Properties

Label 3800.2.d.n.3649.6
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: 6.0.399424.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - 2x^{5} + 3x^{4} - 6x^{3} + 6x^{2} - 8x + 8 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: no (minimal twist has level 760)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 3649.6
Root \(-0.671462 + 1.24464i\) of defining polynomial
Character \(\chi\) \(=\) 3800.3649
Dual form 3800.2.d.n.3649.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+2.34292i q^{3} +1.19656i q^{7} -2.48929 q^{9} +O(q^{10})\) \(q+2.34292i q^{3} +1.19656i q^{7} -2.48929 q^{9} +4.97858 q^{11} +6.63565i q^{13} +1.48929i q^{17} -1.00000 q^{19} -2.80344 q^{21} +0.510711i q^{23} +1.19656i q^{27} +7.88240 q^{29} -2.97858 q^{31} +11.6644i q^{33} -7.14637i q^{37} -15.5468 q^{39} +1.66442 q^{41} +6.39312i q^{43} -9.95715i q^{47} +5.56825 q^{49} -3.48929 q^{51} +11.4219i q^{53} -2.34292i q^{57} +11.8396 q^{59} +3.66442 q^{61} -2.97858i q^{63} +7.61423i q^{67} -1.19656 q^{69} -13.8396i q^{73} +5.95715i q^{77} -12.6858 q^{79} -10.2713 q^{81} +8.68585i q^{83} +18.4679i q^{87} +4.87819 q^{89} -7.93994 q^{91} -6.97858i q^{93} -6.81079i q^{97} -12.3931 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q+O(q^{10}) \) Copy content Toggle raw display \( 6 q - 6 q^{19} - 26 q^{21} + 14 q^{29} + 12 q^{31} - 6 q^{39} - 44 q^{41} - 24 q^{49} - 6 q^{51} - 22 q^{59} - 32 q^{61} + 2 q^{69} - 52 q^{79} - 26 q^{81} + 12 q^{89} + 58 q^{91} - 56 q^{99}+O(q^{100}) \) 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
\(3\) 2.34292i 1.35269i 0.736586 + 0.676344i \(0.236437\pi\)
−0.736586 + 0.676344i \(0.763563\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) 1.19656i 0.452256i 0.974098 + 0.226128i \(0.0726068\pi\)
−0.974098 + 0.226128i \(0.927393\pi\)
\(8\) 0 0
\(9\) −2.48929 −0.829763
\(10\) 0 0
\(11\) 4.97858 1.50110 0.750549 0.660815i \(-0.229789\pi\)
0.750549 + 0.660815i \(0.229789\pi\)
\(12\) 0 0
\(13\) 6.63565i 1.84040i 0.391449 + 0.920200i \(0.371974\pi\)
−0.391449 + 0.920200i \(0.628026\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 1.48929i 0.361206i 0.983556 + 0.180603i \(0.0578048\pi\)
−0.983556 + 0.180603i \(0.942195\pi\)
\(18\) 0 0
\(19\) −1.00000 −0.229416
\(20\) 0 0
\(21\) −2.80344 −0.611761
\(22\) 0 0
\(23\) 0.510711i 0.106491i 0.998581 + 0.0532453i \(0.0169565\pi\)
−0.998581 + 0.0532453i \(0.983043\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 1.19656i 0.230278i
\(28\) 0 0
\(29\) 7.88240 1.46373 0.731863 0.681452i \(-0.238651\pi\)
0.731863 + 0.681452i \(0.238651\pi\)
\(30\) 0 0
\(31\) −2.97858 −0.534968 −0.267484 0.963562i \(-0.586192\pi\)
−0.267484 + 0.963562i \(0.586192\pi\)
\(32\) 0 0
\(33\) 11.6644i 2.03052i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) − 7.14637i − 1.17486i −0.809277 0.587428i \(-0.800141\pi\)
0.809277 0.587428i \(-0.199859\pi\)
\(38\) 0 0
\(39\) −15.5468 −2.48948
\(40\) 0 0
\(41\) 1.66442 0.259939 0.129970 0.991518i \(-0.458512\pi\)
0.129970 + 0.991518i \(0.458512\pi\)
\(42\) 0 0
\(43\) 6.39312i 0.974941i 0.873139 + 0.487470i \(0.162080\pi\)
−0.873139 + 0.487470i \(0.837920\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) − 9.95715i − 1.45240i −0.687483 0.726200i \(-0.741285\pi\)
0.687483 0.726200i \(-0.258715\pi\)
\(48\) 0 0
\(49\) 5.56825 0.795464
\(50\) 0 0
\(51\) −3.48929 −0.488598
\(52\) 0 0
\(53\) 11.4219i 1.56892i 0.620182 + 0.784458i \(0.287059\pi\)
−0.620182 + 0.784458i \(0.712941\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) − 2.34292i − 0.310328i
\(58\) 0 0
\(59\) 11.8396 1.54138 0.770690 0.637211i \(-0.219912\pi\)
0.770690 + 0.637211i \(0.219912\pi\)
\(60\) 0 0
\(61\) 3.66442 0.469181 0.234591 0.972094i \(-0.424625\pi\)
0.234591 + 0.972094i \(0.424625\pi\)
\(62\) 0 0
\(63\) − 2.97858i − 0.375265i
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 7.61423i 0.930226i 0.885251 + 0.465113i \(0.153986\pi\)
−0.885251 + 0.465113i \(0.846014\pi\)
\(68\) 0 0
\(69\) −1.19656 −0.144049
\(70\) 0 0
\(71\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(72\) 0 0
\(73\) − 13.8396i − 1.61980i −0.586570 0.809899i \(-0.699522\pi\)
0.586570 0.809899i \(-0.300478\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 5.95715i 0.678881i
\(78\) 0 0
\(79\) −12.6858 −1.42727 −0.713635 0.700518i \(-0.752952\pi\)
−0.713635 + 0.700518i \(0.752952\pi\)
\(80\) 0 0
\(81\) −10.2713 −1.14126
\(82\) 0 0
\(83\) 8.68585i 0.953395i 0.879067 + 0.476698i \(0.158166\pi\)
−0.879067 + 0.476698i \(0.841834\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 18.4679i 1.97996i
\(88\) 0 0
\(89\) 4.87819 0.517087 0.258544 0.966000i \(-0.416757\pi\)
0.258544 + 0.966000i \(0.416757\pi\)
\(90\) 0 0
\(91\) −7.93994 −0.832332
\(92\) 0 0
\(93\) − 6.97858i − 0.723645i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) − 6.81079i − 0.691531i −0.938321 0.345765i \(-0.887619\pi\)
0.938321 0.345765i \(-0.112381\pi\)
\(98\) 0 0
\(99\) −12.3931 −1.24555
\(100\) 0 0
\(101\) −2.29273 −0.228135 −0.114068 0.993473i \(-0.536388\pi\)
−0.114068 + 0.993473i \(0.536388\pi\)
\(102\) 0 0
\(103\) 6.51806i 0.642243i 0.947038 + 0.321122i \(0.104060\pi\)
−0.947038 + 0.321122i \(0.895940\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 7.71462i 0.745800i 0.927872 + 0.372900i \(0.121637\pi\)
−0.927872 + 0.372900i \(0.878363\pi\)
\(108\) 0 0
\(109\) −15.5468 −1.48912 −0.744558 0.667558i \(-0.767340\pi\)
−0.744558 + 0.667558i \(0.767340\pi\)
\(110\) 0 0
\(111\) 16.7434 1.58921
\(112\) 0 0
\(113\) 0.753250i 0.0708598i 0.999372 + 0.0354299i \(0.0112801\pi\)
−0.999372 + 0.0354299i \(0.988720\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) − 16.5181i − 1.52709i
\(118\) 0 0
\(119\) −1.78202 −0.163357
\(120\) 0 0
\(121\) 13.7862 1.25329
\(122\) 0 0
\(123\) 3.89962i 0.351617i
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) 10.4177i 0.924419i 0.886771 + 0.462210i \(0.152943\pi\)
−0.886771 + 0.462210i \(0.847057\pi\)
\(128\) 0 0
\(129\) −14.9786 −1.31879
\(130\) 0 0
\(131\) −17.9572 −1.56892 −0.784462 0.620177i \(-0.787061\pi\)
−0.784462 + 0.620177i \(0.787061\pi\)
\(132\) 0 0
\(133\) − 1.19656i − 0.103755i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) − 9.83956i − 0.840650i −0.907374 0.420325i \(-0.861916\pi\)
0.907374 0.420325i \(-0.138084\pi\)
\(138\) 0 0
\(139\) 0.978577 0.0830018 0.0415009 0.999138i \(-0.486786\pi\)
0.0415009 + 0.999138i \(0.486786\pi\)
\(140\) 0 0
\(141\) 23.3288 1.96464
\(142\) 0 0
\(143\) 33.0361i 2.76262i
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 13.0460i 1.07601i
\(148\) 0 0
\(149\) 8.24989 0.675857 0.337928 0.941172i \(-0.390274\pi\)
0.337928 + 0.941172i \(0.390274\pi\)
\(150\) 0 0
\(151\) 13.8568 1.12765 0.563824 0.825895i \(-0.309330\pi\)
0.563824 + 0.825895i \(0.309330\pi\)
\(152\) 0 0
\(153\) − 3.70727i − 0.299715i
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) − 15.7073i − 1.25358i −0.779190 0.626788i \(-0.784369\pi\)
0.779190 0.626788i \(-0.215631\pi\)
\(158\) 0 0
\(159\) −26.7606 −2.12225
\(160\) 0 0
\(161\) −0.611096 −0.0481611
\(162\) 0 0
\(163\) − 5.80765i − 0.454891i −0.973791 0.227445i \(-0.926963\pi\)
0.973791 0.227445i \(-0.0730373\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) − 13.8322i − 1.07037i −0.844735 0.535184i \(-0.820242\pi\)
0.844735 0.535184i \(-0.179758\pi\)
\(168\) 0 0
\(169\) −31.0319 −2.38707
\(170\) 0 0
\(171\) 2.48929 0.190361
\(172\) 0 0
\(173\) − 14.8108i − 1.12604i −0.826442 0.563022i \(-0.809639\pi\)
0.826442 0.563022i \(-0.190361\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 27.7392i 2.08500i
\(178\) 0 0
\(179\) −15.6644 −1.17081 −0.585407 0.810740i \(-0.699065\pi\)
−0.585407 + 0.810740i \(0.699065\pi\)
\(180\) 0 0
\(181\) −14.7862 −1.09905 −0.549526 0.835477i \(-0.685192\pi\)
−0.549526 + 0.835477i \(0.685192\pi\)
\(182\) 0 0
\(183\) 8.58546i 0.634656i
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 7.41454i 0.542205i
\(188\) 0 0
\(189\) −1.43175 −0.104144
\(190\) 0 0
\(191\) −6.36748 −0.460735 −0.230367 0.973104i \(-0.573993\pi\)
−0.230367 + 0.973104i \(0.573993\pi\)
\(192\) 0 0
\(193\) 19.5970i 1.41062i 0.708897 + 0.705312i \(0.249193\pi\)
−0.708897 + 0.705312i \(0.750807\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 10.7862i 0.768487i 0.923232 + 0.384244i \(0.125538\pi\)
−0.923232 + 0.384244i \(0.874462\pi\)
\(198\) 0 0
\(199\) −2.80344 −0.198731 −0.0993654 0.995051i \(-0.531681\pi\)
−0.0993654 + 0.995051i \(0.531681\pi\)
\(200\) 0 0
\(201\) −17.8396 −1.25831
\(202\) 0 0
\(203\) 9.43175i 0.661979i
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) − 1.27131i − 0.0883620i
\(208\) 0 0
\(209\) −4.97858 −0.344375
\(210\) 0 0
\(211\) −11.2541 −0.774764 −0.387382 0.921919i \(-0.626621\pi\)
−0.387382 + 0.921919i \(0.626621\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) − 3.56404i − 0.241943i
\(218\) 0 0
\(219\) 32.4250 2.19108
\(220\) 0 0
\(221\) −9.88240 −0.664762
\(222\) 0 0
\(223\) − 17.4966i − 1.17166i −0.810433 0.585831i \(-0.800768\pi\)
0.810433 0.585831i \(-0.199232\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) − 3.84942i − 0.255495i −0.991807 0.127748i \(-0.959225\pi\)
0.991807 0.127748i \(-0.0407747\pi\)
\(228\) 0 0
\(229\) 14.6430 0.967637 0.483818 0.875168i \(-0.339250\pi\)
0.483818 + 0.875168i \(0.339250\pi\)
\(230\) 0 0
\(231\) −13.9572 −0.918313
\(232\) 0 0
\(233\) − 11.9572i − 0.783339i −0.920106 0.391670i \(-0.871898\pi\)
0.920106 0.391670i \(-0.128102\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) − 29.7220i − 1.93065i
\(238\) 0 0
\(239\) −15.5897 −1.00841 −0.504206 0.863583i \(-0.668215\pi\)
−0.504206 + 0.863583i \(0.668215\pi\)
\(240\) 0 0
\(241\) −16.0575 −1.03436 −0.517178 0.855878i \(-0.673018\pi\)
−0.517178 + 0.855878i \(0.673018\pi\)
\(242\) 0 0
\(243\) − 20.4752i − 1.31349i
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) − 6.63565i − 0.422217i
\(248\) 0 0
\(249\) −20.3503 −1.28965
\(250\) 0 0
\(251\) 23.9143 1.50946 0.754729 0.656037i \(-0.227768\pi\)
0.754729 + 0.656037i \(0.227768\pi\)
\(252\) 0 0
\(253\) 2.54262i 0.159853i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) − 24.4324i − 1.52405i −0.647548 0.762025i \(-0.724206\pi\)
0.647548 0.762025i \(-0.275794\pi\)
\(258\) 0 0
\(259\) 8.55104 0.531336
\(260\) 0 0
\(261\) −19.6216 −1.21455
\(262\) 0 0
\(263\) 14.4935i 0.893707i 0.894607 + 0.446854i \(0.147456\pi\)
−0.894607 + 0.446854i \(0.852544\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 11.4292i 0.699458i
\(268\) 0 0
\(269\) 14.0000 0.853595 0.426798 0.904347i \(-0.359642\pi\)
0.426798 + 0.904347i \(0.359642\pi\)
\(270\) 0 0
\(271\) −6.76060 −0.410677 −0.205339 0.978691i \(-0.565830\pi\)
−0.205339 + 0.978691i \(0.565830\pi\)
\(272\) 0 0
\(273\) − 18.6027i − 1.12589i
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) − 25.4292i − 1.52789i −0.645279 0.763947i \(-0.723259\pi\)
0.645279 0.763947i \(-0.276741\pi\)
\(278\) 0 0
\(279\) 7.41454 0.443897
\(280\) 0 0
\(281\) 9.07896 0.541605 0.270803 0.962635i \(-0.412711\pi\)
0.270803 + 0.962635i \(0.412711\pi\)
\(282\) 0 0
\(283\) − 12.0147i − 0.714199i −0.934066 0.357100i \(-0.883766\pi\)
0.934066 0.357100i \(-0.116234\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 1.99158i 0.117559i
\(288\) 0 0
\(289\) 14.7820 0.869531
\(290\) 0 0
\(291\) 15.9572 0.935425
\(292\) 0 0
\(293\) − 23.6718i − 1.38292i −0.722415 0.691460i \(-0.756968\pi\)
0.722415 0.691460i \(-0.243032\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 5.95715i 0.345669i
\(298\) 0 0
\(299\) −3.38890 −0.195985
\(300\) 0 0
\(301\) −7.64973 −0.440923
\(302\) 0 0
\(303\) − 5.37169i − 0.308596i
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) − 16.0246i − 0.914570i −0.889320 0.457285i \(-0.848822\pi\)
0.889320 0.457285i \(-0.151178\pi\)
\(308\) 0 0
\(309\) −15.2713 −0.868754
\(310\) 0 0
\(311\) −27.5468 −1.56204 −0.781019 0.624508i \(-0.785300\pi\)
−0.781019 + 0.624508i \(0.785300\pi\)
\(312\) 0 0
\(313\) − 10.1751i − 0.575133i −0.957761 0.287566i \(-0.907154\pi\)
0.957761 0.287566i \(-0.0928462\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 29.6289i 1.66413i 0.554681 + 0.832063i \(0.312840\pi\)
−0.554681 + 0.832063i \(0.687160\pi\)
\(318\) 0 0
\(319\) 39.2432 2.19719
\(320\) 0 0
\(321\) −18.0748 −1.00883
\(322\) 0 0
\(323\) − 1.48929i − 0.0828662i
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) − 36.4250i − 2.01431i
\(328\) 0 0
\(329\) 11.9143 0.656857
\(330\) 0 0
\(331\) 30.4679 1.67467 0.837333 0.546694i \(-0.184114\pi\)
0.837333 + 0.546694i \(0.184114\pi\)
\(332\) 0 0
\(333\) 17.7894i 0.974851i
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 26.1396i 1.42392i 0.702222 + 0.711958i \(0.252192\pi\)
−0.702222 + 0.711958i \(0.747808\pi\)
\(338\) 0 0
\(339\) −1.76481 −0.0958512
\(340\) 0 0
\(341\) −14.8291 −0.803039
\(342\) 0 0
\(343\) 15.0386i 0.812010i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) − 7.31415i − 0.392644i −0.980539 0.196322i \(-0.937100\pi\)
0.980539 0.196322i \(-0.0628998\pi\)
\(348\) 0 0
\(349\) −0.628308 −0.0336325 −0.0168163 0.999859i \(-0.505353\pi\)
−0.0168163 + 0.999859i \(0.505353\pi\)
\(350\) 0 0
\(351\) −7.93994 −0.423803
\(352\) 0 0
\(353\) 8.23267i 0.438181i 0.975705 + 0.219090i \(0.0703090\pi\)
−0.975705 + 0.219090i \(0.929691\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) − 4.17513i − 0.220972i
\(358\) 0 0
\(359\) −12.1323 −0.640318 −0.320159 0.947364i \(-0.603736\pi\)
−0.320159 + 0.947364i \(0.603736\pi\)
\(360\) 0 0
\(361\) 1.00000 0.0526316
\(362\) 0 0
\(363\) 32.3001i 1.69531i
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 17.9572i 0.937356i 0.883369 + 0.468678i \(0.155270\pi\)
−0.883369 + 0.468678i \(0.844730\pi\)
\(368\) 0 0
\(369\) −4.14323 −0.215688
\(370\) 0 0
\(371\) −13.6669 −0.709552
\(372\) 0 0
\(373\) − 29.6289i − 1.53413i −0.641571 0.767064i \(-0.721717\pi\)
0.641571 0.767064i \(-0.278283\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 52.3049i 2.69384i
\(378\) 0 0
\(379\) 6.80344 0.349469 0.174735 0.984616i \(-0.444093\pi\)
0.174735 + 0.984616i \(0.444093\pi\)
\(380\) 0 0
\(381\) −24.4078 −1.25045
\(382\) 0 0
\(383\) 20.2253i 1.03347i 0.856147 + 0.516733i \(0.172852\pi\)
−0.856147 + 0.516733i \(0.827148\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) − 15.9143i − 0.808970i
\(388\) 0 0
\(389\) −0.393115 −0.0199317 −0.00996587 0.999950i \(-0.503172\pi\)
−0.00996587 + 0.999950i \(0.503172\pi\)
\(390\) 0 0
\(391\) −0.760597 −0.0384650
\(392\) 0 0
\(393\) − 42.0722i − 2.12226i
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) 17.4292i 0.874748i 0.899280 + 0.437374i \(0.144091\pi\)
−0.899280 + 0.437374i \(0.855909\pi\)
\(398\) 0 0
\(399\) 2.80344 0.140348
\(400\) 0 0
\(401\) 13.1281 0.655585 0.327792 0.944750i \(-0.393695\pi\)
0.327792 + 0.944750i \(0.393695\pi\)
\(402\) 0 0
\(403\) − 19.7648i − 0.984555i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) − 35.5787i − 1.76357i
\(408\) 0 0
\(409\) 18.7862 0.928919 0.464460 0.885594i \(-0.346249\pi\)
0.464460 + 0.885594i \(0.346249\pi\)
\(410\) 0 0
\(411\) 23.0533 1.13714
\(412\) 0 0
\(413\) 14.1667i 0.697098i
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 2.29273i 0.112276i
\(418\) 0 0
\(419\) 5.22219 0.255121 0.127560 0.991831i \(-0.459285\pi\)
0.127560 + 0.991831i \(0.459285\pi\)
\(420\) 0 0
\(421\) −7.68164 −0.374380 −0.187190 0.982324i \(-0.559938\pi\)
−0.187190 + 0.982324i \(0.559938\pi\)
\(422\) 0 0
\(423\) 24.7862i 1.20515i
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 4.38469i 0.212190i
\(428\) 0 0
\(429\) −77.4011 −3.73696
\(430\) 0 0
\(431\) −17.7073 −0.852929 −0.426465 0.904504i \(-0.640241\pi\)
−0.426465 + 0.904504i \(0.640241\pi\)
\(432\) 0 0
\(433\) − 17.9901i − 0.864551i −0.901742 0.432275i \(-0.857711\pi\)
0.901742 0.432275i \(-0.142289\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) − 0.510711i − 0.0244306i
\(438\) 0 0
\(439\) −14.8438 −0.708454 −0.354227 0.935159i \(-0.615256\pi\)
−0.354227 + 0.935159i \(0.615256\pi\)
\(440\) 0 0
\(441\) −13.8610 −0.660047
\(442\) 0 0
\(443\) − 22.4078i − 1.06463i −0.846547 0.532314i \(-0.821323\pi\)
0.846547 0.532314i \(-0.178677\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 19.3288i 0.914223i
\(448\) 0 0
\(449\) 40.7434 1.92280 0.961400 0.275156i \(-0.0887295\pi\)
0.961400 + 0.275156i \(0.0887295\pi\)
\(450\) 0 0
\(451\) 8.28646 0.390194
\(452\) 0 0
\(453\) 32.4653i 1.52536i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 8.01721i 0.375029i 0.982262 + 0.187515i \(0.0600432\pi\)
−0.982262 + 0.187515i \(0.939957\pi\)
\(458\) 0 0
\(459\) −1.78202 −0.0831775
\(460\) 0 0
\(461\) −25.1281 −1.17033 −0.585166 0.810914i \(-0.698971\pi\)
−0.585166 + 0.810914i \(0.698971\pi\)
\(462\) 0 0
\(463\) 31.7795i 1.47692i 0.674298 + 0.738459i \(0.264446\pi\)
−0.674298 + 0.738459i \(0.735554\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 20.9210i 0.968110i 0.875038 + 0.484055i \(0.160837\pi\)
−0.875038 + 0.484055i \(0.839163\pi\)
\(468\) 0 0
\(469\) −9.11087 −0.420701
\(470\) 0 0
\(471\) 36.8009 1.69570
\(472\) 0 0
\(473\) 31.8286i 1.46348i
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) − 28.4324i − 1.30183i
\(478\) 0 0
\(479\) −27.7220 −1.26665 −0.633324 0.773886i \(-0.718310\pi\)
−0.633324 + 0.773886i \(0.718310\pi\)
\(480\) 0 0
\(481\) 47.4208 2.16220
\(482\) 0 0
\(483\) − 1.43175i − 0.0651469i
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) 3.20390i 0.145183i 0.997362 + 0.0725914i \(0.0231269\pi\)
−0.997362 + 0.0725914i \(0.976873\pi\)
\(488\) 0 0
\(489\) 13.6069 0.615325
\(490\) 0 0
\(491\) −21.5212 −0.971238 −0.485619 0.874171i \(-0.661406\pi\)
−0.485619 + 0.874171i \(0.661406\pi\)
\(492\) 0 0
\(493\) 11.7392i 0.528706i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) 23.1709 1.03727 0.518637 0.854995i \(-0.326440\pi\)
0.518637 + 0.854995i \(0.326440\pi\)
\(500\) 0 0
\(501\) 32.4078 1.44787
\(502\) 0 0
\(503\) 1.88240i 0.0839322i 0.999119 + 0.0419661i \(0.0133622\pi\)
−0.999119 + 0.0419661i \(0.986638\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) − 72.7054i − 3.22896i
\(508\) 0 0
\(509\) −5.46365 −0.242172 −0.121086 0.992642i \(-0.538638\pi\)
−0.121086 + 0.992642i \(0.538638\pi\)
\(510\) 0 0
\(511\) 16.5598 0.732564
\(512\) 0 0
\(513\) − 1.19656i − 0.0528293i
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) − 49.5725i − 2.18019i
\(518\) 0 0
\(519\) 34.7005 1.52318
\(520\) 0 0
\(521\) 11.2222 0.491653 0.245827 0.969314i \(-0.420941\pi\)
0.245827 + 0.969314i \(0.420941\pi\)
\(522\) 0 0
\(523\) 29.1867i 1.27624i 0.769935 + 0.638122i \(0.220289\pi\)
−0.769935 + 0.638122i \(0.779711\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) − 4.43596i − 0.193233i
\(528\) 0 0
\(529\) 22.7392 0.988660
\(530\) 0 0
\(531\) −29.4721 −1.27898
\(532\) 0 0
\(533\) 11.0445i 0.478392i
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) − 36.7005i − 1.58375i
\(538\) 0 0
\(539\) 27.7220 1.19407
\(540\) 0 0
\(541\) 20.2499 0.870611 0.435305 0.900283i \(-0.356640\pi\)
0.435305 + 0.900283i \(0.356640\pi\)
\(542\) 0 0
\(543\) − 34.6430i − 1.48667i
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 42.8830i 1.83355i 0.399409 + 0.916773i \(0.369215\pi\)
−0.399409 + 0.916773i \(0.630785\pi\)
\(548\) 0 0
\(549\) −9.12181 −0.389309
\(550\) 0 0
\(551\) −7.88240 −0.335802
\(552\) 0 0
\(553\) − 15.1793i − 0.645491i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 28.2583i 1.19734i 0.800995 + 0.598671i \(0.204304\pi\)
−0.800995 + 0.598671i \(0.795696\pi\)
\(558\) 0 0
\(559\) −42.4225 −1.79428
\(560\) 0 0
\(561\) −17.3717 −0.733433
\(562\) 0 0
\(563\) 20.2253i 0.852396i 0.904630 + 0.426198i \(0.140147\pi\)
−0.904630 + 0.426198i \(0.859853\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) − 12.2902i − 0.516140i
\(568\) 0 0
\(569\) 31.1365 1.30531 0.652655 0.757655i \(-0.273655\pi\)
0.652655 + 0.757655i \(0.273655\pi\)
\(570\) 0 0
\(571\) 27.4868 1.15029 0.575143 0.818053i \(-0.304947\pi\)
0.575143 + 0.818053i \(0.304947\pi\)
\(572\) 0 0
\(573\) − 14.9185i − 0.623230i
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) 23.7820i 0.990058i 0.868876 + 0.495029i \(0.164843\pi\)
−0.868876 + 0.495029i \(0.835157\pi\)
\(578\) 0 0
\(579\) −45.9143 −1.90813
\(580\) 0 0
\(581\) −10.3931 −0.431179
\(582\) 0 0
\(583\) 56.8647i 2.35510i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 24.0147i 0.991192i 0.868553 + 0.495596i \(0.165050\pi\)
−0.868553 + 0.495596i \(0.834950\pi\)
\(588\) 0 0
\(589\) 2.97858 0.122730
\(590\) 0 0
\(591\) −25.2713 −1.03952
\(592\) 0 0
\(593\) − 46.6148i − 1.91424i −0.289688 0.957121i \(-0.593551\pi\)
0.289688 0.957121i \(-0.406449\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) − 6.56825i − 0.268821i
\(598\) 0 0
\(599\) 44.6002 1.82231 0.911156 0.412061i \(-0.135191\pi\)
0.911156 + 0.412061i \(0.135191\pi\)
\(600\) 0 0
\(601\) −17.5787 −0.717051 −0.358526 0.933520i \(-0.616720\pi\)
−0.358526 + 0.933520i \(0.616720\pi\)
\(602\) 0 0
\(603\) − 18.9540i − 0.771867i
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) 37.2467i 1.51180i 0.654688 + 0.755899i \(0.272800\pi\)
−0.654688 + 0.755899i \(0.727200\pi\)
\(608\) 0 0
\(609\) −22.0979 −0.895451
\(610\) 0 0
\(611\) 66.0722 2.67300
\(612\) 0 0
\(613\) 14.8866i 0.601265i 0.953740 + 0.300632i \(0.0971977\pi\)
−0.953740 + 0.300632i \(0.902802\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 3.70727i 0.149249i 0.997212 + 0.0746245i \(0.0237758\pi\)
−0.997212 + 0.0746245i \(0.976224\pi\)
\(618\) 0 0
\(619\) −17.7648 −0.714028 −0.357014 0.934099i \(-0.616205\pi\)
−0.357014 + 0.934099i \(0.616205\pi\)
\(620\) 0 0
\(621\) −0.611096 −0.0245224
\(622\) 0 0
\(623\) 5.83704i 0.233856i
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) − 11.6644i − 0.465832i
\(628\) 0 0
\(629\) 10.6430 0.424364
\(630\) 0 0
\(631\) 40.3074 1.60461 0.802307 0.596912i \(-0.203606\pi\)
0.802307 + 0.596912i \(0.203606\pi\)
\(632\) 0 0
\(633\) − 26.3675i − 1.04801i
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 36.9490i 1.46397i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) −27.6728 −1.09301 −0.546506 0.837455i \(-0.684042\pi\)
−0.546506 + 0.837455i \(0.684042\pi\)
\(642\) 0 0
\(643\) − 19.0277i − 0.750379i −0.926948 0.375190i \(-0.877578\pi\)
0.926948 0.375190i \(-0.122422\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) − 32.0403i − 1.25964i −0.776743 0.629818i \(-0.783130\pi\)
0.776743 0.629818i \(-0.216870\pi\)
\(648\) 0 0
\(649\) 58.9442 2.31376
\(650\) 0 0
\(651\) 8.35027 0.327273
\(652\) 0 0
\(653\) − 25.3142i − 0.990619i −0.868716 0.495310i \(-0.835055\pi\)
0.868716 0.495310i \(-0.164945\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 34.4507i 1.34405i
\(658\) 0 0
\(659\) 0.860981 0.0335391 0.0167695 0.999859i \(-0.494662\pi\)
0.0167695 + 0.999859i \(0.494662\pi\)
\(660\) 0 0
\(661\) 4.55356 0.177113 0.0885564 0.996071i \(-0.471775\pi\)
0.0885564 + 0.996071i \(0.471775\pi\)
\(662\) 0 0
\(663\) − 23.1537i − 0.899216i
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 4.02563i 0.155873i
\(668\) 0 0
\(669\) 40.9933 1.58489
\(670\) 0 0
\(671\) 18.2436 0.704287
\(672\) 0 0
\(673\) 11.7465i 0.452795i 0.974035 + 0.226398i \(0.0726948\pi\)
−0.974035 + 0.226398i \(0.927305\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) − 22.2854i − 0.856497i −0.903661 0.428248i \(-0.859131\pi\)
0.903661 0.428248i \(-0.140869\pi\)
\(678\) 0 0
\(679\) 8.14950 0.312749
\(680\) 0 0
\(681\) 9.01890 0.345605
\(682\) 0 0
\(683\) 7.23833i 0.276967i 0.990365 + 0.138483i \(0.0442228\pi\)
−0.990365 + 0.138483i \(0.955777\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) 34.3074i 1.30891i
\(688\) 0 0
\(689\) −75.7917 −2.88743
\(690\) 0 0
\(691\) 32.7434 1.24562 0.622809 0.782374i \(-0.285992\pi\)
0.622809 + 0.782374i \(0.285992\pi\)
\(692\) 0 0
\(693\) − 14.8291i − 0.563310i
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 2.47881i 0.0938915i
\(698\) 0 0
\(699\) 28.0147 1.05961
\(700\) 0 0
\(701\) −29.4208 −1.11121 −0.555604 0.831447i \(-0.687513\pi\)
−0.555604 + 0.831447i \(0.687513\pi\)
\(702\) 0 0
\(703\) 7.14637i 0.269530i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) − 2.74338i − 0.103176i
\(708\) 0 0
\(709\) 5.32885 0.200129 0.100065 0.994981i \(-0.468095\pi\)
0.100065 + 0.994981i \(0.468095\pi\)
\(710\) 0 0
\(711\) 31.5787 1.18429
\(712\) 0 0
\(713\) − 1.52119i − 0.0569691i
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) − 36.5254i − 1.36407i
\(718\) 0 0
\(719\) −8.45317 −0.315250 −0.157625 0.987499i \(-0.550384\pi\)
−0.157625 + 0.987499i \(0.550384\pi\)
\(720\) 0 0
\(721\) −7.79923 −0.290459
\(722\) 0 0
\(723\) − 37.6216i − 1.39916i
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) 2.56825i 0.0952511i 0.998865 + 0.0476256i \(0.0151654\pi\)
−0.998865 + 0.0476256i \(0.984835\pi\)
\(728\) 0 0
\(729\) 17.1579 0.635479
\(730\) 0 0
\(731\) −9.52119 −0.352154
\(732\) 0 0
\(733\) − 35.5212i − 1.31201i −0.754759 0.656003i \(-0.772246\pi\)
0.754759 0.656003i \(-0.227754\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 37.9080i 1.39636i
\(738\) 0 0
\(739\) 37.0508 1.36294 0.681468 0.731848i \(-0.261342\pi\)
0.681468 + 0.731848i \(0.261342\pi\)
\(740\) 0 0
\(741\) 15.5468 0.571127
\(742\) 0 0
\(743\) − 23.2039i − 0.851269i −0.904895 0.425634i \(-0.860051\pi\)
0.904895 0.425634i \(-0.139949\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) − 21.6216i − 0.791092i
\(748\) 0 0
\(749\) −9.23098 −0.337293
\(750\) 0 0
\(751\) 51.2285 1.86935 0.934677 0.355499i \(-0.115689\pi\)
0.934677 + 0.355499i \(0.115689\pi\)
\(752\) 0 0
\(753\) 56.0294i 2.04182i
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) − 43.3717i − 1.57637i −0.615438 0.788185i \(-0.711021\pi\)
0.615438 0.788185i \(-0.288979\pi\)
\(758\) 0 0
\(759\) −5.95715 −0.216231
\(760\) 0 0
\(761\) 24.7753 0.898104 0.449052 0.893506i \(-0.351762\pi\)
0.449052 + 0.893506i \(0.351762\pi\)
\(762\) 0 0
\(763\) − 18.6027i − 0.673462i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 78.5632i 2.83675i
\(768\) 0 0
\(769\) 11.4893 0.414314 0.207157 0.978308i \(-0.433579\pi\)
0.207157 + 0.978308i \(0.433579\pi\)
\(770\) 0 0
\(771\) 57.2432 2.06156
\(772\) 0 0
\(773\) − 27.0863i − 0.974227i −0.873339 0.487113i \(-0.838050\pi\)
0.873339 0.487113i \(-0.161950\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 20.0344i 0.718731i
\(778\) 0 0
\(779\) −1.66442 −0.0596342
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 9.43175i 0.337063i
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) 13.4563i 0.479666i 0.970814 + 0.239833i \(0.0770926\pi\)
−0.970814 + 0.239833i \(0.922907\pi\)
\(788\) 0 0
\(789\) −33.9572 −1.20891
\(790\) 0 0
\(791\) −0.901307 −0.0320468
\(792\) 0 0
\(793\) 24.3158i 0.863481i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 30.4496i 1.07858i 0.842120 + 0.539290i \(0.181307\pi\)
−0.842120 + 0.539290i \(0.818693\pi\)
\(798\) 0 0
\(799\) 14.8291 0.524615
\(800\) 0 0
\(801\) −12.1432 −0.429060
\(802\) 0 0
\(803\) − 68.9013i − 2.43147i
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 32.8009i 1.15465i
\(808\) 0 0
\(809\) 29.7047 1.04436 0.522182 0.852834i \(-0.325118\pi\)
0.522182 + 0.852834i \(0.325118\pi\)
\(810\) 0 0
\(811\) 32.2902 1.13386 0.566931 0.823765i \(-0.308130\pi\)
0.566931 + 0.823765i \(0.308130\pi\)
\(812\) 0 0
\(813\) − 15.8396i − 0.555518i
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) − 6.39312i − 0.223667i
\(818\) 0 0
\(819\) 19.7648 0.690638
\(820\) 0 0
\(821\) 9.06427 0.316345 0.158173 0.987411i \(-0.449440\pi\)
0.158173 + 0.987411i \(0.449440\pi\)
\(822\) 0 0
\(823\) 7.35448i 0.256361i 0.991751 + 0.128181i \(0.0409137\pi\)
−0.991751 + 0.128181i \(0.959086\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 2.02204i 0.0703132i 0.999382 + 0.0351566i \(0.0111930\pi\)
−0.999382 + 0.0351566i \(0.988807\pi\)
\(828\) 0 0
\(829\) 36.3675 1.26309 0.631547 0.775337i \(-0.282420\pi\)
0.631547 + 0.775337i \(0.282420\pi\)
\(830\) 0 0
\(831\) 59.5787 2.06676
\(832\) 0 0
\(833\) 8.29273i 0.287326i
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) − 3.56404i − 0.123191i
\(838\) 0 0
\(839\) −25.8223 −0.891486 −0.445743 0.895161i \(-0.647061\pi\)
−0.445743 + 0.895161i \(0.647061\pi\)
\(840\) 0 0
\(841\) 33.1323 1.14249
\(842\) 0 0
\(843\) 21.2713i 0.732623i
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) 16.4960i 0.566810i
\(848\) 0 0
\(849\) 28.1495 0.966088
\(850\) 0 0
\(851\) 3.64973 0.125111
\(852\) 0 0
\(853\) − 8.54262i − 0.292494i −0.989248 0.146247i \(-0.953281\pi\)
0.989248 0.146247i \(-0.0467194\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 27.8322i 0.950730i 0.879789 + 0.475365i \(0.157684\pi\)
−0.879789 + 0.475365i \(0.842316\pi\)
\(858\) 0 0
\(859\) −33.4868 −1.14255 −0.571277 0.820757i \(-0.693552\pi\)
−0.571277 + 0.820757i \(0.693552\pi\)
\(860\) 0 0
\(861\) −4.66611 −0.159021
\(862\) 0 0
\(863\) 29.2614i 0.996071i 0.867157 + 0.498036i \(0.165945\pi\)
−0.867157 + 0.498036i \(0.834055\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) 34.6331i 1.17620i
\(868\) 0 0
\(869\) −63.1575 −2.14247
\(870\) 0 0
\(871\) −50.5254 −1.71199
\(872\) 0 0
\(873\) 16.9540i 0.573807i
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) − 46.7852i − 1.57982i −0.613221 0.789911i \(-0.710127\pi\)
0.613221 0.789911i \(-0.289873\pi\)
\(878\) 0 0
\(879\) 55.4611 1.87066
\(880\) 0 0
\(881\) −30.7581 −1.03627 −0.518133 0.855300i \(-0.673373\pi\)
−0.518133 + 0.855300i \(0.673373\pi\)
\(882\) 0 0
\(883\) − 0.435961i − 0.0146713i −0.999973 0.00733563i \(-0.997665\pi\)
0.999973 0.00733563i \(-0.00233502\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) − 2.15310i − 0.0722939i −0.999346 0.0361469i \(-0.988492\pi\)
0.999346 0.0361469i \(-0.0115084\pi\)
\(888\) 0 0
\(889\) −12.4653 −0.418074
\(890\) 0 0
\(891\) −51.1365 −1.71314
\(892\) 0 0
\(893\) 9.95715i 0.333203i
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) − 7.93994i − 0.265107i
\(898\) 0 0
\(899\) −23.4783 −0.783047
\(900\) 0 0
\(901\) −17.0105 −0.566701
\(902\) 0 0
\(903\) − 17.9227i − 0.596431i
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) − 36.9002i − 1.22525i −0.790373 0.612626i \(-0.790113\pi\)
0.790373 0.612626i \(-0.209887\pi\)
\(908\) 0 0
\(909\) 5.70727 0.189298
\(910\) 0 0
\(911\) −36.9504 −1.22422 −0.612111 0.790772i \(-0.709679\pi\)
−0.612111 + 0.790772i \(0.709679\pi\)
\(912\) 0 0
\(913\) 43.2432i 1.43114i
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) − 21.4868i − 0.709556i
\(918\) 0 0
\(919\) −19.1537 −0.631823 −0.315911 0.948789i \(-0.602310\pi\)
−0.315911 + 0.948789i \(0.602310\pi\)
\(920\) 0 0
\(921\) 37.5443 1.23713
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) − 16.2253i − 0.532910i
\(928\) 0 0
\(929\) 56.9185 1.86744 0.933718 0.358009i \(-0.116544\pi\)
0.933718 + 0.358009i \(0.116544\pi\)
\(930\) 0 0
\(931\) −5.56825 −0.182492
\(932\) 0 0
\(933\) − 64.5401i − 2.11295i
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) − 24.1176i − 0.787888i −0.919135 0.393944i \(-0.871110\pi\)
0.919135 0.393944i \(-0.128890\pi\)
\(938\) 0 0
\(939\) 23.8396 0.777975
\(940\) 0 0
\(941\) 3.19656 0.104205 0.0521024 0.998642i \(-0.483408\pi\)
0.0521024 + 0.998642i \(0.483408\pi\)
\(942\) 0 0
\(943\) 0.850040i 0.0276811i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 3.63504i 0.118123i 0.998254 + 0.0590614i \(0.0188108\pi\)
−0.998254 + 0.0590614i \(0.981189\pi\)
\(948\) 0 0
\(949\) 91.8345 2.98107
\(950\) 0 0
\(951\) −69.4183 −2.25104
\(952\) 0 0
\(953\) 48.0821i 1.55753i 0.627315 + 0.778766i \(0.284154\pi\)
−0.627315 + 0.778766i \(0.715846\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 91.9437i 2.97212i
\(958\) 0 0
\(959\) 11.7736 0.380189
\(960\) 0 0
\(961\) −22.1281 −0.713809
\(962\) 0 0
\(963\) − 19.2039i − 0.618837i
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) − 23.4637i − 0.754540i −0.926103 0.377270i \(-0.876863\pi\)
0.926103 0.377270i \(-0.123137\pi\)
\(968\) 0 0
\(969\) 3.48929 0.112092
\(970\) 0 0
\(971\) 17.4868 0.561177 0.280589 0.959828i \(-0.409470\pi\)
0.280589 + 0.959828i \(0.409470\pi\)
\(972\) 0 0
\(973\) 1.17092i 0.0375381i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 20.2316i 0.647266i 0.946183 + 0.323633i \(0.104904\pi\)
−0.946183 + 0.323633i \(0.895096\pi\)
\(978\) 0 0
\(979\) 24.2865 0.776199
\(980\) 0 0
\(981\) 38.7005 1.23561
\(982\) 0 0
\(983\) − 28.1249i − 0.897046i −0.893771 0.448523i \(-0.851950\pi\)
0.893771 0.448523i \(-0.148050\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 27.9143i 0.888522i
\(988\) 0 0
\(989\) −3.26504 −0.103822
\(990\) 0 0
\(991\) −33.9718 −1.07915 −0.539576 0.841937i \(-0.681415\pi\)
−0.539576 + 0.841937i \(0.681415\pi\)
\(992\) 0 0
\(993\) 71.3839i 2.26530i
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) − 23.8223i − 0.754461i −0.926119 0.377231i \(-0.876876\pi\)
0.926119 0.377231i \(-0.123124\pi\)
\(998\) 0 0
\(999\) 8.55104 0.270543
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.n.3649.6 6
5.2 odd 4 3800.2.a.w.1.3 3
5.3 odd 4 760.2.a.i.1.1 3
5.4 even 2 inner 3800.2.d.n.3649.1 6
15.8 even 4 6840.2.a.bm.1.2 3
20.3 even 4 1520.2.a.q.1.3 3
20.7 even 4 7600.2.a.bp.1.1 3
40.3 even 4 6080.2.a.br.1.1 3
40.13 odd 4 6080.2.a.bx.1.3 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
760.2.a.i.1.1 3 5.3 odd 4
1520.2.a.q.1.3 3 20.3 even 4
3800.2.a.w.1.3 3 5.2 odd 4
3800.2.d.n.3649.1 6 5.4 even 2 inner
3800.2.d.n.3649.6 6 1.1 even 1 trivial
6080.2.a.br.1.1 3 40.3 even 4
6080.2.a.bx.1.3 3 40.13 odd 4
6840.2.a.bm.1.2 3 15.8 even 4
7600.2.a.bp.1.1 3 20.7 even 4