Properties

Label 2760.2.a.s.1.1
Level $2760$
Weight $2$
Character 2760.1
Self dual yes
Analytic conductor $22.039$
Analytic rank $0$
Dimension $3$
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] = [2760,2,Mod(1,2760)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2760, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 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("2760.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2760 = 2^{3} \cdot 3 \cdot 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2760.a (trivial)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(22.0387109579\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.1436.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - 11x - 12 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-1.28282\) of defining polynomial
Character \(\chi\) \(=\) 2760.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.00000 q^{3} -1.00000 q^{5} -3.78872 q^{7} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{3} -1.00000 q^{5} -3.78872 q^{7} +1.00000 q^{9} -3.07154 q^{11} -5.07154 q^{13} -1.00000 q^{15} +3.28282 q^{17} +5.07154 q^{19} -3.78872 q^{21} +1.00000 q^{23} +1.00000 q^{25} +1.00000 q^{27} +1.28282 q^{29} +3.78872 q^{31} -3.07154 q^{33} +3.78872 q^{35} +3.22307 q^{37} -5.07154 q^{39} -9.42590 q^{41} +12.1431 q^{43} -1.00000 q^{45} +10.0833 q^{47} +7.35436 q^{49} +3.28282 q^{51} +5.84847 q^{53} +3.07154 q^{55} +5.07154 q^{57} -2.29461 q^{59} -3.07154 q^{61} -3.78872 q^{63} +5.07154 q^{65} +3.22307 q^{67} +1.00000 q^{69} +7.28282 q^{71} +9.63719 q^{73} +1.00000 q^{75} +11.6372 q^{77} -8.00000 q^{79} +1.00000 q^{81} +6.71718 q^{83} -3.28282 q^{85} +1.28282 q^{87} +5.01178 q^{89} +19.2146 q^{91} +3.78872 q^{93} -5.07154 q^{95} -1.13130 q^{97} -3.07154 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 3 q^{3} - 3 q^{5} - 2 q^{7} + 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q + 3 q^{3} - 3 q^{5} - 2 q^{7} + 3 q^{9} + 4 q^{11} - 2 q^{13} - 3 q^{15} + 6 q^{17} + 2 q^{19} - 2 q^{21} + 3 q^{23} + 3 q^{25} + 3 q^{27} + 2 q^{31} + 4 q^{33} + 2 q^{35} + 8 q^{37} - 2 q^{39} + 2 q^{41} + 10 q^{43} - 3 q^{45} + 6 q^{47} + 5 q^{49} + 6 q^{51} + 6 q^{53} - 4 q^{55} + 2 q^{57} + 8 q^{59} + 4 q^{61} - 2 q^{63} + 2 q^{65} + 8 q^{67} + 3 q^{69} + 18 q^{71} + 8 q^{73} + 3 q^{75} + 14 q^{77} - 24 q^{79} + 3 q^{81} + 24 q^{83} - 6 q^{85} + 4 q^{89} + 18 q^{91} + 2 q^{93} - 2 q^{95} + 12 q^{97} + 4 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 1.00000 0.577350
\(4\) 0 0
\(5\) −1.00000 −0.447214
\(6\) 0 0
\(7\) −3.78872 −1.43200 −0.716000 0.698100i \(-0.754029\pi\)
−0.716000 + 0.698100i \(0.754029\pi\)
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) −3.07154 −0.926104 −0.463052 0.886331i \(-0.653246\pi\)
−0.463052 + 0.886331i \(0.653246\pi\)
\(12\) 0 0
\(13\) −5.07154 −1.40659 −0.703296 0.710897i \(-0.748289\pi\)
−0.703296 + 0.710897i \(0.748289\pi\)
\(14\) 0 0
\(15\) −1.00000 −0.258199
\(16\) 0 0
\(17\) 3.28282 0.796202 0.398101 0.917342i \(-0.369669\pi\)
0.398101 + 0.917342i \(0.369669\pi\)
\(18\) 0 0
\(19\) 5.07154 1.16349 0.581745 0.813371i \(-0.302370\pi\)
0.581745 + 0.813371i \(0.302370\pi\)
\(20\) 0 0
\(21\) −3.78872 −0.826765
\(22\) 0 0
\(23\) 1.00000 0.208514
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) 1.00000 0.192450
\(28\) 0 0
\(29\) 1.28282 0.238214 0.119107 0.992881i \(-0.461997\pi\)
0.119107 + 0.992881i \(0.461997\pi\)
\(30\) 0 0
\(31\) 3.78872 0.680473 0.340237 0.940340i \(-0.389493\pi\)
0.340237 + 0.940340i \(0.389493\pi\)
\(32\) 0 0
\(33\) −3.07154 −0.534686
\(34\) 0 0
\(35\) 3.78872 0.640410
\(36\) 0 0
\(37\) 3.22307 0.529869 0.264935 0.964266i \(-0.414650\pi\)
0.264935 + 0.964266i \(0.414650\pi\)
\(38\) 0 0
\(39\) −5.07154 −0.812096
\(40\) 0 0
\(41\) −9.42590 −1.47208 −0.736039 0.676939i \(-0.763306\pi\)
−0.736039 + 0.676939i \(0.763306\pi\)
\(42\) 0 0
\(43\) 12.1431 1.85180 0.925901 0.377766i \(-0.123308\pi\)
0.925901 + 0.377766i \(0.123308\pi\)
\(44\) 0 0
\(45\) −1.00000 −0.149071
\(46\) 0 0
\(47\) 10.0833 1.47080 0.735402 0.677631i \(-0.236993\pi\)
0.735402 + 0.677631i \(0.236993\pi\)
\(48\) 0 0
\(49\) 7.35436 1.05062
\(50\) 0 0
\(51\) 3.28282 0.459687
\(52\) 0 0
\(53\) 5.84847 0.803349 0.401675 0.915782i \(-0.368428\pi\)
0.401675 + 0.915782i \(0.368428\pi\)
\(54\) 0 0
\(55\) 3.07154 0.414166
\(56\) 0 0
\(57\) 5.07154 0.671742
\(58\) 0 0
\(59\) −2.29461 −0.298732 −0.149366 0.988782i \(-0.547723\pi\)
−0.149366 + 0.988782i \(0.547723\pi\)
\(60\) 0 0
\(61\) −3.07154 −0.393270 −0.196635 0.980477i \(-0.563001\pi\)
−0.196635 + 0.980477i \(0.563001\pi\)
\(62\) 0 0
\(63\) −3.78872 −0.477333
\(64\) 0 0
\(65\) 5.07154 0.629047
\(66\) 0 0
\(67\) 3.22307 0.393760 0.196880 0.980428i \(-0.436919\pi\)
0.196880 + 0.980428i \(0.436919\pi\)
\(68\) 0 0
\(69\) 1.00000 0.120386
\(70\) 0 0
\(71\) 7.28282 0.864312 0.432156 0.901799i \(-0.357753\pi\)
0.432156 + 0.901799i \(0.357753\pi\)
\(72\) 0 0
\(73\) 9.63719 1.12795 0.563974 0.825793i \(-0.309272\pi\)
0.563974 + 0.825793i \(0.309272\pi\)
\(74\) 0 0
\(75\) 1.00000 0.115470
\(76\) 0 0
\(77\) 11.6372 1.32618
\(78\) 0 0
\(79\) −8.00000 −0.900070 −0.450035 0.893011i \(-0.648589\pi\)
−0.450035 + 0.893011i \(0.648589\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) 6.71718 0.737306 0.368653 0.929567i \(-0.379819\pi\)
0.368653 + 0.929567i \(0.379819\pi\)
\(84\) 0 0
\(85\) −3.28282 −0.356072
\(86\) 0 0
\(87\) 1.28282 0.137533
\(88\) 0 0
\(89\) 5.01178 0.531248 0.265624 0.964077i \(-0.414422\pi\)
0.265624 + 0.964077i \(0.414422\pi\)
\(90\) 0 0
\(91\) 19.2146 2.01424
\(92\) 0 0
\(93\) 3.78872 0.392871
\(94\) 0 0
\(95\) −5.07154 −0.520329
\(96\) 0 0
\(97\) −1.13130 −0.114866 −0.0574328 0.998349i \(-0.518292\pi\)
−0.0574328 + 0.998349i \(0.518292\pi\)
\(98\) 0 0
\(99\) −3.07154 −0.308701
\(100\) 0 0
\(101\) −6.29461 −0.626337 −0.313168 0.949698i \(-0.601390\pi\)
−0.313168 + 0.949698i \(0.601390\pi\)
\(102\) 0 0
\(103\) −12.7087 −1.25223 −0.626114 0.779732i \(-0.715356\pi\)
−0.626114 + 0.779732i \(0.715356\pi\)
\(104\) 0 0
\(105\) 3.78872 0.369741
\(106\) 0 0
\(107\) 18.2946 1.76861 0.884303 0.466913i \(-0.154634\pi\)
0.884303 + 0.466913i \(0.154634\pi\)
\(108\) 0 0
\(109\) −4.50589 −0.431586 −0.215793 0.976439i \(-0.569234\pi\)
−0.215793 + 0.976439i \(0.569234\pi\)
\(110\) 0 0
\(111\) 3.22307 0.305920
\(112\) 0 0
\(113\) 4.71718 0.443755 0.221877 0.975075i \(-0.428782\pi\)
0.221877 + 0.975075i \(0.428782\pi\)
\(114\) 0 0
\(115\) −1.00000 −0.0932505
\(116\) 0 0
\(117\) −5.07154 −0.468864
\(118\) 0 0
\(119\) −12.4377 −1.14016
\(120\) 0 0
\(121\) −1.56565 −0.142332
\(122\) 0 0
\(123\) −9.42590 −0.849905
\(124\) 0 0
\(125\) −1.00000 −0.0894427
\(126\) 0 0
\(127\) −12.0833 −1.07222 −0.536111 0.844148i \(-0.680107\pi\)
−0.536111 + 0.844148i \(0.680107\pi\)
\(128\) 0 0
\(129\) 12.1431 1.06914
\(130\) 0 0
\(131\) 17.1313 1.49677 0.748384 0.663266i \(-0.230830\pi\)
0.748384 + 0.663266i \(0.230830\pi\)
\(132\) 0 0
\(133\) −19.2146 −1.66612
\(134\) 0 0
\(135\) −1.00000 −0.0860663
\(136\) 0 0
\(137\) −2.70873 −0.231422 −0.115711 0.993283i \(-0.536915\pi\)
−0.115711 + 0.993283i \(0.536915\pi\)
\(138\) 0 0
\(139\) −0.211285 −0.0179209 −0.00896047 0.999960i \(-0.502852\pi\)
−0.00896047 + 0.999960i \(0.502852\pi\)
\(140\) 0 0
\(141\) 10.0833 0.849169
\(142\) 0 0
\(143\) 15.5774 1.30265
\(144\) 0 0
\(145\) −1.28282 −0.106533
\(146\) 0 0
\(147\) 7.35436 0.606578
\(148\) 0 0
\(149\) −17.6372 −1.44489 −0.722447 0.691426i \(-0.756983\pi\)
−0.722447 + 0.691426i \(0.756983\pi\)
\(150\) 0 0
\(151\) −13.7205 −1.11656 −0.558280 0.829653i \(-0.688538\pi\)
−0.558280 + 0.829653i \(0.688538\pi\)
\(152\) 0 0
\(153\) 3.28282 0.265401
\(154\) 0 0
\(155\) −3.78872 −0.304317
\(156\) 0 0
\(157\) 13.4857 1.07627 0.538136 0.842858i \(-0.319129\pi\)
0.538136 + 0.842858i \(0.319129\pi\)
\(158\) 0 0
\(159\) 5.84847 0.463814
\(160\) 0 0
\(161\) −3.78872 −0.298593
\(162\) 0 0
\(163\) −7.57743 −0.593510 −0.296755 0.954954i \(-0.595904\pi\)
−0.296755 + 0.954954i \(0.595904\pi\)
\(164\) 0 0
\(165\) 3.07154 0.239119
\(166\) 0 0
\(167\) 14.2028 1.09905 0.549524 0.835478i \(-0.314809\pi\)
0.549524 + 0.835478i \(0.314809\pi\)
\(168\) 0 0
\(169\) 12.7205 0.978501
\(170\) 0 0
\(171\) 5.07154 0.387830
\(172\) 0 0
\(173\) 5.57743 0.424044 0.212022 0.977265i \(-0.431995\pi\)
0.212022 + 0.977265i \(0.431995\pi\)
\(174\) 0 0
\(175\) −3.78872 −0.286400
\(176\) 0 0
\(177\) −2.29461 −0.172473
\(178\) 0 0
\(179\) −14.8518 −1.11008 −0.555038 0.831825i \(-0.687296\pi\)
−0.555038 + 0.831825i \(0.687296\pi\)
\(180\) 0 0
\(181\) 20.8518 1.54990 0.774951 0.632021i \(-0.217774\pi\)
0.774951 + 0.632021i \(0.217774\pi\)
\(182\) 0 0
\(183\) −3.07154 −0.227055
\(184\) 0 0
\(185\) −3.22307 −0.236965
\(186\) 0 0
\(187\) −10.0833 −0.737366
\(188\) 0 0
\(189\) −3.78872 −0.275588
\(190\) 0 0
\(191\) 19.0951 1.38167 0.690837 0.723011i \(-0.257242\pi\)
0.690837 + 0.723011i \(0.257242\pi\)
\(192\) 0 0
\(193\) −8.56565 −0.616569 −0.308284 0.951294i \(-0.599755\pi\)
−0.308284 + 0.951294i \(0.599755\pi\)
\(194\) 0 0
\(195\) 5.07154 0.363180
\(196\) 0 0
\(197\) 3.01178 0.214581 0.107290 0.994228i \(-0.465783\pi\)
0.107290 + 0.994228i \(0.465783\pi\)
\(198\) 0 0
\(199\) −22.8518 −1.61992 −0.809961 0.586484i \(-0.800512\pi\)
−0.809961 + 0.586484i \(0.800512\pi\)
\(200\) 0 0
\(201\) 3.22307 0.227338
\(202\) 0 0
\(203\) −4.86025 −0.341123
\(204\) 0 0
\(205\) 9.42590 0.658334
\(206\) 0 0
\(207\) 1.00000 0.0695048
\(208\) 0 0
\(209\) −15.5774 −1.07751
\(210\) 0 0
\(211\) 8.21128 0.565288 0.282644 0.959225i \(-0.408788\pi\)
0.282644 + 0.959225i \(0.408788\pi\)
\(212\) 0 0
\(213\) 7.28282 0.499011
\(214\) 0 0
\(215\) −12.1431 −0.828151
\(216\) 0 0
\(217\) −14.3544 −0.974438
\(218\) 0 0
\(219\) 9.63719 0.651221
\(220\) 0 0
\(221\) −16.6490 −1.11993
\(222\) 0 0
\(223\) 10.2862 0.688812 0.344406 0.938821i \(-0.388080\pi\)
0.344406 + 0.938821i \(0.388080\pi\)
\(224\) 0 0
\(225\) 1.00000 0.0666667
\(226\) 0 0
\(227\) 23.1549 1.53684 0.768421 0.639945i \(-0.221043\pi\)
0.768421 + 0.639945i \(0.221043\pi\)
\(228\) 0 0
\(229\) −23.7205 −1.56750 −0.783748 0.621079i \(-0.786694\pi\)
−0.783748 + 0.621079i \(0.786694\pi\)
\(230\) 0 0
\(231\) 11.6372 0.765671
\(232\) 0 0
\(233\) 2.70873 0.177455 0.0887273 0.996056i \(-0.471720\pi\)
0.0887273 + 0.996056i \(0.471720\pi\)
\(234\) 0 0
\(235\) −10.0833 −0.657763
\(236\) 0 0
\(237\) −8.00000 −0.519656
\(238\) 0 0
\(239\) 17.0033 1.09985 0.549927 0.835213i \(-0.314656\pi\)
0.549927 + 0.835213i \(0.314656\pi\)
\(240\) 0 0
\(241\) −2.36281 −0.152202 −0.0761011 0.997100i \(-0.524247\pi\)
−0.0761011 + 0.997100i \(0.524247\pi\)
\(242\) 0 0
\(243\) 1.00000 0.0641500
\(244\) 0 0
\(245\) −7.35436 −0.469853
\(246\) 0 0
\(247\) −25.7205 −1.63656
\(248\) 0 0
\(249\) 6.71718 0.425684
\(250\) 0 0
\(251\) 23.1313 1.46003 0.730017 0.683429i \(-0.239512\pi\)
0.730017 + 0.683429i \(0.239512\pi\)
\(252\) 0 0
\(253\) −3.07154 −0.193106
\(254\) 0 0
\(255\) −3.28282 −0.205578
\(256\) 0 0
\(257\) −0.505891 −0.0315566 −0.0157783 0.999876i \(-0.505023\pi\)
−0.0157783 + 0.999876i \(0.505023\pi\)
\(258\) 0 0
\(259\) −12.2113 −0.758772
\(260\) 0 0
\(261\) 1.28282 0.0794048
\(262\) 0 0
\(263\) 1.99155 0.122804 0.0614021 0.998113i \(-0.480443\pi\)
0.0614021 + 0.998113i \(0.480443\pi\)
\(264\) 0 0
\(265\) −5.84847 −0.359269
\(266\) 0 0
\(267\) 5.01178 0.306716
\(268\) 0 0
\(269\) −3.13975 −0.191434 −0.0957168 0.995409i \(-0.530514\pi\)
−0.0957168 + 0.995409i \(0.530514\pi\)
\(270\) 0 0
\(271\) 21.9318 1.33226 0.666131 0.745835i \(-0.267949\pi\)
0.666131 + 0.745835i \(0.267949\pi\)
\(272\) 0 0
\(273\) 19.2146 1.16292
\(274\) 0 0
\(275\) −3.07154 −0.185221
\(276\) 0 0
\(277\) 15.8636 0.953151 0.476575 0.879134i \(-0.341878\pi\)
0.476575 + 0.879134i \(0.341878\pi\)
\(278\) 0 0
\(279\) 3.78872 0.226824
\(280\) 0 0
\(281\) −1.37460 −0.0820015 −0.0410008 0.999159i \(-0.513055\pi\)
−0.0410008 + 0.999159i \(0.513055\pi\)
\(282\) 0 0
\(283\) 12.3544 0.734391 0.367195 0.930144i \(-0.380318\pi\)
0.367195 + 0.930144i \(0.380318\pi\)
\(284\) 0 0
\(285\) −5.07154 −0.300412
\(286\) 0 0
\(287\) 35.7121 2.10802
\(288\) 0 0
\(289\) −6.22307 −0.366063
\(290\) 0 0
\(291\) −1.13130 −0.0663177
\(292\) 0 0
\(293\) −29.7121 −1.73580 −0.867898 0.496742i \(-0.834530\pi\)
−0.867898 + 0.496742i \(0.834530\pi\)
\(294\) 0 0
\(295\) 2.29461 0.133597
\(296\) 0 0
\(297\) −3.07154 −0.178229
\(298\) 0 0
\(299\) −5.07154 −0.293295
\(300\) 0 0
\(301\) −46.0067 −2.65178
\(302\) 0 0
\(303\) −6.29461 −0.361616
\(304\) 0 0
\(305\) 3.07154 0.175876
\(306\) 0 0
\(307\) −30.4890 −1.74010 −0.870049 0.492965i \(-0.835913\pi\)
−0.870049 + 0.492965i \(0.835913\pi\)
\(308\) 0 0
\(309\) −12.7087 −0.722974
\(310\) 0 0
\(311\) 4.44613 0.252117 0.126059 0.992023i \(-0.459767\pi\)
0.126059 + 0.992023i \(0.459767\pi\)
\(312\) 0 0
\(313\) 6.35436 0.359170 0.179585 0.983742i \(-0.442525\pi\)
0.179585 + 0.983742i \(0.442525\pi\)
\(314\) 0 0
\(315\) 3.78872 0.213470
\(316\) 0 0
\(317\) 2.64897 0.148781 0.0743905 0.997229i \(-0.476299\pi\)
0.0743905 + 0.997229i \(0.476299\pi\)
\(318\) 0 0
\(319\) −3.94024 −0.220611
\(320\) 0 0
\(321\) 18.2946 1.02111
\(322\) 0 0
\(323\) 16.6490 0.926373
\(324\) 0 0
\(325\) −5.07154 −0.281318
\(326\) 0 0
\(327\) −4.50589 −0.249176
\(328\) 0 0
\(329\) −38.2028 −2.10619
\(330\) 0 0
\(331\) 4.80050 0.263859 0.131930 0.991259i \(-0.457883\pi\)
0.131930 + 0.991259i \(0.457883\pi\)
\(332\) 0 0
\(333\) 3.22307 0.176623
\(334\) 0 0
\(335\) −3.22307 −0.176095
\(336\) 0 0
\(337\) −14.7323 −0.802519 −0.401260 0.915964i \(-0.631427\pi\)
−0.401260 + 0.915964i \(0.631427\pi\)
\(338\) 0 0
\(339\) 4.71718 0.256202
\(340\) 0 0
\(341\) −11.6372 −0.630189
\(342\) 0 0
\(343\) −1.34258 −0.0724925
\(344\) 0 0
\(345\) −1.00000 −0.0538382
\(346\) 0 0
\(347\) −33.5841 −1.80289 −0.901444 0.432895i \(-0.857492\pi\)
−0.901444 + 0.432895i \(0.857492\pi\)
\(348\) 0 0
\(349\) −6.92001 −0.370420 −0.185210 0.982699i \(-0.559296\pi\)
−0.185210 + 0.982699i \(0.559296\pi\)
\(350\) 0 0
\(351\) −5.07154 −0.270699
\(352\) 0 0
\(353\) 8.38638 0.446362 0.223181 0.974777i \(-0.428356\pi\)
0.223181 + 0.974777i \(0.428356\pi\)
\(354\) 0 0
\(355\) −7.28282 −0.386532
\(356\) 0 0
\(357\) −12.4377 −0.658272
\(358\) 0 0
\(359\) 19.2146 1.01411 0.507054 0.861914i \(-0.330734\pi\)
0.507054 + 0.861914i \(0.330734\pi\)
\(360\) 0 0
\(361\) 6.72051 0.353711
\(362\) 0 0
\(363\) −1.56565 −0.0821752
\(364\) 0 0
\(365\) −9.63719 −0.504433
\(366\) 0 0
\(367\) 16.4974 0.861159 0.430580 0.902553i \(-0.358309\pi\)
0.430580 + 0.902553i \(0.358309\pi\)
\(368\) 0 0
\(369\) −9.42590 −0.490693
\(370\) 0 0
\(371\) −22.1582 −1.15040
\(372\) 0 0
\(373\) −5.45792 −0.282600 −0.141300 0.989967i \(-0.545128\pi\)
−0.141300 + 0.989967i \(0.545128\pi\)
\(374\) 0 0
\(375\) −1.00000 −0.0516398
\(376\) 0 0
\(377\) −6.50589 −0.335070
\(378\) 0 0
\(379\) 25.4175 1.30561 0.652803 0.757527i \(-0.273593\pi\)
0.652803 + 0.757527i \(0.273593\pi\)
\(380\) 0 0
\(381\) −12.0833 −0.619047
\(382\) 0 0
\(383\) 27.7121 1.41602 0.708010 0.706202i \(-0.249593\pi\)
0.708010 + 0.706202i \(0.249593\pi\)
\(384\) 0 0
\(385\) −11.6372 −0.593086
\(386\) 0 0
\(387\) 12.1431 0.617267
\(388\) 0 0
\(389\) 38.4528 1.94963 0.974817 0.223006i \(-0.0715868\pi\)
0.974817 + 0.223006i \(0.0715868\pi\)
\(390\) 0 0
\(391\) 3.28282 0.166020
\(392\) 0 0
\(393\) 17.1313 0.864160
\(394\) 0 0
\(395\) 8.00000 0.402524
\(396\) 0 0
\(397\) 8.28616 0.415870 0.207935 0.978143i \(-0.433326\pi\)
0.207935 + 0.978143i \(0.433326\pi\)
\(398\) 0 0
\(399\) −19.2146 −0.961934
\(400\) 0 0
\(401\) 5.72051 0.285669 0.142834 0.989747i \(-0.454378\pi\)
0.142834 + 0.989747i \(0.454378\pi\)
\(402\) 0 0
\(403\) −19.2146 −0.957148
\(404\) 0 0
\(405\) −1.00000 −0.0496904
\(406\) 0 0
\(407\) −9.89978 −0.490714
\(408\) 0 0
\(409\) −26.4974 −1.31021 −0.655107 0.755536i \(-0.727376\pi\)
−0.655107 + 0.755536i \(0.727376\pi\)
\(410\) 0 0
\(411\) −2.70873 −0.133612
\(412\) 0 0
\(413\) 8.69361 0.427785
\(414\) 0 0
\(415\) −6.71718 −0.329733
\(416\) 0 0
\(417\) −0.211285 −0.0103467
\(418\) 0 0
\(419\) 22.2264 1.08583 0.542915 0.839787i \(-0.317320\pi\)
0.542915 + 0.839787i \(0.317320\pi\)
\(420\) 0 0
\(421\) 18.9351 0.922842 0.461421 0.887181i \(-0.347340\pi\)
0.461421 + 0.887181i \(0.347340\pi\)
\(422\) 0 0
\(423\) 10.0833 0.490268
\(424\) 0 0
\(425\) 3.28282 0.159240
\(426\) 0 0
\(427\) 11.6372 0.563163
\(428\) 0 0
\(429\) 15.5774 0.752085
\(430\) 0 0
\(431\) 10.9882 0.529284 0.264642 0.964347i \(-0.414746\pi\)
0.264642 + 0.964347i \(0.414746\pi\)
\(432\) 0 0
\(433\) 18.3544 0.882054 0.441027 0.897494i \(-0.354614\pi\)
0.441027 + 0.897494i \(0.354614\pi\)
\(434\) 0 0
\(435\) −1.28282 −0.0615067
\(436\) 0 0
\(437\) 5.07154 0.242605
\(438\) 0 0
\(439\) 31.0184 1.48043 0.740215 0.672370i \(-0.234724\pi\)
0.740215 + 0.672370i \(0.234724\pi\)
\(440\) 0 0
\(441\) 7.35436 0.350208
\(442\) 0 0
\(443\) 7.09510 0.337099 0.168549 0.985693i \(-0.446092\pi\)
0.168549 + 0.985693i \(0.446092\pi\)
\(444\) 0 0
\(445\) −5.01178 −0.237581
\(446\) 0 0
\(447\) −17.6372 −0.834210
\(448\) 0 0
\(449\) 35.4326 1.67217 0.836083 0.548603i \(-0.184840\pi\)
0.836083 + 0.548603i \(0.184840\pi\)
\(450\) 0 0
\(451\) 28.9520 1.36330
\(452\) 0 0
\(453\) −13.7205 −0.644646
\(454\) 0 0
\(455\) −19.2146 −0.900795
\(456\) 0 0
\(457\) −0.514342 −0.0240599 −0.0120299 0.999928i \(-0.503829\pi\)
−0.0120299 + 0.999928i \(0.503829\pi\)
\(458\) 0 0
\(459\) 3.28282 0.153229
\(460\) 0 0
\(461\) −16.5892 −0.772637 −0.386318 0.922366i \(-0.626253\pi\)
−0.386318 + 0.922366i \(0.626253\pi\)
\(462\) 0 0
\(463\) 24.6725 1.14663 0.573315 0.819335i \(-0.305657\pi\)
0.573315 + 0.819335i \(0.305657\pi\)
\(464\) 0 0
\(465\) −3.78872 −0.175697
\(466\) 0 0
\(467\) 19.4259 0.898924 0.449462 0.893300i \(-0.351616\pi\)
0.449462 + 0.893300i \(0.351616\pi\)
\(468\) 0 0
\(469\) −12.2113 −0.563865
\(470\) 0 0
\(471\) 13.4857 0.621386
\(472\) 0 0
\(473\) −37.2979 −1.71496
\(474\) 0 0
\(475\) 5.07154 0.232698
\(476\) 0 0
\(477\) 5.84847 0.267783
\(478\) 0 0
\(479\) −19.2146 −0.877938 −0.438969 0.898502i \(-0.644656\pi\)
−0.438969 + 0.898502i \(0.644656\pi\)
\(480\) 0 0
\(481\) −16.3459 −0.745309
\(482\) 0 0
\(483\) −3.78872 −0.172393
\(484\) 0 0
\(485\) 1.13130 0.0513695
\(486\) 0 0
\(487\) −43.5243 −1.97228 −0.986138 0.165927i \(-0.946938\pi\)
−0.986138 + 0.165927i \(0.946938\pi\)
\(488\) 0 0
\(489\) −7.57743 −0.342663
\(490\) 0 0
\(491\) 1.82491 0.0823569 0.0411784 0.999152i \(-0.486889\pi\)
0.0411784 + 0.999152i \(0.486889\pi\)
\(492\) 0 0
\(493\) 4.21128 0.189667
\(494\) 0 0
\(495\) 3.07154 0.138055
\(496\) 0 0
\(497\) −27.5925 −1.23769
\(498\) 0 0
\(499\) 14.6405 0.655400 0.327700 0.944782i \(-0.393727\pi\)
0.327700 + 0.944782i \(0.393727\pi\)
\(500\) 0 0
\(501\) 14.2028 0.634536
\(502\) 0 0
\(503\) −6.88382 −0.306934 −0.153467 0.988154i \(-0.549044\pi\)
−0.153467 + 0.988154i \(0.549044\pi\)
\(504\) 0 0
\(505\) 6.29461 0.280106
\(506\) 0 0
\(507\) 12.7205 0.564938
\(508\) 0 0
\(509\) −28.0000 −1.24108 −0.620539 0.784176i \(-0.713086\pi\)
−0.620539 + 0.784176i \(0.713086\pi\)
\(510\) 0 0
\(511\) −36.5126 −1.61522
\(512\) 0 0
\(513\) 5.07154 0.223914
\(514\) 0 0
\(515\) 12.7087 0.560013
\(516\) 0 0
\(517\) −30.9713 −1.36212
\(518\) 0 0
\(519\) 5.57743 0.244822
\(520\) 0 0
\(521\) 33.9469 1.48724 0.743621 0.668602i \(-0.233107\pi\)
0.743621 + 0.668602i \(0.233107\pi\)
\(522\) 0 0
\(523\) 13.5774 0.593700 0.296850 0.954924i \(-0.404064\pi\)
0.296850 + 0.954924i \(0.404064\pi\)
\(524\) 0 0
\(525\) −3.78872 −0.165353
\(526\) 0 0
\(527\) 12.4377 0.541794
\(528\) 0 0
\(529\) 1.00000 0.0434783
\(530\) 0 0
\(531\) −2.29461 −0.0995774
\(532\) 0 0
\(533\) 47.8038 2.07061
\(534\) 0 0
\(535\) −18.2946 −0.790945
\(536\) 0 0
\(537\) −14.8518 −0.640903
\(538\) 0 0
\(539\) −22.5892 −0.972986
\(540\) 0 0
\(541\) −16.6852 −0.717351 −0.358676 0.933462i \(-0.616772\pi\)
−0.358676 + 0.933462i \(0.616772\pi\)
\(542\) 0 0
\(543\) 20.8518 0.894837
\(544\) 0 0
\(545\) 4.50589 0.193011
\(546\) 0 0
\(547\) 13.4175 0.573689 0.286844 0.957977i \(-0.407394\pi\)
0.286844 + 0.957977i \(0.407394\pi\)
\(548\) 0 0
\(549\) −3.07154 −0.131090
\(550\) 0 0
\(551\) 6.50589 0.277160
\(552\) 0 0
\(553\) 30.3097 1.28890
\(554\) 0 0
\(555\) −3.22307 −0.136812
\(556\) 0 0
\(557\) 35.0269 1.48414 0.742069 0.670324i \(-0.233845\pi\)
0.742069 + 0.670324i \(0.233845\pi\)
\(558\) 0 0
\(559\) −61.5841 −2.60473
\(560\) 0 0
\(561\) −10.0833 −0.425718
\(562\) 0 0
\(563\) −33.7356 −1.42179 −0.710894 0.703300i \(-0.751709\pi\)
−0.710894 + 0.703300i \(0.751709\pi\)
\(564\) 0 0
\(565\) −4.71718 −0.198453
\(566\) 0 0
\(567\) −3.78872 −0.159111
\(568\) 0 0
\(569\) −19.4579 −0.815718 −0.407859 0.913045i \(-0.633725\pi\)
−0.407859 + 0.913045i \(0.633725\pi\)
\(570\) 0 0
\(571\) 39.6843 1.66074 0.830369 0.557215i \(-0.188130\pi\)
0.830369 + 0.557215i \(0.188130\pi\)
\(572\) 0 0
\(573\) 19.0951 0.797709
\(574\) 0 0
\(575\) 1.00000 0.0417029
\(576\) 0 0
\(577\) −37.8636 −1.57628 −0.788141 0.615495i \(-0.788956\pi\)
−0.788141 + 0.615495i \(0.788956\pi\)
\(578\) 0 0
\(579\) −8.56565 −0.355976
\(580\) 0 0
\(581\) −25.4495 −1.05582
\(582\) 0 0
\(583\) −17.9638 −0.743985
\(584\) 0 0
\(585\) 5.07154 0.209682
\(586\) 0 0
\(587\) 9.55387 0.394330 0.197165 0.980370i \(-0.436826\pi\)
0.197165 + 0.980370i \(0.436826\pi\)
\(588\) 0 0
\(589\) 19.2146 0.791725
\(590\) 0 0
\(591\) 3.01178 0.123888
\(592\) 0 0
\(593\) −22.9351 −0.941833 −0.470916 0.882178i \(-0.656077\pi\)
−0.470916 + 0.882178i \(0.656077\pi\)
\(594\) 0 0
\(595\) 12.4377 0.509895
\(596\) 0 0
\(597\) −22.8518 −0.935262
\(598\) 0 0
\(599\) −25.6969 −1.04995 −0.524974 0.851118i \(-0.675925\pi\)
−0.524974 + 0.851118i \(0.675925\pi\)
\(600\) 0 0
\(601\) −17.0800 −0.696707 −0.348354 0.937363i \(-0.613259\pi\)
−0.348354 + 0.937363i \(0.613259\pi\)
\(602\) 0 0
\(603\) 3.22307 0.131253
\(604\) 0 0
\(605\) 1.56565 0.0636526
\(606\) 0 0
\(607\) −24.2028 −0.982363 −0.491181 0.871057i \(-0.663435\pi\)
−0.491181 + 0.871057i \(0.663435\pi\)
\(608\) 0 0
\(609\) −4.86025 −0.196947
\(610\) 0 0
\(611\) −51.1380 −2.06882
\(612\) 0 0
\(613\) 3.73741 0.150953 0.0754763 0.997148i \(-0.475952\pi\)
0.0754763 + 0.997148i \(0.475952\pi\)
\(614\) 0 0
\(615\) 9.42590 0.380089
\(616\) 0 0
\(617\) −14.3182 −0.576428 −0.288214 0.957566i \(-0.593061\pi\)
−0.288214 + 0.957566i \(0.593061\pi\)
\(618\) 0 0
\(619\) −37.4175 −1.50393 −0.751967 0.659201i \(-0.770895\pi\)
−0.751967 + 0.659201i \(0.770895\pi\)
\(620\) 0 0
\(621\) 1.00000 0.0401286
\(622\) 0 0
\(623\) −18.9882 −0.760747
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) −15.5774 −0.622103
\(628\) 0 0
\(629\) 10.5808 0.421883
\(630\) 0 0
\(631\) −42.6725 −1.69877 −0.849383 0.527776i \(-0.823026\pi\)
−0.849383 + 0.527776i \(0.823026\pi\)
\(632\) 0 0
\(633\) 8.21128 0.326369
\(634\) 0 0
\(635\) 12.0833 0.479512
\(636\) 0 0
\(637\) −37.2979 −1.47780
\(638\) 0 0
\(639\) 7.28282 0.288104
\(640\) 0 0
\(641\) 3.39816 0.134219 0.0671097 0.997746i \(-0.478622\pi\)
0.0671097 + 0.997746i \(0.478622\pi\)
\(642\) 0 0
\(643\) 16.5210 0.651525 0.325762 0.945452i \(-0.394379\pi\)
0.325762 + 0.945452i \(0.394379\pi\)
\(644\) 0 0
\(645\) −12.1431 −0.478133
\(646\) 0 0
\(647\) −37.9469 −1.49185 −0.745923 0.666032i \(-0.767992\pi\)
−0.745923 + 0.666032i \(0.767992\pi\)
\(648\) 0 0
\(649\) 7.04797 0.276657
\(650\) 0 0
\(651\) −14.3544 −0.562592
\(652\) 0 0
\(653\) 19.0715 0.746327 0.373163 0.927766i \(-0.378273\pi\)
0.373163 + 0.927766i \(0.378273\pi\)
\(654\) 0 0
\(655\) −17.1313 −0.669375
\(656\) 0 0
\(657\) 9.63719 0.375982
\(658\) 0 0
\(659\) 21.5008 0.837551 0.418776 0.908090i \(-0.362459\pi\)
0.418776 + 0.908090i \(0.362459\pi\)
\(660\) 0 0
\(661\) 2.11951 0.0824395 0.0412197 0.999150i \(-0.486876\pi\)
0.0412197 + 0.999150i \(0.486876\pi\)
\(662\) 0 0
\(663\) −16.6490 −0.646592
\(664\) 0 0
\(665\) 19.2146 0.745111
\(666\) 0 0
\(667\) 1.28282 0.0496711
\(668\) 0 0
\(669\) 10.2862 0.397686
\(670\) 0 0
\(671\) 9.43435 0.364209
\(672\) 0 0
\(673\) 8.03619 0.309772 0.154886 0.987932i \(-0.450499\pi\)
0.154886 + 0.987932i \(0.450499\pi\)
\(674\) 0 0
\(675\) 1.00000 0.0384900
\(676\) 0 0
\(677\) 27.5690 1.05956 0.529781 0.848134i \(-0.322274\pi\)
0.529781 + 0.848134i \(0.322274\pi\)
\(678\) 0 0
\(679\) 4.28616 0.164488
\(680\) 0 0
\(681\) 23.1549 0.887296
\(682\) 0 0
\(683\) −9.35770 −0.358062 −0.179031 0.983843i \(-0.557296\pi\)
−0.179031 + 0.983843i \(0.557296\pi\)
\(684\) 0 0
\(685\) 2.70873 0.103495
\(686\) 0 0
\(687\) −23.7205 −0.904994
\(688\) 0 0
\(689\) −29.6608 −1.12998
\(690\) 0 0
\(691\) −44.8754 −1.70714 −0.853570 0.520979i \(-0.825567\pi\)
−0.853570 + 0.520979i \(0.825567\pi\)
\(692\) 0 0
\(693\) 11.6372 0.442060
\(694\) 0 0
\(695\) 0.211285 0.00801449
\(696\) 0 0
\(697\) −30.9436 −1.17207
\(698\) 0 0
\(699\) 2.70873 0.102453
\(700\) 0 0
\(701\) −28.9116 −1.09197 −0.545987 0.837793i \(-0.683845\pi\)
−0.545987 + 0.837793i \(0.683845\pi\)
\(702\) 0 0
\(703\) 16.3459 0.616498
\(704\) 0 0
\(705\) −10.0833 −0.379760
\(706\) 0 0
\(707\) 23.8485 0.896914
\(708\) 0 0
\(709\) −52.7751 −1.98201 −0.991006 0.133817i \(-0.957277\pi\)
−0.991006 + 0.133817i \(0.957277\pi\)
\(710\) 0 0
\(711\) −8.00000 −0.300023
\(712\) 0 0
\(713\) 3.78872 0.141888
\(714\) 0 0
\(715\) −15.5774 −0.582563
\(716\) 0 0
\(717\) 17.0033 0.635001
\(718\) 0 0
\(719\) −45.4090 −1.69347 −0.846735 0.532015i \(-0.821435\pi\)
−0.846735 + 0.532015i \(0.821435\pi\)
\(720\) 0 0
\(721\) 48.1497 1.79319
\(722\) 0 0
\(723\) −2.36281 −0.0878740
\(724\) 0 0
\(725\) 1.28282 0.0476429
\(726\) 0 0
\(727\) 32.2415 1.19577 0.597886 0.801581i \(-0.296008\pi\)
0.597886 + 0.801581i \(0.296008\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) 39.8636 1.47441
\(732\) 0 0
\(733\) 39.3897 1.45489 0.727446 0.686165i \(-0.240707\pi\)
0.727446 + 0.686165i \(0.240707\pi\)
\(734\) 0 0
\(735\) −7.35436 −0.271270
\(736\) 0 0
\(737\) −9.89978 −0.364663
\(738\) 0 0
\(739\) −23.4857 −0.863934 −0.431967 0.901889i \(-0.642180\pi\)
−0.431967 + 0.901889i \(0.642180\pi\)
\(740\) 0 0
\(741\) −25.7205 −0.944866
\(742\) 0 0
\(743\) −37.1313 −1.36222 −0.681108 0.732183i \(-0.738501\pi\)
−0.681108 + 0.732183i \(0.738501\pi\)
\(744\) 0 0
\(745\) 17.6372 0.646177
\(746\) 0 0
\(747\) 6.71718 0.245769
\(748\) 0 0
\(749\) −69.3131 −2.53264
\(750\) 0 0
\(751\) −14.0833 −0.513908 −0.256954 0.966424i \(-0.582719\pi\)
−0.256954 + 0.966424i \(0.582719\pi\)
\(752\) 0 0
\(753\) 23.1313 0.842951
\(754\) 0 0
\(755\) 13.7205 0.499340
\(756\) 0 0
\(757\) 44.8072 1.62854 0.814272 0.580483i \(-0.197136\pi\)
0.814272 + 0.580483i \(0.197136\pi\)
\(758\) 0 0
\(759\) −3.07154 −0.111490
\(760\) 0 0
\(761\) 3.56898 0.129375 0.0646877 0.997906i \(-0.479395\pi\)
0.0646877 + 0.997906i \(0.479395\pi\)
\(762\) 0 0
\(763\) 17.0715 0.618031
\(764\) 0 0
\(765\) −3.28282 −0.118691
\(766\) 0 0
\(767\) 11.6372 0.420194
\(768\) 0 0
\(769\) −43.6439 −1.57384 −0.786919 0.617057i \(-0.788325\pi\)
−0.786919 + 0.617057i \(0.788325\pi\)
\(770\) 0 0
\(771\) −0.505891 −0.0182192
\(772\) 0 0
\(773\) 13.8805 0.499246 0.249623 0.968343i \(-0.419693\pi\)
0.249623 + 0.968343i \(0.419693\pi\)
\(774\) 0 0
\(775\) 3.78872 0.136095
\(776\) 0 0
\(777\) −12.2113 −0.438077
\(778\) 0 0
\(779\) −47.8038 −1.71275
\(780\) 0 0
\(781\) −22.3695 −0.800443
\(782\) 0 0
\(783\) 1.28282 0.0458444
\(784\) 0 0
\(785\) −13.4857 −0.481324
\(786\) 0 0
\(787\) 37.6523 1.34216 0.671080 0.741385i \(-0.265831\pi\)
0.671080 + 0.741385i \(0.265831\pi\)
\(788\) 0 0
\(789\) 1.99155 0.0709010
\(790\) 0 0
\(791\) −17.8720 −0.635456
\(792\) 0 0
\(793\) 15.5774 0.553171
\(794\) 0 0
\(795\) −5.84847 −0.207424
\(796\) 0 0
\(797\) 2.15153 0.0762110 0.0381055 0.999274i \(-0.487868\pi\)
0.0381055 + 0.999274i \(0.487868\pi\)
\(798\) 0 0
\(799\) 33.1018 1.17106
\(800\) 0 0
\(801\) 5.01178 0.177083
\(802\) 0 0
\(803\) −29.6010 −1.04460
\(804\) 0 0
\(805\) 3.78872 0.133535
\(806\) 0 0
\(807\) −3.13975 −0.110524
\(808\) 0 0
\(809\) 14.5572 0.511804 0.255902 0.966703i \(-0.417628\pi\)
0.255902 + 0.966703i \(0.417628\pi\)
\(810\) 0 0
\(811\) 2.47388 0.0868695 0.0434348 0.999056i \(-0.486170\pi\)
0.0434348 + 0.999056i \(0.486170\pi\)
\(812\) 0 0
\(813\) 21.9318 0.769182
\(814\) 0 0
\(815\) 7.57743 0.265426
\(816\) 0 0
\(817\) 61.5841 2.15455
\(818\) 0 0
\(819\) 19.2146 0.671413
\(820\) 0 0
\(821\) 32.3333 1.12844 0.564220 0.825625i \(-0.309177\pi\)
0.564220 + 0.825625i \(0.309177\pi\)
\(822\) 0 0
\(823\) 14.7087 0.512714 0.256357 0.966582i \(-0.417478\pi\)
0.256357 + 0.966582i \(0.417478\pi\)
\(824\) 0 0
\(825\) −3.07154 −0.106937
\(826\) 0 0
\(827\) 19.5623 0.680248 0.340124 0.940381i \(-0.389531\pi\)
0.340124 + 0.940381i \(0.389531\pi\)
\(828\) 0 0
\(829\) 31.5092 1.09436 0.547180 0.837015i \(-0.315701\pi\)
0.547180 + 0.837015i \(0.315701\pi\)
\(830\) 0 0
\(831\) 15.8636 0.550302
\(832\) 0 0
\(833\) 24.1431 0.836508
\(834\) 0 0
\(835\) −14.2028 −0.491509
\(836\) 0 0
\(837\) 3.78872 0.130957
\(838\) 0 0
\(839\) 20.9949 0.724824 0.362412 0.932018i \(-0.381953\pi\)
0.362412 + 0.932018i \(0.381953\pi\)
\(840\) 0 0
\(841\) −27.3544 −0.943254
\(842\) 0 0
\(843\) −1.37460 −0.0473436
\(844\) 0 0
\(845\) −12.7205 −0.437599
\(846\) 0 0
\(847\) 5.93179 0.203819
\(848\) 0 0
\(849\) 12.3544 0.424001
\(850\) 0 0
\(851\) 3.22307 0.110485
\(852\) 0 0
\(853\) −50.0067 −1.71220 −0.856098 0.516814i \(-0.827118\pi\)
−0.856098 + 0.516814i \(0.827118\pi\)
\(854\) 0 0
\(855\) −5.07154 −0.173443
\(856\) 0 0
\(857\) −13.3939 −0.457526 −0.228763 0.973482i \(-0.573468\pi\)
−0.228763 + 0.973482i \(0.573468\pi\)
\(858\) 0 0
\(859\) −0.633854 −0.0216268 −0.0108134 0.999942i \(-0.503442\pi\)
−0.0108134 + 0.999942i \(0.503442\pi\)
\(860\) 0 0
\(861\) 35.7121 1.21706
\(862\) 0 0
\(863\) −29.1144 −0.991066 −0.495533 0.868589i \(-0.665027\pi\)
−0.495533 + 0.868589i \(0.665027\pi\)
\(864\) 0 0
\(865\) −5.57743 −0.189638
\(866\) 0 0
\(867\) −6.22307 −0.211346
\(868\) 0 0
\(869\) 24.5723 0.833559
\(870\) 0 0
\(871\) −16.3459 −0.553860
\(872\) 0 0
\(873\) −1.13130 −0.0382886
\(874\) 0 0
\(875\) 3.78872 0.128082
\(876\) 0 0
\(877\) 56.3164 1.90167 0.950835 0.309699i \(-0.100228\pi\)
0.950835 + 0.309699i \(0.100228\pi\)
\(878\) 0 0
\(879\) −29.7121 −1.00216
\(880\) 0 0
\(881\) −1.79717 −0.0605480 −0.0302740 0.999542i \(-0.509638\pi\)
−0.0302740 + 0.999542i \(0.509638\pi\)
\(882\) 0 0
\(883\) −47.6843 −1.60471 −0.802353 0.596850i \(-0.796419\pi\)
−0.802353 + 0.596850i \(0.796419\pi\)
\(884\) 0 0
\(885\) 2.29461 0.0771324
\(886\) 0 0
\(887\) 4.87537 0.163699 0.0818494 0.996645i \(-0.473917\pi\)
0.0818494 + 0.996645i \(0.473917\pi\)
\(888\) 0 0
\(889\) 45.7803 1.53542
\(890\) 0 0
\(891\) −3.07154 −0.102900
\(892\) 0 0
\(893\) 51.1380 1.71127
\(894\) 0 0
\(895\) 14.8518 0.496441
\(896\) 0 0
\(897\) −5.07154 −0.169334
\(898\) 0 0
\(899\) 4.86025 0.162099
\(900\) 0 0
\(901\) 19.1995 0.639628
\(902\) 0 0
\(903\) −46.0067 −1.53101
\(904\) 0 0
\(905\) −20.8518 −0.693137
\(906\) 0 0
\(907\) −25.6692 −0.852332 −0.426166 0.904645i \(-0.640136\pi\)
−0.426166 + 0.904645i \(0.640136\pi\)
\(908\) 0 0
\(909\) −6.29461 −0.208779
\(910\) 0 0
\(911\) 17.9764 0.595586 0.297793 0.954630i \(-0.403750\pi\)
0.297793 + 0.954630i \(0.403750\pi\)
\(912\) 0 0
\(913\) −20.6321 −0.682822
\(914\) 0 0
\(915\) 3.07154 0.101542
\(916\) 0 0
\(917\) −64.9056 −2.14337
\(918\) 0 0
\(919\) −32.0000 −1.05558 −0.527791 0.849374i \(-0.676980\pi\)
−0.527791 + 0.849374i \(0.676980\pi\)
\(920\) 0 0
\(921\) −30.4890 −1.00465
\(922\) 0 0
\(923\) −36.9351 −1.21573
\(924\) 0 0
\(925\) 3.22307 0.105974
\(926\) 0 0
\(927\) −12.7087 −0.417409
\(928\) 0 0
\(929\) 12.1582 0.398897 0.199449 0.979908i \(-0.436085\pi\)
0.199449 + 0.979908i \(0.436085\pi\)
\(930\) 0 0
\(931\) 37.2979 1.22239
\(932\) 0 0
\(933\) 4.44613 0.145560
\(934\) 0 0
\(935\) 10.0833 0.329760
\(936\) 0 0
\(937\) −9.97643 −0.325916 −0.162958 0.986633i \(-0.552103\pi\)
−0.162958 + 0.986633i \(0.552103\pi\)
\(938\) 0 0
\(939\) 6.35436 0.207367
\(940\) 0 0
\(941\) 10.7685 0.351042 0.175521 0.984476i \(-0.443839\pi\)
0.175521 + 0.984476i \(0.443839\pi\)
\(942\) 0 0
\(943\) −9.42590 −0.306950
\(944\) 0 0
\(945\) 3.78872 0.123247
\(946\) 0 0
\(947\) 17.8569 0.580272 0.290136 0.956985i \(-0.406299\pi\)
0.290136 + 0.956985i \(0.406299\pi\)
\(948\) 0 0
\(949\) −48.8754 −1.58656
\(950\) 0 0
\(951\) 2.64897 0.0858987
\(952\) 0 0
\(953\) −23.4175 −0.758566 −0.379283 0.925281i \(-0.623829\pi\)
−0.379283 + 0.925281i \(0.623829\pi\)
\(954\) 0 0
\(955\) −19.0951 −0.617903
\(956\) 0 0
\(957\) −3.94024 −0.127370
\(958\) 0 0
\(959\) 10.2626 0.331396
\(960\) 0 0
\(961\) −16.6456 −0.536956
\(962\) 0 0
\(963\) 18.2946 0.589535
\(964\) 0 0
\(965\) 8.56565 0.275738
\(966\) 0 0
\(967\) −32.0833 −1.03173 −0.515865 0.856670i \(-0.672529\pi\)
−0.515865 + 0.856670i \(0.672529\pi\)
\(968\) 0 0
\(969\) 16.6490 0.534842
\(970\) 0 0
\(971\) 55.1615 1.77022 0.885109 0.465384i \(-0.154084\pi\)
0.885109 + 0.465384i \(0.154084\pi\)
\(972\) 0 0
\(973\) 0.800498 0.0256628
\(974\) 0 0
\(975\) −5.07154 −0.162419
\(976\) 0 0
\(977\) 58.7069 1.87820 0.939101 0.343642i \(-0.111661\pi\)
0.939101 + 0.343642i \(0.111661\pi\)
\(978\) 0 0
\(979\) −15.3939 −0.491991
\(980\) 0 0
\(981\) −4.50589 −0.143862
\(982\) 0 0
\(983\) −27.5925 −0.880066 −0.440033 0.897982i \(-0.645033\pi\)
−0.440033 + 0.897982i \(0.645033\pi\)
\(984\) 0 0
\(985\) −3.01178 −0.0959634
\(986\) 0 0
\(987\) −38.2028 −1.21601
\(988\) 0 0
\(989\) 12.1431 0.386127
\(990\) 0 0
\(991\) 52.8307 1.67822 0.839112 0.543959i \(-0.183075\pi\)
0.839112 + 0.543959i \(0.183075\pi\)
\(992\) 0 0
\(993\) 4.80050 0.152339
\(994\) 0 0
\(995\) 22.8518 0.724451
\(996\) 0 0
\(997\) 34.4292 1.09038 0.545192 0.838311i \(-0.316457\pi\)
0.545192 + 0.838311i \(0.316457\pi\)
\(998\) 0 0
\(999\) 3.22307 0.101973
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 2760.2.a.s.1.1 3
3.2 odd 2 8280.2.a.bn.1.1 3
4.3 odd 2 5520.2.a.bw.1.3 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2760.2.a.s.1.1 3 1.1 even 1 trivial
5520.2.a.bw.1.3 3 4.3 odd 2
8280.2.a.bn.1.1 3 3.2 odd 2