.503895i −0.967741 0.251947i \(-0.918929\pi\)
0.967741 0.251947i \(-0.0810710\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 3.69222 0.405273 0.202637 0.979254i \(-0.435049\pi\)
0.202637 + 0.979254i \(0.435049\pi\)
\(84\) 0 0
\(85\) 18.7279 2.03133
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 13.9318i 1.47677i −0.674380 0.738385i \(-0.735589\pi\)
0.674380 0.738385i \(-0.264411\pi\)
\(90\) 0 0
\(91\) 14.4853 + 5.91359i 1.51847 + 0.619913i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 19.0782i 1.95738i
\(96\) 0 0
\(97\) 13.2621i 1.34656i −0.739388 0.673279i \(-0.764885\pi\)
0.739388 0.673279i \(-0.235115\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 5.14639i 0.512084i 0.966666 + 0.256042i \(0.0824186\pi\)
−0.966666 + 0.256042i \(0.917581\pi\)
\(102\) 0 0
\(103\) 11.7279 1.15559 0.577793 0.816183i \(-0.303914\pi\)
0.577793 + 0.816183i \(0.303914\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 5.14639i 0.497520i 0.968565 + 0.248760i \(0.0800230\pi\)
−0.968565 + 0.248760i \(0.919977\pi\)
\(108\) 0 0
\(109\) −11.2426 −1.07685 −0.538425 0.842674i \(-0.680980\pi\)
−0.538425 + 0.842674i \(0.680980\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 5.22158 0.491205 0.245603 0.969371i \(-0.421014\pi\)
0.245603 + 0.969371i \(0.421014\pi\)
\(114\) 0 0
\(115\) 31.9706 2.98127
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 12.6060 + 5.14639i 1.15559 + 0.471768i
\(120\) 0 0
\(121\) 8.72792 0.793447
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 11.8001i 1.05543i
\(126\) 0 0
\(127\) 9.37769i 0.832136i −0.909334 0.416068i \(-0.863408\pi\)
0.909334 0.416068i \(-0.136592\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 8.91380 0.778802 0.389401 0.921068i \(-0.372682\pi\)
0.389401 + 0.921068i \(0.372682\pi\)
\(132\) 0 0
\(133\) −5.24264 + 12.8418i −0.454595 + 1.11352i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −16.2982 −1.39245 −0.696226 0.717823i \(-0.745139\pi\)
−0.696226 + 0.717823i \(0.745139\pi\)
\(138\) 0 0
\(139\) −1.51472 −0.128477 −0.0642384 0.997935i \(-0.520462\pi\)
−0.0642384 + 0.997935i \(0.520462\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) −8.91380 −0.745409
\(144\) 0 0
\(145\) 32.4377i 2.69381i
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 12.6060 1.03273 0.516363 0.856370i \(-0.327286\pi\)
0.516363 + 0.856370i \(0.327286\pi\)
\(150\) 0 0
\(151\) 18.1610i 1.47792i 0.673747 + 0.738962i \(0.264684\pi\)
−0.673747 + 0.738962i \(0.735316\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 11.8001i 0.947808i
\(156\) 0 0
\(157\) 1.01461i 0.0809748i 0.999180 + 0.0404874i \(0.0128911\pi\)
−0.999180 + 0.0404874i \(0.987109\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 21.5198 + 8.78543i 1.69600 + 0.692389i
\(162\) 0 0
\(163\) 2.44949i 0.191859i 0.995388 + 0.0959294i \(0.0305823\pi\)
−0.995388 + 0.0959294i \(0.969418\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −17.8276 −1.37954 −0.689771 0.724028i \(-0.742289\pi\)
−0.689771 + 0.724028i \(0.742289\pi\)
\(168\) 0 0
\(169\) −21.9706 −1.69004
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 13.0488i 0.992085i 0.868298 + 0.496042i \(0.165214\pi\)
−0.868298 + 0.496042i \(0.834786\pi\)
\(174\) 0 0
\(175\) 8.24264 20.1903i 0.623085 1.52624i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 16.6879i 1.24731i 0.781699 + 0.623655i \(0.214353\pi\)
−0.781699 + 0.623655i \(0.785647\pi\)
\(180\) 0 0
\(181\) 0.594346i 0.0441774i −0.999756 0.0220887i \(-0.992968\pi\)
0.999756 0.0220887i \(-0.00703162\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 3.63904i 0.267548i
\(186\) 0 0
\(187\) −7.75736 −0.567274
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 20.3269i 1.47080i −0.677631 0.735402i \(-0.736993\pi\)
0.677631 0.735402i \(-0.263007\pi\)
\(192\) 0 0
\(193\) 14.9706 1.07760 0.538802 0.842432i \(-0.318877\pi\)
0.538802 + 0.842432i \(0.318877\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(198\) 0 0
\(199\) −11.0000 −0.779769 −0.389885 0.920864i \(-0.627485\pi\)
−0.389885 + 0.920864i \(0.627485\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 8.91380 21.8343i 0.625626 1.53246i
\(204\) 0 0
\(205\) −24.2132 −1.69112
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 7.90245i 0.546624i
\(210\) 0 0
\(211\) 1.43488i 0.0987811i −0.998780 0.0493905i \(-0.984272\pi\)
0.998780 0.0493905i \(-0.0157279\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 34.1258 2.32736
\(216\) 0 0
\(217\) −3.24264 + 7.94282i −0.220125 + 0.539193i
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) −30.4336 −2.04719
\(222\) 0 0
\(223\) 17.9706 1.20340 0.601699 0.798723i \(-0.294491\pi\)
0.601699 + 0.798723i \(0.294491\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) −12.6060 −0.836691 −0.418345 0.908288i \(-0.637390\pi\)
−0.418345 + 0.908288i \(0.637390\pi\)
\(228\) 0 0
\(229\) 12.4215i 0.820838i −0.911897 0.410419i \(-0.865383\pi\)
0.911897 0.410419i \(-0.134617\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −5.22158 −0.342077 −0.171039 0.985264i \(-0.554712\pi\)
−0.171039 + 0.985264i \(0.554712\pi\)
\(234\) 0 0
\(235\) 13.4361i 0.876477i
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 3.89766i 0.252119i −0.992023 0.126059i \(-0.959767\pi\)
0.992023 0.126059i \(-0.0402330\pi\)
\(240\) 0 0
\(241\) 6.75412i 0.435071i −0.976052 0.217536i \(-0.930198\pi\)
0.976052 0.217536i \(-0.0698018\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 17.8276 18.1952i 1.13896 1.16245i
\(246\) 0 0
\(247\) 31.0028i 1.97266i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) −7.38443 −0.466101 −0.233051 0.972465i \(-0.574871\pi\)
−0.233051 + 0.972465i \(0.574871\pi\)
\(252\) 0 0
\(253\) −13.2426 −0.832558
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 13.0488i 0.813964i −0.913436 0.406982i \(-0.866581\pi\)
0.913436 0.406982i \(-0.133419\pi\)
\(258\) 0 0
\(259\) −1.00000 + 2.44949i −0.0621370 + 0.152204i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 2.75606i 0.169946i 0.996383 + 0.0849731i \(0.0270804\pi\)
−0.996383 + 0.0849731i \(0.972920\pi\)
\(264\) 0 0
\(265\) 45.8739i 2.81801i
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 22.0929i 1.34703i 0.739175 + 0.673513i \(0.235216\pi\)
−0.739175 + 0.673513i \(0.764784\pi\)
\(270\) 0 0
\(271\) 8.00000 0.485965 0.242983 0.970031i \(-0.421874\pi\)
0.242983 + 0.970031i \(0.421874\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 12.4245i 0.749224i
\(276\) 0 0
\(277\) 10.5147 0.631768 0.315884 0.948798i \(-0.397699\pi\)
0.315884 + 0.948798i \(0.397699\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) −14.1354 −0.843246 −0.421623 0.906771i \(-0.638539\pi\)
−0.421623 + 0.906771i \(0.638539\pi\)
\(282\) 0 0
\(283\) 4.00000 0.237775 0.118888 0.992908i \(-0.462067\pi\)
0.118888 + 0.992908i \(0.462067\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) −16.2982 6.65373i −0.962054 0.392757i
\(288\) 0 0
\(289\) −9.48528 −0.557958
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 6.39511i 0.373606i 0.982397 + 0.186803i \(0.0598126\pi\)
−0.982397 + 0.186803i \(0.940187\pi\)
\(294\) 0 0
\(295\) 32.4377i 1.88860i
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) −51.9534 −3.00454
\(300\) 0 0
\(301\) 22.9706 + 9.37769i 1.32400 + 0.540521i
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) −12.6060 −0.721818
\(306\) 0 0
\(307\) −15.2426 −0.869943 −0.434972 0.900444i \(-0.643242\pi\)
−0.434972 + 0.900444i \(0.643242\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) −8.91380 −0.505455 −0.252728 0.967537i \(-0.581328\pi\)
−0.252728 + 0.967537i \(0.581328\pi\)
\(312\) 0 0
\(313\) 7.94282i 0.448954i 0.974479 + 0.224477i \(0.0720674\pi\)
−0.974479 + 0.224477i \(0.927933\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −14.1354 −0.793922 −0.396961 0.917835i \(-0.629935\pi\)
−0.396961 + 0.917835i \(0.629935\pi\)
\(318\) 0 0
\(319\) 13.4361i 0.752279i
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 26.9806i 1.50124i
\(324\) 0 0
\(325\) 48.7436i 2.70381i
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 3.69222 9.04405i 0.203559 0.498615i
\(330\) 0 0
\(331\) 23.6544i 1.30016i 0.759865 + 0.650081i \(0.225265\pi\)
−0.759865 + 0.650081i \(0.774735\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 39.3474 2.14978
\(336\) 0 0
\(337\) −2.02944 −0.110550 −0.0552752 0.998471i \(-0.517604\pi\)
−0.0552752 + 0.998471i \(0.517604\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 4.88777i 0.264687i
\(342\) 0 0
\(343\) 17.0000 7.34847i 0.917914 0.396780i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 13.9318i 0.747899i −0.927449 0.373949i \(-0.878003\pi\)
0.927449 0.373949i \(-0.121997\pi\)
\(348\) 0 0
\(349\) 10.9867i 0.588102i −0.955790 0.294051i \(-0.904996\pi\)
0.955790 0.294051i \(-0.0950035\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 12.6831i 0.675053i −0.941316 0.337526i \(-0.890410\pi\)
0.941316 0.337526i \(-0.109590\pi\)
\(354\) 0 0
\(355\) −13.2426 −0.702846
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 26.9806i 1.42398i −0.702187 0.711992i \(-0.747793\pi\)
0.702187 0.711992i \(-0.252207\pi\)
\(360\) 0 0
\(361\) 8.48528 0.446594
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 1.52937 0.0800507
\(366\) 0 0
\(367\) 25.0000 1.30499 0.652495 0.757793i \(-0.273722\pi\)
0.652495 + 0.757793i \(0.273722\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 12.6060 30.8783i 0.654472 1.60312i
\(372\) 0 0
\(373\) 20.2132 1.04660 0.523300 0.852149i \(-0.324701\pi\)
0.523300 + 0.852149i \(0.324701\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 52.7126i 2.71484i
\(378\) 0 0
\(379\) 17.1464i 0.880753i 0.897813 + 0.440376i \(0.145155\pi\)
−0.897813 + 0.440376i \(0.854845\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) −23.0492 −1.17776 −0.588879 0.808221i \(-0.700431\pi\)
−0.588879 + 0.808221i \(0.700431\pi\)
\(384\) 0 0
\(385\) −5.48528 + 13.4361i −0.279556 + 0.684769i
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 5.22158 0.264745 0.132372 0.991200i \(-0.457741\pi\)
0.132372 + 0.991200i \(0.457741\pi\)
\(390\) 0 0
\(391\) −45.2132 −2.28653
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 16.2982 0.820053
\(396\) 0 0
\(397\) 31.4231i 1.57708i 0.614983 + 0.788540i \(0.289163\pi\)
−0.614983 + 0.788540i \(0.710837\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −3.69222 −0.184381 −0.0921903 0.995741i \(-0.529387\pi\)
−0.0921903 + 0.995741i \(0.529387\pi\)
\(402\) 0 0
\(403\) 19.1757i 0.955207i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 1.50734i 0.0747161i
\(408\) 0 0
\(409\) 7.94282i 0.392747i 0.980529 + 0.196373i \(0.0629165\pi\)
−0.980529 + 0.196373i \(0.937084\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) −8.91380 + 21.8343i −0.438619 + 1.07439i
\(414\) 0 0
\(415\) 13.4361i 0.659554i
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 5.22158 0.255091 0.127546 0.991833i \(-0.459290\pi\)
0.127546 + 0.991833i \(0.459290\pi\)
\(420\) 0 0
\(421\) −9.97056 −0.485935 −0.242968 0.970034i \(-0.578121\pi\)
−0.242968 + 0.970034i \(0.578121\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 42.4198i 2.05766i
\(426\) 0 0
\(427\) −8.48528 3.46410i −0.410632 0.167640i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 10.0342i 0.483328i −0.970360 0.241664i \(-0.922307\pi\)
0.970360 0.241664i \(-0.0776932\pi\)
\(432\) 0 0
\(433\) 34.4669i 1.65638i −0.560451 0.828188i \(-0.689372\pi\)
0.560451 0.828188i \(-0.310628\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 46.0588i 2.20329i
\(438\) 0 0
\(439\) 1.51472 0.0722936 0.0361468 0.999346i \(-0.488492\pi\)
0.0361468 + 0.999346i \(0.488492\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 31.8684i 1.51411i 0.653349 + 0.757057i \(0.273364\pi\)
−0.653349 + 0.757057i \(0.726636\pi\)
\(444\) 0 0
\(445\) 50.6985 2.40334
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(450\) 0 0
\(451\) 10.0294 0.472268
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) −21.5198 + 52.7126i −1.00886 + 2.47120i
\(456\) 0 0
\(457\) −15.4853 −0.724371 −0.362185 0.932106i \(-0.617969\pi\)
−0.362185 + 0.932106i \(0.617969\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 29.3710i 1.36794i −0.729509 0.683971i \(-0.760251\pi\)
0.729509 0.683971i \(-0.239749\pi\)
\(462\) 0 0
\(463\) 17.5667i 0.816394i −0.912894 0.408197i \(-0.866158\pi\)
0.912894 0.408197i \(-0.133842\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 41.5103 1.92087 0.960433 0.278511i \(-0.0898408\pi\)
0.960433 + 0.278511i \(0.0898408\pi\)
\(468\) 0 0
\(469\) 26.4853 + 10.8126i 1.22298 + 0.499278i
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) −14.1354 −0.649946
\(474\) 0 0
\(475\) −43.2132 −1.98276
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 17.8276 0.814564 0.407282 0.913302i \(-0.366477\pi\)
0.407282 + 0.913302i \(0.366477\pi\)
\(480\) 0 0
\(481\) 5.91359i 0.269637i
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 48.2612 2.19143
\(486\) 0 0
\(487\) 29.5680i 1.33985i −0.742428 0.669926i \(-0.766326\pi\)
0.742428 0.669926i \(-0.233674\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 6.65373i 0.300278i 0.988665 + 0.150139i \(0.0479722\pi\)
−0.988665 + 0.150139i \(0.952028\pi\)
\(492\) 0 0
\(493\) 45.8739i 2.06605i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −8.91380 3.63904i −0.399839 0.163233i
\(498\) 0 0
\(499\) 17.7408i 0.794186i −0.917778 0.397093i \(-0.870019\pi\)
0.917778 0.397093i \(-0.129981\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) −16.2982 −0.726702 −0.363351 0.931652i \(-0.618367\pi\)
−0.363351 + 0.931652i \(0.618367\pi\)
\(504\) 0 0
\(505\) −18.7279 −0.833382
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 5.14639i 0.228110i −0.993474 0.114055i \(-0.963616\pi\)
0.993474 0.114055i \(-0.0363839\pi\)
\(510\) 0 0
\(511\) 1.02944 + 0.420266i 0.0455396 + 0.0185915i
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 42.6784i 1.88064i
\(516\) 0 0
\(517\) 5.56543i 0.244767i
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 23.3416i 1.02261i −0.859398 0.511307i \(-0.829161\pi\)
0.859398 0.511307i \(-0.170839\pi\)
\(522\) 0 0
\(523\) −7.00000 −0.306089 −0.153044 0.988219i \(-0.548908\pi\)
−0.153044 + 0.988219i \(0.548908\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 16.6879i 0.726935i
\(528\) 0 0
\(529\) −54.1838 −2.35582
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 39.3474 1.70433
\(534\) 0 0
\(535\) −18.7279 −0.809679
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) −7.38443 + 7.53671i −0.318070 + 0.324629i
\(540\) 0 0
\(541\) 25.0000 1.07483 0.537417 0.843317i \(-0.319400\pi\)
0.537417 + 0.843317i \(0.319400\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 40.9125i 1.75250i
\(546\) 0 0
\(547\) 13.8564i 0.592457i 0.955117 + 0.296229i \(0.0957290\pi\)
−0.955117 + 0.296229i \(0.904271\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) −46.7319 −1.99084
\(552\) 0 0
\(553\) 10.9706 + 4.47871i 0.466516 + 0.190454i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 43.0396 1.82365 0.911824 0.410581i \(-0.134674\pi\)
0.911824 + 0.410581i \(0.134674\pi\)
\(558\) 0 0
\(559\) −55.4558 −2.34553
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 3.69222 0.155608 0.0778042 0.996969i \(-0.475209\pi\)
0.0778042 + 0.996969i \(0.475209\pi\)
\(564\) 0 0
\(565\) 19.0016i 0.799402i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −25.2120 −1.05694 −0.528472 0.848951i \(-0.677235\pi\)
−0.528472 + 0.848951i \(0.677235\pi\)
\(570\) 0 0
\(571\) 21.3790i 0.894681i 0.894364 + 0.447341i \(0.147629\pi\)
−0.894364 + 0.447341i \(0.852371\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 72.4151i 3.01992i
\(576\) 0 0
\(577\) 0.840532i 0.0349918i −0.999847 0.0174959i \(-0.994431\pi\)
0.999847 0.0174959i \(-0.00556940\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) −3.69222 + 9.04405i −0.153179 + 0.375210i
\(582\) 0 0
\(583\) 19.0016i 0.786965i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 19.9905 0.825094 0.412547 0.910936i \(-0.364639\pi\)
0.412547 + 0.910936i \(0.364639\pi\)
\(588\) 0 0
\(589\) 17.0000 0.700473
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 7.53671i 0.309495i 0.987954 + 0.154748i \(0.0494565\pi\)
−0.987954 + 0.154748i \(0.950544\pi\)
\(594\) 0 0
\(595\) −18.7279 + 45.8739i −0.767770 + 1.88064i
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 19.0782i 0.779514i −0.920918 0.389757i \(-0.872559\pi\)
0.920918 0.389757i \(-0.127441\pi\)
\(600\) 0 0
\(601\) 15.4654i 0.630845i −0.948951 0.315423i \(-0.897854\pi\)
0.948951 0.315423i \(-0.102146\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 31.7613i 1.29128i
\(606\) 0 0
\(607\) −41.9411 −1.70234 −0.851169 0.524892i \(-0.824106\pi\)
−0.851169 + 0.524892i \(0.824106\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 21.8343i 0.883320i
\(612\) 0 0
\(613\) −40.2132 −1.62420 −0.812098 0.583521i \(-0.801675\pi\)
−0.812098 + 0.583521i \(0.801675\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −14.1354 −0.569069 −0.284535 0.958666i \(-0.591839\pi\)
−0.284535 + 0.958666i \(0.591839\pi\)
\(618\) 0 0
\(619\) −28.9411 −1.16324 −0.581621 0.813460i \(-0.697581\pi\)
−0.581621 + 0.813460i \(0.697581\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 34.1258 + 13.9318i 1.36722 + 0.558166i
\(624\) 0 0
\(625\) 1.72792 0.0691169
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 5.14639i 0.205200i
\(630\) 0 0
\(631\) 6.75412i 0.268877i −0.990922 0.134439i \(-0.957077\pi\)
0.990922 0.134439i \(-0.0429231\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 34.1258 1.35424
\(636\) 0 0
\(637\) −28.9706 + 29.5680i −1.14786 + 1.17153i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 37.8181 1.49372 0.746862 0.664979i \(-0.231560\pi\)
0.746862 + 0.664979i \(0.231560\pi\)
\(642\) 0 0
\(643\) 13.7279 0.541376 0.270688 0.962667i \(-0.412749\pi\)
0.270688 + 0.962667i \(0.412749\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 44.5690 1.75219 0.876094 0.482140i \(-0.160140\pi\)
0.876094 + 0.482140i \(0.160140\pi\)
\(648\) 0 0
\(649\) 13.4361i 0.527415i
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 1.52937 0.0598487 0.0299244 0.999552i \(-0.490473\pi\)
0.0299244 + 0.999552i \(0.490473\pi\)
\(654\) 0 0
\(655\) 32.4377i 1.26745i
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 34.8831i 1.35885i −0.733744 0.679426i \(-0.762229\pi\)
0.733744 0.679426i \(-0.237771\pi\)
\(660\) 0 0
\(661\) 43.6705i 1.69859i 0.527921 + 0.849294i \(0.322972\pi\)
−0.527921 + 0.849294i \(0.677028\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) −46.7319 19.0782i −1.81218 0.739821i
\(666\) 0 0
\(667\) 78.3116i 3.03224i
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 5.22158 0.201577
\(672\) 0 0
\(673\) 18.4853 0.712555 0.356278 0.934380i \(-0.384046\pi\)
0.356278 + 0.934380i \(0.384046\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 16.9465i 0.651307i −0.945489 0.325653i \(-0.894416\pi\)
0.945489 0.325653i \(-0.105584\pi\)
\(678\) 0 0
\(679\) 32.4853 + 13.2621i 1.24667 + 0.508951i
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 22.0929i 0.845361i −0.906279 0.422680i \(-0.861089\pi\)
0.906279 0.422680i \(-0.138911\pi\)
\(684\) 0 0
\(685\) 59.3100i 2.26612i
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 74.5468i 2.84001i
\(690\) 0 0
\(691\) −43.9411 −1.67160 −0.835800 0.549035i \(-0.814995\pi\)
−0.835800 + 0.549035i \(0.814995\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 5.51213i 0.209087i
\(696\) 0 0
\(697\) 34.2426 1.29703
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) −31.9630 −1.20723 −0.603613 0.797278i \(-0.706273\pi\)
−0.603613 + 0.797278i \(0.706273\pi\)
\(702\) 0 0
\(703\) 5.24264 0.197730
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) −12.6060 5.14639i −0.474098 0.193550i
\(708\) 0 0
\(709\) 18.7574 0.704447 0.352224 0.935916i \(-0.385426\pi\)
0.352224 + 0.935916i \(0.385426\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 28.4880i 1.06688i
\(714\) 0 0
\(715\) 32.4377i 1.21310i
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) −14.1354 −0.527161 −0.263580 0.964637i \(-0.584903\pi\)
−0.263580 + 0.964637i \(0.584903\pi\)
\(720\) 0 0
\(721\) −11.7279 + 28.7274i −0.436771 + 1.06987i
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 73.4733 2.72873
\(726\) 0 0
\(727\) −26.9706 −1.00028 −0.500141 0.865944i \(-0.666719\pi\)
−0.500141 + 0.865944i \(0.666719\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) −48.2612 −1.78501
\(732\) 0 0
\(733\) 44.8592i 1.65691i −0.560053 0.828457i \(-0.689219\pi\)
0.560053 0.828457i \(-0.310781\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −16.2982 −0.600353
\(738\) 0 0
\(739\) 12.4215i 0.456933i −0.973552 0.228467i \(-0.926629\pi\)
0.973552 0.228467i \(-0.0733712\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 7.90245i 0.289913i −0.989438 0.144956i \(-0.953696\pi\)
0.989438 0.144956i \(-0.0463042\pi\)
\(744\) 0 0
\(745\) 45.8739i 1.68069i
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) −12.6060 5.14639i −0.460614 0.188045i
\(750\) 0 0
\(751\) 2.44949i 0.0893832i 0.999001 + 0.0446916i \(0.0142305\pi\)
−0.999001 + 0.0446916i \(0.985769\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) −66.0888 −2.40522
\(756\) 0 0
\(757\) 34.4853 1.25339 0.626694 0.779265i \(-0.284407\pi\)
0.626694 + 0.779265i \(0.284407\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 3.89766i 0.141290i −0.997502 0.0706451i \(-0.977494\pi\)
0.997502 0.0706451i \(-0.0225058\pi\)
\(762\) 0 0
\(763\) 11.2426 27.5387i 0.407011 0.996969i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 52.7126i 1.90334i
\(768\) 0 0
\(769\) 37.0905i 1.33752i −0.743479 0.668759i \(-0.766826\pi\)
0.743479 0.668759i \(-0.233174\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 15.6978i 0.564610i −0.959325 0.282305i \(-0.908901\pi\)
0.959325 0.282305i \(-0.0910990\pi\)
\(774\) 0 0
\(775\) −26.7279 −0.960095
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 34.8831i 1.24982i
\(780\) 0 0
\(781\) 5.48528 0.196279
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) −3.69222 −0.131781
\(786\) 0 0
\(787\) 21.0294 0.749618 0.374809 0.927102i \(-0.377708\pi\)
0.374809 + 0.927102i \(0.377708\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) −5.22158 + 12.7902i −0.185658 + 0.454768i
\(792\) 0 0
\(793\) 20.4853 0.727454
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 46.0588i 1.63149i 0.578413 + 0.815744i \(0.303672\pi\)
−0.578413 + 0.815744i \(0.696328\pi\)
\(798\) 0 0
\(799\) 19.0016i 0.672227i
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) −0.633484 −0.0223552
\(804\) 0 0
\(805\) −31.9706 + 78.3116i −1.12681 + 2.76012i
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 16.2982 0.573015 0.286508 0.958078i \(-0.407506\pi\)
0.286508 + 0.958078i \(0.407506\pi\)
\(810\) 0 0
\(811\) 8.51472 0.298992 0.149496 0.988762i \(-0.452235\pi\)
0.149496 + 0.988762i \(0.452235\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) −8.91380 −0.312237
\(816\) 0 0
\(817\) 49.1639i 1.72003i
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −41.5103 −1.44872 −0.724359 0.689423i \(-0.757864\pi\)
−0.724359 + 0.689423i \(0.757864\pi\)
\(822\) 0 0
\(823\) 29.7420i 1.03674i 0.855156 + 0.518371i \(0.173461\pi\)
−0.855156 + 0.518371i \(0.826539\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 44.8101i 1.55820i −0.626899 0.779100i \(-0.715676\pi\)
0.626899 0.779100i \(-0.284324\pi\)
\(828\) 0 0
\(829\) 14.8710i 0.516492i 0.966079 + 0.258246i \(0.0831445\pi\)
−0.966079 + 0.258246i \(0.916856\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) −25.2120 + 25.7319i −0.873545 + 0.891558i
\(834\) 0 0
\(835\) 64.8754i 2.24511i
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 11.0767 0.382408 0.191204 0.981550i \(-0.438761\pi\)
0.191204 + 0.981550i \(0.438761\pi\)
\(840\) 0 0
\(841\) 50.4558 1.73986
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 79.9518i 2.75043i
\(846\) 0 0
\(847\) −8.72792 + 21.3790i −0.299895 + 0.734590i
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 8.78543i 0.301161i
\(852\) 0 0
\(853\) 25.9298i 0.887819i 0.896072 + 0.443909i \(0.146409\pi\)
−0.896072 + 0.443909i \(0.853591\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 6.65373i 0.227287i 0.993522 + 0.113643i \(0.0362522\pi\)
−0.993522 + 0.113643i \(0.963748\pi\)
\(858\) 0 0
\(859\) −21.2426 −0.724790 −0.362395 0.932025i \(-0.618041\pi\)
−0.362395 + 0.932025i \(0.618041\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 24.4832i 0.833418i −0.909040 0.416709i \(-0.863183\pi\)
0.909040 0.416709i \(-0.136817\pi\)
\(864\) 0 0
\(865\) −47.4853 −1.61455
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) −6.75095 −0.229010
\(870\) 0 0
\(871\) −63.9411 −2.16656
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 28.9043 + 11.8001i 0.977142 + 0.398917i
\(876\) 0 0
\(877\) −19.5147 −0.658965 −0.329483 0.944162i \(-0.606874\pi\)
−0.329483 + 0.944162i \(0.606874\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 9.15117i 0.308311i 0.988047 + 0.154155i \(0.0492656\pi\)
−0.988047 + 0.154155i \(0.950734\pi\)
\(882\) 0 0
\(883\) 7.52255i 0.253154i −0.991957 0.126577i \(-0.959601\pi\)
0.991957 0.126577i \(-0.0403991\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 49.7906 1.67180 0.835902 0.548878i \(-0.184945\pi\)
0.835902 + 0.548878i \(0.184945\pi\)
\(888\) 0 0
\(889\) 22.9706 + 9.37769i 0.770408 + 0.314518i
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) −19.3570 −0.647756
\(894\) 0 0
\(895\) −60.7279 −2.02991
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) −28.9043 −0.964011
\(900\) 0 0
\(901\) 64.8754i 2.16131i
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 2.16285 0.0718956
\(906\) 0 0
\(907\) 17.9149i 0.594853i 0.954745 + 0.297426i \(0.0961283\pi\)
−0.954745 + 0.297426i \(0.903872\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 28.2294i 0.935281i 0.883919 + 0.467640i \(0.154896\pi\)
−0.883919 + 0.467640i \(0.845104\pi\)
\(912\) 0 0
\(913\) 5.56543i 0.184189i
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) −8.91380 + 21.8343i −0.294360 + 0.721031i
\(918\) 0 0
\(919\) 33.6985i 1.11161i −0.831312 0.555806i \(-0.812410\pi\)
0.831312 0.555806i \(-0.187590\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 21.5198 0.708333
\(924\) 0 0
\(925\) −8.24264 −0.271016
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 26.9806i 0.885206i 0.896718 + 0.442603i \(0.145945\pi\)
−0.896718 + 0.442603i \(0.854055\pi\)
\(930\) 0 0
\(931\) −26.2132 25.6836i −0.859103 0.841746i
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 28.2294i 0.923199i
\(936\) 0 0
\(937\) 14.1026i 0.460712i 0.973106 + 0.230356i \(0.0739890\pi\)
−0.973106 + 0.230356i \(0.926011\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 7.90245i 0.257612i −0.991670 0.128806i \(-0.958885\pi\)
0.991670 0.128806i \(-0.0411145\pi\)
\(942\) 0 0
\(943\) 58.4558 1.90358
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 9.15117i 0.297373i 0.988884 + 0.148687i \(0.0475045\pi\)
−0.988884 + 0.148687i \(0.952495\pi\)
\(948\) 0 0
\(949\) −2.48528 −0.0806756
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 31.9630 1.03538 0.517691 0.855567i \(-0.326792\pi\)
0.517691 + 0.855567i \(0.326792\pi\)
\(954\) 0 0
\(955\) 73.9706 2.39363
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 16.2982 39.9224i 0.526297 1.28916i
\(960\) 0 0
\(961\) −20.4853 −0.660816
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 54.4785i 1.75373i
\(966\) 0 0
\(967\) 21.6251i 0.695418i 0.937603 + 0.347709i \(0.113040\pi\)
−0.937603 + 0.347709i \(0.886960\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 26.7414 0.858172 0.429086 0.903264i \(-0.358836\pi\)
0.429086 + 0.903264i \(0.358836\pi\)
\(972\) 0 0
\(973\) 1.51472 3.71029i 0.0485596 0.118946i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 35.6552 1.14071 0.570356 0.821398i \(-0.306805\pi\)
0.570356 + 0.821398i \(0.306805\pi\)
\(978\) 0 0
\(979\) −21.0000 −0.671163
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) −51.9534 −1.65706 −0.828529 0.559947i \(-0.810822\pi\)
−0.828529 + 0.559947i \(0.810822\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) −82.3871 −2.61976
\(990\) 0 0
\(991\) 37.6849i 1.19710i −0.801086 0.598549i \(-0.795744\pi\)
0.801086 0.598549i \(-0.204256\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 40.0295i 1.26902i
\(996\) 0 0
\(997\) 35.4815i 1.12371i 0.827235 + 0.561856i \(0.189912\pi\)
−0.827235 + 0.561856i \(0.810088\pi\)
\(998\) 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 3024.2.b.u.1567.8 yes 8
3.2 odd 2 inner 3024.2.b.u.1567.2 yes 8
4.3 odd 2 3024.2.b.v.1567.7 yes 8
7.6 odd 2 3024.2.b.v.1567.2 yes 8
12.11 even 2 3024.2.b.v.1567.1 yes 8
21.20 even 2 3024.2.b.v.1567.8 yes 8
28.27 even 2 inner 3024.2.b.u.1567.1 8
84.83 odd 2 inner 3024.2.b.u.1567.7 yes 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
3024.2.b.u.1567.1 8 28.27 even 2 inner
3024.2.b.u.1567.2 yes 8 3.2 odd 2 inner
3024.2.b.u.1567.7 yes 8 84.83 odd 2 inner
3024.2.b.u.1567.8 yes 8 1.1 even 1 trivial
3024.2.b.v.1567.1 yes 8 12.11 even 2
3024.2.b.v.1567.2 yes 8 7.6 odd 2
3024.2.b.v.1567.7 yes 8 4.3 odd 2
3024.2.b.v.1567.8 yes 8 21.20 even 2