Properties

Label 7600.2.a.v.1.1
Level $7600$
Weight $2$
Character 7600.1
Self dual yes
Analytic conductor $60.686$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [7600,2,Mod(1,7600)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(7600, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("7600.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 7600 = 2^{4} \cdot 5^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 7600.a (trivial)

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: yes
Analytic conductor: \(60.6863055362\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{8})^+\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 190)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-1.41421\) of defining polynomial
Character \(\chi\) \(=\) 7600.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-2.41421 q^{3} -1.58579 q^{7} +2.82843 q^{9} +O(q^{10})\) \(q-2.41421 q^{3} -1.58579 q^{7} +2.82843 q^{9} -1.41421 q^{11} +0.171573 q^{13} +1.00000 q^{17} +1.00000 q^{19} +3.82843 q^{21} -9.24264 q^{23} +0.414214 q^{27} -5.82843 q^{29} +2.24264 q^{31} +3.41421 q^{33} +8.48528 q^{37} -0.414214 q^{39} +4.24264 q^{41} -10.2426 q^{43} -4.48528 q^{49} -2.41421 q^{51} -11.4853 q^{53} -2.41421 q^{57} +12.8995 q^{59} +5.75736 q^{61} -4.48528 q^{63} -13.2426 q^{67} +22.3137 q^{69} +10.5858 q^{71} -5.48528 q^{73} +2.24264 q^{77} -10.4853 q^{79} -9.48528 q^{81} +2.48528 q^{83} +14.0711 q^{87} -7.07107 q^{89} -0.272078 q^{91} -5.41421 q^{93} -11.6569 q^{97} -4.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{3} - 6 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{3} - 6 q^{7} + 6 q^{13} + 2 q^{17} + 2 q^{19} + 2 q^{21} - 10 q^{23} - 2 q^{27} - 6 q^{29} - 4 q^{31} + 4 q^{33} + 2 q^{39} - 12 q^{43} + 8 q^{49} - 2 q^{51} - 6 q^{53} - 2 q^{57} + 6 q^{59} + 20 q^{61} + 8 q^{63} - 18 q^{67} + 22 q^{69} + 24 q^{71} + 6 q^{73} - 4 q^{77} - 4 q^{79} - 2 q^{81} - 12 q^{83} + 14 q^{87} - 26 q^{91} - 8 q^{93} - 12 q^{97} - 8 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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
\(3\) −2.41421 −1.39385 −0.696923 0.717146i \(-0.745448\pi\)
−0.696923 + 0.717146i \(0.745448\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) −1.58579 −0.599371 −0.299685 0.954038i \(-0.596882\pi\)
−0.299685 + 0.954038i \(0.596882\pi\)
\(8\) 0 0
\(9\) 2.82843 0.942809
\(10\) 0 0
\(11\) −1.41421 −0.426401 −0.213201 0.977008i \(-0.568389\pi\)
−0.213201 + 0.977008i \(0.568389\pi\)
\(12\) 0 0
\(13\) 0.171573 0.0475858 0.0237929 0.999717i \(-0.492426\pi\)
0.0237929 + 0.999717i \(0.492426\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 1.00000 0.242536 0.121268 0.992620i \(-0.461304\pi\)
0.121268 + 0.992620i \(0.461304\pi\)
\(18\) 0 0
\(19\) 1.00000 0.229416
\(20\) 0 0
\(21\) 3.82843 0.835431
\(22\) 0 0
\(23\) −9.24264 −1.92722 −0.963612 0.267305i \(-0.913867\pi\)
−0.963612 + 0.267305i \(0.913867\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 0.414214 0.0797154
\(28\) 0 0
\(29\) −5.82843 −1.08231 −0.541156 0.840922i \(-0.682013\pi\)
−0.541156 + 0.840922i \(0.682013\pi\)
\(30\) 0 0
\(31\) 2.24264 0.402790 0.201395 0.979510i \(-0.435452\pi\)
0.201395 + 0.979510i \(0.435452\pi\)
\(32\) 0 0
\(33\) 3.41421 0.594338
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 8.48528 1.39497 0.697486 0.716599i \(-0.254302\pi\)
0.697486 + 0.716599i \(0.254302\pi\)
\(38\) 0 0
\(39\) −0.414214 −0.0663273
\(40\) 0 0
\(41\) 4.24264 0.662589 0.331295 0.943527i \(-0.392515\pi\)
0.331295 + 0.943527i \(0.392515\pi\)
\(42\) 0 0
\(43\) −10.2426 −1.56199 −0.780994 0.624538i \(-0.785287\pi\)
−0.780994 + 0.624538i \(0.785287\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(48\) 0 0
\(49\) −4.48528 −0.640754
\(50\) 0 0
\(51\) −2.41421 −0.338058
\(52\) 0 0
\(53\) −11.4853 −1.57762 −0.788812 0.614634i \(-0.789304\pi\)
−0.788812 + 0.614634i \(0.789304\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) −2.41421 −0.319770
\(58\) 0 0
\(59\) 12.8995 1.67937 0.839686 0.543073i \(-0.182739\pi\)
0.839686 + 0.543073i \(0.182739\pi\)
\(60\) 0 0
\(61\) 5.75736 0.737154 0.368577 0.929597i \(-0.379845\pi\)
0.368577 + 0.929597i \(0.379845\pi\)
\(62\) 0 0
\(63\) −4.48528 −0.565092
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) −13.2426 −1.61785 −0.808923 0.587915i \(-0.799949\pi\)
−0.808923 + 0.587915i \(0.799949\pi\)
\(68\) 0 0
\(69\) 22.3137 2.68625
\(70\) 0 0
\(71\) 10.5858 1.25630 0.628151 0.778092i \(-0.283812\pi\)
0.628151 + 0.778092i \(0.283812\pi\)
\(72\) 0 0
\(73\) −5.48528 −0.642004 −0.321002 0.947079i \(-0.604020\pi\)
−0.321002 + 0.947079i \(0.604020\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 2.24264 0.255573
\(78\) 0 0
\(79\) −10.4853 −1.17969 −0.589843 0.807518i \(-0.700810\pi\)
−0.589843 + 0.807518i \(0.700810\pi\)
\(80\) 0 0
\(81\) −9.48528 −1.05392
\(82\) 0 0
\(83\) 2.48528 0.272795 0.136398 0.990654i \(-0.456448\pi\)
0.136398 + 0.990654i \(0.456448\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 14.0711 1.50858
\(88\) 0 0
\(89\) −7.07107 −0.749532 −0.374766 0.927119i \(-0.622277\pi\)
−0.374766 + 0.927119i \(0.622277\pi\)
\(90\) 0 0
\(91\) −0.272078 −0.0285215
\(92\) 0 0
\(93\) −5.41421 −0.561428
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −11.6569 −1.18357 −0.591787 0.806094i \(-0.701577\pi\)
−0.591787 + 0.806094i \(0.701577\pi\)
\(98\) 0 0
\(99\) −4.00000 −0.402015
\(100\) 0 0
\(101\) −1.07107 −0.106575 −0.0532876 0.998579i \(-0.516970\pi\)
−0.0532876 + 0.998579i \(0.516970\pi\)
\(102\) 0 0
\(103\) 4.24264 0.418040 0.209020 0.977911i \(-0.432973\pi\)
0.209020 + 0.977911i \(0.432973\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 5.72792 0.553739 0.276870 0.960908i \(-0.410703\pi\)
0.276870 + 0.960908i \(0.410703\pi\)
\(108\) 0 0
\(109\) 15.9706 1.52970 0.764851 0.644207i \(-0.222812\pi\)
0.764851 + 0.644207i \(0.222812\pi\)
\(110\) 0 0
\(111\) −20.4853 −1.94438
\(112\) 0 0
\(113\) −1.75736 −0.165318 −0.0826592 0.996578i \(-0.526341\pi\)
−0.0826592 + 0.996578i \(0.526341\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 0.485281 0.0448643
\(118\) 0 0
\(119\) −1.58579 −0.145369
\(120\) 0 0
\(121\) −9.00000 −0.818182
\(122\) 0 0
\(123\) −10.2426 −0.923548
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) −14.4853 −1.28536 −0.642680 0.766134i \(-0.722178\pi\)
−0.642680 + 0.766134i \(0.722178\pi\)
\(128\) 0 0
\(129\) 24.7279 2.17717
\(130\) 0 0
\(131\) −16.9706 −1.48272 −0.741362 0.671105i \(-0.765820\pi\)
−0.741362 + 0.671105i \(0.765820\pi\)
\(132\) 0 0
\(133\) −1.58579 −0.137505
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 13.0000 1.11066 0.555332 0.831628i \(-0.312591\pi\)
0.555332 + 0.831628i \(0.312591\pi\)
\(138\) 0 0
\(139\) 12.0000 1.01783 0.508913 0.860818i \(-0.330047\pi\)
0.508913 + 0.860818i \(0.330047\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) −0.242641 −0.0202906
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 10.8284 0.893114
\(148\) 0 0
\(149\) 6.34315 0.519651 0.259825 0.965656i \(-0.416335\pi\)
0.259825 + 0.965656i \(0.416335\pi\)
\(150\) 0 0
\(151\) −6.48528 −0.527765 −0.263882 0.964555i \(-0.585003\pi\)
−0.263882 + 0.964555i \(0.585003\pi\)
\(152\) 0 0
\(153\) 2.82843 0.228665
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 0.343146 0.0273860 0.0136930 0.999906i \(-0.495641\pi\)
0.0136930 + 0.999906i \(0.495641\pi\)
\(158\) 0 0
\(159\) 27.7279 2.19897
\(160\) 0 0
\(161\) 14.6569 1.15512
\(162\) 0 0
\(163\) 1.75736 0.137647 0.0688235 0.997629i \(-0.478075\pi\)
0.0688235 + 0.997629i \(0.478075\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 9.75736 0.755047 0.377524 0.926000i \(-0.376776\pi\)
0.377524 + 0.926000i \(0.376776\pi\)
\(168\) 0 0
\(169\) −12.9706 −0.997736
\(170\) 0 0
\(171\) 2.82843 0.216295
\(172\) 0 0
\(173\) −16.4853 −1.25335 −0.626676 0.779280i \(-0.715585\pi\)
−0.626676 + 0.779280i \(0.715585\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) −31.1421 −2.34079
\(178\) 0 0
\(179\) 0.343146 0.0256479 0.0128240 0.999918i \(-0.495918\pi\)
0.0128240 + 0.999918i \(0.495918\pi\)
\(180\) 0 0
\(181\) 8.48528 0.630706 0.315353 0.948974i \(-0.397877\pi\)
0.315353 + 0.948974i \(0.397877\pi\)
\(182\) 0 0
\(183\) −13.8995 −1.02748
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) −1.41421 −0.103418
\(188\) 0 0
\(189\) −0.656854 −0.0477791
\(190\) 0 0
\(191\) 18.5563 1.34269 0.671345 0.741145i \(-0.265717\pi\)
0.671345 + 0.741145i \(0.265717\pi\)
\(192\) 0 0
\(193\) −11.6569 −0.839079 −0.419539 0.907737i \(-0.637808\pi\)
−0.419539 + 0.907737i \(0.637808\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −20.2426 −1.44223 −0.721114 0.692816i \(-0.756370\pi\)
−0.721114 + 0.692816i \(0.756370\pi\)
\(198\) 0 0
\(199\) −0.757359 −0.0536878 −0.0268439 0.999640i \(-0.508546\pi\)
−0.0268439 + 0.999640i \(0.508546\pi\)
\(200\) 0 0
\(201\) 31.9706 2.25503
\(202\) 0 0
\(203\) 9.24264 0.648706
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) −26.1421 −1.81700
\(208\) 0 0
\(209\) −1.41421 −0.0978232
\(210\) 0 0
\(211\) −5.72792 −0.394326 −0.197163 0.980371i \(-0.563173\pi\)
−0.197163 + 0.980371i \(0.563173\pi\)
\(212\) 0 0
\(213\) −25.5563 −1.75109
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) −3.55635 −0.241421
\(218\) 0 0
\(219\) 13.2426 0.894855
\(220\) 0 0
\(221\) 0.171573 0.0115412
\(222\) 0 0
\(223\) 20.8284 1.39477 0.697387 0.716694i \(-0.254346\pi\)
0.697387 + 0.716694i \(0.254346\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 25.2426 1.67541 0.837706 0.546121i \(-0.183896\pi\)
0.837706 + 0.546121i \(0.183896\pi\)
\(228\) 0 0
\(229\) −18.9706 −1.25361 −0.626805 0.779176i \(-0.715638\pi\)
−0.626805 + 0.779176i \(0.715638\pi\)
\(230\) 0 0
\(231\) −5.41421 −0.356229
\(232\) 0 0
\(233\) 8.97056 0.587681 0.293841 0.955854i \(-0.405067\pi\)
0.293841 + 0.955854i \(0.405067\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 25.3137 1.64430
\(238\) 0 0
\(239\) −12.8995 −0.834399 −0.417199 0.908815i \(-0.636988\pi\)
−0.417199 + 0.908815i \(0.636988\pi\)
\(240\) 0 0
\(241\) −24.9706 −1.60850 −0.804248 0.594294i \(-0.797431\pi\)
−0.804248 + 0.594294i \(0.797431\pi\)
\(242\) 0 0
\(243\) 21.6569 1.38929
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 0.171573 0.0109169
\(248\) 0 0
\(249\) −6.00000 −0.380235
\(250\) 0 0
\(251\) 27.5563 1.73934 0.869671 0.493632i \(-0.164331\pi\)
0.869671 + 0.493632i \(0.164331\pi\)
\(252\) 0 0
\(253\) 13.0711 0.821771
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −4.72792 −0.294920 −0.147460 0.989068i \(-0.547110\pi\)
−0.147460 + 0.989068i \(0.547110\pi\)
\(258\) 0 0
\(259\) −13.4558 −0.836105
\(260\) 0 0
\(261\) −16.4853 −1.02041
\(262\) 0 0
\(263\) −6.97056 −0.429823 −0.214912 0.976633i \(-0.568946\pi\)
−0.214912 + 0.976633i \(0.568946\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 17.0711 1.04473
\(268\) 0 0
\(269\) 28.6274 1.74544 0.872722 0.488217i \(-0.162353\pi\)
0.872722 + 0.488217i \(0.162353\pi\)
\(270\) 0 0
\(271\) 18.7574 1.13943 0.569714 0.821843i \(-0.307054\pi\)
0.569714 + 0.821843i \(0.307054\pi\)
\(272\) 0 0
\(273\) 0.656854 0.0397546
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(278\) 0 0
\(279\) 6.34315 0.379754
\(280\) 0 0
\(281\) −4.24264 −0.253095 −0.126547 0.991961i \(-0.540390\pi\)
−0.126547 + 0.991961i \(0.540390\pi\)
\(282\) 0 0
\(283\) −3.85786 −0.229326 −0.114663 0.993404i \(-0.536579\pi\)
−0.114663 + 0.993404i \(0.536579\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) −6.72792 −0.397137
\(288\) 0 0
\(289\) −16.0000 −0.941176
\(290\) 0 0
\(291\) 28.1421 1.64972
\(292\) 0 0
\(293\) 11.4853 0.670977 0.335489 0.942044i \(-0.391099\pi\)
0.335489 + 0.942044i \(0.391099\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) −0.585786 −0.0339908
\(298\) 0 0
\(299\) −1.58579 −0.0917084
\(300\) 0 0
\(301\) 16.2426 0.936210
\(302\) 0 0
\(303\) 2.58579 0.148550
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) 6.34315 0.362022 0.181011 0.983481i \(-0.442063\pi\)
0.181011 + 0.983481i \(0.442063\pi\)
\(308\) 0 0
\(309\) −10.2426 −0.582683
\(310\) 0 0
\(311\) 13.2426 0.750921 0.375461 0.926838i \(-0.377485\pi\)
0.375461 + 0.926838i \(0.377485\pi\)
\(312\) 0 0
\(313\) 25.9706 1.46794 0.733971 0.679180i \(-0.237665\pi\)
0.733971 + 0.679180i \(0.237665\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 7.48528 0.420415 0.210208 0.977657i \(-0.432586\pi\)
0.210208 + 0.977657i \(0.432586\pi\)
\(318\) 0 0
\(319\) 8.24264 0.461499
\(320\) 0 0
\(321\) −13.8284 −0.771828
\(322\) 0 0
\(323\) 1.00000 0.0556415
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) −38.5563 −2.13217
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) 10.7574 0.591278 0.295639 0.955300i \(-0.404467\pi\)
0.295639 + 0.955300i \(0.404467\pi\)
\(332\) 0 0
\(333\) 24.0000 1.31519
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 33.8995 1.84662 0.923312 0.384052i \(-0.125472\pi\)
0.923312 + 0.384052i \(0.125472\pi\)
\(338\) 0 0
\(339\) 4.24264 0.230429
\(340\) 0 0
\(341\) −3.17157 −0.171750
\(342\) 0 0
\(343\) 18.2132 0.983421
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 22.4853 1.20707 0.603537 0.797335i \(-0.293758\pi\)
0.603537 + 0.797335i \(0.293758\pi\)
\(348\) 0 0
\(349\) 6.00000 0.321173 0.160586 0.987022i \(-0.448662\pi\)
0.160586 + 0.987022i \(0.448662\pi\)
\(350\) 0 0
\(351\) 0.0710678 0.00379332
\(352\) 0 0
\(353\) −19.4853 −1.03710 −0.518548 0.855048i \(-0.673527\pi\)
−0.518548 + 0.855048i \(0.673527\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 3.82843 0.202622
\(358\) 0 0
\(359\) 31.2426 1.64892 0.824462 0.565918i \(-0.191478\pi\)
0.824462 + 0.565918i \(0.191478\pi\)
\(360\) 0 0
\(361\) 1.00000 0.0526316
\(362\) 0 0
\(363\) 21.7279 1.14042
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) −25.4558 −1.32878 −0.664392 0.747384i \(-0.731309\pi\)
−0.664392 + 0.747384i \(0.731309\pi\)
\(368\) 0 0
\(369\) 12.0000 0.624695
\(370\) 0 0
\(371\) 18.2132 0.945582
\(372\) 0 0
\(373\) 9.00000 0.466002 0.233001 0.972476i \(-0.425145\pi\)
0.233001 + 0.972476i \(0.425145\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −1.00000 −0.0515026
\(378\) 0 0
\(379\) −2.75736 −0.141636 −0.0708180 0.997489i \(-0.522561\pi\)
−0.0708180 + 0.997489i \(0.522561\pi\)
\(380\) 0 0
\(381\) 34.9706 1.79160
\(382\) 0 0
\(383\) −3.75736 −0.191992 −0.0959960 0.995382i \(-0.530604\pi\)
−0.0959960 + 0.995382i \(0.530604\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) −28.9706 −1.47266
\(388\) 0 0
\(389\) 37.0711 1.87958 0.939789 0.341756i \(-0.111021\pi\)
0.939789 + 0.341756i \(0.111021\pi\)
\(390\) 0 0
\(391\) −9.24264 −0.467420
\(392\) 0 0
\(393\) 40.9706 2.06669
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) 24.0000 1.20453 0.602263 0.798298i \(-0.294266\pi\)
0.602263 + 0.798298i \(0.294266\pi\)
\(398\) 0 0
\(399\) 3.82843 0.191661
\(400\) 0 0
\(401\) −22.5858 −1.12788 −0.563940 0.825816i \(-0.690715\pi\)
−0.563940 + 0.825816i \(0.690715\pi\)
\(402\) 0 0
\(403\) 0.384776 0.0191671
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) −12.0000 −0.594818
\(408\) 0 0
\(409\) −17.2132 −0.851138 −0.425569 0.904926i \(-0.639926\pi\)
−0.425569 + 0.904926i \(0.639926\pi\)
\(410\) 0 0
\(411\) −31.3848 −1.54810
\(412\) 0 0
\(413\) −20.4558 −1.00657
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) −28.9706 −1.41869
\(418\) 0 0
\(419\) −16.5858 −0.810269 −0.405134 0.914257i \(-0.632775\pi\)
−0.405134 + 0.914257i \(0.632775\pi\)
\(420\) 0 0
\(421\) 2.51472 0.122560 0.0612799 0.998121i \(-0.480482\pi\)
0.0612799 + 0.998121i \(0.480482\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) −9.12994 −0.441829
\(428\) 0 0
\(429\) 0.585786 0.0282820
\(430\) 0 0
\(431\) −30.3848 −1.46358 −0.731792 0.681528i \(-0.761316\pi\)
−0.731792 + 0.681528i \(0.761316\pi\)
\(432\) 0 0
\(433\) 21.5563 1.03593 0.517966 0.855401i \(-0.326689\pi\)
0.517966 + 0.855401i \(0.326689\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −9.24264 −0.442135
\(438\) 0 0
\(439\) 14.2426 0.679764 0.339882 0.940468i \(-0.389613\pi\)
0.339882 + 0.940468i \(0.389613\pi\)
\(440\) 0 0
\(441\) −12.6863 −0.604109
\(442\) 0 0
\(443\) 4.24264 0.201574 0.100787 0.994908i \(-0.467864\pi\)
0.100787 + 0.994908i \(0.467864\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) −15.3137 −0.724314
\(448\) 0 0
\(449\) −5.31371 −0.250769 −0.125385 0.992108i \(-0.540017\pi\)
−0.125385 + 0.992108i \(0.540017\pi\)
\(450\) 0 0
\(451\) −6.00000 −0.282529
\(452\) 0 0
\(453\) 15.6569 0.735623
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −3.00000 −0.140334 −0.0701670 0.997535i \(-0.522353\pi\)
−0.0701670 + 0.997535i \(0.522353\pi\)
\(458\) 0 0
\(459\) 0.414214 0.0193338
\(460\) 0 0
\(461\) 27.5563 1.28343 0.641714 0.766944i \(-0.278224\pi\)
0.641714 + 0.766944i \(0.278224\pi\)
\(462\) 0 0
\(463\) 14.1421 0.657241 0.328620 0.944462i \(-0.393416\pi\)
0.328620 + 0.944462i \(0.393416\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 24.7279 1.14427 0.572136 0.820159i \(-0.306115\pi\)
0.572136 + 0.820159i \(0.306115\pi\)
\(468\) 0 0
\(469\) 21.0000 0.969690
\(470\) 0 0
\(471\) −0.828427 −0.0381719
\(472\) 0 0
\(473\) 14.4853 0.666034
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) −32.4853 −1.48740
\(478\) 0 0
\(479\) −31.1127 −1.42158 −0.710788 0.703407i \(-0.751661\pi\)
−0.710788 + 0.703407i \(0.751661\pi\)
\(480\) 0 0
\(481\) 1.45584 0.0663808
\(482\) 0 0
\(483\) −35.3848 −1.61006
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) 1.79899 0.0815200 0.0407600 0.999169i \(-0.487022\pi\)
0.0407600 + 0.999169i \(0.487022\pi\)
\(488\) 0 0
\(489\) −4.24264 −0.191859
\(490\) 0 0
\(491\) 2.44365 0.110280 0.0551402 0.998479i \(-0.482439\pi\)
0.0551402 + 0.998479i \(0.482439\pi\)
\(492\) 0 0
\(493\) −5.82843 −0.262499
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −16.7868 −0.752991
\(498\) 0 0
\(499\) 34.2426 1.53291 0.766456 0.642297i \(-0.222019\pi\)
0.766456 + 0.642297i \(0.222019\pi\)
\(500\) 0 0
\(501\) −23.5563 −1.05242
\(502\) 0 0
\(503\) −39.7279 −1.77138 −0.885690 0.464277i \(-0.846314\pi\)
−0.885690 + 0.464277i \(0.846314\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 31.3137 1.39069
\(508\) 0 0
\(509\) −4.97056 −0.220316 −0.110158 0.993914i \(-0.535136\pi\)
−0.110158 + 0.993914i \(0.535136\pi\)
\(510\) 0 0
\(511\) 8.69848 0.384798
\(512\) 0 0
\(513\) 0.414214 0.0182880
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) 39.7990 1.74698
\(520\) 0 0
\(521\) 0.686292 0.0300670 0.0150335 0.999887i \(-0.495215\pi\)
0.0150335 + 0.999887i \(0.495215\pi\)
\(522\) 0 0
\(523\) 27.7279 1.21246 0.606229 0.795290i \(-0.292682\pi\)
0.606229 + 0.795290i \(0.292682\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 2.24264 0.0976910
\(528\) 0 0
\(529\) 62.4264 2.71419
\(530\) 0 0
\(531\) 36.4853 1.58333
\(532\) 0 0
\(533\) 0.727922 0.0315298
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) −0.828427 −0.0357493
\(538\) 0 0
\(539\) 6.34315 0.273219
\(540\) 0 0
\(541\) 18.2426 0.784312 0.392156 0.919899i \(-0.371729\pi\)
0.392156 + 0.919899i \(0.371729\pi\)
\(542\) 0 0
\(543\) −20.4853 −0.879108
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) −5.31371 −0.227198 −0.113599 0.993527i \(-0.536238\pi\)
−0.113599 + 0.993527i \(0.536238\pi\)
\(548\) 0 0
\(549\) 16.2843 0.694996
\(550\) 0 0
\(551\) −5.82843 −0.248299
\(552\) 0 0
\(553\) 16.6274 0.707070
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 16.0000 0.677942 0.338971 0.940797i \(-0.389921\pi\)
0.338971 + 0.940797i \(0.389921\pi\)
\(558\) 0 0
\(559\) −1.75736 −0.0743284
\(560\) 0 0
\(561\) 3.41421 0.144148
\(562\) 0 0
\(563\) 28.9706 1.22096 0.610482 0.792030i \(-0.290976\pi\)
0.610482 + 0.792030i \(0.290976\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 15.0416 0.631689
\(568\) 0 0
\(569\) 28.2843 1.18574 0.592869 0.805299i \(-0.297995\pi\)
0.592869 + 0.805299i \(0.297995\pi\)
\(570\) 0 0
\(571\) −6.24264 −0.261246 −0.130623 0.991432i \(-0.541698\pi\)
−0.130623 + 0.991432i \(0.541698\pi\)
\(572\) 0 0
\(573\) −44.7990 −1.87150
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) 2.31371 0.0963209 0.0481605 0.998840i \(-0.484664\pi\)
0.0481605 + 0.998840i \(0.484664\pi\)
\(578\) 0 0
\(579\) 28.1421 1.16955
\(580\) 0 0
\(581\) −3.94113 −0.163505
\(582\) 0 0
\(583\) 16.2426 0.672701
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −21.7574 −0.898022 −0.449011 0.893526i \(-0.648224\pi\)
−0.449011 + 0.893526i \(0.648224\pi\)
\(588\) 0 0
\(589\) 2.24264 0.0924064
\(590\) 0 0
\(591\) 48.8701 2.01025
\(592\) 0 0
\(593\) 22.0000 0.903432 0.451716 0.892162i \(-0.350812\pi\)
0.451716 + 0.892162i \(0.350812\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 1.82843 0.0748325
\(598\) 0 0
\(599\) −27.2132 −1.11190 −0.555951 0.831215i \(-0.687646\pi\)
−0.555951 + 0.831215i \(0.687646\pi\)
\(600\) 0 0
\(601\) −11.7574 −0.479593 −0.239796 0.970823i \(-0.577081\pi\)
−0.239796 + 0.970823i \(0.577081\pi\)
\(602\) 0 0
\(603\) −37.4558 −1.52532
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) −8.82843 −0.358335 −0.179167 0.983819i \(-0.557340\pi\)
−0.179167 + 0.983819i \(0.557340\pi\)
\(608\) 0 0
\(609\) −22.3137 −0.904197
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) 47.6985 1.92652 0.963262 0.268564i \(-0.0865491\pi\)
0.963262 + 0.268564i \(0.0865491\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 12.4853 0.502639 0.251319 0.967904i \(-0.419136\pi\)
0.251319 + 0.967904i \(0.419136\pi\)
\(618\) 0 0
\(619\) −16.2426 −0.652847 −0.326423 0.945224i \(-0.605844\pi\)
−0.326423 + 0.945224i \(0.605844\pi\)
\(620\) 0 0
\(621\) −3.82843 −0.153629
\(622\) 0 0
\(623\) 11.2132 0.449248
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) 3.41421 0.136351
\(628\) 0 0
\(629\) 8.48528 0.338330
\(630\) 0 0
\(631\) 5.02944 0.200219 0.100109 0.994976i \(-0.468081\pi\)
0.100109 + 0.994976i \(0.468081\pi\)
\(632\) 0 0
\(633\) 13.8284 0.549631
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) −0.769553 −0.0304908
\(638\) 0 0
\(639\) 29.9411 1.18445
\(640\) 0 0
\(641\) 30.0416 1.18657 0.593287 0.804991i \(-0.297830\pi\)
0.593287 + 0.804991i \(0.297830\pi\)
\(642\) 0 0
\(643\) −2.48528 −0.0980099 −0.0490050 0.998799i \(-0.515605\pi\)
−0.0490050 + 0.998799i \(0.515605\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 27.2426 1.07102 0.535509 0.844529i \(-0.320120\pi\)
0.535509 + 0.844529i \(0.320120\pi\)
\(648\) 0 0
\(649\) −18.2426 −0.716086
\(650\) 0 0
\(651\) 8.58579 0.336504
\(652\) 0 0
\(653\) 4.97056 0.194513 0.0972566 0.995259i \(-0.468993\pi\)
0.0972566 + 0.995259i \(0.468993\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) −15.5147 −0.605287
\(658\) 0 0
\(659\) 0.899495 0.0350393 0.0175197 0.999847i \(-0.494423\pi\)
0.0175197 + 0.999847i \(0.494423\pi\)
\(660\) 0 0
\(661\) 32.4558 1.26239 0.631193 0.775626i \(-0.282566\pi\)
0.631193 + 0.775626i \(0.282566\pi\)
\(662\) 0 0
\(663\) −0.414214 −0.0160867
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 53.8701 2.08586
\(668\) 0 0
\(669\) −50.2843 −1.94410
\(670\) 0 0
\(671\) −8.14214 −0.314324
\(672\) 0 0
\(673\) −12.0000 −0.462566 −0.231283 0.972887i \(-0.574292\pi\)
−0.231283 + 0.972887i \(0.574292\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 26.9411 1.03543 0.517716 0.855553i \(-0.326782\pi\)
0.517716 + 0.855553i \(0.326782\pi\)
\(678\) 0 0
\(679\) 18.4853 0.709400
\(680\) 0 0
\(681\) −60.9411 −2.33527
\(682\) 0 0
\(683\) −12.0000 −0.459167 −0.229584 0.973289i \(-0.573736\pi\)
−0.229584 + 0.973289i \(0.573736\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) 45.7990 1.74734
\(688\) 0 0
\(689\) −1.97056 −0.0750725
\(690\) 0 0
\(691\) −5.51472 −0.209790 −0.104895 0.994483i \(-0.533451\pi\)
−0.104895 + 0.994483i \(0.533451\pi\)
\(692\) 0 0
\(693\) 6.34315 0.240956
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 4.24264 0.160701
\(698\) 0 0
\(699\) −21.6569 −0.819137
\(700\) 0 0
\(701\) −16.9706 −0.640969 −0.320485 0.947254i \(-0.603846\pi\)
−0.320485 + 0.947254i \(0.603846\pi\)
\(702\) 0 0
\(703\) 8.48528 0.320028
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 1.69848 0.0638781
\(708\) 0 0
\(709\) −34.0000 −1.27690 −0.638448 0.769665i \(-0.720423\pi\)
−0.638448 + 0.769665i \(0.720423\pi\)
\(710\) 0 0
\(711\) −29.6569 −1.11222
\(712\) 0 0
\(713\) −20.7279 −0.776267
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 31.1421 1.16302
\(718\) 0 0
\(719\) 24.8995 0.928594 0.464297 0.885679i \(-0.346307\pi\)
0.464297 + 0.885679i \(0.346307\pi\)
\(720\) 0 0
\(721\) −6.72792 −0.250561
\(722\) 0 0
\(723\) 60.2843 2.24200
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) 15.7279 0.583316 0.291658 0.956523i \(-0.405793\pi\)
0.291658 + 0.956523i \(0.405793\pi\)
\(728\) 0 0
\(729\) −23.8284 −0.882534
\(730\) 0 0
\(731\) −10.2426 −0.378838
\(732\) 0 0
\(733\) 12.0000 0.443230 0.221615 0.975134i \(-0.428867\pi\)
0.221615 + 0.975134i \(0.428867\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 18.7279 0.689852
\(738\) 0 0
\(739\) −26.7279 −0.983203 −0.491601 0.870820i \(-0.663588\pi\)
−0.491601 + 0.870820i \(0.663588\pi\)
\(740\) 0 0
\(741\) −0.414214 −0.0152165
\(742\) 0 0
\(743\) 18.7279 0.687061 0.343530 0.939142i \(-0.388377\pi\)
0.343530 + 0.939142i \(0.388377\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 7.02944 0.257194
\(748\) 0 0
\(749\) −9.08326 −0.331895
\(750\) 0 0
\(751\) 1.27208 0.0464188 0.0232094 0.999731i \(-0.492612\pi\)
0.0232094 + 0.999731i \(0.492612\pi\)
\(752\) 0 0
\(753\) −66.5269 −2.42438
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 23.6569 0.859823 0.429911 0.902871i \(-0.358545\pi\)
0.429911 + 0.902871i \(0.358545\pi\)
\(758\) 0 0
\(759\) −31.5563 −1.14542
\(760\) 0 0
\(761\) 10.0294 0.363567 0.181783 0.983339i \(-0.441813\pi\)
0.181783 + 0.983339i \(0.441813\pi\)
\(762\) 0 0
\(763\) −25.3259 −0.916859
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 2.21320 0.0799141
\(768\) 0 0
\(769\) 14.4558 0.521291 0.260646 0.965435i \(-0.416065\pi\)
0.260646 + 0.965435i \(0.416065\pi\)
\(770\) 0 0
\(771\) 11.4142 0.411073
\(772\) 0 0
\(773\) −19.9706 −0.718291 −0.359146 0.933282i \(-0.616932\pi\)
−0.359146 + 0.933282i \(0.616932\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 32.4853 1.16540
\(778\) 0 0
\(779\) 4.24264 0.152008
\(780\) 0 0
\(781\) −14.9706 −0.535689
\(782\) 0 0
\(783\) −2.41421 −0.0862770
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) −35.1838 −1.25417 −0.627083 0.778953i \(-0.715751\pi\)
−0.627083 + 0.778953i \(0.715751\pi\)
\(788\) 0 0
\(789\) 16.8284 0.599108
\(790\) 0 0
\(791\) 2.78680 0.0990871
\(792\) 0 0
\(793\) 0.987807 0.0350780
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −30.5147 −1.08089 −0.540443 0.841380i \(-0.681743\pi\)
−0.540443 + 0.841380i \(0.681743\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 0 0
\(801\) −20.0000 −0.706665
\(802\) 0 0
\(803\) 7.75736 0.273751
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) −69.1127 −2.43288
\(808\) 0 0
\(809\) 10.7990 0.379672 0.189836 0.981816i \(-0.439204\pi\)
0.189836 + 0.981816i \(0.439204\pi\)
\(810\) 0 0
\(811\) 6.69848 0.235216 0.117608 0.993060i \(-0.462477\pi\)
0.117608 + 0.993060i \(0.462477\pi\)
\(812\) 0 0
\(813\) −45.2843 −1.58819
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) −10.2426 −0.358345
\(818\) 0 0
\(819\) −0.769553 −0.0268903
\(820\) 0 0
\(821\) 17.6569 0.616228 0.308114 0.951349i \(-0.400302\pi\)
0.308114 + 0.951349i \(0.400302\pi\)
\(822\) 0 0
\(823\) −9.72792 −0.339094 −0.169547 0.985522i \(-0.554230\pi\)
−0.169547 + 0.985522i \(0.554230\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −38.6985 −1.34568 −0.672839 0.739789i \(-0.734925\pi\)
−0.672839 + 0.739789i \(0.734925\pi\)
\(828\) 0 0
\(829\) −40.4558 −1.40509 −0.702545 0.711640i \(-0.747953\pi\)
−0.702545 + 0.711640i \(0.747953\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) −4.48528 −0.155406
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 0.928932 0.0321086
\(838\) 0 0
\(839\) 22.6274 0.781185 0.390593 0.920564i \(-0.372270\pi\)
0.390593 + 0.920564i \(0.372270\pi\)
\(840\) 0 0
\(841\) 4.97056 0.171399
\(842\) 0 0
\(843\) 10.2426 0.352775
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) 14.2721 0.490394
\(848\) 0 0
\(849\) 9.31371 0.319646
\(850\) 0 0
\(851\) −78.4264 −2.68842
\(852\) 0 0
\(853\) 39.9411 1.36756 0.683779 0.729689i \(-0.260335\pi\)
0.683779 + 0.729689i \(0.260335\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(858\) 0 0
\(859\) 20.7279 0.707228 0.353614 0.935392i \(-0.384953\pi\)
0.353614 + 0.935392i \(0.384953\pi\)
\(860\) 0 0
\(861\) 16.2426 0.553548
\(862\) 0 0
\(863\) 24.7279 0.841748 0.420874 0.907119i \(-0.361723\pi\)
0.420874 + 0.907119i \(0.361723\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) 38.6274 1.31186
\(868\) 0 0
\(869\) 14.8284 0.503020
\(870\) 0 0
\(871\) −2.27208 −0.0769864
\(872\) 0 0
\(873\) −32.9706 −1.11588
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) −47.1421 −1.59188 −0.795938 0.605378i \(-0.793022\pi\)
−0.795938 + 0.605378i \(0.793022\pi\)
\(878\) 0 0
\(879\) −27.7279 −0.935240
\(880\) 0 0
\(881\) 39.5980 1.33409 0.667045 0.745018i \(-0.267559\pi\)
0.667045 + 0.745018i \(0.267559\pi\)
\(882\) 0 0
\(883\) 2.48528 0.0836364 0.0418182 0.999125i \(-0.486685\pi\)
0.0418182 + 0.999125i \(0.486685\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 2.78680 0.0935715 0.0467857 0.998905i \(-0.485102\pi\)
0.0467857 + 0.998905i \(0.485102\pi\)
\(888\) 0 0
\(889\) 22.9706 0.770408
\(890\) 0 0
\(891\) 13.4142 0.449393
\(892\) 0 0
\(893\) 0 0
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 3.82843 0.127827
\(898\) 0 0
\(899\) −13.0711 −0.435945
\(900\) 0 0
\(901\) −11.4853 −0.382630
\(902\) 0 0
\(903\) −39.2132 −1.30493
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) 20.6985 0.687282 0.343641 0.939101i \(-0.388340\pi\)
0.343641 + 0.939101i \(0.388340\pi\)
\(908\) 0 0
\(909\) −3.02944 −0.100480
\(910\) 0 0
\(911\) 16.2843 0.539522 0.269761 0.962927i \(-0.413055\pi\)
0.269761 + 0.962927i \(0.413055\pi\)
\(912\) 0 0
\(913\) −3.51472 −0.116320
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 26.9117 0.888702
\(918\) 0 0
\(919\) −36.2132 −1.19456 −0.597282 0.802032i \(-0.703753\pi\)
−0.597282 + 0.802032i \(0.703753\pi\)
\(920\) 0 0
\(921\) −15.3137 −0.504604
\(922\) 0 0
\(923\) 1.81623 0.0597821
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) 12.0000 0.394132
\(928\) 0 0
\(929\) −17.8284 −0.584932 −0.292466 0.956276i \(-0.594476\pi\)
−0.292466 + 0.956276i \(0.594476\pi\)
\(930\) 0 0
\(931\) −4.48528 −0.146999
\(932\) 0 0
\(933\) −31.9706 −1.04667
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 27.0000 0.882052 0.441026 0.897494i \(-0.354615\pi\)
0.441026 + 0.897494i \(0.354615\pi\)
\(938\) 0 0
\(939\) −62.6985 −2.04609
\(940\) 0 0
\(941\) 24.8579 0.810343 0.405172 0.914241i \(-0.367212\pi\)
0.405172 + 0.914241i \(0.367212\pi\)
\(942\) 0 0
\(943\) −39.2132 −1.27696
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 12.0000 0.389948 0.194974 0.980808i \(-0.437538\pi\)
0.194974 + 0.980808i \(0.437538\pi\)
\(948\) 0 0
\(949\) −0.941125 −0.0305502
\(950\) 0 0
\(951\) −18.0711 −0.585995
\(952\) 0 0
\(953\) −6.72792 −0.217939 −0.108969 0.994045i \(-0.534755\pi\)
−0.108969 + 0.994045i \(0.534755\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) −19.8995 −0.643259
\(958\) 0 0
\(959\) −20.6152 −0.665700
\(960\) 0 0
\(961\) −25.9706 −0.837760
\(962\) 0 0
\(963\) 16.2010 0.522070
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) 19.1127 0.614623 0.307311 0.951609i \(-0.400571\pi\)
0.307311 + 0.951609i \(0.400571\pi\)
\(968\) 0 0
\(969\) −2.41421 −0.0775557
\(970\) 0 0
\(971\) −30.3431 −0.973758 −0.486879 0.873469i \(-0.661865\pi\)
−0.486879 + 0.873469i \(0.661865\pi\)
\(972\) 0 0
\(973\) −19.0294 −0.610056
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −8.24264 −0.263705 −0.131853 0.991269i \(-0.542093\pi\)
−0.131853 + 0.991269i \(0.542093\pi\)
\(978\) 0 0
\(979\) 10.0000 0.319601
\(980\) 0 0
\(981\) 45.1716 1.44222
\(982\) 0 0
\(983\) 0.544156 0.0173559 0.00867794 0.999962i \(-0.497238\pi\)
0.00867794 + 0.999962i \(0.497238\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 94.6690 3.01030
\(990\) 0 0
\(991\) −22.2426 −0.706561 −0.353280 0.935517i \(-0.614934\pi\)
−0.353280 + 0.935517i \(0.614934\pi\)
\(992\) 0 0
\(993\) −25.9706 −0.824151
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) 23.2721 0.737034 0.368517 0.929621i \(-0.379866\pi\)
0.368517 + 0.929621i \(0.379866\pi\)
\(998\) 0 0
\(999\) 3.51472 0.111201
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 7600.2.a.v.1.1 2
4.3 odd 2 950.2.a.f.1.2 2
5.2 odd 4 1520.2.d.e.609.4 4
5.3 odd 4 1520.2.d.e.609.1 4
5.4 even 2 7600.2.a.bg.1.2 2
12.11 even 2 8550.2.a.cb.1.1 2
20.3 even 4 190.2.b.a.39.4 yes 4
20.7 even 4 190.2.b.a.39.1 4
20.19 odd 2 950.2.a.g.1.1 2
60.23 odd 4 1710.2.d.c.1369.2 4
60.47 odd 4 1710.2.d.c.1369.4 4
60.59 even 2 8550.2.a.bn.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
190.2.b.a.39.1 4 20.7 even 4
190.2.b.a.39.4 yes 4 20.3 even 4
950.2.a.f.1.2 2 4.3 odd 2
950.2.a.g.1.1 2 20.19 odd 2
1520.2.d.e.609.1 4 5.3 odd 4
1520.2.d.e.609.4 4 5.2 odd 4
1710.2.d.c.1369.2 4 60.23 odd 4
1710.2.d.c.1369.4 4 60.47 odd 4
7600.2.a.v.1.1 2 1.1 even 1 trivial
7600.2.a.bg.1.2 2 5.4 even 2
8550.2.a.bn.1.2 2 60.59 even 2
8550.2.a.cb.1.1 2 12.11 even 2