Properties

Label 1452.4.a.q.1.3
Level $1452$
Weight $4$
Character 1452.1
Self dual yes
Analytic conductor $85.671$
Analytic rank $0$
Dimension $4$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1452,4,Mod(1,1452)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1452, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1452.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1452 = 2^{2} \cdot 3 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1452.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: \(85.6707733283\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: 4.4.20959101.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 2x^{3} - 90x^{2} + 91x + 2026 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{5}\cdot 3 \)
Twist minimal: yes
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(7.73893\) of defining polynomial
Character \(\chi\) \(=\) 1452.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+3.00000 q^{3} +16.3041 q^{5} -33.7213 q^{7} +9.00000 q^{9} +O(q^{10})\) \(q+3.00000 q^{3} +16.3041 q^{5} -33.7213 q^{7} +9.00000 q^{9} +53.1459 q^{13} +48.9124 q^{15} -86.8671 q^{17} +154.310 q^{19} -101.164 q^{21} +185.345 q^{23} +140.825 q^{25} +27.0000 q^{27} -115.460 q^{29} -193.474 q^{31} -549.796 q^{35} -123.825 q^{37} +159.438 q^{39} +144.054 q^{41} -317.788 q^{43} +146.737 q^{45} +492.387 q^{47} +794.124 q^{49} -260.601 q^{51} +31.4380 q^{53} +462.929 q^{57} +202.691 q^{59} -284.067 q^{61} -303.491 q^{63} +866.497 q^{65} +275.299 q^{67} +556.036 q^{69} +646.046 q^{71} -38.8492 q^{73} +422.474 q^{75} +1016.77 q^{79} +81.0000 q^{81} +1068.82 q^{83} -1416.29 q^{85} -346.381 q^{87} -1052.08 q^{89} -1792.15 q^{91} -580.423 q^{93} +2515.89 q^{95} +292.876 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 12 q^{3} + 12 q^{5} + 36 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 12 q^{3} + 12 q^{5} + 36 q^{9} + 36 q^{15} + 156 q^{23} + 244 q^{25} + 108 q^{27} + 184 q^{31} - 176 q^{37} + 108 q^{45} + 852 q^{47} + 1580 q^{49} + 924 q^{53} - 360 q^{59} - 176 q^{67} + 468 q^{69} + 3276 q^{71} + 732 q^{75} + 324 q^{81} - 3144 q^{89} - 144 q^{91} + 552 q^{93} + 2768 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Coefficient data

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



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 3.00000 0.577350
\(4\) 0 0
\(5\) 16.3041 1.45829 0.729143 0.684361i \(-0.239919\pi\)
0.729143 + 0.684361i \(0.239919\pi\)
\(6\) 0 0
\(7\) −33.7213 −1.82078 −0.910389 0.413754i \(-0.864217\pi\)
−0.910389 + 0.413754i \(0.864217\pi\)
\(8\) 0 0
\(9\) 9.00000 0.333333
\(10\) 0 0
\(11\) 0 0
\(12\) 0 0
\(13\) 53.1459 1.13385 0.566923 0.823771i \(-0.308134\pi\)
0.566923 + 0.823771i \(0.308134\pi\)
\(14\) 0 0
\(15\) 48.9124 0.841942
\(16\) 0 0
\(17\) −86.8671 −1.23932 −0.619658 0.784872i \(-0.712729\pi\)
−0.619658 + 0.784872i \(0.712729\pi\)
\(18\) 0 0
\(19\) 154.310 1.86321 0.931607 0.363467i \(-0.118407\pi\)
0.931607 + 0.363467i \(0.118407\pi\)
\(20\) 0 0
\(21\) −101.164 −1.05123
\(22\) 0 0
\(23\) 185.345 1.68031 0.840157 0.542344i \(-0.182463\pi\)
0.840157 + 0.542344i \(0.182463\pi\)
\(24\) 0 0
\(25\) 140.825 1.12660
\(26\) 0 0
\(27\) 27.0000 0.192450
\(28\) 0 0
\(29\) −115.460 −0.739327 −0.369663 0.929166i \(-0.620527\pi\)
−0.369663 + 0.929166i \(0.620527\pi\)
\(30\) 0 0
\(31\) −193.474 −1.12094 −0.560468 0.828176i \(-0.689379\pi\)
−0.560468 + 0.828176i \(0.689379\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −549.796 −2.65521
\(36\) 0 0
\(37\) −123.825 −0.550180 −0.275090 0.961418i \(-0.588708\pi\)
−0.275090 + 0.961418i \(0.588708\pi\)
\(38\) 0 0
\(39\) 159.438 0.654627
\(40\) 0 0
\(41\) 144.054 0.548718 0.274359 0.961627i \(-0.411534\pi\)
0.274359 + 0.961627i \(0.411534\pi\)
\(42\) 0 0
\(43\) −317.788 −1.12703 −0.563514 0.826106i \(-0.690551\pi\)
−0.563514 + 0.826106i \(0.690551\pi\)
\(44\) 0 0
\(45\) 146.737 0.486095
\(46\) 0 0
\(47\) 492.387 1.52813 0.764064 0.645141i \(-0.223201\pi\)
0.764064 + 0.645141i \(0.223201\pi\)
\(48\) 0 0
\(49\) 794.124 2.31523
\(50\) 0 0
\(51\) −260.601 −0.715519
\(52\) 0 0
\(53\) 31.4380 0.0814781 0.0407390 0.999170i \(-0.487029\pi\)
0.0407390 + 0.999170i \(0.487029\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 462.929 1.07573
\(58\) 0 0
\(59\) 202.691 0.447256 0.223628 0.974675i \(-0.428210\pi\)
0.223628 + 0.974675i \(0.428210\pi\)
\(60\) 0 0
\(61\) −284.067 −0.596246 −0.298123 0.954527i \(-0.596361\pi\)
−0.298123 + 0.954527i \(0.596361\pi\)
\(62\) 0 0
\(63\) −303.491 −0.606926
\(64\) 0 0
\(65\) 866.497 1.65347
\(66\) 0 0
\(67\) 275.299 0.501987 0.250994 0.967989i \(-0.419243\pi\)
0.250994 + 0.967989i \(0.419243\pi\)
\(68\) 0 0
\(69\) 556.036 0.970129
\(70\) 0 0
\(71\) 646.046 1.07988 0.539941 0.841703i \(-0.318447\pi\)
0.539941 + 0.841703i \(0.318447\pi\)
\(72\) 0 0
\(73\) −38.8492 −0.0622870 −0.0311435 0.999515i \(-0.509915\pi\)
−0.0311435 + 0.999515i \(0.509915\pi\)
\(74\) 0 0
\(75\) 422.474 0.650442
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 1016.77 1.44804 0.724020 0.689779i \(-0.242292\pi\)
0.724020 + 0.689779i \(0.242292\pi\)
\(80\) 0 0
\(81\) 81.0000 0.111111
\(82\) 0 0
\(83\) 1068.82 1.41348 0.706740 0.707474i \(-0.250165\pi\)
0.706740 + 0.707474i \(0.250165\pi\)
\(84\) 0 0
\(85\) −1416.29 −1.80728
\(86\) 0 0
\(87\) −346.381 −0.426850
\(88\) 0 0
\(89\) −1052.08 −1.25304 −0.626520 0.779405i \(-0.715521\pi\)
−0.626520 + 0.779405i \(0.715521\pi\)
\(90\) 0 0
\(91\) −1792.15 −2.06448
\(92\) 0 0
\(93\) −580.423 −0.647173
\(94\) 0 0
\(95\) 2515.89 2.71710
\(96\) 0 0
\(97\) 292.876 0.306568 0.153284 0.988182i \(-0.451015\pi\)
0.153284 + 0.988182i \(0.451015\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) −605.896 −0.596920 −0.298460 0.954422i \(-0.596473\pi\)
−0.298460 + 0.954422i \(0.596473\pi\)
\(102\) 0 0
\(103\) 34.8760 0.0333634 0.0166817 0.999861i \(-0.494690\pi\)
0.0166817 + 0.999861i \(0.494690\pi\)
\(104\) 0 0
\(105\) −1649.39 −1.53299
\(106\) 0 0
\(107\) 1620.80 1.46438 0.732188 0.681103i \(-0.238499\pi\)
0.732188 + 0.681103i \(0.238499\pi\)
\(108\) 0 0
\(109\) 592.686 0.520817 0.260408 0.965499i \(-0.416143\pi\)
0.260408 + 0.965499i \(0.416143\pi\)
\(110\) 0 0
\(111\) −371.474 −0.317647
\(112\) 0 0
\(113\) 1954.17 1.62684 0.813418 0.581680i \(-0.197604\pi\)
0.813418 + 0.581680i \(0.197604\pi\)
\(114\) 0 0
\(115\) 3021.90 2.45038
\(116\) 0 0
\(117\) 478.313 0.377949
\(118\) 0 0
\(119\) 2929.27 2.25652
\(120\) 0 0
\(121\) 0 0
\(122\) 0 0
\(123\) 432.162 0.316802
\(124\) 0 0
\(125\) 258.010 0.184617
\(126\) 0 0
\(127\) 477.226 0.333440 0.166720 0.986004i \(-0.446682\pi\)
0.166720 + 0.986004i \(0.446682\pi\)
\(128\) 0 0
\(129\) −953.364 −0.650690
\(130\) 0 0
\(131\) 1475.65 0.984187 0.492094 0.870542i \(-0.336232\pi\)
0.492094 + 0.870542i \(0.336232\pi\)
\(132\) 0 0
\(133\) −5203.52 −3.39250
\(134\) 0 0
\(135\) 440.212 0.280647
\(136\) 0 0
\(137\) −1261.54 −0.786718 −0.393359 0.919385i \(-0.628687\pi\)
−0.393359 + 0.919385i \(0.628687\pi\)
\(138\) 0 0
\(139\) −732.699 −0.447099 −0.223549 0.974693i \(-0.571764\pi\)
−0.223549 + 0.974693i \(0.571764\pi\)
\(140\) 0 0
\(141\) 1477.16 0.882265
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) −1882.48 −1.07815
\(146\) 0 0
\(147\) 2382.37 1.33670
\(148\) 0 0
\(149\) 1247.99 0.686172 0.343086 0.939304i \(-0.388528\pi\)
0.343086 + 0.939304i \(0.388528\pi\)
\(150\) 0 0
\(151\) −201.548 −0.108621 −0.0543104 0.998524i \(-0.517296\pi\)
−0.0543104 + 0.998524i \(0.517296\pi\)
\(152\) 0 0
\(153\) −781.804 −0.413105
\(154\) 0 0
\(155\) −3154.43 −1.63465
\(156\) 0 0
\(157\) −259.197 −0.131759 −0.0658795 0.997828i \(-0.520985\pi\)
−0.0658795 + 0.997828i \(0.520985\pi\)
\(158\) 0 0
\(159\) 94.3139 0.0470414
\(160\) 0 0
\(161\) −6250.09 −3.05948
\(162\) 0 0
\(163\) −1061.55 −0.510103 −0.255052 0.966927i \(-0.582092\pi\)
−0.255052 + 0.966927i \(0.582092\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 1015.99 0.470775 0.235387 0.971902i \(-0.424364\pi\)
0.235387 + 0.971902i \(0.424364\pi\)
\(168\) 0 0
\(169\) 627.482 0.285609
\(170\) 0 0
\(171\) 1388.79 0.621071
\(172\) 0 0
\(173\) 575.128 0.252753 0.126376 0.991982i \(-0.459665\pi\)
0.126376 + 0.991982i \(0.459665\pi\)
\(174\) 0 0
\(175\) −4748.79 −2.05129
\(176\) 0 0
\(177\) 608.073 0.258223
\(178\) 0 0
\(179\) 236.112 0.0985913 0.0492957 0.998784i \(-0.484302\pi\)
0.0492957 + 0.998784i \(0.484302\pi\)
\(180\) 0 0
\(181\) 2542.07 1.04393 0.521964 0.852968i \(-0.325200\pi\)
0.521964 + 0.852968i \(0.325200\pi\)
\(182\) 0 0
\(183\) −852.201 −0.344243
\(184\) 0 0
\(185\) −2018.86 −0.802320
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) −910.474 −0.350409
\(190\) 0 0
\(191\) 4201.91 1.59183 0.795916 0.605407i \(-0.206990\pi\)
0.795916 + 0.605407i \(0.206990\pi\)
\(192\) 0 0
\(193\) −2652.94 −0.989446 −0.494723 0.869051i \(-0.664731\pi\)
−0.494723 + 0.869051i \(0.664731\pi\)
\(194\) 0 0
\(195\) 2599.49 0.954633
\(196\) 0 0
\(197\) −1934.24 −0.699536 −0.349768 0.936836i \(-0.613740\pi\)
−0.349768 + 0.936836i \(0.613740\pi\)
\(198\) 0 0
\(199\) −21.6061 −0.00769658 −0.00384829 0.999993i \(-0.501225\pi\)
−0.00384829 + 0.999993i \(0.501225\pi\)
\(200\) 0 0
\(201\) 825.898 0.289823
\(202\) 0 0
\(203\) 3893.47 1.34615
\(204\) 0 0
\(205\) 2348.67 0.800188
\(206\) 0 0
\(207\) 1668.11 0.560104
\(208\) 0 0
\(209\) 0 0
\(210\) 0 0
\(211\) −295.102 −0.0962829 −0.0481414 0.998841i \(-0.515330\pi\)
−0.0481414 + 0.998841i \(0.515330\pi\)
\(212\) 0 0
\(213\) 1938.14 0.623470
\(214\) 0 0
\(215\) −5181.26 −1.64353
\(216\) 0 0
\(217\) 6524.20 2.04098
\(218\) 0 0
\(219\) −116.548 −0.0359614
\(220\) 0 0
\(221\) −4616.63 −1.40519
\(222\) 0 0
\(223\) 369.547 0.110972 0.0554859 0.998459i \(-0.482329\pi\)
0.0554859 + 0.998459i \(0.482329\pi\)
\(224\) 0 0
\(225\) 1267.42 0.375533
\(226\) 0 0
\(227\) 2258.55 0.660374 0.330187 0.943916i \(-0.392888\pi\)
0.330187 + 0.943916i \(0.392888\pi\)
\(228\) 0 0
\(229\) −1791.55 −0.516982 −0.258491 0.966014i \(-0.583225\pi\)
−0.258491 + 0.966014i \(0.583225\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −788.326 −0.221652 −0.110826 0.993840i \(-0.535350\pi\)
−0.110826 + 0.993840i \(0.535350\pi\)
\(234\) 0 0
\(235\) 8027.94 2.22845
\(236\) 0 0
\(237\) 3050.30 0.836026
\(238\) 0 0
\(239\) −6848.37 −1.85349 −0.926745 0.375690i \(-0.877406\pi\)
−0.926745 + 0.375690i \(0.877406\pi\)
\(240\) 0 0
\(241\) 7339.42 1.96172 0.980858 0.194723i \(-0.0623808\pi\)
0.980858 + 0.194723i \(0.0623808\pi\)
\(242\) 0 0
\(243\) 243.000 0.0641500
\(244\) 0 0
\(245\) 12947.5 3.37627
\(246\) 0 0
\(247\) 8200.92 2.11260
\(248\) 0 0
\(249\) 3206.47 0.816073
\(250\) 0 0
\(251\) −5193.08 −1.30591 −0.652957 0.757395i \(-0.726472\pi\)
−0.652957 + 0.757395i \(0.726472\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 0 0
\(255\) −4248.88 −1.04343
\(256\) 0 0
\(257\) 705.153 0.171153 0.0855764 0.996332i \(-0.472727\pi\)
0.0855764 + 0.996332i \(0.472727\pi\)
\(258\) 0 0
\(259\) 4175.53 1.00176
\(260\) 0 0
\(261\) −1039.14 −0.246442
\(262\) 0 0
\(263\) −3039.26 −0.712582 −0.356291 0.934375i \(-0.615959\pi\)
−0.356291 + 0.934375i \(0.615959\pi\)
\(264\) 0 0
\(265\) 512.569 0.118818
\(266\) 0 0
\(267\) −3156.25 −0.723443
\(268\) 0 0
\(269\) 4226.68 0.958012 0.479006 0.877812i \(-0.340997\pi\)
0.479006 + 0.877812i \(0.340997\pi\)
\(270\) 0 0
\(271\) −2472.52 −0.554225 −0.277113 0.960837i \(-0.589378\pi\)
−0.277113 + 0.960837i \(0.589378\pi\)
\(272\) 0 0
\(273\) −5376.44 −1.19193
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) 6471.06 1.40364 0.701820 0.712355i \(-0.252371\pi\)
0.701820 + 0.712355i \(0.252371\pi\)
\(278\) 0 0
\(279\) −1741.27 −0.373646
\(280\) 0 0
\(281\) −1531.75 −0.325184 −0.162592 0.986693i \(-0.551985\pi\)
−0.162592 + 0.986693i \(0.551985\pi\)
\(282\) 0 0
\(283\) −2807.25 −0.589661 −0.294830 0.955550i \(-0.595263\pi\)
−0.294830 + 0.955550i \(0.595263\pi\)
\(284\) 0 0
\(285\) 7547.66 1.56872
\(286\) 0 0
\(287\) −4857.68 −0.999093
\(288\) 0 0
\(289\) 2632.90 0.535904
\(290\) 0 0
\(291\) 878.628 0.176997
\(292\) 0 0
\(293\) 5098.74 1.01663 0.508313 0.861172i \(-0.330269\pi\)
0.508313 + 0.861172i \(0.330269\pi\)
\(294\) 0 0
\(295\) 3304.70 0.652228
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 9850.34 1.90522
\(300\) 0 0
\(301\) 10716.2 2.05207
\(302\) 0 0
\(303\) −1817.69 −0.344632
\(304\) 0 0
\(305\) −4631.46 −0.869498
\(306\) 0 0
\(307\) 9653.45 1.79463 0.897315 0.441390i \(-0.145514\pi\)
0.897315 + 0.441390i \(0.145514\pi\)
\(308\) 0 0
\(309\) 104.628 0.0192624
\(310\) 0 0
\(311\) 244.373 0.0445566 0.0222783 0.999752i \(-0.492908\pi\)
0.0222783 + 0.999752i \(0.492908\pi\)
\(312\) 0 0
\(313\) −8398.23 −1.51660 −0.758301 0.651905i \(-0.773970\pi\)
−0.758301 + 0.651905i \(0.773970\pi\)
\(314\) 0 0
\(315\) −4948.17 −0.885071
\(316\) 0 0
\(317\) −3155.60 −0.559105 −0.279553 0.960130i \(-0.590186\pi\)
−0.279553 + 0.960130i \(0.590186\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0 0
\(321\) 4862.39 0.845458
\(322\) 0 0
\(323\) −13404.4 −2.30911
\(324\) 0 0
\(325\) 7484.26 1.27739
\(326\) 0 0
\(327\) 1778.06 0.300694
\(328\) 0 0
\(329\) −16603.9 −2.78238
\(330\) 0 0
\(331\) 5406.39 0.897771 0.448886 0.893589i \(-0.351821\pi\)
0.448886 + 0.893589i \(0.351821\pi\)
\(332\) 0 0
\(333\) −1114.42 −0.183393
\(334\) 0 0
\(335\) 4488.52 0.732041
\(336\) 0 0
\(337\) 6676.20 1.07916 0.539578 0.841936i \(-0.318584\pi\)
0.539578 + 0.841936i \(0.318584\pi\)
\(338\) 0 0
\(339\) 5862.50 0.939254
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) −15212.5 −2.39474
\(344\) 0 0
\(345\) 9065.69 1.41473
\(346\) 0 0
\(347\) 4196.04 0.649151 0.324575 0.945860i \(-0.394779\pi\)
0.324575 + 0.945860i \(0.394779\pi\)
\(348\) 0 0
\(349\) −10258.7 −1.57346 −0.786728 0.617300i \(-0.788227\pi\)
−0.786728 + 0.617300i \(0.788227\pi\)
\(350\) 0 0
\(351\) 1434.94 0.218209
\(352\) 0 0
\(353\) −6526.66 −0.984077 −0.492038 0.870573i \(-0.663748\pi\)
−0.492038 + 0.870573i \(0.663748\pi\)
\(354\) 0 0
\(355\) 10533.2 1.57478
\(356\) 0 0
\(357\) 8787.81 1.30280
\(358\) 0 0
\(359\) 3351.62 0.492734 0.246367 0.969177i \(-0.420763\pi\)
0.246367 + 0.969177i \(0.420763\pi\)
\(360\) 0 0
\(361\) 16952.5 2.47157
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) −633.402 −0.0908323
\(366\) 0 0
\(367\) −2089.40 −0.297182 −0.148591 0.988899i \(-0.547474\pi\)
−0.148591 + 0.988899i \(0.547474\pi\)
\(368\) 0 0
\(369\) 1296.48 0.182906
\(370\) 0 0
\(371\) −1060.13 −0.148353
\(372\) 0 0
\(373\) −12579.7 −1.74626 −0.873129 0.487490i \(-0.837913\pi\)
−0.873129 + 0.487490i \(0.837913\pi\)
\(374\) 0 0
\(375\) 774.029 0.106589
\(376\) 0 0
\(377\) −6136.25 −0.838283
\(378\) 0 0
\(379\) 1900.41 0.257566 0.128783 0.991673i \(-0.458893\pi\)
0.128783 + 0.991673i \(0.458893\pi\)
\(380\) 0 0
\(381\) 1431.68 0.192512
\(382\) 0 0
\(383\) 4761.49 0.635250 0.317625 0.948216i \(-0.397115\pi\)
0.317625 + 0.948216i \(0.397115\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) −2860.09 −0.375676
\(388\) 0 0
\(389\) −1205.71 −0.157152 −0.0785758 0.996908i \(-0.525037\pi\)
−0.0785758 + 0.996908i \(0.525037\pi\)
\(390\) 0 0
\(391\) −16100.4 −2.08244
\(392\) 0 0
\(393\) 4426.96 0.568221
\(394\) 0 0
\(395\) 16577.5 2.11166
\(396\) 0 0
\(397\) −9233.40 −1.16728 −0.583641 0.812012i \(-0.698373\pi\)
−0.583641 + 0.812012i \(0.698373\pi\)
\(398\) 0 0
\(399\) −15610.6 −1.95866
\(400\) 0 0
\(401\) −7551.12 −0.940362 −0.470181 0.882570i \(-0.655811\pi\)
−0.470181 + 0.882570i \(0.655811\pi\)
\(402\) 0 0
\(403\) −10282.4 −1.27097
\(404\) 0 0
\(405\) 1320.63 0.162032
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) −2495.37 −0.301683 −0.150841 0.988558i \(-0.548198\pi\)
−0.150841 + 0.988558i \(0.548198\pi\)
\(410\) 0 0
\(411\) −3784.61 −0.454212
\(412\) 0 0
\(413\) −6835.00 −0.814354
\(414\) 0 0
\(415\) 17426.3 2.06126
\(416\) 0 0
\(417\) −2198.10 −0.258133
\(418\) 0 0
\(419\) −12820.8 −1.49484 −0.747419 0.664353i \(-0.768707\pi\)
−0.747419 + 0.664353i \(0.768707\pi\)
\(420\) 0 0
\(421\) −8805.52 −1.01937 −0.509685 0.860361i \(-0.670238\pi\)
−0.509685 + 0.860361i \(0.670238\pi\)
\(422\) 0 0
\(423\) 4431.48 0.509376
\(424\) 0 0
\(425\) −12233.0 −1.39621
\(426\) 0 0
\(427\) 9579.09 1.08563
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) −1570.46 −0.175514 −0.0877569 0.996142i \(-0.527970\pi\)
−0.0877569 + 0.996142i \(0.527970\pi\)
\(432\) 0 0
\(433\) 1070.06 0.118762 0.0593808 0.998235i \(-0.481087\pi\)
0.0593808 + 0.998235i \(0.481087\pi\)
\(434\) 0 0
\(435\) −5647.45 −0.622470
\(436\) 0 0
\(437\) 28600.6 3.13078
\(438\) 0 0
\(439\) −1816.77 −0.197516 −0.0987580 0.995111i \(-0.531487\pi\)
−0.0987580 + 0.995111i \(0.531487\pi\)
\(440\) 0 0
\(441\) 7147.12 0.771743
\(442\) 0 0
\(443\) 8255.74 0.885422 0.442711 0.896664i \(-0.354017\pi\)
0.442711 + 0.896664i \(0.354017\pi\)
\(444\) 0 0
\(445\) −17153.3 −1.82729
\(446\) 0 0
\(447\) 3743.98 0.396162
\(448\) 0 0
\(449\) −11952.0 −1.25623 −0.628115 0.778120i \(-0.716173\pi\)
−0.628115 + 0.778120i \(0.716173\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 0 0
\(453\) −604.643 −0.0627122
\(454\) 0 0
\(455\) −29219.4 −3.01061
\(456\) 0 0
\(457\) 13270.0 1.35830 0.679152 0.733997i \(-0.262348\pi\)
0.679152 + 0.733997i \(0.262348\pi\)
\(458\) 0 0
\(459\) −2345.41 −0.238506
\(460\) 0 0
\(461\) 3022.63 0.305375 0.152687 0.988275i \(-0.451207\pi\)
0.152687 + 0.988275i \(0.451207\pi\)
\(462\) 0 0
\(463\) 13286.5 1.33364 0.666822 0.745217i \(-0.267654\pi\)
0.666822 + 0.745217i \(0.267654\pi\)
\(464\) 0 0
\(465\) −9463.30 −0.943763
\(466\) 0 0
\(467\) −5910.67 −0.585681 −0.292840 0.956161i \(-0.594600\pi\)
−0.292840 + 0.956161i \(0.594600\pi\)
\(468\) 0 0
\(469\) −9283.44 −0.914007
\(470\) 0 0
\(471\) −777.591 −0.0760711
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) 21730.6 2.09909
\(476\) 0 0
\(477\) 282.942 0.0271594
\(478\) 0 0
\(479\) 4904.02 0.467788 0.233894 0.972262i \(-0.424853\pi\)
0.233894 + 0.972262i \(0.424853\pi\)
\(480\) 0 0
\(481\) −6580.78 −0.623820
\(482\) 0 0
\(483\) −18750.3 −1.76639
\(484\) 0 0
\(485\) 4775.09 0.447063
\(486\) 0 0
\(487\) 4244.52 0.394944 0.197472 0.980309i \(-0.436727\pi\)
0.197472 + 0.980309i \(0.436727\pi\)
\(488\) 0 0
\(489\) −3184.64 −0.294508
\(490\) 0 0
\(491\) −14886.1 −1.36823 −0.684116 0.729373i \(-0.739812\pi\)
−0.684116 + 0.729373i \(0.739812\pi\)
\(492\) 0 0
\(493\) 10029.7 0.916259
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −21785.5 −1.96622
\(498\) 0 0
\(499\) −5354.31 −0.480344 −0.240172 0.970730i \(-0.577204\pi\)
−0.240172 + 0.970730i \(0.577204\pi\)
\(500\) 0 0
\(501\) 3047.96 0.271802
\(502\) 0 0
\(503\) −17927.6 −1.58917 −0.794583 0.607155i \(-0.792311\pi\)
−0.794583 + 0.607155i \(0.792311\pi\)
\(504\) 0 0
\(505\) −9878.61 −0.870480
\(506\) 0 0
\(507\) 1882.45 0.164896
\(508\) 0 0
\(509\) 5090.72 0.443305 0.221652 0.975126i \(-0.428855\pi\)
0.221652 + 0.975126i \(0.428855\pi\)
\(510\) 0 0
\(511\) 1310.04 0.113411
\(512\) 0 0
\(513\) 4166.36 0.358576
\(514\) 0 0
\(515\) 568.622 0.0486534
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) 1725.39 0.145927
\(520\) 0 0
\(521\) −17575.7 −1.47794 −0.738971 0.673738i \(-0.764688\pi\)
−0.738971 + 0.673738i \(0.764688\pi\)
\(522\) 0 0
\(523\) −4537.13 −0.379340 −0.189670 0.981848i \(-0.560742\pi\)
−0.189670 + 0.981848i \(0.560742\pi\)
\(524\) 0 0
\(525\) −14246.4 −1.18431
\(526\) 0 0
\(527\) 16806.6 1.38919
\(528\) 0 0
\(529\) 22185.9 1.82345
\(530\) 0 0
\(531\) 1824.22 0.149085
\(532\) 0 0
\(533\) 7655.87 0.622162
\(534\) 0 0
\(535\) 26425.7 2.13548
\(536\) 0 0
\(537\) 708.336 0.0569217
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) 8227.07 0.653806 0.326903 0.945058i \(-0.393995\pi\)
0.326903 + 0.945058i \(0.393995\pi\)
\(542\) 0 0
\(543\) 7626.22 0.602712
\(544\) 0 0
\(545\) 9663.24 0.759500
\(546\) 0 0
\(547\) −3564.34 −0.278611 −0.139305 0.990249i \(-0.544487\pi\)
−0.139305 + 0.990249i \(0.544487\pi\)
\(548\) 0 0
\(549\) −2556.60 −0.198749
\(550\) 0 0
\(551\) −17816.7 −1.37752
\(552\) 0 0
\(553\) −34286.6 −2.63656
\(554\) 0 0
\(555\) −6056.57 −0.463220
\(556\) 0 0
\(557\) 12892.9 0.980769 0.490385 0.871506i \(-0.336856\pi\)
0.490385 + 0.871506i \(0.336856\pi\)
\(558\) 0 0
\(559\) −16889.1 −1.27788
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) −2220.93 −0.166254 −0.0831268 0.996539i \(-0.526491\pi\)
−0.0831268 + 0.996539i \(0.526491\pi\)
\(564\) 0 0
\(565\) 31861.0 2.37239
\(566\) 0 0
\(567\) −2731.42 −0.202309
\(568\) 0 0
\(569\) 1949.45 0.143630 0.0718149 0.997418i \(-0.477121\pi\)
0.0718149 + 0.997418i \(0.477121\pi\)
\(570\) 0 0
\(571\) −10794.1 −0.791099 −0.395550 0.918445i \(-0.629446\pi\)
−0.395550 + 0.918445i \(0.629446\pi\)
\(572\) 0 0
\(573\) 12605.7 0.919044
\(574\) 0 0
\(575\) 26101.2 1.89304
\(576\) 0 0
\(577\) −10610.9 −0.765576 −0.382788 0.923836i \(-0.625036\pi\)
−0.382788 + 0.923836i \(0.625036\pi\)
\(578\) 0 0
\(579\) −7958.83 −0.571257
\(580\) 0 0
\(581\) −36042.1 −2.57363
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) 7798.47 0.551158
\(586\) 0 0
\(587\) −20780.1 −1.46114 −0.730569 0.682839i \(-0.760745\pi\)
−0.730569 + 0.682839i \(0.760745\pi\)
\(588\) 0 0
\(589\) −29855.0 −2.08854
\(590\) 0 0
\(591\) −5802.71 −0.403877
\(592\) 0 0
\(593\) 25940.7 1.79638 0.898192 0.439604i \(-0.144881\pi\)
0.898192 + 0.439604i \(0.144881\pi\)
\(594\) 0 0
\(595\) 47759.2 3.29065
\(596\) 0 0
\(597\) −64.8184 −0.00444362
\(598\) 0 0
\(599\) −18108.3 −1.23520 −0.617601 0.786491i \(-0.711895\pi\)
−0.617601 + 0.786491i \(0.711895\pi\)
\(600\) 0 0
\(601\) 6214.64 0.421798 0.210899 0.977508i \(-0.432361\pi\)
0.210899 + 0.977508i \(0.432361\pi\)
\(602\) 0 0
\(603\) 2477.69 0.167329
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) −29021.6 −1.94061 −0.970305 0.241886i \(-0.922234\pi\)
−0.970305 + 0.241886i \(0.922234\pi\)
\(608\) 0 0
\(609\) 11680.4 0.777200
\(610\) 0 0
\(611\) 26168.3 1.73266
\(612\) 0 0
\(613\) −13485.4 −0.888533 −0.444266 0.895895i \(-0.646536\pi\)
−0.444266 + 0.895895i \(0.646536\pi\)
\(614\) 0 0
\(615\) 7046.02 0.461989
\(616\) 0 0
\(617\) 27940.5 1.82309 0.911543 0.411205i \(-0.134892\pi\)
0.911543 + 0.411205i \(0.134892\pi\)
\(618\) 0 0
\(619\) −1525.96 −0.0990846 −0.0495423 0.998772i \(-0.515776\pi\)
−0.0495423 + 0.998772i \(0.515776\pi\)
\(620\) 0 0
\(621\) 5004.33 0.323376
\(622\) 0 0
\(623\) 35477.6 2.28151
\(624\) 0 0
\(625\) −13396.5 −0.857374
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 10756.3 0.681847
\(630\) 0 0
\(631\) −20683.5 −1.30491 −0.652454 0.757829i \(-0.726260\pi\)
−0.652454 + 0.757829i \(0.726260\pi\)
\(632\) 0 0
\(633\) −885.307 −0.0555890
\(634\) 0 0
\(635\) 7780.75 0.486252
\(636\) 0 0
\(637\) 42204.4 2.62512
\(638\) 0 0
\(639\) 5814.42 0.359960
\(640\) 0 0
\(641\) 7771.73 0.478884 0.239442 0.970911i \(-0.423035\pi\)
0.239442 + 0.970911i \(0.423035\pi\)
\(642\) 0 0
\(643\) 6090.22 0.373522 0.186761 0.982405i \(-0.440201\pi\)
0.186761 + 0.982405i \(0.440201\pi\)
\(644\) 0 0
\(645\) −15543.8 −0.948893
\(646\) 0 0
\(647\) −2607.46 −0.158439 −0.0792194 0.996857i \(-0.525243\pi\)
−0.0792194 + 0.996857i \(0.525243\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) 19572.6 1.17836
\(652\) 0 0
\(653\) −1635.49 −0.0980116 −0.0490058 0.998798i \(-0.515605\pi\)
−0.0490058 + 0.998798i \(0.515605\pi\)
\(654\) 0 0
\(655\) 24059.3 1.43523
\(656\) 0 0
\(657\) −349.643 −0.0207623
\(658\) 0 0
\(659\) −19797.0 −1.17023 −0.585115 0.810950i \(-0.698951\pi\)
−0.585115 + 0.810950i \(0.698951\pi\)
\(660\) 0 0
\(661\) −24158.2 −1.42155 −0.710775 0.703419i \(-0.751656\pi\)
−0.710775 + 0.703419i \(0.751656\pi\)
\(662\) 0 0
\(663\) −13849.9 −0.811289
\(664\) 0 0
\(665\) −84838.9 −4.94723
\(666\) 0 0
\(667\) −21400.1 −1.24230
\(668\) 0 0
\(669\) 1108.64 0.0640696
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) 32107.6 1.83902 0.919508 0.393072i \(-0.128588\pi\)
0.919508 + 0.393072i \(0.128588\pi\)
\(674\) 0 0
\(675\) 3802.27 0.216814
\(676\) 0 0
\(677\) −7581.35 −0.430392 −0.215196 0.976571i \(-0.569039\pi\)
−0.215196 + 0.976571i \(0.569039\pi\)
\(678\) 0 0
\(679\) −9876.15 −0.558191
\(680\) 0 0
\(681\) 6775.64 0.381267
\(682\) 0 0
\(683\) 27647.5 1.54890 0.774452 0.632632i \(-0.218026\pi\)
0.774452 + 0.632632i \(0.218026\pi\)
\(684\) 0 0
\(685\) −20568.3 −1.14726
\(686\) 0 0
\(687\) −5374.64 −0.298479
\(688\) 0 0
\(689\) 1670.80 0.0923837
\(690\) 0 0
\(691\) −21015.7 −1.15698 −0.578491 0.815688i \(-0.696358\pi\)
−0.578491 + 0.815688i \(0.696358\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −11946.0 −0.651998
\(696\) 0 0
\(697\) −12513.5 −0.680035
\(698\) 0 0
\(699\) −2364.98 −0.127971
\(700\) 0 0
\(701\) 14656.3 0.789673 0.394837 0.918751i \(-0.370801\pi\)
0.394837 + 0.918751i \(0.370801\pi\)
\(702\) 0 0
\(703\) −19107.4 −1.02510
\(704\) 0 0
\(705\) 24083.8 1.28659
\(706\) 0 0
\(707\) 20431.6 1.08686
\(708\) 0 0
\(709\) 19186.6 1.01632 0.508159 0.861263i \(-0.330326\pi\)
0.508159 + 0.861263i \(0.330326\pi\)
\(710\) 0 0
\(711\) 9150.89 0.482680
\(712\) 0 0
\(713\) −35859.6 −1.88352
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) −20545.1 −1.07011
\(718\) 0 0
\(719\) −31817.1 −1.65031 −0.825157 0.564903i \(-0.808914\pi\)
−0.825157 + 0.564903i \(0.808914\pi\)
\(720\) 0 0
\(721\) −1176.06 −0.0607473
\(722\) 0 0
\(723\) 22018.3 1.13260
\(724\) 0 0
\(725\) −16259.7 −0.832924
\(726\) 0 0
\(727\) 9774.72 0.498658 0.249329 0.968419i \(-0.419790\pi\)
0.249329 + 0.968419i \(0.419790\pi\)
\(728\) 0 0
\(729\) 729.000 0.0370370
\(730\) 0 0
\(731\) 27605.3 1.39674
\(732\) 0 0
\(733\) −6778.73 −0.341580 −0.170790 0.985307i \(-0.554632\pi\)
−0.170790 + 0.985307i \(0.554632\pi\)
\(734\) 0 0
\(735\) 38842.5 1.94929
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) 32832.8 1.63434 0.817169 0.576398i \(-0.195542\pi\)
0.817169 + 0.576398i \(0.195542\pi\)
\(740\) 0 0
\(741\) 24602.8 1.21971
\(742\) 0 0
\(743\) −29759.2 −1.46939 −0.734697 0.678395i \(-0.762676\pi\)
−0.734697 + 0.678395i \(0.762676\pi\)
\(744\) 0 0
\(745\) 20347.5 1.00064
\(746\) 0 0
\(747\) 9619.42 0.471160
\(748\) 0 0
\(749\) −54655.3 −2.66630
\(750\) 0 0
\(751\) −30899.8 −1.50140 −0.750698 0.660646i \(-0.770282\pi\)
−0.750698 + 0.660646i \(0.770282\pi\)
\(752\) 0 0
\(753\) −15579.3 −0.753970
\(754\) 0 0
\(755\) −3286.06 −0.158400
\(756\) 0 0
\(757\) −21704.5 −1.04209 −0.521045 0.853529i \(-0.674458\pi\)
−0.521045 + 0.853529i \(0.674458\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −1133.62 −0.0539996 −0.0269998 0.999635i \(-0.508595\pi\)
−0.0269998 + 0.999635i \(0.508595\pi\)
\(762\) 0 0
\(763\) −19986.1 −0.948292
\(764\) 0 0
\(765\) −12746.6 −0.602426
\(766\) 0 0
\(767\) 10772.2 0.507120
\(768\) 0 0
\(769\) −19846.8 −0.930680 −0.465340 0.885132i \(-0.654068\pi\)
−0.465340 + 0.885132i \(0.654068\pi\)
\(770\) 0 0
\(771\) 2115.46 0.0988151
\(772\) 0 0
\(773\) −10933.5 −0.508732 −0.254366 0.967108i \(-0.581867\pi\)
−0.254366 + 0.967108i \(0.581867\pi\)
\(774\) 0 0
\(775\) −27246.0 −1.26285
\(776\) 0 0
\(777\) 12526.6 0.578364
\(778\) 0 0
\(779\) 22228.9 1.02238
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) −3117.43 −0.142283
\(784\) 0 0
\(785\) −4225.98 −0.192142
\(786\) 0 0
\(787\) 19794.4 0.896560 0.448280 0.893893i \(-0.352037\pi\)
0.448280 + 0.893893i \(0.352037\pi\)
\(788\) 0 0
\(789\) −9117.79 −0.411409
\(790\) 0 0
\(791\) −65896.9 −2.96211
\(792\) 0 0
\(793\) −15097.0 −0.676052
\(794\) 0 0
\(795\) 1537.71 0.0685998
\(796\) 0 0
\(797\) −21767.8 −0.967445 −0.483723 0.875221i \(-0.660716\pi\)
−0.483723 + 0.875221i \(0.660716\pi\)
\(798\) 0 0
\(799\) −42772.2 −1.89383
\(800\) 0 0
\(801\) −9468.74 −0.417680
\(802\) 0 0
\(803\) 0 0
\(804\) 0 0
\(805\) −101902. −4.46159
\(806\) 0 0
\(807\) 12680.0 0.553108
\(808\) 0 0
\(809\) −31676.7 −1.37663 −0.688315 0.725412i \(-0.741649\pi\)
−0.688315 + 0.725412i \(0.741649\pi\)
\(810\) 0 0
\(811\) −77.5097 −0.00335602 −0.00167801 0.999999i \(-0.500534\pi\)
−0.00167801 + 0.999999i \(0.500534\pi\)
\(812\) 0 0
\(813\) −7417.57 −0.319982
\(814\) 0 0
\(815\) −17307.6 −0.743876
\(816\) 0 0
\(817\) −49037.8 −2.09990
\(818\) 0 0
\(819\) −16129.3 −0.688161
\(820\) 0 0
\(821\) −39100.3 −1.66213 −0.831066 0.556174i \(-0.812269\pi\)
−0.831066 + 0.556174i \(0.812269\pi\)
\(822\) 0 0
\(823\) 31066.5 1.31581 0.657905 0.753101i \(-0.271443\pi\)
0.657905 + 0.753101i \(0.271443\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −16497.9 −0.693698 −0.346849 0.937921i \(-0.612748\pi\)
−0.346849 + 0.937921i \(0.612748\pi\)
\(828\) 0 0
\(829\) 7309.01 0.306215 0.153108 0.988210i \(-0.451072\pi\)
0.153108 + 0.988210i \(0.451072\pi\)
\(830\) 0 0
\(831\) 19413.2 0.810392
\(832\) 0 0
\(833\) −68983.3 −2.86930
\(834\) 0 0
\(835\) 16564.8 0.686524
\(836\) 0 0
\(837\) −5223.81 −0.215724
\(838\) 0 0
\(839\) 8200.25 0.337430 0.168715 0.985665i \(-0.446038\pi\)
0.168715 + 0.985665i \(0.446038\pi\)
\(840\) 0 0
\(841\) −11057.9 −0.453396
\(842\) 0 0
\(843\) −4595.26 −0.187745
\(844\) 0 0
\(845\) 10230.6 0.416499
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) −8421.76 −0.340441
\(850\) 0 0
\(851\) −22950.4 −0.924475
\(852\) 0 0
\(853\) 21020.3 0.843753 0.421877 0.906653i \(-0.361372\pi\)
0.421877 + 0.906653i \(0.361372\pi\)
\(854\) 0 0
\(855\) 22643.0 0.905700
\(856\) 0 0
\(857\) 37976.8 1.51373 0.756863 0.653574i \(-0.226731\pi\)
0.756863 + 0.653574i \(0.226731\pi\)
\(858\) 0 0
\(859\) 36137.8 1.43540 0.717698 0.696355i \(-0.245196\pi\)
0.717698 + 0.696355i \(0.245196\pi\)
\(860\) 0 0
\(861\) −14573.0 −0.576827
\(862\) 0 0
\(863\) −36186.9 −1.42737 −0.713683 0.700468i \(-0.752974\pi\)
−0.713683 + 0.700468i \(0.752974\pi\)
\(864\) 0 0
\(865\) 9376.97 0.368586
\(866\) 0 0
\(867\) 7898.69 0.309404
\(868\) 0 0
\(869\) 0 0
\(870\) 0 0
\(871\) 14631.0 0.569177
\(872\) 0 0
\(873\) 2635.88 0.102189
\(874\) 0 0
\(875\) −8700.42 −0.336146
\(876\) 0 0
\(877\) 30124.1 1.15988 0.579942 0.814658i \(-0.303075\pi\)
0.579942 + 0.814658i \(0.303075\pi\)
\(878\) 0 0
\(879\) 15296.2 0.586950
\(880\) 0 0
\(881\) 10032.5 0.383659 0.191829 0.981428i \(-0.438558\pi\)
0.191829 + 0.981428i \(0.438558\pi\)
\(882\) 0 0
\(883\) −46545.3 −1.77392 −0.886962 0.461842i \(-0.847189\pi\)
−0.886962 + 0.461842i \(0.847189\pi\)
\(884\) 0 0
\(885\) 9914.10 0.376564
\(886\) 0 0
\(887\) −29706.4 −1.12451 −0.562256 0.826963i \(-0.690067\pi\)
−0.562256 + 0.826963i \(0.690067\pi\)
\(888\) 0 0
\(889\) −16092.7 −0.607121
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 75980.0 2.84723
\(894\) 0 0
\(895\) 3849.60 0.143774
\(896\) 0 0
\(897\) 29551.0 1.09998
\(898\) 0 0
\(899\) 22338.7 0.828738
\(900\) 0 0
\(901\) −2730.93 −0.100977
\(902\) 0 0
\(903\) 32148.7 1.18476
\(904\) 0 0
\(905\) 41446.3 1.52234
\(906\) 0 0
\(907\) 38251.8 1.40036 0.700181 0.713965i \(-0.253102\pi\)
0.700181 + 0.713965i \(0.253102\pi\)
\(908\) 0 0
\(909\) −5453.06 −0.198973
\(910\) 0 0
\(911\) 47235.6 1.71788 0.858939 0.512078i \(-0.171124\pi\)
0.858939 + 0.512078i \(0.171124\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) 0 0
\(915\) −13894.4 −0.502005
\(916\) 0 0
\(917\) −49760.9 −1.79199
\(918\) 0 0
\(919\) −22275.1 −0.799552 −0.399776 0.916613i \(-0.630912\pi\)
−0.399776 + 0.916613i \(0.630912\pi\)
\(920\) 0 0
\(921\) 28960.4 1.03613
\(922\) 0 0
\(923\) 34334.7 1.22442
\(924\) 0 0
\(925\) −17437.6 −0.619832
\(926\) 0 0
\(927\) 313.884 0.0111211
\(928\) 0 0
\(929\) 24873.4 0.878439 0.439219 0.898380i \(-0.355255\pi\)
0.439219 + 0.898380i \(0.355255\pi\)
\(930\) 0 0
\(931\) 122541. 4.31377
\(932\) 0 0
\(933\) 733.118 0.0257248
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 15097.8 0.526385 0.263192 0.964743i \(-0.415225\pi\)
0.263192 + 0.964743i \(0.415225\pi\)
\(938\) 0 0
\(939\) −25194.7 −0.875610
\(940\) 0 0
\(941\) 27258.1 0.944304 0.472152 0.881517i \(-0.343477\pi\)
0.472152 + 0.881517i \(0.343477\pi\)
\(942\) 0 0
\(943\) 26699.7 0.922018
\(944\) 0 0
\(945\) −14844.5 −0.510996
\(946\) 0 0
\(947\) −49780.6 −1.70819 −0.854093 0.520120i \(-0.825887\pi\)
−0.854093 + 0.520120i \(0.825887\pi\)
\(948\) 0 0
\(949\) −2064.67 −0.0706239
\(950\) 0 0
\(951\) −9466.81 −0.322800
\(952\) 0 0
\(953\) −10844.7 −0.368619 −0.184309 0.982868i \(-0.559005\pi\)
−0.184309 + 0.982868i \(0.559005\pi\)
\(954\) 0 0
\(955\) 68508.6 2.32135
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 42540.6 1.43244
\(960\) 0 0
\(961\) 7641.35 0.256499
\(962\) 0 0
\(963\) 14587.2 0.488125
\(964\) 0 0
\(965\) −43254.0 −1.44290
\(966\) 0 0
\(967\) 43404.2 1.44342 0.721709 0.692196i \(-0.243357\pi\)
0.721709 + 0.692196i \(0.243357\pi\)
\(968\) 0 0
\(969\) −40213.3 −1.33317
\(970\) 0 0
\(971\) 28093.0 0.928473 0.464236 0.885711i \(-0.346329\pi\)
0.464236 + 0.885711i \(0.346329\pi\)
\(972\) 0 0
\(973\) 24707.5 0.814067
\(974\) 0 0
\(975\) 22452.8 0.737502
\(976\) 0 0
\(977\) −42152.9 −1.38034 −0.690169 0.723649i \(-0.742464\pi\)
−0.690169 + 0.723649i \(0.742464\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 0 0
\(981\) 5334.18 0.173606
\(982\) 0 0
\(983\) 35695.2 1.15819 0.579095 0.815260i \(-0.303406\pi\)
0.579095 + 0.815260i \(0.303406\pi\)
\(984\) 0 0
\(985\) −31536.0 −1.02012
\(986\) 0 0
\(987\) −49811.7 −1.60641
\(988\) 0 0
\(989\) −58900.6 −1.89376
\(990\) 0 0
\(991\) 2018.61 0.0647057 0.0323528 0.999477i \(-0.489700\pi\)
0.0323528 + 0.999477i \(0.489700\pi\)
\(992\) 0 0
\(993\) 16219.2 0.518328
\(994\) 0 0
\(995\) −352.269 −0.0112238
\(996\) 0 0
\(997\) −26002.4 −0.825981 −0.412990 0.910735i \(-0.635516\pi\)
−0.412990 + 0.910735i \(0.635516\pi\)
\(998\) 0 0
\(999\) −3343.27 −0.105882
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 1452.4.a.q.1.3 4
11.10 odd 2 inner 1452.4.a.q.1.4 yes 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1452.4.a.q.1.3 4 1.1 even 1 trivial
1452.4.a.q.1.4 yes 4 11.10 odd 2 inner