Properties

Label 1911.2.a.j.1.1
Level $1911$
Weight $2$
Character 1911.1
Self dual yes
Analytic conductor $15.259$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1911,2,Mod(1,1911)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1911, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("1911.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1911 = 3 \cdot 7^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1911.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: \(15.2594118263\)
Analytic rank: \(1\)
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: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-1.41421\) of defining polynomial
Character \(\chi\) \(=\) 1911.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.41421 q^{2} +1.00000 q^{3} -1.00000 q^{5} -1.41421 q^{6} +2.82843 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-1.41421 q^{2} +1.00000 q^{3} -1.00000 q^{5} -1.41421 q^{6} +2.82843 q^{8} +1.00000 q^{9} +1.41421 q^{10} -2.00000 q^{11} +1.00000 q^{13} -1.00000 q^{15} -4.00000 q^{16} +1.41421 q^{17} -1.41421 q^{18} -3.58579 q^{19} +2.82843 q^{22} +0.414214 q^{23} +2.82843 q^{24} -4.00000 q^{25} -1.41421 q^{26} +1.00000 q^{27} +1.24264 q^{29} +1.41421 q^{30} -0.414214 q^{31} -2.00000 q^{33} -2.00000 q^{34} -4.00000 q^{37} +5.07107 q^{38} +1.00000 q^{39} -2.82843 q^{40} +5.17157 q^{41} +3.82843 q^{43} -1.00000 q^{45} -0.585786 q^{46} -4.65685 q^{47} -4.00000 q^{48} +5.65685 q^{50} +1.41421 q^{51} +2.41421 q^{53} -1.41421 q^{54} +2.00000 q^{55} -3.58579 q^{57} -1.75736 q^{58} +7.65685 q^{59} -11.6569 q^{61} +0.585786 q^{62} +8.00000 q^{64} -1.00000 q^{65} +2.82843 q^{66} -4.58579 q^{67} +0.414214 q^{69} -6.82843 q^{71} +2.82843 q^{72} -5.58579 q^{73} +5.65685 q^{74} -4.00000 q^{75} -1.41421 q^{78} +1.48528 q^{79} +4.00000 q^{80} +1.00000 q^{81} -7.31371 q^{82} -5.48528 q^{83} -1.41421 q^{85} -5.41421 q^{86} +1.24264 q^{87} -5.65685 q^{88} +2.65685 q^{89} +1.41421 q^{90} -0.414214 q^{93} +6.58579 q^{94} +3.58579 q^{95} -16.8995 q^{97} -2.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{3} - 2 q^{5} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{3} - 2 q^{5} + 2 q^{9} - 4 q^{11} + 2 q^{13} - 2 q^{15} - 8 q^{16} - 10 q^{19} - 2 q^{23} - 8 q^{25} + 2 q^{27} - 6 q^{29} + 2 q^{31} - 4 q^{33} - 4 q^{34} - 8 q^{37} - 4 q^{38} + 2 q^{39} + 16 q^{41} + 2 q^{43} - 2 q^{45} - 4 q^{46} + 2 q^{47} - 8 q^{48} + 2 q^{53} + 4 q^{55} - 10 q^{57} - 12 q^{58} + 4 q^{59} - 12 q^{61} + 4 q^{62} + 16 q^{64} - 2 q^{65} - 12 q^{67} - 2 q^{69} - 8 q^{71} - 14 q^{73} - 8 q^{75} - 14 q^{79} + 8 q^{80} + 2 q^{81} + 8 q^{82} + 6 q^{83} - 8 q^{86} - 6 q^{87} - 6 q^{89} + 2 q^{93} + 16 q^{94} + 10 q^{95} - 14 q^{97} - 4 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.41421 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(3\) 1.00000 0.577350
\(4\) 0 0
\(5\) −1.00000 −0.447214 −0.223607 0.974679i \(-0.571783\pi\)
−0.223607 + 0.974679i \(0.571783\pi\)
\(6\) −1.41421 −0.577350
\(7\) 0 0
\(8\) 2.82843 1.00000
\(9\) 1.00000 0.333333
\(10\) 1.41421 0.447214
\(11\) −2.00000 −0.603023 −0.301511 0.953463i \(-0.597491\pi\)
−0.301511 + 0.953463i \(0.597491\pi\)
\(12\) 0 0
\(13\) 1.00000 0.277350
\(14\) 0 0
\(15\) −1.00000 −0.258199
\(16\) −4.00000 −1.00000
\(17\) 1.41421 0.342997 0.171499 0.985184i \(-0.445139\pi\)
0.171499 + 0.985184i \(0.445139\pi\)
\(18\) −1.41421 −0.333333
\(19\) −3.58579 −0.822636 −0.411318 0.911492i \(-0.634931\pi\)
−0.411318 + 0.911492i \(0.634931\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 2.82843 0.603023
\(23\) 0.414214 0.0863695 0.0431847 0.999067i \(-0.486250\pi\)
0.0431847 + 0.999067i \(0.486250\pi\)
\(24\) 2.82843 0.577350
\(25\) −4.00000 −0.800000
\(26\) −1.41421 −0.277350
\(27\) 1.00000 0.192450
\(28\) 0 0
\(29\) 1.24264 0.230753 0.115376 0.993322i \(-0.463193\pi\)
0.115376 + 0.993322i \(0.463193\pi\)
\(30\) 1.41421 0.258199
\(31\) −0.414214 −0.0743950 −0.0371975 0.999308i \(-0.511843\pi\)
−0.0371975 + 0.999308i \(0.511843\pi\)
\(32\) 0 0
\(33\) −2.00000 −0.348155
\(34\) −2.00000 −0.342997
\(35\) 0 0
\(36\) 0 0
\(37\) −4.00000 −0.657596 −0.328798 0.944400i \(-0.606644\pi\)
−0.328798 + 0.944400i \(0.606644\pi\)
\(38\) 5.07107 0.822636
\(39\) 1.00000 0.160128
\(40\) −2.82843 −0.447214
\(41\) 5.17157 0.807664 0.403832 0.914833i \(-0.367678\pi\)
0.403832 + 0.914833i \(0.367678\pi\)
\(42\) 0 0
\(43\) 3.82843 0.583830 0.291915 0.956444i \(-0.405708\pi\)
0.291915 + 0.956444i \(0.405708\pi\)
\(44\) 0 0
\(45\) −1.00000 −0.149071
\(46\) −0.585786 −0.0863695
\(47\) −4.65685 −0.679272 −0.339636 0.940557i \(-0.610304\pi\)
−0.339636 + 0.940557i \(0.610304\pi\)
\(48\) −4.00000 −0.577350
\(49\) 0 0
\(50\) 5.65685 0.800000
\(51\) 1.41421 0.198030
\(52\) 0 0
\(53\) 2.41421 0.331618 0.165809 0.986158i \(-0.446977\pi\)
0.165809 + 0.986158i \(0.446977\pi\)
\(54\) −1.41421 −0.192450
\(55\) 2.00000 0.269680
\(56\) 0 0
\(57\) −3.58579 −0.474949
\(58\) −1.75736 −0.230753
\(59\) 7.65685 0.996838 0.498419 0.866936i \(-0.333914\pi\)
0.498419 + 0.866936i \(0.333914\pi\)
\(60\) 0 0
\(61\) −11.6569 −1.49251 −0.746254 0.665662i \(-0.768149\pi\)
−0.746254 + 0.665662i \(0.768149\pi\)
\(62\) 0.585786 0.0743950
\(63\) 0 0
\(64\) 8.00000 1.00000
\(65\) −1.00000 −0.124035
\(66\) 2.82843 0.348155
\(67\) −4.58579 −0.560243 −0.280121 0.959965i \(-0.590375\pi\)
−0.280121 + 0.959965i \(0.590375\pi\)
\(68\) 0 0
\(69\) 0.414214 0.0498655
\(70\) 0 0
\(71\) −6.82843 −0.810385 −0.405193 0.914231i \(-0.632796\pi\)
−0.405193 + 0.914231i \(0.632796\pi\)
\(72\) 2.82843 0.333333
\(73\) −5.58579 −0.653767 −0.326883 0.945065i \(-0.605998\pi\)
−0.326883 + 0.945065i \(0.605998\pi\)
\(74\) 5.65685 0.657596
\(75\) −4.00000 −0.461880
\(76\) 0 0
\(77\) 0 0
\(78\) −1.41421 −0.160128
\(79\) 1.48528 0.167107 0.0835536 0.996503i \(-0.473373\pi\)
0.0835536 + 0.996503i \(0.473373\pi\)
\(80\) 4.00000 0.447214
\(81\) 1.00000 0.111111
\(82\) −7.31371 −0.807664
\(83\) −5.48528 −0.602088 −0.301044 0.953610i \(-0.597335\pi\)
−0.301044 + 0.953610i \(0.597335\pi\)
\(84\) 0 0
\(85\) −1.41421 −0.153393
\(86\) −5.41421 −0.583830
\(87\) 1.24264 0.133225
\(88\) −5.65685 −0.603023
\(89\) 2.65685 0.281626 0.140813 0.990036i \(-0.455028\pi\)
0.140813 + 0.990036i \(0.455028\pi\)
\(90\) 1.41421 0.149071
\(91\) 0 0
\(92\) 0 0
\(93\) −0.414214 −0.0429519
\(94\) 6.58579 0.679272
\(95\) 3.58579 0.367894
\(96\) 0 0
\(97\) −16.8995 −1.71588 −0.857942 0.513747i \(-0.828257\pi\)
−0.857942 + 0.513747i \(0.828257\pi\)
\(98\) 0 0
\(99\) −2.00000 −0.201008
\(100\) 0 0
\(101\) 7.65685 0.761885 0.380943 0.924599i \(-0.375599\pi\)
0.380943 + 0.924599i \(0.375599\pi\)
\(102\) −2.00000 −0.198030
\(103\) 4.48528 0.441948 0.220974 0.975280i \(-0.429076\pi\)
0.220974 + 0.975280i \(0.429076\pi\)
\(104\) 2.82843 0.277350
\(105\) 0 0
\(106\) −3.41421 −0.331618
\(107\) −16.4853 −1.59369 −0.796846 0.604182i \(-0.793500\pi\)
−0.796846 + 0.604182i \(0.793500\pi\)
\(108\) 0 0
\(109\) −17.0711 −1.63511 −0.817556 0.575849i \(-0.804672\pi\)
−0.817556 + 0.575849i \(0.804672\pi\)
\(110\) −2.82843 −0.269680
\(111\) −4.00000 −0.379663
\(112\) 0 0
\(113\) −14.8995 −1.40163 −0.700813 0.713345i \(-0.747179\pi\)
−0.700813 + 0.713345i \(0.747179\pi\)
\(114\) 5.07107 0.474949
\(115\) −0.414214 −0.0386256
\(116\) 0 0
\(117\) 1.00000 0.0924500
\(118\) −10.8284 −0.996838
\(119\) 0 0
\(120\) −2.82843 −0.258199
\(121\) −7.00000 −0.636364
\(122\) 16.4853 1.49251
\(123\) 5.17157 0.466305
\(124\) 0 0
\(125\) 9.00000 0.804984
\(126\) 0 0
\(127\) −5.31371 −0.471515 −0.235758 0.971812i \(-0.575757\pi\)
−0.235758 + 0.971812i \(0.575757\pi\)
\(128\) −11.3137 −1.00000
\(129\) 3.82843 0.337074
\(130\) 1.41421 0.124035
\(131\) 12.2426 1.06964 0.534822 0.844965i \(-0.320379\pi\)
0.534822 + 0.844965i \(0.320379\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 6.48528 0.560243
\(135\) −1.00000 −0.0860663
\(136\) 4.00000 0.342997
\(137\) −3.51472 −0.300283 −0.150141 0.988665i \(-0.547973\pi\)
−0.150141 + 0.988665i \(0.547973\pi\)
\(138\) −0.585786 −0.0498655
\(139\) −10.9706 −0.930511 −0.465255 0.885176i \(-0.654038\pi\)
−0.465255 + 0.885176i \(0.654038\pi\)
\(140\) 0 0
\(141\) −4.65685 −0.392178
\(142\) 9.65685 0.810385
\(143\) −2.00000 −0.167248
\(144\) −4.00000 −0.333333
\(145\) −1.24264 −0.103196
\(146\) 7.89949 0.653767
\(147\) 0 0
\(148\) 0 0
\(149\) −20.1421 −1.65011 −0.825054 0.565054i \(-0.808855\pi\)
−0.825054 + 0.565054i \(0.808855\pi\)
\(150\) 5.65685 0.461880
\(151\) −10.8284 −0.881205 −0.440602 0.897702i \(-0.645235\pi\)
−0.440602 + 0.897702i \(0.645235\pi\)
\(152\) −10.1421 −0.822636
\(153\) 1.41421 0.114332
\(154\) 0 0
\(155\) 0.414214 0.0332704
\(156\) 0 0
\(157\) −2.24264 −0.178982 −0.0894911 0.995988i \(-0.528524\pi\)
−0.0894911 + 0.995988i \(0.528524\pi\)
\(158\) −2.10051 −0.167107
\(159\) 2.41421 0.191460
\(160\) 0 0
\(161\) 0 0
\(162\) −1.41421 −0.111111
\(163\) 3.75736 0.294299 0.147150 0.989114i \(-0.452990\pi\)
0.147150 + 0.989114i \(0.452990\pi\)
\(164\) 0 0
\(165\) 2.00000 0.155700
\(166\) 7.75736 0.602088
\(167\) 17.9706 1.39060 0.695302 0.718718i \(-0.255271\pi\)
0.695302 + 0.718718i \(0.255271\pi\)
\(168\) 0 0
\(169\) 1.00000 0.0769231
\(170\) 2.00000 0.153393
\(171\) −3.58579 −0.274212
\(172\) 0 0
\(173\) 14.8284 1.12738 0.563692 0.825985i \(-0.309380\pi\)
0.563692 + 0.825985i \(0.309380\pi\)
\(174\) −1.75736 −0.133225
\(175\) 0 0
\(176\) 8.00000 0.603023
\(177\) 7.65685 0.575524
\(178\) −3.75736 −0.281626
\(179\) 12.7574 0.953530 0.476765 0.879031i \(-0.341809\pi\)
0.476765 + 0.879031i \(0.341809\pi\)
\(180\) 0 0
\(181\) 8.82843 0.656212 0.328106 0.944641i \(-0.393590\pi\)
0.328106 + 0.944641i \(0.393590\pi\)
\(182\) 0 0
\(183\) −11.6569 −0.861699
\(184\) 1.17157 0.0863695
\(185\) 4.00000 0.294086
\(186\) 0.585786 0.0429519
\(187\) −2.82843 −0.206835
\(188\) 0 0
\(189\) 0 0
\(190\) −5.07107 −0.367894
\(191\) 6.34315 0.458974 0.229487 0.973312i \(-0.426295\pi\)
0.229487 + 0.973312i \(0.426295\pi\)
\(192\) 8.00000 0.577350
\(193\) −4.58579 −0.330092 −0.165046 0.986286i \(-0.552777\pi\)
−0.165046 + 0.986286i \(0.552777\pi\)
\(194\) 23.8995 1.71588
\(195\) −1.00000 −0.0716115
\(196\) 0 0
\(197\) −20.8284 −1.48396 −0.741982 0.670420i \(-0.766114\pi\)
−0.741982 + 0.670420i \(0.766114\pi\)
\(198\) 2.82843 0.201008
\(199\) −7.41421 −0.525580 −0.262790 0.964853i \(-0.584643\pi\)
−0.262790 + 0.964853i \(0.584643\pi\)
\(200\) −11.3137 −0.800000
\(201\) −4.58579 −0.323456
\(202\) −10.8284 −0.761885
\(203\) 0 0
\(204\) 0 0
\(205\) −5.17157 −0.361198
\(206\) −6.34315 −0.441948
\(207\) 0.414214 0.0287898
\(208\) −4.00000 −0.277350
\(209\) 7.17157 0.496068
\(210\) 0 0
\(211\) −7.14214 −0.491685 −0.245842 0.969310i \(-0.579065\pi\)
−0.245842 + 0.969310i \(0.579065\pi\)
\(212\) 0 0
\(213\) −6.82843 −0.467876
\(214\) 23.3137 1.59369
\(215\) −3.82843 −0.261097
\(216\) 2.82843 0.192450
\(217\) 0 0
\(218\) 24.1421 1.63511
\(219\) −5.58579 −0.377452
\(220\) 0 0
\(221\) 1.41421 0.0951303
\(222\) 5.65685 0.379663
\(223\) −22.0711 −1.47799 −0.738994 0.673712i \(-0.764699\pi\)
−0.738994 + 0.673712i \(0.764699\pi\)
\(224\) 0 0
\(225\) −4.00000 −0.266667
\(226\) 21.0711 1.40163
\(227\) −10.0000 −0.663723 −0.331862 0.943328i \(-0.607677\pi\)
−0.331862 + 0.943328i \(0.607677\pi\)
\(228\) 0 0
\(229\) 18.4853 1.22154 0.610771 0.791807i \(-0.290860\pi\)
0.610771 + 0.791807i \(0.290860\pi\)
\(230\) 0.585786 0.0386256
\(231\) 0 0
\(232\) 3.51472 0.230753
\(233\) −29.7279 −1.94754 −0.973770 0.227533i \(-0.926934\pi\)
−0.973770 + 0.227533i \(0.926934\pi\)
\(234\) −1.41421 −0.0924500
\(235\) 4.65685 0.303780
\(236\) 0 0
\(237\) 1.48528 0.0964794
\(238\) 0 0
\(239\) 4.58579 0.296630 0.148315 0.988940i \(-0.452615\pi\)
0.148315 + 0.988940i \(0.452615\pi\)
\(240\) 4.00000 0.258199
\(241\) −18.0711 −1.16406 −0.582030 0.813167i \(-0.697741\pi\)
−0.582030 + 0.813167i \(0.697741\pi\)
\(242\) 9.89949 0.636364
\(243\) 1.00000 0.0641500
\(244\) 0 0
\(245\) 0 0
\(246\) −7.31371 −0.466305
\(247\) −3.58579 −0.228158
\(248\) −1.17157 −0.0743950
\(249\) −5.48528 −0.347616
\(250\) −12.7279 −0.804984
\(251\) 1.51472 0.0956082 0.0478041 0.998857i \(-0.484778\pi\)
0.0478041 + 0.998857i \(0.484778\pi\)
\(252\) 0 0
\(253\) −0.828427 −0.0520828
\(254\) 7.51472 0.471515
\(255\) −1.41421 −0.0885615
\(256\) 0 0
\(257\) 18.4853 1.15308 0.576540 0.817069i \(-0.304402\pi\)
0.576540 + 0.817069i \(0.304402\pi\)
\(258\) −5.41421 −0.337074
\(259\) 0 0
\(260\) 0 0
\(261\) 1.24264 0.0769175
\(262\) −17.3137 −1.06964
\(263\) 7.58579 0.467760 0.233880 0.972266i \(-0.424858\pi\)
0.233880 + 0.972266i \(0.424858\pi\)
\(264\) −5.65685 −0.348155
\(265\) −2.41421 −0.148304
\(266\) 0 0
\(267\) 2.65685 0.162597
\(268\) 0 0
\(269\) −13.2132 −0.805623 −0.402812 0.915283i \(-0.631967\pi\)
−0.402812 + 0.915283i \(0.631967\pi\)
\(270\) 1.41421 0.0860663
\(271\) 2.97056 0.180449 0.0902244 0.995921i \(-0.471242\pi\)
0.0902244 + 0.995921i \(0.471242\pi\)
\(272\) −5.65685 −0.342997
\(273\) 0 0
\(274\) 4.97056 0.300283
\(275\) 8.00000 0.482418
\(276\) 0 0
\(277\) −23.6274 −1.41963 −0.709817 0.704386i \(-0.751222\pi\)
−0.709817 + 0.704386i \(0.751222\pi\)
\(278\) 15.5147 0.930511
\(279\) −0.414214 −0.0247983
\(280\) 0 0
\(281\) −14.5858 −0.870115 −0.435058 0.900403i \(-0.643272\pi\)
−0.435058 + 0.900403i \(0.643272\pi\)
\(282\) 6.58579 0.392178
\(283\) −1.17157 −0.0696428 −0.0348214 0.999394i \(-0.511086\pi\)
−0.0348214 + 0.999394i \(0.511086\pi\)
\(284\) 0 0
\(285\) 3.58579 0.212404
\(286\) 2.82843 0.167248
\(287\) 0 0
\(288\) 0 0
\(289\) −15.0000 −0.882353
\(290\) 1.75736 0.103196
\(291\) −16.8995 −0.990666
\(292\) 0 0
\(293\) 10.6569 0.622580 0.311290 0.950315i \(-0.399239\pi\)
0.311290 + 0.950315i \(0.399239\pi\)
\(294\) 0 0
\(295\) −7.65685 −0.445799
\(296\) −11.3137 −0.657596
\(297\) −2.00000 −0.116052
\(298\) 28.4853 1.65011
\(299\) 0.414214 0.0239546
\(300\) 0 0
\(301\) 0 0
\(302\) 15.3137 0.881205
\(303\) 7.65685 0.439875
\(304\) 14.3431 0.822636
\(305\) 11.6569 0.667470
\(306\) −2.00000 −0.114332
\(307\) −18.4142 −1.05095 −0.525477 0.850808i \(-0.676113\pi\)
−0.525477 + 0.850808i \(0.676113\pi\)
\(308\) 0 0
\(309\) 4.48528 0.255159
\(310\) −0.585786 −0.0332704
\(311\) −7.41421 −0.420421 −0.210211 0.977656i \(-0.567415\pi\)
−0.210211 + 0.977656i \(0.567415\pi\)
\(312\) 2.82843 0.160128
\(313\) 25.7990 1.45825 0.729123 0.684383i \(-0.239928\pi\)
0.729123 + 0.684383i \(0.239928\pi\)
\(314\) 3.17157 0.178982
\(315\) 0 0
\(316\) 0 0
\(317\) 17.7574 0.997353 0.498676 0.866788i \(-0.333820\pi\)
0.498676 + 0.866788i \(0.333820\pi\)
\(318\) −3.41421 −0.191460
\(319\) −2.48528 −0.139149
\(320\) −8.00000 −0.447214
\(321\) −16.4853 −0.920119
\(322\) 0 0
\(323\) −5.07107 −0.282162
\(324\) 0 0
\(325\) −4.00000 −0.221880
\(326\) −5.31371 −0.294299
\(327\) −17.0711 −0.944032
\(328\) 14.6274 0.807664
\(329\) 0 0
\(330\) −2.82843 −0.155700
\(331\) −14.9706 −0.822857 −0.411428 0.911442i \(-0.634970\pi\)
−0.411428 + 0.911442i \(0.634970\pi\)
\(332\) 0 0
\(333\) −4.00000 −0.219199
\(334\) −25.4142 −1.39060
\(335\) 4.58579 0.250548
\(336\) 0 0
\(337\) 28.7990 1.56878 0.784390 0.620267i \(-0.212976\pi\)
0.784390 + 0.620267i \(0.212976\pi\)
\(338\) −1.41421 −0.0769231
\(339\) −14.8995 −0.809229
\(340\) 0 0
\(341\) 0.828427 0.0448618
\(342\) 5.07107 0.274212
\(343\) 0 0
\(344\) 10.8284 0.583830
\(345\) −0.414214 −0.0223005
\(346\) −20.9706 −1.12738
\(347\) 21.4558 1.15181 0.575905 0.817517i \(-0.304650\pi\)
0.575905 + 0.817517i \(0.304650\pi\)
\(348\) 0 0
\(349\) 19.0416 1.01928 0.509638 0.860389i \(-0.329779\pi\)
0.509638 + 0.860389i \(0.329779\pi\)
\(350\) 0 0
\(351\) 1.00000 0.0533761
\(352\) 0 0
\(353\) −15.7990 −0.840895 −0.420448 0.907317i \(-0.638127\pi\)
−0.420448 + 0.907317i \(0.638127\pi\)
\(354\) −10.8284 −0.575524
\(355\) 6.82843 0.362415
\(356\) 0 0
\(357\) 0 0
\(358\) −18.0416 −0.953530
\(359\) 14.7279 0.777310 0.388655 0.921383i \(-0.372940\pi\)
0.388655 + 0.921383i \(0.372940\pi\)
\(360\) −2.82843 −0.149071
\(361\) −6.14214 −0.323270
\(362\) −12.4853 −0.656212
\(363\) −7.00000 −0.367405
\(364\) 0 0
\(365\) 5.58579 0.292373
\(366\) 16.4853 0.861699
\(367\) 1.41421 0.0738213 0.0369107 0.999319i \(-0.488248\pi\)
0.0369107 + 0.999319i \(0.488248\pi\)
\(368\) −1.65685 −0.0863695
\(369\) 5.17157 0.269221
\(370\) −5.65685 −0.294086
\(371\) 0 0
\(372\) 0 0
\(373\) 29.1716 1.51045 0.755223 0.655467i \(-0.227528\pi\)
0.755223 + 0.655467i \(0.227528\pi\)
\(374\) 4.00000 0.206835
\(375\) 9.00000 0.464758
\(376\) −13.1716 −0.679272
\(377\) 1.24264 0.0639993
\(378\) 0 0
\(379\) −8.34315 −0.428559 −0.214279 0.976772i \(-0.568740\pi\)
−0.214279 + 0.976772i \(0.568740\pi\)
\(380\) 0 0
\(381\) −5.31371 −0.272230
\(382\) −8.97056 −0.458974
\(383\) 2.97056 0.151789 0.0758943 0.997116i \(-0.475819\pi\)
0.0758943 + 0.997116i \(0.475819\pi\)
\(384\) −11.3137 −0.577350
\(385\) 0 0
\(386\) 6.48528 0.330092
\(387\) 3.82843 0.194610
\(388\) 0 0
\(389\) 10.4853 0.531625 0.265812 0.964025i \(-0.414360\pi\)
0.265812 + 0.964025i \(0.414360\pi\)
\(390\) 1.41421 0.0716115
\(391\) 0.585786 0.0296245
\(392\) 0 0
\(393\) 12.2426 0.617560
\(394\) 29.4558 1.48396
\(395\) −1.48528 −0.0747326
\(396\) 0 0
\(397\) −12.4142 −0.623052 −0.311526 0.950238i \(-0.600840\pi\)
−0.311526 + 0.950238i \(0.600840\pi\)
\(398\) 10.4853 0.525580
\(399\) 0 0
\(400\) 16.0000 0.800000
\(401\) −20.5858 −1.02801 −0.514003 0.857789i \(-0.671838\pi\)
−0.514003 + 0.857789i \(0.671838\pi\)
\(402\) 6.48528 0.323456
\(403\) −0.414214 −0.0206334
\(404\) 0 0
\(405\) −1.00000 −0.0496904
\(406\) 0 0
\(407\) 8.00000 0.396545
\(408\) 4.00000 0.198030
\(409\) 15.7279 0.777696 0.388848 0.921302i \(-0.372873\pi\)
0.388848 + 0.921302i \(0.372873\pi\)
\(410\) 7.31371 0.361198
\(411\) −3.51472 −0.173368
\(412\) 0 0
\(413\) 0 0
\(414\) −0.585786 −0.0287898
\(415\) 5.48528 0.269262
\(416\) 0 0
\(417\) −10.9706 −0.537231
\(418\) −10.1421 −0.496068
\(419\) 16.1421 0.788595 0.394297 0.918983i \(-0.370988\pi\)
0.394297 + 0.918983i \(0.370988\pi\)
\(420\) 0 0
\(421\) 5.21320 0.254076 0.127038 0.991898i \(-0.459453\pi\)
0.127038 + 0.991898i \(0.459453\pi\)
\(422\) 10.1005 0.491685
\(423\) −4.65685 −0.226424
\(424\) 6.82843 0.331618
\(425\) −5.65685 −0.274398
\(426\) 9.65685 0.467876
\(427\) 0 0
\(428\) 0 0
\(429\) −2.00000 −0.0965609
\(430\) 5.41421 0.261097
\(431\) −26.9706 −1.29913 −0.649563 0.760308i \(-0.725048\pi\)
−0.649563 + 0.760308i \(0.725048\pi\)
\(432\) −4.00000 −0.192450
\(433\) 18.7279 0.900006 0.450003 0.893027i \(-0.351423\pi\)
0.450003 + 0.893027i \(0.351423\pi\)
\(434\) 0 0
\(435\) −1.24264 −0.0595801
\(436\) 0 0
\(437\) −1.48528 −0.0710506
\(438\) 7.89949 0.377452
\(439\) 38.2843 1.82721 0.913604 0.406604i \(-0.133287\pi\)
0.913604 + 0.406604i \(0.133287\pi\)
\(440\) 5.65685 0.269680
\(441\) 0 0
\(442\) −2.00000 −0.0951303
\(443\) −3.24264 −0.154063 −0.0770313 0.997029i \(-0.524544\pi\)
−0.0770313 + 0.997029i \(0.524544\pi\)
\(444\) 0 0
\(445\) −2.65685 −0.125947
\(446\) 31.2132 1.47799
\(447\) −20.1421 −0.952690
\(448\) 0 0
\(449\) 1.55635 0.0734487 0.0367243 0.999325i \(-0.488308\pi\)
0.0367243 + 0.999325i \(0.488308\pi\)
\(450\) 5.65685 0.266667
\(451\) −10.3431 −0.487040
\(452\) 0 0
\(453\) −10.8284 −0.508764
\(454\) 14.1421 0.663723
\(455\) 0 0
\(456\) −10.1421 −0.474949
\(457\) 39.0711 1.82767 0.913834 0.406089i \(-0.133108\pi\)
0.913834 + 0.406089i \(0.133108\pi\)
\(458\) −26.1421 −1.22154
\(459\) 1.41421 0.0660098
\(460\) 0 0
\(461\) 10.3431 0.481728 0.240864 0.970559i \(-0.422569\pi\)
0.240864 + 0.970559i \(0.422569\pi\)
\(462\) 0 0
\(463\) −10.9289 −0.507911 −0.253955 0.967216i \(-0.581732\pi\)
−0.253955 + 0.967216i \(0.581732\pi\)
\(464\) −4.97056 −0.230753
\(465\) 0.414214 0.0192087
\(466\) 42.0416 1.94754
\(467\) 5.55635 0.257117 0.128559 0.991702i \(-0.458965\pi\)
0.128559 + 0.991702i \(0.458965\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) −6.58579 −0.303780
\(471\) −2.24264 −0.103335
\(472\) 21.6569 0.996838
\(473\) −7.65685 −0.352063
\(474\) −2.10051 −0.0964794
\(475\) 14.3431 0.658109
\(476\) 0 0
\(477\) 2.41421 0.110539
\(478\) −6.48528 −0.296630
\(479\) −8.79899 −0.402036 −0.201018 0.979588i \(-0.564425\pi\)
−0.201018 + 0.979588i \(0.564425\pi\)
\(480\) 0 0
\(481\) −4.00000 −0.182384
\(482\) 25.5563 1.16406
\(483\) 0 0
\(484\) 0 0
\(485\) 16.8995 0.767367
\(486\) −1.41421 −0.0641500
\(487\) 37.5980 1.70373 0.851864 0.523764i \(-0.175473\pi\)
0.851864 + 0.523764i \(0.175473\pi\)
\(488\) −32.9706 −1.49251
\(489\) 3.75736 0.169914
\(490\) 0 0
\(491\) 26.9706 1.21716 0.608582 0.793491i \(-0.291739\pi\)
0.608582 + 0.793491i \(0.291739\pi\)
\(492\) 0 0
\(493\) 1.75736 0.0791475
\(494\) 5.07107 0.228158
\(495\) 2.00000 0.0898933
\(496\) 1.65685 0.0743950
\(497\) 0 0
\(498\) 7.75736 0.347616
\(499\) 38.1838 1.70934 0.854670 0.519172i \(-0.173759\pi\)
0.854670 + 0.519172i \(0.173759\pi\)
\(500\) 0 0
\(501\) 17.9706 0.802866
\(502\) −2.14214 −0.0956082
\(503\) −35.9411 −1.60254 −0.801268 0.598306i \(-0.795841\pi\)
−0.801268 + 0.598306i \(0.795841\pi\)
\(504\) 0 0
\(505\) −7.65685 −0.340726
\(506\) 1.17157 0.0520828
\(507\) 1.00000 0.0444116
\(508\) 0 0
\(509\) 1.14214 0.0506243 0.0253121 0.999680i \(-0.491942\pi\)
0.0253121 + 0.999680i \(0.491942\pi\)
\(510\) 2.00000 0.0885615
\(511\) 0 0
\(512\) 22.6274 1.00000
\(513\) −3.58579 −0.158316
\(514\) −26.1421 −1.15308
\(515\) −4.48528 −0.197645
\(516\) 0 0
\(517\) 9.31371 0.409616
\(518\) 0 0
\(519\) 14.8284 0.650896
\(520\) −2.82843 −0.124035
\(521\) 4.00000 0.175243 0.0876216 0.996154i \(-0.472073\pi\)
0.0876216 + 0.996154i \(0.472073\pi\)
\(522\) −1.75736 −0.0769175
\(523\) 27.9411 1.22178 0.610890 0.791715i \(-0.290812\pi\)
0.610890 + 0.791715i \(0.290812\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) −10.7279 −0.467760
\(527\) −0.585786 −0.0255173
\(528\) 8.00000 0.348155
\(529\) −22.8284 −0.992540
\(530\) 3.41421 0.148304
\(531\) 7.65685 0.332279
\(532\) 0 0
\(533\) 5.17157 0.224006
\(534\) −3.75736 −0.162597
\(535\) 16.4853 0.712721
\(536\) −12.9706 −0.560243
\(537\) 12.7574 0.550521
\(538\) 18.6863 0.805623
\(539\) 0 0
\(540\) 0 0
\(541\) −11.3137 −0.486414 −0.243207 0.969974i \(-0.578199\pi\)
−0.243207 + 0.969974i \(0.578199\pi\)
\(542\) −4.20101 −0.180449
\(543\) 8.82843 0.378864
\(544\) 0 0
\(545\) 17.0711 0.731244
\(546\) 0 0
\(547\) 36.1127 1.54407 0.772034 0.635582i \(-0.219240\pi\)
0.772034 + 0.635582i \(0.219240\pi\)
\(548\) 0 0
\(549\) −11.6569 −0.497502
\(550\) −11.3137 −0.482418
\(551\) −4.45584 −0.189825
\(552\) 1.17157 0.0498655
\(553\) 0 0
\(554\) 33.4142 1.41963
\(555\) 4.00000 0.169791
\(556\) 0 0
\(557\) 39.1127 1.65726 0.828629 0.559798i \(-0.189121\pi\)
0.828629 + 0.559798i \(0.189121\pi\)
\(558\) 0.585786 0.0247983
\(559\) 3.82843 0.161925
\(560\) 0 0
\(561\) −2.82843 −0.119416
\(562\) 20.6274 0.870115
\(563\) 29.4142 1.23966 0.619831 0.784736i \(-0.287201\pi\)
0.619831 + 0.784736i \(0.287201\pi\)
\(564\) 0 0
\(565\) 14.8995 0.626826
\(566\) 1.65685 0.0696428
\(567\) 0 0
\(568\) −19.3137 −0.810385
\(569\) 27.5269 1.15399 0.576994 0.816748i \(-0.304226\pi\)
0.576994 + 0.816748i \(0.304226\pi\)
\(570\) −5.07107 −0.212404
\(571\) −0.798990 −0.0334367 −0.0167183 0.999860i \(-0.505322\pi\)
−0.0167183 + 0.999860i \(0.505322\pi\)
\(572\) 0 0
\(573\) 6.34315 0.264989
\(574\) 0 0
\(575\) −1.65685 −0.0690956
\(576\) 8.00000 0.333333
\(577\) 10.6863 0.444876 0.222438 0.974947i \(-0.428598\pi\)
0.222438 + 0.974947i \(0.428598\pi\)
\(578\) 21.2132 0.882353
\(579\) −4.58579 −0.190579
\(580\) 0 0
\(581\) 0 0
\(582\) 23.8995 0.990666
\(583\) −4.82843 −0.199973
\(584\) −15.7990 −0.653767
\(585\) −1.00000 −0.0413449
\(586\) −15.0711 −0.622580
\(587\) −42.9411 −1.77237 −0.886185 0.463332i \(-0.846654\pi\)
−0.886185 + 0.463332i \(0.846654\pi\)
\(588\) 0 0
\(589\) 1.48528 0.0612000
\(590\) 10.8284 0.445799
\(591\) −20.8284 −0.856767
\(592\) 16.0000 0.657596
\(593\) −11.9706 −0.491572 −0.245786 0.969324i \(-0.579046\pi\)
−0.245786 + 0.969324i \(0.579046\pi\)
\(594\) 2.82843 0.116052
\(595\) 0 0
\(596\) 0 0
\(597\) −7.41421 −0.303444
\(598\) −0.585786 −0.0239546
\(599\) −4.89949 −0.200188 −0.100094 0.994978i \(-0.531914\pi\)
−0.100094 + 0.994978i \(0.531914\pi\)
\(600\) −11.3137 −0.461880
\(601\) 45.2132 1.84429 0.922143 0.386850i \(-0.126437\pi\)
0.922143 + 0.386850i \(0.126437\pi\)
\(602\) 0 0
\(603\) −4.58579 −0.186748
\(604\) 0 0
\(605\) 7.00000 0.284590
\(606\) −10.8284 −0.439875
\(607\) −29.3137 −1.18981 −0.594903 0.803797i \(-0.702810\pi\)
−0.594903 + 0.803797i \(0.702810\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) −16.4853 −0.667470
\(611\) −4.65685 −0.188396
\(612\) 0 0
\(613\) 13.6985 0.553277 0.276638 0.960974i \(-0.410780\pi\)
0.276638 + 0.960974i \(0.410780\pi\)
\(614\) 26.0416 1.05095
\(615\) −5.17157 −0.208538
\(616\) 0 0
\(617\) −0.142136 −0.00572216 −0.00286108 0.999996i \(-0.500911\pi\)
−0.00286108 + 0.999996i \(0.500911\pi\)
\(618\) −6.34315 −0.255159
\(619\) 23.7990 0.956562 0.478281 0.878207i \(-0.341260\pi\)
0.478281 + 0.878207i \(0.341260\pi\)
\(620\) 0 0
\(621\) 0.414214 0.0166218
\(622\) 10.4853 0.420421
\(623\) 0 0
\(624\) −4.00000 −0.160128
\(625\) 11.0000 0.440000
\(626\) −36.4853 −1.45825
\(627\) 7.17157 0.286405
\(628\) 0 0
\(629\) −5.65685 −0.225554
\(630\) 0 0
\(631\) −45.7990 −1.82323 −0.911614 0.411046i \(-0.865163\pi\)
−0.911614 + 0.411046i \(0.865163\pi\)
\(632\) 4.20101 0.167107
\(633\) −7.14214 −0.283874
\(634\) −25.1127 −0.997353
\(635\) 5.31371 0.210868
\(636\) 0 0
\(637\) 0 0
\(638\) 3.51472 0.139149
\(639\) −6.82843 −0.270128
\(640\) 11.3137 0.447214
\(641\) −38.0711 −1.50372 −0.751858 0.659325i \(-0.770842\pi\)
−0.751858 + 0.659325i \(0.770842\pi\)
\(642\) 23.3137 0.920119
\(643\) −33.1716 −1.30816 −0.654080 0.756426i \(-0.726944\pi\)
−0.654080 + 0.756426i \(0.726944\pi\)
\(644\) 0 0
\(645\) −3.82843 −0.150744
\(646\) 7.17157 0.282162
\(647\) 38.2426 1.50347 0.751737 0.659463i \(-0.229216\pi\)
0.751737 + 0.659463i \(0.229216\pi\)
\(648\) 2.82843 0.111111
\(649\) −15.3137 −0.601116
\(650\) 5.65685 0.221880
\(651\) 0 0
\(652\) 0 0
\(653\) −40.8284 −1.59774 −0.798870 0.601504i \(-0.794568\pi\)
−0.798870 + 0.601504i \(0.794568\pi\)
\(654\) 24.1421 0.944032
\(655\) −12.2426 −0.478360
\(656\) −20.6863 −0.807664
\(657\) −5.58579 −0.217922
\(658\) 0 0
\(659\) −25.2426 −0.983314 −0.491657 0.870789i \(-0.663609\pi\)
−0.491657 + 0.870789i \(0.663609\pi\)
\(660\) 0 0
\(661\) −12.8995 −0.501732 −0.250866 0.968022i \(-0.580715\pi\)
−0.250866 + 0.968022i \(0.580715\pi\)
\(662\) 21.1716 0.822857
\(663\) 1.41421 0.0549235
\(664\) −15.5147 −0.602088
\(665\) 0 0
\(666\) 5.65685 0.219199
\(667\) 0.514719 0.0199300
\(668\) 0 0
\(669\) −22.0711 −0.853317
\(670\) −6.48528 −0.250548
\(671\) 23.3137 0.900016
\(672\) 0 0
\(673\) −27.8284 −1.07271 −0.536354 0.843993i \(-0.680199\pi\)
−0.536354 + 0.843993i \(0.680199\pi\)
\(674\) −40.7279 −1.56878
\(675\) −4.00000 −0.153960
\(676\) 0 0
\(677\) −23.6985 −0.910807 −0.455403 0.890285i \(-0.650505\pi\)
−0.455403 + 0.890285i \(0.650505\pi\)
\(678\) 21.0711 0.809229
\(679\) 0 0
\(680\) −4.00000 −0.153393
\(681\) −10.0000 −0.383201
\(682\) −1.17157 −0.0448618
\(683\) 36.9706 1.41464 0.707320 0.706894i \(-0.249904\pi\)
0.707320 + 0.706894i \(0.249904\pi\)
\(684\) 0 0
\(685\) 3.51472 0.134290
\(686\) 0 0
\(687\) 18.4853 0.705257
\(688\) −15.3137 −0.583830
\(689\) 2.41421 0.0919742
\(690\) 0.585786 0.0223005
\(691\) −21.8701 −0.831976 −0.415988 0.909370i \(-0.636564\pi\)
−0.415988 + 0.909370i \(0.636564\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) −30.3431 −1.15181
\(695\) 10.9706 0.416137
\(696\) 3.51472 0.133225
\(697\) 7.31371 0.277026
\(698\) −26.9289 −1.01928
\(699\) −29.7279 −1.12441
\(700\) 0 0
\(701\) −17.0416 −0.643654 −0.321827 0.946799i \(-0.604297\pi\)
−0.321827 + 0.946799i \(0.604297\pi\)
\(702\) −1.41421 −0.0533761
\(703\) 14.3431 0.540962
\(704\) −16.0000 −0.603023
\(705\) 4.65685 0.175387
\(706\) 22.3431 0.840895
\(707\) 0 0
\(708\) 0 0
\(709\) −17.1127 −0.642681 −0.321340 0.946964i \(-0.604133\pi\)
−0.321340 + 0.946964i \(0.604133\pi\)
\(710\) −9.65685 −0.362415
\(711\) 1.48528 0.0557024
\(712\) 7.51472 0.281626
\(713\) −0.171573 −0.00642545
\(714\) 0 0
\(715\) 2.00000 0.0747958
\(716\) 0 0
\(717\) 4.58579 0.171259
\(718\) −20.8284 −0.777310
\(719\) 43.8995 1.63717 0.818587 0.574382i \(-0.194758\pi\)
0.818587 + 0.574382i \(0.194758\pi\)
\(720\) 4.00000 0.149071
\(721\) 0 0
\(722\) 8.68629 0.323270
\(723\) −18.0711 −0.672070
\(724\) 0 0
\(725\) −4.97056 −0.184602
\(726\) 9.89949 0.367405
\(727\) −20.2426 −0.750758 −0.375379 0.926871i \(-0.622487\pi\)
−0.375379 + 0.926871i \(0.622487\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) −7.89949 −0.292373
\(731\) 5.41421 0.200252
\(732\) 0 0
\(733\) 28.0711 1.03683 0.518414 0.855130i \(-0.326523\pi\)
0.518414 + 0.855130i \(0.326523\pi\)
\(734\) −2.00000 −0.0738213
\(735\) 0 0
\(736\) 0 0
\(737\) 9.17157 0.337839
\(738\) −7.31371 −0.269221
\(739\) 28.1421 1.03523 0.517613 0.855615i \(-0.326821\pi\)
0.517613 + 0.855615i \(0.326821\pi\)
\(740\) 0 0
\(741\) −3.58579 −0.131727
\(742\) 0 0
\(743\) −23.4142 −0.858984 −0.429492 0.903071i \(-0.641307\pi\)
−0.429492 + 0.903071i \(0.641307\pi\)
\(744\) −1.17157 −0.0429519
\(745\) 20.1421 0.737951
\(746\) −41.2548 −1.51045
\(747\) −5.48528 −0.200696
\(748\) 0 0
\(749\) 0 0
\(750\) −12.7279 −0.464758
\(751\) 10.3137 0.376353 0.188176 0.982135i \(-0.439742\pi\)
0.188176 + 0.982135i \(0.439742\pi\)
\(752\) 18.6274 0.679272
\(753\) 1.51472 0.0551994
\(754\) −1.75736 −0.0639993
\(755\) 10.8284 0.394087
\(756\) 0 0
\(757\) 7.48528 0.272057 0.136029 0.990705i \(-0.456566\pi\)
0.136029 + 0.990705i \(0.456566\pi\)
\(758\) 11.7990 0.428559
\(759\) −0.828427 −0.0300700
\(760\) 10.1421 0.367894
\(761\) −17.1421 −0.621402 −0.310701 0.950508i \(-0.600564\pi\)
−0.310701 + 0.950508i \(0.600564\pi\)
\(762\) 7.51472 0.272230
\(763\) 0 0
\(764\) 0 0
\(765\) −1.41421 −0.0511310
\(766\) −4.20101 −0.151789
\(767\) 7.65685 0.276473
\(768\) 0 0
\(769\) −27.2426 −0.982395 −0.491197 0.871048i \(-0.663441\pi\)
−0.491197 + 0.871048i \(0.663441\pi\)
\(770\) 0 0
\(771\) 18.4853 0.665731
\(772\) 0 0
\(773\) −11.5147 −0.414156 −0.207078 0.978324i \(-0.566395\pi\)
−0.207078 + 0.978324i \(0.566395\pi\)
\(774\) −5.41421 −0.194610
\(775\) 1.65685 0.0595160
\(776\) −47.7990 −1.71588
\(777\) 0 0
\(778\) −14.8284 −0.531625
\(779\) −18.5442 −0.664413
\(780\) 0 0
\(781\) 13.6569 0.488681
\(782\) −0.828427 −0.0296245
\(783\) 1.24264 0.0444084
\(784\) 0 0
\(785\) 2.24264 0.0800433
\(786\) −17.3137 −0.617560
\(787\) −8.61522 −0.307100 −0.153550 0.988141i \(-0.549071\pi\)
−0.153550 + 0.988141i \(0.549071\pi\)
\(788\) 0 0
\(789\) 7.58579 0.270061
\(790\) 2.10051 0.0747326
\(791\) 0 0
\(792\) −5.65685 −0.201008
\(793\) −11.6569 −0.413947
\(794\) 17.5563 0.623052
\(795\) −2.41421 −0.0856233
\(796\) 0 0
\(797\) −9.27208 −0.328434 −0.164217 0.986424i \(-0.552510\pi\)
−0.164217 + 0.986424i \(0.552510\pi\)
\(798\) 0 0
\(799\) −6.58579 −0.232988
\(800\) 0 0
\(801\) 2.65685 0.0938753
\(802\) 29.1127 1.02801
\(803\) 11.1716 0.394236
\(804\) 0 0
\(805\) 0 0
\(806\) 0.585786 0.0206334
\(807\) −13.2132 −0.465127
\(808\) 21.6569 0.761885
\(809\) −2.12994 −0.0748848 −0.0374424 0.999299i \(-0.511921\pi\)
−0.0374424 + 0.999299i \(0.511921\pi\)
\(810\) 1.41421 0.0496904
\(811\) −21.9411 −0.770457 −0.385229 0.922821i \(-0.625877\pi\)
−0.385229 + 0.922821i \(0.625877\pi\)
\(812\) 0 0
\(813\) 2.97056 0.104182
\(814\) −11.3137 −0.396545
\(815\) −3.75736 −0.131615
\(816\) −5.65685 −0.198030
\(817\) −13.7279 −0.480279
\(818\) −22.2426 −0.777696
\(819\) 0 0
\(820\) 0 0
\(821\) 1.61522 0.0563717 0.0281858 0.999603i \(-0.491027\pi\)
0.0281858 + 0.999603i \(0.491027\pi\)
\(822\) 4.97056 0.173368
\(823\) −54.0833 −1.88522 −0.942612 0.333890i \(-0.891639\pi\)
−0.942612 + 0.333890i \(0.891639\pi\)
\(824\) 12.6863 0.441948
\(825\) 8.00000 0.278524
\(826\) 0 0
\(827\) −15.2132 −0.529015 −0.264507 0.964384i \(-0.585209\pi\)
−0.264507 + 0.964384i \(0.585209\pi\)
\(828\) 0 0
\(829\) 10.8701 0.377533 0.188766 0.982022i \(-0.439551\pi\)
0.188766 + 0.982022i \(0.439551\pi\)
\(830\) −7.75736 −0.269262
\(831\) −23.6274 −0.819626
\(832\) 8.00000 0.277350
\(833\) 0 0
\(834\) 15.5147 0.537231
\(835\) −17.9706 −0.621897
\(836\) 0 0
\(837\) −0.414214 −0.0143173
\(838\) −22.8284 −0.788595
\(839\) −53.5980 −1.85041 −0.925204 0.379470i \(-0.876106\pi\)
−0.925204 + 0.379470i \(0.876106\pi\)
\(840\) 0 0
\(841\) −27.4558 −0.946753
\(842\) −7.37258 −0.254076
\(843\) −14.5858 −0.502361
\(844\) 0 0
\(845\) −1.00000 −0.0344010
\(846\) 6.58579 0.226424
\(847\) 0 0
\(848\) −9.65685 −0.331618
\(849\) −1.17157 −0.0402083
\(850\) 8.00000 0.274398
\(851\) −1.65685 −0.0567962
\(852\) 0 0
\(853\) −0.129942 −0.00444914 −0.00222457 0.999998i \(-0.500708\pi\)
−0.00222457 + 0.999998i \(0.500708\pi\)
\(854\) 0 0
\(855\) 3.58579 0.122631
\(856\) −46.6274 −1.59369
\(857\) 23.1716 0.791526 0.395763 0.918353i \(-0.370480\pi\)
0.395763 + 0.918353i \(0.370480\pi\)
\(858\) 2.82843 0.0965609
\(859\) 52.7279 1.79905 0.899527 0.436866i \(-0.143912\pi\)
0.899527 + 0.436866i \(0.143912\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 38.1421 1.29913
\(863\) 45.9411 1.56385 0.781927 0.623370i \(-0.214237\pi\)
0.781927 + 0.623370i \(0.214237\pi\)
\(864\) 0 0
\(865\) −14.8284 −0.504182
\(866\) −26.4853 −0.900006
\(867\) −15.0000 −0.509427
\(868\) 0 0
\(869\) −2.97056 −0.100769
\(870\) 1.75736 0.0595801
\(871\) −4.58579 −0.155383
\(872\) −48.2843 −1.63511
\(873\) −16.8995 −0.571961
\(874\) 2.10051 0.0710506
\(875\) 0 0
\(876\) 0 0
\(877\) 18.3848 0.620810 0.310405 0.950604i \(-0.399535\pi\)
0.310405 + 0.950604i \(0.399535\pi\)
\(878\) −54.1421 −1.82721
\(879\) 10.6569 0.359447
\(880\) −8.00000 −0.269680
\(881\) 10.5269 0.354661 0.177330 0.984151i \(-0.443254\pi\)
0.177330 + 0.984151i \(0.443254\pi\)
\(882\) 0 0
\(883\) 33.6569 1.13264 0.566322 0.824184i \(-0.308366\pi\)
0.566322 + 0.824184i \(0.308366\pi\)
\(884\) 0 0
\(885\) −7.65685 −0.257382
\(886\) 4.58579 0.154063
\(887\) 40.2426 1.35122 0.675608 0.737261i \(-0.263881\pi\)
0.675608 + 0.737261i \(0.263881\pi\)
\(888\) −11.3137 −0.379663
\(889\) 0 0
\(890\) 3.75736 0.125947
\(891\) −2.00000 −0.0670025
\(892\) 0 0
\(893\) 16.6985 0.558793
\(894\) 28.4853 0.952690
\(895\) −12.7574 −0.426431
\(896\) 0 0
\(897\) 0.414214 0.0138302
\(898\) −2.20101 −0.0734487
\(899\) −0.514719 −0.0171668
\(900\) 0 0
\(901\) 3.41421 0.113744
\(902\) 14.6274 0.487040
\(903\) 0 0
\(904\) −42.1421 −1.40163
\(905\) −8.82843 −0.293467
\(906\) 15.3137 0.508764
\(907\) −13.2843 −0.441097 −0.220548 0.975376i \(-0.570785\pi\)
−0.220548 + 0.975376i \(0.570785\pi\)
\(908\) 0 0
\(909\) 7.65685 0.253962
\(910\) 0 0
\(911\) 14.6152 0.484224 0.242112 0.970248i \(-0.422160\pi\)
0.242112 + 0.970248i \(0.422160\pi\)
\(912\) 14.3431 0.474949
\(913\) 10.9706 0.363073
\(914\) −55.2548 −1.82767
\(915\) 11.6569 0.385364
\(916\) 0 0
\(917\) 0 0
\(918\) −2.00000 −0.0660098
\(919\) −17.1127 −0.564496 −0.282248 0.959342i \(-0.591080\pi\)
−0.282248 + 0.959342i \(0.591080\pi\)
\(920\) −1.17157 −0.0386256
\(921\) −18.4142 −0.606769
\(922\) −14.6274 −0.481728
\(923\) −6.82843 −0.224760
\(924\) 0 0
\(925\) 16.0000 0.526077
\(926\) 15.4558 0.507911
\(927\) 4.48528 0.147316
\(928\) 0 0
\(929\) −25.9706 −0.852067 −0.426033 0.904707i \(-0.640089\pi\)
−0.426033 + 0.904707i \(0.640089\pi\)
\(930\) −0.585786 −0.0192087
\(931\) 0 0
\(932\) 0 0
\(933\) −7.41421 −0.242730
\(934\) −7.85786 −0.257117
\(935\) 2.82843 0.0924995
\(936\) 2.82843 0.0924500
\(937\) −50.8284 −1.66049 −0.830246 0.557397i \(-0.811800\pi\)
−0.830246 + 0.557397i \(0.811800\pi\)
\(938\) 0 0
\(939\) 25.7990 0.841918
\(940\) 0 0
\(941\) −17.0000 −0.554184 −0.277092 0.960843i \(-0.589371\pi\)
−0.277092 + 0.960843i \(0.589371\pi\)
\(942\) 3.17157 0.103335
\(943\) 2.14214 0.0697575
\(944\) −30.6274 −0.996838
\(945\) 0 0
\(946\) 10.8284 0.352063
\(947\) 28.7279 0.933532 0.466766 0.884381i \(-0.345419\pi\)
0.466766 + 0.884381i \(0.345419\pi\)
\(948\) 0 0
\(949\) −5.58579 −0.181322
\(950\) −20.2843 −0.658109
\(951\) 17.7574 0.575822
\(952\) 0 0
\(953\) −31.9289 −1.03428 −0.517140 0.855901i \(-0.673003\pi\)
−0.517140 + 0.855901i \(0.673003\pi\)
\(954\) −3.41421 −0.110539
\(955\) −6.34315 −0.205259
\(956\) 0 0
\(957\) −2.48528 −0.0803377
\(958\) 12.4437 0.402036
\(959\) 0 0
\(960\) −8.00000 −0.258199
\(961\) −30.8284 −0.994465
\(962\) 5.65685 0.182384
\(963\) −16.4853 −0.531231
\(964\) 0 0
\(965\) 4.58579 0.147622
\(966\) 0 0
\(967\) 17.5147 0.563235 0.281618 0.959527i \(-0.409129\pi\)
0.281618 + 0.959527i \(0.409129\pi\)
\(968\) −19.7990 −0.636364
\(969\) −5.07107 −0.162906
\(970\) −23.8995 −0.767367
\(971\) −4.34315 −0.139378 −0.0696891 0.997569i \(-0.522201\pi\)
−0.0696891 + 0.997569i \(0.522201\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) −53.1716 −1.70373
\(975\) −4.00000 −0.128103
\(976\) 46.6274 1.49251
\(977\) 15.6569 0.500907 0.250454 0.968129i \(-0.419420\pi\)
0.250454 + 0.968129i \(0.419420\pi\)
\(978\) −5.31371 −0.169914
\(979\) −5.31371 −0.169827
\(980\) 0 0
\(981\) −17.0711 −0.545037
\(982\) −38.1421 −1.21716
\(983\) 34.4558 1.09897 0.549485 0.835503i \(-0.314824\pi\)
0.549485 + 0.835503i \(0.314824\pi\)
\(984\) 14.6274 0.466305
\(985\) 20.8284 0.663649
\(986\) −2.48528 −0.0791475
\(987\) 0 0
\(988\) 0 0
\(989\) 1.58579 0.0504251
\(990\) −2.82843 −0.0898933
\(991\) 47.9411 1.52290 0.761450 0.648224i \(-0.224488\pi\)
0.761450 + 0.648224i \(0.224488\pi\)
\(992\) 0 0
\(993\) −14.9706 −0.475076
\(994\) 0 0
\(995\) 7.41421 0.235046
\(996\) 0 0
\(997\) −51.1127 −1.61876 −0.809378 0.587288i \(-0.800195\pi\)
−0.809378 + 0.587288i \(0.800195\pi\)
\(998\) −54.0000 −1.70934
\(999\) −4.00000 −0.126554
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 1911.2.a.j.1.1 yes 2
3.2 odd 2 5733.2.a.r.1.2 2
7.6 odd 2 1911.2.a.i.1.1 2
21.20 even 2 5733.2.a.q.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1911.2.a.i.1.1 2 7.6 odd 2
1911.2.a.j.1.1 yes 2 1.1 even 1 trivial
5733.2.a.q.1.2 2 21.20 even 2
5733.2.a.r.1.2 2 3.2 odd 2