Properties

Label 76.4.a.b.1.1
Level $76$
Weight $4$
Character 76.1
Self dual yes
Analytic conductor $4.484$
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] = [76,4,Mod(1,76)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(76, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("76.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 76 = 2^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 76.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: \(4.48414516044\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.35529.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 52x + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(0.0768183\) of defining polynomial
Character \(\chi\) \(=\) 76.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-8.47932 q^{3} -5.55614 q^{5} +32.0355 q^{7} +44.8989 q^{9} +O(q^{10})\) \(q-8.47932 q^{3} -5.55614 q^{5} +32.0355 q^{7} +44.8989 q^{9} +54.0539 q^{11} -62.2771 q^{13} +47.1123 q^{15} +75.3355 q^{17} +19.0000 q^{19} -271.639 q^{21} +114.160 q^{23} -94.1293 q^{25} -151.770 q^{27} +171.178 q^{29} +59.9028 q^{31} -458.340 q^{33} -177.993 q^{35} +156.792 q^{37} +528.067 q^{39} +163.384 q^{41} -240.326 q^{43} -249.464 q^{45} -292.877 q^{47} +683.271 q^{49} -638.794 q^{51} -214.708 q^{53} -300.331 q^{55} -161.107 q^{57} +137.343 q^{59} -3.76203 q^{61} +1438.36 q^{63} +346.020 q^{65} +842.158 q^{67} -968.001 q^{69} +1090.78 q^{71} -954.887 q^{73} +798.153 q^{75} +1731.64 q^{77} -363.816 q^{79} +74.6391 q^{81} +131.966 q^{83} -418.574 q^{85} -1451.47 q^{87} +55.5599 q^{89} -1995.07 q^{91} -507.935 q^{93} -105.567 q^{95} -470.657 q^{97} +2426.96 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + q^{3} + 9 q^{5} + 44 q^{7} + 60 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q + q^{3} + 9 q^{5} + 44 q^{7} + 60 q^{9} + 79 q^{11} - 11 q^{13} + 90 q^{15} + 82 q^{17} + 57 q^{19} - 213 q^{21} + 103 q^{23} - 210 q^{25} - 107 q^{27} - 93 q^{29} - 116 q^{31} - 664 q^{33} - 93 q^{35} - 466 q^{37} + 337 q^{39} - 188 q^{41} - 11 q^{43} - 387 q^{45} + 163 q^{47} + 69 q^{49} + 203 q^{51} + 197 q^{53} + 231 q^{55} + 19 q^{57} + 1381 q^{59} - 405 q^{61} + 1547 q^{63} + 1188 q^{65} + 943 q^{67} - 1339 q^{69} + 1052 q^{71} - 580 q^{73} - 131 q^{75} + 1855 q^{77} - 1402 q^{79} + 495 q^{81} + 1802 q^{83} - 1245 q^{85} - 3159 q^{87} + 1966 q^{89} - 1723 q^{91} - 224 q^{93} + 171 q^{95} + 48 q^{97} - 455 q^{99}+O(q^{100}) \) 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
\(3\) −8.47932 −1.63185 −0.815923 0.578161i \(-0.803771\pi\)
−0.815923 + 0.578161i \(0.803771\pi\)
\(4\) 0 0
\(5\) −5.55614 −0.496956 −0.248478 0.968638i \(-0.579930\pi\)
−0.248478 + 0.968638i \(0.579930\pi\)
\(6\) 0 0
\(7\) 32.0355 1.72975 0.864876 0.501985i \(-0.167397\pi\)
0.864876 + 0.501985i \(0.167397\pi\)
\(8\) 0 0
\(9\) 44.8989 1.66292
\(10\) 0 0
\(11\) 54.0539 1.48162 0.740812 0.671713i \(-0.234441\pi\)
0.740812 + 0.671713i \(0.234441\pi\)
\(12\) 0 0
\(13\) −62.2771 −1.32866 −0.664329 0.747440i \(-0.731283\pi\)
−0.664329 + 0.747440i \(0.731283\pi\)
\(14\) 0 0
\(15\) 47.1123 0.810956
\(16\) 0 0
\(17\) 75.3355 1.07480 0.537398 0.843329i \(-0.319407\pi\)
0.537398 + 0.843329i \(0.319407\pi\)
\(18\) 0 0
\(19\) 19.0000 0.229416
\(20\) 0 0
\(21\) −271.639 −2.82269
\(22\) 0 0
\(23\) 114.160 1.03496 0.517479 0.855696i \(-0.326870\pi\)
0.517479 + 0.855696i \(0.326870\pi\)
\(24\) 0 0
\(25\) −94.1293 −0.753035
\(26\) 0 0
\(27\) −151.770 −1.08179
\(28\) 0 0
\(29\) 171.178 1.09610 0.548051 0.836445i \(-0.315370\pi\)
0.548051 + 0.836445i \(0.315370\pi\)
\(30\) 0 0
\(31\) 59.9028 0.347060 0.173530 0.984829i \(-0.444483\pi\)
0.173530 + 0.984829i \(0.444483\pi\)
\(32\) 0 0
\(33\) −458.340 −2.41778
\(34\) 0 0
\(35\) −177.993 −0.859611
\(36\) 0 0
\(37\) 156.792 0.696660 0.348330 0.937372i \(-0.386749\pi\)
0.348330 + 0.937372i \(0.386749\pi\)
\(38\) 0 0
\(39\) 528.067 2.16816
\(40\) 0 0
\(41\) 163.384 0.622349 0.311175 0.950353i \(-0.399278\pi\)
0.311175 + 0.950353i \(0.399278\pi\)
\(42\) 0 0
\(43\) −240.326 −0.852310 −0.426155 0.904650i \(-0.640132\pi\)
−0.426155 + 0.904650i \(0.640132\pi\)
\(44\) 0 0
\(45\) −249.464 −0.826399
\(46\) 0 0
\(47\) −292.877 −0.908947 −0.454474 0.890760i \(-0.650173\pi\)
−0.454474 + 0.890760i \(0.650173\pi\)
\(48\) 0 0
\(49\) 683.271 1.99204
\(50\) 0 0
\(51\) −638.794 −1.75390
\(52\) 0 0
\(53\) −214.708 −0.556461 −0.278230 0.960514i \(-0.589748\pi\)
−0.278230 + 0.960514i \(0.589748\pi\)
\(54\) 0 0
\(55\) −300.331 −0.736302
\(56\) 0 0
\(57\) −161.107 −0.374371
\(58\) 0 0
\(59\) 137.343 0.303061 0.151530 0.988453i \(-0.451580\pi\)
0.151530 + 0.988453i \(0.451580\pi\)
\(60\) 0 0
\(61\) −3.76203 −0.00789637 −0.00394818 0.999992i \(-0.501257\pi\)
−0.00394818 + 0.999992i \(0.501257\pi\)
\(62\) 0 0
\(63\) 1438.36 2.87644
\(64\) 0 0
\(65\) 346.020 0.660285
\(66\) 0 0
\(67\) 842.158 1.53561 0.767805 0.640683i \(-0.221349\pi\)
0.767805 + 0.640683i \(0.221349\pi\)
\(68\) 0 0
\(69\) −968.001 −1.68889
\(70\) 0 0
\(71\) 1090.78 1.82327 0.911633 0.411005i \(-0.134822\pi\)
0.911633 + 0.411005i \(0.134822\pi\)
\(72\) 0 0
\(73\) −954.887 −1.53097 −0.765487 0.643452i \(-0.777502\pi\)
−0.765487 + 0.643452i \(0.777502\pi\)
\(74\) 0 0
\(75\) 798.153 1.22884
\(76\) 0 0
\(77\) 1731.64 2.56284
\(78\) 0 0
\(79\) −363.816 −0.518133 −0.259066 0.965860i \(-0.583415\pi\)
−0.259066 + 0.965860i \(0.583415\pi\)
\(80\) 0 0
\(81\) 74.6391 0.102386
\(82\) 0 0
\(83\) 131.966 0.174520 0.0872598 0.996186i \(-0.472189\pi\)
0.0872598 + 0.996186i \(0.472189\pi\)
\(84\) 0 0
\(85\) −418.574 −0.534127
\(86\) 0 0
\(87\) −1451.47 −1.78867
\(88\) 0 0
\(89\) 55.5599 0.0661723 0.0330861 0.999453i \(-0.489466\pi\)
0.0330861 + 0.999453i \(0.489466\pi\)
\(90\) 0 0
\(91\) −1995.07 −2.29825
\(92\) 0 0
\(93\) −507.935 −0.566348
\(94\) 0 0
\(95\) −105.567 −0.114010
\(96\) 0 0
\(97\) −470.657 −0.492659 −0.246330 0.969186i \(-0.579225\pi\)
−0.246330 + 0.969186i \(0.579225\pi\)
\(98\) 0 0
\(99\) 2426.96 2.46382
\(100\) 0 0
\(101\) −1444.43 −1.42303 −0.711517 0.702669i \(-0.751991\pi\)
−0.711517 + 0.702669i \(0.751991\pi\)
\(102\) 0 0
\(103\) −1013.94 −0.969965 −0.484983 0.874524i \(-0.661174\pi\)
−0.484983 + 0.874524i \(0.661174\pi\)
\(104\) 0 0
\(105\) 1509.26 1.40275
\(106\) 0 0
\(107\) −1636.57 −1.47863 −0.739314 0.673361i \(-0.764850\pi\)
−0.739314 + 0.673361i \(0.764850\pi\)
\(108\) 0 0
\(109\) −87.1564 −0.0765878 −0.0382939 0.999267i \(-0.512192\pi\)
−0.0382939 + 0.999267i \(0.512192\pi\)
\(110\) 0 0
\(111\) −1329.49 −1.13684
\(112\) 0 0
\(113\) 524.144 0.436348 0.218174 0.975910i \(-0.429990\pi\)
0.218174 + 0.975910i \(0.429990\pi\)
\(114\) 0 0
\(115\) −634.290 −0.514329
\(116\) 0 0
\(117\) −2796.17 −2.20945
\(118\) 0 0
\(119\) 2413.41 1.85913
\(120\) 0 0
\(121\) 1590.82 1.19521
\(122\) 0 0
\(123\) −1385.39 −1.01558
\(124\) 0 0
\(125\) 1217.51 0.871181
\(126\) 0 0
\(127\) −795.847 −0.556063 −0.278032 0.960572i \(-0.589682\pi\)
−0.278032 + 0.960572i \(0.589682\pi\)
\(128\) 0 0
\(129\) 2037.80 1.39084
\(130\) 0 0
\(131\) 1550.55 1.03414 0.517069 0.855944i \(-0.327023\pi\)
0.517069 + 0.855944i \(0.327023\pi\)
\(132\) 0 0
\(133\) 608.674 0.396832
\(134\) 0 0
\(135\) 843.257 0.537600
\(136\) 0 0
\(137\) −853.926 −0.532524 −0.266262 0.963901i \(-0.585789\pi\)
−0.266262 + 0.963901i \(0.585789\pi\)
\(138\) 0 0
\(139\) 3041.73 1.85609 0.928044 0.372472i \(-0.121490\pi\)
0.928044 + 0.372472i \(0.121490\pi\)
\(140\) 0 0
\(141\) 2483.40 1.48326
\(142\) 0 0
\(143\) −3366.32 −1.96857
\(144\) 0 0
\(145\) −951.089 −0.544715
\(146\) 0 0
\(147\) −5793.67 −3.25071
\(148\) 0 0
\(149\) −1981.04 −1.08921 −0.544607 0.838692i \(-0.683321\pi\)
−0.544607 + 0.838692i \(0.683321\pi\)
\(150\) 0 0
\(151\) −1753.69 −0.945121 −0.472561 0.881298i \(-0.656670\pi\)
−0.472561 + 0.881298i \(0.656670\pi\)
\(152\) 0 0
\(153\) 3382.48 1.78730
\(154\) 0 0
\(155\) −332.828 −0.172473
\(156\) 0 0
\(157\) −727.595 −0.369862 −0.184931 0.982751i \(-0.559206\pi\)
−0.184931 + 0.982751i \(0.559206\pi\)
\(158\) 0 0
\(159\) 1820.58 0.908058
\(160\) 0 0
\(161\) 3657.18 1.79022
\(162\) 0 0
\(163\) −937.248 −0.450374 −0.225187 0.974316i \(-0.572299\pi\)
−0.225187 + 0.974316i \(0.572299\pi\)
\(164\) 0 0
\(165\) 2546.60 1.20153
\(166\) 0 0
\(167\) 3051.88 1.41414 0.707070 0.707144i \(-0.250017\pi\)
0.707070 + 0.707144i \(0.250017\pi\)
\(168\) 0 0
\(169\) 1681.43 0.765331
\(170\) 0 0
\(171\) 853.079 0.381500
\(172\) 0 0
\(173\) 2044.95 0.898698 0.449349 0.893356i \(-0.351656\pi\)
0.449349 + 0.893356i \(0.351656\pi\)
\(174\) 0 0
\(175\) −3015.48 −1.30256
\(176\) 0 0
\(177\) −1164.58 −0.494548
\(178\) 0 0
\(179\) −846.452 −0.353446 −0.176723 0.984261i \(-0.556550\pi\)
−0.176723 + 0.984261i \(0.556550\pi\)
\(180\) 0 0
\(181\) 2698.93 1.10834 0.554171 0.832403i \(-0.313035\pi\)
0.554171 + 0.832403i \(0.313035\pi\)
\(182\) 0 0
\(183\) 31.8994 0.0128857
\(184\) 0 0
\(185\) −871.157 −0.346210
\(186\) 0 0
\(187\) 4072.17 1.59244
\(188\) 0 0
\(189\) −4862.03 −1.87122
\(190\) 0 0
\(191\) 2247.79 0.851540 0.425770 0.904831i \(-0.360003\pi\)
0.425770 + 0.904831i \(0.360003\pi\)
\(192\) 0 0
\(193\) 1859.21 0.693416 0.346708 0.937973i \(-0.387300\pi\)
0.346708 + 0.937973i \(0.387300\pi\)
\(194\) 0 0
\(195\) −2934.01 −1.07748
\(196\) 0 0
\(197\) −2426.97 −0.877740 −0.438870 0.898551i \(-0.644621\pi\)
−0.438870 + 0.898551i \(0.644621\pi\)
\(198\) 0 0
\(199\) −3158.42 −1.12510 −0.562549 0.826764i \(-0.690179\pi\)
−0.562549 + 0.826764i \(0.690179\pi\)
\(200\) 0 0
\(201\) −7140.92 −2.50588
\(202\) 0 0
\(203\) 5483.77 1.89598
\(204\) 0 0
\(205\) −907.785 −0.309280
\(206\) 0 0
\(207\) 5125.67 1.72106
\(208\) 0 0
\(209\) 1027.02 0.339908
\(210\) 0 0
\(211\) 3223.50 1.05173 0.525864 0.850569i \(-0.323742\pi\)
0.525864 + 0.850569i \(0.323742\pi\)
\(212\) 0 0
\(213\) −9249.08 −2.97529
\(214\) 0 0
\(215\) 1335.28 0.423561
\(216\) 0 0
\(217\) 1919.01 0.600327
\(218\) 0 0
\(219\) 8096.79 2.49831
\(220\) 0 0
\(221\) −4691.67 −1.42804
\(222\) 0 0
\(223\) −5068.34 −1.52198 −0.760989 0.648765i \(-0.775286\pi\)
−0.760989 + 0.648765i \(0.775286\pi\)
\(224\) 0 0
\(225\) −4226.30 −1.25224
\(226\) 0 0
\(227\) −6443.84 −1.88411 −0.942054 0.335462i \(-0.891108\pi\)
−0.942054 + 0.335462i \(0.891108\pi\)
\(228\) 0 0
\(229\) −3897.61 −1.12472 −0.562361 0.826892i \(-0.690107\pi\)
−0.562361 + 0.826892i \(0.690107\pi\)
\(230\) 0 0
\(231\) −14683.1 −4.18216
\(232\) 0 0
\(233\) 2901.11 0.815699 0.407849 0.913049i \(-0.366279\pi\)
0.407849 + 0.913049i \(0.366279\pi\)
\(234\) 0 0
\(235\) 1627.27 0.451707
\(236\) 0 0
\(237\) 3084.91 0.845513
\(238\) 0 0
\(239\) −1420.60 −0.384482 −0.192241 0.981348i \(-0.561576\pi\)
−0.192241 + 0.981348i \(0.561576\pi\)
\(240\) 0 0
\(241\) 6149.55 1.64368 0.821841 0.569716i \(-0.192947\pi\)
0.821841 + 0.569716i \(0.192947\pi\)
\(242\) 0 0
\(243\) 3464.91 0.914708
\(244\) 0 0
\(245\) −3796.35 −0.989958
\(246\) 0 0
\(247\) −1183.26 −0.304815
\(248\) 0 0
\(249\) −1118.98 −0.284789
\(250\) 0 0
\(251\) 3772.54 0.948688 0.474344 0.880340i \(-0.342685\pi\)
0.474344 + 0.880340i \(0.342685\pi\)
\(252\) 0 0
\(253\) 6170.80 1.53342
\(254\) 0 0
\(255\) 3549.23 0.871612
\(256\) 0 0
\(257\) −5930.61 −1.43946 −0.719730 0.694254i \(-0.755735\pi\)
−0.719730 + 0.694254i \(0.755735\pi\)
\(258\) 0 0
\(259\) 5022.90 1.20505
\(260\) 0 0
\(261\) 7685.70 1.82273
\(262\) 0 0
\(263\) 4559.86 1.06910 0.534549 0.845137i \(-0.320481\pi\)
0.534549 + 0.845137i \(0.320481\pi\)
\(264\) 0 0
\(265\) 1192.95 0.276537
\(266\) 0 0
\(267\) −471.110 −0.107983
\(268\) 0 0
\(269\) −4130.57 −0.936227 −0.468114 0.883668i \(-0.655066\pi\)
−0.468114 + 0.883668i \(0.655066\pi\)
\(270\) 0 0
\(271\) −569.640 −0.127687 −0.0638435 0.997960i \(-0.520336\pi\)
−0.0638435 + 0.997960i \(0.520336\pi\)
\(272\) 0 0
\(273\) 16916.9 3.75039
\(274\) 0 0
\(275\) −5088.05 −1.11571
\(276\) 0 0
\(277\) −3463.48 −0.751265 −0.375633 0.926769i \(-0.622575\pi\)
−0.375633 + 0.926769i \(0.622575\pi\)
\(278\) 0 0
\(279\) 2689.57 0.577133
\(280\) 0 0
\(281\) −2951.55 −0.626601 −0.313300 0.949654i \(-0.601435\pi\)
−0.313300 + 0.949654i \(0.601435\pi\)
\(282\) 0 0
\(283\) −3104.67 −0.652133 −0.326066 0.945347i \(-0.605723\pi\)
−0.326066 + 0.945347i \(0.605723\pi\)
\(284\) 0 0
\(285\) 895.133 0.186046
\(286\) 0 0
\(287\) 5234.09 1.07651
\(288\) 0 0
\(289\) 762.433 0.155187
\(290\) 0 0
\(291\) 3990.85 0.803944
\(292\) 0 0
\(293\) −1466.23 −0.292348 −0.146174 0.989259i \(-0.546696\pi\)
−0.146174 + 0.989259i \(0.546696\pi\)
\(294\) 0 0
\(295\) −763.099 −0.150608
\(296\) 0 0
\(297\) −8203.77 −1.60280
\(298\) 0 0
\(299\) −7109.56 −1.37511
\(300\) 0 0
\(301\) −7698.95 −1.47429
\(302\) 0 0
\(303\) 12247.8 2.32217
\(304\) 0 0
\(305\) 20.9023 0.00392415
\(306\) 0 0
\(307\) 2044.70 0.380121 0.190061 0.981772i \(-0.439132\pi\)
0.190061 + 0.981772i \(0.439132\pi\)
\(308\) 0 0
\(309\) 8597.51 1.58283
\(310\) 0 0
\(311\) −4002.96 −0.729862 −0.364931 0.931035i \(-0.618907\pi\)
−0.364931 + 0.931035i \(0.618907\pi\)
\(312\) 0 0
\(313\) −10350.4 −1.86913 −0.934564 0.355794i \(-0.884211\pi\)
−0.934564 + 0.355794i \(0.884211\pi\)
\(314\) 0 0
\(315\) −7991.70 −1.42947
\(316\) 0 0
\(317\) −7084.90 −1.25529 −0.627646 0.778499i \(-0.715981\pi\)
−0.627646 + 0.778499i \(0.715981\pi\)
\(318\) 0 0
\(319\) 9252.83 1.62401
\(320\) 0 0
\(321\) 13877.0 2.41289
\(322\) 0 0
\(323\) 1431.37 0.246575
\(324\) 0 0
\(325\) 5862.10 1.00053
\(326\) 0 0
\(327\) 739.027 0.124979
\(328\) 0 0
\(329\) −9382.45 −1.57225
\(330\) 0 0
\(331\) −7837.56 −1.30148 −0.650742 0.759299i \(-0.725542\pi\)
−0.650742 + 0.759299i \(0.725542\pi\)
\(332\) 0 0
\(333\) 7039.78 1.15849
\(334\) 0 0
\(335\) −4679.14 −0.763131
\(336\) 0 0
\(337\) 1418.55 0.229298 0.114649 0.993406i \(-0.463426\pi\)
0.114649 + 0.993406i \(0.463426\pi\)
\(338\) 0 0
\(339\) −4444.38 −0.712052
\(340\) 0 0
\(341\) 3237.98 0.514212
\(342\) 0 0
\(343\) 10900.7 1.71599
\(344\) 0 0
\(345\) 5378.35 0.839306
\(346\) 0 0
\(347\) 5558.78 0.859974 0.429987 0.902835i \(-0.358518\pi\)
0.429987 + 0.902835i \(0.358518\pi\)
\(348\) 0 0
\(349\) −1077.94 −0.165331 −0.0826656 0.996577i \(-0.526343\pi\)
−0.0826656 + 0.996577i \(0.526343\pi\)
\(350\) 0 0
\(351\) 9451.81 1.43732
\(352\) 0 0
\(353\) 4498.87 0.678331 0.339166 0.940727i \(-0.389855\pi\)
0.339166 + 0.940727i \(0.389855\pi\)
\(354\) 0 0
\(355\) −6060.53 −0.906083
\(356\) 0 0
\(357\) −20464.0 −3.03382
\(358\) 0 0
\(359\) −9060.02 −1.33195 −0.665974 0.745975i \(-0.731984\pi\)
−0.665974 + 0.745975i \(0.731984\pi\)
\(360\) 0 0
\(361\) 361.000 0.0526316
\(362\) 0 0
\(363\) −13489.1 −1.95039
\(364\) 0 0
\(365\) 5305.48 0.760827
\(366\) 0 0
\(367\) 2859.88 0.406770 0.203385 0.979099i \(-0.434806\pi\)
0.203385 + 0.979099i \(0.434806\pi\)
\(368\) 0 0
\(369\) 7335.76 1.03492
\(370\) 0 0
\(371\) −6878.27 −0.962539
\(372\) 0 0
\(373\) 335.799 0.0466140 0.0233070 0.999728i \(-0.492580\pi\)
0.0233070 + 0.999728i \(0.492580\pi\)
\(374\) 0 0
\(375\) −10323.7 −1.42163
\(376\) 0 0
\(377\) −10660.5 −1.45634
\(378\) 0 0
\(379\) 3507.13 0.475328 0.237664 0.971347i \(-0.423618\pi\)
0.237664 + 0.971347i \(0.423618\pi\)
\(380\) 0 0
\(381\) 6748.24 0.907410
\(382\) 0 0
\(383\) 5680.18 0.757816 0.378908 0.925434i \(-0.376300\pi\)
0.378908 + 0.925434i \(0.376300\pi\)
\(384\) 0 0
\(385\) −9621.23 −1.27362
\(386\) 0 0
\(387\) −10790.4 −1.41732
\(388\) 0 0
\(389\) 3884.56 0.506311 0.253155 0.967426i \(-0.418532\pi\)
0.253155 + 0.967426i \(0.418532\pi\)
\(390\) 0 0
\(391\) 8600.32 1.11237
\(392\) 0 0
\(393\) −13147.6 −1.68755
\(394\) 0 0
\(395\) 2021.41 0.257489
\(396\) 0 0
\(397\) −1328.04 −0.167890 −0.0839450 0.996470i \(-0.526752\pi\)
−0.0839450 + 0.996470i \(0.526752\pi\)
\(398\) 0 0
\(399\) −5161.14 −0.647569
\(400\) 0 0
\(401\) 2023.72 0.252020 0.126010 0.992029i \(-0.459783\pi\)
0.126010 + 0.992029i \(0.459783\pi\)
\(402\) 0 0
\(403\) −3730.57 −0.461124
\(404\) 0 0
\(405\) −414.705 −0.0508811
\(406\) 0 0
\(407\) 8475.21 1.03219
\(408\) 0 0
\(409\) 3409.21 0.412163 0.206081 0.978535i \(-0.433929\pi\)
0.206081 + 0.978535i \(0.433929\pi\)
\(410\) 0 0
\(411\) 7240.71 0.868997
\(412\) 0 0
\(413\) 4399.86 0.524220
\(414\) 0 0
\(415\) −733.220 −0.0867286
\(416\) 0 0
\(417\) −25791.8 −3.02885
\(418\) 0 0
\(419\) 6306.05 0.735253 0.367626 0.929974i \(-0.380171\pi\)
0.367626 + 0.929974i \(0.380171\pi\)
\(420\) 0 0
\(421\) −1989.12 −0.230271 −0.115135 0.993350i \(-0.536730\pi\)
−0.115135 + 0.993350i \(0.536730\pi\)
\(422\) 0 0
\(423\) −13149.9 −1.51151
\(424\) 0 0
\(425\) −7091.28 −0.809359
\(426\) 0 0
\(427\) −120.518 −0.0136588
\(428\) 0 0
\(429\) 28544.1 3.21240
\(430\) 0 0
\(431\) −10688.9 −1.19459 −0.597293 0.802023i \(-0.703757\pi\)
−0.597293 + 0.802023i \(0.703757\pi\)
\(432\) 0 0
\(433\) 3581.70 0.397519 0.198759 0.980048i \(-0.436309\pi\)
0.198759 + 0.980048i \(0.436309\pi\)
\(434\) 0 0
\(435\) 8064.59 0.888890
\(436\) 0 0
\(437\) 2169.04 0.237436
\(438\) 0 0
\(439\) −14447.8 −1.57074 −0.785370 0.619027i \(-0.787527\pi\)
−0.785370 + 0.619027i \(0.787527\pi\)
\(440\) 0 0
\(441\) 30678.1 3.31261
\(442\) 0 0
\(443\) 14046.0 1.50643 0.753214 0.657776i \(-0.228503\pi\)
0.753214 + 0.657776i \(0.228503\pi\)
\(444\) 0 0
\(445\) −308.698 −0.0328847
\(446\) 0 0
\(447\) 16797.8 1.77743
\(448\) 0 0
\(449\) −13791.0 −1.44952 −0.724761 0.689000i \(-0.758050\pi\)
−0.724761 + 0.689000i \(0.758050\pi\)
\(450\) 0 0
\(451\) 8831.55 0.922087
\(452\) 0 0
\(453\) 14870.1 1.54229
\(454\) 0 0
\(455\) 11084.9 1.14213
\(456\) 0 0
\(457\) 15109.6 1.54661 0.773303 0.634036i \(-0.218603\pi\)
0.773303 + 0.634036i \(0.218603\pi\)
\(458\) 0 0
\(459\) −11433.7 −1.16270
\(460\) 0 0
\(461\) −7351.36 −0.742705 −0.371352 0.928492i \(-0.621106\pi\)
−0.371352 + 0.928492i \(0.621106\pi\)
\(462\) 0 0
\(463\) −4793.20 −0.481120 −0.240560 0.970634i \(-0.577331\pi\)
−0.240560 + 0.970634i \(0.577331\pi\)
\(464\) 0 0
\(465\) 2822.16 0.281450
\(466\) 0 0
\(467\) 1821.63 0.180503 0.0902515 0.995919i \(-0.471233\pi\)
0.0902515 + 0.995919i \(0.471233\pi\)
\(468\) 0 0
\(469\) 26978.9 2.65623
\(470\) 0 0
\(471\) 6169.51 0.603558
\(472\) 0 0
\(473\) −12990.5 −1.26280
\(474\) 0 0
\(475\) −1788.46 −0.172758
\(476\) 0 0
\(477\) −9640.15 −0.925350
\(478\) 0 0
\(479\) −366.616 −0.0349710 −0.0174855 0.999847i \(-0.505566\pi\)
−0.0174855 + 0.999847i \(0.505566\pi\)
\(480\) 0 0
\(481\) −9764.54 −0.925623
\(482\) 0 0
\(483\) −31010.4 −2.92137
\(484\) 0 0
\(485\) 2615.04 0.244830
\(486\) 0 0
\(487\) −3086.31 −0.287175 −0.143587 0.989638i \(-0.545864\pi\)
−0.143587 + 0.989638i \(0.545864\pi\)
\(488\) 0 0
\(489\) 7947.23 0.734941
\(490\) 0 0
\(491\) 8230.71 0.756511 0.378255 0.925701i \(-0.376524\pi\)
0.378255 + 0.925701i \(0.376524\pi\)
\(492\) 0 0
\(493\) 12895.8 1.17809
\(494\) 0 0
\(495\) −13484.5 −1.22441
\(496\) 0 0
\(497\) 34943.7 3.15380
\(498\) 0 0
\(499\) 12825.9 1.15063 0.575315 0.817932i \(-0.304879\pi\)
0.575315 + 0.817932i \(0.304879\pi\)
\(500\) 0 0
\(501\) −25877.8 −2.30766
\(502\) 0 0
\(503\) −731.374 −0.0648318 −0.0324159 0.999474i \(-0.510320\pi\)
−0.0324159 + 0.999474i \(0.510320\pi\)
\(504\) 0 0
\(505\) 8025.47 0.707185
\(506\) 0 0
\(507\) −14257.4 −1.24890
\(508\) 0 0
\(509\) −13268.0 −1.15539 −0.577695 0.816252i \(-0.696048\pi\)
−0.577695 + 0.816252i \(0.696048\pi\)
\(510\) 0 0
\(511\) −30590.2 −2.64820
\(512\) 0 0
\(513\) −2883.63 −0.248179
\(514\) 0 0
\(515\) 5633.59 0.482030
\(516\) 0 0
\(517\) −15831.1 −1.34672
\(518\) 0 0
\(519\) −17339.8 −1.46654
\(520\) 0 0
\(521\) −12804.7 −1.07675 −0.538374 0.842706i \(-0.680961\pi\)
−0.538374 + 0.842706i \(0.680961\pi\)
\(522\) 0 0
\(523\) −3115.61 −0.260490 −0.130245 0.991482i \(-0.541576\pi\)
−0.130245 + 0.991482i \(0.541576\pi\)
\(524\) 0 0
\(525\) 25569.2 2.12558
\(526\) 0 0
\(527\) 4512.80 0.373018
\(528\) 0 0
\(529\) 865.561 0.0711401
\(530\) 0 0
\(531\) 6166.56 0.503966
\(532\) 0 0
\(533\) −10175.1 −0.826889
\(534\) 0 0
\(535\) 9093.00 0.734813
\(536\) 0 0
\(537\) 7177.34 0.576769
\(538\) 0 0
\(539\) 36933.4 2.95146
\(540\) 0 0
\(541\) 664.570 0.0528135 0.0264068 0.999651i \(-0.491593\pi\)
0.0264068 + 0.999651i \(0.491593\pi\)
\(542\) 0 0
\(543\) −22885.1 −1.80864
\(544\) 0 0
\(545\) 484.253 0.0380608
\(546\) 0 0
\(547\) −11510.2 −0.899709 −0.449854 0.893102i \(-0.648524\pi\)
−0.449854 + 0.893102i \(0.648524\pi\)
\(548\) 0 0
\(549\) −168.911 −0.0131310
\(550\) 0 0
\(551\) 3252.38 0.251463
\(552\) 0 0
\(553\) −11655.0 −0.896241
\(554\) 0 0
\(555\) 7386.82 0.564961
\(556\) 0 0
\(557\) 13369.8 1.01705 0.508525 0.861047i \(-0.330191\pi\)
0.508525 + 0.861047i \(0.330191\pi\)
\(558\) 0 0
\(559\) 14966.8 1.13243
\(560\) 0 0
\(561\) −34529.3 −2.59862
\(562\) 0 0
\(563\) 9379.69 0.702143 0.351072 0.936349i \(-0.385817\pi\)
0.351072 + 0.936349i \(0.385817\pi\)
\(564\) 0 0
\(565\) −2912.21 −0.216846
\(566\) 0 0
\(567\) 2391.10 0.177102
\(568\) 0 0
\(569\) 3929.70 0.289528 0.144764 0.989466i \(-0.453758\pi\)
0.144764 + 0.989466i \(0.453758\pi\)
\(570\) 0 0
\(571\) −19626.1 −1.43840 −0.719202 0.694802i \(-0.755492\pi\)
−0.719202 + 0.694802i \(0.755492\pi\)
\(572\) 0 0
\(573\) −19059.7 −1.38958
\(574\) 0 0
\(575\) −10745.8 −0.779360
\(576\) 0 0
\(577\) −4740.81 −0.342049 −0.171025 0.985267i \(-0.554708\pi\)
−0.171025 + 0.985267i \(0.554708\pi\)
\(578\) 0 0
\(579\) −15764.9 −1.13155
\(580\) 0 0
\(581\) 4227.58 0.301876
\(582\) 0 0
\(583\) −11605.8 −0.824465
\(584\) 0 0
\(585\) 15535.9 1.09800
\(586\) 0 0
\(587\) 22436.3 1.57759 0.788796 0.614656i \(-0.210705\pi\)
0.788796 + 0.614656i \(0.210705\pi\)
\(588\) 0 0
\(589\) 1138.15 0.0796210
\(590\) 0 0
\(591\) 20579.1 1.43234
\(592\) 0 0
\(593\) −16921.2 −1.17179 −0.585893 0.810389i \(-0.699256\pi\)
−0.585893 + 0.810389i \(0.699256\pi\)
\(594\) 0 0
\(595\) −13409.2 −0.923907
\(596\) 0 0
\(597\) 26781.3 1.83599
\(598\) 0 0
\(599\) −5903.16 −0.402665 −0.201333 0.979523i \(-0.564527\pi\)
−0.201333 + 0.979523i \(0.564527\pi\)
\(600\) 0 0
\(601\) −7880.39 −0.534855 −0.267428 0.963578i \(-0.586174\pi\)
−0.267428 + 0.963578i \(0.586174\pi\)
\(602\) 0 0
\(603\) 37811.9 2.55360
\(604\) 0 0
\(605\) −8838.82 −0.593966
\(606\) 0 0
\(607\) 15127.6 1.01155 0.505774 0.862666i \(-0.331207\pi\)
0.505774 + 0.862666i \(0.331207\pi\)
\(608\) 0 0
\(609\) −46498.6 −3.09395
\(610\) 0 0
\(611\) 18239.5 1.20768
\(612\) 0 0
\(613\) 11225.2 0.739610 0.369805 0.929109i \(-0.379425\pi\)
0.369805 + 0.929109i \(0.379425\pi\)
\(614\) 0 0
\(615\) 7697.40 0.504698
\(616\) 0 0
\(617\) 3338.12 0.217808 0.108904 0.994052i \(-0.465266\pi\)
0.108904 + 0.994052i \(0.465266\pi\)
\(618\) 0 0
\(619\) 10482.7 0.680669 0.340334 0.940305i \(-0.389460\pi\)
0.340334 + 0.940305i \(0.389460\pi\)
\(620\) 0 0
\(621\) −17326.1 −1.11960
\(622\) 0 0
\(623\) 1779.89 0.114462
\(624\) 0 0
\(625\) 5001.50 0.320096
\(626\) 0 0
\(627\) −8708.46 −0.554677
\(628\) 0 0
\(629\) 11812.0 0.748768
\(630\) 0 0
\(631\) −27061.1 −1.70727 −0.853633 0.520875i \(-0.825606\pi\)
−0.853633 + 0.520875i \(0.825606\pi\)
\(632\) 0 0
\(633\) −27333.1 −1.71626
\(634\) 0 0
\(635\) 4421.84 0.276339
\(636\) 0 0
\(637\) −42552.1 −2.64674
\(638\) 0 0
\(639\) 48974.8 3.03195
\(640\) 0 0
\(641\) −4000.76 −0.246522 −0.123261 0.992374i \(-0.539335\pi\)
−0.123261 + 0.992374i \(0.539335\pi\)
\(642\) 0 0
\(643\) 5874.39 0.360285 0.180142 0.983641i \(-0.442344\pi\)
0.180142 + 0.983641i \(0.442344\pi\)
\(644\) 0 0
\(645\) −11322.3 −0.691186
\(646\) 0 0
\(647\) −23424.1 −1.42333 −0.711666 0.702518i \(-0.752059\pi\)
−0.711666 + 0.702518i \(0.752059\pi\)
\(648\) 0 0
\(649\) 7423.94 0.449022
\(650\) 0 0
\(651\) −16271.9 −0.979642
\(652\) 0 0
\(653\) −8096.54 −0.485210 −0.242605 0.970125i \(-0.578002\pi\)
−0.242605 + 0.970125i \(0.578002\pi\)
\(654\) 0 0
\(655\) −8615.07 −0.513921
\(656\) 0 0
\(657\) −42873.3 −2.54589
\(658\) 0 0
\(659\) 25418.1 1.50250 0.751250 0.660017i \(-0.229451\pi\)
0.751250 + 0.660017i \(0.229451\pi\)
\(660\) 0 0
\(661\) 13421.9 0.789788 0.394894 0.918727i \(-0.370781\pi\)
0.394894 + 0.918727i \(0.370781\pi\)
\(662\) 0 0
\(663\) 39782.2 2.33033
\(664\) 0 0
\(665\) −3381.88 −0.197208
\(666\) 0 0
\(667\) 19541.7 1.13442
\(668\) 0 0
\(669\) 42976.1 2.48363
\(670\) 0 0
\(671\) −203.352 −0.0116994
\(672\) 0 0
\(673\) 6406.65 0.366951 0.183476 0.983024i \(-0.441265\pi\)
0.183476 + 0.983024i \(0.441265\pi\)
\(674\) 0 0
\(675\) 14286.0 0.814622
\(676\) 0 0
\(677\) 21258.3 1.20683 0.603414 0.797428i \(-0.293806\pi\)
0.603414 + 0.797428i \(0.293806\pi\)
\(678\) 0 0
\(679\) −15077.7 −0.852179
\(680\) 0 0
\(681\) 54639.3 3.07457
\(682\) 0 0
\(683\) 7604.89 0.426052 0.213026 0.977047i \(-0.431668\pi\)
0.213026 + 0.977047i \(0.431668\pi\)
\(684\) 0 0
\(685\) 4744.53 0.264641
\(686\) 0 0
\(687\) 33049.1 1.83537
\(688\) 0 0
\(689\) 13371.4 0.739346
\(690\) 0 0
\(691\) 773.527 0.0425852 0.0212926 0.999773i \(-0.493222\pi\)
0.0212926 + 0.999773i \(0.493222\pi\)
\(692\) 0 0
\(693\) 77748.7 4.26180
\(694\) 0 0
\(695\) −16900.3 −0.922394
\(696\) 0 0
\(697\) 12308.6 0.668899
\(698\) 0 0
\(699\) −24599.4 −1.33110
\(700\) 0 0
\(701\) −17527.6 −0.944376 −0.472188 0.881498i \(-0.656536\pi\)
−0.472188 + 0.881498i \(0.656536\pi\)
\(702\) 0 0
\(703\) 2979.05 0.159825
\(704\) 0 0
\(705\) −13798.1 −0.737116
\(706\) 0 0
\(707\) −46273.0 −2.46150
\(708\) 0 0
\(709\) −18974.0 −1.00506 −0.502528 0.864561i \(-0.667597\pi\)
−0.502528 + 0.864561i \(0.667597\pi\)
\(710\) 0 0
\(711\) −16334.9 −0.861614
\(712\) 0 0
\(713\) 6838.51 0.359193
\(714\) 0 0
\(715\) 18703.7 0.978293
\(716\) 0 0
\(717\) 12045.8 0.627416
\(718\) 0 0
\(719\) 18481.4 0.958610 0.479305 0.877648i \(-0.340889\pi\)
0.479305 + 0.877648i \(0.340889\pi\)
\(720\) 0 0
\(721\) −32482.0 −1.67780
\(722\) 0 0
\(723\) −52144.0 −2.68224
\(724\) 0 0
\(725\) −16112.9 −0.825403
\(726\) 0 0
\(727\) 7059.21 0.360126 0.180063 0.983655i \(-0.442370\pi\)
0.180063 + 0.983655i \(0.442370\pi\)
\(728\) 0 0
\(729\) −31395.3 −1.59505
\(730\) 0 0
\(731\) −18105.1 −0.916060
\(732\) 0 0
\(733\) −1633.99 −0.0823369 −0.0411684 0.999152i \(-0.513108\pi\)
−0.0411684 + 0.999152i \(0.513108\pi\)
\(734\) 0 0
\(735\) 32190.4 1.61546
\(736\) 0 0
\(737\) 45521.9 2.27520
\(738\) 0 0
\(739\) −26039.2 −1.29617 −0.648083 0.761569i \(-0.724429\pi\)
−0.648083 + 0.761569i \(0.724429\pi\)
\(740\) 0 0
\(741\) 10033.3 0.497411
\(742\) 0 0
\(743\) −33333.8 −1.64589 −0.822947 0.568119i \(-0.807671\pi\)
−0.822947 + 0.568119i \(0.807671\pi\)
\(744\) 0 0
\(745\) 11006.9 0.541291
\(746\) 0 0
\(747\) 5925.12 0.290212
\(748\) 0 0
\(749\) −52428.2 −2.55766
\(750\) 0 0
\(751\) 15487.3 0.752517 0.376259 0.926515i \(-0.377210\pi\)
0.376259 + 0.926515i \(0.377210\pi\)
\(752\) 0 0
\(753\) −31988.6 −1.54811
\(754\) 0 0
\(755\) 9743.75 0.469684
\(756\) 0 0
\(757\) 20858.3 1.00147 0.500733 0.865602i \(-0.333064\pi\)
0.500733 + 0.865602i \(0.333064\pi\)
\(758\) 0 0
\(759\) −52324.2 −2.50230
\(760\) 0 0
\(761\) 32715.3 1.55838 0.779191 0.626786i \(-0.215630\pi\)
0.779191 + 0.626786i \(0.215630\pi\)
\(762\) 0 0
\(763\) −2792.10 −0.132478
\(764\) 0 0
\(765\) −18793.5 −0.888210
\(766\) 0 0
\(767\) −8553.34 −0.402664
\(768\) 0 0
\(769\) −37273.2 −1.74786 −0.873932 0.486049i \(-0.838438\pi\)
−0.873932 + 0.486049i \(0.838438\pi\)
\(770\) 0 0
\(771\) 50287.5 2.34898
\(772\) 0 0
\(773\) −13735.3 −0.639098 −0.319549 0.947570i \(-0.603531\pi\)
−0.319549 + 0.947570i \(0.603531\pi\)
\(774\) 0 0
\(775\) −5638.61 −0.261348
\(776\) 0 0
\(777\) −42590.8 −1.96646
\(778\) 0 0
\(779\) 3104.30 0.142777
\(780\) 0 0
\(781\) 58960.9 2.70139
\(782\) 0 0
\(783\) −25979.7 −1.18575
\(784\) 0 0
\(785\) 4042.62 0.183805
\(786\) 0 0
\(787\) 18769.2 0.850127 0.425064 0.905163i \(-0.360252\pi\)
0.425064 + 0.905163i \(0.360252\pi\)
\(788\) 0 0
\(789\) −38664.5 −1.74460
\(790\) 0 0
\(791\) 16791.2 0.754773
\(792\) 0 0
\(793\) 234.288 0.0104916
\(794\) 0 0
\(795\) −10115.4 −0.451265
\(796\) 0 0
\(797\) 34274.2 1.52328 0.761639 0.648001i \(-0.224395\pi\)
0.761639 + 0.648001i \(0.224395\pi\)
\(798\) 0 0
\(799\) −22064.0 −0.976933
\(800\) 0 0
\(801\) 2494.58 0.110039
\(802\) 0 0
\(803\) −51615.3 −2.26833
\(804\) 0 0
\(805\) −20319.8 −0.889662
\(806\) 0 0
\(807\) 35024.4 1.52778
\(808\) 0 0
\(809\) 33407.2 1.45184 0.725918 0.687781i \(-0.241415\pi\)
0.725918 + 0.687781i \(0.241415\pi\)
\(810\) 0 0
\(811\) 13794.5 0.597275 0.298638 0.954367i \(-0.403468\pi\)
0.298638 + 0.954367i \(0.403468\pi\)
\(812\) 0 0
\(813\) 4830.16 0.208366
\(814\) 0 0
\(815\) 5207.48 0.223816
\(816\) 0 0
\(817\) −4566.19 −0.195533
\(818\) 0 0
\(819\) −89576.6 −3.82181
\(820\) 0 0
\(821\) −20672.7 −0.878784 −0.439392 0.898296i \(-0.644806\pi\)
−0.439392 + 0.898296i \(0.644806\pi\)
\(822\) 0 0
\(823\) 13796.6 0.584347 0.292174 0.956365i \(-0.405622\pi\)
0.292174 + 0.956365i \(0.405622\pi\)
\(824\) 0 0
\(825\) 43143.2 1.82067
\(826\) 0 0
\(827\) −23226.2 −0.976605 −0.488303 0.872674i \(-0.662384\pi\)
−0.488303 + 0.872674i \(0.662384\pi\)
\(828\) 0 0
\(829\) −9821.70 −0.411486 −0.205743 0.978606i \(-0.565961\pi\)
−0.205743 + 0.978606i \(0.565961\pi\)
\(830\) 0 0
\(831\) 29368.0 1.22595
\(832\) 0 0
\(833\) 51474.5 2.14104
\(834\) 0 0
\(835\) −16956.7 −0.702765
\(836\) 0 0
\(837\) −9091.46 −0.375444
\(838\) 0 0
\(839\) 12633.7 0.519862 0.259931 0.965627i \(-0.416300\pi\)
0.259931 + 0.965627i \(0.416300\pi\)
\(840\) 0 0
\(841\) 4912.91 0.201439
\(842\) 0 0
\(843\) 25027.2 1.02252
\(844\) 0 0
\(845\) −9342.27 −0.380336
\(846\) 0 0
\(847\) 50962.7 2.06741
\(848\) 0 0
\(849\) 26325.5 1.06418
\(850\) 0 0
\(851\) 17899.4 0.721015
\(852\) 0 0
\(853\) −1749.34 −0.0702182 −0.0351091 0.999383i \(-0.511178\pi\)
−0.0351091 + 0.999383i \(0.511178\pi\)
\(854\) 0 0
\(855\) −4739.82 −0.189589
\(856\) 0 0
\(857\) 21953.5 0.875048 0.437524 0.899207i \(-0.355856\pi\)
0.437524 + 0.899207i \(0.355856\pi\)
\(858\) 0 0
\(859\) 25690.7 1.02044 0.510219 0.860044i \(-0.329564\pi\)
0.510219 + 0.860044i \(0.329564\pi\)
\(860\) 0 0
\(861\) −44381.5 −1.75670
\(862\) 0 0
\(863\) −46400.0 −1.83022 −0.915108 0.403209i \(-0.867895\pi\)
−0.915108 + 0.403209i \(0.867895\pi\)
\(864\) 0 0
\(865\) −11362.0 −0.446613
\(866\) 0 0
\(867\) −6464.91 −0.253241
\(868\) 0 0
\(869\) −19665.7 −0.767677
\(870\) 0 0
\(871\) −52447.1 −2.04030
\(872\) 0 0
\(873\) −21132.0 −0.819254
\(874\) 0 0
\(875\) 39003.6 1.50693
\(876\) 0 0
\(877\) 24330.7 0.936818 0.468409 0.883512i \(-0.344827\pi\)
0.468409 + 0.883512i \(0.344827\pi\)
\(878\) 0 0
\(879\) 12432.6 0.477068
\(880\) 0 0
\(881\) 31103.2 1.18944 0.594719 0.803934i \(-0.297263\pi\)
0.594719 + 0.803934i \(0.297263\pi\)
\(882\) 0 0
\(883\) 22128.2 0.843344 0.421672 0.906748i \(-0.361443\pi\)
0.421672 + 0.906748i \(0.361443\pi\)
\(884\) 0 0
\(885\) 6470.56 0.245769
\(886\) 0 0
\(887\) −20897.9 −0.791076 −0.395538 0.918450i \(-0.629442\pi\)
−0.395538 + 0.918450i \(0.629442\pi\)
\(888\) 0 0
\(889\) −25495.3 −0.961852
\(890\) 0 0
\(891\) 4034.53 0.151697
\(892\) 0 0
\(893\) −5564.67 −0.208527
\(894\) 0 0
\(895\) 4703.00 0.175647
\(896\) 0 0
\(897\) 60284.3 2.24396
\(898\) 0 0
\(899\) 10254.0 0.380413
\(900\) 0 0
\(901\) −16175.1 −0.598082
\(902\) 0 0
\(903\) 65281.8 2.40581
\(904\) 0 0
\(905\) −14995.6 −0.550798
\(906\) 0 0
\(907\) −45089.0 −1.65067 −0.825334 0.564645i \(-0.809013\pi\)
−0.825334 + 0.564645i \(0.809013\pi\)
\(908\) 0 0
\(909\) −64853.4 −2.36639
\(910\) 0 0
\(911\) 26499.7 0.963749 0.481874 0.876240i \(-0.339956\pi\)
0.481874 + 0.876240i \(0.339956\pi\)
\(912\) 0 0
\(913\) 7133.26 0.258572
\(914\) 0 0
\(915\) −177.238 −0.00640360
\(916\) 0 0
\(917\) 49672.6 1.78880
\(918\) 0 0
\(919\) −19159.5 −0.687718 −0.343859 0.939021i \(-0.611734\pi\)
−0.343859 + 0.939021i \(0.611734\pi\)
\(920\) 0 0
\(921\) −17337.7 −0.620299
\(922\) 0 0
\(923\) −67930.6 −2.42250
\(924\) 0 0
\(925\) −14758.7 −0.524609
\(926\) 0 0
\(927\) −45524.7 −1.61298
\(928\) 0 0
\(929\) 4702.46 0.166074 0.0830370 0.996546i \(-0.473538\pi\)
0.0830370 + 0.996546i \(0.473538\pi\)
\(930\) 0 0
\(931\) 12982.1 0.457006
\(932\) 0 0
\(933\) 33942.4 1.19102
\(934\) 0 0
\(935\) −22625.6 −0.791374
\(936\) 0 0
\(937\) 37839.1 1.31926 0.659632 0.751589i \(-0.270712\pi\)
0.659632 + 0.751589i \(0.270712\pi\)
\(938\) 0 0
\(939\) 87764.1 3.05013
\(940\) 0 0
\(941\) 6041.39 0.209292 0.104646 0.994510i \(-0.466629\pi\)
0.104646 + 0.994510i \(0.466629\pi\)
\(942\) 0 0
\(943\) 18652.0 0.644106
\(944\) 0 0
\(945\) 27014.1 0.929914
\(946\) 0 0
\(947\) −1829.18 −0.0627670 −0.0313835 0.999507i \(-0.509991\pi\)
−0.0313835 + 0.999507i \(0.509991\pi\)
\(948\) 0 0
\(949\) 59467.6 2.03414
\(950\) 0 0
\(951\) 60075.1 2.04844
\(952\) 0 0
\(953\) 2153.53 0.0732001 0.0366000 0.999330i \(-0.488347\pi\)
0.0366000 + 0.999330i \(0.488347\pi\)
\(954\) 0 0
\(955\) −12489.0 −0.423178
\(956\) 0 0
\(957\) −78457.7 −2.65013
\(958\) 0 0
\(959\) −27355.9 −0.921135
\(960\) 0 0
\(961\) −26202.7 −0.879550
\(962\) 0 0
\(963\) −73480.1 −2.45884
\(964\) 0 0
\(965\) −10330.1 −0.344597
\(966\) 0 0
\(967\) 56520.9 1.87962 0.939808 0.341703i \(-0.111004\pi\)
0.939808 + 0.341703i \(0.111004\pi\)
\(968\) 0 0
\(969\) −12137.1 −0.402373
\(970\) 0 0
\(971\) −27512.1 −0.909275 −0.454637 0.890677i \(-0.650231\pi\)
−0.454637 + 0.890677i \(0.650231\pi\)
\(972\) 0 0
\(973\) 97443.2 3.21057
\(974\) 0 0
\(975\) −49706.6 −1.63270
\(976\) 0 0
\(977\) −27370.9 −0.896286 −0.448143 0.893962i \(-0.647914\pi\)
−0.448143 + 0.893962i \(0.647914\pi\)
\(978\) 0 0
\(979\) 3003.23 0.0980424
\(980\) 0 0
\(981\) −3913.22 −0.127359
\(982\) 0 0
\(983\) 29069.1 0.943194 0.471597 0.881814i \(-0.343678\pi\)
0.471597 + 0.881814i \(0.343678\pi\)
\(984\) 0 0
\(985\) 13484.6 0.436198
\(986\) 0 0
\(987\) 79556.8 2.56568
\(988\) 0 0
\(989\) −27435.6 −0.882106
\(990\) 0 0
\(991\) 19874.2 0.637057 0.318529 0.947913i \(-0.396811\pi\)
0.318529 + 0.947913i \(0.396811\pi\)
\(992\) 0 0
\(993\) 66457.1 2.12382
\(994\) 0 0
\(995\) 17548.6 0.559124
\(996\) 0 0
\(997\) 10809.0 0.343355 0.171677 0.985153i \(-0.445081\pi\)
0.171677 + 0.985153i \(0.445081\pi\)
\(998\) 0 0
\(999\) −23796.3 −0.753637
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 76.4.a.b.1.1 3
3.2 odd 2 684.4.a.i.1.3 3
4.3 odd 2 304.4.a.h.1.3 3
5.2 odd 4 1900.4.c.c.1749.6 6
5.3 odd 4 1900.4.c.c.1749.1 6
5.4 even 2 1900.4.a.c.1.3 3
8.3 odd 2 1216.4.a.v.1.1 3
8.5 even 2 1216.4.a.t.1.3 3
19.18 odd 2 1444.4.a.e.1.3 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
76.4.a.b.1.1 3 1.1 even 1 trivial
304.4.a.h.1.3 3 4.3 odd 2
684.4.a.i.1.3 3 3.2 odd 2
1216.4.a.t.1.3 3 8.5 even 2
1216.4.a.v.1.1 3 8.3 odd 2
1444.4.a.e.1.3 3 19.18 odd 2
1900.4.a.c.1.3 3 5.4 even 2
1900.4.c.c.1749.1 6 5.3 odd 4
1900.4.c.c.1749.6 6 5.2 odd 4