Properties

Label 2960.2.a.ba.1.1
Level $2960$
Weight $2$
Character 2960.1
Self dual yes
Analytic conductor $23.636$
Analytic rank $1$
Dimension $5$
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] = [2960,2,Mod(1,2960)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2960, 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("2960.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2960 = 2^{4} \cdot 5 \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2960.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: \(23.6357189983\)
Analytic rank: \(1\)
Dimension: \(5\)
Coefficient field: 5.5.368464.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{5} - 2x^{4} - 6x^{3} + 6x^{2} + 6x - 4 \) 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 185)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(3.14884\) of defining polynomial
Character \(\chi\) \(=\) 2960.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-2.51369 q^{3} +1.00000 q^{5} -0.281955 q^{7} +3.31863 q^{9} +O(q^{10})\) \(q-2.51369 q^{3} +1.00000 q^{5} -0.281955 q^{7} +3.31863 q^{9} +0.160490 q^{11} +1.61752 q^{13} -2.51369 q^{15} -2.88783 q^{17} -4.15216 q^{19} +0.708747 q^{21} +2.52202 q^{23} +1.00000 q^{25} -0.800936 q^{27} +2.70640 q^{29} -4.41915 q^{31} -0.403421 q^{33} -0.281955 q^{35} -1.00000 q^{37} -4.06595 q^{39} -11.1646 q^{41} +5.88897 q^{43} +3.31863 q^{45} +13.3827 q^{47} -6.92050 q^{49} +7.25911 q^{51} -6.86689 q^{53} +0.160490 q^{55} +10.4372 q^{57} +5.42986 q^{59} +13.3251 q^{61} -0.935704 q^{63} +1.61752 q^{65} -1.09956 q^{67} -6.33957 q^{69} -15.5022 q^{71} +0.701106 q^{73} -2.51369 q^{75} -0.0452508 q^{77} +1.24549 q^{79} -7.94259 q^{81} -0.157450 q^{83} -2.88783 q^{85} -6.80304 q^{87} -7.21783 q^{89} -0.456069 q^{91} +11.1084 q^{93} -4.15216 q^{95} +6.79246 q^{97} +0.532605 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5 q + q^{3} + 5 q^{5} - 7 q^{7} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 5 q + q^{3} + 5 q^{5} - 7 q^{7} + 2 q^{9} - 7 q^{11} + 2 q^{13} + q^{15} - 8 q^{17} - 14 q^{19} - 9 q^{21} - 2 q^{23} + 5 q^{25} + 7 q^{27} + 2 q^{29} - 8 q^{31} - 21 q^{33} - 7 q^{35} - 5 q^{37} - 12 q^{39} - 9 q^{41} - 14 q^{43} + 2 q^{45} + 5 q^{47} + 2 q^{49} - 10 q^{51} - 15 q^{53} - 7 q^{55} + 4 q^{57} - 12 q^{59} + 12 q^{61} - 24 q^{63} + 2 q^{65} + 2 q^{67} - 30 q^{69} - 13 q^{71} - 5 q^{73} + q^{75} - 7 q^{77} - 36 q^{79} + 21 q^{81} + 9 q^{83} - 8 q^{85} + 6 q^{87} - 16 q^{89} + 4 q^{91} - 16 q^{93} - 14 q^{95} - 18 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) −2.51369 −1.45128 −0.725639 0.688075i \(-0.758456\pi\)
−0.725639 + 0.688075i \(0.758456\pi\)
\(4\) 0 0
\(5\) 1.00000 0.447214
\(6\) 0 0
\(7\) −0.281955 −0.106569 −0.0532845 0.998579i \(-0.516969\pi\)
−0.0532845 + 0.998579i \(0.516969\pi\)
\(8\) 0 0
\(9\) 3.31863 1.10621
\(10\) 0 0
\(11\) 0.160490 0.0483894 0.0241947 0.999707i \(-0.492298\pi\)
0.0241947 + 0.999707i \(0.492298\pi\)
\(12\) 0 0
\(13\) 1.61752 0.448620 0.224310 0.974518i \(-0.427987\pi\)
0.224310 + 0.974518i \(0.427987\pi\)
\(14\) 0 0
\(15\) −2.51369 −0.649032
\(16\) 0 0
\(17\) −2.88783 −0.700402 −0.350201 0.936675i \(-0.613887\pi\)
−0.350201 + 0.936675i \(0.613887\pi\)
\(18\) 0 0
\(19\) −4.15216 −0.952570 −0.476285 0.879291i \(-0.658017\pi\)
−0.476285 + 0.879291i \(0.658017\pi\)
\(20\) 0 0
\(21\) 0.708747 0.154661
\(22\) 0 0
\(23\) 2.52202 0.525878 0.262939 0.964813i \(-0.415308\pi\)
0.262939 + 0.964813i \(0.415308\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) −0.800936 −0.154140
\(28\) 0 0
\(29\) 2.70640 0.502565 0.251283 0.967914i \(-0.419148\pi\)
0.251283 + 0.967914i \(0.419148\pi\)
\(30\) 0 0
\(31\) −4.41915 −0.793703 −0.396851 0.917883i \(-0.629897\pi\)
−0.396851 + 0.917883i \(0.629897\pi\)
\(32\) 0 0
\(33\) −0.403421 −0.0702265
\(34\) 0 0
\(35\) −0.281955 −0.0476591
\(36\) 0 0
\(37\) −1.00000 −0.164399
\(38\) 0 0
\(39\) −4.06595 −0.651073
\(40\) 0 0
\(41\) −11.1646 −1.74361 −0.871807 0.489850i \(-0.837051\pi\)
−0.871807 + 0.489850i \(0.837051\pi\)
\(42\) 0 0
\(43\) 5.88897 0.898060 0.449030 0.893517i \(-0.351770\pi\)
0.449030 + 0.893517i \(0.351770\pi\)
\(44\) 0 0
\(45\) 3.31863 0.494712
\(46\) 0 0
\(47\) 13.3827 1.95206 0.976032 0.217626i \(-0.0698311\pi\)
0.976032 + 0.217626i \(0.0698311\pi\)
\(48\) 0 0
\(49\) −6.92050 −0.988643
\(50\) 0 0
\(51\) 7.25911 1.01648
\(52\) 0 0
\(53\) −6.86689 −0.943240 −0.471620 0.881802i \(-0.656331\pi\)
−0.471620 + 0.881802i \(0.656331\pi\)
\(54\) 0 0
\(55\) 0.160490 0.0216404
\(56\) 0 0
\(57\) 10.4372 1.38244
\(58\) 0 0
\(59\) 5.42986 0.706908 0.353454 0.935452i \(-0.385007\pi\)
0.353454 + 0.935452i \(0.385007\pi\)
\(60\) 0 0
\(61\) 13.3251 1.70610 0.853050 0.521830i \(-0.174750\pi\)
0.853050 + 0.521830i \(0.174750\pi\)
\(62\) 0 0
\(63\) −0.935704 −0.117888
\(64\) 0 0
\(65\) 1.61752 0.200629
\(66\) 0 0
\(67\) −1.09956 −0.134332 −0.0671660 0.997742i \(-0.521396\pi\)
−0.0671660 + 0.997742i \(0.521396\pi\)
\(68\) 0 0
\(69\) −6.33957 −0.763195
\(70\) 0 0
\(71\) −15.5022 −1.83977 −0.919887 0.392183i \(-0.871720\pi\)
−0.919887 + 0.392183i \(0.871720\pi\)
\(72\) 0 0
\(73\) 0.701106 0.0820582 0.0410291 0.999158i \(-0.486936\pi\)
0.0410291 + 0.999158i \(0.486936\pi\)
\(74\) 0 0
\(75\) −2.51369 −0.290256
\(76\) 0 0
\(77\) −0.0452508 −0.00515681
\(78\) 0 0
\(79\) 1.24549 0.140128 0.0700640 0.997542i \(-0.477680\pi\)
0.0700640 + 0.997542i \(0.477680\pi\)
\(80\) 0 0
\(81\) −7.94259 −0.882510
\(82\) 0 0
\(83\) −0.157450 −0.0172823 −0.00864117 0.999963i \(-0.502751\pi\)
−0.00864117 + 0.999963i \(0.502751\pi\)
\(84\) 0 0
\(85\) −2.88783 −0.313229
\(86\) 0 0
\(87\) −6.80304 −0.729363
\(88\) 0 0
\(89\) −7.21783 −0.765089 −0.382544 0.923937i \(-0.624952\pi\)
−0.382544 + 0.923937i \(0.624952\pi\)
\(90\) 0 0
\(91\) −0.456069 −0.0478090
\(92\) 0 0
\(93\) 11.1084 1.15188
\(94\) 0 0
\(95\) −4.15216 −0.426002
\(96\) 0 0
\(97\) 6.79246 0.689669 0.344835 0.938663i \(-0.387935\pi\)
0.344835 + 0.938663i \(0.387935\pi\)
\(98\) 0 0
\(99\) 0.532605 0.0535288
\(100\) 0 0
\(101\) 5.41515 0.538827 0.269414 0.963025i \(-0.413170\pi\)
0.269414 + 0.963025i \(0.413170\pi\)
\(102\) 0 0
\(103\) 6.64784 0.655032 0.327516 0.944846i \(-0.393789\pi\)
0.327516 + 0.944846i \(0.393789\pi\)
\(104\) 0 0
\(105\) 0.708747 0.0691666
\(106\) 0 0
\(107\) −5.22219 −0.504848 −0.252424 0.967617i \(-0.581228\pi\)
−0.252424 + 0.967617i \(0.581228\pi\)
\(108\) 0 0
\(109\) −2.37345 −0.227336 −0.113668 0.993519i \(-0.536260\pi\)
−0.113668 + 0.993519i \(0.536260\pi\)
\(110\) 0 0
\(111\) 2.51369 0.238589
\(112\) 0 0
\(113\) 17.7642 1.67112 0.835559 0.549401i \(-0.185144\pi\)
0.835559 + 0.549401i \(0.185144\pi\)
\(114\) 0 0
\(115\) 2.52202 0.235180
\(116\) 0 0
\(117\) 5.36796 0.496268
\(118\) 0 0
\(119\) 0.814239 0.0746412
\(120\) 0 0
\(121\) −10.9742 −0.997658
\(122\) 0 0
\(123\) 28.0643 2.53047
\(124\) 0 0
\(125\) 1.00000 0.0894427
\(126\) 0 0
\(127\) −19.2934 −1.71202 −0.856008 0.516963i \(-0.827062\pi\)
−0.856008 + 0.516963i \(0.827062\pi\)
\(128\) 0 0
\(129\) −14.8030 −1.30334
\(130\) 0 0
\(131\) −14.8127 −1.29419 −0.647096 0.762408i \(-0.724017\pi\)
−0.647096 + 0.762408i \(0.724017\pi\)
\(132\) 0 0
\(133\) 1.17072 0.101514
\(134\) 0 0
\(135\) −0.800936 −0.0689336
\(136\) 0 0
\(137\) −22.5368 −1.92545 −0.962725 0.270482i \(-0.912817\pi\)
−0.962725 + 0.270482i \(0.912817\pi\)
\(138\) 0 0
\(139\) −14.3710 −1.21894 −0.609468 0.792811i \(-0.708617\pi\)
−0.609468 + 0.792811i \(0.708617\pi\)
\(140\) 0 0
\(141\) −33.6399 −2.83299
\(142\) 0 0
\(143\) 0.259596 0.0217085
\(144\) 0 0
\(145\) 2.70640 0.224754
\(146\) 0 0
\(147\) 17.3960 1.43480
\(148\) 0 0
\(149\) 0.782096 0.0640718 0.0320359 0.999487i \(-0.489801\pi\)
0.0320359 + 0.999487i \(0.489801\pi\)
\(150\) 0 0
\(151\) −2.52865 −0.205779 −0.102889 0.994693i \(-0.532809\pi\)
−0.102889 + 0.994693i \(0.532809\pi\)
\(152\) 0 0
\(153\) −9.58365 −0.774792
\(154\) 0 0
\(155\) −4.41915 −0.354955
\(156\) 0 0
\(157\) −19.4789 −1.55459 −0.777294 0.629138i \(-0.783408\pi\)
−0.777294 + 0.629138i \(0.783408\pi\)
\(158\) 0 0
\(159\) 17.2612 1.36890
\(160\) 0 0
\(161\) −0.711097 −0.0560423
\(162\) 0 0
\(163\) −16.1867 −1.26784 −0.633918 0.773400i \(-0.718555\pi\)
−0.633918 + 0.773400i \(0.718555\pi\)
\(164\) 0 0
\(165\) −0.403421 −0.0314063
\(166\) 0 0
\(167\) −13.3864 −1.03587 −0.517937 0.855419i \(-0.673300\pi\)
−0.517937 + 0.855419i \(0.673300\pi\)
\(168\) 0 0
\(169\) −10.3836 −0.798740
\(170\) 0 0
\(171\) −13.7795 −1.05374
\(172\) 0 0
\(173\) 21.3160 1.62062 0.810312 0.585999i \(-0.199298\pi\)
0.810312 + 0.585999i \(0.199298\pi\)
\(174\) 0 0
\(175\) −0.281955 −0.0213138
\(176\) 0 0
\(177\) −13.6490 −1.02592
\(178\) 0 0
\(179\) 0.279850 0.0209169 0.0104585 0.999945i \(-0.496671\pi\)
0.0104585 + 0.999945i \(0.496671\pi\)
\(180\) 0 0
\(181\) 10.1663 0.755652 0.377826 0.925877i \(-0.376672\pi\)
0.377826 + 0.925877i \(0.376672\pi\)
\(182\) 0 0
\(183\) −33.4951 −2.47603
\(184\) 0 0
\(185\) −1.00000 −0.0735215
\(186\) 0 0
\(187\) −0.463467 −0.0338920
\(188\) 0 0
\(189\) 0.225828 0.0164266
\(190\) 0 0
\(191\) 9.40464 0.680496 0.340248 0.940336i \(-0.389489\pi\)
0.340248 + 0.940336i \(0.389489\pi\)
\(192\) 0 0
\(193\) −12.2852 −0.884310 −0.442155 0.896939i \(-0.645786\pi\)
−0.442155 + 0.896939i \(0.645786\pi\)
\(194\) 0 0
\(195\) −4.06595 −0.291169
\(196\) 0 0
\(197\) −16.8525 −1.20069 −0.600346 0.799741i \(-0.704970\pi\)
−0.600346 + 0.799741i \(0.704970\pi\)
\(198\) 0 0
\(199\) −21.2801 −1.50851 −0.754254 0.656582i \(-0.772001\pi\)
−0.754254 + 0.656582i \(0.772001\pi\)
\(200\) 0 0
\(201\) 2.76394 0.194953
\(202\) 0 0
\(203\) −0.763083 −0.0535579
\(204\) 0 0
\(205\) −11.1646 −0.779768
\(206\) 0 0
\(207\) 8.36965 0.581731
\(208\) 0 0
\(209\) −0.666378 −0.0460943
\(210\) 0 0
\(211\) −5.24034 −0.360760 −0.180380 0.983597i \(-0.557733\pi\)
−0.180380 + 0.983597i \(0.557733\pi\)
\(212\) 0 0
\(213\) 38.9677 2.67003
\(214\) 0 0
\(215\) 5.88897 0.401625
\(216\) 0 0
\(217\) 1.24600 0.0845841
\(218\) 0 0
\(219\) −1.76236 −0.119089
\(220\) 0 0
\(221\) −4.67114 −0.314215
\(222\) 0 0
\(223\) 12.0192 0.804863 0.402432 0.915450i \(-0.368165\pi\)
0.402432 + 0.915450i \(0.368165\pi\)
\(224\) 0 0
\(225\) 3.31863 0.221242
\(226\) 0 0
\(227\) 10.8369 0.719272 0.359636 0.933093i \(-0.382901\pi\)
0.359636 + 0.933093i \(0.382901\pi\)
\(228\) 0 0
\(229\) −15.1312 −0.999901 −0.499950 0.866054i \(-0.666648\pi\)
−0.499950 + 0.866054i \(0.666648\pi\)
\(230\) 0 0
\(231\) 0.113746 0.00748397
\(232\) 0 0
\(233\) −20.1614 −1.32082 −0.660410 0.750905i \(-0.729617\pi\)
−0.660410 + 0.750905i \(0.729617\pi\)
\(234\) 0 0
\(235\) 13.3827 0.872990
\(236\) 0 0
\(237\) −3.13076 −0.203365
\(238\) 0 0
\(239\) −4.65751 −0.301270 −0.150635 0.988589i \(-0.548132\pi\)
−0.150635 + 0.988589i \(0.548132\pi\)
\(240\) 0 0
\(241\) 2.01197 0.129602 0.0648011 0.997898i \(-0.479359\pi\)
0.0648011 + 0.997898i \(0.479359\pi\)
\(242\) 0 0
\(243\) 22.3680 1.43491
\(244\) 0 0
\(245\) −6.92050 −0.442135
\(246\) 0 0
\(247\) −6.71621 −0.427342
\(248\) 0 0
\(249\) 0.395779 0.0250815
\(250\) 0 0
\(251\) −27.0033 −1.70443 −0.852217 0.523188i \(-0.824743\pi\)
−0.852217 + 0.523188i \(0.824743\pi\)
\(252\) 0 0
\(253\) 0.404758 0.0254469
\(254\) 0 0
\(255\) 7.25911 0.454583
\(256\) 0 0
\(257\) −9.85270 −0.614595 −0.307297 0.951614i \(-0.599425\pi\)
−0.307297 + 0.951614i \(0.599425\pi\)
\(258\) 0 0
\(259\) 0.281955 0.0175198
\(260\) 0 0
\(261\) 8.98153 0.555943
\(262\) 0 0
\(263\) −5.07097 −0.312689 −0.156345 0.987703i \(-0.549971\pi\)
−0.156345 + 0.987703i \(0.549971\pi\)
\(264\) 0 0
\(265\) −6.86689 −0.421830
\(266\) 0 0
\(267\) 18.1434 1.11036
\(268\) 0 0
\(269\) 21.6998 1.32306 0.661530 0.749919i \(-0.269908\pi\)
0.661530 + 0.749919i \(0.269908\pi\)
\(270\) 0 0
\(271\) −22.9216 −1.39239 −0.696195 0.717852i \(-0.745125\pi\)
−0.696195 + 0.717852i \(0.745125\pi\)
\(272\) 0 0
\(273\) 1.14642 0.0693842
\(274\) 0 0
\(275\) 0.160490 0.00967788
\(276\) 0 0
\(277\) −0.611022 −0.0367128 −0.0183564 0.999832i \(-0.505843\pi\)
−0.0183564 + 0.999832i \(0.505843\pi\)
\(278\) 0 0
\(279\) −14.6655 −0.878002
\(280\) 0 0
\(281\) 9.25171 0.551911 0.275955 0.961170i \(-0.411006\pi\)
0.275955 + 0.961170i \(0.411006\pi\)
\(282\) 0 0
\(283\) 22.9836 1.36623 0.683117 0.730309i \(-0.260624\pi\)
0.683117 + 0.730309i \(0.260624\pi\)
\(284\) 0 0
\(285\) 10.4372 0.618248
\(286\) 0 0
\(287\) 3.14791 0.185815
\(288\) 0 0
\(289\) −8.66043 −0.509437
\(290\) 0 0
\(291\) −17.0741 −1.00090
\(292\) 0 0
\(293\) 7.62462 0.445435 0.222717 0.974883i \(-0.428507\pi\)
0.222717 + 0.974883i \(0.428507\pi\)
\(294\) 0 0
\(295\) 5.42986 0.316139
\(296\) 0 0
\(297\) −0.128542 −0.00745875
\(298\) 0 0
\(299\) 4.07943 0.235919
\(300\) 0 0
\(301\) −1.66043 −0.0957053
\(302\) 0 0
\(303\) −13.6120 −0.781988
\(304\) 0 0
\(305\) 13.3251 0.762991
\(306\) 0 0
\(307\) 3.09745 0.176781 0.0883904 0.996086i \(-0.471828\pi\)
0.0883904 + 0.996086i \(0.471828\pi\)
\(308\) 0 0
\(309\) −16.7106 −0.950633
\(310\) 0 0
\(311\) 11.1070 0.629818 0.314909 0.949122i \(-0.398026\pi\)
0.314909 + 0.949122i \(0.398026\pi\)
\(312\) 0 0
\(313\) −1.80559 −0.102058 −0.0510289 0.998697i \(-0.516250\pi\)
−0.0510289 + 0.998697i \(0.516250\pi\)
\(314\) 0 0
\(315\) −0.935704 −0.0527210
\(316\) 0 0
\(317\) −2.53528 −0.142395 −0.0711977 0.997462i \(-0.522682\pi\)
−0.0711977 + 0.997462i \(0.522682\pi\)
\(318\) 0 0
\(319\) 0.434348 0.0243188
\(320\) 0 0
\(321\) 13.1270 0.732676
\(322\) 0 0
\(323\) 11.9907 0.667182
\(324\) 0 0
\(325\) 1.61752 0.0897241
\(326\) 0 0
\(327\) 5.96612 0.329927
\(328\) 0 0
\(329\) −3.77331 −0.208030
\(330\) 0 0
\(331\) 10.2147 0.561449 0.280724 0.959788i \(-0.409425\pi\)
0.280724 + 0.959788i \(0.409425\pi\)
\(332\) 0 0
\(333\) −3.31863 −0.181860
\(334\) 0 0
\(335\) −1.09956 −0.0600751
\(336\) 0 0
\(337\) −0.365331 −0.0199009 −0.00995043 0.999950i \(-0.503167\pi\)
−0.00995043 + 0.999950i \(0.503167\pi\)
\(338\) 0 0
\(339\) −44.6537 −2.42526
\(340\) 0 0
\(341\) −0.709227 −0.0384068
\(342\) 0 0
\(343\) 3.92496 0.211928
\(344\) 0 0
\(345\) −6.33957 −0.341311
\(346\) 0 0
\(347\) −19.5941 −1.05187 −0.525934 0.850526i \(-0.676284\pi\)
−0.525934 + 0.850526i \(0.676284\pi\)
\(348\) 0 0
\(349\) 31.7079 1.69729 0.848644 0.528965i \(-0.177420\pi\)
0.848644 + 0.528965i \(0.177420\pi\)
\(350\) 0 0
\(351\) −1.29553 −0.0691504
\(352\) 0 0
\(353\) −21.0549 −1.12064 −0.560319 0.828277i \(-0.689322\pi\)
−0.560319 + 0.828277i \(0.689322\pi\)
\(354\) 0 0
\(355\) −15.5022 −0.822772
\(356\) 0 0
\(357\) −2.04674 −0.108325
\(358\) 0 0
\(359\) −2.50782 −0.132358 −0.0661788 0.997808i \(-0.521081\pi\)
−0.0661788 + 0.997808i \(0.521081\pi\)
\(360\) 0 0
\(361\) −1.75959 −0.0926100
\(362\) 0 0
\(363\) 27.5858 1.44788
\(364\) 0 0
\(365\) 0.701106 0.0366975
\(366\) 0 0
\(367\) −35.5047 −1.85333 −0.926666 0.375886i \(-0.877338\pi\)
−0.926666 + 0.375886i \(0.877338\pi\)
\(368\) 0 0
\(369\) −37.0511 −1.92880
\(370\) 0 0
\(371\) 1.93615 0.100520
\(372\) 0 0
\(373\) 23.2671 1.20472 0.602362 0.798223i \(-0.294226\pi\)
0.602362 + 0.798223i \(0.294226\pi\)
\(374\) 0 0
\(375\) −2.51369 −0.129806
\(376\) 0 0
\(377\) 4.37766 0.225461
\(378\) 0 0
\(379\) −4.89632 −0.251507 −0.125754 0.992062i \(-0.540135\pi\)
−0.125754 + 0.992062i \(0.540135\pi\)
\(380\) 0 0
\(381\) 48.4977 2.48461
\(382\) 0 0
\(383\) 12.6187 0.644784 0.322392 0.946606i \(-0.395513\pi\)
0.322392 + 0.946606i \(0.395513\pi\)
\(384\) 0 0
\(385\) −0.0452508 −0.00230620
\(386\) 0 0
\(387\) 19.5433 0.993443
\(388\) 0 0
\(389\) −10.1847 −0.516386 −0.258193 0.966093i \(-0.583127\pi\)
−0.258193 + 0.966093i \(0.583127\pi\)
\(390\) 0 0
\(391\) −7.28317 −0.368326
\(392\) 0 0
\(393\) 37.2345 1.87823
\(394\) 0 0
\(395\) 1.24549 0.0626672
\(396\) 0 0
\(397\) −13.1107 −0.658008 −0.329004 0.944329i \(-0.606713\pi\)
−0.329004 + 0.944329i \(0.606713\pi\)
\(398\) 0 0
\(399\) −2.94283 −0.147326
\(400\) 0 0
\(401\) 17.6608 0.881940 0.440970 0.897522i \(-0.354635\pi\)
0.440970 + 0.897522i \(0.354635\pi\)
\(402\) 0 0
\(403\) −7.14808 −0.356071
\(404\) 0 0
\(405\) −7.94259 −0.394670
\(406\) 0 0
\(407\) −0.160490 −0.00795517
\(408\) 0 0
\(409\) −25.6108 −1.26637 −0.633187 0.773999i \(-0.718254\pi\)
−0.633187 + 0.773999i \(0.718254\pi\)
\(410\) 0 0
\(411\) 56.6505 2.79436
\(412\) 0 0
\(413\) −1.53098 −0.0753345
\(414\) 0 0
\(415\) −0.157450 −0.00772889
\(416\) 0 0
\(417\) 36.1243 1.76901
\(418\) 0 0
\(419\) −20.3511 −0.994216 −0.497108 0.867689i \(-0.665605\pi\)
−0.497108 + 0.867689i \(0.665605\pi\)
\(420\) 0 0
\(421\) −13.0941 −0.638167 −0.319084 0.947727i \(-0.603375\pi\)
−0.319084 + 0.947727i \(0.603375\pi\)
\(422\) 0 0
\(423\) 44.4122 2.15939
\(424\) 0 0
\(425\) −2.88783 −0.140080
\(426\) 0 0
\(427\) −3.75707 −0.181817
\(428\) 0 0
\(429\) −0.652543 −0.0315051
\(430\) 0 0
\(431\) −16.3248 −0.786337 −0.393169 0.919466i \(-0.628621\pi\)
−0.393169 + 0.919466i \(0.628621\pi\)
\(432\) 0 0
\(433\) 28.7539 1.38183 0.690913 0.722938i \(-0.257209\pi\)
0.690913 + 0.722938i \(0.257209\pi\)
\(434\) 0 0
\(435\) −6.80304 −0.326181
\(436\) 0 0
\(437\) −10.4718 −0.500935
\(438\) 0 0
\(439\) 10.1331 0.483625 0.241813 0.970323i \(-0.422258\pi\)
0.241813 + 0.970323i \(0.422258\pi\)
\(440\) 0 0
\(441\) −22.9666 −1.09365
\(442\) 0 0
\(443\) 3.66136 0.173956 0.0869782 0.996210i \(-0.472279\pi\)
0.0869782 + 0.996210i \(0.472279\pi\)
\(444\) 0 0
\(445\) −7.21783 −0.342158
\(446\) 0 0
\(447\) −1.96595 −0.0929861
\(448\) 0 0
\(449\) −6.81556 −0.321646 −0.160823 0.986983i \(-0.551415\pi\)
−0.160823 + 0.986983i \(0.551415\pi\)
\(450\) 0 0
\(451\) −1.79180 −0.0843724
\(452\) 0 0
\(453\) 6.35624 0.298642
\(454\) 0 0
\(455\) −0.456069 −0.0213809
\(456\) 0 0
\(457\) 32.7873 1.53372 0.766862 0.641812i \(-0.221817\pi\)
0.766862 + 0.641812i \(0.221817\pi\)
\(458\) 0 0
\(459\) 2.31297 0.107960
\(460\) 0 0
\(461\) −8.73107 −0.406647 −0.203323 0.979112i \(-0.565174\pi\)
−0.203323 + 0.979112i \(0.565174\pi\)
\(462\) 0 0
\(463\) 9.22462 0.428704 0.214352 0.976756i \(-0.431236\pi\)
0.214352 + 0.976756i \(0.431236\pi\)
\(464\) 0 0
\(465\) 11.1084 0.515138
\(466\) 0 0
\(467\) −9.83042 −0.454898 −0.227449 0.973790i \(-0.573038\pi\)
−0.227449 + 0.973790i \(0.573038\pi\)
\(468\) 0 0
\(469\) 0.310025 0.0143156
\(470\) 0 0
\(471\) 48.9639 2.25614
\(472\) 0 0
\(473\) 0.945118 0.0434566
\(474\) 0 0
\(475\) −4.15216 −0.190514
\(476\) 0 0
\(477\) −22.7887 −1.04342
\(478\) 0 0
\(479\) 30.0355 1.37236 0.686178 0.727434i \(-0.259287\pi\)
0.686178 + 0.727434i \(0.259287\pi\)
\(480\) 0 0
\(481\) −1.61752 −0.0737527
\(482\) 0 0
\(483\) 1.78748 0.0813329
\(484\) 0 0
\(485\) 6.79246 0.308430
\(486\) 0 0
\(487\) −41.5018 −1.88063 −0.940314 0.340308i \(-0.889469\pi\)
−0.940314 + 0.340308i \(0.889469\pi\)
\(488\) 0 0
\(489\) 40.6882 1.83998
\(490\) 0 0
\(491\) 29.3046 1.32250 0.661249 0.750167i \(-0.270027\pi\)
0.661249 + 0.750167i \(0.270027\pi\)
\(492\) 0 0
\(493\) −7.81562 −0.351998
\(494\) 0 0
\(495\) 0.532605 0.0239388
\(496\) 0 0
\(497\) 4.37093 0.196063
\(498\) 0 0
\(499\) 24.5959 1.10106 0.550532 0.834814i \(-0.314425\pi\)
0.550532 + 0.834814i \(0.314425\pi\)
\(500\) 0 0
\(501\) 33.6494 1.50334
\(502\) 0 0
\(503\) −10.0315 −0.447281 −0.223640 0.974672i \(-0.571794\pi\)
−0.223640 + 0.974672i \(0.571794\pi\)
\(504\) 0 0
\(505\) 5.41515 0.240971
\(506\) 0 0
\(507\) 26.1012 1.15919
\(508\) 0 0
\(509\) −21.3474 −0.946208 −0.473104 0.881007i \(-0.656867\pi\)
−0.473104 + 0.881007i \(0.656867\pi\)
\(510\) 0 0
\(511\) −0.197680 −0.00874486
\(512\) 0 0
\(513\) 3.32561 0.146829
\(514\) 0 0
\(515\) 6.64784 0.292939
\(516\) 0 0
\(517\) 2.14778 0.0944593
\(518\) 0 0
\(519\) −53.5817 −2.35198
\(520\) 0 0
\(521\) 40.4370 1.77158 0.885790 0.464087i \(-0.153618\pi\)
0.885790 + 0.464087i \(0.153618\pi\)
\(522\) 0 0
\(523\) 13.5202 0.591198 0.295599 0.955312i \(-0.404481\pi\)
0.295599 + 0.955312i \(0.404481\pi\)
\(524\) 0 0
\(525\) 0.708747 0.0309323
\(526\) 0 0
\(527\) 12.7618 0.555911
\(528\) 0 0
\(529\) −16.6394 −0.723453
\(530\) 0 0
\(531\) 18.0197 0.781989
\(532\) 0 0
\(533\) −18.0590 −0.782221
\(534\) 0 0
\(535\) −5.22219 −0.225775
\(536\) 0 0
\(537\) −0.703455 −0.0303563
\(538\) 0 0
\(539\) −1.11067 −0.0478399
\(540\) 0 0
\(541\) −27.8160 −1.19590 −0.597952 0.801532i \(-0.704019\pi\)
−0.597952 + 0.801532i \(0.704019\pi\)
\(542\) 0 0
\(543\) −25.5548 −1.09666
\(544\) 0 0
\(545\) −2.37345 −0.101668
\(546\) 0 0
\(547\) −6.83360 −0.292184 −0.146092 0.989271i \(-0.546670\pi\)
−0.146092 + 0.989271i \(0.546670\pi\)
\(548\) 0 0
\(549\) 44.2209 1.88730
\(550\) 0 0
\(551\) −11.2374 −0.478729
\(552\) 0 0
\(553\) −0.351171 −0.0149333
\(554\) 0 0
\(555\) 2.51369 0.106700
\(556\) 0 0
\(557\) 22.3389 0.946530 0.473265 0.880920i \(-0.343075\pi\)
0.473265 + 0.880920i \(0.343075\pi\)
\(558\) 0 0
\(559\) 9.52555 0.402888
\(560\) 0 0
\(561\) 1.16501 0.0491868
\(562\) 0 0
\(563\) 33.2527 1.40143 0.700717 0.713440i \(-0.252864\pi\)
0.700717 + 0.713440i \(0.252864\pi\)
\(564\) 0 0
\(565\) 17.7642 0.747346
\(566\) 0 0
\(567\) 2.23945 0.0940482
\(568\) 0 0
\(569\) 40.1937 1.68501 0.842504 0.538691i \(-0.181081\pi\)
0.842504 + 0.538691i \(0.181081\pi\)
\(570\) 0 0
\(571\) 3.49128 0.146106 0.0730528 0.997328i \(-0.476726\pi\)
0.0730528 + 0.997328i \(0.476726\pi\)
\(572\) 0 0
\(573\) −23.6403 −0.987589
\(574\) 0 0
\(575\) 2.52202 0.105176
\(576\) 0 0
\(577\) 32.1865 1.33994 0.669972 0.742387i \(-0.266306\pi\)
0.669972 + 0.742387i \(0.266306\pi\)
\(578\) 0 0
\(579\) 30.8812 1.28338
\(580\) 0 0
\(581\) 0.0443937 0.00184176
\(582\) 0 0
\(583\) −1.10206 −0.0456428
\(584\) 0 0
\(585\) 5.36796 0.221938
\(586\) 0 0
\(587\) 36.5196 1.50733 0.753663 0.657261i \(-0.228285\pi\)
0.753663 + 0.657261i \(0.228285\pi\)
\(588\) 0 0
\(589\) 18.3490 0.756058
\(590\) 0 0
\(591\) 42.3619 1.74254
\(592\) 0 0
\(593\) −8.93223 −0.366803 −0.183401 0.983038i \(-0.558711\pi\)
−0.183401 + 0.983038i \(0.558711\pi\)
\(594\) 0 0
\(595\) 0.814239 0.0333805
\(596\) 0 0
\(597\) 53.4916 2.18927
\(598\) 0 0
\(599\) −35.2408 −1.43990 −0.719949 0.694027i \(-0.755835\pi\)
−0.719949 + 0.694027i \(0.755835\pi\)
\(600\) 0 0
\(601\) −11.3066 −0.461205 −0.230603 0.973048i \(-0.574070\pi\)
−0.230603 + 0.973048i \(0.574070\pi\)
\(602\) 0 0
\(603\) −3.64902 −0.148599
\(604\) 0 0
\(605\) −10.9742 −0.446166
\(606\) 0 0
\(607\) −24.7839 −1.00595 −0.502974 0.864302i \(-0.667761\pi\)
−0.502974 + 0.864302i \(0.667761\pi\)
\(608\) 0 0
\(609\) 1.91815 0.0777274
\(610\) 0 0
\(611\) 21.6468 0.875736
\(612\) 0 0
\(613\) −3.23397 −0.130619 −0.0653093 0.997865i \(-0.520803\pi\)
−0.0653093 + 0.997865i \(0.520803\pi\)
\(614\) 0 0
\(615\) 28.0643 1.13166
\(616\) 0 0
\(617\) 24.0009 0.966242 0.483121 0.875554i \(-0.339503\pi\)
0.483121 + 0.875554i \(0.339503\pi\)
\(618\) 0 0
\(619\) 24.5590 0.987111 0.493555 0.869714i \(-0.335697\pi\)
0.493555 + 0.869714i \(0.335697\pi\)
\(620\) 0 0
\(621\) −2.01998 −0.0810589
\(622\) 0 0
\(623\) 2.03510 0.0815348
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) 1.67507 0.0668957
\(628\) 0 0
\(629\) 2.88783 0.115145
\(630\) 0 0
\(631\) −31.2104 −1.24247 −0.621234 0.783625i \(-0.713368\pi\)
−0.621234 + 0.783625i \(0.713368\pi\)
\(632\) 0 0
\(633\) 13.1726 0.523563
\(634\) 0 0
\(635\) −19.2934 −0.765637
\(636\) 0 0
\(637\) −11.1941 −0.443525
\(638\) 0 0
\(639\) −51.4461 −2.03518
\(640\) 0 0
\(641\) 12.5272 0.494795 0.247397 0.968914i \(-0.420425\pi\)
0.247397 + 0.968914i \(0.420425\pi\)
\(642\) 0 0
\(643\) 5.76315 0.227276 0.113638 0.993522i \(-0.463750\pi\)
0.113638 + 0.993522i \(0.463750\pi\)
\(644\) 0 0
\(645\) −14.8030 −0.582869
\(646\) 0 0
\(647\) −2.59926 −0.102188 −0.0510938 0.998694i \(-0.516271\pi\)
−0.0510938 + 0.998694i \(0.516271\pi\)
\(648\) 0 0
\(649\) 0.871436 0.0342069
\(650\) 0 0
\(651\) −3.13206 −0.122755
\(652\) 0 0
\(653\) −4.79108 −0.187489 −0.0937446 0.995596i \(-0.529884\pi\)
−0.0937446 + 0.995596i \(0.529884\pi\)
\(654\) 0 0
\(655\) −14.8127 −0.578781
\(656\) 0 0
\(657\) 2.32671 0.0907736
\(658\) 0 0
\(659\) −37.8970 −1.47626 −0.738128 0.674661i \(-0.764290\pi\)
−0.738128 + 0.674661i \(0.764290\pi\)
\(660\) 0 0
\(661\) 5.57407 0.216806 0.108403 0.994107i \(-0.465426\pi\)
0.108403 + 0.994107i \(0.465426\pi\)
\(662\) 0 0
\(663\) 11.7418 0.456013
\(664\) 0 0
\(665\) 1.17072 0.0453986
\(666\) 0 0
\(667\) 6.82559 0.264288
\(668\) 0 0
\(669\) −30.2125 −1.16808
\(670\) 0 0
\(671\) 2.13853 0.0825571
\(672\) 0 0
\(673\) −15.9396 −0.614425 −0.307212 0.951641i \(-0.599396\pi\)
−0.307212 + 0.951641i \(0.599396\pi\)
\(674\) 0 0
\(675\) −0.800936 −0.0308280
\(676\) 0 0
\(677\) −35.9932 −1.38333 −0.691665 0.722218i \(-0.743123\pi\)
−0.691665 + 0.722218i \(0.743123\pi\)
\(678\) 0 0
\(679\) −1.91517 −0.0734974
\(680\) 0 0
\(681\) −27.2406 −1.04386
\(682\) 0 0
\(683\) 42.4996 1.62620 0.813101 0.582122i \(-0.197777\pi\)
0.813101 + 0.582122i \(0.197777\pi\)
\(684\) 0 0
\(685\) −22.5368 −0.861087
\(686\) 0 0
\(687\) 38.0352 1.45113
\(688\) 0 0
\(689\) −11.1074 −0.423157
\(690\) 0 0
\(691\) 6.62722 0.252111 0.126056 0.992023i \(-0.459768\pi\)
0.126056 + 0.992023i \(0.459768\pi\)
\(692\) 0 0
\(693\) −0.150171 −0.00570452
\(694\) 0 0
\(695\) −14.3710 −0.545124
\(696\) 0 0
\(697\) 32.2414 1.22123
\(698\) 0 0
\(699\) 50.6796 1.91688
\(700\) 0 0
\(701\) −39.6044 −1.49584 −0.747919 0.663790i \(-0.768947\pi\)
−0.747919 + 0.663790i \(0.768947\pi\)
\(702\) 0 0
\(703\) 4.15216 0.156602
\(704\) 0 0
\(705\) −33.6399 −1.26695
\(706\) 0 0
\(707\) −1.52683 −0.0574223
\(708\) 0 0
\(709\) 15.7832 0.592750 0.296375 0.955072i \(-0.404222\pi\)
0.296375 + 0.955072i \(0.404222\pi\)
\(710\) 0 0
\(711\) 4.13331 0.155011
\(712\) 0 0
\(713\) −11.1452 −0.417391
\(714\) 0 0
\(715\) 0.259596 0.00970833
\(716\) 0 0
\(717\) 11.7075 0.437226
\(718\) 0 0
\(719\) 22.8483 0.852098 0.426049 0.904700i \(-0.359905\pi\)
0.426049 + 0.904700i \(0.359905\pi\)
\(720\) 0 0
\(721\) −1.87439 −0.0698061
\(722\) 0 0
\(723\) −5.05746 −0.188089
\(724\) 0 0
\(725\) 2.70640 0.100513
\(726\) 0 0
\(727\) 30.8290 1.14339 0.571693 0.820468i \(-0.306287\pi\)
0.571693 + 0.820468i \(0.306287\pi\)
\(728\) 0 0
\(729\) −32.3984 −1.19994
\(730\) 0 0
\(731\) −17.0064 −0.629003
\(732\) 0 0
\(733\) −2.87159 −0.106064 −0.0530322 0.998593i \(-0.516889\pi\)
−0.0530322 + 0.998593i \(0.516889\pi\)
\(734\) 0 0
\(735\) 17.3960 0.641661
\(736\) 0 0
\(737\) −0.176467 −0.00650025
\(738\) 0 0
\(739\) 19.5586 0.719473 0.359736 0.933054i \(-0.382867\pi\)
0.359736 + 0.933054i \(0.382867\pi\)
\(740\) 0 0
\(741\) 16.8825 0.620193
\(742\) 0 0
\(743\) 7.67284 0.281489 0.140745 0.990046i \(-0.455050\pi\)
0.140745 + 0.990046i \(0.455050\pi\)
\(744\) 0 0
\(745\) 0.782096 0.0286538
\(746\) 0 0
\(747\) −0.522517 −0.0191179
\(748\) 0 0
\(749\) 1.47242 0.0538012
\(750\) 0 0
\(751\) 10.8710 0.396690 0.198345 0.980132i \(-0.436443\pi\)
0.198345 + 0.980132i \(0.436443\pi\)
\(752\) 0 0
\(753\) 67.8779 2.47361
\(754\) 0 0
\(755\) −2.52865 −0.0920270
\(756\) 0 0
\(757\) 7.45938 0.271116 0.135558 0.990769i \(-0.456717\pi\)
0.135558 + 0.990769i \(0.456717\pi\)
\(758\) 0 0
\(759\) −1.01744 −0.0369306
\(760\) 0 0
\(761\) 45.8693 1.66276 0.831380 0.555704i \(-0.187551\pi\)
0.831380 + 0.555704i \(0.187551\pi\)
\(762\) 0 0
\(763\) 0.669207 0.0242269
\(764\) 0 0
\(765\) −9.58365 −0.346497
\(766\) 0 0
\(767\) 8.78293 0.317133
\(768\) 0 0
\(769\) −39.9990 −1.44240 −0.721201 0.692726i \(-0.756410\pi\)
−0.721201 + 0.692726i \(0.756410\pi\)
\(770\) 0 0
\(771\) 24.7666 0.891948
\(772\) 0 0
\(773\) 25.9576 0.933629 0.466815 0.884355i \(-0.345402\pi\)
0.466815 + 0.884355i \(0.345402\pi\)
\(774\) 0 0
\(775\) −4.41915 −0.158741
\(776\) 0 0
\(777\) −0.708747 −0.0254262
\(778\) 0 0
\(779\) 46.3571 1.66091
\(780\) 0 0
\(781\) −2.48794 −0.0890256
\(782\) 0 0
\(783\) −2.16765 −0.0774655
\(784\) 0 0
\(785\) −19.4789 −0.695233
\(786\) 0 0
\(787\) 14.4524 0.515174 0.257587 0.966255i \(-0.417073\pi\)
0.257587 + 0.966255i \(0.417073\pi\)
\(788\) 0 0
\(789\) 12.7468 0.453799
\(790\) 0 0
\(791\) −5.00871 −0.178089
\(792\) 0 0
\(793\) 21.5536 0.765391
\(794\) 0 0
\(795\) 17.2612 0.612192
\(796\) 0 0
\(797\) 2.88581 0.102221 0.0511103 0.998693i \(-0.483724\pi\)
0.0511103 + 0.998693i \(0.483724\pi\)
\(798\) 0 0
\(799\) −38.6469 −1.36723
\(800\) 0 0
\(801\) −23.9533 −0.846349
\(802\) 0 0
\(803\) 0.112520 0.00397075
\(804\) 0 0
\(805\) −0.711097 −0.0250629
\(806\) 0 0
\(807\) −54.5465 −1.92013
\(808\) 0 0
\(809\) 6.66435 0.234306 0.117153 0.993114i \(-0.462623\pi\)
0.117153 + 0.993114i \(0.462623\pi\)
\(810\) 0 0
\(811\) −17.2301 −0.605031 −0.302515 0.953144i \(-0.597826\pi\)
−0.302515 + 0.953144i \(0.597826\pi\)
\(812\) 0 0
\(813\) 57.6179 2.02075
\(814\) 0 0
\(815\) −16.1867 −0.566994
\(816\) 0 0
\(817\) −24.4519 −0.855465
\(818\) 0 0
\(819\) −1.51352 −0.0528868
\(820\) 0 0
\(821\) 4.96739 0.173363 0.0866816 0.996236i \(-0.472374\pi\)
0.0866816 + 0.996236i \(0.472374\pi\)
\(822\) 0 0
\(823\) −17.5539 −0.611892 −0.305946 0.952049i \(-0.598973\pi\)
−0.305946 + 0.952049i \(0.598973\pi\)
\(824\) 0 0
\(825\) −0.403421 −0.0140453
\(826\) 0 0
\(827\) 39.3418 1.36805 0.684024 0.729459i \(-0.260228\pi\)
0.684024 + 0.729459i \(0.260228\pi\)
\(828\) 0 0
\(829\) −39.4784 −1.37114 −0.685571 0.728006i \(-0.740447\pi\)
−0.685571 + 0.728006i \(0.740447\pi\)
\(830\) 0 0
\(831\) 1.53592 0.0532805
\(832\) 0 0
\(833\) 19.9852 0.692448
\(834\) 0 0
\(835\) −13.3864 −0.463257
\(836\) 0 0
\(837\) 3.53946 0.122341
\(838\) 0 0
\(839\) 21.3840 0.738259 0.369129 0.929378i \(-0.379656\pi\)
0.369129 + 0.929378i \(0.379656\pi\)
\(840\) 0 0
\(841\) −21.6754 −0.747428
\(842\) 0 0
\(843\) −23.2559 −0.800977
\(844\) 0 0
\(845\) −10.3836 −0.357207
\(846\) 0 0
\(847\) 3.09424 0.106319
\(848\) 0 0
\(849\) −57.7737 −1.98279
\(850\) 0 0
\(851\) −2.52202 −0.0864538
\(852\) 0 0
\(853\) −18.4715 −0.632451 −0.316226 0.948684i \(-0.602416\pi\)
−0.316226 + 0.948684i \(0.602416\pi\)
\(854\) 0 0
\(855\) −13.7795 −0.471248
\(856\) 0 0
\(857\) 32.6073 1.11385 0.556923 0.830564i \(-0.311982\pi\)
0.556923 + 0.830564i \(0.311982\pi\)
\(858\) 0 0
\(859\) −33.3456 −1.13774 −0.568869 0.822428i \(-0.692619\pi\)
−0.568869 + 0.822428i \(0.692619\pi\)
\(860\) 0 0
\(861\) −7.91286 −0.269670
\(862\) 0 0
\(863\) −40.3578 −1.37380 −0.686898 0.726754i \(-0.741028\pi\)
−0.686898 + 0.726754i \(0.741028\pi\)
\(864\) 0 0
\(865\) 21.3160 0.724765
\(866\) 0 0
\(867\) 21.7696 0.739335
\(868\) 0 0
\(869\) 0.199887 0.00678072
\(870\) 0 0
\(871\) −1.77856 −0.0602641
\(872\) 0 0
\(873\) 22.5416 0.762919
\(874\) 0 0
\(875\) −0.281955 −0.00953182
\(876\) 0 0
\(877\) −16.5768 −0.559758 −0.279879 0.960035i \(-0.590294\pi\)
−0.279879 + 0.960035i \(0.590294\pi\)
\(878\) 0 0
\(879\) −19.1659 −0.646450
\(880\) 0 0
\(881\) 14.1872 0.477978 0.238989 0.971022i \(-0.423184\pi\)
0.238989 + 0.971022i \(0.423184\pi\)
\(882\) 0 0
\(883\) −56.2999 −1.89464 −0.947321 0.320286i \(-0.896221\pi\)
−0.947321 + 0.320286i \(0.896221\pi\)
\(884\) 0 0
\(885\) −13.6490 −0.458806
\(886\) 0 0
\(887\) −33.2070 −1.11498 −0.557491 0.830183i \(-0.688236\pi\)
−0.557491 + 0.830183i \(0.688236\pi\)
\(888\) 0 0
\(889\) 5.43988 0.182448
\(890\) 0 0
\(891\) −1.27470 −0.0427041
\(892\) 0 0
\(893\) −55.5670 −1.85948
\(894\) 0 0
\(895\) 0.279850 0.00935434
\(896\) 0 0
\(897\) −10.2544 −0.342385
\(898\) 0 0
\(899\) −11.9600 −0.398888
\(900\) 0 0
\(901\) 19.8304 0.660647
\(902\) 0 0
\(903\) 4.17379 0.138895
\(904\) 0 0
\(905\) 10.1663 0.337938
\(906\) 0 0
\(907\) −55.0317 −1.82730 −0.913650 0.406503i \(-0.866748\pi\)
−0.913650 + 0.406503i \(0.866748\pi\)
\(908\) 0 0
\(909\) 17.9709 0.596056
\(910\) 0 0
\(911\) 26.8532 0.889688 0.444844 0.895608i \(-0.353259\pi\)
0.444844 + 0.895608i \(0.353259\pi\)
\(912\) 0 0
\(913\) −0.0252690 −0.000836282 0
\(914\) 0 0
\(915\) −33.4951 −1.10731
\(916\) 0 0
\(917\) 4.17652 0.137921
\(918\) 0 0
\(919\) 32.2839 1.06495 0.532473 0.846447i \(-0.321263\pi\)
0.532473 + 0.846447i \(0.321263\pi\)
\(920\) 0 0
\(921\) −7.78602 −0.256558
\(922\) 0 0
\(923\) −25.0752 −0.825361
\(924\) 0 0
\(925\) −1.00000 −0.0328798
\(926\) 0 0
\(927\) 22.0617 0.724602
\(928\) 0 0
\(929\) 19.8991 0.652867 0.326434 0.945220i \(-0.394153\pi\)
0.326434 + 0.945220i \(0.394153\pi\)
\(930\) 0 0
\(931\) 28.7350 0.941752
\(932\) 0 0
\(933\) −27.9194 −0.914041
\(934\) 0 0
\(935\) −0.463467 −0.0151570
\(936\) 0 0
\(937\) −0.121019 −0.00395351 −0.00197676 0.999998i \(-0.500629\pi\)
−0.00197676 + 0.999998i \(0.500629\pi\)
\(938\) 0 0
\(939\) 4.53869 0.148114
\(940\) 0 0
\(941\) −50.1748 −1.63565 −0.817826 0.575466i \(-0.804821\pi\)
−0.817826 + 0.575466i \(0.804821\pi\)
\(942\) 0 0
\(943\) −28.1573 −0.916927
\(944\) 0 0
\(945\) 0.225828 0.00734618
\(946\) 0 0
\(947\) 8.20738 0.266704 0.133352 0.991069i \(-0.457426\pi\)
0.133352 + 0.991069i \(0.457426\pi\)
\(948\) 0 0
\(949\) 1.13406 0.0368130
\(950\) 0 0
\(951\) 6.37290 0.206656
\(952\) 0 0
\(953\) −17.2409 −0.558489 −0.279244 0.960220i \(-0.590084\pi\)
−0.279244 + 0.960220i \(0.590084\pi\)
\(954\) 0 0
\(955\) 9.40464 0.304327
\(956\) 0 0
\(957\) −1.09182 −0.0352934
\(958\) 0 0
\(959\) 6.35437 0.205193
\(960\) 0 0
\(961\) −11.4711 −0.370036
\(962\) 0 0
\(963\) −17.3305 −0.558468
\(964\) 0 0
\(965\) −12.2852 −0.395476
\(966\) 0 0
\(967\) −12.1538 −0.390840 −0.195420 0.980720i \(-0.562607\pi\)
−0.195420 + 0.980720i \(0.562607\pi\)
\(968\) 0 0
\(969\) −30.1410 −0.968267
\(970\) 0 0
\(971\) 23.6672 0.759517 0.379758 0.925086i \(-0.376007\pi\)
0.379758 + 0.925086i \(0.376007\pi\)
\(972\) 0 0
\(973\) 4.05199 0.129901
\(974\) 0 0
\(975\) −4.06595 −0.130215
\(976\) 0 0
\(977\) 30.6954 0.982033 0.491017 0.871150i \(-0.336625\pi\)
0.491017 + 0.871150i \(0.336625\pi\)
\(978\) 0 0
\(979\) −1.15839 −0.0370222
\(980\) 0 0
\(981\) −7.87661 −0.251481
\(982\) 0 0
\(983\) 53.3677 1.70216 0.851082 0.525032i \(-0.175947\pi\)
0.851082 + 0.525032i \(0.175947\pi\)
\(984\) 0 0
\(985\) −16.8525 −0.536965
\(986\) 0 0
\(987\) 9.48494 0.301909
\(988\) 0 0
\(989\) 14.8521 0.472270
\(990\) 0 0
\(991\) 58.4497 1.85672 0.928358 0.371687i \(-0.121221\pi\)
0.928358 + 0.371687i \(0.121221\pi\)
\(992\) 0 0
\(993\) −25.6765 −0.814819
\(994\) 0 0
\(995\) −21.2801 −0.674626
\(996\) 0 0
\(997\) −60.1177 −1.90395 −0.951973 0.306182i \(-0.900948\pi\)
−0.951973 + 0.306182i \(0.900948\pi\)
\(998\) 0 0
\(999\) 0.800936 0.0253405
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 2960.2.a.ba.1.1 5
4.3 odd 2 185.2.a.d.1.2 5
12.11 even 2 1665.2.a.q.1.4 5
20.3 even 4 925.2.b.g.149.7 10
20.7 even 4 925.2.b.g.149.4 10
20.19 odd 2 925.2.a.h.1.4 5
28.27 even 2 9065.2.a.j.1.2 5
60.59 even 2 8325.2.a.cc.1.2 5
148.147 odd 2 6845.2.a.g.1.4 5
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
185.2.a.d.1.2 5 4.3 odd 2
925.2.a.h.1.4 5 20.19 odd 2
925.2.b.g.149.4 10 20.7 even 4
925.2.b.g.149.7 10 20.3 even 4
1665.2.a.q.1.4 5 12.11 even 2
2960.2.a.ba.1.1 5 1.1 even 1 trivial
6845.2.a.g.1.4 5 148.147 odd 2
8325.2.a.cc.1.2 5 60.59 even 2
9065.2.a.j.1.2 5 28.27 even 2