Properties

Label 1904.2.a.m.1.3
Level $1904$
Weight $2$
Character 1904.1
Self dual yes
Analytic conductor $15.204$
Analytic rank $1$
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] = [1904,2,Mod(1,1904)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1904, 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("1904.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1904 = 2^{4} \cdot 7 \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1904.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: \(15.2035165449\)
Analytic rank: \(1\)
Dimension: \(3\)
Coefficient field: 3.3.229.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - 4x - 1 \) 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 952)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(2.11491\) of defining polynomial
Character \(\chi\) \(=\) 1904.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.47283 q^{3} -1.11491 q^{5} -1.00000 q^{7} -0.830760 q^{9} +O(q^{10})\) \(q+1.47283 q^{3} -1.11491 q^{5} -1.00000 q^{7} -0.830760 q^{9} +2.22982 q^{11} -1.64207 q^{15} -1.00000 q^{17} -4.94567 q^{19} -1.47283 q^{21} -9.17548 q^{23} -3.75698 q^{25} -5.64207 q^{27} +0.945668 q^{29} +1.11491 q^{31} +3.28415 q^{33} +1.11491 q^{35} -4.94567 q^{37} +10.2709 q^{41} +11.2166 q^{43} +0.926221 q^{45} -9.89134 q^{47} +1.00000 q^{49} -1.47283 q^{51} +7.83076 q^{53} -2.48604 q^{55} -7.28415 q^{57} -13.7438 q^{59} +0.904539 q^{61} +0.830760 q^{63} -9.91302 q^{67} -13.5140 q^{69} -14.6894 q^{71} +0.756981 q^{73} -5.53341 q^{75} -2.22982 q^{77} -8.00000 q^{79} -5.81756 q^{81} +11.9736 q^{83} +1.11491 q^{85} +1.39281 q^{87} +11.1755 q^{89} +1.64207 q^{93} +5.51396 q^{95} +10.0606 q^{97} -1.85244 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - q^{3} + 3 q^{5} - 3 q^{7} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q - q^{3} + 3 q^{5} - 3 q^{7} + 2 q^{9} - 6 q^{11} - 4 q^{15} - 3 q^{17} - 4 q^{19} + q^{21} - 4 q^{23} - 4 q^{25} - 16 q^{27} - 8 q^{29} - 3 q^{31} + 8 q^{33} - 3 q^{35} - 4 q^{37} + 9 q^{41} + q^{43} - 8 q^{47} + 3 q^{49} + q^{51} + 19 q^{53} - 22 q^{55} - 20 q^{57} - 14 q^{59} + q^{61} - 2 q^{63} - 7 q^{67} - 26 q^{69} - 6 q^{71} - 5 q^{73} + 6 q^{75} + 6 q^{77} - 24 q^{79} + 7 q^{81} - 4 q^{83} - 3 q^{85} + 24 q^{87} + 10 q^{89} + 4 q^{93} + 2 q^{95} + 13 q^{97}+O(q^{100}) \) 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
\(3\) 1.47283 0.850341 0.425171 0.905113i \(-0.360214\pi\)
0.425171 + 0.905113i \(0.360214\pi\)
\(4\) 0 0
\(5\) −1.11491 −0.498602 −0.249301 0.968426i \(-0.580201\pi\)
−0.249301 + 0.968426i \(0.580201\pi\)
\(6\) 0 0
\(7\) −1.00000 −0.377964
\(8\) 0 0
\(9\) −0.830760 −0.276920
\(10\) 0 0
\(11\) 2.22982 0.672315 0.336157 0.941806i \(-0.390873\pi\)
0.336157 + 0.941806i \(0.390873\pi\)
\(12\) 0 0
\(13\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(14\) 0 0
\(15\) −1.64207 −0.423982
\(16\) 0 0
\(17\) −1.00000 −0.242536
\(18\) 0 0
\(19\) −4.94567 −1.13461 −0.567307 0.823506i \(-0.692015\pi\)
−0.567307 + 0.823506i \(0.692015\pi\)
\(20\) 0 0
\(21\) −1.47283 −0.321399
\(22\) 0 0
\(23\) −9.17548 −1.91322 −0.956610 0.291371i \(-0.905889\pi\)
−0.956610 + 0.291371i \(0.905889\pi\)
\(24\) 0 0
\(25\) −3.75698 −0.751396
\(26\) 0 0
\(27\) −5.64207 −1.08582
\(28\) 0 0
\(29\) 0.945668 0.175606 0.0878031 0.996138i \(-0.472015\pi\)
0.0878031 + 0.996138i \(0.472015\pi\)
\(30\) 0 0
\(31\) 1.11491 0.200243 0.100122 0.994975i \(-0.468077\pi\)
0.100122 + 0.994975i \(0.468077\pi\)
\(32\) 0 0
\(33\) 3.28415 0.571697
\(34\) 0 0
\(35\) 1.11491 0.188454
\(36\) 0 0
\(37\) −4.94567 −0.813063 −0.406531 0.913637i \(-0.633262\pi\)
−0.406531 + 0.913637i \(0.633262\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 10.2709 1.60405 0.802026 0.597289i \(-0.203756\pi\)
0.802026 + 0.597289i \(0.203756\pi\)
\(42\) 0 0
\(43\) 11.2166 1.71052 0.855259 0.518201i \(-0.173398\pi\)
0.855259 + 0.518201i \(0.173398\pi\)
\(44\) 0 0
\(45\) 0.926221 0.138073
\(46\) 0 0
\(47\) −9.89134 −1.44280 −0.721400 0.692519i \(-0.756501\pi\)
−0.721400 + 0.692519i \(0.756501\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) −1.47283 −0.206238
\(52\) 0 0
\(53\) 7.83076 1.07564 0.537819 0.843060i \(-0.319248\pi\)
0.537819 + 0.843060i \(0.319248\pi\)
\(54\) 0 0
\(55\) −2.48604 −0.335217
\(56\) 0 0
\(57\) −7.28415 −0.964809
\(58\) 0 0
\(59\) −13.7438 −1.78929 −0.894644 0.446780i \(-0.852571\pi\)
−0.894644 + 0.446780i \(0.852571\pi\)
\(60\) 0 0
\(61\) 0.904539 0.115814 0.0579072 0.998322i \(-0.481557\pi\)
0.0579072 + 0.998322i \(0.481557\pi\)
\(62\) 0 0
\(63\) 0.830760 0.104666
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) −9.91302 −1.21107 −0.605534 0.795820i \(-0.707040\pi\)
−0.605534 + 0.795820i \(0.707040\pi\)
\(68\) 0 0
\(69\) −13.5140 −1.62689
\(70\) 0 0
\(71\) −14.6894 −1.74332 −0.871658 0.490114i \(-0.836955\pi\)
−0.871658 + 0.490114i \(0.836955\pi\)
\(72\) 0 0
\(73\) 0.756981 0.0885979 0.0442990 0.999018i \(-0.485895\pi\)
0.0442990 + 0.999018i \(0.485895\pi\)
\(74\) 0 0
\(75\) −5.53341 −0.638943
\(76\) 0 0
\(77\) −2.22982 −0.254111
\(78\) 0 0
\(79\) −8.00000 −0.900070 −0.450035 0.893011i \(-0.648589\pi\)
−0.450035 + 0.893011i \(0.648589\pi\)
\(80\) 0 0
\(81\) −5.81756 −0.646395
\(82\) 0 0
\(83\) 11.9736 1.31427 0.657136 0.753772i \(-0.271768\pi\)
0.657136 + 0.753772i \(0.271768\pi\)
\(84\) 0 0
\(85\) 1.11491 0.120929
\(86\) 0 0
\(87\) 1.39281 0.149325
\(88\) 0 0
\(89\) 11.1755 1.18460 0.592299 0.805718i \(-0.298220\pi\)
0.592299 + 0.805718i \(0.298220\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 1.64207 0.170275
\(94\) 0 0
\(95\) 5.51396 0.565721
\(96\) 0 0
\(97\) 10.0606 1.02150 0.510748 0.859730i \(-0.329368\pi\)
0.510748 + 0.859730i \(0.329368\pi\)
\(98\) 0 0
\(99\) −1.85244 −0.186177
\(100\) 0 0
\(101\) −18.6894 −1.85967 −0.929835 0.367978i \(-0.880050\pi\)
−0.929835 + 0.367978i \(0.880050\pi\)
\(102\) 0 0
\(103\) −12.1212 −1.19433 −0.597166 0.802118i \(-0.703707\pi\)
−0.597166 + 0.802118i \(0.703707\pi\)
\(104\) 0 0
\(105\) 1.64207 0.160250
\(106\) 0 0
\(107\) 1.89134 0.182842 0.0914212 0.995812i \(-0.470859\pi\)
0.0914212 + 0.995812i \(0.470859\pi\)
\(108\) 0 0
\(109\) −14.0000 −1.34096 −0.670478 0.741929i \(-0.733911\pi\)
−0.670478 + 0.741929i \(0.733911\pi\)
\(110\) 0 0
\(111\) −7.28415 −0.691381
\(112\) 0 0
\(113\) −3.85244 −0.362407 −0.181204 0.983446i \(-0.557999\pi\)
−0.181204 + 0.983446i \(0.557999\pi\)
\(114\) 0 0
\(115\) 10.2298 0.953935
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 1.00000 0.0916698
\(120\) 0 0
\(121\) −6.02792 −0.547993
\(122\) 0 0
\(123\) 15.1274 1.36399
\(124\) 0 0
\(125\) 9.76322 0.873249
\(126\) 0 0
\(127\) 21.2361 1.88440 0.942198 0.335057i \(-0.108756\pi\)
0.942198 + 0.335057i \(0.108756\pi\)
\(128\) 0 0
\(129\) 16.5202 1.45452
\(130\) 0 0
\(131\) 4.60719 0.402532 0.201266 0.979537i \(-0.435494\pi\)
0.201266 + 0.979537i \(0.435494\pi\)
\(132\) 0 0
\(133\) 4.94567 0.428844
\(134\) 0 0
\(135\) 6.29039 0.541391
\(136\) 0 0
\(137\) 0.379608 0.0324321 0.0162160 0.999869i \(-0.494838\pi\)
0.0162160 + 0.999869i \(0.494838\pi\)
\(138\) 0 0
\(139\) −6.06058 −0.514051 −0.257026 0.966405i \(-0.582742\pi\)
−0.257026 + 0.966405i \(0.582742\pi\)
\(140\) 0 0
\(141\) −14.5683 −1.22687
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) −1.05433 −0.0875575
\(146\) 0 0
\(147\) 1.47283 0.121477
\(148\) 0 0
\(149\) −10.1887 −0.834690 −0.417345 0.908748i \(-0.637039\pi\)
−0.417345 + 0.908748i \(0.637039\pi\)
\(150\) 0 0
\(151\) 8.18869 0.666386 0.333193 0.942859i \(-0.391874\pi\)
0.333193 + 0.942859i \(0.391874\pi\)
\(152\) 0 0
\(153\) 0.830760 0.0671630
\(154\) 0 0
\(155\) −1.24302 −0.0998417
\(156\) 0 0
\(157\) −14.2687 −1.13877 −0.569383 0.822072i \(-0.692818\pi\)
−0.569383 + 0.822072i \(0.692818\pi\)
\(158\) 0 0
\(159\) 11.5334 0.914659
\(160\) 0 0
\(161\) 9.17548 0.723129
\(162\) 0 0
\(163\) −15.7438 −1.23315 −0.616574 0.787297i \(-0.711480\pi\)
−0.616574 + 0.787297i \(0.711480\pi\)
\(164\) 0 0
\(165\) −3.66152 −0.285049
\(166\) 0 0
\(167\) 0.648317 0.0501683 0.0250841 0.999685i \(-0.492015\pi\)
0.0250841 + 0.999685i \(0.492015\pi\)
\(168\) 0 0
\(169\) −13.0000 −1.00000
\(170\) 0 0
\(171\) 4.10866 0.314197
\(172\) 0 0
\(173\) 15.9325 1.21132 0.605661 0.795722i \(-0.292909\pi\)
0.605661 + 0.795722i \(0.292909\pi\)
\(174\) 0 0
\(175\) 3.75698 0.284001
\(176\) 0 0
\(177\) −20.2423 −1.52150
\(178\) 0 0
\(179\) 17.0257 1.27256 0.636280 0.771458i \(-0.280472\pi\)
0.636280 + 0.771458i \(0.280472\pi\)
\(180\) 0 0
\(181\) 5.63511 0.418855 0.209427 0.977824i \(-0.432840\pi\)
0.209427 + 0.977824i \(0.432840\pi\)
\(182\) 0 0
\(183\) 1.33224 0.0984817
\(184\) 0 0
\(185\) 5.51396 0.405395
\(186\) 0 0
\(187\) −2.22982 −0.163060
\(188\) 0 0
\(189\) 5.64207 0.410400
\(190\) 0 0
\(191\) −2.88509 −0.208758 −0.104379 0.994538i \(-0.533285\pi\)
−0.104379 + 0.994538i \(0.533285\pi\)
\(192\) 0 0
\(193\) 17.5264 1.26158 0.630791 0.775953i \(-0.282731\pi\)
0.630791 + 0.775953i \(0.282731\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 14.9193 1.06295 0.531477 0.847073i \(-0.321637\pi\)
0.531477 + 0.847073i \(0.321637\pi\)
\(198\) 0 0
\(199\) 16.6483 1.18017 0.590084 0.807342i \(-0.299095\pi\)
0.590084 + 0.807342i \(0.299095\pi\)
\(200\) 0 0
\(201\) −14.6002 −1.02982
\(202\) 0 0
\(203\) −0.945668 −0.0663729
\(204\) 0 0
\(205\) −11.4512 −0.799783
\(206\) 0 0
\(207\) 7.62263 0.529809
\(208\) 0 0
\(209\) −11.0279 −0.762818
\(210\) 0 0
\(211\) −7.40530 −0.509802 −0.254901 0.966967i \(-0.582043\pi\)
−0.254901 + 0.966967i \(0.582043\pi\)
\(212\) 0 0
\(213\) −21.6351 −1.48241
\(214\) 0 0
\(215\) −12.5055 −0.852867
\(216\) 0 0
\(217\) −1.11491 −0.0756849
\(218\) 0 0
\(219\) 1.11491 0.0753385
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) −27.1491 −1.81804 −0.909018 0.416756i \(-0.863167\pi\)
−0.909018 + 0.416756i \(0.863167\pi\)
\(224\) 0 0
\(225\) 3.12115 0.208077
\(226\) 0 0
\(227\) 4.50772 0.299188 0.149594 0.988748i \(-0.452203\pi\)
0.149594 + 0.988748i \(0.452203\pi\)
\(228\) 0 0
\(229\) 10.2298 0.676005 0.338003 0.941145i \(-0.390249\pi\)
0.338003 + 0.941145i \(0.390249\pi\)
\(230\) 0 0
\(231\) −3.28415 −0.216081
\(232\) 0 0
\(233\) 11.5529 0.756853 0.378426 0.925631i \(-0.376465\pi\)
0.378426 + 0.925631i \(0.376465\pi\)
\(234\) 0 0
\(235\) 11.0279 0.719382
\(236\) 0 0
\(237\) −11.7827 −0.765367
\(238\) 0 0
\(239\) 30.7570 1.98950 0.994752 0.102317i \(-0.0326256\pi\)
0.994752 + 0.102317i \(0.0326256\pi\)
\(240\) 0 0
\(241\) −0.967349 −0.0623124 −0.0311562 0.999515i \(-0.509919\pi\)
−0.0311562 + 0.999515i \(0.509919\pi\)
\(242\) 0 0
\(243\) 8.35793 0.536161
\(244\) 0 0
\(245\) −1.11491 −0.0712288
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) 17.6351 1.11758
\(250\) 0 0
\(251\) −0.0264075 −0.00166683 −0.000833413 1.00000i \(-0.500265\pi\)
−0.000833413 1.00000i \(0.500265\pi\)
\(252\) 0 0
\(253\) −20.4596 −1.28629
\(254\) 0 0
\(255\) 1.64207 0.102831
\(256\) 0 0
\(257\) 14.7159 0.917950 0.458975 0.888449i \(-0.348217\pi\)
0.458975 + 0.888449i \(0.348217\pi\)
\(258\) 0 0
\(259\) 4.94567 0.307309
\(260\) 0 0
\(261\) −0.785623 −0.0486289
\(262\) 0 0
\(263\) −24.9193 −1.53659 −0.768294 0.640097i \(-0.778894\pi\)
−0.768294 + 0.640097i \(0.778894\pi\)
\(264\) 0 0
\(265\) −8.73057 −0.536315
\(266\) 0 0
\(267\) 16.4596 1.00731
\(268\) 0 0
\(269\) −23.9861 −1.46246 −0.731229 0.682133i \(-0.761053\pi\)
−0.731229 + 0.682133i \(0.761053\pi\)
\(270\) 0 0
\(271\) 15.0668 0.915244 0.457622 0.889147i \(-0.348701\pi\)
0.457622 + 0.889147i \(0.348701\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −8.37737 −0.505175
\(276\) 0 0
\(277\) 0.350966 0.0210875 0.0105437 0.999944i \(-0.496644\pi\)
0.0105437 + 0.999944i \(0.496644\pi\)
\(278\) 0 0
\(279\) −0.926221 −0.0554514
\(280\) 0 0
\(281\) 5.30583 0.316519 0.158260 0.987398i \(-0.449412\pi\)
0.158260 + 0.987398i \(0.449412\pi\)
\(282\) 0 0
\(283\) −11.6134 −0.690347 −0.345173 0.938539i \(-0.612180\pi\)
−0.345173 + 0.938539i \(0.612180\pi\)
\(284\) 0 0
\(285\) 8.12115 0.481055
\(286\) 0 0
\(287\) −10.2709 −0.606275
\(288\) 0 0
\(289\) 1.00000 0.0588235
\(290\) 0 0
\(291\) 14.8176 0.868621
\(292\) 0 0
\(293\) 20.2423 1.18257 0.591284 0.806463i \(-0.298621\pi\)
0.591284 + 0.806463i \(0.298621\pi\)
\(294\) 0 0
\(295\) 15.3230 0.892142
\(296\) 0 0
\(297\) −12.5808 −0.730011
\(298\) 0 0
\(299\) 0 0
\(300\) 0 0
\(301\) −11.2166 −0.646515
\(302\) 0 0
\(303\) −27.5264 −1.58135
\(304\) 0 0
\(305\) −1.00848 −0.0577453
\(306\) 0 0
\(307\) 1.20189 0.0685955 0.0342978 0.999412i \(-0.489081\pi\)
0.0342978 + 0.999412i \(0.489081\pi\)
\(308\) 0 0
\(309\) −17.8524 −1.01559
\(310\) 0 0
\(311\) 10.2081 0.578850 0.289425 0.957201i \(-0.406536\pi\)
0.289425 + 0.957201i \(0.406536\pi\)
\(312\) 0 0
\(313\) 18.7375 1.05911 0.529554 0.848276i \(-0.322359\pi\)
0.529554 + 0.848276i \(0.322359\pi\)
\(314\) 0 0
\(315\) −0.926221 −0.0521866
\(316\) 0 0
\(317\) −10.4985 −0.589656 −0.294828 0.955550i \(-0.595262\pi\)
−0.294828 + 0.955550i \(0.595262\pi\)
\(318\) 0 0
\(319\) 2.10866 0.118063
\(320\) 0 0
\(321\) 2.78562 0.155478
\(322\) 0 0
\(323\) 4.94567 0.275184
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) −20.6197 −1.14027
\(328\) 0 0
\(329\) 9.89134 0.545327
\(330\) 0 0
\(331\) −27.0062 −1.48440 −0.742199 0.670180i \(-0.766217\pi\)
−0.742199 + 0.670180i \(0.766217\pi\)
\(332\) 0 0
\(333\) 4.10866 0.225153
\(334\) 0 0
\(335\) 11.0521 0.603841
\(336\) 0 0
\(337\) −10.4985 −0.571891 −0.285946 0.958246i \(-0.592308\pi\)
−0.285946 + 0.958246i \(0.592308\pi\)
\(338\) 0 0
\(339\) −5.67401 −0.308170
\(340\) 0 0
\(341\) 2.48604 0.134626
\(342\) 0 0
\(343\) −1.00000 −0.0539949
\(344\) 0 0
\(345\) 15.0668 0.811170
\(346\) 0 0
\(347\) −15.0668 −0.808829 −0.404415 0.914576i \(-0.632525\pi\)
−0.404415 + 0.914576i \(0.632525\pi\)
\(348\) 0 0
\(349\) 7.86493 0.421000 0.210500 0.977594i \(-0.432491\pi\)
0.210500 + 0.977594i \(0.432491\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 29.4053 1.56509 0.782543 0.622596i \(-0.213922\pi\)
0.782543 + 0.622596i \(0.213922\pi\)
\(354\) 0 0
\(355\) 16.3774 0.869221
\(356\) 0 0
\(357\) 1.47283 0.0779506
\(358\) 0 0
\(359\) 5.91302 0.312077 0.156039 0.987751i \(-0.450128\pi\)
0.156039 + 0.987751i \(0.450128\pi\)
\(360\) 0 0
\(361\) 5.45963 0.287349
\(362\) 0 0
\(363\) −8.87813 −0.465981
\(364\) 0 0
\(365\) −0.843964 −0.0441751
\(366\) 0 0
\(367\) 15.6762 0.818293 0.409147 0.912469i \(-0.365826\pi\)
0.409147 + 0.912469i \(0.365826\pi\)
\(368\) 0 0
\(369\) −8.53269 −0.444194
\(370\) 0 0
\(371\) −7.83076 −0.406553
\(372\) 0 0
\(373\) −9.81131 −0.508011 −0.254005 0.967203i \(-0.581748\pi\)
−0.254005 + 0.967203i \(0.581748\pi\)
\(374\) 0 0
\(375\) 14.3796 0.742560
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) −11.6615 −0.599012 −0.299506 0.954094i \(-0.596822\pi\)
−0.299506 + 0.954094i \(0.596822\pi\)
\(380\) 0 0
\(381\) 31.2772 1.60238
\(382\) 0 0
\(383\) 27.6087 1.41074 0.705369 0.708840i \(-0.250781\pi\)
0.705369 + 0.708840i \(0.250781\pi\)
\(384\) 0 0
\(385\) 2.48604 0.126700
\(386\) 0 0
\(387\) −9.31832 −0.473677
\(388\) 0 0
\(389\) 11.8308 0.599843 0.299922 0.953964i \(-0.403039\pi\)
0.299922 + 0.953964i \(0.403039\pi\)
\(390\) 0 0
\(391\) 9.17548 0.464024
\(392\) 0 0
\(393\) 6.78562 0.342289
\(394\) 0 0
\(395\) 8.91926 0.448777
\(396\) 0 0
\(397\) −22.7111 −1.13984 −0.569919 0.821701i \(-0.693026\pi\)
−0.569919 + 0.821701i \(0.693026\pi\)
\(398\) 0 0
\(399\) 7.28415 0.364663
\(400\) 0 0
\(401\) 37.6087 1.87809 0.939045 0.343795i \(-0.111713\pi\)
0.939045 + 0.343795i \(0.111713\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0 0
\(405\) 6.48604 0.322294
\(406\) 0 0
\(407\) −11.0279 −0.546634
\(408\) 0 0
\(409\) −22.5808 −1.11655 −0.558274 0.829657i \(-0.688536\pi\)
−0.558274 + 0.829657i \(0.688536\pi\)
\(410\) 0 0
\(411\) 0.559099 0.0275783
\(412\) 0 0
\(413\) 13.7438 0.676287
\(414\) 0 0
\(415\) −13.3494 −0.655299
\(416\) 0 0
\(417\) −8.92622 −0.437119
\(418\) 0 0
\(419\) −24.2904 −1.18666 −0.593332 0.804958i \(-0.702188\pi\)
−0.593332 + 0.804958i \(0.702188\pi\)
\(420\) 0 0
\(421\) 11.5676 0.563769 0.281885 0.959448i \(-0.409040\pi\)
0.281885 + 0.959448i \(0.409040\pi\)
\(422\) 0 0
\(423\) 8.21733 0.399540
\(424\) 0 0
\(425\) 3.75698 0.182240
\(426\) 0 0
\(427\) −0.904539 −0.0437737
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 22.9457 1.10525 0.552627 0.833429i \(-0.313625\pi\)
0.552627 + 0.833429i \(0.313625\pi\)
\(432\) 0 0
\(433\) 19.8913 0.955917 0.477958 0.878383i \(-0.341377\pi\)
0.477958 + 0.878383i \(0.341377\pi\)
\(434\) 0 0
\(435\) −1.55286 −0.0744538
\(436\) 0 0
\(437\) 45.3789 2.17077
\(438\) 0 0
\(439\) −19.8936 −0.949468 −0.474734 0.880129i \(-0.657456\pi\)
−0.474734 + 0.880129i \(0.657456\pi\)
\(440\) 0 0
\(441\) −0.830760 −0.0395600
\(442\) 0 0
\(443\) 15.7827 0.749857 0.374929 0.927054i \(-0.377667\pi\)
0.374929 + 0.927054i \(0.377667\pi\)
\(444\) 0 0
\(445\) −12.4596 −0.590643
\(446\) 0 0
\(447\) −15.0062 −0.709771
\(448\) 0 0
\(449\) −41.2702 −1.94766 −0.973831 0.227273i \(-0.927019\pi\)
−0.973831 + 0.227273i \(0.927019\pi\)
\(450\) 0 0
\(451\) 22.9023 1.07843
\(452\) 0 0
\(453\) 12.0606 0.566655
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −36.9923 −1.73043 −0.865214 0.501403i \(-0.832817\pi\)
−0.865214 + 0.501403i \(0.832817\pi\)
\(458\) 0 0
\(459\) 5.64207 0.263349
\(460\) 0 0
\(461\) 10.3649 0.482741 0.241370 0.970433i \(-0.422403\pi\)
0.241370 + 0.970433i \(0.422403\pi\)
\(462\) 0 0
\(463\) −12.2709 −0.570279 −0.285140 0.958486i \(-0.592040\pi\)
−0.285140 + 0.958486i \(0.592040\pi\)
\(464\) 0 0
\(465\) −1.83076 −0.0848995
\(466\) 0 0
\(467\) −8.86341 −0.410150 −0.205075 0.978746i \(-0.565744\pi\)
−0.205075 + 0.978746i \(0.565744\pi\)
\(468\) 0 0
\(469\) 9.91302 0.457741
\(470\) 0 0
\(471\) −21.0154 −0.968340
\(472\) 0 0
\(473\) 25.0110 1.15001
\(474\) 0 0
\(475\) 18.5808 0.852545
\(476\) 0 0
\(477\) −6.50548 −0.297866
\(478\) 0 0
\(479\) 4.56606 0.208629 0.104314 0.994544i \(-0.466735\pi\)
0.104314 + 0.994544i \(0.466735\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 0 0
\(483\) 13.5140 0.614907
\(484\) 0 0
\(485\) −11.2166 −0.509320
\(486\) 0 0
\(487\) −23.8649 −1.08142 −0.540712 0.841208i \(-0.681845\pi\)
−0.540712 + 0.841208i \(0.681845\pi\)
\(488\) 0 0
\(489\) −23.1880 −1.04860
\(490\) 0 0
\(491\) 7.00624 0.316187 0.158094 0.987424i \(-0.449465\pi\)
0.158094 + 0.987424i \(0.449465\pi\)
\(492\) 0 0
\(493\) −0.945668 −0.0425907
\(494\) 0 0
\(495\) 2.06530 0.0928284
\(496\) 0 0
\(497\) 14.6894 0.658912
\(498\) 0 0
\(499\) −10.2687 −0.459691 −0.229845 0.973227i \(-0.573822\pi\)
−0.229845 + 0.973227i \(0.573822\pi\)
\(500\) 0 0
\(501\) 0.954863 0.0426601
\(502\) 0 0
\(503\) −8.53413 −0.380518 −0.190259 0.981734i \(-0.560933\pi\)
−0.190259 + 0.981734i \(0.560933\pi\)
\(504\) 0 0
\(505\) 20.8370 0.927234
\(506\) 0 0
\(507\) −19.1468 −0.850341
\(508\) 0 0
\(509\) −23.8216 −1.05587 −0.527936 0.849284i \(-0.677034\pi\)
−0.527936 + 0.849284i \(0.677034\pi\)
\(510\) 0 0
\(511\) −0.756981 −0.0334869
\(512\) 0 0
\(513\) 27.9038 1.23198
\(514\) 0 0
\(515\) 13.5140 0.595496
\(516\) 0 0
\(517\) −22.0558 −0.970015
\(518\) 0 0
\(519\) 23.4659 1.03004
\(520\) 0 0
\(521\) −3.48532 −0.152695 −0.0763473 0.997081i \(-0.524326\pi\)
−0.0763473 + 0.997081i \(0.524326\pi\)
\(522\) 0 0
\(523\) 23.2144 1.01509 0.507547 0.861624i \(-0.330552\pi\)
0.507547 + 0.861624i \(0.330552\pi\)
\(524\) 0 0
\(525\) 5.53341 0.241498
\(526\) 0 0
\(527\) −1.11491 −0.0485661
\(528\) 0 0
\(529\) 61.1895 2.66041
\(530\) 0 0
\(531\) 11.4178 0.495490
\(532\) 0 0
\(533\) 0 0
\(534\) 0 0
\(535\) −2.10866 −0.0911655
\(536\) 0 0
\(537\) 25.0760 1.08211
\(538\) 0 0
\(539\) 2.22982 0.0960449
\(540\) 0 0
\(541\) −9.08074 −0.390411 −0.195206 0.980762i \(-0.562537\pi\)
−0.195206 + 0.980762i \(0.562537\pi\)
\(542\) 0 0
\(543\) 8.29959 0.356169
\(544\) 0 0
\(545\) 15.6087 0.668603
\(546\) 0 0
\(547\) −5.34945 −0.228726 −0.114363 0.993439i \(-0.536483\pi\)
−0.114363 + 0.993439i \(0.536483\pi\)
\(548\) 0 0
\(549\) −0.751455 −0.0320713
\(550\) 0 0
\(551\) −4.67696 −0.199245
\(552\) 0 0
\(553\) 8.00000 0.340195
\(554\) 0 0
\(555\) 8.12115 0.344724
\(556\) 0 0
\(557\) −13.9472 −0.590961 −0.295481 0.955349i \(-0.595480\pi\)
−0.295481 + 0.955349i \(0.595480\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 0 0
\(561\) −3.28415 −0.138657
\(562\) 0 0
\(563\) −8.43322 −0.355418 −0.177709 0.984083i \(-0.556869\pi\)
−0.177709 + 0.984083i \(0.556869\pi\)
\(564\) 0 0
\(565\) 4.29512 0.180697
\(566\) 0 0
\(567\) 5.81756 0.244314
\(568\) 0 0
\(569\) −5.89357 −0.247071 −0.123536 0.992340i \(-0.539423\pi\)
−0.123536 + 0.992340i \(0.539423\pi\)
\(570\) 0 0
\(571\) −40.2857 −1.68590 −0.842951 0.537990i \(-0.819184\pi\)
−0.842951 + 0.537990i \(0.819184\pi\)
\(572\) 0 0
\(573\) −4.24926 −0.177515
\(574\) 0 0
\(575\) 34.4721 1.43759
\(576\) 0 0
\(577\) 27.1616 1.13075 0.565375 0.824834i \(-0.308731\pi\)
0.565375 + 0.824834i \(0.308731\pi\)
\(578\) 0 0
\(579\) 25.8135 1.07277
\(580\) 0 0
\(581\) −11.9736 −0.496748
\(582\) 0 0
\(583\) 17.4611 0.723167
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −10.2423 −0.422745 −0.211373 0.977406i \(-0.567793\pi\)
−0.211373 + 0.977406i \(0.567793\pi\)
\(588\) 0 0
\(589\) −5.51396 −0.227199
\(590\) 0 0
\(591\) 21.9736 0.903873
\(592\) 0 0
\(593\) 27.9736 1.14874 0.574369 0.818597i \(-0.305248\pi\)
0.574369 + 0.818597i \(0.305248\pi\)
\(594\) 0 0
\(595\) −1.11491 −0.0457068
\(596\) 0 0
\(597\) 24.5202 1.00355
\(598\) 0 0
\(599\) −6.00223 −0.245245 −0.122622 0.992453i \(-0.539130\pi\)
−0.122622 + 0.992453i \(0.539130\pi\)
\(600\) 0 0
\(601\) −19.8913 −0.811385 −0.405692 0.914010i \(-0.632970\pi\)
−0.405692 + 0.914010i \(0.632970\pi\)
\(602\) 0 0
\(603\) 8.23534 0.335369
\(604\) 0 0
\(605\) 6.72058 0.273230
\(606\) 0 0
\(607\) 29.4784 1.19649 0.598245 0.801313i \(-0.295865\pi\)
0.598245 + 0.801313i \(0.295865\pi\)
\(608\) 0 0
\(609\) −1.39281 −0.0564396
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) −36.4115 −1.47065 −0.735324 0.677716i \(-0.762970\pi\)
−0.735324 + 0.677716i \(0.762970\pi\)
\(614\) 0 0
\(615\) −16.8656 −0.680088
\(616\) 0 0
\(617\) −7.51396 −0.302501 −0.151250 0.988495i \(-0.548330\pi\)
−0.151250 + 0.988495i \(0.548330\pi\)
\(618\) 0 0
\(619\) −21.7438 −0.873956 −0.436978 0.899472i \(-0.643951\pi\)
−0.436978 + 0.899472i \(0.643951\pi\)
\(620\) 0 0
\(621\) 51.7688 2.07741
\(622\) 0 0
\(623\) −11.1755 −0.447736
\(624\) 0 0
\(625\) 7.89981 0.315993
\(626\) 0 0
\(627\) −16.2423 −0.648655
\(628\) 0 0
\(629\) 4.94567 0.197197
\(630\) 0 0
\(631\) 9.57454 0.381156 0.190578 0.981672i \(-0.438964\pi\)
0.190578 + 0.981672i \(0.438964\pi\)
\(632\) 0 0
\(633\) −10.9068 −0.433505
\(634\) 0 0
\(635\) −23.6762 −0.939563
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) 12.2034 0.482759
\(640\) 0 0
\(641\) 22.8759 0.903544 0.451772 0.892134i \(-0.350792\pi\)
0.451772 + 0.892134i \(0.350792\pi\)
\(642\) 0 0
\(643\) −28.2834 −1.11539 −0.557695 0.830046i \(-0.688314\pi\)
−0.557695 + 0.830046i \(0.688314\pi\)
\(644\) 0 0
\(645\) −18.4185 −0.725228
\(646\) 0 0
\(647\) 10.0125 0.393631 0.196816 0.980440i \(-0.436940\pi\)
0.196816 + 0.980440i \(0.436940\pi\)
\(648\) 0 0
\(649\) −30.6461 −1.20296
\(650\) 0 0
\(651\) −1.64207 −0.0643579
\(652\) 0 0
\(653\) −2.45963 −0.0962528 −0.0481264 0.998841i \(-0.515325\pi\)
−0.0481264 + 0.998841i \(0.515325\pi\)
\(654\) 0 0
\(655\) −5.13659 −0.200703
\(656\) 0 0
\(657\) −0.628870 −0.0245346
\(658\) 0 0
\(659\) −33.0187 −1.28623 −0.643114 0.765771i \(-0.722358\pi\)
−0.643114 + 0.765771i \(0.722358\pi\)
\(660\) 0 0
\(661\) 5.97359 0.232346 0.116173 0.993229i \(-0.462937\pi\)
0.116173 + 0.993229i \(0.462937\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) −5.51396 −0.213822
\(666\) 0 0
\(667\) −8.67696 −0.335973
\(668\) 0 0
\(669\) −39.9861 −1.54595
\(670\) 0 0
\(671\) 2.01696 0.0778637
\(672\) 0 0
\(673\) 6.29512 0.242659 0.121329 0.992612i \(-0.461284\pi\)
0.121329 + 0.992612i \(0.461284\pi\)
\(674\) 0 0
\(675\) 21.1972 0.815879
\(676\) 0 0
\(677\) 46.5544 1.78923 0.894615 0.446838i \(-0.147450\pi\)
0.894615 + 0.446838i \(0.147450\pi\)
\(678\) 0 0
\(679\) −10.0606 −0.386089
\(680\) 0 0
\(681\) 6.63912 0.254412
\(682\) 0 0
\(683\) −43.9861 −1.68308 −0.841540 0.540194i \(-0.818351\pi\)
−0.841540 + 0.540194i \(0.818351\pi\)
\(684\) 0 0
\(685\) −0.423228 −0.0161707
\(686\) 0 0
\(687\) 15.0668 0.574835
\(688\) 0 0
\(689\) 0 0
\(690\) 0 0
\(691\) 9.17772 0.349137 0.174568 0.984645i \(-0.444147\pi\)
0.174568 + 0.984645i \(0.444147\pi\)
\(692\) 0 0
\(693\) 1.85244 0.0703684
\(694\) 0 0
\(695\) 6.75698 0.256307
\(696\) 0 0
\(697\) −10.2709 −0.389040
\(698\) 0 0
\(699\) 17.0154 0.643583
\(700\) 0 0
\(701\) −7.59622 −0.286905 −0.143453 0.989657i \(-0.545820\pi\)
−0.143453 + 0.989657i \(0.545820\pi\)
\(702\) 0 0
\(703\) 24.4596 0.922512
\(704\) 0 0
\(705\) 16.2423 0.611720
\(706\) 0 0
\(707\) 18.6894 0.702889
\(708\) 0 0
\(709\) 7.63511 0.286743 0.143371 0.989669i \(-0.454206\pi\)
0.143371 + 0.989669i \(0.454206\pi\)
\(710\) 0 0
\(711\) 6.64608 0.249248
\(712\) 0 0
\(713\) −10.2298 −0.383110
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 45.2999 1.69176
\(718\) 0 0
\(719\) 13.6204 0.507955 0.253977 0.967210i \(-0.418261\pi\)
0.253977 + 0.967210i \(0.418261\pi\)
\(720\) 0 0
\(721\) 12.1212 0.451415
\(722\) 0 0
\(723\) −1.42474 −0.0529868
\(724\) 0 0
\(725\) −3.55286 −0.131950
\(726\) 0 0
\(727\) −10.9457 −0.405952 −0.202976 0.979184i \(-0.565061\pi\)
−0.202976 + 0.979184i \(0.565061\pi\)
\(728\) 0 0
\(729\) 29.7625 1.10232
\(730\) 0 0
\(731\) −11.2166 −0.414861
\(732\) 0 0
\(733\) 7.28415 0.269046 0.134523 0.990910i \(-0.457050\pi\)
0.134523 + 0.990910i \(0.457050\pi\)
\(734\) 0 0
\(735\) −1.64207 −0.0605688
\(736\) 0 0
\(737\) −22.1042 −0.814218
\(738\) 0 0
\(739\) 50.3268 1.85130 0.925651 0.378380i \(-0.123519\pi\)
0.925651 + 0.378380i \(0.123519\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) −14.9317 −0.547793 −0.273896 0.961759i \(-0.588313\pi\)
−0.273896 + 0.961759i \(0.588313\pi\)
\(744\) 0 0
\(745\) 11.3594 0.416178
\(746\) 0 0
\(747\) −9.94719 −0.363948
\(748\) 0 0
\(749\) −1.89134 −0.0691079
\(750\) 0 0
\(751\) −29.9347 −1.09233 −0.546166 0.837677i \(-0.683913\pi\)
−0.546166 + 0.837677i \(0.683913\pi\)
\(752\) 0 0
\(753\) −0.0388938 −0.00141737
\(754\) 0 0
\(755\) −9.12963 −0.332261
\(756\) 0 0
\(757\) −6.48827 −0.235820 −0.117910 0.993024i \(-0.537619\pi\)
−0.117910 + 0.993024i \(0.537619\pi\)
\(758\) 0 0
\(759\) −30.1336 −1.09378
\(760\) 0 0
\(761\) −28.2159 −1.02283 −0.511413 0.859335i \(-0.670878\pi\)
−0.511413 + 0.859335i \(0.670878\pi\)
\(762\) 0 0
\(763\) 14.0000 0.506834
\(764\) 0 0
\(765\) −0.926221 −0.0334876
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) −21.8260 −0.787067 −0.393533 0.919310i \(-0.628747\pi\)
−0.393533 + 0.919310i \(0.628747\pi\)
\(770\) 0 0
\(771\) 21.6740 0.780570
\(772\) 0 0
\(773\) 26.5683 0.955595 0.477798 0.878470i \(-0.341435\pi\)
0.477798 + 0.878470i \(0.341435\pi\)
\(774\) 0 0
\(775\) −4.18869 −0.150462
\(776\) 0 0
\(777\) 7.28415 0.261317
\(778\) 0 0
\(779\) −50.7967 −1.81998
\(780\) 0 0
\(781\) −32.7547 −1.17206
\(782\) 0 0
\(783\) −5.33553 −0.190676
\(784\) 0 0
\(785\) 15.9083 0.567791
\(786\) 0 0
\(787\) −4.98903 −0.177840 −0.0889199 0.996039i \(-0.528342\pi\)
−0.0889199 + 0.996039i \(0.528342\pi\)
\(788\) 0 0
\(789\) −36.7019 −1.30662
\(790\) 0 0
\(791\) 3.85244 0.136977
\(792\) 0 0
\(793\) 0 0
\(794\) 0 0
\(795\) −12.8587 −0.456051
\(796\) 0 0
\(797\) −47.0963 −1.66824 −0.834118 0.551587i \(-0.814023\pi\)
−0.834118 + 0.551587i \(0.814023\pi\)
\(798\) 0 0
\(799\) 9.89134 0.349930
\(800\) 0 0
\(801\) −9.28415 −0.328039
\(802\) 0 0
\(803\) 1.68793 0.0595657
\(804\) 0 0
\(805\) −10.2298 −0.360554
\(806\) 0 0
\(807\) −35.3275 −1.24359
\(808\) 0 0
\(809\) −10.2423 −0.360100 −0.180050 0.983657i \(-0.557626\pi\)
−0.180050 + 0.983657i \(0.557626\pi\)
\(810\) 0 0
\(811\) 2.98680 0.104881 0.0524403 0.998624i \(-0.483300\pi\)
0.0524403 + 0.998624i \(0.483300\pi\)
\(812\) 0 0
\(813\) 22.1909 0.778270
\(814\) 0 0
\(815\) 17.5529 0.614850
\(816\) 0 0
\(817\) −55.4736 −1.94078
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −52.7283 −1.84023 −0.920116 0.391646i \(-0.871906\pi\)
−0.920116 + 0.391646i \(0.871906\pi\)
\(822\) 0 0
\(823\) 6.68945 0.233179 0.116590 0.993180i \(-0.462804\pi\)
0.116590 + 0.993180i \(0.462804\pi\)
\(824\) 0 0
\(825\) −12.3385 −0.429571
\(826\) 0 0
\(827\) −41.5140 −1.44358 −0.721791 0.692111i \(-0.756681\pi\)
−0.721791 + 0.692111i \(0.756681\pi\)
\(828\) 0 0
\(829\) −48.4596 −1.68307 −0.841536 0.540201i \(-0.818348\pi\)
−0.841536 + 0.540201i \(0.818348\pi\)
\(830\) 0 0
\(831\) 0.516914 0.0179316
\(832\) 0 0
\(833\) −1.00000 −0.0346479
\(834\) 0 0
\(835\) −0.722813 −0.0250140
\(836\) 0 0
\(837\) −6.29039 −0.217428
\(838\) 0 0
\(839\) 9.64903 0.333122 0.166561 0.986031i \(-0.446734\pi\)
0.166561 + 0.986031i \(0.446734\pi\)
\(840\) 0 0
\(841\) −28.1057 −0.969162
\(842\) 0 0
\(843\) 7.81460 0.269149
\(844\) 0 0
\(845\) 14.4938 0.498602
\(846\) 0 0
\(847\) 6.02792 0.207122
\(848\) 0 0
\(849\) −17.1047 −0.587030
\(850\) 0 0
\(851\) 45.3789 1.55557
\(852\) 0 0
\(853\) −35.3619 −1.21077 −0.605385 0.795933i \(-0.706981\pi\)
−0.605385 + 0.795933i \(0.706981\pi\)
\(854\) 0 0
\(855\) −4.58078 −0.156659
\(856\) 0 0
\(857\) 29.0451 0.992163 0.496081 0.868276i \(-0.334772\pi\)
0.496081 + 0.868276i \(0.334772\pi\)
\(858\) 0 0
\(859\) 43.7996 1.49442 0.747212 0.664586i \(-0.231392\pi\)
0.747212 + 0.664586i \(0.231392\pi\)
\(860\) 0 0
\(861\) −15.1274 −0.515540
\(862\) 0 0
\(863\) 3.39754 0.115654 0.0578268 0.998327i \(-0.481583\pi\)
0.0578268 + 0.998327i \(0.481583\pi\)
\(864\) 0 0
\(865\) −17.7632 −0.603968
\(866\) 0 0
\(867\) 1.47283 0.0500201
\(868\) 0 0
\(869\) −17.8385 −0.605130
\(870\) 0 0
\(871\) 0 0
\(872\) 0 0
\(873\) −8.35793 −0.282873
\(874\) 0 0
\(875\) −9.76322 −0.330057
\(876\) 0 0
\(877\) −29.0529 −0.981047 −0.490523 0.871428i \(-0.663194\pi\)
−0.490523 + 0.871428i \(0.663194\pi\)
\(878\) 0 0
\(879\) 29.8135 1.00559
\(880\) 0 0
\(881\) −11.4923 −0.387185 −0.193592 0.981082i \(-0.562014\pi\)
−0.193592 + 0.981082i \(0.562014\pi\)
\(882\) 0 0
\(883\) −25.1927 −0.847802 −0.423901 0.905709i \(-0.639340\pi\)
−0.423901 + 0.905709i \(0.639340\pi\)
\(884\) 0 0
\(885\) 22.5683 0.758625
\(886\) 0 0
\(887\) 35.1957 1.18176 0.590878 0.806761i \(-0.298781\pi\)
0.590878 + 0.806761i \(0.298781\pi\)
\(888\) 0 0
\(889\) −21.2361 −0.712235
\(890\) 0 0
\(891\) −12.9721 −0.434581
\(892\) 0 0
\(893\) 48.9193 1.63702
\(894\) 0 0
\(895\) −18.9821 −0.634501
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 1.05433 0.0351639
\(900\) 0 0
\(901\) −7.83076 −0.260880
\(902\) 0 0
\(903\) −16.5202 −0.549758
\(904\) 0 0
\(905\) −6.28263 −0.208842
\(906\) 0 0
\(907\) 25.1227 0.834184 0.417092 0.908864i \(-0.363049\pi\)
0.417092 + 0.908864i \(0.363049\pi\)
\(908\) 0 0
\(909\) 15.5264 0.514980
\(910\) 0 0
\(911\) −22.5110 −0.745823 −0.372912 0.927867i \(-0.621641\pi\)
−0.372912 + 0.927867i \(0.621641\pi\)
\(912\) 0 0
\(913\) 26.6989 0.883605
\(914\) 0 0
\(915\) −1.48532 −0.0491032
\(916\) 0 0
\(917\) −4.60719 −0.152143
\(918\) 0 0
\(919\) −13.5676 −0.447553 −0.223777 0.974640i \(-0.571839\pi\)
−0.223777 + 0.974640i \(0.571839\pi\)
\(920\) 0 0
\(921\) 1.77018 0.0583296
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) 18.5808 0.610932
\(926\) 0 0
\(927\) 10.0698 0.330735
\(928\) 0 0
\(929\) −11.9519 −0.392129 −0.196065 0.980591i \(-0.562816\pi\)
−0.196065 + 0.980591i \(0.562816\pi\)
\(930\) 0 0
\(931\) −4.94567 −0.162088
\(932\) 0 0
\(933\) 15.0349 0.492220
\(934\) 0 0
\(935\) 2.48604 0.0813021
\(936\) 0 0
\(937\) 53.0140 1.73189 0.865946 0.500138i \(-0.166717\pi\)
0.865946 + 0.500138i \(0.166717\pi\)
\(938\) 0 0
\(939\) 27.5973 0.900603
\(940\) 0 0
\(941\) −27.1204 −0.884101 −0.442050 0.896990i \(-0.645749\pi\)
−0.442050 + 0.896990i \(0.645749\pi\)
\(942\) 0 0
\(943\) −94.2409 −3.06890
\(944\) 0 0
\(945\) −6.29039 −0.204626
\(946\) 0 0
\(947\) 44.5808 1.44868 0.724340 0.689443i \(-0.242144\pi\)
0.724340 + 0.689443i \(0.242144\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) 0 0
\(951\) −15.4626 −0.501409
\(952\) 0 0
\(953\) −51.8991 −1.68118 −0.840588 0.541675i \(-0.817791\pi\)
−0.840588 + 0.541675i \(0.817791\pi\)
\(954\) 0 0
\(955\) 3.21661 0.104087
\(956\) 0 0
\(957\) 3.10571 0.100393
\(958\) 0 0
\(959\) −0.379608 −0.0122582
\(960\) 0 0
\(961\) −29.7570 −0.959903
\(962\) 0 0
\(963\) −1.57125 −0.0506327
\(964\) 0 0
\(965\) −19.5404 −0.629027
\(966\) 0 0
\(967\) −37.2680 −1.19846 −0.599229 0.800578i \(-0.704526\pi\)
−0.599229 + 0.800578i \(0.704526\pi\)
\(968\) 0 0
\(969\) 7.28415 0.234001
\(970\) 0 0
\(971\) 3.05433 0.0980182 0.0490091 0.998798i \(-0.484394\pi\)
0.0490091 + 0.998798i \(0.484394\pi\)
\(972\) 0 0
\(973\) 6.06058 0.194293
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −47.3961 −1.51634 −0.758168 0.652059i \(-0.773905\pi\)
−0.758168 + 0.652059i \(0.773905\pi\)
\(978\) 0 0
\(979\) 24.9193 0.796423
\(980\) 0 0
\(981\) 11.6306 0.371338
\(982\) 0 0
\(983\) 51.1957 1.63289 0.816445 0.577423i \(-0.195942\pi\)
0.816445 + 0.577423i \(0.195942\pi\)
\(984\) 0 0
\(985\) −16.6336 −0.529990
\(986\) 0 0
\(987\) 14.5683 0.463714
\(988\) 0 0
\(989\) −102.918 −3.27260
\(990\) 0 0
\(991\) −16.9332 −0.537900 −0.268950 0.963154i \(-0.586677\pi\)
−0.268950 + 0.963154i \(0.586677\pi\)
\(992\) 0 0
\(993\) −39.7757 −1.26224
\(994\) 0 0
\(995\) −18.5613 −0.588434
\(996\) 0 0
\(997\) −53.3069 −1.68825 −0.844123 0.536150i \(-0.819878\pi\)
−0.844123 + 0.536150i \(0.819878\pi\)
\(998\) 0 0
\(999\) 27.9038 0.882838
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 1904.2.a.m.1.3 3
4.3 odd 2 952.2.a.f.1.1 3
8.3 odd 2 7616.2.a.bc.1.3 3
8.5 even 2 7616.2.a.be.1.1 3
12.11 even 2 8568.2.a.x.1.3 3
28.27 even 2 6664.2.a.j.1.3 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
952.2.a.f.1.1 3 4.3 odd 2
1904.2.a.m.1.3 3 1.1 even 1 trivial
6664.2.a.j.1.3 3 28.27 even 2
7616.2.a.bc.1.3 3 8.3 odd 2
7616.2.a.be.1.1 3 8.5 even 2
8568.2.a.x.1.3 3 12.11 even 2