Properties

Label 3800.2.d.j.3649.2
Level $3800$
Weight $2$
Character 3800.3649
Analytic conductor $30.343$
Analytic rank $0$
Dimension $6$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [3800,2,Mod(3649,3800)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3800, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 1, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("3800.3649");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3800 = 2^{3} \cdot 5^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3800.d (of order \(2\), degree \(1\), 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: \(30.3431527681\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: 6.0.59105344.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} + 21x^{4} + 116x^{2} + 64 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 152)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 3649.2
Root \(0.786802i\) of defining polynomial
Character \(\chi\) \(=\) 3800.3649
Dual form 3800.2.d.j.3649.5

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-3.08387i q^{3} -4.29707i q^{7} -6.51027 q^{9} +O(q^{10})\) \(q-3.08387i q^{3} -4.29707i q^{7} -6.51027 q^{9} -1.21320 q^{11} +5.08387i q^{13} +2.29707i q^{17} +1.00000 q^{19} -13.2516 q^{21} +7.67801i q^{23} +10.8252i q^{27} +0.489731 q^{29} +3.74135i q^{33} +2.00000i q^{37} +15.6780 q^{39} -4.16774 q^{41} +12.9545i q^{43} +5.80734i q^{47} -11.4648 q^{49} +7.08387 q^{51} +1.93667i q^{53} -3.08387i q^{57} +11.0839 q^{59} -5.38094 q^{61} +27.9751i q^{63} -2.48973i q^{67} +23.6780 q^{69} -11.7413 q^{71} -8.46482i q^{73} +5.21320i q^{77} -1.83226 q^{79} +13.8528 q^{81} -7.02054i q^{83} -1.51027i q^{87} -13.7413 q^{89} +21.8458 q^{91} +3.57360i q^{97} +7.89825 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 24 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q - 24 q^{9} - 10 q^{11} + 6 q^{19} - 18 q^{21} + 18 q^{29} + 38 q^{39} + 16 q^{41} - 10 q^{49} + 22 q^{51} + 46 q^{59} + 6 q^{61} + 86 q^{69} - 24 q^{71} - 52 q^{79} + 94 q^{81} - 36 q^{89} + 34 q^{91} + 102 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3800\mathbb{Z}\right)^\times\).

\(n\) \(401\) \(951\) \(1901\) \(1977\)
\(\chi(n)\) \(1\) \(1\) \(1\) \(-1\)

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
\(3\) − 3.08387i − 1.78047i −0.455497 0.890237i \(-0.650538\pi\)
0.455497 0.890237i \(-0.349462\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) − 4.29707i − 1.62414i −0.583560 0.812070i \(-0.698341\pi\)
0.583560 0.812070i \(-0.301659\pi\)
\(8\) 0 0
\(9\) −6.51027 −2.17009
\(10\) 0 0
\(11\) −1.21320 −0.365793 −0.182897 0.983132i \(-0.558547\pi\)
−0.182897 + 0.983132i \(0.558547\pi\)
\(12\) 0 0
\(13\) 5.08387i 1.41001i 0.709201 + 0.705006i \(0.249056\pi\)
−0.709201 + 0.705006i \(0.750944\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 2.29707i 0.557121i 0.960419 + 0.278561i \(0.0898573\pi\)
−0.960419 + 0.278561i \(0.910143\pi\)
\(18\) 0 0
\(19\) 1.00000 0.229416
\(20\) 0 0
\(21\) −13.2516 −2.89174
\(22\) 0 0
\(23\) 7.67801i 1.60098i 0.599348 + 0.800488i \(0.295426\pi\)
−0.599348 + 0.800488i \(0.704574\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 10.8252i 2.08331i
\(28\) 0 0
\(29\) 0.489731 0.0909408 0.0454704 0.998966i \(-0.485521\pi\)
0.0454704 + 0.998966i \(0.485521\pi\)
\(30\) 0 0
\(31\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(32\) 0 0
\(33\) 3.74135i 0.651285i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 2.00000i 0.328798i 0.986394 + 0.164399i \(0.0525685\pi\)
−0.986394 + 0.164399i \(0.947432\pi\)
\(38\) 0 0
\(39\) 15.6780 2.51049
\(40\) 0 0
\(41\) −4.16774 −0.650892 −0.325446 0.945561i \(-0.605514\pi\)
−0.325446 + 0.945561i \(0.605514\pi\)
\(42\) 0 0
\(43\) 12.9545i 1.97555i 0.155887 + 0.987775i \(0.450176\pi\)
−0.155887 + 0.987775i \(0.549824\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 5.80734i 0.847087i 0.905876 + 0.423544i \(0.139214\pi\)
−0.905876 + 0.423544i \(0.860786\pi\)
\(48\) 0 0
\(49\) −11.4648 −1.63783
\(50\) 0 0
\(51\) 7.08387 0.991941
\(52\) 0 0
\(53\) 1.93667i 0.266021i 0.991115 + 0.133011i \(0.0424645\pi\)
−0.991115 + 0.133011i \(0.957536\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) − 3.08387i − 0.408469i
\(58\) 0 0
\(59\) 11.0839 1.44300 0.721499 0.692416i \(-0.243454\pi\)
0.721499 + 0.692416i \(0.243454\pi\)
\(60\) 0 0
\(61\) −5.38094 −0.688959 −0.344480 0.938794i \(-0.611945\pi\)
−0.344480 + 0.938794i \(0.611945\pi\)
\(62\) 0 0
\(63\) 27.9751i 3.52453i
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) − 2.48973i − 0.304169i −0.988367 0.152085i \(-0.951401\pi\)
0.988367 0.152085i \(-0.0485986\pi\)
\(68\) 0 0
\(69\) 23.6780 2.85050
\(70\) 0 0
\(71\) −11.7413 −1.39344 −0.696721 0.717342i \(-0.745358\pi\)
−0.696721 + 0.717342i \(0.745358\pi\)
\(72\) 0 0
\(73\) − 8.46482i − 0.990732i −0.868684 0.495366i \(-0.835034\pi\)
0.868684 0.495366i \(-0.164966\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 5.21320i 0.594099i
\(78\) 0 0
\(79\) −1.83226 −0.206145 −0.103072 0.994674i \(-0.532867\pi\)
−0.103072 + 0.994674i \(0.532867\pi\)
\(80\) 0 0
\(81\) 13.8528 1.53920
\(82\) 0 0
\(83\) − 7.02054i − 0.770604i −0.922791 0.385302i \(-0.874097\pi\)
0.922791 0.385302i \(-0.125903\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) − 1.51027i − 0.161918i
\(88\) 0 0
\(89\) −13.7413 −1.45658 −0.728290 0.685269i \(-0.759685\pi\)
−0.728290 + 0.685269i \(0.759685\pi\)
\(90\) 0 0
\(91\) 21.8458 2.29006
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 3.57360i 0.362844i 0.983405 + 0.181422i \(0.0580701\pi\)
−0.983405 + 0.181422i \(0.941930\pi\)
\(98\) 0 0
\(99\) 7.89825 0.793804
\(100\) 0 0
\(101\) −6.33549 −0.630405 −0.315202 0.949024i \(-0.602072\pi\)
−0.315202 + 0.949024i \(0.602072\pi\)
\(102\) 0 0
\(103\) − 5.44693i − 0.536702i −0.963321 0.268351i \(-0.913521\pi\)
0.963321 0.268351i \(-0.0864788\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) − 3.08387i − 0.298129i −0.988827 0.149065i \(-0.952374\pi\)
0.988827 0.149065i \(-0.0476262\pi\)
\(108\) 0 0
\(109\) −8.10441 −0.776262 −0.388131 0.921604i \(-0.626879\pi\)
−0.388131 + 0.921604i \(0.626879\pi\)
\(110\) 0 0
\(111\) 6.16774 0.585416
\(112\) 0 0
\(113\) 14.3355i 1.34857i 0.738471 + 0.674285i \(0.235548\pi\)
−0.738471 + 0.674285i \(0.764452\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) − 33.0974i − 3.05985i
\(118\) 0 0
\(119\) 9.87067 0.904843
\(120\) 0 0
\(121\) −9.52815 −0.866195
\(122\) 0 0
\(123\) 12.8528i 1.15890i
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) 3.74135i 0.331991i 0.986127 + 0.165995i \(0.0530837\pi\)
−0.986127 + 0.165995i \(0.946916\pi\)
\(128\) 0 0
\(129\) 39.9502 3.51742
\(130\) 0 0
\(131\) −12.9545 −1.13184 −0.565922 0.824459i \(-0.691480\pi\)
−0.565922 + 0.824459i \(0.691480\pi\)
\(132\) 0 0
\(133\) − 4.29707i − 0.372603i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 6.63256i 0.566658i 0.959023 + 0.283329i \(0.0914388\pi\)
−0.959023 + 0.283329i \(0.908561\pi\)
\(138\) 0 0
\(139\) 4.23374 0.359101 0.179550 0.983749i \(-0.442536\pi\)
0.179550 + 0.983749i \(0.442536\pi\)
\(140\) 0 0
\(141\) 17.9091 1.50822
\(142\) 0 0
\(143\) − 6.16774i − 0.515773i
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 35.3560i 2.91612i
\(148\) 0 0
\(149\) 9.63959 0.789706 0.394853 0.918744i \(-0.370795\pi\)
0.394853 + 0.918744i \(0.370795\pi\)
\(150\) 0 0
\(151\) −5.44693 −0.443265 −0.221633 0.975130i \(-0.571139\pi\)
−0.221633 + 0.975130i \(0.571139\pi\)
\(152\) 0 0
\(153\) − 14.9545i − 1.20900i
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) − 14.0000i − 1.11732i −0.829396 0.558661i \(-0.811315\pi\)
0.829396 0.558661i \(-0.188685\pi\)
\(158\) 0 0
\(159\) 5.97243 0.473644
\(160\) 0 0
\(161\) 32.9930 2.60021
\(162\) 0 0
\(163\) − 16.3355i − 1.27949i −0.768585 0.639747i \(-0.779039\pi\)
0.768585 0.639747i \(-0.220961\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 13.5736i 1.05036i 0.850992 + 0.525178i \(0.176001\pi\)
−0.850992 + 0.525178i \(0.823999\pi\)
\(168\) 0 0
\(169\) −12.8458 −0.988135
\(170\) 0 0
\(171\) −6.51027 −0.497853
\(172\) 0 0
\(173\) 18.3355i 1.39402i 0.717061 + 0.697011i \(0.245487\pi\)
−0.717061 + 0.697011i \(0.754513\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) − 34.1812i − 2.56922i
\(178\) 0 0
\(179\) −19.9502 −1.49115 −0.745573 0.666424i \(-0.767824\pi\)
−0.745573 + 0.666424i \(0.767824\pi\)
\(180\) 0 0
\(181\) −6.85279 −0.509364 −0.254682 0.967025i \(-0.581971\pi\)
−0.254682 + 0.967025i \(0.581971\pi\)
\(182\) 0 0
\(183\) 16.5941i 1.22667i
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) − 2.78680i − 0.203791i
\(188\) 0 0
\(189\) 46.5167 3.38359
\(190\) 0 0
\(191\) 19.7798 1.43121 0.715607 0.698503i \(-0.246150\pi\)
0.715607 + 0.698503i \(0.246150\pi\)
\(192\) 0 0
\(193\) − 9.61468i − 0.692080i −0.938220 0.346040i \(-0.887526\pi\)
0.938220 0.346040i \(-0.112474\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) − 4.16774i − 0.296940i −0.988917 0.148470i \(-0.952565\pi\)
0.988917 0.148470i \(-0.0474348\pi\)
\(198\) 0 0
\(199\) −5.48535 −0.388846 −0.194423 0.980918i \(-0.562283\pi\)
−0.194423 + 0.980918i \(0.562283\pi\)
\(200\) 0 0
\(201\) −7.67801 −0.541565
\(202\) 0 0
\(203\) − 2.10441i − 0.147701i
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) − 49.9859i − 3.47426i
\(208\) 0 0
\(209\) −1.21320 −0.0839187
\(210\) 0 0
\(211\) −1.89559 −0.130498 −0.0652489 0.997869i \(-0.520784\pi\)
−0.0652489 + 0.997869i \(0.520784\pi\)
\(212\) 0 0
\(213\) 36.2088i 2.48099i
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) −26.1044 −1.76397
\(220\) 0 0
\(221\) −11.6780 −0.785548
\(222\) 0 0
\(223\) 3.74135i 0.250539i 0.992123 + 0.125270i \(0.0399796\pi\)
−0.992123 + 0.125270i \(0.960020\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) − 19.6780i − 1.30608i −0.757325 0.653038i \(-0.773494\pi\)
0.757325 0.653038i \(-0.226506\pi\)
\(228\) 0 0
\(229\) −21.7164 −1.43506 −0.717531 0.696526i \(-0.754728\pi\)
−0.717531 + 0.696526i \(0.754728\pi\)
\(230\) 0 0
\(231\) 16.0768 1.05778
\(232\) 0 0
\(233\) − 10.5692i − 0.692413i −0.938158 0.346206i \(-0.887470\pi\)
0.938158 0.346206i \(-0.112530\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 5.65044i 0.367036i
\(238\) 0 0
\(239\) 16.0384 1.03744 0.518720 0.854944i \(-0.326409\pi\)
0.518720 + 0.854944i \(0.326409\pi\)
\(240\) 0 0
\(241\) −17.0205 −1.09639 −0.548195 0.836351i \(-0.684685\pi\)
−0.548195 + 0.836351i \(0.684685\pi\)
\(242\) 0 0
\(243\) − 10.2446i − 0.657190i
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 5.08387i 0.323479i
\(248\) 0 0
\(249\) −21.6504 −1.37204
\(250\) 0 0
\(251\) −19.1223 −1.20699 −0.603494 0.797367i \(-0.706225\pi\)
−0.603494 + 0.797367i \(0.706225\pi\)
\(252\) 0 0
\(253\) − 9.31495i − 0.585626i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 0.553066i 0.0344993i 0.999851 + 0.0172497i \(0.00549101\pi\)
−0.999851 + 0.0172497i \(0.994509\pi\)
\(258\) 0 0
\(259\) 8.59414 0.534014
\(260\) 0 0
\(261\) −3.18828 −0.197350
\(262\) 0 0
\(263\) 7.04545i 0.434441i 0.976123 + 0.217221i \(0.0696991\pi\)
−0.976123 + 0.217221i \(0.930301\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 42.3766i 2.59340i
\(268\) 0 0
\(269\) −12.7619 −0.778106 −0.389053 0.921215i \(-0.627198\pi\)
−0.389053 + 0.921215i \(0.627198\pi\)
\(270\) 0 0
\(271\) 15.6780 0.952371 0.476186 0.879345i \(-0.342019\pi\)
0.476186 + 0.879345i \(0.342019\pi\)
\(272\) 0 0
\(273\) − 67.3695i − 4.07739i
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) 5.97508i 0.359008i 0.983757 + 0.179504i \(0.0574493\pi\)
−0.983757 + 0.179504i \(0.942551\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 30.2088 1.80211 0.901054 0.433708i \(-0.142795\pi\)
0.901054 + 0.433708i \(0.142795\pi\)
\(282\) 0 0
\(283\) − 1.21320i − 0.0721171i −0.999350 0.0360586i \(-0.988520\pi\)
0.999350 0.0360586i \(-0.0114803\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 17.9091i 1.05714i
\(288\) 0 0
\(289\) 11.7235 0.689616
\(290\) 0 0
\(291\) 11.0205 0.646035
\(292\) 0 0
\(293\) 0.104410i 0.00609969i 0.999995 + 0.00304984i \(0.000970797\pi\)
−0.999995 + 0.00304984i \(0.999029\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) − 13.1331i − 0.762062i
\(298\) 0 0
\(299\) −39.0340 −2.25740
\(300\) 0 0
\(301\) 55.6666 3.20857
\(302\) 0 0
\(303\) 19.5378i 1.12242i
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) − 0.852793i − 0.0486715i −0.999704 0.0243357i \(-0.992253\pi\)
0.999704 0.0243357i \(-0.00774707\pi\)
\(308\) 0 0
\(309\) −16.7976 −0.955585
\(310\) 0 0
\(311\) −6.12933 −0.347562 −0.173781 0.984784i \(-0.555599\pi\)
−0.173781 + 0.984784i \(0.555599\pi\)
\(312\) 0 0
\(313\) − 3.17478i − 0.179449i −0.995967 0.0897246i \(-0.971401\pi\)
0.995967 0.0897246i \(-0.0285987\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 20.6986i 1.16255i 0.813708 + 0.581273i \(0.197445\pi\)
−0.813708 + 0.581273i \(0.802555\pi\)
\(318\) 0 0
\(319\) −0.594141 −0.0332655
\(320\) 0 0
\(321\) −9.51027 −0.530811
\(322\) 0 0
\(323\) 2.29707i 0.127812i
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) 24.9930i 1.38211i
\(328\) 0 0
\(329\) 24.9545 1.37579
\(330\) 0 0
\(331\) 9.84576 0.541172 0.270586 0.962696i \(-0.412783\pi\)
0.270586 + 0.962696i \(0.412783\pi\)
\(332\) 0 0
\(333\) − 13.0205i − 0.713521i
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 18.9296i 1.03116i 0.856841 + 0.515581i \(0.172424\pi\)
−0.856841 + 0.515581i \(0.827576\pi\)
\(338\) 0 0
\(339\) 44.2088 2.40109
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) 19.1856i 1.03593i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 6.78680i 0.364335i 0.983268 + 0.182167i \(0.0583112\pi\)
−0.983268 + 0.182167i \(0.941689\pi\)
\(348\) 0 0
\(349\) −6.23374 −0.333684 −0.166842 0.985984i \(-0.553357\pi\)
−0.166842 + 0.985984i \(0.553357\pi\)
\(350\) 0 0
\(351\) −55.0340 −2.93750
\(352\) 0 0
\(353\) 31.7191i 1.68824i 0.536157 + 0.844118i \(0.319876\pi\)
−0.536157 + 0.844118i \(0.680124\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) − 30.4399i − 1.61105i
\(358\) 0 0
\(359\) 23.3944 1.23471 0.617356 0.786684i \(-0.288204\pi\)
0.617356 + 0.786684i \(0.288204\pi\)
\(360\) 0 0
\(361\) 1.00000 0.0526316
\(362\) 0 0
\(363\) 29.3836i 1.54224i
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 20.3355i 1.06150i 0.847527 + 0.530752i \(0.178090\pi\)
−0.847527 + 0.530752i \(0.821910\pi\)
\(368\) 0 0
\(369\) 27.1331 1.41249
\(370\) 0 0
\(371\) 8.32199 0.432056
\(372\) 0 0
\(373\) − 36.9021i − 1.91072i −0.295450 0.955358i \(-0.595470\pi\)
0.295450 0.955358i \(-0.404530\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 2.48973i 0.128228i
\(378\) 0 0
\(379\) 15.9367 0.818611 0.409306 0.912397i \(-0.365771\pi\)
0.409306 + 0.912397i \(0.365771\pi\)
\(380\) 0 0
\(381\) 11.5378 0.591101
\(382\) 0 0
\(383\) − 19.0205i − 0.971904i −0.873985 0.485952i \(-0.838473\pi\)
0.873985 0.485952i \(-0.161527\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) − 84.3376i − 4.28712i
\(388\) 0 0
\(389\) −2.61906 −0.132791 −0.0663957 0.997793i \(-0.521150\pi\)
−0.0663957 + 0.997793i \(0.521150\pi\)
\(390\) 0 0
\(391\) −17.6369 −0.891938
\(392\) 0 0
\(393\) 39.9502i 2.01522i
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) 31.1634i 1.56404i 0.623251 + 0.782022i \(0.285812\pi\)
−0.623251 + 0.782022i \(0.714188\pi\)
\(398\) 0 0
\(399\) −13.2516 −0.663411
\(400\) 0 0
\(401\) 21.7413 1.08571 0.542856 0.839826i \(-0.317343\pi\)
0.542856 + 0.839826i \(0.317343\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) − 2.42640i − 0.120272i
\(408\) 0 0
\(409\) 23.2651 1.15039 0.575193 0.818018i \(-0.304927\pi\)
0.575193 + 0.818018i \(0.304927\pi\)
\(410\) 0 0
\(411\) 20.4540 1.00892
\(412\) 0 0
\(413\) − 47.6282i − 2.34363i
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) − 13.0563i − 0.639370i
\(418\) 0 0
\(419\) −8.33549 −0.407215 −0.203608 0.979053i \(-0.565267\pi\)
−0.203608 + 0.979053i \(0.565267\pi\)
\(420\) 0 0
\(421\) 0.748383 0.0364740 0.0182370 0.999834i \(-0.494195\pi\)
0.0182370 + 0.999834i \(0.494195\pi\)
\(422\) 0 0
\(423\) − 37.8073i − 1.83826i
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 23.1223i 1.11897i
\(428\) 0 0
\(429\) −19.0205 −0.918320
\(430\) 0 0
\(431\) 31.3560 1.51037 0.755183 0.655514i \(-0.227548\pi\)
0.755183 + 0.655514i \(0.227548\pi\)
\(432\) 0 0
\(433\) 9.48270i 0.455709i 0.973695 + 0.227855i \(0.0731711\pi\)
−0.973695 + 0.227855i \(0.926829\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 7.67801i 0.367289i
\(438\) 0 0
\(439\) 31.9502 1.52490 0.762449 0.647048i \(-0.223997\pi\)
0.762449 + 0.647048i \(0.223997\pi\)
\(440\) 0 0
\(441\) 74.6390 3.55424
\(442\) 0 0
\(443\) 2.91878i 0.138676i 0.997593 + 0.0693378i \(0.0220886\pi\)
−0.997593 + 0.0693378i \(0.977911\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) − 29.7273i − 1.40605i
\(448\) 0 0
\(449\) −0.0909069 −0.00429016 −0.00214508 0.999998i \(-0.500683\pi\)
−0.00214508 + 0.999998i \(0.500683\pi\)
\(450\) 0 0
\(451\) 5.05630 0.238092
\(452\) 0 0
\(453\) 16.7976i 0.789222i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) − 19.1499i − 0.895793i −0.894085 0.447896i \(-0.852173\pi\)
0.894085 0.447896i \(-0.147827\pi\)
\(458\) 0 0
\(459\) −24.8663 −1.16066
\(460\) 0 0
\(461\) 33.9253 1.58006 0.790028 0.613070i \(-0.210066\pi\)
0.790028 + 0.613070i \(0.210066\pi\)
\(462\) 0 0
\(463\) 27.3809i 1.27250i 0.771483 + 0.636250i \(0.219515\pi\)
−0.771483 + 0.636250i \(0.780485\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) − 17.2132i − 0.796532i −0.917270 0.398266i \(-0.869612\pi\)
0.917270 0.398266i \(-0.130388\pi\)
\(468\) 0 0
\(469\) −10.6986 −0.494013
\(470\) 0 0
\(471\) −43.1742 −1.98936
\(472\) 0 0
\(473\) − 15.7164i − 0.722642i
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) − 12.6082i − 0.577290i
\(478\) 0 0
\(479\) −15.4827 −0.707422 −0.353711 0.935355i \(-0.615080\pi\)
−0.353711 + 0.935355i \(0.615080\pi\)
\(480\) 0 0
\(481\) −10.1677 −0.463609
\(482\) 0 0
\(483\) − 101.746i − 4.62961i
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) − 30.1179i − 1.36477i −0.730992 0.682386i \(-0.760942\pi\)
0.730992 0.682386i \(-0.239058\pi\)
\(488\) 0 0
\(489\) −50.3766 −2.27811
\(490\) 0 0
\(491\) −18.2944 −0.825615 −0.412808 0.910818i \(-0.635452\pi\)
−0.412808 + 0.910818i \(0.635452\pi\)
\(492\) 0 0
\(493\) 1.12495i 0.0506651i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 50.4534i 2.26314i
\(498\) 0 0
\(499\) −10.5282 −0.471305 −0.235652 0.971837i \(-0.575723\pi\)
−0.235652 + 0.971837i \(0.575723\pi\)
\(500\) 0 0
\(501\) 41.8593 1.87013
\(502\) 0 0
\(503\) 37.2018i 1.65875i 0.558696 + 0.829373i \(0.311302\pi\)
−0.558696 + 0.829373i \(0.688698\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 39.6147i 1.75935i
\(508\) 0 0
\(509\) 20.4264 0.905384 0.452692 0.891667i \(-0.350464\pi\)
0.452692 + 0.891667i \(0.350464\pi\)
\(510\) 0 0
\(511\) −36.3739 −1.60909
\(512\) 0 0
\(513\) 10.8252i 0.477945i
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) − 7.04545i − 0.309859i
\(518\) 0 0
\(519\) 56.5443 2.48202
\(520\) 0 0
\(521\) −37.8235 −1.65708 −0.828539 0.559932i \(-0.810827\pi\)
−0.828539 + 0.559932i \(0.810827\pi\)
\(522\) 0 0
\(523\) 13.0428i 0.570322i 0.958480 + 0.285161i \(0.0920470\pi\)
−0.958480 + 0.285161i \(0.907953\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) −35.9519 −1.56313
\(530\) 0 0
\(531\) −72.1590 −3.13143
\(532\) 0 0
\(533\) − 21.1883i − 0.917766i
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 61.5238i 2.65495i
\(538\) 0 0
\(539\) 13.9091 0.599107
\(540\) 0 0
\(541\) −27.6754 −1.18986 −0.594928 0.803779i \(-0.702820\pi\)
−0.594928 + 0.803779i \(0.702820\pi\)
\(542\) 0 0
\(543\) 21.1331i 0.906910i
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) − 12.0000i − 0.513083i −0.966533 0.256541i \(-0.917417\pi\)
0.966533 0.256541i \(-0.0825830\pi\)
\(548\) 0 0
\(549\) 35.0314 1.49510
\(550\) 0 0
\(551\) 0.489731 0.0208633
\(552\) 0 0
\(553\) 7.87333i 0.334808i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 36.2695i 1.53679i 0.639977 + 0.768394i \(0.278944\pi\)
−0.639977 + 0.768394i \(0.721056\pi\)
\(558\) 0 0
\(559\) −65.8593 −2.78555
\(560\) 0 0
\(561\) −8.59414 −0.362845
\(562\) 0 0
\(563\) 13.9091i 0.586198i 0.956082 + 0.293099i \(0.0946866\pi\)
−0.956082 + 0.293099i \(0.905313\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) − 59.5264i − 2.49987i
\(568\) 0 0
\(569\) −29.7413 −1.24682 −0.623411 0.781894i \(-0.714254\pi\)
−0.623411 + 0.781894i \(0.714254\pi\)
\(570\) 0 0
\(571\) −36.6710 −1.53463 −0.767316 0.641269i \(-0.778408\pi\)
−0.767316 + 0.641269i \(0.778408\pi\)
\(572\) 0 0
\(573\) − 60.9983i − 2.54824i
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) − 20.4648i − 0.851961i −0.904732 0.425981i \(-0.859929\pi\)
0.904732 0.425981i \(-0.140071\pi\)
\(578\) 0 0
\(579\) −29.6504 −1.23223
\(580\) 0 0
\(581\) −30.1677 −1.25157
\(582\) 0 0
\(583\) − 2.34956i − 0.0973088i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) − 18.2748i − 0.754282i −0.926156 0.377141i \(-0.876907\pi\)
0.926156 0.377141i \(-0.123093\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) 0 0
\(591\) −12.8528 −0.528693
\(592\) 0 0
\(593\) 31.5238i 1.29453i 0.762267 + 0.647263i \(0.224086\pi\)
−0.762267 + 0.647263i \(0.775914\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 16.9161i 0.692331i
\(598\) 0 0
\(599\) 26.3766 1.07772 0.538859 0.842396i \(-0.318856\pi\)
0.538859 + 0.842396i \(0.318856\pi\)
\(600\) 0 0
\(601\) −27.3971 −1.11755 −0.558776 0.829319i \(-0.688729\pi\)
−0.558776 + 0.829319i \(0.688729\pi\)
\(602\) 0 0
\(603\) 16.2088i 0.660074i
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) − 33.9859i − 1.37945i −0.724073 0.689723i \(-0.757732\pi\)
0.724073 0.689723i \(-0.242268\pi\)
\(608\) 0 0
\(609\) −6.48973 −0.262977
\(610\) 0 0
\(611\) −29.5238 −1.19440
\(612\) 0 0
\(613\) 3.08653i 0.124664i 0.998055 + 0.0623319i \(0.0198537\pi\)
−0.998055 + 0.0623319i \(0.980146\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 26.1018i 1.05082i 0.850850 + 0.525409i \(0.176087\pi\)
−0.850850 + 0.525409i \(0.823913\pi\)
\(618\) 0 0
\(619\) −29.0616 −1.16808 −0.584042 0.811723i \(-0.698530\pi\)
−0.584042 + 0.811723i \(0.698530\pi\)
\(620\) 0 0
\(621\) −83.1162 −3.33534
\(622\) 0 0
\(623\) 59.0475i 2.36569i
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) 3.74135i 0.149415i
\(628\) 0 0
\(629\) −4.59414 −0.183180
\(630\) 0 0
\(631\) 20.1018 0.800238 0.400119 0.916463i \(-0.368969\pi\)
0.400119 + 0.916463i \(0.368969\pi\)
\(632\) 0 0
\(633\) 5.84576i 0.232348i
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) − 58.2857i − 2.30936i
\(638\) 0 0
\(639\) 76.4393 3.02389
\(640\) 0 0
\(641\) 24.7619 0.978036 0.489018 0.872274i \(-0.337355\pi\)
0.489018 + 0.872274i \(0.337355\pi\)
\(642\) 0 0
\(643\) − 11.8431i − 0.467046i −0.972351 0.233523i \(-0.924975\pi\)
0.972351 0.233523i \(-0.0750255\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) − 37.4854i − 1.47370i −0.676056 0.736851i \(-0.736312\pi\)
0.676056 0.736851i \(-0.263688\pi\)
\(648\) 0 0
\(649\) −13.4469 −0.527838
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) − 34.8279i − 1.36292i −0.731855 0.681460i \(-0.761345\pi\)
0.731855 0.681460i \(-0.238655\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 55.1082i 2.14998i
\(658\) 0 0
\(659\) −28.2722 −1.10133 −0.550663 0.834727i \(-0.685625\pi\)
−0.550663 + 0.834727i \(0.685625\pi\)
\(660\) 0 0
\(661\) 2.58064 0.100375 0.0501876 0.998740i \(-0.484018\pi\)
0.0501876 + 0.998740i \(0.484018\pi\)
\(662\) 0 0
\(663\) 36.0135i 1.39865i
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 3.76016i 0.145594i
\(668\) 0 0
\(669\) 11.5378 0.446079
\(670\) 0 0
\(671\) 6.52815 0.252016
\(672\) 0 0
\(673\) 50.5443i 1.94834i 0.225816 + 0.974170i \(0.427495\pi\)
−0.225816 + 0.974170i \(0.572505\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) − 0.825221i − 0.0317158i −0.999874 0.0158579i \(-0.994952\pi\)
0.999874 0.0158579i \(-0.00504794\pi\)
\(678\) 0 0
\(679\) 15.3560 0.589310
\(680\) 0 0
\(681\) −60.6845 −2.32543
\(682\) 0 0
\(683\) − 12.0000i − 0.459167i −0.973289 0.229584i \(-0.926264\pi\)
0.973289 0.229584i \(-0.0737364\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) 66.9707i 2.55509i
\(688\) 0 0
\(689\) −9.84576 −0.375094
\(690\) 0 0
\(691\) 0.0249160 0.000947850 0 0.000473925 1.00000i \(-0.499849\pi\)
0.000473925 1.00000i \(0.499849\pi\)
\(692\) 0 0
\(693\) − 33.9393i − 1.28925i
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) − 9.57360i − 0.362626i
\(698\) 0 0
\(699\) −32.5941 −1.23282
\(700\) 0 0
\(701\) 29.3560 1.10876 0.554381 0.832263i \(-0.312955\pi\)
0.554381 + 0.832263i \(0.312955\pi\)
\(702\) 0 0
\(703\) 2.00000i 0.0754314i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 27.2240i 1.02387i
\(708\) 0 0
\(709\) 17.4827 0.656576 0.328288 0.944578i \(-0.393528\pi\)
0.328288 + 0.944578i \(0.393528\pi\)
\(710\) 0 0
\(711\) 11.9285 0.447353
\(712\) 0 0
\(713\) 0 0
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) − 49.4604i − 1.84713i
\(718\) 0 0
\(719\) −18.5915 −0.693345 −0.346673 0.937986i \(-0.612689\pi\)
−0.346673 + 0.937986i \(0.612689\pi\)
\(720\) 0 0
\(721\) −23.4059 −0.871680
\(722\) 0 0
\(723\) 52.4892i 1.95209i
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) 8.50589i 0.315466i 0.987482 + 0.157733i \(0.0504185\pi\)
−0.987482 + 0.157733i \(0.949581\pi\)
\(728\) 0 0
\(729\) 9.96539 0.369089
\(730\) 0 0
\(731\) −29.7575 −1.10062
\(732\) 0 0
\(733\) 24.5032i 0.905048i 0.891752 + 0.452524i \(0.149476\pi\)
−0.891752 + 0.452524i \(0.850524\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 3.02054i 0.111263i
\(738\) 0 0
\(739\) 20.9047 0.768992 0.384496 0.923127i \(-0.374375\pi\)
0.384496 + 0.923127i \(0.374375\pi\)
\(740\) 0 0
\(741\) 15.6780 0.575946
\(742\) 0 0
\(743\) 21.5238i 0.789631i 0.918761 + 0.394815i \(0.129191\pi\)
−0.918761 + 0.394815i \(0.870809\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 45.7056i 1.67228i
\(748\) 0 0
\(749\) −13.2516 −0.484204
\(750\) 0 0
\(751\) 2.37656 0.0867221 0.0433610 0.999059i \(-0.486193\pi\)
0.0433610 + 0.999059i \(0.486193\pi\)
\(752\) 0 0
\(753\) 58.9707i 2.14901i
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) − 46.9545i − 1.70659i −0.521427 0.853296i \(-0.674600\pi\)
0.521427 0.853296i \(-0.325400\pi\)
\(758\) 0 0
\(759\) −28.7261 −1.04269
\(760\) 0 0
\(761\) −5.44428 −0.197355 −0.0986775 0.995119i \(-0.531461\pi\)
−0.0986775 + 0.995119i \(0.531461\pi\)
\(762\) 0 0
\(763\) 34.8252i 1.26076i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 56.3490i 2.03464i
\(768\) 0 0
\(769\) −8.92697 −0.321915 −0.160957 0.986961i \(-0.551458\pi\)
−0.160957 + 0.986961i \(0.551458\pi\)
\(770\) 0 0
\(771\) 1.70559 0.0614252
\(772\) 0 0
\(773\) − 17.8047i − 0.640390i −0.947352 0.320195i \(-0.896252\pi\)
0.947352 0.320195i \(-0.103748\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) − 26.5032i − 0.950798i
\(778\) 0 0
\(779\) −4.16774 −0.149325
\(780\) 0 0
\(781\) 14.2446 0.509711
\(782\) 0 0
\(783\) 5.30145i 0.189458i
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) − 13.3836i − 0.477074i −0.971133 0.238537i \(-0.923332\pi\)
0.971133 0.238537i \(-0.0766678\pi\)
\(788\) 0 0
\(789\) 21.7273 0.773512
\(790\) 0 0
\(791\) 61.6006 2.19027
\(792\) 0 0
\(793\) − 27.3560i − 0.971441i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) − 28.5666i − 1.01188i −0.862569 0.505940i \(-0.831146\pi\)
0.862569 0.505940i \(-0.168854\pi\)
\(798\) 0 0
\(799\) −13.3399 −0.471931
\(800\) 0 0
\(801\) 89.4599 3.16091
\(802\) 0 0
\(803\) 10.2695i 0.362403i
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 39.3560i 1.38540i
\(808\) 0 0
\(809\) −45.2625 −1.59134 −0.795672 0.605728i \(-0.792882\pi\)
−0.795672 + 0.605728i \(0.792882\pi\)
\(810\) 0 0
\(811\) −49.4551 −1.73660 −0.868302 0.496036i \(-0.834789\pi\)
−0.868302 + 0.496036i \(0.834789\pi\)
\(812\) 0 0
\(813\) − 48.3490i − 1.69567i
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 12.9545i 0.453222i
\(818\) 0 0
\(819\) −142.222 −4.96963
\(820\) 0 0
\(821\) −41.7933 −1.45860 −0.729298 0.684197i \(-0.760153\pi\)
−0.729298 + 0.684197i \(0.760153\pi\)
\(822\) 0 0
\(823\) − 8.68239i − 0.302649i −0.988484 0.151325i \(-0.951646\pi\)
0.988484 0.151325i \(-0.0483539\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) − 13.1196i − 0.456214i −0.973636 0.228107i \(-0.926746\pi\)
0.973636 0.228107i \(-0.0732537\pi\)
\(828\) 0 0
\(829\) 8.83053 0.306697 0.153349 0.988172i \(-0.450994\pi\)
0.153349 + 0.988172i \(0.450994\pi\)
\(830\) 0 0
\(831\) 18.4264 0.639205
\(832\) 0 0
\(833\) − 26.3355i − 0.912471i
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) −4.80296 −0.165817 −0.0829083 0.996557i \(-0.526421\pi\)
−0.0829083 + 0.996557i \(0.526421\pi\)
\(840\) 0 0
\(841\) −28.7602 −0.991730
\(842\) 0 0
\(843\) − 93.1601i − 3.20861i
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) 40.9431i 1.40682i
\(848\) 0 0
\(849\) −3.74135 −0.128403
\(850\) 0 0
\(851\) −15.3560 −0.526398
\(852\) 0 0
\(853\) − 2.51730i − 0.0861908i −0.999071 0.0430954i \(-0.986278\pi\)
0.999071 0.0430954i \(-0.0137219\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 37.3560i 1.27606i 0.770013 + 0.638029i \(0.220250\pi\)
−0.770013 + 0.638029i \(0.779750\pi\)
\(858\) 0 0
\(859\) −40.6461 −1.38683 −0.693413 0.720540i \(-0.743894\pi\)
−0.693413 + 0.720540i \(0.743894\pi\)
\(860\) 0 0
\(861\) 55.2294 1.88221
\(862\) 0 0
\(863\) − 7.40586i − 0.252098i −0.992024 0.126049i \(-0.959770\pi\)
0.992024 0.126049i \(-0.0402297\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) − 36.1537i − 1.22784i
\(868\) 0 0
\(869\) 2.22289 0.0754063
\(870\) 0 0
\(871\) 12.6575 0.428882
\(872\) 0 0
\(873\) − 23.2651i − 0.787405i
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) − 50.5308i − 1.70630i −0.521662 0.853152i \(-0.674688\pi\)
0.521662 0.853152i \(-0.325312\pi\)
\(878\) 0 0
\(879\) 0.321987 0.0108603
\(880\) 0 0
\(881\) 33.4631 1.12740 0.563700 0.825980i \(-0.309377\pi\)
0.563700 + 0.825980i \(0.309377\pi\)
\(882\) 0 0
\(883\) 30.0162i 1.01012i 0.863083 + 0.505062i \(0.168530\pi\)
−0.863083 + 0.505062i \(0.831470\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) − 19.0704i − 0.640320i −0.947363 0.320160i \(-0.896263\pi\)
0.947363 0.320160i \(-0.103737\pi\)
\(888\) 0 0
\(889\) 16.0768 0.539200
\(890\) 0 0
\(891\) −16.8062 −0.563028
\(892\) 0 0
\(893\) 5.80734i 0.194335i
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 120.376i 4.01924i
\(898\) 0 0
\(899\) 0 0
\(900\) 0 0
\(901\) −4.44866 −0.148206
\(902\) 0 0
\(903\) − 171.669i − 5.71278i
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) 48.0135i 1.59426i 0.603806 + 0.797131i \(0.293650\pi\)
−0.603806 + 0.797131i \(0.706350\pi\)
\(908\) 0 0
\(909\) 41.2457 1.36803
\(910\) 0 0
\(911\) −46.2446 −1.53215 −0.766076 0.642750i \(-0.777793\pi\)
−0.766076 + 0.642750i \(0.777793\pi\)
\(912\) 0 0
\(913\) 8.51730i 0.281882i
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 55.6666i 1.83827i
\(918\) 0 0
\(919\) −41.5103 −1.36930 −0.684649 0.728873i \(-0.740044\pi\)
−0.684649 + 0.728873i \(0.740044\pi\)
\(920\) 0 0
\(921\) −2.62990 −0.0866583
\(922\) 0 0
\(923\) − 59.6915i − 1.96477i
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) 35.4610i 1.16469i
\(928\) 0 0
\(929\) −55.4551 −1.81942 −0.909712 0.415240i \(-0.863698\pi\)
−0.909712 + 0.415240i \(0.863698\pi\)
\(930\) 0 0
\(931\) −11.4648 −0.375744
\(932\) 0 0
\(933\) 18.9021i 0.618826i
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 6.08825i 0.198894i 0.995043 + 0.0994472i \(0.0317075\pi\)
−0.995043 + 0.0994472i \(0.968293\pi\)
\(938\) 0 0
\(939\) −9.79061 −0.319505
\(940\) 0 0
\(941\) −11.3836 −0.371095 −0.185547 0.982635i \(-0.559406\pi\)
−0.185547 + 0.982635i \(0.559406\pi\)
\(942\) 0 0
\(943\) − 32.0000i − 1.04206i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 38.3766i 1.24707i 0.781795 + 0.623535i \(0.214304\pi\)
−0.781795 + 0.623535i \(0.785696\pi\)
\(948\) 0 0
\(949\) 43.0340 1.39694
\(950\) 0 0
\(951\) 63.8317 2.06988
\(952\) 0 0
\(953\) 30.2857i 0.981049i 0.871427 + 0.490524i \(0.163195\pi\)
−0.871427 + 0.490524i \(0.836805\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 1.83226i 0.0592284i
\(958\) 0 0
\(959\) 28.5006 0.920332
\(960\) 0 0
\(961\) −31.0000 −1.00000
\(962\) 0 0
\(963\) 20.0768i 0.646967i
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) 24.6710i 0.793365i 0.917956 + 0.396683i \(0.129839\pi\)
−0.917956 + 0.396683i \(0.870161\pi\)
\(968\) 0 0
\(969\) 7.08387 0.227567
\(970\) 0 0
\(971\) 20.0000 0.641831 0.320915 0.947108i \(-0.396010\pi\)
0.320915 + 0.947108i \(0.396010\pi\)
\(972\) 0 0
\(973\) − 18.1927i − 0.583230i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 21.4059i 0.684834i 0.939548 + 0.342417i \(0.111246\pi\)
−0.939548 + 0.342417i \(0.888754\pi\)
\(978\) 0 0
\(979\) 16.6710 0.532807
\(980\) 0 0
\(981\) 52.7619 1.68456
\(982\) 0 0
\(983\) 38.1677i 1.21736i 0.793415 + 0.608681i \(0.208301\pi\)
−0.793415 + 0.608681i \(0.791699\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) − 76.9566i − 2.44956i
\(988\) 0 0
\(989\) −99.4652 −3.16281
\(990\) 0 0
\(991\) 16.4675 0.523106 0.261553 0.965189i \(-0.415765\pi\)
0.261553 + 0.965189i \(0.415765\pi\)
\(992\) 0 0
\(993\) − 30.3631i − 0.963543i
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) − 13.8431i − 0.438415i −0.975678 0.219208i \(-0.929653\pi\)
0.975678 0.219208i \(-0.0703472\pi\)
\(998\) 0 0
\(999\) −21.6504 −0.684990
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 3800.2.d.j.3649.2 6
5.2 odd 4 152.2.a.c.1.1 3
5.3 odd 4 3800.2.a.r.1.3 3
5.4 even 2 inner 3800.2.d.j.3649.5 6
15.2 even 4 1368.2.a.n.1.2 3
20.3 even 4 7600.2.a.bv.1.1 3
20.7 even 4 304.2.a.g.1.3 3
35.27 even 4 7448.2.a.bf.1.3 3
40.27 even 4 1216.2.a.v.1.1 3
40.37 odd 4 1216.2.a.u.1.3 3
60.47 odd 4 2736.2.a.bd.1.2 3
95.37 even 4 2888.2.a.o.1.3 3
380.227 odd 4 5776.2.a.bp.1.1 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
152.2.a.c.1.1 3 5.2 odd 4
304.2.a.g.1.3 3 20.7 even 4
1216.2.a.u.1.3 3 40.37 odd 4
1216.2.a.v.1.1 3 40.27 even 4
1368.2.a.n.1.2 3 15.2 even 4
2736.2.a.bd.1.2 3 60.47 odd 4
2888.2.a.o.1.3 3 95.37 even 4
3800.2.a.r.1.3 3 5.3 odd 4
3800.2.d.j.3649.2 6 1.1 even 1 trivial
3800.2.d.j.3649.5 6 5.4 even 2 inner
5776.2.a.bp.1.1 3 380.227 odd 4
7448.2.a.bf.1.3 3 35.27 even 4
7600.2.a.bv.1.1 3 20.3 even 4