Properties

Label 1856.4.a.bi.1.6
Level $1856$
Weight $4$
Character 1856.1
Self dual yes
Analytic conductor $109.508$
Analytic rank $0$
Dimension $10$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1856,4,Mod(1,1856)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1856, 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("1856.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1856 = 2^{6} \cdot 29 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1856.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: \(109.507544971\)
Analytic rank: \(0\)
Dimension: \(10\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{10} - 4 x^{9} - 135 x^{8} + 788 x^{7} + 3323 x^{6} - 26136 x^{5} + 2315 x^{4} + 188664 x^{3} - 265632 x^{2} - 18800 x + 128592 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{14} \)
Twist minimal: no (minimal twist has level 928)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.6
Root \(-0.612154\) of defining polynomial
Character \(\chi\) \(=\) 1856.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+0.0474689 q^{3} -5.61587 q^{5} +9.39051 q^{7} -26.9977 q^{9} +O(q^{10})\) \(q+0.0474689 q^{3} -5.61587 q^{5} +9.39051 q^{7} -26.9977 q^{9} -7.05123 q^{11} -11.5402 q^{13} -0.266579 q^{15} +43.2804 q^{17} +94.6522 q^{19} +0.445757 q^{21} -183.781 q^{23} -93.4620 q^{25} -2.56321 q^{27} +29.0000 q^{29} +299.700 q^{31} -0.334714 q^{33} -52.7359 q^{35} -116.770 q^{37} -0.547802 q^{39} -461.690 q^{41} -120.966 q^{43} +151.616 q^{45} +115.282 q^{47} -254.818 q^{49} +2.05447 q^{51} +1.62495 q^{53} +39.5988 q^{55} +4.49303 q^{57} +167.219 q^{59} +753.613 q^{61} -253.523 q^{63} +64.8085 q^{65} +495.939 q^{67} -8.72386 q^{69} -394.893 q^{71} +35.7036 q^{73} -4.43654 q^{75} -66.2147 q^{77} +659.203 q^{79} +728.817 q^{81} -1437.34 q^{83} -243.057 q^{85} +1.37660 q^{87} +171.190 q^{89} -108.369 q^{91} +14.2264 q^{93} -531.554 q^{95} -750.095 q^{97} +190.367 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q + 4 q^{5} + 50 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 10 q + 4 q^{5} + 50 q^{9} + 220 q^{13} + 76 q^{17} + 104 q^{21} + 446 q^{25} + 290 q^{29} - 1120 q^{33} + 1708 q^{37} - 980 q^{41} - 348 q^{45} + 1146 q^{49} + 44 q^{53} - 40 q^{57} + 1492 q^{61} - 2016 q^{65} + 2328 q^{69} - 3100 q^{73} + 3016 q^{77} - 2174 q^{81} + 9440 q^{85} + 636 q^{89} + 2536 q^{93} + 620 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Coefficient data

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



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 0.0474689 0.00913539 0.00456769 0.999990i \(-0.498546\pi\)
0.00456769 + 0.999990i \(0.498546\pi\)
\(4\) 0 0
\(5\) −5.61587 −0.502299 −0.251149 0.967948i \(-0.580809\pi\)
−0.251149 + 0.967948i \(0.580809\pi\)
\(6\) 0 0
\(7\) 9.39051 0.507040 0.253520 0.967330i \(-0.418412\pi\)
0.253520 + 0.967330i \(0.418412\pi\)
\(8\) 0 0
\(9\) −26.9977 −0.999917
\(10\) 0 0
\(11\) −7.05123 −0.193275 −0.0966375 0.995320i \(-0.530809\pi\)
−0.0966375 + 0.995320i \(0.530809\pi\)
\(12\) 0 0
\(13\) −11.5402 −0.246207 −0.123103 0.992394i \(-0.539285\pi\)
−0.123103 + 0.992394i \(0.539285\pi\)
\(14\) 0 0
\(15\) −0.266579 −0.00458870
\(16\) 0 0
\(17\) 43.2804 0.617473 0.308737 0.951148i \(-0.400094\pi\)
0.308737 + 0.951148i \(0.400094\pi\)
\(18\) 0 0
\(19\) 94.6522 1.14288 0.571439 0.820644i \(-0.306385\pi\)
0.571439 + 0.820644i \(0.306385\pi\)
\(20\) 0 0
\(21\) 0.445757 0.00463201
\(22\) 0 0
\(23\) −183.781 −1.66613 −0.833064 0.553177i \(-0.813415\pi\)
−0.833064 + 0.553177i \(0.813415\pi\)
\(24\) 0 0
\(25\) −93.4620 −0.747696
\(26\) 0 0
\(27\) −2.56321 −0.0182700
\(28\) 0 0
\(29\) 29.0000 0.185695
\(30\) 0 0
\(31\) 299.700 1.73638 0.868189 0.496233i \(-0.165284\pi\)
0.868189 + 0.496233i \(0.165284\pi\)
\(32\) 0 0
\(33\) −0.334714 −0.00176564
\(34\) 0 0
\(35\) −52.7359 −0.254686
\(36\) 0 0
\(37\) −116.770 −0.518834 −0.259417 0.965765i \(-0.583530\pi\)
−0.259417 + 0.965765i \(0.583530\pi\)
\(38\) 0 0
\(39\) −0.547802 −0.00224919
\(40\) 0 0
\(41\) −461.690 −1.75863 −0.879316 0.476239i \(-0.842000\pi\)
−0.879316 + 0.476239i \(0.842000\pi\)
\(42\) 0 0
\(43\) −120.966 −0.429003 −0.214502 0.976724i \(-0.568813\pi\)
−0.214502 + 0.976724i \(0.568813\pi\)
\(44\) 0 0
\(45\) 151.616 0.502257
\(46\) 0 0
\(47\) 115.282 0.357779 0.178890 0.983869i \(-0.442750\pi\)
0.178890 + 0.983869i \(0.442750\pi\)
\(48\) 0 0
\(49\) −254.818 −0.742910
\(50\) 0 0
\(51\) 2.05447 0.00564086
\(52\) 0 0
\(53\) 1.62495 0.00421139 0.00210570 0.999998i \(-0.499330\pi\)
0.00210570 + 0.999998i \(0.499330\pi\)
\(54\) 0 0
\(55\) 39.5988 0.0970818
\(56\) 0 0
\(57\) 4.49303 0.0104406
\(58\) 0 0
\(59\) 167.219 0.368984 0.184492 0.982834i \(-0.440936\pi\)
0.184492 + 0.982834i \(0.440936\pi\)
\(60\) 0 0
\(61\) 753.613 1.58181 0.790904 0.611940i \(-0.209611\pi\)
0.790904 + 0.611940i \(0.209611\pi\)
\(62\) 0 0
\(63\) −253.523 −0.506998
\(64\) 0 0
\(65\) 64.8085 0.123669
\(66\) 0 0
\(67\) 495.939 0.904308 0.452154 0.891940i \(-0.350656\pi\)
0.452154 + 0.891940i \(0.350656\pi\)
\(68\) 0 0
\(69\) −8.72386 −0.0152207
\(70\) 0 0
\(71\) −394.893 −0.660074 −0.330037 0.943968i \(-0.607061\pi\)
−0.330037 + 0.943968i \(0.607061\pi\)
\(72\) 0 0
\(73\) 35.7036 0.0572438 0.0286219 0.999590i \(-0.490888\pi\)
0.0286219 + 0.999590i \(0.490888\pi\)
\(74\) 0 0
\(75\) −4.43654 −0.00683049
\(76\) 0 0
\(77\) −66.2147 −0.0979982
\(78\) 0 0
\(79\) 659.203 0.938811 0.469406 0.882983i \(-0.344468\pi\)
0.469406 + 0.882983i \(0.344468\pi\)
\(80\) 0 0
\(81\) 728.817 0.999750
\(82\) 0 0
\(83\) −1437.34 −1.90082 −0.950410 0.311001i \(-0.899336\pi\)
−0.950410 + 0.311001i \(0.899336\pi\)
\(84\) 0 0
\(85\) −243.057 −0.310156
\(86\) 0 0
\(87\) 1.37660 0.00169640
\(88\) 0 0
\(89\) 171.190 0.203889 0.101944 0.994790i \(-0.467494\pi\)
0.101944 + 0.994790i \(0.467494\pi\)
\(90\) 0 0
\(91\) −108.369 −0.124837
\(92\) 0 0
\(93\) 14.2264 0.0158625
\(94\) 0 0
\(95\) −531.554 −0.574067
\(96\) 0 0
\(97\) −750.095 −0.785161 −0.392581 0.919718i \(-0.628418\pi\)
−0.392581 + 0.919718i \(0.628418\pi\)
\(98\) 0 0
\(99\) 190.367 0.193259
\(100\) 0 0
\(101\) 1587.54 1.56402 0.782008 0.623268i \(-0.214196\pi\)
0.782008 + 0.623268i \(0.214196\pi\)
\(102\) 0 0
\(103\) 764.687 0.731523 0.365762 0.930709i \(-0.380809\pi\)
0.365762 + 0.930709i \(0.380809\pi\)
\(104\) 0 0
\(105\) −2.50332 −0.00232665
\(106\) 0 0
\(107\) 1366.26 1.23440 0.617200 0.786806i \(-0.288267\pi\)
0.617200 + 0.786806i \(0.288267\pi\)
\(108\) 0 0
\(109\) 1652.02 1.45169 0.725847 0.687856i \(-0.241448\pi\)
0.725847 + 0.687856i \(0.241448\pi\)
\(110\) 0 0
\(111\) −5.54294 −0.00473975
\(112\) 0 0
\(113\) 1170.07 0.974077 0.487038 0.873381i \(-0.338077\pi\)
0.487038 + 0.873381i \(0.338077\pi\)
\(114\) 0 0
\(115\) 1032.09 0.836894
\(116\) 0 0
\(117\) 311.561 0.246186
\(118\) 0 0
\(119\) 406.425 0.313084
\(120\) 0 0
\(121\) −1281.28 −0.962645
\(122\) 0 0
\(123\) −21.9159 −0.0160658
\(124\) 0 0
\(125\) 1226.85 0.877866
\(126\) 0 0
\(127\) −2522.35 −1.76238 −0.881189 0.472764i \(-0.843256\pi\)
−0.881189 + 0.472764i \(0.843256\pi\)
\(128\) 0 0
\(129\) −5.74212 −0.00391911
\(130\) 0 0
\(131\) −2718.35 −1.81300 −0.906501 0.422204i \(-0.861257\pi\)
−0.906501 + 0.422204i \(0.861257\pi\)
\(132\) 0 0
\(133\) 888.833 0.579485
\(134\) 0 0
\(135\) 14.3947 0.00917701
\(136\) 0 0
\(137\) −272.519 −0.169948 −0.0849738 0.996383i \(-0.527081\pi\)
−0.0849738 + 0.996383i \(0.527081\pi\)
\(138\) 0 0
\(139\) 2354.00 1.43643 0.718213 0.695823i \(-0.244960\pi\)
0.718213 + 0.695823i \(0.244960\pi\)
\(140\) 0 0
\(141\) 5.47231 0.00326845
\(142\) 0 0
\(143\) 81.3729 0.0475856
\(144\) 0 0
\(145\) −162.860 −0.0932746
\(146\) 0 0
\(147\) −12.0959 −0.00678678
\(148\) 0 0
\(149\) 2439.00 1.34101 0.670505 0.741905i \(-0.266077\pi\)
0.670505 + 0.741905i \(0.266077\pi\)
\(150\) 0 0
\(151\) 2649.49 1.42790 0.713949 0.700198i \(-0.246905\pi\)
0.713949 + 0.700198i \(0.246905\pi\)
\(152\) 0 0
\(153\) −1168.47 −0.617421
\(154\) 0 0
\(155\) −1683.08 −0.872181
\(156\) 0 0
\(157\) 1504.16 0.764619 0.382309 0.924034i \(-0.375129\pi\)
0.382309 + 0.924034i \(0.375129\pi\)
\(158\) 0 0
\(159\) 0.0771345 3.84727e−5 0
\(160\) 0 0
\(161\) −1725.80 −0.844793
\(162\) 0 0
\(163\) −2016.09 −0.968788 −0.484394 0.874850i \(-0.660960\pi\)
−0.484394 + 0.874850i \(0.660960\pi\)
\(164\) 0 0
\(165\) 1.87971 0.000886880 0
\(166\) 0 0
\(167\) −3547.68 −1.64388 −0.821939 0.569576i \(-0.807107\pi\)
−0.821939 + 0.569576i \(0.807107\pi\)
\(168\) 0 0
\(169\) −2063.82 −0.939382
\(170\) 0 0
\(171\) −2555.40 −1.14278
\(172\) 0 0
\(173\) 1916.32 0.842168 0.421084 0.907022i \(-0.361650\pi\)
0.421084 + 0.907022i \(0.361650\pi\)
\(174\) 0 0
\(175\) −877.656 −0.379112
\(176\) 0 0
\(177\) 7.93770 0.00337081
\(178\) 0 0
\(179\) 2799.09 1.16879 0.584395 0.811469i \(-0.301332\pi\)
0.584395 + 0.811469i \(0.301332\pi\)
\(180\) 0 0
\(181\) −1347.05 −0.553180 −0.276590 0.960988i \(-0.589204\pi\)
−0.276590 + 0.960988i \(0.589204\pi\)
\(182\) 0 0
\(183\) 35.7732 0.0144504
\(184\) 0 0
\(185\) 655.765 0.260610
\(186\) 0 0
\(187\) −305.180 −0.119342
\(188\) 0 0
\(189\) −24.0699 −0.00926363
\(190\) 0 0
\(191\) −2439.45 −0.924149 −0.462074 0.886841i \(-0.652895\pi\)
−0.462074 + 0.886841i \(0.652895\pi\)
\(192\) 0 0
\(193\) 3779.81 1.40972 0.704862 0.709344i \(-0.251009\pi\)
0.704862 + 0.709344i \(0.251009\pi\)
\(194\) 0 0
\(195\) 3.07639 0.00112977
\(196\) 0 0
\(197\) 3645.34 1.31838 0.659188 0.751979i \(-0.270900\pi\)
0.659188 + 0.751979i \(0.270900\pi\)
\(198\) 0 0
\(199\) 4042.15 1.43990 0.719950 0.694026i \(-0.244165\pi\)
0.719950 + 0.694026i \(0.244165\pi\)
\(200\) 0 0
\(201\) 23.5417 0.00826120
\(202\) 0 0
\(203\) 272.325 0.0941550
\(204\) 0 0
\(205\) 2592.79 0.883359
\(206\) 0 0
\(207\) 4961.67 1.66599
\(208\) 0 0
\(209\) −667.414 −0.220890
\(210\) 0 0
\(211\) 4454.20 1.45327 0.726634 0.687024i \(-0.241083\pi\)
0.726634 + 0.687024i \(0.241083\pi\)
\(212\) 0 0
\(213\) −18.7451 −0.00603003
\(214\) 0 0
\(215\) 679.329 0.215488
\(216\) 0 0
\(217\) 2814.34 0.880414
\(218\) 0 0
\(219\) 1.69481 0.000522944 0
\(220\) 0 0
\(221\) −499.466 −0.152026
\(222\) 0 0
\(223\) −843.600 −0.253326 −0.126663 0.991946i \(-0.540427\pi\)
−0.126663 + 0.991946i \(0.540427\pi\)
\(224\) 0 0
\(225\) 2523.26 0.747633
\(226\) 0 0
\(227\) 128.667 0.0376208 0.0188104 0.999823i \(-0.494012\pi\)
0.0188104 + 0.999823i \(0.494012\pi\)
\(228\) 0 0
\(229\) 1875.55 0.541221 0.270610 0.962689i \(-0.412774\pi\)
0.270610 + 0.962689i \(0.412774\pi\)
\(230\) 0 0
\(231\) −3.14314 −0.000895252 0
\(232\) 0 0
\(233\) −732.103 −0.205844 −0.102922 0.994689i \(-0.532819\pi\)
−0.102922 + 0.994689i \(0.532819\pi\)
\(234\) 0 0
\(235\) −647.409 −0.179712
\(236\) 0 0
\(237\) 31.2916 0.00857641
\(238\) 0 0
\(239\) 3892.20 1.05341 0.526706 0.850047i \(-0.323427\pi\)
0.526706 + 0.850047i \(0.323427\pi\)
\(240\) 0 0
\(241\) 3897.87 1.04184 0.520920 0.853605i \(-0.325589\pi\)
0.520920 + 0.853605i \(0.325589\pi\)
\(242\) 0 0
\(243\) 103.803 0.0274031
\(244\) 0 0
\(245\) 1431.03 0.373163
\(246\) 0 0
\(247\) −1092.31 −0.281384
\(248\) 0 0
\(249\) −68.2287 −0.0173647
\(250\) 0 0
\(251\) 5746.49 1.44508 0.722540 0.691329i \(-0.242974\pi\)
0.722540 + 0.691329i \(0.242974\pi\)
\(252\) 0 0
\(253\) 1295.88 0.322021
\(254\) 0 0
\(255\) −11.5377 −0.00283340
\(256\) 0 0
\(257\) 4983.07 1.20948 0.604738 0.796424i \(-0.293278\pi\)
0.604738 + 0.796424i \(0.293278\pi\)
\(258\) 0 0
\(259\) −1096.53 −0.263070
\(260\) 0 0
\(261\) −782.935 −0.185680
\(262\) 0 0
\(263\) 530.122 0.124292 0.0621459 0.998067i \(-0.480206\pi\)
0.0621459 + 0.998067i \(0.480206\pi\)
\(264\) 0 0
\(265\) −9.12550 −0.00211538
\(266\) 0 0
\(267\) 8.12619 0.00186260
\(268\) 0 0
\(269\) −2205.16 −0.499817 −0.249908 0.968269i \(-0.580401\pi\)
−0.249908 + 0.968269i \(0.580401\pi\)
\(270\) 0 0
\(271\) 324.628 0.0727666 0.0363833 0.999338i \(-0.488416\pi\)
0.0363833 + 0.999338i \(0.488416\pi\)
\(272\) 0 0
\(273\) −5.14415 −0.00114043
\(274\) 0 0
\(275\) 659.022 0.144511
\(276\) 0 0
\(277\) 8591.36 1.86356 0.931778 0.363028i \(-0.118257\pi\)
0.931778 + 0.363028i \(0.118257\pi\)
\(278\) 0 0
\(279\) −8091.23 −1.73623
\(280\) 0 0
\(281\) 5176.79 1.09901 0.549504 0.835491i \(-0.314817\pi\)
0.549504 + 0.835491i \(0.314817\pi\)
\(282\) 0 0
\(283\) −8331.74 −1.75007 −0.875036 0.484058i \(-0.839162\pi\)
−0.875036 + 0.484058i \(0.839162\pi\)
\(284\) 0 0
\(285\) −25.2323 −0.00524432
\(286\) 0 0
\(287\) −4335.51 −0.891697
\(288\) 0 0
\(289\) −3039.81 −0.618727
\(290\) 0 0
\(291\) −35.6062 −0.00717275
\(292\) 0 0
\(293\) −5137.07 −1.02427 −0.512135 0.858905i \(-0.671145\pi\)
−0.512135 + 0.858905i \(0.671145\pi\)
\(294\) 0 0
\(295\) −939.080 −0.185340
\(296\) 0 0
\(297\) 18.0738 0.00353114
\(298\) 0 0
\(299\) 2120.87 0.410212
\(300\) 0 0
\(301\) −1135.93 −0.217522
\(302\) 0 0
\(303\) 75.3585 0.0142879
\(304\) 0 0
\(305\) −4232.20 −0.794540
\(306\) 0 0
\(307\) −8143.50 −1.51392 −0.756961 0.653460i \(-0.773317\pi\)
−0.756961 + 0.653460i \(0.773317\pi\)
\(308\) 0 0
\(309\) 36.2989 0.00668275
\(310\) 0 0
\(311\) 4108.03 0.749020 0.374510 0.927223i \(-0.377811\pi\)
0.374510 + 0.927223i \(0.377811\pi\)
\(312\) 0 0
\(313\) 3593.05 0.648853 0.324426 0.945911i \(-0.394829\pi\)
0.324426 + 0.945911i \(0.394829\pi\)
\(314\) 0 0
\(315\) 1423.75 0.254664
\(316\) 0 0
\(317\) −819.529 −0.145203 −0.0726015 0.997361i \(-0.523130\pi\)
−0.0726015 + 0.997361i \(0.523130\pi\)
\(318\) 0 0
\(319\) −204.486 −0.0358903
\(320\) 0 0
\(321\) 64.8546 0.0112767
\(322\) 0 0
\(323\) 4096.58 0.705697
\(324\) 0 0
\(325\) 1078.57 0.184088
\(326\) 0 0
\(327\) 78.4195 0.0132618
\(328\) 0 0
\(329\) 1082.56 0.181408
\(330\) 0 0
\(331\) −826.130 −0.137185 −0.0685924 0.997645i \(-0.521851\pi\)
−0.0685924 + 0.997645i \(0.521851\pi\)
\(332\) 0 0
\(333\) 3152.52 0.518791
\(334\) 0 0
\(335\) −2785.13 −0.454233
\(336\) 0 0
\(337\) 6421.60 1.03800 0.519001 0.854773i \(-0.326304\pi\)
0.519001 + 0.854773i \(0.326304\pi\)
\(338\) 0 0
\(339\) 55.5418 0.00889857
\(340\) 0 0
\(341\) −2113.25 −0.335599
\(342\) 0 0
\(343\) −5613.82 −0.883725
\(344\) 0 0
\(345\) 48.9921 0.00764535
\(346\) 0 0
\(347\) −1996.34 −0.308845 −0.154423 0.988005i \(-0.549352\pi\)
−0.154423 + 0.988005i \(0.549352\pi\)
\(348\) 0 0
\(349\) 5157.51 0.791046 0.395523 0.918456i \(-0.370563\pi\)
0.395523 + 0.918456i \(0.370563\pi\)
\(350\) 0 0
\(351\) 29.5801 0.00449820
\(352\) 0 0
\(353\) 8738.74 1.31761 0.658804 0.752314i \(-0.271062\pi\)
0.658804 + 0.752314i \(0.271062\pi\)
\(354\) 0 0
\(355\) 2217.67 0.331554
\(356\) 0 0
\(357\) 19.2926 0.00286014
\(358\) 0 0
\(359\) 9664.38 1.42080 0.710399 0.703799i \(-0.248515\pi\)
0.710399 + 0.703799i \(0.248515\pi\)
\(360\) 0 0
\(361\) 2100.03 0.306172
\(362\) 0 0
\(363\) −60.8209 −0.00879414
\(364\) 0 0
\(365\) −200.507 −0.0287535
\(366\) 0 0
\(367\) 3092.89 0.439912 0.219956 0.975510i \(-0.429409\pi\)
0.219956 + 0.975510i \(0.429409\pi\)
\(368\) 0 0
\(369\) 12464.6 1.75849
\(370\) 0 0
\(371\) 15.2591 0.00213535
\(372\) 0 0
\(373\) −9050.45 −1.25634 −0.628170 0.778076i \(-0.716196\pi\)
−0.628170 + 0.778076i \(0.716196\pi\)
\(374\) 0 0
\(375\) 58.2374 0.00801964
\(376\) 0 0
\(377\) −334.667 −0.0457194
\(378\) 0 0
\(379\) 9830.13 1.33230 0.666148 0.745820i \(-0.267942\pi\)
0.666148 + 0.745820i \(0.267942\pi\)
\(380\) 0 0
\(381\) −119.733 −0.0161000
\(382\) 0 0
\(383\) 10042.5 1.33980 0.669902 0.742449i \(-0.266336\pi\)
0.669902 + 0.742449i \(0.266336\pi\)
\(384\) 0 0
\(385\) 371.853 0.0492244
\(386\) 0 0
\(387\) 3265.81 0.428967
\(388\) 0 0
\(389\) 5075.42 0.661526 0.330763 0.943714i \(-0.392694\pi\)
0.330763 + 0.943714i \(0.392694\pi\)
\(390\) 0 0
\(391\) −7954.10 −1.02879
\(392\) 0 0
\(393\) −129.037 −0.0165625
\(394\) 0 0
\(395\) −3702.00 −0.471564
\(396\) 0 0
\(397\) −9074.11 −1.14715 −0.573573 0.819155i \(-0.694443\pi\)
−0.573573 + 0.819155i \(0.694443\pi\)
\(398\) 0 0
\(399\) 42.1919 0.00529382
\(400\) 0 0
\(401\) −3865.42 −0.481372 −0.240686 0.970603i \(-0.577372\pi\)
−0.240686 + 0.970603i \(0.577372\pi\)
\(402\) 0 0
\(403\) −3458.61 −0.427508
\(404\) 0 0
\(405\) −4092.95 −0.502173
\(406\) 0 0
\(407\) 823.371 0.100278
\(408\) 0 0
\(409\) 1170.84 0.141551 0.0707757 0.997492i \(-0.477453\pi\)
0.0707757 + 0.997492i \(0.477453\pi\)
\(410\) 0 0
\(411\) −12.9361 −0.00155254
\(412\) 0 0
\(413\) 1570.27 0.187090
\(414\) 0 0
\(415\) 8071.89 0.954779
\(416\) 0 0
\(417\) 111.742 0.0131223
\(418\) 0 0
\(419\) −12741.7 −1.48562 −0.742810 0.669502i \(-0.766507\pi\)
−0.742810 + 0.669502i \(0.766507\pi\)
\(420\) 0 0
\(421\) 2207.04 0.255497 0.127749 0.991807i \(-0.459225\pi\)
0.127749 + 0.991807i \(0.459225\pi\)
\(422\) 0 0
\(423\) −3112.36 −0.357749
\(424\) 0 0
\(425\) −4045.07 −0.461682
\(426\) 0 0
\(427\) 7076.82 0.802040
\(428\) 0 0
\(429\) 3.86268 0.000434713 0
\(430\) 0 0
\(431\) −10293.5 −1.15039 −0.575196 0.818015i \(-0.695074\pi\)
−0.575196 + 0.818015i \(0.695074\pi\)
\(432\) 0 0
\(433\) 12374.3 1.37337 0.686687 0.726954i \(-0.259064\pi\)
0.686687 + 0.726954i \(0.259064\pi\)
\(434\) 0 0
\(435\) −7.73080 −0.000852099 0
\(436\) 0 0
\(437\) −17395.2 −1.90418
\(438\) 0 0
\(439\) 2087.57 0.226958 0.113479 0.993540i \(-0.463801\pi\)
0.113479 + 0.993540i \(0.463801\pi\)
\(440\) 0 0
\(441\) 6879.52 0.742848
\(442\) 0 0
\(443\) −6401.81 −0.686590 −0.343295 0.939228i \(-0.611543\pi\)
−0.343295 + 0.939228i \(0.611543\pi\)
\(444\) 0 0
\(445\) −961.380 −0.102413
\(446\) 0 0
\(447\) 115.777 0.0122507
\(448\) 0 0
\(449\) −7560.57 −0.794666 −0.397333 0.917674i \(-0.630064\pi\)
−0.397333 + 0.917674i \(0.630064\pi\)
\(450\) 0 0
\(451\) 3255.48 0.339900
\(452\) 0 0
\(453\) 125.768 0.0130444
\(454\) 0 0
\(455\) 608.585 0.0627053
\(456\) 0 0
\(457\) 11624.1 1.18983 0.594913 0.803790i \(-0.297186\pi\)
0.594913 + 0.803790i \(0.297186\pi\)
\(458\) 0 0
\(459\) −110.937 −0.0112812
\(460\) 0 0
\(461\) −2487.08 −0.251269 −0.125635 0.992077i \(-0.540097\pi\)
−0.125635 + 0.992077i \(0.540097\pi\)
\(462\) 0 0
\(463\) −10489.7 −1.05291 −0.526453 0.850204i \(-0.676478\pi\)
−0.526453 + 0.850204i \(0.676478\pi\)
\(464\) 0 0
\(465\) −79.8938 −0.00796771
\(466\) 0 0
\(467\) −5870.85 −0.581735 −0.290868 0.956763i \(-0.593944\pi\)
−0.290868 + 0.956763i \(0.593944\pi\)
\(468\) 0 0
\(469\) 4657.12 0.458520
\(470\) 0 0
\(471\) 71.4009 0.00698509
\(472\) 0 0
\(473\) 852.959 0.0829156
\(474\) 0 0
\(475\) −8846.38 −0.854526
\(476\) 0 0
\(477\) −43.8700 −0.00421104
\(478\) 0 0
\(479\) −11566.3 −1.10329 −0.551647 0.834078i \(-0.686000\pi\)
−0.551647 + 0.834078i \(0.686000\pi\)
\(480\) 0 0
\(481\) 1347.55 0.127740
\(482\) 0 0
\(483\) −81.9216 −0.00771752
\(484\) 0 0
\(485\) 4212.44 0.394386
\(486\) 0 0
\(487\) 17667.5 1.64392 0.821962 0.569542i \(-0.192879\pi\)
0.821962 + 0.569542i \(0.192879\pi\)
\(488\) 0 0
\(489\) −95.7015 −0.00885025
\(490\) 0 0
\(491\) 5200.51 0.477996 0.238998 0.971020i \(-0.423181\pi\)
0.238998 + 0.971020i \(0.423181\pi\)
\(492\) 0 0
\(493\) 1255.13 0.114662
\(494\) 0 0
\(495\) −1069.08 −0.0970737
\(496\) 0 0
\(497\) −3708.25 −0.334684
\(498\) 0 0
\(499\) 19925.1 1.78752 0.893758 0.448550i \(-0.148060\pi\)
0.893758 + 0.448550i \(0.148060\pi\)
\(500\) 0 0
\(501\) −168.404 −0.0150175
\(502\) 0 0
\(503\) −14124.6 −1.25206 −0.626030 0.779799i \(-0.715321\pi\)
−0.626030 + 0.779799i \(0.715321\pi\)
\(504\) 0 0
\(505\) −8915.39 −0.785604
\(506\) 0 0
\(507\) −97.9674 −0.00858162
\(508\) 0 0
\(509\) 9959.76 0.867305 0.433653 0.901080i \(-0.357224\pi\)
0.433653 + 0.901080i \(0.357224\pi\)
\(510\) 0 0
\(511\) 335.275 0.0290249
\(512\) 0 0
\(513\) −242.614 −0.0208804
\(514\) 0 0
\(515\) −4294.39 −0.367443
\(516\) 0 0
\(517\) −812.880 −0.0691498
\(518\) 0 0
\(519\) 90.9655 0.00769353
\(520\) 0 0
\(521\) 1656.26 0.139275 0.0696374 0.997572i \(-0.477816\pi\)
0.0696374 + 0.997572i \(0.477816\pi\)
\(522\) 0 0
\(523\) −7994.22 −0.668380 −0.334190 0.942506i \(-0.608463\pi\)
−0.334190 + 0.942506i \(0.608463\pi\)
\(524\) 0 0
\(525\) −41.6613 −0.00346333
\(526\) 0 0
\(527\) 12971.1 1.07217
\(528\) 0 0
\(529\) 21608.4 1.77598
\(530\) 0 0
\(531\) −4514.53 −0.368953
\(532\) 0 0
\(533\) 5328.02 0.432987
\(534\) 0 0
\(535\) −7672.71 −0.620038
\(536\) 0 0
\(537\) 132.869 0.0106774
\(538\) 0 0
\(539\) 1796.78 0.143586
\(540\) 0 0
\(541\) −18621.0 −1.47981 −0.739905 0.672711i \(-0.765130\pi\)
−0.739905 + 0.672711i \(0.765130\pi\)
\(542\) 0 0
\(543\) −63.9431 −0.00505352
\(544\) 0 0
\(545\) −9277.52 −0.729184
\(546\) 0 0
\(547\) 19421.9 1.51814 0.759069 0.651010i \(-0.225654\pi\)
0.759069 + 0.651010i \(0.225654\pi\)
\(548\) 0 0
\(549\) −20345.9 −1.58168
\(550\) 0 0
\(551\) 2744.91 0.212227
\(552\) 0 0
\(553\) 6190.25 0.476015
\(554\) 0 0
\(555\) 31.1284 0.00238077
\(556\) 0 0
\(557\) 2156.17 0.164021 0.0820105 0.996631i \(-0.473866\pi\)
0.0820105 + 0.996631i \(0.473866\pi\)
\(558\) 0 0
\(559\) 1395.98 0.105623
\(560\) 0 0
\(561\) −14.4866 −0.00109024
\(562\) 0 0
\(563\) 16030.3 1.20000 0.599998 0.800001i \(-0.295168\pi\)
0.599998 + 0.800001i \(0.295168\pi\)
\(564\) 0 0
\(565\) −6570.95 −0.489278
\(566\) 0 0
\(567\) 6843.97 0.506913
\(568\) 0 0
\(569\) 2928.95 0.215796 0.107898 0.994162i \(-0.465588\pi\)
0.107898 + 0.994162i \(0.465588\pi\)
\(570\) 0 0
\(571\) 3742.27 0.274272 0.137136 0.990552i \(-0.456210\pi\)
0.137136 + 0.990552i \(0.456210\pi\)
\(572\) 0 0
\(573\) −115.798 −0.00844246
\(574\) 0 0
\(575\) 17176.5 1.24576
\(576\) 0 0
\(577\) −8924.35 −0.643892 −0.321946 0.946758i \(-0.604337\pi\)
−0.321946 + 0.946758i \(0.604337\pi\)
\(578\) 0 0
\(579\) 179.423 0.0128784
\(580\) 0 0
\(581\) −13497.3 −0.963792
\(582\) 0 0
\(583\) −11.4579 −0.000813957 0
\(584\) 0 0
\(585\) −1749.68 −0.123659
\(586\) 0 0
\(587\) 310.349 0.0218219 0.0109110 0.999940i \(-0.496527\pi\)
0.0109110 + 0.999940i \(0.496527\pi\)
\(588\) 0 0
\(589\) 28367.3 1.98447
\(590\) 0 0
\(591\) 173.040 0.0120439
\(592\) 0 0
\(593\) 27310.8 1.89127 0.945634 0.325233i \(-0.105443\pi\)
0.945634 + 0.325233i \(0.105443\pi\)
\(594\) 0 0
\(595\) −2282.43 −0.157262
\(596\) 0 0
\(597\) 191.876 0.0131541
\(598\) 0 0
\(599\) 3123.84 0.213083 0.106542 0.994308i \(-0.466022\pi\)
0.106542 + 0.994308i \(0.466022\pi\)
\(600\) 0 0
\(601\) 3130.50 0.212472 0.106236 0.994341i \(-0.466120\pi\)
0.106236 + 0.994341i \(0.466120\pi\)
\(602\) 0 0
\(603\) −13389.2 −0.904232
\(604\) 0 0
\(605\) 7195.51 0.483535
\(606\) 0 0
\(607\) −9839.02 −0.657914 −0.328957 0.944345i \(-0.606697\pi\)
−0.328957 + 0.944345i \(0.606697\pi\)
\(608\) 0 0
\(609\) 12.9270 0.000860142 0
\(610\) 0 0
\(611\) −1330.38 −0.0880876
\(612\) 0 0
\(613\) −1326.74 −0.0874172 −0.0437086 0.999044i \(-0.513917\pi\)
−0.0437086 + 0.999044i \(0.513917\pi\)
\(614\) 0 0
\(615\) 123.077 0.00806983
\(616\) 0 0
\(617\) −18555.3 −1.21071 −0.605357 0.795954i \(-0.706970\pi\)
−0.605357 + 0.795954i \(0.706970\pi\)
\(618\) 0 0
\(619\) −8583.21 −0.557332 −0.278666 0.960388i \(-0.589892\pi\)
−0.278666 + 0.960388i \(0.589892\pi\)
\(620\) 0 0
\(621\) 471.069 0.0304402
\(622\) 0 0
\(623\) 1607.56 0.103380
\(624\) 0 0
\(625\) 4792.89 0.306745
\(626\) 0 0
\(627\) −31.6814 −0.00201792
\(628\) 0 0
\(629\) −5053.85 −0.320366
\(630\) 0 0
\(631\) 13298.9 0.839017 0.419508 0.907751i \(-0.362202\pi\)
0.419508 + 0.907751i \(0.362202\pi\)
\(632\) 0 0
\(633\) 211.436 0.0132762
\(634\) 0 0
\(635\) 14165.2 0.885241
\(636\) 0 0
\(637\) 2940.66 0.182910
\(638\) 0 0
\(639\) 10661.2 0.660018
\(640\) 0 0
\(641\) −25074.9 −1.54508 −0.772542 0.634964i \(-0.781015\pi\)
−0.772542 + 0.634964i \(0.781015\pi\)
\(642\) 0 0
\(643\) −2108.13 −0.129295 −0.0646475 0.997908i \(-0.520592\pi\)
−0.0646475 + 0.997908i \(0.520592\pi\)
\(644\) 0 0
\(645\) 32.2470 0.00196857
\(646\) 0 0
\(647\) −19874.2 −1.20763 −0.603813 0.797126i \(-0.706353\pi\)
−0.603813 + 0.797126i \(0.706353\pi\)
\(648\) 0 0
\(649\) −1179.10 −0.0713154
\(650\) 0 0
\(651\) 133.593 0.00804292
\(652\) 0 0
\(653\) 22531.8 1.35029 0.675143 0.737687i \(-0.264082\pi\)
0.675143 + 0.737687i \(0.264082\pi\)
\(654\) 0 0
\(655\) 15265.9 0.910669
\(656\) 0 0
\(657\) −963.918 −0.0572390
\(658\) 0 0
\(659\) −26356.5 −1.55797 −0.778984 0.627043i \(-0.784265\pi\)
−0.778984 + 0.627043i \(0.784265\pi\)
\(660\) 0 0
\(661\) −22193.4 −1.30593 −0.652966 0.757387i \(-0.726476\pi\)
−0.652966 + 0.757387i \(0.726476\pi\)
\(662\) 0 0
\(663\) −23.7091 −0.00138882
\(664\) 0 0
\(665\) −4991.57 −0.291075
\(666\) 0 0
\(667\) −5329.64 −0.309392
\(668\) 0 0
\(669\) −40.0448 −0.00231423
\(670\) 0 0
\(671\) −5313.90 −0.305724
\(672\) 0 0
\(673\) −1255.70 −0.0719223 −0.0359611 0.999353i \(-0.511449\pi\)
−0.0359611 + 0.999353i \(0.511449\pi\)
\(674\) 0 0
\(675\) 239.563 0.0136604
\(676\) 0 0
\(677\) 5002.40 0.283985 0.141992 0.989868i \(-0.454649\pi\)
0.141992 + 0.989868i \(0.454649\pi\)
\(678\) 0 0
\(679\) −7043.78 −0.398108
\(680\) 0 0
\(681\) 6.10768 0.000343681 0
\(682\) 0 0
\(683\) 2701.21 0.151331 0.0756653 0.997133i \(-0.475892\pi\)
0.0756653 + 0.997133i \(0.475892\pi\)
\(684\) 0 0
\(685\) 1530.43 0.0853645
\(686\) 0 0
\(687\) 89.0301 0.00494426
\(688\) 0 0
\(689\) −18.7523 −0.00103687
\(690\) 0 0
\(691\) −11622.9 −0.639879 −0.319939 0.947438i \(-0.603663\pi\)
−0.319939 + 0.947438i \(0.603663\pi\)
\(692\) 0 0
\(693\) 1787.65 0.0979900
\(694\) 0 0
\(695\) −13219.7 −0.721515
\(696\) 0 0
\(697\) −19982.1 −1.08591
\(698\) 0 0
\(699\) −34.7521 −0.00188046
\(700\) 0 0
\(701\) 222.521 0.0119893 0.00599465 0.999982i \(-0.498092\pi\)
0.00599465 + 0.999982i \(0.498092\pi\)
\(702\) 0 0
\(703\) −11052.5 −0.592964
\(704\) 0 0
\(705\) −30.7318 −0.00164174
\(706\) 0 0
\(707\) 14907.8 0.793019
\(708\) 0 0
\(709\) −4632.13 −0.245364 −0.122682 0.992446i \(-0.539150\pi\)
−0.122682 + 0.992446i \(0.539150\pi\)
\(710\) 0 0
\(711\) −17797.0 −0.938733
\(712\) 0 0
\(713\) −55079.1 −2.89303
\(714\) 0 0
\(715\) −456.980 −0.0239022
\(716\) 0 0
\(717\) 184.758 0.00962333
\(718\) 0 0
\(719\) −29009.9 −1.50471 −0.752356 0.658757i \(-0.771083\pi\)
−0.752356 + 0.658757i \(0.771083\pi\)
\(720\) 0 0
\(721\) 7180.81 0.370912
\(722\) 0 0
\(723\) 185.027 0.00951762
\(724\) 0 0
\(725\) −2710.40 −0.138844
\(726\) 0 0
\(727\) 19106.3 0.974708 0.487354 0.873204i \(-0.337962\pi\)
0.487354 + 0.873204i \(0.337962\pi\)
\(728\) 0 0
\(729\) −19673.1 −0.999499
\(730\) 0 0
\(731\) −5235.46 −0.264898
\(732\) 0 0
\(733\) 5209.20 0.262492 0.131246 0.991350i \(-0.458102\pi\)
0.131246 + 0.991350i \(0.458102\pi\)
\(734\) 0 0
\(735\) 67.9292 0.00340899
\(736\) 0 0
\(737\) −3496.98 −0.174780
\(738\) 0 0
\(739\) 318.081 0.0158333 0.00791664 0.999969i \(-0.497480\pi\)
0.00791664 + 0.999969i \(0.497480\pi\)
\(740\) 0 0
\(741\) −51.8507 −0.00257056
\(742\) 0 0
\(743\) 23407.1 1.15575 0.577876 0.816124i \(-0.303882\pi\)
0.577876 + 0.816124i \(0.303882\pi\)
\(744\) 0 0
\(745\) −13697.1 −0.673588
\(746\) 0 0
\(747\) 38804.8 1.90066
\(748\) 0 0
\(749\) 12829.8 0.625891
\(750\) 0 0
\(751\) 6268.91 0.304601 0.152301 0.988334i \(-0.451332\pi\)
0.152301 + 0.988334i \(0.451332\pi\)
\(752\) 0 0
\(753\) 272.779 0.0132014
\(754\) 0 0
\(755\) −14879.2 −0.717231
\(756\) 0 0
\(757\) −31403.2 −1.50775 −0.753877 0.657016i \(-0.771818\pi\)
−0.753877 + 0.657016i \(0.771818\pi\)
\(758\) 0 0
\(759\) 61.5140 0.00294179
\(760\) 0 0
\(761\) −31552.8 −1.50301 −0.751504 0.659729i \(-0.770671\pi\)
−0.751504 + 0.659729i \(0.770671\pi\)
\(762\) 0 0
\(763\) 15513.3 0.736067
\(764\) 0 0
\(765\) 6562.00 0.310130
\(766\) 0 0
\(767\) −1929.75 −0.0908463
\(768\) 0 0
\(769\) −3893.63 −0.182585 −0.0912924 0.995824i \(-0.529100\pi\)
−0.0912924 + 0.995824i \(0.529100\pi\)
\(770\) 0 0
\(771\) 236.541 0.0110490
\(772\) 0 0
\(773\) −11899.3 −0.553673 −0.276837 0.960917i \(-0.589286\pi\)
−0.276837 + 0.960917i \(0.589286\pi\)
\(774\) 0 0
\(775\) −28010.6 −1.29828
\(776\) 0 0
\(777\) −52.0510 −0.00240324
\(778\) 0 0
\(779\) −43700.0 −2.00990
\(780\) 0 0
\(781\) 2784.48 0.127576
\(782\) 0 0
\(783\) −74.3332 −0.00339266
\(784\) 0 0
\(785\) −8447.18 −0.384067
\(786\) 0 0
\(787\) −17713.4 −0.802308 −0.401154 0.916011i \(-0.631391\pi\)
−0.401154 + 0.916011i \(0.631391\pi\)
\(788\) 0 0
\(789\) 25.1643 0.00113545
\(790\) 0 0
\(791\) 10987.5 0.493896
\(792\) 0 0
\(793\) −8696.88 −0.389452
\(794\) 0 0
\(795\) −0.433177 −1.93248e−5 0
\(796\) 0 0
\(797\) 28725.3 1.27667 0.638333 0.769761i \(-0.279624\pi\)
0.638333 + 0.769761i \(0.279624\pi\)
\(798\) 0 0
\(799\) 4989.46 0.220919
\(800\) 0 0
\(801\) −4621.74 −0.203872
\(802\) 0 0
\(803\) −251.755 −0.0110638
\(804\) 0 0
\(805\) 9691.85 0.424339
\(806\) 0 0
\(807\) −104.676 −0.00456602
\(808\) 0 0
\(809\) −34175.7 −1.48524 −0.742618 0.669716i \(-0.766416\pi\)
−0.742618 + 0.669716i \(0.766416\pi\)
\(810\) 0 0
\(811\) −24008.9 −1.03954 −0.519770 0.854306i \(-0.673982\pi\)
−0.519770 + 0.854306i \(0.673982\pi\)
\(812\) 0 0
\(813\) 15.4097 0.000664752 0
\(814\) 0 0
\(815\) 11322.1 0.486621
\(816\) 0 0
\(817\) −11449.7 −0.490299
\(818\) 0 0
\(819\) 2925.71 0.124826
\(820\) 0 0
\(821\) −25997.4 −1.10513 −0.552567 0.833468i \(-0.686352\pi\)
−0.552567 + 0.833468i \(0.686352\pi\)
\(822\) 0 0
\(823\) 1210.94 0.0512889 0.0256444 0.999671i \(-0.491836\pi\)
0.0256444 + 0.999671i \(0.491836\pi\)
\(824\) 0 0
\(825\) 31.2830 0.00132016
\(826\) 0 0
\(827\) 36601.6 1.53901 0.769506 0.638639i \(-0.220502\pi\)
0.769506 + 0.638639i \(0.220502\pi\)
\(828\) 0 0
\(829\) −32943.6 −1.38019 −0.690096 0.723718i \(-0.742432\pi\)
−0.690096 + 0.723718i \(0.742432\pi\)
\(830\) 0 0
\(831\) 407.822 0.0170243
\(832\) 0 0
\(833\) −11028.6 −0.458727
\(834\) 0 0
\(835\) 19923.3 0.825718
\(836\) 0 0
\(837\) −768.195 −0.0317237
\(838\) 0 0
\(839\) 16403.5 0.674986 0.337493 0.941328i \(-0.390421\pi\)
0.337493 + 0.941328i \(0.390421\pi\)
\(840\) 0 0
\(841\) 841.000 0.0344828
\(842\) 0 0
\(843\) 245.736 0.0100399
\(844\) 0 0
\(845\) 11590.2 0.471851
\(846\) 0 0
\(847\) −12031.9 −0.488099
\(848\) 0 0
\(849\) −395.498 −0.0159876
\(850\) 0 0
\(851\) 21460.1 0.864443
\(852\) 0 0
\(853\) −4307.51 −0.172903 −0.0864515 0.996256i \(-0.527553\pi\)
−0.0864515 + 0.996256i \(0.527553\pi\)
\(854\) 0 0
\(855\) 14350.8 0.574019
\(856\) 0 0
\(857\) −32132.7 −1.28079 −0.640393 0.768048i \(-0.721228\pi\)
−0.640393 + 0.768048i \(0.721228\pi\)
\(858\) 0 0
\(859\) −14761.1 −0.586313 −0.293157 0.956064i \(-0.594706\pi\)
−0.293157 + 0.956064i \(0.594706\pi\)
\(860\) 0 0
\(861\) −205.802 −0.00814600
\(862\) 0 0
\(863\) 7303.62 0.288086 0.144043 0.989571i \(-0.453990\pi\)
0.144043 + 0.989571i \(0.453990\pi\)
\(864\) 0 0
\(865\) −10761.8 −0.423020
\(866\) 0 0
\(867\) −144.296 −0.00565231
\(868\) 0 0
\(869\) −4648.19 −0.181449
\(870\) 0 0
\(871\) −5723.26 −0.222647
\(872\) 0 0
\(873\) 20250.9 0.785096
\(874\) 0 0
\(875\) 11520.8 0.445113
\(876\) 0 0
\(877\) 4634.99 0.178464 0.0892318 0.996011i \(-0.471559\pi\)
0.0892318 + 0.996011i \(0.471559\pi\)
\(878\) 0 0
\(879\) −243.851 −0.00935710
\(880\) 0 0
\(881\) −37912.2 −1.44982 −0.724911 0.688842i \(-0.758119\pi\)
−0.724911 + 0.688842i \(0.758119\pi\)
\(882\) 0 0
\(883\) 15949.9 0.607877 0.303939 0.952692i \(-0.401698\pi\)
0.303939 + 0.952692i \(0.401698\pi\)
\(884\) 0 0
\(885\) −44.5771 −0.00169316
\(886\) 0 0
\(887\) −2511.51 −0.0950712 −0.0475356 0.998870i \(-0.515137\pi\)
−0.0475356 + 0.998870i \(0.515137\pi\)
\(888\) 0 0
\(889\) −23686.1 −0.893597
\(890\) 0 0
\(891\) −5139.06 −0.193227
\(892\) 0 0
\(893\) 10911.7 0.408898
\(894\) 0 0
\(895\) −15719.3 −0.587082
\(896\) 0 0
\(897\) 100.676 0.00374744
\(898\) 0 0
\(899\) 8691.30 0.322437
\(900\) 0 0
\(901\) 70.3284 0.00260042
\(902\) 0 0
\(903\) −53.9214 −0.00198715
\(904\) 0 0
\(905\) 7564.87 0.277862
\(906\) 0 0
\(907\) 17661.5 0.646570 0.323285 0.946302i \(-0.395213\pi\)
0.323285 + 0.946302i \(0.395213\pi\)
\(908\) 0 0
\(909\) −42859.9 −1.56389
\(910\) 0 0
\(911\) −14133.2 −0.514002 −0.257001 0.966411i \(-0.582734\pi\)
−0.257001 + 0.966411i \(0.582734\pi\)
\(912\) 0 0
\(913\) 10135.0 0.367381
\(914\) 0 0
\(915\) −200.898 −0.00725844
\(916\) 0 0
\(917\) −25526.7 −0.919265
\(918\) 0 0
\(919\) −52721.7 −1.89241 −0.946207 0.323561i \(-0.895120\pi\)
−0.946207 + 0.323561i \(0.895120\pi\)
\(920\) 0 0
\(921\) −386.563 −0.0138303
\(922\) 0 0
\(923\) 4557.17 0.162515
\(924\) 0 0
\(925\) 10913.5 0.387930
\(926\) 0 0
\(927\) −20644.8 −0.731462
\(928\) 0 0
\(929\) −12210.5 −0.431232 −0.215616 0.976478i \(-0.569176\pi\)
−0.215616 + 0.976478i \(0.569176\pi\)
\(930\) 0 0
\(931\) −24119.1 −0.849057
\(932\) 0 0
\(933\) 195.004 0.00684259
\(934\) 0 0
\(935\) 1713.85 0.0599454
\(936\) 0 0
\(937\) 31399.0 1.09473 0.547365 0.836894i \(-0.315631\pi\)
0.547365 + 0.836894i \(0.315631\pi\)
\(938\) 0 0
\(939\) 170.558 0.00592752
\(940\) 0 0
\(941\) 1297.72 0.0449568 0.0224784 0.999747i \(-0.492844\pi\)
0.0224784 + 0.999747i \(0.492844\pi\)
\(942\) 0 0
\(943\) 84849.8 2.93010
\(944\) 0 0
\(945\) 135.173 0.00465311
\(946\) 0 0
\(947\) −37731.8 −1.29474 −0.647370 0.762176i \(-0.724131\pi\)
−0.647370 + 0.762176i \(0.724131\pi\)
\(948\) 0 0
\(949\) −412.029 −0.0140938
\(950\) 0 0
\(951\) −38.9021 −0.00132649
\(952\) 0 0
\(953\) 1230.12 0.0418126 0.0209063 0.999781i \(-0.493345\pi\)
0.0209063 + 0.999781i \(0.493345\pi\)
\(954\) 0 0
\(955\) 13699.6 0.464199
\(956\) 0 0
\(957\) −9.70670 −0.000327872 0
\(958\) 0 0
\(959\) −2559.09 −0.0861703
\(960\) 0 0
\(961\) 60029.2 2.01501
\(962\) 0 0
\(963\) −36885.8 −1.23430
\(964\) 0 0
\(965\) −21226.9 −0.708103
\(966\) 0 0
\(967\) 45125.5 1.50066 0.750329 0.661064i \(-0.229895\pi\)
0.750329 + 0.661064i \(0.229895\pi\)
\(968\) 0 0
\(969\) 194.460 0.00644682
\(970\) 0 0
\(971\) 17836.7 0.589502 0.294751 0.955574i \(-0.404763\pi\)
0.294751 + 0.955574i \(0.404763\pi\)
\(972\) 0 0
\(973\) 22105.2 0.728326
\(974\) 0 0
\(975\) 51.1987 0.00168171
\(976\) 0 0
\(977\) −28809.1 −0.943384 −0.471692 0.881763i \(-0.656357\pi\)
−0.471692 + 0.881763i \(0.656357\pi\)
\(978\) 0 0
\(979\) −1207.10 −0.0394066
\(980\) 0 0
\(981\) −44600.8 −1.45157
\(982\) 0 0
\(983\) 47625.6 1.54529 0.772646 0.634837i \(-0.218933\pi\)
0.772646 + 0.634837i \(0.218933\pi\)
\(984\) 0 0
\(985\) −20471.8 −0.662218
\(986\) 0 0
\(987\) 51.3878 0.00165724
\(988\) 0 0
\(989\) 22231.2 0.714774
\(990\) 0 0
\(991\) −24042.4 −0.770670 −0.385335 0.922777i \(-0.625914\pi\)
−0.385335 + 0.922777i \(0.625914\pi\)
\(992\) 0 0
\(993\) −39.2154 −0.00125324
\(994\) 0 0
\(995\) −22700.2 −0.723260
\(996\) 0 0
\(997\) 24316.1 0.772415 0.386208 0.922412i \(-0.373785\pi\)
0.386208 + 0.922412i \(0.373785\pi\)
\(998\) 0 0
\(999\) 299.306 0.00947910
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 1856.4.a.bi.1.6 10
4.3 odd 2 inner 1856.4.a.bi.1.5 10
8.3 odd 2 928.4.a.f.1.6 yes 10
8.5 even 2 928.4.a.f.1.5 10
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
928.4.a.f.1.5 10 8.5 even 2
928.4.a.f.1.6 yes 10 8.3 odd 2
1856.4.a.bi.1.5 10 4.3 odd 2 inner
1856.4.a.bi.1.6 10 1.1 even 1 trivial