Properties

Label 2800.2.a.br.1.2
Level $2800$
Weight $2$
Character 2800.1
Self dual yes
Analytic conductor $22.358$
Analytic rank $0$
Dimension $3$
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] = [2800,2,Mod(1,2800)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2800, 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("2800.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2800 = 2^{4} \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2800.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: \(22.3581125660\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.568.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 6x - 2 \) 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 280)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-0.363328\) of defining polynomial
Character \(\chi\) \(=\) 2800.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-0.363328 q^{3} -1.00000 q^{7} -2.86799 q^{9} +O(q^{10})\) \(q-0.363328 q^{3} -1.00000 q^{7} -2.86799 q^{9} -5.14134 q^{11} -4.64600 q^{13} +3.86799 q^{17} +0.778008 q^{19} +0.363328 q^{21} -5.00933 q^{23} +2.13201 q^{27} +9.42401 q^{29} -4.72666 q^{31} +1.86799 q^{33} +6.00000 q^{37} +1.68802 q^{39} -1.00933 q^{41} +7.00933 q^{43} +11.4240 q^{47} +1.00000 q^{49} -1.40535 q^{51} +7.55602 q^{53} -0.282672 q^{57} -12.5140 q^{59} +11.5047 q^{61} +2.86799 q^{63} +11.7360 q^{67} +1.82003 q^{69} -2.72666 q^{71} +5.00933 q^{73} +5.14134 q^{77} -5.68802 q^{79} +7.82936 q^{81} +4.67531 q^{83} -3.42401 q^{87} -2.82936 q^{89} +4.64600 q^{91} +1.71733 q^{93} -1.58532 q^{97} +14.7453 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + q^{3} - 3 q^{7} + 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q + q^{3} - 3 q^{7} + 4 q^{9} - 7 q^{11} + 5 q^{13} - q^{17} - 4 q^{19} - q^{21} + 6 q^{23} + 19 q^{27} + 3 q^{29} - 10 q^{31} - 7 q^{33} + 18 q^{37} + 5 q^{39} + 18 q^{41} + 9 q^{47} + 3 q^{49} - 21 q^{51} + 10 q^{53} + 16 q^{57} - 6 q^{59} + 24 q^{61} - 4 q^{63} + 10 q^{67} + 18 q^{69} - 4 q^{71} - 6 q^{73} + 7 q^{77} - 17 q^{79} + 15 q^{81} + 12 q^{83} + 15 q^{87} - 5 q^{91} + 22 q^{93} - 9 q^{97} - 2 q^{99}+O(q^{100}) \) 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
\(3\) −0.363328 −0.209768 −0.104884 0.994484i \(-0.533447\pi\)
−0.104884 + 0.994484i \(0.533447\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) −1.00000 −0.377964
\(8\) 0 0
\(9\) −2.86799 −0.955998
\(10\) 0 0
\(11\) −5.14134 −1.55017 −0.775086 0.631856i \(-0.782293\pi\)
−0.775086 + 0.631856i \(0.782293\pi\)
\(12\) 0 0
\(13\) −4.64600 −1.28857 −0.644284 0.764786i \(-0.722845\pi\)
−0.644284 + 0.764786i \(0.722845\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 3.86799 0.938126 0.469063 0.883165i \(-0.344592\pi\)
0.469063 + 0.883165i \(0.344592\pi\)
\(18\) 0 0
\(19\) 0.778008 0.178487 0.0892436 0.996010i \(-0.471555\pi\)
0.0892436 + 0.996010i \(0.471555\pi\)
\(20\) 0 0
\(21\) 0.363328 0.0792847
\(22\) 0 0
\(23\) −5.00933 −1.04452 −0.522259 0.852787i \(-0.674910\pi\)
−0.522259 + 0.852787i \(0.674910\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 2.13201 0.410305
\(28\) 0 0
\(29\) 9.42401 1.74999 0.874997 0.484128i \(-0.160863\pi\)
0.874997 + 0.484128i \(0.160863\pi\)
\(30\) 0 0
\(31\) −4.72666 −0.848933 −0.424466 0.905444i \(-0.639538\pi\)
−0.424466 + 0.905444i \(0.639538\pi\)
\(32\) 0 0
\(33\) 1.86799 0.325176
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 6.00000 0.986394 0.493197 0.869918i \(-0.335828\pi\)
0.493197 + 0.869918i \(0.335828\pi\)
\(38\) 0 0
\(39\) 1.68802 0.270300
\(40\) 0 0
\(41\) −1.00933 −0.157631 −0.0788153 0.996889i \(-0.525114\pi\)
−0.0788153 + 0.996889i \(0.525114\pi\)
\(42\) 0 0
\(43\) 7.00933 1.06891 0.534456 0.845196i \(-0.320516\pi\)
0.534456 + 0.845196i \(0.320516\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 11.4240 1.66636 0.833181 0.553000i \(-0.186517\pi\)
0.833181 + 0.553000i \(0.186517\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) −1.40535 −0.196788
\(52\) 0 0
\(53\) 7.55602 1.03790 0.518949 0.854805i \(-0.326323\pi\)
0.518949 + 0.854805i \(0.326323\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) −0.282672 −0.0374409
\(58\) 0 0
\(59\) −12.5140 −1.62918 −0.814592 0.580035i \(-0.803039\pi\)
−0.814592 + 0.580035i \(0.803039\pi\)
\(60\) 0 0
\(61\) 11.5047 1.47302 0.736511 0.676426i \(-0.236472\pi\)
0.736511 + 0.676426i \(0.236472\pi\)
\(62\) 0 0
\(63\) 2.86799 0.361333
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 11.7360 1.43378 0.716889 0.697187i \(-0.245565\pi\)
0.716889 + 0.697187i \(0.245565\pi\)
\(68\) 0 0
\(69\) 1.82003 0.219106
\(70\) 0 0
\(71\) −2.72666 −0.323595 −0.161797 0.986824i \(-0.551729\pi\)
−0.161797 + 0.986824i \(0.551729\pi\)
\(72\) 0 0
\(73\) 5.00933 0.586298 0.293149 0.956067i \(-0.405297\pi\)
0.293149 + 0.956067i \(0.405297\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 5.14134 0.585910
\(78\) 0 0
\(79\) −5.68802 −0.639953 −0.319976 0.947426i \(-0.603675\pi\)
−0.319976 + 0.947426i \(0.603675\pi\)
\(80\) 0 0
\(81\) 7.82936 0.869929
\(82\) 0 0
\(83\) 4.67531 0.513181 0.256591 0.966520i \(-0.417401\pi\)
0.256591 + 0.966520i \(0.417401\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) −3.42401 −0.367092
\(88\) 0 0
\(89\) −2.82936 −0.299911 −0.149956 0.988693i \(-0.547913\pi\)
−0.149956 + 0.988693i \(0.547913\pi\)
\(90\) 0 0
\(91\) 4.64600 0.487033
\(92\) 0 0
\(93\) 1.71733 0.178079
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −1.58532 −0.160965 −0.0804824 0.996756i \(-0.525646\pi\)
−0.0804824 + 0.996756i \(0.525646\pi\)
\(98\) 0 0
\(99\) 14.7453 1.48196
\(100\) 0 0
\(101\) −9.06068 −0.901571 −0.450786 0.892632i \(-0.648856\pi\)
−0.450786 + 0.892632i \(0.648856\pi\)
\(102\) 0 0
\(103\) −5.14134 −0.506591 −0.253295 0.967389i \(-0.581514\pi\)
−0.253295 + 0.967389i \(0.581514\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(108\) 0 0
\(109\) −3.97070 −0.380324 −0.190162 0.981753i \(-0.560901\pi\)
−0.190162 + 0.981753i \(0.560901\pi\)
\(110\) 0 0
\(111\) −2.17997 −0.206914
\(112\) 0 0
\(113\) 4.54669 0.427716 0.213858 0.976865i \(-0.431397\pi\)
0.213858 + 0.976865i \(0.431397\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 13.3247 1.23187
\(118\) 0 0
\(119\) −3.86799 −0.354578
\(120\) 0 0
\(121\) 15.4333 1.40303
\(122\) 0 0
\(123\) 0.366718 0.0330658
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) −16.1214 −1.43054 −0.715270 0.698849i \(-0.753696\pi\)
−0.715270 + 0.698849i \(0.753696\pi\)
\(128\) 0 0
\(129\) −2.54669 −0.224223
\(130\) 0 0
\(131\) −0.0513514 −0.00448659 −0.00224330 0.999997i \(-0.500714\pi\)
−0.00224330 + 0.999997i \(0.500714\pi\)
\(132\) 0 0
\(133\) −0.778008 −0.0674618
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −14.0187 −1.19769 −0.598847 0.800863i \(-0.704374\pi\)
−0.598847 + 0.800863i \(0.704374\pi\)
\(138\) 0 0
\(139\) −12.6167 −1.07013 −0.535067 0.844810i \(-0.679714\pi\)
−0.535067 + 0.844810i \(0.679714\pi\)
\(140\) 0 0
\(141\) −4.15066 −0.349549
\(142\) 0 0
\(143\) 23.8867 1.99750
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) −0.363328 −0.0299668
\(148\) 0 0
\(149\) −11.4533 −0.938292 −0.469146 0.883121i \(-0.655438\pi\)
−0.469146 + 0.883121i \(0.655438\pi\)
\(150\) 0 0
\(151\) −5.86799 −0.477530 −0.238765 0.971077i \(-0.576743\pi\)
−0.238765 + 0.971077i \(0.576743\pi\)
\(152\) 0 0
\(153\) −11.0934 −0.896846
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) −5.78734 −0.461880 −0.230940 0.972968i \(-0.574180\pi\)
−0.230940 + 0.972968i \(0.574180\pi\)
\(158\) 0 0
\(159\) −2.74531 −0.217718
\(160\) 0 0
\(161\) 5.00933 0.394790
\(162\) 0 0
\(163\) −3.27334 −0.256388 −0.128194 0.991749i \(-0.540918\pi\)
−0.128194 + 0.991749i \(0.540918\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 24.8773 1.92506 0.962532 0.271166i \(-0.0874094\pi\)
0.962532 + 0.271166i \(0.0874094\pi\)
\(168\) 0 0
\(169\) 8.58532 0.660409
\(170\) 0 0
\(171\) −2.23132 −0.170633
\(172\) 0 0
\(173\) 6.62734 0.503868 0.251934 0.967744i \(-0.418933\pi\)
0.251934 + 0.967744i \(0.418933\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 4.54669 0.341750
\(178\) 0 0
\(179\) 10.0187 0.748830 0.374415 0.927261i \(-0.377844\pi\)
0.374415 + 0.927261i \(0.377844\pi\)
\(180\) 0 0
\(181\) 1.78734 0.132852 0.0664258 0.997791i \(-0.478840\pi\)
0.0664258 + 0.997791i \(0.478840\pi\)
\(182\) 0 0
\(183\) −4.17997 −0.308992
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) −19.8867 −1.45426
\(188\) 0 0
\(189\) −2.13201 −0.155081
\(190\) 0 0
\(191\) 8.59465 0.621887 0.310943 0.950428i \(-0.399355\pi\)
0.310943 + 0.950428i \(0.399355\pi\)
\(192\) 0 0
\(193\) −5.17064 −0.372191 −0.186095 0.982532i \(-0.559583\pi\)
−0.186095 + 0.982532i \(0.559583\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 23.9160 1.70394 0.851971 0.523590i \(-0.175407\pi\)
0.851971 + 0.523590i \(0.175407\pi\)
\(198\) 0 0
\(199\) 15.3107 1.08534 0.542672 0.839945i \(-0.317413\pi\)
0.542672 + 0.839945i \(0.317413\pi\)
\(200\) 0 0
\(201\) −4.26401 −0.300760
\(202\) 0 0
\(203\) −9.42401 −0.661436
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 14.3667 0.998556
\(208\) 0 0
\(209\) −4.00000 −0.276686
\(210\) 0 0
\(211\) 1.03863 0.0715025 0.0357512 0.999361i \(-0.488618\pi\)
0.0357512 + 0.999361i \(0.488618\pi\)
\(212\) 0 0
\(213\) 0.990671 0.0678797
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 4.72666 0.320866
\(218\) 0 0
\(219\) −1.82003 −0.122986
\(220\) 0 0
\(221\) −17.9707 −1.20884
\(222\) 0 0
\(223\) 9.86799 0.660810 0.330405 0.943839i \(-0.392815\pi\)
0.330405 + 0.943839i \(0.392815\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 21.6553 1.43731 0.718657 0.695364i \(-0.244757\pi\)
0.718657 + 0.695364i \(0.244757\pi\)
\(228\) 0 0
\(229\) 20.2313 1.33692 0.668462 0.743747i \(-0.266953\pi\)
0.668462 + 0.743747i \(0.266953\pi\)
\(230\) 0 0
\(231\) −1.86799 −0.122905
\(232\) 0 0
\(233\) −5.45331 −0.357258 −0.178629 0.983916i \(-0.557166\pi\)
−0.178629 + 0.983916i \(0.557166\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 2.06662 0.134241
\(238\) 0 0
\(239\) 11.5853 0.749392 0.374696 0.927148i \(-0.377747\pi\)
0.374696 + 0.927148i \(0.377747\pi\)
\(240\) 0 0
\(241\) −5.00933 −0.322679 −0.161340 0.986899i \(-0.551581\pi\)
−0.161340 + 0.986899i \(0.551581\pi\)
\(242\) 0 0
\(243\) −9.24065 −0.592788
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) −3.61462 −0.229993
\(248\) 0 0
\(249\) −1.69867 −0.107649
\(250\) 0 0
\(251\) 23.4206 1.47830 0.739148 0.673543i \(-0.235228\pi\)
0.739148 + 0.673543i \(0.235228\pi\)
\(252\) 0 0
\(253\) 25.7546 1.61918
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 13.0093 0.811500 0.405750 0.913984i \(-0.367010\pi\)
0.405750 + 0.913984i \(0.367010\pi\)
\(258\) 0 0
\(259\) −6.00000 −0.372822
\(260\) 0 0
\(261\) −27.0280 −1.67299
\(262\) 0 0
\(263\) −6.01866 −0.371126 −0.185563 0.982632i \(-0.559411\pi\)
−0.185563 + 0.982632i \(0.559411\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 1.02799 0.0629117
\(268\) 0 0
\(269\) 3.24065 0.197586 0.0987929 0.995108i \(-0.468502\pi\)
0.0987929 + 0.995108i \(0.468502\pi\)
\(270\) 0 0
\(271\) 29.1307 1.76956 0.884782 0.466006i \(-0.154307\pi\)
0.884782 + 0.466006i \(0.154307\pi\)
\(272\) 0 0
\(273\) −1.68802 −0.102164
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) −4.44398 −0.267013 −0.133507 0.991048i \(-0.542624\pi\)
−0.133507 + 0.991048i \(0.542624\pi\)
\(278\) 0 0
\(279\) 13.5560 0.811577
\(280\) 0 0
\(281\) −24.6974 −1.47332 −0.736660 0.676263i \(-0.763598\pi\)
−0.736660 + 0.676263i \(0.763598\pi\)
\(282\) 0 0
\(283\) −7.73937 −0.460058 −0.230029 0.973184i \(-0.573882\pi\)
−0.230029 + 0.973184i \(0.573882\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 1.00933 0.0595788
\(288\) 0 0
\(289\) −2.03863 −0.119920
\(290\) 0 0
\(291\) 0.575992 0.0337652
\(292\) 0 0
\(293\) 25.1086 1.46686 0.733431 0.679764i \(-0.237918\pi\)
0.733431 + 0.679764i \(0.237918\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) −10.9614 −0.636043
\(298\) 0 0
\(299\) 23.2733 1.34593
\(300\) 0 0
\(301\) −7.00933 −0.404011
\(302\) 0 0
\(303\) 3.29200 0.189121
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) 33.9193 1.93588 0.967940 0.251183i \(-0.0808196\pi\)
0.967940 + 0.251183i \(0.0808196\pi\)
\(308\) 0 0
\(309\) 1.86799 0.106266
\(310\) 0 0
\(311\) −0.565344 −0.0320577 −0.0160289 0.999872i \(-0.505102\pi\)
−0.0160289 + 0.999872i \(0.505102\pi\)
\(312\) 0 0
\(313\) −27.3400 −1.54535 −0.772673 0.634804i \(-0.781081\pi\)
−0.772673 + 0.634804i \(0.781081\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 22.7267 1.27646 0.638228 0.769847i \(-0.279668\pi\)
0.638228 + 0.769847i \(0.279668\pi\)
\(318\) 0 0
\(319\) −48.4520 −2.71279
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 3.00933 0.167444
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) 1.44267 0.0797796
\(328\) 0 0
\(329\) −11.4240 −0.629826
\(330\) 0 0
\(331\) −0.462642 −0.0254291 −0.0127145 0.999919i \(-0.504047\pi\)
−0.0127145 + 0.999919i \(0.504047\pi\)
\(332\) 0 0
\(333\) −17.2080 −0.942990
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) −10.5653 −0.575531 −0.287765 0.957701i \(-0.592912\pi\)
−0.287765 + 0.957701i \(0.592912\pi\)
\(338\) 0 0
\(339\) −1.65194 −0.0897211
\(340\) 0 0
\(341\) 24.3013 1.31599
\(342\) 0 0
\(343\) −1.00000 −0.0539949
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −31.8760 −1.71119 −0.855597 0.517643i \(-0.826810\pi\)
−0.855597 + 0.517643i \(0.826810\pi\)
\(348\) 0 0
\(349\) −9.62602 −0.515269 −0.257635 0.966242i \(-0.582943\pi\)
−0.257635 + 0.966242i \(0.582943\pi\)
\(350\) 0 0
\(351\) −9.90531 −0.528706
\(352\) 0 0
\(353\) −26.2534 −1.39733 −0.698663 0.715451i \(-0.746221\pi\)
−0.698663 + 0.715451i \(0.746221\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 1.40535 0.0743791
\(358\) 0 0
\(359\) 4.56534 0.240950 0.120475 0.992716i \(-0.461558\pi\)
0.120475 + 0.992716i \(0.461558\pi\)
\(360\) 0 0
\(361\) −18.3947 −0.968142
\(362\) 0 0
\(363\) −5.60737 −0.294310
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) −3.79073 −0.197874 −0.0989371 0.995094i \(-0.531544\pi\)
−0.0989371 + 0.995094i \(0.531544\pi\)
\(368\) 0 0
\(369\) 2.89475 0.150695
\(370\) 0 0
\(371\) −7.55602 −0.392289
\(372\) 0 0
\(373\) 20.7453 1.07415 0.537076 0.843534i \(-0.319529\pi\)
0.537076 + 0.843534i \(0.319529\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −43.7839 −2.25499
\(378\) 0 0
\(379\) −3.00933 −0.154579 −0.0772894 0.997009i \(-0.524627\pi\)
−0.0772894 + 0.997009i \(0.524627\pi\)
\(380\) 0 0
\(381\) 5.85735 0.300081
\(382\) 0 0
\(383\) −7.00933 −0.358160 −0.179080 0.983835i \(-0.557312\pi\)
−0.179080 + 0.983835i \(0.557312\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) −20.1027 −1.02188
\(388\) 0 0
\(389\) −19.7653 −1.00214 −0.501070 0.865407i \(-0.667060\pi\)
−0.501070 + 0.865407i \(0.667060\pi\)
\(390\) 0 0
\(391\) −19.3760 −0.979889
\(392\) 0 0
\(393\) 0.0186574 0.000941142 0
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) −9.37266 −0.470400 −0.235200 0.971947i \(-0.575575\pi\)
−0.235200 + 0.971947i \(0.575575\pi\)
\(398\) 0 0
\(399\) 0.282672 0.0141513
\(400\) 0 0
\(401\) −16.1507 −0.806526 −0.403263 0.915084i \(-0.632124\pi\)
−0.403263 + 0.915084i \(0.632124\pi\)
\(402\) 0 0
\(403\) 21.9600 1.09391
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) −30.8480 −1.52908
\(408\) 0 0
\(409\) 15.2920 0.756141 0.378070 0.925777i \(-0.376588\pi\)
0.378070 + 0.925777i \(0.376588\pi\)
\(410\) 0 0
\(411\) 5.09337 0.251238
\(412\) 0 0
\(413\) 12.5140 0.615773
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 4.58400 0.224480
\(418\) 0 0
\(419\) 33.1820 1.62105 0.810524 0.585705i \(-0.199182\pi\)
0.810524 + 0.585705i \(0.199182\pi\)
\(420\) 0 0
\(421\) 27.8094 1.35535 0.677673 0.735363i \(-0.262988\pi\)
0.677673 + 0.735363i \(0.262988\pi\)
\(422\) 0 0
\(423\) −32.7640 −1.59304
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) −11.5047 −0.556750
\(428\) 0 0
\(429\) −8.67869 −0.419011
\(430\) 0 0
\(431\) 21.5454 1.03780 0.518902 0.854834i \(-0.326341\pi\)
0.518902 + 0.854834i \(0.326341\pi\)
\(432\) 0 0
\(433\) −36.0187 −1.73095 −0.865473 0.500955i \(-0.832982\pi\)
−0.865473 + 0.500955i \(0.832982\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −3.89730 −0.186433
\(438\) 0 0
\(439\) 21.1893 1.01131 0.505655 0.862736i \(-0.331251\pi\)
0.505655 + 0.862736i \(0.331251\pi\)
\(440\) 0 0
\(441\) −2.86799 −0.136571
\(442\) 0 0
\(443\) 11.1120 0.527949 0.263974 0.964530i \(-0.414967\pi\)
0.263974 + 0.964530i \(0.414967\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 4.16131 0.196823
\(448\) 0 0
\(449\) 0.961367 0.0453697 0.0226848 0.999743i \(-0.492779\pi\)
0.0226848 + 0.999743i \(0.492779\pi\)
\(450\) 0 0
\(451\) 5.18930 0.244355
\(452\) 0 0
\(453\) 2.13201 0.100170
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 28.2241 1.32027 0.660133 0.751149i \(-0.270500\pi\)
0.660133 + 0.751149i \(0.270500\pi\)
\(458\) 0 0
\(459\) 8.24659 0.384918
\(460\) 0 0
\(461\) 26.0887 1.21507 0.607535 0.794293i \(-0.292158\pi\)
0.607535 + 0.794293i \(0.292158\pi\)
\(462\) 0 0
\(463\) 2.90663 0.135082 0.0675412 0.997716i \(-0.478485\pi\)
0.0675412 + 0.997716i \(0.478485\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −12.2020 −0.564642 −0.282321 0.959320i \(-0.591104\pi\)
−0.282321 + 0.959320i \(0.591104\pi\)
\(468\) 0 0
\(469\) −11.7360 −0.541917
\(470\) 0 0
\(471\) 2.10270 0.0968874
\(472\) 0 0
\(473\) −36.0373 −1.65700
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) −21.6706 −0.992228
\(478\) 0 0
\(479\) 17.6519 0.806538 0.403269 0.915082i \(-0.367874\pi\)
0.403269 + 0.915082i \(0.367874\pi\)
\(480\) 0 0
\(481\) −27.8760 −1.27104
\(482\) 0 0
\(483\) −1.82003 −0.0828143
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) 1.57467 0.0713553 0.0356776 0.999363i \(-0.488641\pi\)
0.0356776 + 0.999363i \(0.488641\pi\)
\(488\) 0 0
\(489\) 1.18930 0.0537819
\(490\) 0 0
\(491\) 3.26270 0.147243 0.0736217 0.997286i \(-0.476544\pi\)
0.0736217 + 0.997286i \(0.476544\pi\)
\(492\) 0 0
\(493\) 36.4520 1.64172
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 2.72666 0.122307
\(498\) 0 0
\(499\) −42.5293 −1.90387 −0.951936 0.306298i \(-0.900910\pi\)
−0.951936 + 0.306298i \(0.900910\pi\)
\(500\) 0 0
\(501\) −9.03863 −0.403816
\(502\) 0 0
\(503\) 18.6974 0.833674 0.416837 0.908981i \(-0.363139\pi\)
0.416837 + 0.908981i \(0.363139\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) −3.11929 −0.138533
\(508\) 0 0
\(509\) 20.2313 0.896738 0.448369 0.893849i \(-0.352005\pi\)
0.448369 + 0.893849i \(0.352005\pi\)
\(510\) 0 0
\(511\) −5.00933 −0.221600
\(512\) 0 0
\(513\) 1.65872 0.0732342
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) −58.7347 −2.58315
\(518\) 0 0
\(519\) −2.40790 −0.105695
\(520\) 0 0
\(521\) 21.8387 0.956770 0.478385 0.878150i \(-0.341222\pi\)
0.478385 + 0.878150i \(0.341222\pi\)
\(522\) 0 0
\(523\) 20.4554 0.894451 0.447226 0.894421i \(-0.352412\pi\)
0.447226 + 0.894421i \(0.352412\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −18.2827 −0.796406
\(528\) 0 0
\(529\) 2.09337 0.0910163
\(530\) 0 0
\(531\) 35.8900 1.55750
\(532\) 0 0
\(533\) 4.68934 0.203118
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) −3.64006 −0.157080
\(538\) 0 0
\(539\) −5.14134 −0.221453
\(540\) 0 0
\(541\) 27.3400 1.17544 0.587718 0.809066i \(-0.300026\pi\)
0.587718 + 0.809066i \(0.300026\pi\)
\(542\) 0 0
\(543\) −0.649390 −0.0278680
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 22.6867 0.970013 0.485007 0.874510i \(-0.338817\pi\)
0.485007 + 0.874510i \(0.338817\pi\)
\(548\) 0 0
\(549\) −32.9953 −1.40820
\(550\) 0 0
\(551\) 7.33195 0.312352
\(552\) 0 0
\(553\) 5.68802 0.241879
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −12.7453 −0.540036 −0.270018 0.962855i \(-0.587030\pi\)
−0.270018 + 0.962855i \(0.587030\pi\)
\(558\) 0 0
\(559\) −32.5653 −1.37737
\(560\) 0 0
\(561\) 7.22538 0.305056
\(562\) 0 0
\(563\) 5.20333 0.219294 0.109647 0.993971i \(-0.465028\pi\)
0.109647 + 0.993971i \(0.465028\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) −7.82936 −0.328802
\(568\) 0 0
\(569\) −7.37605 −0.309220 −0.154610 0.987976i \(-0.549412\pi\)
−0.154610 + 0.987976i \(0.549412\pi\)
\(570\) 0 0
\(571\) −15.8973 −0.665281 −0.332641 0.943054i \(-0.607940\pi\)
−0.332641 + 0.943054i \(0.607940\pi\)
\(572\) 0 0
\(573\) −3.12268 −0.130452
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) −19.9707 −0.831391 −0.415695 0.909504i \(-0.636462\pi\)
−0.415695 + 0.909504i \(0.636462\pi\)
\(578\) 0 0
\(579\) 1.87864 0.0780736
\(580\) 0 0
\(581\) −4.67531 −0.193964
\(582\) 0 0
\(583\) −38.8480 −1.60892
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 13.8060 0.569834 0.284917 0.958552i \(-0.408034\pi\)
0.284917 + 0.958552i \(0.408034\pi\)
\(588\) 0 0
\(589\) −3.67738 −0.151524
\(590\) 0 0
\(591\) −8.68934 −0.357432
\(592\) 0 0
\(593\) 20.0666 0.824037 0.412019 0.911175i \(-0.364824\pi\)
0.412019 + 0.911175i \(0.364824\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) −5.56279 −0.227670
\(598\) 0 0
\(599\) −46.0267 −1.88060 −0.940299 0.340349i \(-0.889455\pi\)
−0.940299 + 0.340349i \(0.889455\pi\)
\(600\) 0 0
\(601\) −1.37605 −0.0561301 −0.0280651 0.999606i \(-0.508935\pi\)
−0.0280651 + 0.999606i \(0.508935\pi\)
\(602\) 0 0
\(603\) −33.6587 −1.37069
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) −18.7933 −0.762796 −0.381398 0.924411i \(-0.624557\pi\)
−0.381398 + 0.924411i \(0.624557\pi\)
\(608\) 0 0
\(609\) 3.42401 0.138748
\(610\) 0 0
\(611\) −53.0759 −2.14722
\(612\) 0 0
\(613\) 24.8480 1.00360 0.501801 0.864983i \(-0.332671\pi\)
0.501801 + 0.864983i \(0.332671\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 43.1680 1.73788 0.868939 0.494919i \(-0.164802\pi\)
0.868939 + 0.494919i \(0.164802\pi\)
\(618\) 0 0
\(619\) −31.9486 −1.28412 −0.642062 0.766652i \(-0.721921\pi\)
−0.642062 + 0.766652i \(0.721921\pi\)
\(620\) 0 0
\(621\) −10.6799 −0.428571
\(622\) 0 0
\(623\) 2.82936 0.113356
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) 1.45331 0.0580397
\(628\) 0 0
\(629\) 23.2080 0.925362
\(630\) 0 0
\(631\) 8.59465 0.342148 0.171074 0.985258i \(-0.445276\pi\)
0.171074 + 0.985258i \(0.445276\pi\)
\(632\) 0 0
\(633\) −0.377365 −0.0149989
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) −4.64600 −0.184081
\(638\) 0 0
\(639\) 7.82003 0.309356
\(640\) 0 0
\(641\) 33.2266 1.31237 0.656186 0.754599i \(-0.272169\pi\)
0.656186 + 0.754599i \(0.272169\pi\)
\(642\) 0 0
\(643\) −17.4940 −0.689897 −0.344948 0.938622i \(-0.612104\pi\)
−0.344948 + 0.938622i \(0.612104\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −9.91595 −0.389836 −0.194918 0.980820i \(-0.562444\pi\)
−0.194918 + 0.980820i \(0.562444\pi\)
\(648\) 0 0
\(649\) 64.3386 2.52551
\(650\) 0 0
\(651\) −1.71733 −0.0673074
\(652\) 0 0
\(653\) 14.6240 0.572280 0.286140 0.958188i \(-0.407628\pi\)
0.286140 + 0.958188i \(0.407628\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) −14.3667 −0.560499
\(658\) 0 0
\(659\) −20.2534 −0.788959 −0.394480 0.918905i \(-0.629075\pi\)
−0.394480 + 0.918905i \(0.629075\pi\)
\(660\) 0 0
\(661\) −12.4299 −0.483469 −0.241734 0.970342i \(-0.577716\pi\)
−0.241734 + 0.970342i \(0.577716\pi\)
\(662\) 0 0
\(663\) 6.52926 0.253575
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) −47.2080 −1.82790
\(668\) 0 0
\(669\) −3.58532 −0.138616
\(670\) 0 0
\(671\) −59.1493 −2.28344
\(672\) 0 0
\(673\) −47.6774 −1.83783 −0.918914 0.394458i \(-0.870932\pi\)
−0.918914 + 0.394458i \(0.870932\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 6.00339 0.230729 0.115364 0.993323i \(-0.463196\pi\)
0.115364 + 0.993323i \(0.463196\pi\)
\(678\) 0 0
\(679\) 1.58532 0.0608390
\(680\) 0 0
\(681\) −7.86799 −0.301502
\(682\) 0 0
\(683\) 28.5653 1.09302 0.546511 0.837452i \(-0.315956\pi\)
0.546511 + 0.837452i \(0.315956\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) −7.35061 −0.280443
\(688\) 0 0
\(689\) −35.1053 −1.33740
\(690\) 0 0
\(691\) −47.8247 −1.81934 −0.909668 0.415337i \(-0.863664\pi\)
−0.909668 + 0.415337i \(0.863664\pi\)
\(692\) 0 0
\(693\) −14.7453 −0.560128
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) −3.90408 −0.147877
\(698\) 0 0
\(699\) 1.98134 0.0749413
\(700\) 0 0
\(701\) −4.80005 −0.181296 −0.0906478 0.995883i \(-0.528894\pi\)
−0.0906478 + 0.995883i \(0.528894\pi\)
\(702\) 0 0
\(703\) 4.66805 0.176059
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 9.06068 0.340762
\(708\) 0 0
\(709\) 11.7653 0.441855 0.220927 0.975290i \(-0.429092\pi\)
0.220927 + 0.975290i \(0.429092\pi\)
\(710\) 0 0
\(711\) 16.3132 0.611793
\(712\) 0 0
\(713\) 23.6774 0.886725
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) −4.20927 −0.157198
\(718\) 0 0
\(719\) 40.7640 1.52024 0.760120 0.649783i \(-0.225140\pi\)
0.760120 + 0.649783i \(0.225140\pi\)
\(720\) 0 0
\(721\) 5.14134 0.191473
\(722\) 0 0
\(723\) 1.82003 0.0676877
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) −22.1214 −0.820436 −0.410218 0.911988i \(-0.634547\pi\)
−0.410218 + 0.911988i \(0.634547\pi\)
\(728\) 0 0
\(729\) −20.1307 −0.745581
\(730\) 0 0
\(731\) 27.1120 1.00277
\(732\) 0 0
\(733\) 13.0500 0.482014 0.241007 0.970523i \(-0.422522\pi\)
0.241007 + 0.970523i \(0.422522\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −60.3386 −2.22260
\(738\) 0 0
\(739\) 34.5734 1.27180 0.635901 0.771771i \(-0.280629\pi\)
0.635901 + 0.771771i \(0.280629\pi\)
\(740\) 0 0
\(741\) 1.31330 0.0482451
\(742\) 0 0
\(743\) 3.55602 0.130458 0.0652288 0.997870i \(-0.479222\pi\)
0.0652288 + 0.997870i \(0.479222\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) −13.4087 −0.490600
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) 23.6226 0.862002 0.431001 0.902351i \(-0.358161\pi\)
0.431001 + 0.902351i \(0.358161\pi\)
\(752\) 0 0
\(753\) −8.50937 −0.310099
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 15.1893 0.552064 0.276032 0.961148i \(-0.410980\pi\)
0.276032 + 0.961148i \(0.410980\pi\)
\(758\) 0 0
\(759\) −9.35739 −0.339652
\(760\) 0 0
\(761\) 34.4626 1.24927 0.624635 0.780917i \(-0.285248\pi\)
0.624635 + 0.780917i \(0.285248\pi\)
\(762\) 0 0
\(763\) 3.97070 0.143749
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 58.1400 2.09931
\(768\) 0 0
\(769\) 40.3854 1.45633 0.728167 0.685400i \(-0.240373\pi\)
0.728167 + 0.685400i \(0.240373\pi\)
\(770\) 0 0
\(771\) −4.72666 −0.170226
\(772\) 0 0
\(773\) −10.7674 −0.387275 −0.193638 0.981073i \(-0.562029\pi\)
−0.193638 + 0.981073i \(0.562029\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 2.17997 0.0782060
\(778\) 0 0
\(779\) −0.785266 −0.0281351
\(780\) 0 0
\(781\) 14.0187 0.501627
\(782\) 0 0
\(783\) 20.0921 0.718031
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) 9.19269 0.327684 0.163842 0.986487i \(-0.447611\pi\)
0.163842 + 0.986487i \(0.447611\pi\)
\(788\) 0 0
\(789\) 2.18675 0.0778503
\(790\) 0 0
\(791\) −4.54669 −0.161662
\(792\) 0 0
\(793\) −53.4507 −1.89809
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 21.1086 0.747706 0.373853 0.927488i \(-0.378036\pi\)
0.373853 + 0.927488i \(0.378036\pi\)
\(798\) 0 0
\(799\) 44.1880 1.56326
\(800\) 0 0
\(801\) 8.11458 0.286715
\(802\) 0 0
\(803\) −25.7546 −0.908862
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) −1.17742 −0.0414471
\(808\) 0 0
\(809\) −27.8280 −0.978382 −0.489191 0.872177i \(-0.662708\pi\)
−0.489191 + 0.872177i \(0.662708\pi\)
\(810\) 0 0
\(811\) 11.0607 0.388393 0.194197 0.980963i \(-0.437790\pi\)
0.194197 + 0.980963i \(0.437790\pi\)
\(812\) 0 0
\(813\) −10.5840 −0.371197
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 5.45331 0.190787
\(818\) 0 0
\(819\) −13.3247 −0.465603
\(820\) 0 0
\(821\) −17.8867 −0.624248 −0.312124 0.950041i \(-0.601041\pi\)
−0.312124 + 0.950041i \(0.601041\pi\)
\(822\) 0 0
\(823\) −52.8667 −1.84282 −0.921408 0.388596i \(-0.872960\pi\)
−0.921408 + 0.388596i \(0.872960\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −36.3013 −1.26232 −0.631160 0.775652i \(-0.717421\pi\)
−0.631160 + 0.775652i \(0.717421\pi\)
\(828\) 0 0
\(829\) −0.759350 −0.0263733 −0.0131867 0.999913i \(-0.504198\pi\)
−0.0131867 + 0.999913i \(0.504198\pi\)
\(830\) 0 0
\(831\) 1.61462 0.0560107
\(832\) 0 0
\(833\) 3.86799 0.134018
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) −10.0773 −0.348321
\(838\) 0 0
\(839\) 43.3107 1.49525 0.747625 0.664121i \(-0.231194\pi\)
0.747625 + 0.664121i \(0.231194\pi\)
\(840\) 0 0
\(841\) 59.8119 2.06248
\(842\) 0 0
\(843\) 8.97325 0.309055
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) −15.4333 −0.530296
\(848\) 0 0
\(849\) 2.81193 0.0965053
\(850\) 0 0
\(851\) −30.0560 −1.03031
\(852\) 0 0
\(853\) −40.3713 −1.38229 −0.691144 0.722717i \(-0.742893\pi\)
−0.691144 + 0.722717i \(0.742893\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 13.4720 0.460194 0.230097 0.973168i \(-0.426096\pi\)
0.230097 + 0.973168i \(0.426096\pi\)
\(858\) 0 0
\(859\) 34.7313 1.18502 0.592508 0.805565i \(-0.298138\pi\)
0.592508 + 0.805565i \(0.298138\pi\)
\(860\) 0 0
\(861\) −0.366718 −0.0124977
\(862\) 0 0
\(863\) 14.2454 0.484918 0.242459 0.970162i \(-0.422046\pi\)
0.242459 + 0.970162i \(0.422046\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) 0.740693 0.0251553
\(868\) 0 0
\(869\) 29.2440 0.992036
\(870\) 0 0
\(871\) −54.5254 −1.84752
\(872\) 0 0
\(873\) 4.54669 0.153882
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 19.7546 0.667067 0.333533 0.942738i \(-0.391759\pi\)
0.333533 + 0.942738i \(0.391759\pi\)
\(878\) 0 0
\(879\) −9.12268 −0.307700
\(880\) 0 0
\(881\) −9.53058 −0.321093 −0.160547 0.987028i \(-0.551326\pi\)
−0.160547 + 0.987028i \(0.551326\pi\)
\(882\) 0 0
\(883\) −51.1867 −1.72257 −0.861284 0.508124i \(-0.830339\pi\)
−0.861284 + 0.508124i \(0.830339\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) −24.4626 −0.821375 −0.410688 0.911776i \(-0.634711\pi\)
−0.410688 + 0.911776i \(0.634711\pi\)
\(888\) 0 0
\(889\) 16.1214 0.540693
\(890\) 0 0
\(891\) −40.2534 −1.34854
\(892\) 0 0
\(893\) 8.88797 0.297425
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) −8.45586 −0.282333
\(898\) 0 0
\(899\) −44.5441 −1.48563
\(900\) 0 0
\(901\) 29.2266 0.973680
\(902\) 0 0
\(903\) 2.54669 0.0847484
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) 8.09592 0.268821 0.134410 0.990926i \(-0.457086\pi\)
0.134410 + 0.990926i \(0.457086\pi\)
\(908\) 0 0
\(909\) 25.9860 0.861900
\(910\) 0 0
\(911\) 59.5093 1.97163 0.985815 0.167834i \(-0.0536772\pi\)
0.985815 + 0.167834i \(0.0536772\pi\)
\(912\) 0 0
\(913\) −24.0373 −0.795519
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 0.0513514 0.00169577
\(918\) 0 0
\(919\) −51.7067 −1.70565 −0.852823 0.522200i \(-0.825111\pi\)
−0.852823 + 0.522200i \(0.825111\pi\)
\(920\) 0 0
\(921\) −12.3239 −0.406085
\(922\) 0 0
\(923\) 12.6680 0.416974
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) 14.7453 0.484300
\(928\) 0 0
\(929\) −3.76142 −0.123408 −0.0617041 0.998094i \(-0.519654\pi\)
−0.0617041 + 0.998094i \(0.519654\pi\)
\(930\) 0 0
\(931\) 0.778008 0.0254982
\(932\) 0 0
\(933\) 0.205406 0.00672468
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 21.1014 0.689352 0.344676 0.938722i \(-0.387989\pi\)
0.344676 + 0.938722i \(0.387989\pi\)
\(938\) 0 0
\(939\) 9.93338 0.324164
\(940\) 0 0
\(941\) −1.72873 −0.0563549 −0.0281775 0.999603i \(-0.508970\pi\)
−0.0281775 + 0.999603i \(0.508970\pi\)
\(942\) 0 0
\(943\) 5.05606 0.164648
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 24.7267 0.803508 0.401754 0.915748i \(-0.368401\pi\)
0.401754 + 0.915748i \(0.368401\pi\)
\(948\) 0 0
\(949\) −23.2733 −0.755485
\(950\) 0 0
\(951\) −8.25724 −0.267759
\(952\) 0 0
\(953\) 1.18930 0.0385251 0.0192626 0.999814i \(-0.493868\pi\)
0.0192626 + 0.999814i \(0.493868\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 17.6040 0.569056
\(958\) 0 0
\(959\) 14.0187 0.452686
\(960\) 0 0
\(961\) −8.65872 −0.279314
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) 23.3293 0.750220 0.375110 0.926980i \(-0.377605\pi\)
0.375110 + 0.926980i \(0.377605\pi\)
\(968\) 0 0
\(969\) −1.09337 −0.0351242
\(970\) 0 0
\(971\) 4.61670 0.148157 0.0740784 0.997252i \(-0.476398\pi\)
0.0740784 + 0.997252i \(0.476398\pi\)
\(972\) 0 0
\(973\) 12.6167 0.404473
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −19.1120 −0.611448 −0.305724 0.952120i \(-0.598898\pi\)
−0.305724 + 0.952120i \(0.598898\pi\)
\(978\) 0 0
\(979\) 14.5467 0.464914
\(980\) 0 0
\(981\) 11.3879 0.363588
\(982\) 0 0
\(983\) 52.6506 1.67929 0.839647 0.543132i \(-0.182762\pi\)
0.839647 + 0.543132i \(0.182762\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 4.15066 0.132117
\(988\) 0 0
\(989\) −35.1120 −1.11650
\(990\) 0 0
\(991\) −52.4227 −1.66526 −0.832631 0.553828i \(-0.813166\pi\)
−0.832631 + 0.553828i \(0.813166\pi\)
\(992\) 0 0
\(993\) 0.168091 0.00533420
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) 19.7580 0.625743 0.312872 0.949795i \(-0.398709\pi\)
0.312872 + 0.949795i \(0.398709\pi\)
\(998\) 0 0
\(999\) 12.7920 0.404722
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 2800.2.a.br.1.2 3
4.3 odd 2 1400.2.a.s.1.2 3
5.2 odd 4 560.2.g.f.449.4 6
5.3 odd 4 560.2.g.f.449.3 6
5.4 even 2 2800.2.a.bq.1.2 3
15.2 even 4 5040.2.t.y.1009.5 6
15.8 even 4 5040.2.t.y.1009.6 6
20.3 even 4 280.2.g.b.169.4 yes 6
20.7 even 4 280.2.g.b.169.3 6
20.19 odd 2 1400.2.a.t.1.2 3
28.27 even 2 9800.2.a.cg.1.2 3
40.3 even 4 2240.2.g.l.449.3 6
40.13 odd 4 2240.2.g.m.449.4 6
40.27 even 4 2240.2.g.l.449.4 6
40.37 odd 4 2240.2.g.m.449.3 6
60.23 odd 4 2520.2.t.g.1009.6 6
60.47 odd 4 2520.2.t.g.1009.5 6
140.27 odd 4 1960.2.g.c.1569.4 6
140.83 odd 4 1960.2.g.c.1569.3 6
140.139 even 2 9800.2.a.cd.1.2 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
280.2.g.b.169.3 6 20.7 even 4
280.2.g.b.169.4 yes 6 20.3 even 4
560.2.g.f.449.3 6 5.3 odd 4
560.2.g.f.449.4 6 5.2 odd 4
1400.2.a.s.1.2 3 4.3 odd 2
1400.2.a.t.1.2 3 20.19 odd 2
1960.2.g.c.1569.3 6 140.83 odd 4
1960.2.g.c.1569.4 6 140.27 odd 4
2240.2.g.l.449.3 6 40.3 even 4
2240.2.g.l.449.4 6 40.27 even 4
2240.2.g.m.449.3 6 40.37 odd 4
2240.2.g.m.449.4 6 40.13 odd 4
2520.2.t.g.1009.5 6 60.47 odd 4
2520.2.t.g.1009.6 6 60.23 odd 4
2800.2.a.bq.1.2 3 5.4 even 2
2800.2.a.br.1.2 3 1.1 even 1 trivial
5040.2.t.y.1009.5 6 15.2 even 4
5040.2.t.y.1009.6 6 15.8 even 4
9800.2.a.cd.1.2 3 140.139 even 2
9800.2.a.cg.1.2 3 28.27 even 2