Properties

Label 2496.4.a.ca.1.2
Level $2496$
Weight $4$
Character 2496.1
Self dual yes
Analytic conductor $147.269$
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] = [2496,4,Mod(1,2496)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2496, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 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("2496.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2496 = 2^{6} \cdot 3 \cdot 13 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 2496.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: \(147.268767374\)
Analytic rank: \(1\)
Dimension: \(5\)
Coefficient field: \(\mathbb{Q}[x]/(x^{5} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{5} - 20x^{3} - 33x^{2} + 17x + 40 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{8} \)
Twist minimal: no (minimal twist has level 1248)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(5.05502\) of defining polynomial
Character \(\chi\) \(=\) 2496.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-3.00000 q^{3} -2.77633 q^{5} -5.23038 q^{7} +9.00000 q^{9} -26.0282 q^{11} +13.0000 q^{13} +8.32898 q^{15} +4.06350 q^{17} +73.9991 q^{19} +15.6911 q^{21} -145.163 q^{23} -117.292 q^{25} -27.0000 q^{27} +259.573 q^{29} +78.5918 q^{31} +78.0845 q^{33} +14.5213 q^{35} -99.8291 q^{37} -39.0000 q^{39} +346.320 q^{41} +137.732 q^{43} -24.9869 q^{45} +56.7561 q^{47} -315.643 q^{49} -12.1905 q^{51} -173.382 q^{53} +72.2627 q^{55} -221.997 q^{57} -675.923 q^{59} +798.265 q^{61} -47.0734 q^{63} -36.0923 q^{65} +292.584 q^{67} +435.489 q^{69} -333.464 q^{71} +243.292 q^{73} +351.876 q^{75} +136.137 q^{77} -670.246 q^{79} +81.0000 q^{81} +786.662 q^{83} -11.2816 q^{85} -778.719 q^{87} -317.044 q^{89} -67.9950 q^{91} -235.775 q^{93} -205.446 q^{95} +886.328 q^{97} -234.253 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5 q - 15 q^{3} + 10 q^{5} - 14 q^{7} + 45 q^{9} + 22 q^{11} + 65 q^{13} - 30 q^{15} - 34 q^{17} - 90 q^{19} + 42 q^{21} - 96 q^{23} + 107 q^{25} - 135 q^{27} + 54 q^{29} - 378 q^{31} - 66 q^{33} + 84 q^{35}+ \cdots + 198 q^{99}+O(q^{100}) \) 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
\(3\) −3.00000 −0.577350
\(4\) 0 0
\(5\) −2.77633 −0.248322 −0.124161 0.992262i \(-0.539624\pi\)
−0.124161 + 0.992262i \(0.539624\pi\)
\(6\) 0 0
\(7\) −5.23038 −0.282414 −0.141207 0.989980i \(-0.545098\pi\)
−0.141207 + 0.989980i \(0.545098\pi\)
\(8\) 0 0
\(9\) 9.00000 0.333333
\(10\) 0 0
\(11\) −26.0282 −0.713435 −0.356717 0.934212i \(-0.616104\pi\)
−0.356717 + 0.934212i \(0.616104\pi\)
\(12\) 0 0
\(13\) 13.0000 0.277350
\(14\) 0 0
\(15\) 8.32898 0.143369
\(16\) 0 0
\(17\) 4.06350 0.0579731 0.0289866 0.999580i \(-0.490772\pi\)
0.0289866 + 0.999580i \(0.490772\pi\)
\(18\) 0 0
\(19\) 73.9991 0.893503 0.446751 0.894658i \(-0.352581\pi\)
0.446751 + 0.894658i \(0.352581\pi\)
\(20\) 0 0
\(21\) 15.6911 0.163052
\(22\) 0 0
\(23\) −145.163 −1.31603 −0.658013 0.753006i \(-0.728603\pi\)
−0.658013 + 0.753006i \(0.728603\pi\)
\(24\) 0 0
\(25\) −117.292 −0.938336
\(26\) 0 0
\(27\) −27.0000 −0.192450
\(28\) 0 0
\(29\) 259.573 1.66212 0.831060 0.556183i \(-0.187735\pi\)
0.831060 + 0.556183i \(0.187735\pi\)
\(30\) 0 0
\(31\) 78.5918 0.455339 0.227669 0.973738i \(-0.426889\pi\)
0.227669 + 0.973738i \(0.426889\pi\)
\(32\) 0 0
\(33\) 78.0845 0.411902
\(34\) 0 0
\(35\) 14.5213 0.0701297
\(36\) 0 0
\(37\) −99.8291 −0.443562 −0.221781 0.975096i \(-0.571187\pi\)
−0.221781 + 0.975096i \(0.571187\pi\)
\(38\) 0 0
\(39\) −39.0000 −0.160128
\(40\) 0 0
\(41\) 346.320 1.31917 0.659586 0.751629i \(-0.270732\pi\)
0.659586 + 0.751629i \(0.270732\pi\)
\(42\) 0 0
\(43\) 137.732 0.488463 0.244231 0.969717i \(-0.421464\pi\)
0.244231 + 0.969717i \(0.421464\pi\)
\(44\) 0 0
\(45\) −24.9869 −0.0827741
\(46\) 0 0
\(47\) 56.7561 0.176143 0.0880716 0.996114i \(-0.471930\pi\)
0.0880716 + 0.996114i \(0.471930\pi\)
\(48\) 0 0
\(49\) −315.643 −0.920242
\(50\) 0 0
\(51\) −12.1905 −0.0334708
\(52\) 0 0
\(53\) −173.382 −0.449355 −0.224677 0.974433i \(-0.572133\pi\)
−0.224677 + 0.974433i \(0.572133\pi\)
\(54\) 0 0
\(55\) 72.2627 0.177162
\(56\) 0 0
\(57\) −221.997 −0.515864
\(58\) 0 0
\(59\) −675.923 −1.49149 −0.745743 0.666234i \(-0.767905\pi\)
−0.745743 + 0.666234i \(0.767905\pi\)
\(60\) 0 0
\(61\) 798.265 1.67553 0.837765 0.546030i \(-0.183862\pi\)
0.837765 + 0.546030i \(0.183862\pi\)
\(62\) 0 0
\(63\) −47.0734 −0.0941380
\(64\) 0 0
\(65\) −36.0923 −0.0688722
\(66\) 0 0
\(67\) 292.584 0.533504 0.266752 0.963765i \(-0.414050\pi\)
0.266752 + 0.963765i \(0.414050\pi\)
\(68\) 0 0
\(69\) 435.489 0.759808
\(70\) 0 0
\(71\) −333.464 −0.557393 −0.278697 0.960379i \(-0.589902\pi\)
−0.278697 + 0.960379i \(0.589902\pi\)
\(72\) 0 0
\(73\) 243.292 0.390072 0.195036 0.980796i \(-0.437518\pi\)
0.195036 + 0.980796i \(0.437518\pi\)
\(74\) 0 0
\(75\) 351.876 0.541749
\(76\) 0 0
\(77\) 136.137 0.201484
\(78\) 0 0
\(79\) −670.246 −0.954539 −0.477270 0.878757i \(-0.658373\pi\)
−0.477270 + 0.878757i \(0.658373\pi\)
\(80\) 0 0
\(81\) 81.0000 0.111111
\(82\) 0 0
\(83\) 786.662 1.04033 0.520165 0.854066i \(-0.325870\pi\)
0.520165 + 0.854066i \(0.325870\pi\)
\(84\) 0 0
\(85\) −11.2816 −0.0143960
\(86\) 0 0
\(87\) −778.719 −0.959625
\(88\) 0 0
\(89\) −317.044 −0.377602 −0.188801 0.982015i \(-0.560460\pi\)
−0.188801 + 0.982015i \(0.560460\pi\)
\(90\) 0 0
\(91\) −67.9950 −0.0783276
\(92\) 0 0
\(93\) −235.775 −0.262890
\(94\) 0 0
\(95\) −205.446 −0.221877
\(96\) 0 0
\(97\) 886.328 0.927762 0.463881 0.885897i \(-0.346456\pi\)
0.463881 + 0.885897i \(0.346456\pi\)
\(98\) 0 0
\(99\) −234.253 −0.237812
\(100\) 0 0
\(101\) −207.528 −0.204454 −0.102227 0.994761i \(-0.532597\pi\)
−0.102227 + 0.994761i \(0.532597\pi\)
\(102\) 0 0
\(103\) 1738.86 1.66345 0.831724 0.555189i \(-0.187354\pi\)
0.831724 + 0.555189i \(0.187354\pi\)
\(104\) 0 0
\(105\) −43.5638 −0.0404894
\(106\) 0 0
\(107\) 1248.91 1.12838 0.564189 0.825645i \(-0.309189\pi\)
0.564189 + 0.825645i \(0.309189\pi\)
\(108\) 0 0
\(109\) 1775.18 1.55992 0.779961 0.625828i \(-0.215239\pi\)
0.779961 + 0.625828i \(0.215239\pi\)
\(110\) 0 0
\(111\) 299.487 0.256091
\(112\) 0 0
\(113\) −1385.03 −1.15304 −0.576518 0.817085i \(-0.695589\pi\)
−0.576518 + 0.817085i \(0.695589\pi\)
\(114\) 0 0
\(115\) 403.020 0.326799
\(116\) 0 0
\(117\) 117.000 0.0924500
\(118\) 0 0
\(119\) −21.2537 −0.0163724
\(120\) 0 0
\(121\) −653.535 −0.491011
\(122\) 0 0
\(123\) −1038.96 −0.761624
\(124\) 0 0
\(125\) 672.682 0.481332
\(126\) 0 0
\(127\) −276.727 −0.193351 −0.0966755 0.995316i \(-0.530821\pi\)
−0.0966755 + 0.995316i \(0.530821\pi\)
\(128\) 0 0
\(129\) −413.196 −0.282014
\(130\) 0 0
\(131\) −2293.51 −1.52966 −0.764829 0.644233i \(-0.777177\pi\)
−0.764829 + 0.644233i \(0.777177\pi\)
\(132\) 0 0
\(133\) −387.043 −0.252338
\(134\) 0 0
\(135\) 74.9608 0.0477896
\(136\) 0 0
\(137\) −2639.75 −1.64620 −0.823098 0.567900i \(-0.807756\pi\)
−0.823098 + 0.567900i \(0.807756\pi\)
\(138\) 0 0
\(139\) −1371.48 −0.836888 −0.418444 0.908243i \(-0.637424\pi\)
−0.418444 + 0.908243i \(0.637424\pi\)
\(140\) 0 0
\(141\) −170.268 −0.101696
\(142\) 0 0
\(143\) −338.366 −0.197871
\(144\) 0 0
\(145\) −720.659 −0.412741
\(146\) 0 0
\(147\) 946.929 0.531302
\(148\) 0 0
\(149\) 1674.18 0.920498 0.460249 0.887790i \(-0.347760\pi\)
0.460249 + 0.887790i \(0.347760\pi\)
\(150\) 0 0
\(151\) −2426.56 −1.30775 −0.653876 0.756602i \(-0.726858\pi\)
−0.653876 + 0.756602i \(0.726858\pi\)
\(152\) 0 0
\(153\) 36.5715 0.0193244
\(154\) 0 0
\(155\) −218.197 −0.113071
\(156\) 0 0
\(157\) 1290.62 0.656069 0.328035 0.944666i \(-0.393614\pi\)
0.328035 + 0.944666i \(0.393614\pi\)
\(158\) 0 0
\(159\) 520.145 0.259435
\(160\) 0 0
\(161\) 759.259 0.371664
\(162\) 0 0
\(163\) −3787.85 −1.82017 −0.910083 0.414426i \(-0.863982\pi\)
−0.910083 + 0.414426i \(0.863982\pi\)
\(164\) 0 0
\(165\) −216.788 −0.102284
\(166\) 0 0
\(167\) 1004.20 0.465314 0.232657 0.972559i \(-0.425258\pi\)
0.232657 + 0.972559i \(0.425258\pi\)
\(168\) 0 0
\(169\) 169.000 0.0769231
\(170\) 0 0
\(171\) 665.992 0.297834
\(172\) 0 0
\(173\) −12.1058 −0.00532015 −0.00266008 0.999996i \(-0.500847\pi\)
−0.00266008 + 0.999996i \(0.500847\pi\)
\(174\) 0 0
\(175\) 613.482 0.264999
\(176\) 0 0
\(177\) 2027.77 0.861110
\(178\) 0 0
\(179\) 263.726 0.110122 0.0550610 0.998483i \(-0.482465\pi\)
0.0550610 + 0.998483i \(0.482465\pi\)
\(180\) 0 0
\(181\) 1941.94 0.797478 0.398739 0.917064i \(-0.369448\pi\)
0.398739 + 0.917064i \(0.369448\pi\)
\(182\) 0 0
\(183\) −2394.80 −0.967368
\(184\) 0 0
\(185\) 277.158 0.110146
\(186\) 0 0
\(187\) −105.765 −0.0413601
\(188\) 0 0
\(189\) 141.220 0.0543506
\(190\) 0 0
\(191\) 1925.11 0.729297 0.364649 0.931145i \(-0.381189\pi\)
0.364649 + 0.931145i \(0.381189\pi\)
\(192\) 0 0
\(193\) −3317.81 −1.23742 −0.618708 0.785621i \(-0.712343\pi\)
−0.618708 + 0.785621i \(0.712343\pi\)
\(194\) 0 0
\(195\) 108.277 0.0397634
\(196\) 0 0
\(197\) 4282.47 1.54880 0.774399 0.632697i \(-0.218052\pi\)
0.774399 + 0.632697i \(0.218052\pi\)
\(198\) 0 0
\(199\) −1219.10 −0.434271 −0.217135 0.976142i \(-0.569671\pi\)
−0.217135 + 0.976142i \(0.569671\pi\)
\(200\) 0 0
\(201\) −877.751 −0.308019
\(202\) 0 0
\(203\) −1357.67 −0.469406
\(204\) 0 0
\(205\) −961.496 −0.327580
\(206\) 0 0
\(207\) −1306.47 −0.438675
\(208\) 0 0
\(209\) −1926.06 −0.637456
\(210\) 0 0
\(211\) −4956.66 −1.61721 −0.808603 0.588355i \(-0.799776\pi\)
−0.808603 + 0.588355i \(0.799776\pi\)
\(212\) 0 0
\(213\) 1000.39 0.321811
\(214\) 0 0
\(215\) −382.389 −0.121296
\(216\) 0 0
\(217\) −411.065 −0.128594
\(218\) 0 0
\(219\) −729.877 −0.225208
\(220\) 0 0
\(221\) 52.8255 0.0160789
\(222\) 0 0
\(223\) 1918.63 0.576148 0.288074 0.957608i \(-0.406985\pi\)
0.288074 + 0.957608i \(0.406985\pi\)
\(224\) 0 0
\(225\) −1055.63 −0.312779
\(226\) 0 0
\(227\) 1268.36 0.370853 0.185427 0.982658i \(-0.440633\pi\)
0.185427 + 0.982658i \(0.440633\pi\)
\(228\) 0 0
\(229\) 2481.05 0.715950 0.357975 0.933731i \(-0.383467\pi\)
0.357975 + 0.933731i \(0.383467\pi\)
\(230\) 0 0
\(231\) −408.412 −0.116327
\(232\) 0 0
\(233\) 5103.45 1.43493 0.717464 0.696595i \(-0.245303\pi\)
0.717464 + 0.696595i \(0.245303\pi\)
\(234\) 0 0
\(235\) −157.574 −0.0437403
\(236\) 0 0
\(237\) 2010.74 0.551103
\(238\) 0 0
\(239\) −6552.57 −1.77343 −0.886716 0.462314i \(-0.847019\pi\)
−0.886716 + 0.462314i \(0.847019\pi\)
\(240\) 0 0
\(241\) 2398.64 0.641120 0.320560 0.947228i \(-0.396129\pi\)
0.320560 + 0.947228i \(0.396129\pi\)
\(242\) 0 0
\(243\) −243.000 −0.0641500
\(244\) 0 0
\(245\) 876.329 0.228517
\(246\) 0 0
\(247\) 961.988 0.247813
\(248\) 0 0
\(249\) −2359.99 −0.600635
\(250\) 0 0
\(251\) −2899.21 −0.729069 −0.364535 0.931190i \(-0.618772\pi\)
−0.364535 + 0.931190i \(0.618772\pi\)
\(252\) 0 0
\(253\) 3778.33 0.938899
\(254\) 0 0
\(255\) 33.8448 0.00831155
\(256\) 0 0
\(257\) −5784.67 −1.40404 −0.702019 0.712158i \(-0.747718\pi\)
−0.702019 + 0.712158i \(0.747718\pi\)
\(258\) 0 0
\(259\) 522.144 0.125268
\(260\) 0 0
\(261\) 2336.16 0.554040
\(262\) 0 0
\(263\) −5346.03 −1.25342 −0.626712 0.779251i \(-0.715600\pi\)
−0.626712 + 0.779251i \(0.715600\pi\)
\(264\) 0 0
\(265\) 481.364 0.111585
\(266\) 0 0
\(267\) 951.132 0.218009
\(268\) 0 0
\(269\) −5693.67 −1.29052 −0.645259 0.763964i \(-0.723250\pi\)
−0.645259 + 0.763964i \(0.723250\pi\)
\(270\) 0 0
\(271\) −496.345 −0.111258 −0.0556288 0.998452i \(-0.517716\pi\)
−0.0556288 + 0.998452i \(0.517716\pi\)
\(272\) 0 0
\(273\) 203.985 0.0452224
\(274\) 0 0
\(275\) 3052.89 0.669442
\(276\) 0 0
\(277\) −2832.42 −0.614380 −0.307190 0.951648i \(-0.599389\pi\)
−0.307190 + 0.951648i \(0.599389\pi\)
\(278\) 0 0
\(279\) 707.326 0.151780
\(280\) 0 0
\(281\) −5362.12 −1.13835 −0.569176 0.822216i \(-0.692738\pi\)
−0.569176 + 0.822216i \(0.692738\pi\)
\(282\) 0 0
\(283\) −3330.32 −0.699530 −0.349765 0.936838i \(-0.613739\pi\)
−0.349765 + 0.936838i \(0.613739\pi\)
\(284\) 0 0
\(285\) 616.337 0.128100
\(286\) 0 0
\(287\) −1811.38 −0.372553
\(288\) 0 0
\(289\) −4896.49 −0.996639
\(290\) 0 0
\(291\) −2658.98 −0.535644
\(292\) 0 0
\(293\) 939.941 0.187413 0.0937064 0.995600i \(-0.470128\pi\)
0.0937064 + 0.995600i \(0.470128\pi\)
\(294\) 0 0
\(295\) 1876.58 0.370369
\(296\) 0 0
\(297\) 702.760 0.137301
\(298\) 0 0
\(299\) −1887.12 −0.365000
\(300\) 0 0
\(301\) −720.390 −0.137949
\(302\) 0 0
\(303\) 622.585 0.118041
\(304\) 0 0
\(305\) −2216.25 −0.416072
\(306\) 0 0
\(307\) −366.403 −0.0681165 −0.0340582 0.999420i \(-0.510843\pi\)
−0.0340582 + 0.999420i \(0.510843\pi\)
\(308\) 0 0
\(309\) −5216.59 −0.960392
\(310\) 0 0
\(311\) −3116.58 −0.568247 −0.284124 0.958788i \(-0.591703\pi\)
−0.284124 + 0.958788i \(0.591703\pi\)
\(312\) 0 0
\(313\) 2360.35 0.426246 0.213123 0.977025i \(-0.431636\pi\)
0.213123 + 0.977025i \(0.431636\pi\)
\(314\) 0 0
\(315\) 130.691 0.0233766
\(316\) 0 0
\(317\) 1279.56 0.226711 0.113356 0.993554i \(-0.463840\pi\)
0.113356 + 0.993554i \(0.463840\pi\)
\(318\) 0 0
\(319\) −6756.20 −1.18581
\(320\) 0 0
\(321\) −3746.73 −0.651470
\(322\) 0 0
\(323\) 300.695 0.0517991
\(324\) 0 0
\(325\) −1524.80 −0.260248
\(326\) 0 0
\(327\) −5325.54 −0.900622
\(328\) 0 0
\(329\) −296.856 −0.0497453
\(330\) 0 0
\(331\) −10696.4 −1.77621 −0.888107 0.459638i \(-0.847979\pi\)
−0.888107 + 0.459638i \(0.847979\pi\)
\(332\) 0 0
\(333\) −898.462 −0.147854
\(334\) 0 0
\(335\) −812.308 −0.132481
\(336\) 0 0
\(337\) −1046.80 −0.169208 −0.0846038 0.996415i \(-0.526962\pi\)
−0.0846038 + 0.996415i \(0.526962\pi\)
\(338\) 0 0
\(339\) 4155.10 0.665705
\(340\) 0 0
\(341\) −2045.60 −0.324855
\(342\) 0 0
\(343\) 3444.95 0.542303
\(344\) 0 0
\(345\) −1209.06 −0.188677
\(346\) 0 0
\(347\) −6330.72 −0.979397 −0.489698 0.871892i \(-0.662893\pi\)
−0.489698 + 0.871892i \(0.662893\pi\)
\(348\) 0 0
\(349\) −5009.55 −0.768353 −0.384176 0.923260i \(-0.625515\pi\)
−0.384176 + 0.923260i \(0.625515\pi\)
\(350\) 0 0
\(351\) −351.000 −0.0533761
\(352\) 0 0
\(353\) 1846.27 0.278376 0.139188 0.990266i \(-0.455551\pi\)
0.139188 + 0.990266i \(0.455551\pi\)
\(354\) 0 0
\(355\) 925.806 0.138413
\(356\) 0 0
\(357\) 63.7610 0.00945263
\(358\) 0 0
\(359\) −2967.18 −0.436217 −0.218109 0.975925i \(-0.569989\pi\)
−0.218109 + 0.975925i \(0.569989\pi\)
\(360\) 0 0
\(361\) −1383.14 −0.201653
\(362\) 0 0
\(363\) 1960.61 0.283485
\(364\) 0 0
\(365\) −675.459 −0.0968634
\(366\) 0 0
\(367\) −1714.49 −0.243858 −0.121929 0.992539i \(-0.538908\pi\)
−0.121929 + 0.992539i \(0.538908\pi\)
\(368\) 0 0
\(369\) 3116.88 0.439724
\(370\) 0 0
\(371\) 906.852 0.126904
\(372\) 0 0
\(373\) 12101.2 1.67983 0.839917 0.542715i \(-0.182604\pi\)
0.839917 + 0.542715i \(0.182604\pi\)
\(374\) 0 0
\(375\) −2018.05 −0.277897
\(376\) 0 0
\(377\) 3374.45 0.460989
\(378\) 0 0
\(379\) −8525.01 −1.15541 −0.577705 0.816245i \(-0.696052\pi\)
−0.577705 + 0.816245i \(0.696052\pi\)
\(380\) 0 0
\(381\) 830.182 0.111631
\(382\) 0 0
\(383\) −2563.79 −0.342045 −0.171023 0.985267i \(-0.554707\pi\)
−0.171023 + 0.985267i \(0.554707\pi\)
\(384\) 0 0
\(385\) −377.961 −0.0500330
\(386\) 0 0
\(387\) 1239.59 0.162821
\(388\) 0 0
\(389\) −10963.3 −1.42895 −0.714475 0.699661i \(-0.753334\pi\)
−0.714475 + 0.699661i \(0.753334\pi\)
\(390\) 0 0
\(391\) −589.870 −0.0762942
\(392\) 0 0
\(393\) 6880.54 0.883149
\(394\) 0 0
\(395\) 1860.82 0.237033
\(396\) 0 0
\(397\) −7970.57 −1.00764 −0.503818 0.863810i \(-0.668072\pi\)
−0.503818 + 0.863810i \(0.668072\pi\)
\(398\) 0 0
\(399\) 1161.13 0.145687
\(400\) 0 0
\(401\) 972.347 0.121089 0.0605445 0.998165i \(-0.480716\pi\)
0.0605445 + 0.998165i \(0.480716\pi\)
\(402\) 0 0
\(403\) 1021.69 0.126288
\(404\) 0 0
\(405\) −224.883 −0.0275914
\(406\) 0 0
\(407\) 2598.37 0.316453
\(408\) 0 0
\(409\) 13183.5 1.59384 0.796920 0.604085i \(-0.206461\pi\)
0.796920 + 0.604085i \(0.206461\pi\)
\(410\) 0 0
\(411\) 7919.24 0.950431
\(412\) 0 0
\(413\) 3535.33 0.421216
\(414\) 0 0
\(415\) −2184.03 −0.258337
\(416\) 0 0
\(417\) 4114.44 0.483177
\(418\) 0 0
\(419\) 4346.49 0.506778 0.253389 0.967365i \(-0.418455\pi\)
0.253389 + 0.967365i \(0.418455\pi\)
\(420\) 0 0
\(421\) 2930.12 0.339205 0.169603 0.985513i \(-0.445752\pi\)
0.169603 + 0.985513i \(0.445752\pi\)
\(422\) 0 0
\(423\) 510.805 0.0587144
\(424\) 0 0
\(425\) −476.616 −0.0543983
\(426\) 0 0
\(427\) −4175.23 −0.473193
\(428\) 0 0
\(429\) 1015.10 0.114241
\(430\) 0 0
\(431\) −14506.2 −1.62121 −0.810604 0.585595i \(-0.800861\pi\)
−0.810604 + 0.585595i \(0.800861\pi\)
\(432\) 0 0
\(433\) 3672.96 0.407647 0.203823 0.979008i \(-0.434663\pi\)
0.203823 + 0.979008i \(0.434663\pi\)
\(434\) 0 0
\(435\) 2161.98 0.238296
\(436\) 0 0
\(437\) −10741.9 −1.17587
\(438\) 0 0
\(439\) 17003.5 1.84860 0.924299 0.381669i \(-0.124650\pi\)
0.924299 + 0.381669i \(0.124650\pi\)
\(440\) 0 0
\(441\) −2840.79 −0.306747
\(442\) 0 0
\(443\) 14173.8 1.52013 0.760066 0.649846i \(-0.225167\pi\)
0.760066 + 0.649846i \(0.225167\pi\)
\(444\) 0 0
\(445\) 880.218 0.0937671
\(446\) 0 0
\(447\) −5022.54 −0.531450
\(448\) 0 0
\(449\) −12102.9 −1.27209 −0.636046 0.771651i \(-0.719431\pi\)
−0.636046 + 0.771651i \(0.719431\pi\)
\(450\) 0 0
\(451\) −9014.06 −0.941143
\(452\) 0 0
\(453\) 7279.68 0.755031
\(454\) 0 0
\(455\) 188.776 0.0194505
\(456\) 0 0
\(457\) −3341.74 −0.342056 −0.171028 0.985266i \(-0.554709\pi\)
−0.171028 + 0.985266i \(0.554709\pi\)
\(458\) 0 0
\(459\) −109.714 −0.0111569
\(460\) 0 0
\(461\) −7857.61 −0.793851 −0.396926 0.917851i \(-0.629923\pi\)
−0.396926 + 0.917851i \(0.629923\pi\)
\(462\) 0 0
\(463\) 7149.73 0.717659 0.358829 0.933403i \(-0.383176\pi\)
0.358829 + 0.933403i \(0.383176\pi\)
\(464\) 0 0
\(465\) 654.590 0.0652814
\(466\) 0 0
\(467\) −11893.0 −1.17846 −0.589230 0.807965i \(-0.700569\pi\)
−0.589230 + 0.807965i \(0.700569\pi\)
\(468\) 0 0
\(469\) −1530.32 −0.150669
\(470\) 0 0
\(471\) −3871.87 −0.378782
\(472\) 0 0
\(473\) −3584.91 −0.348487
\(474\) 0 0
\(475\) −8679.50 −0.838406
\(476\) 0 0
\(477\) −1560.43 −0.149785
\(478\) 0 0
\(479\) −12678.9 −1.20942 −0.604710 0.796446i \(-0.706711\pi\)
−0.604710 + 0.796446i \(0.706711\pi\)
\(480\) 0 0
\(481\) −1297.78 −0.123022
\(482\) 0 0
\(483\) −2277.78 −0.214581
\(484\) 0 0
\(485\) −2460.74 −0.230384
\(486\) 0 0
\(487\) −8701.84 −0.809688 −0.404844 0.914386i \(-0.632674\pi\)
−0.404844 + 0.914386i \(0.632674\pi\)
\(488\) 0 0
\(489\) 11363.5 1.05087
\(490\) 0 0
\(491\) −5212.84 −0.479128 −0.239564 0.970881i \(-0.577005\pi\)
−0.239564 + 0.970881i \(0.577005\pi\)
\(492\) 0 0
\(493\) 1054.77 0.0963583
\(494\) 0 0
\(495\) 650.364 0.0590539
\(496\) 0 0
\(497\) 1744.14 0.157416
\(498\) 0 0
\(499\) −4138.31 −0.371255 −0.185627 0.982620i \(-0.559432\pi\)
−0.185627 + 0.982620i \(0.559432\pi\)
\(500\) 0 0
\(501\) −3012.60 −0.268649
\(502\) 0 0
\(503\) −17161.5 −1.52126 −0.760630 0.649185i \(-0.775110\pi\)
−0.760630 + 0.649185i \(0.775110\pi\)
\(504\) 0 0
\(505\) 576.166 0.0507704
\(506\) 0 0
\(507\) −507.000 −0.0444116
\(508\) 0 0
\(509\) −16707.7 −1.45492 −0.727461 0.686149i \(-0.759300\pi\)
−0.727461 + 0.686149i \(0.759300\pi\)
\(510\) 0 0
\(511\) −1272.51 −0.110162
\(512\) 0 0
\(513\) −1997.97 −0.171955
\(514\) 0 0
\(515\) −4827.65 −0.413071
\(516\) 0 0
\(517\) −1477.26 −0.125667
\(518\) 0 0
\(519\) 36.3174 0.00307159
\(520\) 0 0
\(521\) 8042.87 0.676324 0.338162 0.941088i \(-0.390195\pi\)
0.338162 + 0.941088i \(0.390195\pi\)
\(522\) 0 0
\(523\) −3521.49 −0.294425 −0.147212 0.989105i \(-0.547030\pi\)
−0.147212 + 0.989105i \(0.547030\pi\)
\(524\) 0 0
\(525\) −1840.45 −0.152997
\(526\) 0 0
\(527\) 319.358 0.0263974
\(528\) 0 0
\(529\) 8905.34 0.731926
\(530\) 0 0
\(531\) −6083.30 −0.497162
\(532\) 0 0
\(533\) 4502.15 0.365872
\(534\) 0 0
\(535\) −3467.38 −0.280202
\(536\) 0 0
\(537\) −791.179 −0.0635790
\(538\) 0 0
\(539\) 8215.61 0.656533
\(540\) 0 0
\(541\) −20824.9 −1.65496 −0.827480 0.561495i \(-0.810226\pi\)
−0.827480 + 0.561495i \(0.810226\pi\)
\(542\) 0 0
\(543\) −5825.83 −0.460424
\(544\) 0 0
\(545\) −4928.49 −0.387363
\(546\) 0 0
\(547\) −8219.38 −0.642478 −0.321239 0.946998i \(-0.604099\pi\)
−0.321239 + 0.946998i \(0.604099\pi\)
\(548\) 0 0
\(549\) 7184.39 0.558510
\(550\) 0 0
\(551\) 19208.1 1.48511
\(552\) 0 0
\(553\) 3505.64 0.269575
\(554\) 0 0
\(555\) −831.475 −0.0635931
\(556\) 0 0
\(557\) 644.376 0.0490181 0.0245091 0.999700i \(-0.492198\pi\)
0.0245091 + 0.999700i \(0.492198\pi\)
\(558\) 0 0
\(559\) 1790.51 0.135475
\(560\) 0 0
\(561\) 317.296 0.0238792
\(562\) 0 0
\(563\) 4735.04 0.354455 0.177227 0.984170i \(-0.443287\pi\)
0.177227 + 0.984170i \(0.443287\pi\)
\(564\) 0 0
\(565\) 3845.31 0.286324
\(566\) 0 0
\(567\) −423.661 −0.0313793
\(568\) 0 0
\(569\) 3403.32 0.250746 0.125373 0.992110i \(-0.459987\pi\)
0.125373 + 0.992110i \(0.459987\pi\)
\(570\) 0 0
\(571\) −12107.4 −0.887351 −0.443676 0.896187i \(-0.646326\pi\)
−0.443676 + 0.896187i \(0.646326\pi\)
\(572\) 0 0
\(573\) −5775.32 −0.421060
\(574\) 0 0
\(575\) 17026.5 1.23488
\(576\) 0 0
\(577\) 3208.73 0.231510 0.115755 0.993278i \(-0.463071\pi\)
0.115755 + 0.993278i \(0.463071\pi\)
\(578\) 0 0
\(579\) 9953.43 0.714422
\(580\) 0 0
\(581\) −4114.54 −0.293804
\(582\) 0 0
\(583\) 4512.80 0.320585
\(584\) 0 0
\(585\) −324.830 −0.0229574
\(586\) 0 0
\(587\) −13630.6 −0.958427 −0.479213 0.877698i \(-0.659078\pi\)
−0.479213 + 0.877698i \(0.659078\pi\)
\(588\) 0 0
\(589\) 5815.72 0.406846
\(590\) 0 0
\(591\) −12847.4 −0.894199
\(592\) 0 0
\(593\) 3392.81 0.234951 0.117476 0.993076i \(-0.462520\pi\)
0.117476 + 0.993076i \(0.462520\pi\)
\(594\) 0 0
\(595\) 59.0071 0.00406564
\(596\) 0 0
\(597\) 3657.31 0.250726
\(598\) 0 0
\(599\) −2245.25 −0.153153 −0.0765763 0.997064i \(-0.524399\pi\)
−0.0765763 + 0.997064i \(0.524399\pi\)
\(600\) 0 0
\(601\) −7307.86 −0.495997 −0.247998 0.968760i \(-0.579773\pi\)
−0.247998 + 0.968760i \(0.579773\pi\)
\(602\) 0 0
\(603\) 2633.25 0.177835
\(604\) 0 0
\(605\) 1814.43 0.121929
\(606\) 0 0
\(607\) 8567.55 0.572893 0.286447 0.958096i \(-0.407526\pi\)
0.286447 + 0.958096i \(0.407526\pi\)
\(608\) 0 0
\(609\) 4073.00 0.271012
\(610\) 0 0
\(611\) 737.830 0.0488533
\(612\) 0 0
\(613\) −20298.2 −1.33741 −0.668707 0.743526i \(-0.733152\pi\)
−0.668707 + 0.743526i \(0.733152\pi\)
\(614\) 0 0
\(615\) 2884.49 0.189128
\(616\) 0 0
\(617\) −7172.25 −0.467980 −0.233990 0.972239i \(-0.575178\pi\)
−0.233990 + 0.972239i \(0.575178\pi\)
\(618\) 0 0
\(619\) −2861.42 −0.185800 −0.0928998 0.995675i \(-0.529614\pi\)
−0.0928998 + 0.995675i \(0.529614\pi\)
\(620\) 0 0
\(621\) 3919.40 0.253269
\(622\) 0 0
\(623\) 1658.26 0.106640
\(624\) 0 0
\(625\) 12793.9 0.818811
\(626\) 0 0
\(627\) 5778.18 0.368035
\(628\) 0 0
\(629\) −405.656 −0.0257147
\(630\) 0 0
\(631\) 30554.9 1.92769 0.963843 0.266469i \(-0.0858570\pi\)
0.963843 + 0.266469i \(0.0858570\pi\)
\(632\) 0 0
\(633\) 14870.0 0.933694
\(634\) 0 0
\(635\) 768.285 0.0480133
\(636\) 0 0
\(637\) −4103.36 −0.255229
\(638\) 0 0
\(639\) −3001.18 −0.185798
\(640\) 0 0
\(641\) 18473.6 1.13832 0.569161 0.822226i \(-0.307268\pi\)
0.569161 + 0.822226i \(0.307268\pi\)
\(642\) 0 0
\(643\) 22801.7 1.39846 0.699232 0.714895i \(-0.253526\pi\)
0.699232 + 0.714895i \(0.253526\pi\)
\(644\) 0 0
\(645\) 1147.17 0.0700304
\(646\) 0 0
\(647\) −11212.0 −0.681280 −0.340640 0.940194i \(-0.610644\pi\)
−0.340640 + 0.940194i \(0.610644\pi\)
\(648\) 0 0
\(649\) 17593.0 1.06408
\(650\) 0 0
\(651\) 1233.20 0.0742438
\(652\) 0 0
\(653\) −1885.87 −0.113017 −0.0565084 0.998402i \(-0.517997\pi\)
−0.0565084 + 0.998402i \(0.517997\pi\)
\(654\) 0 0
\(655\) 6367.55 0.379848
\(656\) 0 0
\(657\) 2189.63 0.130024
\(658\) 0 0
\(659\) −525.205 −0.0310456 −0.0155228 0.999880i \(-0.504941\pi\)
−0.0155228 + 0.999880i \(0.504941\pi\)
\(660\) 0 0
\(661\) 8036.59 0.472900 0.236450 0.971644i \(-0.424016\pi\)
0.236450 + 0.971644i \(0.424016\pi\)
\(662\) 0 0
\(663\) −158.476 −0.00928313
\(664\) 0 0
\(665\) 1074.56 0.0626611
\(666\) 0 0
\(667\) −37680.4 −2.18739
\(668\) 0 0
\(669\) −5755.89 −0.332639
\(670\) 0 0
\(671\) −20777.4 −1.19538
\(672\) 0 0
\(673\) 19505.9 1.11723 0.558616 0.829426i \(-0.311332\pi\)
0.558616 + 0.829426i \(0.311332\pi\)
\(674\) 0 0
\(675\) 3166.88 0.180583
\(676\) 0 0
\(677\) 12847.5 0.729347 0.364673 0.931136i \(-0.381181\pi\)
0.364673 + 0.931136i \(0.381181\pi\)
\(678\) 0 0
\(679\) −4635.83 −0.262013
\(680\) 0 0
\(681\) −3805.07 −0.214112
\(682\) 0 0
\(683\) −12286.7 −0.688342 −0.344171 0.938907i \(-0.611840\pi\)
−0.344171 + 0.938907i \(0.611840\pi\)
\(684\) 0 0
\(685\) 7328.80 0.408787
\(686\) 0 0
\(687\) −7443.15 −0.413354
\(688\) 0 0
\(689\) −2253.96 −0.124629
\(690\) 0 0
\(691\) 24058.5 1.32450 0.662250 0.749283i \(-0.269602\pi\)
0.662250 + 0.749283i \(0.269602\pi\)
\(692\) 0 0
\(693\) 1225.23 0.0671613
\(694\) 0 0
\(695\) 3807.68 0.207818
\(696\) 0 0
\(697\) 1407.27 0.0764765
\(698\) 0 0
\(699\) −15310.4 −0.828456
\(700\) 0 0
\(701\) −28829.0 −1.55329 −0.776644 0.629940i \(-0.783080\pi\)
−0.776644 + 0.629940i \(0.783080\pi\)
\(702\) 0 0
\(703\) −7387.26 −0.396324
\(704\) 0 0
\(705\) 472.721 0.0252535
\(706\) 0 0
\(707\) 1085.45 0.0577406
\(708\) 0 0
\(709\) −4146.60 −0.219646 −0.109823 0.993951i \(-0.535028\pi\)
−0.109823 + 0.993951i \(0.535028\pi\)
\(710\) 0 0
\(711\) −6032.22 −0.318180
\(712\) 0 0
\(713\) −11408.6 −0.599238
\(714\) 0 0
\(715\) 939.415 0.0491358
\(716\) 0 0
\(717\) 19657.7 1.02389
\(718\) 0 0
\(719\) −9681.67 −0.502177 −0.251089 0.967964i \(-0.580789\pi\)
−0.251089 + 0.967964i \(0.580789\pi\)
\(720\) 0 0
\(721\) −9094.91 −0.469781
\(722\) 0 0
\(723\) −7195.91 −0.370151
\(724\) 0 0
\(725\) −30445.8 −1.55963
\(726\) 0 0
\(727\) 10092.3 0.514860 0.257430 0.966297i \(-0.417124\pi\)
0.257430 + 0.966297i \(0.417124\pi\)
\(728\) 0 0
\(729\) 729.000 0.0370370
\(730\) 0 0
\(731\) 559.673 0.0283177
\(732\) 0 0
\(733\) 7775.50 0.391807 0.195904 0.980623i \(-0.437236\pi\)
0.195904 + 0.980623i \(0.437236\pi\)
\(734\) 0 0
\(735\) −2628.99 −0.131934
\(736\) 0 0
\(737\) −7615.41 −0.380621
\(738\) 0 0
\(739\) −2699.84 −0.134392 −0.0671958 0.997740i \(-0.521405\pi\)
−0.0671958 + 0.997740i \(0.521405\pi\)
\(740\) 0 0
\(741\) −2885.96 −0.143075
\(742\) 0 0
\(743\) −14952.1 −0.738279 −0.369139 0.929374i \(-0.620348\pi\)
−0.369139 + 0.929374i \(0.620348\pi\)
\(744\) 0 0
\(745\) −4648.07 −0.228580
\(746\) 0 0
\(747\) 7079.96 0.346777
\(748\) 0 0
\(749\) −6532.27 −0.318670
\(750\) 0 0
\(751\) 21169.6 1.02861 0.514307 0.857606i \(-0.328049\pi\)
0.514307 + 0.857606i \(0.328049\pi\)
\(752\) 0 0
\(753\) 8697.62 0.420928
\(754\) 0 0
\(755\) 6736.92 0.324744
\(756\) 0 0
\(757\) 10096.0 0.484736 0.242368 0.970184i \(-0.422076\pi\)
0.242368 + 0.970184i \(0.422076\pi\)
\(758\) 0 0
\(759\) −11335.0 −0.542074
\(760\) 0 0
\(761\) 31395.8 1.49553 0.747764 0.663965i \(-0.231127\pi\)
0.747764 + 0.663965i \(0.231127\pi\)
\(762\) 0 0
\(763\) −9284.88 −0.440544
\(764\) 0 0
\(765\) −101.534 −0.00479867
\(766\) 0 0
\(767\) −8787.00 −0.413664
\(768\) 0 0
\(769\) −16798.8 −0.787750 −0.393875 0.919164i \(-0.628866\pi\)
−0.393875 + 0.919164i \(0.628866\pi\)
\(770\) 0 0
\(771\) 17354.0 0.810622
\(772\) 0 0
\(773\) 18854.0 0.877271 0.438635 0.898665i \(-0.355462\pi\)
0.438635 + 0.898665i \(0.355462\pi\)
\(774\) 0 0
\(775\) −9218.19 −0.427261
\(776\) 0 0
\(777\) −1566.43 −0.0723237
\(778\) 0 0
\(779\) 25627.3 1.17868
\(780\) 0 0
\(781\) 8679.46 0.397664
\(782\) 0 0
\(783\) −7008.47 −0.319875
\(784\) 0 0
\(785\) −3583.19 −0.162917
\(786\) 0 0
\(787\) −12420.9 −0.562587 −0.281294 0.959622i \(-0.590763\pi\)
−0.281294 + 0.959622i \(0.590763\pi\)
\(788\) 0 0
\(789\) 16038.1 0.723665
\(790\) 0 0
\(791\) 7244.25 0.325633
\(792\) 0 0
\(793\) 10377.4 0.464709
\(794\) 0 0
\(795\) −1444.09 −0.0644235
\(796\) 0 0
\(797\) −34417.0 −1.52963 −0.764814 0.644251i \(-0.777169\pi\)
−0.764814 + 0.644251i \(0.777169\pi\)
\(798\) 0 0
\(799\) 230.628 0.0102116
\(800\) 0 0
\(801\) −2853.40 −0.125867
\(802\) 0 0
\(803\) −6332.45 −0.278291
\(804\) 0 0
\(805\) −2107.95 −0.0922925
\(806\) 0 0
\(807\) 17081.0 0.745081
\(808\) 0 0
\(809\) −21745.5 −0.945033 −0.472516 0.881322i \(-0.656654\pi\)
−0.472516 + 0.881322i \(0.656654\pi\)
\(810\) 0 0
\(811\) −16487.2 −0.713863 −0.356932 0.934131i \(-0.616177\pi\)
−0.356932 + 0.934131i \(0.616177\pi\)
\(812\) 0 0
\(813\) 1489.03 0.0642346
\(814\) 0 0
\(815\) 10516.3 0.451988
\(816\) 0 0
\(817\) 10192.0 0.436443
\(818\) 0 0
\(819\) −611.955 −0.0261092
\(820\) 0 0
\(821\) −12274.1 −0.521767 −0.260883 0.965370i \(-0.584014\pi\)
−0.260883 + 0.965370i \(0.584014\pi\)
\(822\) 0 0
\(823\) 25913.7 1.09756 0.548782 0.835966i \(-0.315092\pi\)
0.548782 + 0.835966i \(0.315092\pi\)
\(824\) 0 0
\(825\) −9158.68 −0.386502
\(826\) 0 0
\(827\) 19617.2 0.824855 0.412428 0.910990i \(-0.364681\pi\)
0.412428 + 0.910990i \(0.364681\pi\)
\(828\) 0 0
\(829\) 40662.5 1.70358 0.851790 0.523883i \(-0.175517\pi\)
0.851790 + 0.523883i \(0.175517\pi\)
\(830\) 0 0
\(831\) 8497.25 0.354713
\(832\) 0 0
\(833\) −1282.62 −0.0533493
\(834\) 0 0
\(835\) −2787.99 −0.115548
\(836\) 0 0
\(837\) −2121.98 −0.0876300
\(838\) 0 0
\(839\) 16709.7 0.687584 0.343792 0.939046i \(-0.388289\pi\)
0.343792 + 0.939046i \(0.388289\pi\)
\(840\) 0 0
\(841\) 42989.1 1.76264
\(842\) 0 0
\(843\) 16086.3 0.657228
\(844\) 0 0
\(845\) −469.199 −0.0191017
\(846\) 0 0
\(847\) 3418.24 0.138668
\(848\) 0 0
\(849\) 9990.96 0.403874
\(850\) 0 0
\(851\) 14491.5 0.583740
\(852\) 0 0
\(853\) 18833.6 0.755978 0.377989 0.925810i \(-0.376616\pi\)
0.377989 + 0.925810i \(0.376616\pi\)
\(854\) 0 0
\(855\) −1849.01 −0.0739589
\(856\) 0 0
\(857\) −26784.6 −1.06761 −0.533807 0.845607i \(-0.679239\pi\)
−0.533807 + 0.845607i \(0.679239\pi\)
\(858\) 0 0
\(859\) 10246.9 0.407007 0.203504 0.979074i \(-0.434767\pi\)
0.203504 + 0.979074i \(0.434767\pi\)
\(860\) 0 0
\(861\) 5434.15 0.215093
\(862\) 0 0
\(863\) −5000.15 −0.197227 −0.0986136 0.995126i \(-0.531441\pi\)
−0.0986136 + 0.995126i \(0.531441\pi\)
\(864\) 0 0
\(865\) 33.6096 0.00132111
\(866\) 0 0
\(867\) 14689.5 0.575410
\(868\) 0 0
\(869\) 17445.3 0.681002
\(870\) 0 0
\(871\) 3803.59 0.147967
\(872\) 0 0
\(873\) 7976.95 0.309254
\(874\) 0 0
\(875\) −3518.38 −0.135935
\(876\) 0 0
\(877\) −16648.8 −0.641039 −0.320519 0.947242i \(-0.603857\pi\)
−0.320519 + 0.947242i \(0.603857\pi\)
\(878\) 0 0
\(879\) −2819.82 −0.108203
\(880\) 0 0
\(881\) −16304.0 −0.623491 −0.311746 0.950166i \(-0.600914\pi\)
−0.311746 + 0.950166i \(0.600914\pi\)
\(882\) 0 0
\(883\) 17857.7 0.680589 0.340295 0.940319i \(-0.389473\pi\)
0.340295 + 0.940319i \(0.389473\pi\)
\(884\) 0 0
\(885\) −5629.75 −0.213833
\(886\) 0 0
\(887\) 46797.7 1.77149 0.885745 0.464172i \(-0.153648\pi\)
0.885745 + 0.464172i \(0.153648\pi\)
\(888\) 0 0
\(889\) 1447.39 0.0546050
\(890\) 0 0
\(891\) −2108.28 −0.0792705
\(892\) 0 0
\(893\) 4199.90 0.157384
\(894\) 0 0
\(895\) −732.191 −0.0273458
\(896\) 0 0
\(897\) 5661.36 0.210733
\(898\) 0 0
\(899\) 20400.3 0.756828
\(900\) 0 0
\(901\) −704.536 −0.0260505
\(902\) 0 0
\(903\) 2161.17 0.0796448
\(904\) 0 0
\(905\) −5391.47 −0.198032
\(906\) 0 0
\(907\) 52238.0 1.91239 0.956193 0.292738i \(-0.0945662\pi\)
0.956193 + 0.292738i \(0.0945662\pi\)
\(908\) 0 0
\(909\) −1867.75 −0.0681513
\(910\) 0 0
\(911\) −31047.7 −1.12915 −0.564576 0.825381i \(-0.690960\pi\)
−0.564576 + 0.825381i \(0.690960\pi\)
\(912\) 0 0
\(913\) −20475.4 −0.742208
\(914\) 0 0
\(915\) 6648.74 0.240219
\(916\) 0 0
\(917\) 11996.0 0.431997
\(918\) 0 0
\(919\) 13876.1 0.498075 0.249038 0.968494i \(-0.419886\pi\)
0.249038 + 0.968494i \(0.419886\pi\)
\(920\) 0 0
\(921\) 1099.21 0.0393271
\(922\) 0 0
\(923\) −4335.03 −0.154593
\(924\) 0 0
\(925\) 11709.2 0.416211
\(926\) 0 0
\(927\) 15649.8 0.554483
\(928\) 0 0
\(929\) −21424.9 −0.756650 −0.378325 0.925673i \(-0.623500\pi\)
−0.378325 + 0.925673i \(0.623500\pi\)
\(930\) 0 0
\(931\) −23357.3 −0.822239
\(932\) 0 0
\(933\) 9349.73 0.328078
\(934\) 0 0
\(935\) 293.639 0.0102706
\(936\) 0 0
\(937\) −4812.13 −0.167775 −0.0838876 0.996475i \(-0.526734\pi\)
−0.0838876 + 0.996475i \(0.526734\pi\)
\(938\) 0 0
\(939\) −7081.06 −0.246093
\(940\) 0 0
\(941\) 48232.7 1.67093 0.835464 0.549546i \(-0.185199\pi\)
0.835464 + 0.549546i \(0.185199\pi\)
\(942\) 0 0
\(943\) −50272.8 −1.73606
\(944\) 0 0
\(945\) −392.074 −0.0134965
\(946\) 0 0
\(947\) 33990.6 1.16636 0.583181 0.812342i \(-0.301808\pi\)
0.583181 + 0.812342i \(0.301808\pi\)
\(948\) 0 0
\(949\) 3162.80 0.108186
\(950\) 0 0
\(951\) −3838.69 −0.130892
\(952\) 0 0
\(953\) −12836.1 −0.436309 −0.218154 0.975914i \(-0.570004\pi\)
−0.218154 + 0.975914i \(0.570004\pi\)
\(954\) 0 0
\(955\) −5344.72 −0.181101
\(956\) 0 0
\(957\) 20268.6 0.684630
\(958\) 0 0
\(959\) 13806.9 0.464909
\(960\) 0 0
\(961\) −23614.3 −0.792667
\(962\) 0 0
\(963\) 11240.2 0.376126
\(964\) 0 0
\(965\) 9211.32 0.307278
\(966\) 0 0
\(967\) −5141.56 −0.170984 −0.0854920 0.996339i \(-0.527246\pi\)
−0.0854920 + 0.996339i \(0.527246\pi\)
\(968\) 0 0
\(969\) −902.085 −0.0299063
\(970\) 0 0
\(971\) 5891.05 0.194699 0.0973496 0.995250i \(-0.468964\pi\)
0.0973496 + 0.995250i \(0.468964\pi\)
\(972\) 0 0
\(973\) 7173.36 0.236349
\(974\) 0 0
\(975\) 4574.39 0.150254
\(976\) 0 0
\(977\) −10105.7 −0.330922 −0.165461 0.986216i \(-0.552911\pi\)
−0.165461 + 0.986216i \(0.552911\pi\)
\(978\) 0 0
\(979\) 8252.07 0.269395
\(980\) 0 0
\(981\) 15976.6 0.519974
\(982\) 0 0
\(983\) −7957.89 −0.258207 −0.129103 0.991631i \(-0.541210\pi\)
−0.129103 + 0.991631i \(0.541210\pi\)
\(984\) 0 0
\(985\) −11889.5 −0.384601
\(986\) 0 0
\(987\) 890.568 0.0287205
\(988\) 0 0
\(989\) −19993.6 −0.642830
\(990\) 0 0
\(991\) −16996.7 −0.544822 −0.272411 0.962181i \(-0.587821\pi\)
−0.272411 + 0.962181i \(0.587821\pi\)
\(992\) 0 0
\(993\) 32089.2 1.02550
\(994\) 0 0
\(995\) 3384.63 0.107839
\(996\) 0 0
\(997\) −46632.0 −1.48130 −0.740648 0.671894i \(-0.765481\pi\)
−0.740648 + 0.671894i \(0.765481\pi\)
\(998\) 0 0
\(999\) 2695.39 0.0853636
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 2496.4.a.ca.1.2 5
4.3 odd 2 2496.4.a.cf.1.2 5
8.3 odd 2 1248.4.a.g.1.4 5
8.5 even 2 1248.4.a.l.1.4 yes 5
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1248.4.a.g.1.4 5 8.3 odd 2
1248.4.a.l.1.4 yes 5 8.5 even 2
2496.4.a.ca.1.2 5 1.1 even 1 trivial
2496.4.a.cf.1.2 5 4.3 odd 2