Properties

Label 784.4.a.bb.1.2
Level $784$
Weight $4$
Character 784.1
Self dual yes
Analytic conductor $46.257$
Analytic rank $1$
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] = [784,4,Mod(1,784)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(784, 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("784.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 784 = 2^{4} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 784.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: \(46.2574974445\)
Analytic rank: \(1\)
Dimension: \(3\)
Coefficient field: 3.3.1929.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 10x + 13 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{3} \)
Twist minimal: no (minimal twist has level 56)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(1.36922\) of defining polynomial
Character \(\chi\) \(=\) 784.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-0.261560 q^{3} -19.5009 q^{5} -26.9316 q^{9} +O(q^{10})\) \(q-0.261560 q^{3} -19.5009 q^{5} -26.9316 q^{9} +56.2875 q^{11} +66.0260 q^{13} +5.10067 q^{15} +18.8372 q^{17} -80.8106 q^{19} +89.3559 q^{23} +255.287 q^{25} +14.1063 q^{27} +104.118 q^{29} -148.437 q^{31} -14.7226 q^{33} +13.8119 q^{37} -17.2697 q^{39} -174.403 q^{41} -205.353 q^{43} +525.191 q^{45} -116.634 q^{47} -4.92706 q^{51} -316.249 q^{53} -1097.66 q^{55} +21.1368 q^{57} -539.585 q^{59} -145.313 q^{61} -1287.57 q^{65} -476.094 q^{67} -23.3719 q^{69} -131.763 q^{71} -349.608 q^{73} -66.7728 q^{75} +806.906 q^{79} +723.463 q^{81} -233.648 q^{83} -367.344 q^{85} -27.2332 q^{87} +535.483 q^{89} +38.8252 q^{93} +1575.88 q^{95} -80.7221 q^{97} -1515.91 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 7 q^{3} + 3 q^{5} + 18 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q - 7 q^{3} + 3 q^{5} + 18 q^{9} + 3 q^{11} + 26 q^{13} - 127 q^{15} + 31 q^{17} - 89 q^{19} + 201 q^{23} + 300 q^{25} - 469 q^{27} + 190 q^{29} - 339 q^{31} + 105 q^{33} - 535 q^{37} + 134 q^{39} - 58 q^{41} - 268 q^{43} + 1410 q^{45} - 205 q^{47} + 965 q^{51} + 757 q^{53} - 1653 q^{55} + 261 q^{57} - 1799 q^{59} - 625 q^{61} - 1750 q^{65} + 495 q^{67} - 973 q^{69} - 640 q^{71} + 443 q^{73} - 1484 q^{75} - 79 q^{79} + 2523 q^{81} - 2372 q^{83} - 977 q^{85} - 910 q^{87} - 821 q^{89} - 1321 q^{93} + 1327 q^{95} + 342 q^{97} - 2310 q^{99}+O(q^{100}) \) 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
\(3\) −0.261560 −0.0503372 −0.0251686 0.999683i \(-0.508012\pi\)
−0.0251686 + 0.999683i \(0.508012\pi\)
\(4\) 0 0
\(5\) −19.5009 −1.74422 −0.872109 0.489312i \(-0.837248\pi\)
−0.872109 + 0.489312i \(0.837248\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) 0 0
\(9\) −26.9316 −0.997466
\(10\) 0 0
\(11\) 56.2875 1.54285 0.771424 0.636322i \(-0.219545\pi\)
0.771424 + 0.636322i \(0.219545\pi\)
\(12\) 0 0
\(13\) 66.0260 1.40864 0.704319 0.709883i \(-0.251252\pi\)
0.704319 + 0.709883i \(0.251252\pi\)
\(14\) 0 0
\(15\) 5.10067 0.0877991
\(16\) 0 0
\(17\) 18.8372 0.268747 0.134373 0.990931i \(-0.457098\pi\)
0.134373 + 0.990931i \(0.457098\pi\)
\(18\) 0 0
\(19\) −80.8106 −0.975749 −0.487875 0.872914i \(-0.662228\pi\)
−0.487875 + 0.872914i \(0.662228\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 89.3559 0.810087 0.405043 0.914297i \(-0.367256\pi\)
0.405043 + 0.914297i \(0.367256\pi\)
\(24\) 0 0
\(25\) 255.287 2.04230
\(26\) 0 0
\(27\) 14.1063 0.100547
\(28\) 0 0
\(29\) 104.118 0.666700 0.333350 0.942803i \(-0.391821\pi\)
0.333350 + 0.942803i \(0.391821\pi\)
\(30\) 0 0
\(31\) −148.437 −0.860002 −0.430001 0.902828i \(-0.641487\pi\)
−0.430001 + 0.902828i \(0.641487\pi\)
\(32\) 0 0
\(33\) −14.7226 −0.0776627
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 13.8119 0.0613693 0.0306847 0.999529i \(-0.490231\pi\)
0.0306847 + 0.999529i \(0.490231\pi\)
\(38\) 0 0
\(39\) −17.2697 −0.0709070
\(40\) 0 0
\(41\) −174.403 −0.664323 −0.332161 0.943223i \(-0.607778\pi\)
−0.332161 + 0.943223i \(0.607778\pi\)
\(42\) 0 0
\(43\) −205.353 −0.728279 −0.364139 0.931344i \(-0.618637\pi\)
−0.364139 + 0.931344i \(0.618637\pi\)
\(44\) 0 0
\(45\) 525.191 1.73980
\(46\) 0 0
\(47\) −116.634 −0.361976 −0.180988 0.983485i \(-0.557930\pi\)
−0.180988 + 0.983485i \(0.557930\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 0 0
\(51\) −4.92706 −0.0135280
\(52\) 0 0
\(53\) −316.249 −0.819625 −0.409813 0.912170i \(-0.634406\pi\)
−0.409813 + 0.912170i \(0.634406\pi\)
\(54\) 0 0
\(55\) −1097.66 −2.69106
\(56\) 0 0
\(57\) 21.1368 0.0491165
\(58\) 0 0
\(59\) −539.585 −1.19064 −0.595322 0.803487i \(-0.702976\pi\)
−0.595322 + 0.803487i \(0.702976\pi\)
\(60\) 0 0
\(61\) −145.313 −0.305007 −0.152503 0.988303i \(-0.548734\pi\)
−0.152503 + 0.988303i \(0.548734\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −1287.57 −2.45697
\(66\) 0 0
\(67\) −476.094 −0.868122 −0.434061 0.900883i \(-0.642920\pi\)
−0.434061 + 0.900883i \(0.642920\pi\)
\(68\) 0 0
\(69\) −23.3719 −0.0407775
\(70\) 0 0
\(71\) −131.763 −0.220245 −0.110123 0.993918i \(-0.535124\pi\)
−0.110123 + 0.993918i \(0.535124\pi\)
\(72\) 0 0
\(73\) −349.608 −0.560528 −0.280264 0.959923i \(-0.590422\pi\)
−0.280264 + 0.959923i \(0.590422\pi\)
\(74\) 0 0
\(75\) −66.7728 −0.102803
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 806.906 1.14917 0.574583 0.818447i \(-0.305164\pi\)
0.574583 + 0.818447i \(0.305164\pi\)
\(80\) 0 0
\(81\) 723.463 0.992405
\(82\) 0 0
\(83\) −233.648 −0.308990 −0.154495 0.987994i \(-0.549375\pi\)
−0.154495 + 0.987994i \(0.549375\pi\)
\(84\) 0 0
\(85\) −367.344 −0.468753
\(86\) 0 0
\(87\) −27.2332 −0.0335599
\(88\) 0 0
\(89\) 535.483 0.637765 0.318882 0.947794i \(-0.396693\pi\)
0.318882 + 0.947794i \(0.396693\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 38.8252 0.0432901
\(94\) 0 0
\(95\) 1575.88 1.70192
\(96\) 0 0
\(97\) −80.7221 −0.0844958 −0.0422479 0.999107i \(-0.513452\pi\)
−0.0422479 + 0.999107i \(0.513452\pi\)
\(98\) 0 0
\(99\) −1515.91 −1.53894
\(100\) 0 0
\(101\) 173.472 0.170902 0.0854509 0.996342i \(-0.472767\pi\)
0.0854509 + 0.996342i \(0.472767\pi\)
\(102\) 0 0
\(103\) −195.751 −0.187262 −0.0936308 0.995607i \(-0.529847\pi\)
−0.0936308 + 0.995607i \(0.529847\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −2048.92 −1.85118 −0.925591 0.378524i \(-0.876432\pi\)
−0.925591 + 0.378524i \(0.876432\pi\)
\(108\) 0 0
\(109\) 1557.94 1.36902 0.684512 0.729001i \(-0.260015\pi\)
0.684512 + 0.729001i \(0.260015\pi\)
\(110\) 0 0
\(111\) −3.61264 −0.00308916
\(112\) 0 0
\(113\) −652.982 −0.543605 −0.271802 0.962353i \(-0.587620\pi\)
−0.271802 + 0.962353i \(0.587620\pi\)
\(114\) 0 0
\(115\) −1742.53 −1.41297
\(116\) 0 0
\(117\) −1778.18 −1.40507
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) 1837.28 1.38038
\(122\) 0 0
\(123\) 45.6170 0.0334402
\(124\) 0 0
\(125\) −2540.72 −1.81799
\(126\) 0 0
\(127\) 819.069 0.572288 0.286144 0.958187i \(-0.407626\pi\)
0.286144 + 0.958187i \(0.407626\pi\)
\(128\) 0 0
\(129\) 53.7120 0.0366595
\(130\) 0 0
\(131\) −2428.30 −1.61955 −0.809777 0.586737i \(-0.800412\pi\)
−0.809777 + 0.586737i \(0.800412\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) −275.087 −0.175376
\(136\) 0 0
\(137\) 100.337 0.0625719 0.0312859 0.999510i \(-0.490040\pi\)
0.0312859 + 0.999510i \(0.490040\pi\)
\(138\) 0 0
\(139\) −634.299 −0.387054 −0.193527 0.981095i \(-0.561993\pi\)
−0.193527 + 0.981095i \(0.561993\pi\)
\(140\) 0 0
\(141\) 30.5069 0.0182209
\(142\) 0 0
\(143\) 3716.44 2.17332
\(144\) 0 0
\(145\) −2030.41 −1.16287
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) −2105.19 −1.15747 −0.578737 0.815514i \(-0.696454\pi\)
−0.578737 + 0.815514i \(0.696454\pi\)
\(150\) 0 0
\(151\) 2868.57 1.54596 0.772982 0.634428i \(-0.218764\pi\)
0.772982 + 0.634428i \(0.218764\pi\)
\(152\) 0 0
\(153\) −507.316 −0.268066
\(154\) 0 0
\(155\) 2894.66 1.50003
\(156\) 0 0
\(157\) −3087.39 −1.56943 −0.784714 0.619858i \(-0.787190\pi\)
−0.784714 + 0.619858i \(0.787190\pi\)
\(158\) 0 0
\(159\) 82.7180 0.0412577
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) 1011.12 0.485869 0.242935 0.970043i \(-0.421890\pi\)
0.242935 + 0.970043i \(0.421890\pi\)
\(164\) 0 0
\(165\) 287.104 0.135461
\(166\) 0 0
\(167\) −1052.69 −0.487784 −0.243892 0.969802i \(-0.578424\pi\)
−0.243892 + 0.969802i \(0.578424\pi\)
\(168\) 0 0
\(169\) 2162.43 0.984264
\(170\) 0 0
\(171\) 2176.36 0.973277
\(172\) 0 0
\(173\) 306.144 0.134542 0.0672708 0.997735i \(-0.478571\pi\)
0.0672708 + 0.997735i \(0.478571\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 141.134 0.0599337
\(178\) 0 0
\(179\) 188.964 0.0789040 0.0394520 0.999221i \(-0.487439\pi\)
0.0394520 + 0.999221i \(0.487439\pi\)
\(180\) 0 0
\(181\) 468.520 0.192402 0.0962011 0.995362i \(-0.469331\pi\)
0.0962011 + 0.995362i \(0.469331\pi\)
\(182\) 0 0
\(183\) 38.0081 0.0153532
\(184\) 0 0
\(185\) −269.345 −0.107041
\(186\) 0 0
\(187\) 1060.30 0.414635
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −4200.00 −1.59111 −0.795553 0.605884i \(-0.792820\pi\)
−0.795553 + 0.605884i \(0.792820\pi\)
\(192\) 0 0
\(193\) −2925.26 −1.09101 −0.545505 0.838108i \(-0.683662\pi\)
−0.545505 + 0.838108i \(0.683662\pi\)
\(194\) 0 0
\(195\) 336.776 0.123677
\(196\) 0 0
\(197\) −3402.39 −1.23051 −0.615254 0.788329i \(-0.710947\pi\)
−0.615254 + 0.788329i \(0.710947\pi\)
\(198\) 0 0
\(199\) −98.3992 −0.0350519 −0.0175260 0.999846i \(-0.505579\pi\)
−0.0175260 + 0.999846i \(0.505579\pi\)
\(200\) 0 0
\(201\) 124.527 0.0436989
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 3401.03 1.15872
\(206\) 0 0
\(207\) −2406.50 −0.808034
\(208\) 0 0
\(209\) −4548.63 −1.50543
\(210\) 0 0
\(211\) −2692.24 −0.878396 −0.439198 0.898390i \(-0.644737\pi\)
−0.439198 + 0.898390i \(0.644737\pi\)
\(212\) 0 0
\(213\) 34.4640 0.0110865
\(214\) 0 0
\(215\) 4004.57 1.27028
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) 91.4435 0.0282154
\(220\) 0 0
\(221\) 1243.75 0.378567
\(222\) 0 0
\(223\) 6075.65 1.82447 0.912233 0.409672i \(-0.134357\pi\)
0.912233 + 0.409672i \(0.134357\pi\)
\(224\) 0 0
\(225\) −6875.28 −2.03712
\(226\) 0 0
\(227\) −2991.15 −0.874581 −0.437290 0.899320i \(-0.644062\pi\)
−0.437290 + 0.899320i \(0.644062\pi\)
\(228\) 0 0
\(229\) 6181.17 1.78368 0.891841 0.452349i \(-0.149414\pi\)
0.891841 + 0.452349i \(0.149414\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 2913.44 0.819166 0.409583 0.912273i \(-0.365674\pi\)
0.409583 + 0.912273i \(0.365674\pi\)
\(234\) 0 0
\(235\) 2274.48 0.631366
\(236\) 0 0
\(237\) −211.054 −0.0578458
\(238\) 0 0
\(239\) −3407.25 −0.922163 −0.461081 0.887358i \(-0.652538\pi\)
−0.461081 + 0.887358i \(0.652538\pi\)
\(240\) 0 0
\(241\) −244.124 −0.0652508 −0.0326254 0.999468i \(-0.510387\pi\)
−0.0326254 + 0.999468i \(0.510387\pi\)
\(242\) 0 0
\(243\) −570.100 −0.150502
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) −5335.60 −1.37448
\(248\) 0 0
\(249\) 61.1129 0.0155537
\(250\) 0 0
\(251\) −4795.49 −1.20593 −0.602966 0.797767i \(-0.706014\pi\)
−0.602966 + 0.797767i \(0.706014\pi\)
\(252\) 0 0
\(253\) 5029.62 1.24984
\(254\) 0 0
\(255\) 96.0824 0.0235957
\(256\) 0 0
\(257\) −4782.35 −1.16076 −0.580380 0.814346i \(-0.697096\pi\)
−0.580380 + 0.814346i \(0.697096\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) −2804.07 −0.665011
\(262\) 0 0
\(263\) −7649.05 −1.79339 −0.896693 0.442652i \(-0.854038\pi\)
−0.896693 + 0.442652i \(0.854038\pi\)
\(264\) 0 0
\(265\) 6167.15 1.42960
\(266\) 0 0
\(267\) −140.061 −0.0321033
\(268\) 0 0
\(269\) −4200.71 −0.952125 −0.476062 0.879412i \(-0.657936\pi\)
−0.476062 + 0.879412i \(0.657936\pi\)
\(270\) 0 0
\(271\) −5063.22 −1.13494 −0.567470 0.823394i \(-0.692078\pi\)
−0.567470 + 0.823394i \(0.692078\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 14369.5 3.15095
\(276\) 0 0
\(277\) −5346.39 −1.15969 −0.579843 0.814728i \(-0.696886\pi\)
−0.579843 + 0.814728i \(0.696886\pi\)
\(278\) 0 0
\(279\) 3997.64 0.857823
\(280\) 0 0
\(281\) 6367.90 1.35188 0.675938 0.736958i \(-0.263739\pi\)
0.675938 + 0.736958i \(0.263739\pi\)
\(282\) 0 0
\(283\) −2124.38 −0.446224 −0.223112 0.974793i \(-0.571622\pi\)
−0.223112 + 0.974793i \(0.571622\pi\)
\(284\) 0 0
\(285\) −412.188 −0.0856699
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) −4558.16 −0.927775
\(290\) 0 0
\(291\) 21.1137 0.00425328
\(292\) 0 0
\(293\) 5921.90 1.18075 0.590377 0.807128i \(-0.298979\pi\)
0.590377 + 0.807128i \(0.298979\pi\)
\(294\) 0 0
\(295\) 10522.4 2.07674
\(296\) 0 0
\(297\) 794.011 0.155129
\(298\) 0 0
\(299\) 5899.81 1.14112
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) −45.3733 −0.00860273
\(304\) 0 0
\(305\) 2833.74 0.531998
\(306\) 0 0
\(307\) −6630.60 −1.23267 −0.616333 0.787486i \(-0.711382\pi\)
−0.616333 + 0.787486i \(0.711382\pi\)
\(308\) 0 0
\(309\) 51.2007 0.00942623
\(310\) 0 0
\(311\) 2181.45 0.397746 0.198873 0.980025i \(-0.436272\pi\)
0.198873 + 0.980025i \(0.436272\pi\)
\(312\) 0 0
\(313\) 8996.44 1.62463 0.812314 0.583220i \(-0.198208\pi\)
0.812314 + 0.583220i \(0.198208\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −314.569 −0.0557348 −0.0278674 0.999612i \(-0.508872\pi\)
−0.0278674 + 0.999612i \(0.508872\pi\)
\(318\) 0 0
\(319\) 5860.57 1.02862
\(320\) 0 0
\(321\) 535.915 0.0931834
\(322\) 0 0
\(323\) −1522.25 −0.262229
\(324\) 0 0
\(325\) 16855.6 2.87686
\(326\) 0 0
\(327\) −407.495 −0.0689129
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) 7006.07 1.16341 0.581704 0.813400i \(-0.302386\pi\)
0.581704 + 0.813400i \(0.302386\pi\)
\(332\) 0 0
\(333\) −371.977 −0.0612138
\(334\) 0 0
\(335\) 9284.29 1.51419
\(336\) 0 0
\(337\) 4697.83 0.759368 0.379684 0.925116i \(-0.376033\pi\)
0.379684 + 0.925116i \(0.376033\pi\)
\(338\) 0 0
\(339\) 170.794 0.0273636
\(340\) 0 0
\(341\) −8355.15 −1.32685
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) 455.775 0.0711249
\(346\) 0 0
\(347\) −8094.65 −1.25229 −0.626143 0.779708i \(-0.715368\pi\)
−0.626143 + 0.779708i \(0.715368\pi\)
\(348\) 0 0
\(349\) 10531.7 1.61532 0.807662 0.589646i \(-0.200733\pi\)
0.807662 + 0.589646i \(0.200733\pi\)
\(350\) 0 0
\(351\) 931.385 0.141634
\(352\) 0 0
\(353\) 11986.5 1.80730 0.903648 0.428276i \(-0.140879\pi\)
0.903648 + 0.428276i \(0.140879\pi\)
\(354\) 0 0
\(355\) 2569.51 0.384155
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −8073.90 −1.18698 −0.593488 0.804843i \(-0.702250\pi\)
−0.593488 + 0.804843i \(0.702250\pi\)
\(360\) 0 0
\(361\) −328.641 −0.0479138
\(362\) 0 0
\(363\) −480.560 −0.0694845
\(364\) 0 0
\(365\) 6817.69 0.977683
\(366\) 0 0
\(367\) 2904.85 0.413166 0.206583 0.978429i \(-0.433766\pi\)
0.206583 + 0.978429i \(0.433766\pi\)
\(368\) 0 0
\(369\) 4696.96 0.662640
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) 5116.71 0.710277 0.355139 0.934814i \(-0.384434\pi\)
0.355139 + 0.934814i \(0.384434\pi\)
\(374\) 0 0
\(375\) 664.550 0.0915126
\(376\) 0 0
\(377\) 6874.52 0.939140
\(378\) 0 0
\(379\) −7841.91 −1.06283 −0.531414 0.847112i \(-0.678339\pi\)
−0.531414 + 0.847112i \(0.678339\pi\)
\(380\) 0 0
\(381\) −214.236 −0.0288074
\(382\) 0 0
\(383\) −4959.17 −0.661623 −0.330812 0.943697i \(-0.607322\pi\)
−0.330812 + 0.943697i \(0.607322\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 5530.47 0.726433
\(388\) 0 0
\(389\) 2188.94 0.285306 0.142653 0.989773i \(-0.454437\pi\)
0.142653 + 0.989773i \(0.454437\pi\)
\(390\) 0 0
\(391\) 1683.22 0.217708
\(392\) 0 0
\(393\) 635.146 0.0815239
\(394\) 0 0
\(395\) −15735.4 −2.00439
\(396\) 0 0
\(397\) 6953.85 0.879103 0.439551 0.898217i \(-0.355137\pi\)
0.439551 + 0.898217i \(0.355137\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −3768.06 −0.469246 −0.234623 0.972086i \(-0.575386\pi\)
−0.234623 + 0.972086i \(0.575386\pi\)
\(402\) 0 0
\(403\) −9800.69 −1.21143
\(404\) 0 0
\(405\) −14108.2 −1.73097
\(406\) 0 0
\(407\) 777.438 0.0946835
\(408\) 0 0
\(409\) −16138.5 −1.95110 −0.975548 0.219787i \(-0.929464\pi\)
−0.975548 + 0.219787i \(0.929464\pi\)
\(410\) 0 0
\(411\) −26.2441 −0.00314969
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) 4556.36 0.538946
\(416\) 0 0
\(417\) 165.907 0.0194832
\(418\) 0 0
\(419\) 9445.77 1.10133 0.550664 0.834727i \(-0.314375\pi\)
0.550664 + 0.834727i \(0.314375\pi\)
\(420\) 0 0
\(421\) −11555.5 −1.33772 −0.668858 0.743390i \(-0.733217\pi\)
−0.668858 + 0.743390i \(0.733217\pi\)
\(422\) 0 0
\(423\) 3141.15 0.361059
\(424\) 0 0
\(425\) 4808.89 0.548860
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) −972.071 −0.109399
\(430\) 0 0
\(431\) −9497.23 −1.06141 −0.530703 0.847558i \(-0.678072\pi\)
−0.530703 + 0.847558i \(0.678072\pi\)
\(432\) 0 0
\(433\) 10826.5 1.20160 0.600798 0.799401i \(-0.294850\pi\)
0.600798 + 0.799401i \(0.294850\pi\)
\(434\) 0 0
\(435\) 531.073 0.0585357
\(436\) 0 0
\(437\) −7220.91 −0.790442
\(438\) 0 0
\(439\) 12635.7 1.37373 0.686867 0.726783i \(-0.258985\pi\)
0.686867 + 0.726783i \(0.258985\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 4875.92 0.522939 0.261470 0.965212i \(-0.415793\pi\)
0.261470 + 0.965212i \(0.415793\pi\)
\(444\) 0 0
\(445\) −10442.4 −1.11240
\(446\) 0 0
\(447\) 550.632 0.0582640
\(448\) 0 0
\(449\) −5294.85 −0.556524 −0.278262 0.960505i \(-0.589758\pi\)
−0.278262 + 0.960505i \(0.589758\pi\)
\(450\) 0 0
\(451\) −9816.74 −1.02495
\(452\) 0 0
\(453\) −750.302 −0.0778196
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 1899.61 0.194442 0.0972211 0.995263i \(-0.469005\pi\)
0.0972211 + 0.995263i \(0.469005\pi\)
\(458\) 0 0
\(459\) 265.724 0.0270217
\(460\) 0 0
\(461\) 4665.78 0.471382 0.235691 0.971828i \(-0.424265\pi\)
0.235691 + 0.971828i \(0.424265\pi\)
\(462\) 0 0
\(463\) −1960.56 −0.196792 −0.0983962 0.995147i \(-0.531371\pi\)
−0.0983962 + 0.995147i \(0.531371\pi\)
\(464\) 0 0
\(465\) −757.127 −0.0755074
\(466\) 0 0
\(467\) −10206.4 −1.01134 −0.505671 0.862726i \(-0.668755\pi\)
−0.505671 + 0.862726i \(0.668755\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 807.537 0.0790007
\(472\) 0 0
\(473\) −11558.8 −1.12362
\(474\) 0 0
\(475\) −20629.9 −1.99277
\(476\) 0 0
\(477\) 8517.08 0.817548
\(478\) 0 0
\(479\) 10961.2 1.04558 0.522789 0.852462i \(-0.324892\pi\)
0.522789 + 0.852462i \(0.324892\pi\)
\(480\) 0 0
\(481\) 911.945 0.0864472
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 1574.16 0.147379
\(486\) 0 0
\(487\) 17409.7 1.61994 0.809970 0.586471i \(-0.199483\pi\)
0.809970 + 0.586471i \(0.199483\pi\)
\(488\) 0 0
\(489\) −264.467 −0.0244573
\(490\) 0 0
\(491\) 5282.62 0.485543 0.242771 0.970084i \(-0.421944\pi\)
0.242771 + 0.970084i \(0.421944\pi\)
\(492\) 0 0
\(493\) 1961.30 0.179174
\(494\) 0 0
\(495\) 29561.7 2.68424
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) −18977.1 −1.70246 −0.851232 0.524789i \(-0.824144\pi\)
−0.851232 + 0.524789i \(0.824144\pi\)
\(500\) 0 0
\(501\) 275.343 0.0245537
\(502\) 0 0
\(503\) 8525.73 0.755753 0.377876 0.925856i \(-0.376654\pi\)
0.377876 + 0.925856i \(0.376654\pi\)
\(504\) 0 0
\(505\) −3382.86 −0.298090
\(506\) 0 0
\(507\) −565.604 −0.0495451
\(508\) 0 0
\(509\) −10235.7 −0.891333 −0.445666 0.895199i \(-0.647033\pi\)
−0.445666 + 0.895199i \(0.647033\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) −1139.94 −0.0981086
\(514\) 0 0
\(515\) 3817.34 0.326625
\(516\) 0 0
\(517\) −6565.07 −0.558474
\(518\) 0 0
\(519\) −80.0750 −0.00677245
\(520\) 0 0
\(521\) −1592.37 −0.133902 −0.0669511 0.997756i \(-0.521327\pi\)
−0.0669511 + 0.997756i \(0.521327\pi\)
\(522\) 0 0
\(523\) −14008.0 −1.17118 −0.585589 0.810608i \(-0.699137\pi\)
−0.585589 + 0.810608i \(0.699137\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −2796.14 −0.231123
\(528\) 0 0
\(529\) −4182.52 −0.343759
\(530\) 0 0
\(531\) 14531.9 1.18763
\(532\) 0 0
\(533\) −11515.2 −0.935791
\(534\) 0 0
\(535\) 39955.9 3.22887
\(536\) 0 0
\(537\) −49.4254 −0.00397181
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) 8928.02 0.709511 0.354755 0.934959i \(-0.384564\pi\)
0.354755 + 0.934959i \(0.384564\pi\)
\(542\) 0 0
\(543\) −122.546 −0.00968500
\(544\) 0 0
\(545\) −30381.3 −2.38788
\(546\) 0 0
\(547\) −9736.48 −0.761063 −0.380532 0.924768i \(-0.624259\pi\)
−0.380532 + 0.924768i \(0.624259\pi\)
\(548\) 0 0
\(549\) 3913.51 0.304234
\(550\) 0 0
\(551\) −8413.88 −0.650532
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) 70.4500 0.00538817
\(556\) 0 0
\(557\) 4432.76 0.337203 0.168602 0.985684i \(-0.446075\pi\)
0.168602 + 0.985684i \(0.446075\pi\)
\(558\) 0 0
\(559\) −13558.6 −1.02588
\(560\) 0 0
\(561\) −277.332 −0.0208716
\(562\) 0 0
\(563\) 1402.96 0.105022 0.0525112 0.998620i \(-0.483277\pi\)
0.0525112 + 0.998620i \(0.483277\pi\)
\(564\) 0 0
\(565\) 12733.8 0.948165
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 5798.54 0.427218 0.213609 0.976919i \(-0.431478\pi\)
0.213609 + 0.976919i \(0.431478\pi\)
\(570\) 0 0
\(571\) −13761.8 −1.00861 −0.504303 0.863527i \(-0.668250\pi\)
−0.504303 + 0.863527i \(0.668250\pi\)
\(572\) 0 0
\(573\) 1098.55 0.0800919
\(574\) 0 0
\(575\) 22811.4 1.65444
\(576\) 0 0
\(577\) 7420.50 0.535389 0.267695 0.963504i \(-0.413738\pi\)
0.267695 + 0.963504i \(0.413738\pi\)
\(578\) 0 0
\(579\) 765.131 0.0549184
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) −17800.9 −1.26456
\(584\) 0 0
\(585\) 34676.3 2.45075
\(586\) 0 0
\(587\) 19685.1 1.38414 0.692072 0.721828i \(-0.256698\pi\)
0.692072 + 0.721828i \(0.256698\pi\)
\(588\) 0 0
\(589\) 11995.3 0.839146
\(590\) 0 0
\(591\) 889.928 0.0619404
\(592\) 0 0
\(593\) −24079.3 −1.66749 −0.833743 0.552153i \(-0.813807\pi\)
−0.833743 + 0.552153i \(0.813807\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 25.7373 0.00176442
\(598\) 0 0
\(599\) 17966.4 1.22552 0.612759 0.790270i \(-0.290060\pi\)
0.612759 + 0.790270i \(0.290060\pi\)
\(600\) 0 0
\(601\) −12893.6 −0.875111 −0.437555 0.899192i \(-0.644156\pi\)
−0.437555 + 0.899192i \(0.644156\pi\)
\(602\) 0 0
\(603\) 12822.0 0.865922
\(604\) 0 0
\(605\) −35828.8 −2.40768
\(606\) 0 0
\(607\) −25200.5 −1.68510 −0.842550 0.538619i \(-0.818946\pi\)
−0.842550 + 0.538619i \(0.818946\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) −7700.90 −0.509894
\(612\) 0 0
\(613\) 12814.9 0.844357 0.422178 0.906513i \(-0.361266\pi\)
0.422178 + 0.906513i \(0.361266\pi\)
\(614\) 0 0
\(615\) −889.574 −0.0583270
\(616\) 0 0
\(617\) 7597.02 0.495696 0.247848 0.968799i \(-0.420277\pi\)
0.247848 + 0.968799i \(0.420277\pi\)
\(618\) 0 0
\(619\) −22412.8 −1.45533 −0.727664 0.685934i \(-0.759394\pi\)
−0.727664 + 0.685934i \(0.759394\pi\)
\(620\) 0 0
\(621\) 1260.49 0.0814518
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) 17635.5 1.12867
\(626\) 0 0
\(627\) 1189.74 0.0757793
\(628\) 0 0
\(629\) 260.178 0.0164928
\(630\) 0 0
\(631\) −8993.43 −0.567389 −0.283695 0.958915i \(-0.591560\pi\)
−0.283695 + 0.958915i \(0.591560\pi\)
\(632\) 0 0
\(633\) 704.183 0.0442160
\(634\) 0 0
\(635\) −15972.6 −0.998195
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) 3548.59 0.219687
\(640\) 0 0
\(641\) −18152.7 −1.11855 −0.559273 0.828983i \(-0.688920\pi\)
−0.559273 + 0.828983i \(0.688920\pi\)
\(642\) 0 0
\(643\) −25767.3 −1.58034 −0.790172 0.612885i \(-0.790009\pi\)
−0.790172 + 0.612885i \(0.790009\pi\)
\(644\) 0 0
\(645\) −1047.44 −0.0639422
\(646\) 0 0
\(647\) −6962.52 −0.423068 −0.211534 0.977371i \(-0.567846\pi\)
−0.211534 + 0.977371i \(0.567846\pi\)
\(648\) 0 0
\(649\) −30371.9 −1.83698
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −12157.8 −0.728595 −0.364297 0.931283i \(-0.618691\pi\)
−0.364297 + 0.931283i \(0.618691\pi\)
\(654\) 0 0
\(655\) 47354.2 2.82486
\(656\) 0 0
\(657\) 9415.50 0.559108
\(658\) 0 0
\(659\) 3853.23 0.227770 0.113885 0.993494i \(-0.463670\pi\)
0.113885 + 0.993494i \(0.463670\pi\)
\(660\) 0 0
\(661\) 4579.02 0.269445 0.134723 0.990883i \(-0.456986\pi\)
0.134723 + 0.990883i \(0.456986\pi\)
\(662\) 0 0
\(663\) −325.314 −0.0190560
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 9303.60 0.540085
\(668\) 0 0
\(669\) −1589.15 −0.0918386
\(670\) 0 0
\(671\) −8179.31 −0.470579
\(672\) 0 0
\(673\) −21847.0 −1.25132 −0.625661 0.780095i \(-0.715171\pi\)
−0.625661 + 0.780095i \(0.715171\pi\)
\(674\) 0 0
\(675\) 3601.16 0.205347
\(676\) 0 0
\(677\) −4823.04 −0.273803 −0.136901 0.990585i \(-0.543714\pi\)
−0.136901 + 0.990585i \(0.543714\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 782.366 0.0440240
\(682\) 0 0
\(683\) 16222.7 0.908850 0.454425 0.890785i \(-0.349845\pi\)
0.454425 + 0.890785i \(0.349845\pi\)
\(684\) 0 0
\(685\) −1956.66 −0.109139
\(686\) 0 0
\(687\) −1616.75 −0.0897856
\(688\) 0 0
\(689\) −20880.6 −1.15456
\(690\) 0 0
\(691\) 25356.2 1.39594 0.697971 0.716126i \(-0.254086\pi\)
0.697971 + 0.716126i \(0.254086\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 12369.4 0.675107
\(696\) 0 0
\(697\) −3285.28 −0.178535
\(698\) 0 0
\(699\) −762.039 −0.0412345
\(700\) 0 0
\(701\) −11538.7 −0.621701 −0.310850 0.950459i \(-0.600614\pi\)
−0.310850 + 0.950459i \(0.600614\pi\)
\(702\) 0 0
\(703\) −1116.15 −0.0598810
\(704\) 0 0
\(705\) −594.914 −0.0317812
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) −33771.6 −1.78888 −0.894442 0.447185i \(-0.852427\pi\)
−0.894442 + 0.447185i \(0.852427\pi\)
\(710\) 0 0
\(711\) −21731.3 −1.14625
\(712\) 0 0
\(713\) −13263.7 −0.696676
\(714\) 0 0
\(715\) −72474.0 −3.79074
\(716\) 0 0
\(717\) 891.201 0.0464191
\(718\) 0 0
\(719\) 14047.9 0.728647 0.364324 0.931272i \(-0.381300\pi\)
0.364324 + 0.931272i \(0.381300\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) 63.8532 0.00328454
\(724\) 0 0
\(725\) 26580.1 1.36160
\(726\) 0 0
\(727\) 10726.8 0.547226 0.273613 0.961840i \(-0.411781\pi\)
0.273613 + 0.961840i \(0.411781\pi\)
\(728\) 0 0
\(729\) −19384.4 −0.984829
\(730\) 0 0
\(731\) −3868.27 −0.195723
\(732\) 0 0
\(733\) 10039.3 0.505880 0.252940 0.967482i \(-0.418603\pi\)
0.252940 + 0.967482i \(0.418603\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −26798.2 −1.33938
\(738\) 0 0
\(739\) −6984.31 −0.347661 −0.173831 0.984776i \(-0.555615\pi\)
−0.173831 + 0.984776i \(0.555615\pi\)
\(740\) 0 0
\(741\) 1395.58 0.0691874
\(742\) 0 0
\(743\) 20657.2 1.01997 0.509986 0.860183i \(-0.329651\pi\)
0.509986 + 0.860183i \(0.329651\pi\)
\(744\) 0 0
\(745\) 41053.1 2.01889
\(746\) 0 0
\(747\) 6292.51 0.308207
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) −29782.2 −1.44709 −0.723547 0.690275i \(-0.757489\pi\)
−0.723547 + 0.690275i \(0.757489\pi\)
\(752\) 0 0
\(753\) 1254.31 0.0607032
\(754\) 0 0
\(755\) −55939.7 −2.69650
\(756\) 0 0
\(757\) −17157.0 −0.823754 −0.411877 0.911240i \(-0.635127\pi\)
−0.411877 + 0.911240i \(0.635127\pi\)
\(758\) 0 0
\(759\) −1315.55 −0.0629135
\(760\) 0 0
\(761\) 38481.4 1.83305 0.916523 0.399981i \(-0.130983\pi\)
0.916523 + 0.399981i \(0.130983\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) 9893.14 0.467565
\(766\) 0 0
\(767\) −35626.6 −1.67719
\(768\) 0 0
\(769\) 17934.4 0.841002 0.420501 0.907292i \(-0.361854\pi\)
0.420501 + 0.907292i \(0.361854\pi\)
\(770\) 0 0
\(771\) 1250.87 0.0584294
\(772\) 0 0
\(773\) 11443.2 0.532451 0.266226 0.963911i \(-0.414223\pi\)
0.266226 + 0.963911i \(0.414223\pi\)
\(774\) 0 0
\(775\) −37894.0 −1.75638
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 14093.7 0.648213
\(780\) 0 0
\(781\) −7416.62 −0.339805
\(782\) 0 0
\(783\) 1468.73 0.0670347
\(784\) 0 0
\(785\) 60207.0 2.73743
\(786\) 0 0
\(787\) −11578.5 −0.524431 −0.262216 0.965009i \(-0.584453\pi\)
−0.262216 + 0.965009i \(0.584453\pi\)
\(788\) 0 0
\(789\) 2000.69 0.0902741
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) −9594.43 −0.429645
\(794\) 0 0
\(795\) −1613.08 −0.0719623
\(796\) 0 0
\(797\) −36578.7 −1.62570 −0.812850 0.582473i \(-0.802085\pi\)
−0.812850 + 0.582473i \(0.802085\pi\)
\(798\) 0 0
\(799\) −2197.07 −0.0972800
\(800\) 0 0
\(801\) −14421.4 −0.636149
\(802\) 0 0
\(803\) −19678.6 −0.864809
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 1098.74 0.0479273
\(808\) 0 0
\(809\) 5876.16 0.255370 0.127685 0.991815i \(-0.459245\pi\)
0.127685 + 0.991815i \(0.459245\pi\)
\(810\) 0 0
\(811\) −3272.77 −0.141705 −0.0708523 0.997487i \(-0.522572\pi\)
−0.0708523 + 0.997487i \(0.522572\pi\)
\(812\) 0 0
\(813\) 1324.33 0.0571297
\(814\) 0 0
\(815\) −19717.7 −0.847462
\(816\) 0 0
\(817\) 16594.7 0.710617
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 11801.6 0.501679 0.250840 0.968029i \(-0.419293\pi\)
0.250840 + 0.968029i \(0.419293\pi\)
\(822\) 0 0
\(823\) −9636.61 −0.408154 −0.204077 0.978955i \(-0.565419\pi\)
−0.204077 + 0.978955i \(0.565419\pi\)
\(824\) 0 0
\(825\) −3758.48 −0.158610
\(826\) 0 0
\(827\) 28192.3 1.18542 0.592710 0.805416i \(-0.298058\pi\)
0.592710 + 0.805416i \(0.298058\pi\)
\(828\) 0 0
\(829\) 24633.1 1.03202 0.516010 0.856583i \(-0.327417\pi\)
0.516010 + 0.856583i \(0.327417\pi\)
\(830\) 0 0
\(831\) 1398.40 0.0583754
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) 20528.5 0.850801
\(836\) 0 0
\(837\) −2093.90 −0.0864706
\(838\) 0 0
\(839\) −3682.43 −0.151527 −0.0757637 0.997126i \(-0.524139\pi\)
−0.0757637 + 0.997126i \(0.524139\pi\)
\(840\) 0 0
\(841\) −13548.4 −0.555511
\(842\) 0 0
\(843\) −1665.59 −0.0680497
\(844\) 0 0
\(845\) −42169.4 −1.71677
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) 555.654 0.0224617
\(850\) 0 0
\(851\) 1234.18 0.0497145
\(852\) 0 0
\(853\) 790.015 0.0317112 0.0158556 0.999874i \(-0.494953\pi\)
0.0158556 + 0.999874i \(0.494953\pi\)
\(854\) 0 0
\(855\) −42441.1 −1.69761
\(856\) 0 0
\(857\) 15450.3 0.615835 0.307918 0.951413i \(-0.400368\pi\)
0.307918 + 0.951413i \(0.400368\pi\)
\(858\) 0 0
\(859\) 19601.6 0.778579 0.389289 0.921116i \(-0.372721\pi\)
0.389289 + 0.921116i \(0.372721\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) −38572.6 −1.52147 −0.760734 0.649064i \(-0.775161\pi\)
−0.760734 + 0.649064i \(0.775161\pi\)
\(864\) 0 0
\(865\) −5970.10 −0.234670
\(866\) 0 0
\(867\) 1192.23 0.0467016
\(868\) 0 0
\(869\) 45418.7 1.77299
\(870\) 0 0
\(871\) −31434.6 −1.22287
\(872\) 0 0
\(873\) 2173.97 0.0842817
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) −6319.27 −0.243314 −0.121657 0.992572i \(-0.538821\pi\)
−0.121657 + 0.992572i \(0.538821\pi\)
\(878\) 0 0
\(879\) −1548.93 −0.0594359
\(880\) 0 0
\(881\) 15224.6 0.582213 0.291106 0.956691i \(-0.405977\pi\)
0.291106 + 0.956691i \(0.405977\pi\)
\(882\) 0 0
\(883\) 33244.2 1.26700 0.633498 0.773744i \(-0.281618\pi\)
0.633498 + 0.773744i \(0.281618\pi\)
\(884\) 0 0
\(885\) −2752.24 −0.104537
\(886\) 0 0
\(887\) 15940.1 0.603401 0.301700 0.953403i \(-0.402446\pi\)
0.301700 + 0.953403i \(0.402446\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) 40721.9 1.53113
\(892\) 0 0
\(893\) 9425.31 0.353198
\(894\) 0 0
\(895\) −3684.97 −0.137626
\(896\) 0 0
\(897\) −1543.15 −0.0574408
\(898\) 0 0
\(899\) −15455.0 −0.573364
\(900\) 0 0
\(901\) −5957.25 −0.220272
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) −9136.58 −0.335591
\(906\) 0 0
\(907\) 22038.3 0.806804 0.403402 0.915023i \(-0.367828\pi\)
0.403402 + 0.915023i \(0.367828\pi\)
\(908\) 0 0
\(909\) −4671.87 −0.170469
\(910\) 0 0
\(911\) −2241.91 −0.0815342 −0.0407671 0.999169i \(-0.512980\pi\)
−0.0407671 + 0.999169i \(0.512980\pi\)
\(912\) 0 0
\(913\) −13151.5 −0.476725
\(914\) 0 0
\(915\) −741.193 −0.0267793
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) 35585.4 1.27732 0.638658 0.769491i \(-0.279490\pi\)
0.638658 + 0.769491i \(0.279490\pi\)
\(920\) 0 0
\(921\) 1734.30 0.0620490
\(922\) 0 0
\(923\) −8699.79 −0.310246
\(924\) 0 0
\(925\) 3526.00 0.125334
\(926\) 0 0
\(927\) 5271.89 0.186787
\(928\) 0 0
\(929\) −45914.2 −1.62153 −0.810763 0.585375i \(-0.800947\pi\)
−0.810763 + 0.585375i \(0.800947\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 0 0
\(933\) −570.581 −0.0200214
\(934\) 0 0
\(935\) −20676.9 −0.723214
\(936\) 0 0
\(937\) 42963.4 1.49792 0.748962 0.662613i \(-0.230553\pi\)
0.748962 + 0.662613i \(0.230553\pi\)
\(938\) 0 0
\(939\) −2353.11 −0.0817793
\(940\) 0 0
\(941\) 21420.7 0.742077 0.371039 0.928617i \(-0.379002\pi\)
0.371039 + 0.928617i \(0.379002\pi\)
\(942\) 0 0
\(943\) −15584.0 −0.538159
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 6523.67 0.223855 0.111928 0.993716i \(-0.464298\pi\)
0.111928 + 0.993716i \(0.464298\pi\)
\(948\) 0 0
\(949\) −23083.2 −0.789581
\(950\) 0 0
\(951\) 82.2786 0.00280554
\(952\) 0 0
\(953\) −32537.1 −1.10596 −0.552979 0.833195i \(-0.686509\pi\)
−0.552979 + 0.833195i \(0.686509\pi\)
\(954\) 0 0
\(955\) 81904.0 2.77524
\(956\) 0 0
\(957\) −1532.89 −0.0517777
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) −7757.48 −0.260397
\(962\) 0 0
\(963\) 55180.7 1.84649
\(964\) 0 0
\(965\) 57045.4 1.90296
\(966\) 0 0
\(967\) −9013.22 −0.299737 −0.149868 0.988706i \(-0.547885\pi\)
−0.149868 + 0.988706i \(0.547885\pi\)
\(968\) 0 0
\(969\) 398.159 0.0131999
\(970\) 0 0
\(971\) −47164.6 −1.55879 −0.779395 0.626533i \(-0.784473\pi\)
−0.779395 + 0.626533i \(0.784473\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) −4408.74 −0.144813
\(976\) 0 0
\(977\) 29115.2 0.953407 0.476703 0.879064i \(-0.341832\pi\)
0.476703 + 0.879064i \(0.341832\pi\)
\(978\) 0 0
\(979\) 30141.0 0.983974
\(980\) 0 0
\(981\) −41957.8 −1.36556
\(982\) 0 0
\(983\) −38771.1 −1.25799 −0.628996 0.777409i \(-0.716534\pi\)
−0.628996 + 0.777409i \(0.716534\pi\)
\(984\) 0 0
\(985\) 66349.8 2.14627
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) −18349.5 −0.589969
\(990\) 0 0
\(991\) 18648.2 0.597760 0.298880 0.954291i \(-0.403387\pi\)
0.298880 + 0.954291i \(0.403387\pi\)
\(992\) 0 0
\(993\) −1832.51 −0.0585628
\(994\) 0 0
\(995\) 1918.88 0.0611382
\(996\) 0 0
\(997\) −16939.6 −0.538097 −0.269049 0.963127i \(-0.586709\pi\)
−0.269049 + 0.963127i \(0.586709\pi\)
\(998\) 0 0
\(999\) 194.836 0.00617050
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 784.4.a.bb.1.2 3
4.3 odd 2 392.4.a.l.1.2 3
7.3 odd 6 112.4.i.e.65.2 6
7.5 odd 6 112.4.i.e.81.2 6
7.6 odd 2 784.4.a.be.1.2 3
28.3 even 6 56.4.i.b.9.2 6
28.11 odd 6 392.4.i.m.177.2 6
28.19 even 6 56.4.i.b.25.2 yes 6
28.23 odd 6 392.4.i.m.361.2 6
28.27 even 2 392.4.a.i.1.2 3
56.3 even 6 448.4.i.j.65.2 6
56.5 odd 6 448.4.i.m.193.2 6
56.19 even 6 448.4.i.j.193.2 6
56.45 odd 6 448.4.i.m.65.2 6
84.47 odd 6 504.4.s.h.361.3 6
84.59 odd 6 504.4.s.h.289.3 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
56.4.i.b.9.2 6 28.3 even 6
56.4.i.b.25.2 yes 6 28.19 even 6
112.4.i.e.65.2 6 7.3 odd 6
112.4.i.e.81.2 6 7.5 odd 6
392.4.a.i.1.2 3 28.27 even 2
392.4.a.l.1.2 3 4.3 odd 2
392.4.i.m.177.2 6 28.11 odd 6
392.4.i.m.361.2 6 28.23 odd 6
448.4.i.j.65.2 6 56.3 even 6
448.4.i.j.193.2 6 56.19 even 6
448.4.i.m.65.2 6 56.45 odd 6
448.4.i.m.193.2 6 56.5 odd 6
504.4.s.h.289.3 6 84.59 odd 6
504.4.s.h.361.3 6 84.47 odd 6
784.4.a.bb.1.2 3 1.1 even 1 trivial
784.4.a.be.1.2 3 7.6 odd 2