Properties

Label 2312.4.a.q.1.18
Level $2312$
Weight $4$
Character 2312.1
Self dual yes
Analytic conductor $136.412$
Analytic rank $0$
Dimension $18$
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] = [2312,4,Mod(1,2312)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2312, 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("2312.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2312 = 2^{3} \cdot 17^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 2312.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: \(136.412415933\)
Analytic rank: \(0\)
Dimension: \(18\)
Coefficient field: \(\mathbb{Q}[x]/(x^{18} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{18} - 294 x^{16} - 14 x^{15} + 34371 x^{14} + 2670 x^{13} - 2054705 x^{12} - 160284 x^{11} + 67981059 x^{10} + 2824200 x^{9} - 1279285428 x^{8} + \cdots - 176969301147 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{15}\cdot 17^{2} \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.18
Root \(-9.04549\) of defining polynomial
Character \(\chi\) \(=\) 2312.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+10.0455 q^{3} -13.8476 q^{5} +18.1085 q^{7} +73.9118 q^{9} +O(q^{10})\) \(q+10.0455 q^{3} -13.8476 q^{5} +18.1085 q^{7} +73.9118 q^{9} -46.2930 q^{11} +65.8265 q^{13} -139.106 q^{15} -19.0550 q^{19} +181.909 q^{21} +194.354 q^{23} +66.7550 q^{25} +471.252 q^{27} -118.661 q^{29} +4.17687 q^{31} -465.036 q^{33} -250.759 q^{35} +270.949 q^{37} +661.259 q^{39} -31.6127 q^{41} +9.51835 q^{43} -1023.50 q^{45} -471.665 q^{47} -15.0824 q^{49} -96.9167 q^{53} +641.045 q^{55} -191.416 q^{57} +854.508 q^{59} -171.489 q^{61} +1338.43 q^{63} -911.536 q^{65} -8.33083 q^{67} +1952.39 q^{69} +51.0425 q^{71} +946.835 q^{73} +670.587 q^{75} -838.297 q^{77} +16.0587 q^{79} +2738.34 q^{81} +1243.38 q^{83} -1192.01 q^{87} -1364.62 q^{89} +1192.02 q^{91} +41.9587 q^{93} +263.865 q^{95} +1166.56 q^{97} -3421.60 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 18 q + 18 q^{3} + 51 q^{7} + 120 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 18 q + 18 q^{3} + 51 q^{7} + 120 q^{9} + 132 q^{11} + 30 q^{13} + 102 q^{15} + 66 q^{19} + 144 q^{21} + 153 q^{23} + 306 q^{25} + 768 q^{27} + 51 q^{29} + 303 q^{31} + 525 q^{33} - 255 q^{35} + 717 q^{37} - 216 q^{39} - 393 q^{41} - 390 q^{43} + 558 q^{45} - 633 q^{47} + 1443 q^{49} + 1275 q^{53} + 1539 q^{55} + 810 q^{57} - 204 q^{59} + 534 q^{61} + 2556 q^{63} - 2127 q^{65} - 405 q^{67} + 2547 q^{69} - 426 q^{71} + 1149 q^{73} + 2226 q^{75} - 357 q^{77} + 1053 q^{79} + 2802 q^{81} + 66 q^{83} + 2487 q^{87} - 4119 q^{89} + 6090 q^{91} + 606 q^{93} + 2109 q^{95} + 2349 q^{97} + 1428 q^{99}+O(q^{100}) \) 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
\(3\) 10.0455 1.93326 0.966628 0.256186i \(-0.0824658\pi\)
0.966628 + 0.256186i \(0.0824658\pi\)
\(4\) 0 0
\(5\) −13.8476 −1.23856 −0.619282 0.785169i \(-0.712576\pi\)
−0.619282 + 0.785169i \(0.712576\pi\)
\(6\) 0 0
\(7\) 18.1085 0.977767 0.488883 0.872349i \(-0.337404\pi\)
0.488883 + 0.872349i \(0.337404\pi\)
\(8\) 0 0
\(9\) 73.9118 2.73748
\(10\) 0 0
\(11\) −46.2930 −1.26890 −0.634448 0.772965i \(-0.718773\pi\)
−0.634448 + 0.772965i \(0.718773\pi\)
\(12\) 0 0
\(13\) 65.8265 1.40438 0.702192 0.711988i \(-0.252205\pi\)
0.702192 + 0.711988i \(0.252205\pi\)
\(14\) 0 0
\(15\) −139.106 −2.39446
\(16\) 0 0
\(17\) 0 0
\(18\) 0 0
\(19\) −19.0550 −0.230079 −0.115040 0.993361i \(-0.536700\pi\)
−0.115040 + 0.993361i \(0.536700\pi\)
\(20\) 0 0
\(21\) 181.909 1.89027
\(22\) 0 0
\(23\) 194.354 1.76199 0.880993 0.473128i \(-0.156875\pi\)
0.880993 + 0.473128i \(0.156875\pi\)
\(24\) 0 0
\(25\) 66.7550 0.534040
\(26\) 0 0
\(27\) 471.252 3.35898
\(28\) 0 0
\(29\) −118.661 −0.759823 −0.379911 0.925023i \(-0.624046\pi\)
−0.379911 + 0.925023i \(0.624046\pi\)
\(30\) 0 0
\(31\) 4.17687 0.0241996 0.0120998 0.999927i \(-0.496148\pi\)
0.0120998 + 0.999927i \(0.496148\pi\)
\(32\) 0 0
\(33\) −465.036 −2.45310
\(34\) 0 0
\(35\) −250.759 −1.21103
\(36\) 0 0
\(37\) 270.949 1.20388 0.601942 0.798540i \(-0.294394\pi\)
0.601942 + 0.798540i \(0.294394\pi\)
\(38\) 0 0
\(39\) 661.259 2.71503
\(40\) 0 0
\(41\) −31.6127 −0.120416 −0.0602082 0.998186i \(-0.519176\pi\)
−0.0602082 + 0.998186i \(0.519176\pi\)
\(42\) 0 0
\(43\) 9.51835 0.0337566 0.0168783 0.999858i \(-0.494627\pi\)
0.0168783 + 0.999858i \(0.494627\pi\)
\(44\) 0 0
\(45\) −1023.50 −3.39054
\(46\) 0 0
\(47\) −471.665 −1.46382 −0.731909 0.681402i \(-0.761371\pi\)
−0.731909 + 0.681402i \(0.761371\pi\)
\(48\) 0 0
\(49\) −15.0824 −0.0439721
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −96.9167 −0.251180 −0.125590 0.992082i \(-0.540082\pi\)
−0.125590 + 0.992082i \(0.540082\pi\)
\(54\) 0 0
\(55\) 641.045 1.57161
\(56\) 0 0
\(57\) −191.416 −0.444802
\(58\) 0 0
\(59\) 854.508 1.88555 0.942775 0.333429i \(-0.108206\pi\)
0.942775 + 0.333429i \(0.108206\pi\)
\(60\) 0 0
\(61\) −171.489 −0.359950 −0.179975 0.983671i \(-0.557602\pi\)
−0.179975 + 0.983671i \(0.557602\pi\)
\(62\) 0 0
\(63\) 1338.43 2.67661
\(64\) 0 0
\(65\) −911.536 −1.73942
\(66\) 0 0
\(67\) −8.33083 −0.0151906 −0.00759532 0.999971i \(-0.502418\pi\)
−0.00759532 + 0.999971i \(0.502418\pi\)
\(68\) 0 0
\(69\) 1952.39 3.40637
\(70\) 0 0
\(71\) 51.0425 0.0853187 0.0426594 0.999090i \(-0.486417\pi\)
0.0426594 + 0.999090i \(0.486417\pi\)
\(72\) 0 0
\(73\) 946.835 1.51806 0.759032 0.651053i \(-0.225673\pi\)
0.759032 + 0.651053i \(0.225673\pi\)
\(74\) 0 0
\(75\) 670.587 1.03244
\(76\) 0 0
\(77\) −838.297 −1.24069
\(78\) 0 0
\(79\) 16.0587 0.0228702 0.0114351 0.999935i \(-0.496360\pi\)
0.0114351 + 0.999935i \(0.496360\pi\)
\(80\) 0 0
\(81\) 2738.34 3.75630
\(82\) 0 0
\(83\) 1243.38 1.64432 0.822161 0.569254i \(-0.192768\pi\)
0.822161 + 0.569254i \(0.192768\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) −1192.01 −1.46893
\(88\) 0 0
\(89\) −1364.62 −1.62527 −0.812635 0.582773i \(-0.801968\pi\)
−0.812635 + 0.582773i \(0.801968\pi\)
\(90\) 0 0
\(91\) 1192.02 1.37316
\(92\) 0 0
\(93\) 41.9587 0.0467840
\(94\) 0 0
\(95\) 263.865 0.284968
\(96\) 0 0
\(97\) 1166.56 1.22110 0.610550 0.791978i \(-0.290949\pi\)
0.610550 + 0.791978i \(0.290949\pi\)
\(98\) 0 0
\(99\) −3421.60 −3.47357
\(100\) 0 0
\(101\) −647.569 −0.637976 −0.318988 0.947759i \(-0.603343\pi\)
−0.318988 + 0.947759i \(0.603343\pi\)
\(102\) 0 0
\(103\) −167.539 −0.160273 −0.0801363 0.996784i \(-0.525536\pi\)
−0.0801363 + 0.996784i \(0.525536\pi\)
\(104\) 0 0
\(105\) −2518.99 −2.34122
\(106\) 0 0
\(107\) −541.412 −0.489161 −0.244581 0.969629i \(-0.578650\pi\)
−0.244581 + 0.969629i \(0.578650\pi\)
\(108\) 0 0
\(109\) 212.961 0.187137 0.0935685 0.995613i \(-0.470173\pi\)
0.0935685 + 0.995613i \(0.470173\pi\)
\(110\) 0 0
\(111\) 2721.81 2.32741
\(112\) 0 0
\(113\) 888.160 0.739390 0.369695 0.929153i \(-0.379462\pi\)
0.369695 + 0.929153i \(0.379462\pi\)
\(114\) 0 0
\(115\) −2691.34 −2.18233
\(116\) 0 0
\(117\) 4865.36 3.84447
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) 812.042 0.610099
\(122\) 0 0
\(123\) −317.565 −0.232796
\(124\) 0 0
\(125\) 806.551 0.577121
\(126\) 0 0
\(127\) 530.718 0.370816 0.185408 0.982662i \(-0.440639\pi\)
0.185408 + 0.982662i \(0.440639\pi\)
\(128\) 0 0
\(129\) 95.6165 0.0652602
\(130\) 0 0
\(131\) 644.014 0.429525 0.214763 0.976666i \(-0.431102\pi\)
0.214763 + 0.976666i \(0.431102\pi\)
\(132\) 0 0
\(133\) −345.057 −0.224964
\(134\) 0 0
\(135\) −6525.70 −4.16032
\(136\) 0 0
\(137\) 2424.52 1.51198 0.755988 0.654585i \(-0.227157\pi\)
0.755988 + 0.654585i \(0.227157\pi\)
\(138\) 0 0
\(139\) −2242.69 −1.36850 −0.684252 0.729246i \(-0.739871\pi\)
−0.684252 + 0.729246i \(0.739871\pi\)
\(140\) 0 0
\(141\) −4738.11 −2.82994
\(142\) 0 0
\(143\) −3047.31 −1.78202
\(144\) 0 0
\(145\) 1643.17 0.941089
\(146\) 0 0
\(147\) −151.510 −0.0850092
\(148\) 0 0
\(149\) −455.171 −0.250262 −0.125131 0.992140i \(-0.539935\pi\)
−0.125131 + 0.992140i \(0.539935\pi\)
\(150\) 0 0
\(151\) −746.455 −0.402289 −0.201145 0.979562i \(-0.564466\pi\)
−0.201145 + 0.979562i \(0.564466\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) −57.8395 −0.0299728
\(156\) 0 0
\(157\) 1565.75 0.795926 0.397963 0.917401i \(-0.369717\pi\)
0.397963 + 0.917401i \(0.369717\pi\)
\(158\) 0 0
\(159\) −973.576 −0.485595
\(160\) 0 0
\(161\) 3519.47 1.72281
\(162\) 0 0
\(163\) 1510.00 0.725596 0.362798 0.931868i \(-0.381822\pi\)
0.362798 + 0.931868i \(0.381822\pi\)
\(164\) 0 0
\(165\) 6439.61 3.03832
\(166\) 0 0
\(167\) −961.994 −0.445756 −0.222878 0.974846i \(-0.571545\pi\)
−0.222878 + 0.974846i \(0.571545\pi\)
\(168\) 0 0
\(169\) 2136.13 0.972292
\(170\) 0 0
\(171\) −1408.39 −0.629837
\(172\) 0 0
\(173\) 756.380 0.332407 0.166204 0.986091i \(-0.446849\pi\)
0.166204 + 0.986091i \(0.446849\pi\)
\(174\) 0 0
\(175\) 1208.83 0.522167
\(176\) 0 0
\(177\) 8583.95 3.64525
\(178\) 0 0
\(179\) 390.689 0.163137 0.0815683 0.996668i \(-0.474007\pi\)
0.0815683 + 0.996668i \(0.474007\pi\)
\(180\) 0 0
\(181\) −1969.73 −0.808888 −0.404444 0.914563i \(-0.632535\pi\)
−0.404444 + 0.914563i \(0.632535\pi\)
\(182\) 0 0
\(183\) −1722.70 −0.695876
\(184\) 0 0
\(185\) −3751.98 −1.49109
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) 8533.67 3.28430
\(190\) 0 0
\(191\) 422.510 0.160061 0.0800307 0.996792i \(-0.474498\pi\)
0.0800307 + 0.996792i \(0.474498\pi\)
\(192\) 0 0
\(193\) 1790.64 0.667839 0.333919 0.942602i \(-0.391629\pi\)
0.333919 + 0.942602i \(0.391629\pi\)
\(194\) 0 0
\(195\) −9156.83 −3.36274
\(196\) 0 0
\(197\) 2580.69 0.933331 0.466666 0.884434i \(-0.345455\pi\)
0.466666 + 0.884434i \(0.345455\pi\)
\(198\) 0 0
\(199\) −1201.62 −0.428043 −0.214022 0.976829i \(-0.568656\pi\)
−0.214022 + 0.976829i \(0.568656\pi\)
\(200\) 0 0
\(201\) −83.6873 −0.0293674
\(202\) 0 0
\(203\) −2148.78 −0.742929
\(204\) 0 0
\(205\) 437.759 0.149143
\(206\) 0 0
\(207\) 14365.1 4.82340
\(208\) 0 0
\(209\) 882.112 0.291947
\(210\) 0 0
\(211\) 5001.53 1.63185 0.815924 0.578160i \(-0.196229\pi\)
0.815924 + 0.578160i \(0.196229\pi\)
\(212\) 0 0
\(213\) 512.747 0.164943
\(214\) 0 0
\(215\) −131.806 −0.0418097
\(216\) 0 0
\(217\) 75.6369 0.0236616
\(218\) 0 0
\(219\) 9511.42 2.93481
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) 5961.70 1.79025 0.895123 0.445819i \(-0.147088\pi\)
0.895123 + 0.445819i \(0.147088\pi\)
\(224\) 0 0
\(225\) 4933.99 1.46192
\(226\) 0 0
\(227\) 5697.72 1.66595 0.832976 0.553309i \(-0.186635\pi\)
0.832976 + 0.553309i \(0.186635\pi\)
\(228\) 0 0
\(229\) 5833.82 1.68345 0.841724 0.539909i \(-0.181541\pi\)
0.841724 + 0.539909i \(0.181541\pi\)
\(230\) 0 0
\(231\) −8421.10 −2.39856
\(232\) 0 0
\(233\) −2251.11 −0.632942 −0.316471 0.948602i \(-0.602498\pi\)
−0.316471 + 0.948602i \(0.602498\pi\)
\(234\) 0 0
\(235\) 6531.42 1.81303
\(236\) 0 0
\(237\) 161.318 0.0442140
\(238\) 0 0
\(239\) −2723.39 −0.737076 −0.368538 0.929613i \(-0.620142\pi\)
−0.368538 + 0.929613i \(0.620142\pi\)
\(240\) 0 0
\(241\) 2762.12 0.738272 0.369136 0.929375i \(-0.379654\pi\)
0.369136 + 0.929375i \(0.379654\pi\)
\(242\) 0 0
\(243\) 14784.2 3.90290
\(244\) 0 0
\(245\) 208.855 0.0544622
\(246\) 0 0
\(247\) −1254.32 −0.323120
\(248\) 0 0
\(249\) 12490.4 3.17890
\(250\) 0 0
\(251\) −2857.56 −0.718596 −0.359298 0.933223i \(-0.616984\pi\)
−0.359298 + 0.933223i \(0.616984\pi\)
\(252\) 0 0
\(253\) −8997.25 −2.23578
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −5620.28 −1.36414 −0.682069 0.731288i \(-0.738920\pi\)
−0.682069 + 0.731288i \(0.738920\pi\)
\(258\) 0 0
\(259\) 4906.47 1.17712
\(260\) 0 0
\(261\) −8770.48 −2.08000
\(262\) 0 0
\(263\) −3064.10 −0.718404 −0.359202 0.933260i \(-0.616951\pi\)
−0.359202 + 0.933260i \(0.616951\pi\)
\(264\) 0 0
\(265\) 1342.06 0.311102
\(266\) 0 0
\(267\) −13708.2 −3.14206
\(268\) 0 0
\(269\) −1166.34 −0.264360 −0.132180 0.991226i \(-0.542198\pi\)
−0.132180 + 0.991226i \(0.542198\pi\)
\(270\) 0 0
\(271\) −8580.83 −1.92343 −0.961713 0.274058i \(-0.911634\pi\)
−0.961713 + 0.274058i \(0.911634\pi\)
\(272\) 0 0
\(273\) 11974.4 2.65467
\(274\) 0 0
\(275\) −3090.29 −0.677642
\(276\) 0 0
\(277\) −6463.59 −1.40202 −0.701010 0.713152i \(-0.747267\pi\)
−0.701010 + 0.713152i \(0.747267\pi\)
\(278\) 0 0
\(279\) 308.720 0.0662459
\(280\) 0 0
\(281\) −5531.65 −1.17434 −0.587172 0.809462i \(-0.699759\pi\)
−0.587172 + 0.809462i \(0.699759\pi\)
\(282\) 0 0
\(283\) 1092.62 0.229504 0.114752 0.993394i \(-0.463393\pi\)
0.114752 + 0.993394i \(0.463393\pi\)
\(284\) 0 0
\(285\) 2650.65 0.550916
\(286\) 0 0
\(287\) −572.458 −0.117739
\(288\) 0 0
\(289\) 0 0
\(290\) 0 0
\(291\) 11718.7 2.36070
\(292\) 0 0
\(293\) −3169.97 −0.632052 −0.316026 0.948750i \(-0.602349\pi\)
−0.316026 + 0.948750i \(0.602349\pi\)
\(294\) 0 0
\(295\) −11832.9 −2.33537
\(296\) 0 0
\(297\) −21815.7 −4.26220
\(298\) 0 0
\(299\) 12793.7 2.47450
\(300\) 0 0
\(301\) 172.363 0.0330061
\(302\) 0 0
\(303\) −6505.15 −1.23337
\(304\) 0 0
\(305\) 2374.71 0.445821
\(306\) 0 0
\(307\) −2627.50 −0.488467 −0.244234 0.969716i \(-0.578536\pi\)
−0.244234 + 0.969716i \(0.578536\pi\)
\(308\) 0 0
\(309\) −1683.01 −0.309848
\(310\) 0 0
\(311\) −3985.35 −0.726650 −0.363325 0.931662i \(-0.618359\pi\)
−0.363325 + 0.931662i \(0.618359\pi\)
\(312\) 0 0
\(313\) 3505.16 0.632983 0.316491 0.948595i \(-0.397495\pi\)
0.316491 + 0.948595i \(0.397495\pi\)
\(314\) 0 0
\(315\) −18534.0 −3.31516
\(316\) 0 0
\(317\) 1649.79 0.292308 0.146154 0.989262i \(-0.453311\pi\)
0.146154 + 0.989262i \(0.453311\pi\)
\(318\) 0 0
\(319\) 5493.19 0.964137
\(320\) 0 0
\(321\) −5438.75 −0.945674
\(322\) 0 0
\(323\) 0 0
\(324\) 0 0
\(325\) 4394.25 0.749997
\(326\) 0 0
\(327\) 2139.29 0.361783
\(328\) 0 0
\(329\) −8541.15 −1.43127
\(330\) 0 0
\(331\) 4301.09 0.714228 0.357114 0.934061i \(-0.383761\pi\)
0.357114 + 0.934061i \(0.383761\pi\)
\(332\) 0 0
\(333\) 20026.3 3.29560
\(334\) 0 0
\(335\) 115.362 0.0188146
\(336\) 0 0
\(337\) 6835.65 1.10493 0.552465 0.833536i \(-0.313687\pi\)
0.552465 + 0.833536i \(0.313687\pi\)
\(338\) 0 0
\(339\) 8922.00 1.42943
\(340\) 0 0
\(341\) −193.360 −0.0307068
\(342\) 0 0
\(343\) −6484.33 −1.02076
\(344\) 0 0
\(345\) −27035.8 −4.21901
\(346\) 0 0
\(347\) −5608.80 −0.867712 −0.433856 0.900982i \(-0.642847\pi\)
−0.433856 + 0.900982i \(0.642847\pi\)
\(348\) 0 0
\(349\) 8300.72 1.27314 0.636572 0.771217i \(-0.280352\pi\)
0.636572 + 0.771217i \(0.280352\pi\)
\(350\) 0 0
\(351\) 31020.9 4.71730
\(352\) 0 0
\(353\) −1181.71 −0.178176 −0.0890880 0.996024i \(-0.528395\pi\)
−0.0890880 + 0.996024i \(0.528395\pi\)
\(354\) 0 0
\(355\) −706.814 −0.105673
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −10863.5 −1.59709 −0.798546 0.601934i \(-0.794397\pi\)
−0.798546 + 0.601934i \(0.794397\pi\)
\(360\) 0 0
\(361\) −6495.91 −0.947063
\(362\) 0 0
\(363\) 8157.36 1.17948
\(364\) 0 0
\(365\) −13111.4 −1.88022
\(366\) 0 0
\(367\) 3724.19 0.529704 0.264852 0.964289i \(-0.414677\pi\)
0.264852 + 0.964289i \(0.414677\pi\)
\(368\) 0 0
\(369\) −2336.55 −0.329637
\(370\) 0 0
\(371\) −1755.02 −0.245595
\(372\) 0 0
\(373\) 9169.41 1.27285 0.636426 0.771338i \(-0.280412\pi\)
0.636426 + 0.771338i \(0.280412\pi\)
\(374\) 0 0
\(375\) 8102.20 1.11572
\(376\) 0 0
\(377\) −7811.06 −1.06708
\(378\) 0 0
\(379\) 5969.10 0.809004 0.404502 0.914537i \(-0.367445\pi\)
0.404502 + 0.914537i \(0.367445\pi\)
\(380\) 0 0
\(381\) 5331.32 0.716881
\(382\) 0 0
\(383\) −3025.44 −0.403637 −0.201818 0.979423i \(-0.564685\pi\)
−0.201818 + 0.979423i \(0.564685\pi\)
\(384\) 0 0
\(385\) 11608.4 1.53667
\(386\) 0 0
\(387\) 703.519 0.0924080
\(388\) 0 0
\(389\) 11596.8 1.51152 0.755758 0.654851i \(-0.227269\pi\)
0.755758 + 0.654851i \(0.227269\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0 0
\(393\) 6469.44 0.830382
\(394\) 0 0
\(395\) −222.374 −0.0283263
\(396\) 0 0
\(397\) −1680.42 −0.212438 −0.106219 0.994343i \(-0.533874\pi\)
−0.106219 + 0.994343i \(0.533874\pi\)
\(398\) 0 0
\(399\) −3466.26 −0.434913
\(400\) 0 0
\(401\) −8962.57 −1.11613 −0.558066 0.829796i \(-0.688457\pi\)
−0.558066 + 0.829796i \(0.688457\pi\)
\(402\) 0 0
\(403\) 274.949 0.0339855
\(404\) 0 0
\(405\) −37919.4 −4.65241
\(406\) 0 0
\(407\) −12543.0 −1.52760
\(408\) 0 0
\(409\) −7158.94 −0.865494 −0.432747 0.901516i \(-0.642456\pi\)
−0.432747 + 0.901516i \(0.642456\pi\)
\(410\) 0 0
\(411\) 24355.5 2.92304
\(412\) 0 0
\(413\) 15473.9 1.84363
\(414\) 0 0
\(415\) −17217.8 −2.03660
\(416\) 0 0
\(417\) −22528.9 −2.64567
\(418\) 0 0
\(419\) 14813.8 1.72722 0.863608 0.504164i \(-0.168199\pi\)
0.863608 + 0.504164i \(0.168199\pi\)
\(420\) 0 0
\(421\) −7260.01 −0.840454 −0.420227 0.907419i \(-0.638050\pi\)
−0.420227 + 0.907419i \(0.638050\pi\)
\(422\) 0 0
\(423\) −34861.7 −4.00717
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) −3105.42 −0.351947
\(428\) 0 0
\(429\) −30611.7 −3.44510
\(430\) 0 0
\(431\) −2370.15 −0.264887 −0.132443 0.991191i \(-0.542282\pi\)
−0.132443 + 0.991191i \(0.542282\pi\)
\(432\) 0 0
\(433\) −3056.05 −0.339178 −0.169589 0.985515i \(-0.554244\pi\)
−0.169589 + 0.985515i \(0.554244\pi\)
\(434\) 0 0
\(435\) 16506.5 1.81936
\(436\) 0 0
\(437\) −3703.42 −0.405397
\(438\) 0 0
\(439\) 2584.92 0.281029 0.140514 0.990079i \(-0.455124\pi\)
0.140514 + 0.990079i \(0.455124\pi\)
\(440\) 0 0
\(441\) −1114.77 −0.120372
\(442\) 0 0
\(443\) −9354.57 −1.00327 −0.501635 0.865079i \(-0.667268\pi\)
−0.501635 + 0.865079i \(0.667268\pi\)
\(444\) 0 0
\(445\) 18896.6 2.01300
\(446\) 0 0
\(447\) −4572.42 −0.483821
\(448\) 0 0
\(449\) 2103.48 0.221090 0.110545 0.993871i \(-0.464740\pi\)
0.110545 + 0.993871i \(0.464740\pi\)
\(450\) 0 0
\(451\) 1463.45 0.152796
\(452\) 0 0
\(453\) −7498.51 −0.777728
\(454\) 0 0
\(455\) −16506.6 −1.70075
\(456\) 0 0
\(457\) 8869.30 0.907852 0.453926 0.891039i \(-0.350023\pi\)
0.453926 + 0.891039i \(0.350023\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) −16376.3 −1.65449 −0.827244 0.561843i \(-0.810092\pi\)
−0.827244 + 0.561843i \(0.810092\pi\)
\(462\) 0 0
\(463\) −6611.44 −0.663628 −0.331814 0.943345i \(-0.607661\pi\)
−0.331814 + 0.943345i \(0.607661\pi\)
\(464\) 0 0
\(465\) −581.026 −0.0579450
\(466\) 0 0
\(467\) 1589.77 0.157528 0.0787640 0.996893i \(-0.474903\pi\)
0.0787640 + 0.996893i \(0.474903\pi\)
\(468\) 0 0
\(469\) −150.859 −0.0148529
\(470\) 0 0
\(471\) 15728.7 1.53873
\(472\) 0 0
\(473\) −440.633 −0.0428337
\(474\) 0 0
\(475\) −1272.01 −0.122872
\(476\) 0 0
\(477\) −7163.29 −0.687599
\(478\) 0 0
\(479\) 317.589 0.0302944 0.0151472 0.999885i \(-0.495178\pi\)
0.0151472 + 0.999885i \(0.495178\pi\)
\(480\) 0 0
\(481\) 17835.6 1.69071
\(482\) 0 0
\(483\) 35354.8 3.33064
\(484\) 0 0
\(485\) −16154.1 −1.51241
\(486\) 0 0
\(487\) −12005.6 −1.11710 −0.558549 0.829472i \(-0.688642\pi\)
−0.558549 + 0.829472i \(0.688642\pi\)
\(488\) 0 0
\(489\) 15168.7 1.40276
\(490\) 0 0
\(491\) −12537.5 −1.15236 −0.576179 0.817324i \(-0.695457\pi\)
−0.576179 + 0.817324i \(0.695457\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) 47380.8 4.30224
\(496\) 0 0
\(497\) 924.303 0.0834218
\(498\) 0 0
\(499\) −7088.85 −0.635953 −0.317977 0.948099i \(-0.603003\pi\)
−0.317977 + 0.948099i \(0.603003\pi\)
\(500\) 0 0
\(501\) −9663.70 −0.861761
\(502\) 0 0
\(503\) −13957.4 −1.23724 −0.618618 0.785692i \(-0.712307\pi\)
−0.618618 + 0.785692i \(0.712307\pi\)
\(504\) 0 0
\(505\) 8967.25 0.790173
\(506\) 0 0
\(507\) 21458.4 1.87969
\(508\) 0 0
\(509\) −21328.4 −1.85729 −0.928647 0.370964i \(-0.879027\pi\)
−0.928647 + 0.370964i \(0.879027\pi\)
\(510\) 0 0
\(511\) 17145.8 1.48431
\(512\) 0 0
\(513\) −8979.70 −0.772833
\(514\) 0 0
\(515\) 2320.00 0.198508
\(516\) 0 0
\(517\) 21834.8 1.85744
\(518\) 0 0
\(519\) 7598.20 0.642628
\(520\) 0 0
\(521\) −2588.63 −0.217677 −0.108839 0.994059i \(-0.534713\pi\)
−0.108839 + 0.994059i \(0.534713\pi\)
\(522\) 0 0
\(523\) 20409.3 1.70638 0.853191 0.521599i \(-0.174664\pi\)
0.853191 + 0.521599i \(0.174664\pi\)
\(524\) 0 0
\(525\) 12143.3 1.00948
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) 25606.6 2.10460
\(530\) 0 0
\(531\) 63158.3 5.16165
\(532\) 0 0
\(533\) −2080.95 −0.169111
\(534\) 0 0
\(535\) 7497.23 0.605857
\(536\) 0 0
\(537\) 3924.66 0.315385
\(538\) 0 0
\(539\) 698.211 0.0557960
\(540\) 0 0
\(541\) 4762.23 0.378455 0.189227 0.981933i \(-0.439402\pi\)
0.189227 + 0.981933i \(0.439402\pi\)
\(542\) 0 0
\(543\) −19786.9 −1.56379
\(544\) 0 0
\(545\) −2948.98 −0.231781
\(546\) 0 0
\(547\) −16094.3 −1.25803 −0.629013 0.777395i \(-0.716541\pi\)
−0.629013 + 0.777395i \(0.716541\pi\)
\(548\) 0 0
\(549\) −12675.1 −0.985355
\(550\) 0 0
\(551\) 2261.09 0.174820
\(552\) 0 0
\(553\) 290.800 0.0223618
\(554\) 0 0
\(555\) −37690.5 −2.88265
\(556\) 0 0
\(557\) −6505.51 −0.494879 −0.247439 0.968903i \(-0.579589\pi\)
−0.247439 + 0.968903i \(0.579589\pi\)
\(558\) 0 0
\(559\) 626.560 0.0474072
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) −4057.45 −0.303732 −0.151866 0.988401i \(-0.548528\pi\)
−0.151866 + 0.988401i \(0.548528\pi\)
\(564\) 0 0
\(565\) −12298.9 −0.915782
\(566\) 0 0
\(567\) 49587.2 3.67278
\(568\) 0 0
\(569\) 14345.2 1.05691 0.528456 0.848961i \(-0.322771\pi\)
0.528456 + 0.848961i \(0.322771\pi\)
\(570\) 0 0
\(571\) 760.937 0.0557692 0.0278846 0.999611i \(-0.491123\pi\)
0.0278846 + 0.999611i \(0.491123\pi\)
\(572\) 0 0
\(573\) 4244.32 0.309440
\(574\) 0 0
\(575\) 12974.1 0.940972
\(576\) 0 0
\(577\) −3549.57 −0.256101 −0.128051 0.991768i \(-0.540872\pi\)
−0.128051 + 0.991768i \(0.540872\pi\)
\(578\) 0 0
\(579\) 17987.8 1.29110
\(580\) 0 0
\(581\) 22515.8 1.60776
\(582\) 0 0
\(583\) 4486.57 0.318721
\(584\) 0 0
\(585\) −67373.3 −4.76162
\(586\) 0 0
\(587\) 3839.58 0.269977 0.134988 0.990847i \(-0.456900\pi\)
0.134988 + 0.990847i \(0.456900\pi\)
\(588\) 0 0
\(589\) −79.5902 −0.00556784
\(590\) 0 0
\(591\) 25924.2 1.80437
\(592\) 0 0
\(593\) 22417.1 1.55238 0.776189 0.630500i \(-0.217150\pi\)
0.776189 + 0.630500i \(0.217150\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) −12070.9 −0.827517
\(598\) 0 0
\(599\) 5574.76 0.380265 0.190132 0.981758i \(-0.439108\pi\)
0.190132 + 0.981758i \(0.439108\pi\)
\(600\) 0 0
\(601\) −25055.3 −1.70054 −0.850272 0.526344i \(-0.823562\pi\)
−0.850272 + 0.526344i \(0.823562\pi\)
\(602\) 0 0
\(603\) −615.747 −0.0415840
\(604\) 0 0
\(605\) −11244.8 −0.755647
\(606\) 0 0
\(607\) −1371.15 −0.0916857 −0.0458428 0.998949i \(-0.514597\pi\)
−0.0458428 + 0.998949i \(0.514597\pi\)
\(608\) 0 0
\(609\) −21585.5 −1.43627
\(610\) 0 0
\(611\) −31048.1 −2.05576
\(612\) 0 0
\(613\) −4685.73 −0.308736 −0.154368 0.988013i \(-0.549334\pi\)
−0.154368 + 0.988013i \(0.549334\pi\)
\(614\) 0 0
\(615\) 4397.50 0.288332
\(616\) 0 0
\(617\) 24700.7 1.61169 0.805846 0.592125i \(-0.201711\pi\)
0.805846 + 0.592125i \(0.201711\pi\)
\(618\) 0 0
\(619\) 18824.1 1.22230 0.611151 0.791514i \(-0.290707\pi\)
0.611151 + 0.791514i \(0.290707\pi\)
\(620\) 0 0
\(621\) 91590.0 5.91849
\(622\) 0 0
\(623\) −24711.1 −1.58914
\(624\) 0 0
\(625\) −19513.1 −1.24884
\(626\) 0 0
\(627\) 8861.24 0.564408
\(628\) 0 0
\(629\) 0 0
\(630\) 0 0
\(631\) −15571.9 −0.982420 −0.491210 0.871041i \(-0.663445\pi\)
−0.491210 + 0.871041i \(0.663445\pi\)
\(632\) 0 0
\(633\) 50242.8 3.15478
\(634\) 0 0
\(635\) −7349.15 −0.459279
\(636\) 0 0
\(637\) −992.823 −0.0617536
\(638\) 0 0
\(639\) 3772.64 0.233558
\(640\) 0 0
\(641\) −27035.7 −1.66591 −0.832954 0.553342i \(-0.813352\pi\)
−0.832954 + 0.553342i \(0.813352\pi\)
\(642\) 0 0
\(643\) −19405.7 −1.19018 −0.595090 0.803659i \(-0.702883\pi\)
−0.595090 + 0.803659i \(0.702883\pi\)
\(644\) 0 0
\(645\) −1324.06 −0.0808289
\(646\) 0 0
\(647\) −20930.2 −1.27179 −0.635897 0.771774i \(-0.719370\pi\)
−0.635897 + 0.771774i \(0.719370\pi\)
\(648\) 0 0
\(649\) −39557.8 −2.39257
\(650\) 0 0
\(651\) 759.809 0.0457439
\(652\) 0 0
\(653\) 5277.55 0.316273 0.158137 0.987417i \(-0.449451\pi\)
0.158137 + 0.987417i \(0.449451\pi\)
\(654\) 0 0
\(655\) −8918.03 −0.531994
\(656\) 0 0
\(657\) 69982.3 4.15566
\(658\) 0 0
\(659\) −7954.54 −0.470205 −0.235102 0.971971i \(-0.575543\pi\)
−0.235102 + 0.971971i \(0.575543\pi\)
\(660\) 0 0
\(661\) 11489.7 0.676094 0.338047 0.941129i \(-0.390234\pi\)
0.338047 + 0.941129i \(0.390234\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 4778.20 0.278632
\(666\) 0 0
\(667\) −23062.4 −1.33880
\(668\) 0 0
\(669\) 59888.2 3.46100
\(670\) 0 0
\(671\) 7938.76 0.456740
\(672\) 0 0
\(673\) −1213.00 −0.0694766 −0.0347383 0.999396i \(-0.511060\pi\)
−0.0347383 + 0.999396i \(0.511060\pi\)
\(674\) 0 0
\(675\) 31458.5 1.79383
\(676\) 0 0
\(677\) 8069.13 0.458083 0.229041 0.973417i \(-0.426441\pi\)
0.229041 + 0.973417i \(0.426441\pi\)
\(678\) 0 0
\(679\) 21124.7 1.19395
\(680\) 0 0
\(681\) 57236.4 3.22071
\(682\) 0 0
\(683\) 12540.1 0.702537 0.351268 0.936275i \(-0.385750\pi\)
0.351268 + 0.936275i \(0.385750\pi\)
\(684\) 0 0
\(685\) −33573.7 −1.87268
\(686\) 0 0
\(687\) 58603.5 3.25453
\(688\) 0 0
\(689\) −6379.69 −0.352753
\(690\) 0 0
\(691\) −6511.86 −0.358499 −0.179250 0.983804i \(-0.557367\pi\)
−0.179250 + 0.983804i \(0.557367\pi\)
\(692\) 0 0
\(693\) −61960.1 −3.39635
\(694\) 0 0
\(695\) 31055.7 1.69498
\(696\) 0 0
\(697\) 0 0
\(698\) 0 0
\(699\) −22613.5 −1.22364
\(700\) 0 0
\(701\) −8151.13 −0.439178 −0.219589 0.975592i \(-0.570472\pi\)
−0.219589 + 0.975592i \(0.570472\pi\)
\(702\) 0 0
\(703\) −5162.92 −0.276989
\(704\) 0 0
\(705\) 65611.3 3.50505
\(706\) 0 0
\(707\) −11726.5 −0.623791
\(708\) 0 0
\(709\) 19489.6 1.03237 0.516184 0.856478i \(-0.327352\pi\)
0.516184 + 0.856478i \(0.327352\pi\)
\(710\) 0 0
\(711\) 1186.93 0.0626068
\(712\) 0 0
\(713\) 811.794 0.0426394
\(714\) 0 0
\(715\) 42197.8 2.20714
\(716\) 0 0
\(717\) −27357.7 −1.42496
\(718\) 0 0
\(719\) −9234.85 −0.479001 −0.239501 0.970896i \(-0.576984\pi\)
−0.239501 + 0.970896i \(0.576984\pi\)
\(720\) 0 0
\(721\) −3033.87 −0.156709
\(722\) 0 0
\(723\) 27746.8 1.42727
\(724\) 0 0
\(725\) −7921.24 −0.405776
\(726\) 0 0
\(727\) 21248.2 1.08398 0.541989 0.840386i \(-0.317671\pi\)
0.541989 + 0.840386i \(0.317671\pi\)
\(728\) 0 0
\(729\) 74578.9 3.78900
\(730\) 0 0
\(731\) 0 0
\(732\) 0 0
\(733\) −33363.0 −1.68116 −0.840579 0.541689i \(-0.817785\pi\)
−0.840579 + 0.541689i \(0.817785\pi\)
\(734\) 0 0
\(735\) 2098.05 0.105289
\(736\) 0 0
\(737\) 385.659 0.0192754
\(738\) 0 0
\(739\) 15082.8 0.750784 0.375392 0.926866i \(-0.377508\pi\)
0.375392 + 0.926866i \(0.377508\pi\)
\(740\) 0 0
\(741\) −12600.3 −0.624673
\(742\) 0 0
\(743\) 1869.09 0.0922883 0.0461442 0.998935i \(-0.485307\pi\)
0.0461442 + 0.998935i \(0.485307\pi\)
\(744\) 0 0
\(745\) 6303.01 0.309966
\(746\) 0 0
\(747\) 91900.6 4.50129
\(748\) 0 0
\(749\) −9804.15 −0.478286
\(750\) 0 0
\(751\) −20231.4 −0.983026 −0.491513 0.870870i \(-0.663556\pi\)
−0.491513 + 0.870870i \(0.663556\pi\)
\(752\) 0 0
\(753\) −28705.6 −1.38923
\(754\) 0 0
\(755\) 10336.6 0.498261
\(756\) 0 0
\(757\) −4216.92 −0.202466 −0.101233 0.994863i \(-0.532279\pi\)
−0.101233 + 0.994863i \(0.532279\pi\)
\(758\) 0 0
\(759\) −90381.8 −4.32233
\(760\) 0 0
\(761\) −7638.77 −0.363870 −0.181935 0.983311i \(-0.558236\pi\)
−0.181935 + 0.983311i \(0.558236\pi\)
\(762\) 0 0
\(763\) 3856.40 0.182976
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 56249.3 2.64804
\(768\) 0 0
\(769\) −32633.5 −1.53029 −0.765145 0.643858i \(-0.777333\pi\)
−0.765145 + 0.643858i \(0.777333\pi\)
\(770\) 0 0
\(771\) −56458.4 −2.63723
\(772\) 0 0
\(773\) −5125.84 −0.238504 −0.119252 0.992864i \(-0.538050\pi\)
−0.119252 + 0.992864i \(0.538050\pi\)
\(774\) 0 0
\(775\) 278.827 0.0129236
\(776\) 0 0
\(777\) 49287.9 2.27567
\(778\) 0 0
\(779\) 602.379 0.0277053
\(780\) 0 0
\(781\) −2362.91 −0.108261
\(782\) 0 0
\(783\) −55919.4 −2.55223
\(784\) 0 0
\(785\) −21681.8 −0.985806
\(786\) 0 0
\(787\) 16716.9 0.757169 0.378584 0.925567i \(-0.376411\pi\)
0.378584 + 0.925567i \(0.376411\pi\)
\(788\) 0 0
\(789\) −30780.4 −1.38886
\(790\) 0 0
\(791\) 16083.2 0.722951
\(792\) 0 0
\(793\) −11288.5 −0.505508
\(794\) 0 0
\(795\) 13481.7 0.601440
\(796\) 0 0
\(797\) 16430.8 0.730248 0.365124 0.930959i \(-0.381027\pi\)
0.365124 + 0.930959i \(0.381027\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 0 0
\(801\) −100861. −4.44914
\(802\) 0 0
\(803\) −43831.9 −1.92627
\(804\) 0 0
\(805\) −48736.0 −2.13381
\(806\) 0 0
\(807\) −11716.4 −0.511075
\(808\) 0 0
\(809\) 13144.4 0.571237 0.285619 0.958343i \(-0.407801\pi\)
0.285619 + 0.958343i \(0.407801\pi\)
\(810\) 0 0
\(811\) 21400.7 0.926608 0.463304 0.886199i \(-0.346664\pi\)
0.463304 + 0.886199i \(0.346664\pi\)
\(812\) 0 0
\(813\) −86198.7 −3.71847
\(814\) 0 0
\(815\) −20909.8 −0.898696
\(816\) 0 0
\(817\) −181.372 −0.00776671
\(818\) 0 0
\(819\) 88104.3 3.75899
\(820\) 0 0
\(821\) −2678.59 −0.113865 −0.0569327 0.998378i \(-0.518132\pi\)
−0.0569327 + 0.998378i \(0.518132\pi\)
\(822\) 0 0
\(823\) 2939.29 0.124493 0.0622463 0.998061i \(-0.480174\pi\)
0.0622463 + 0.998061i \(0.480174\pi\)
\(824\) 0 0
\(825\) −31043.5 −1.31005
\(826\) 0 0
\(827\) −4542.99 −0.191022 −0.0955110 0.995428i \(-0.530449\pi\)
−0.0955110 + 0.995428i \(0.530449\pi\)
\(828\) 0 0
\(829\) −28672.1 −1.20124 −0.600618 0.799536i \(-0.705079\pi\)
−0.600618 + 0.799536i \(0.705079\pi\)
\(830\) 0 0
\(831\) −64929.9 −2.71046
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) 13321.3 0.552098
\(836\) 0 0
\(837\) 1968.36 0.0812862
\(838\) 0 0
\(839\) 20351.7 0.837449 0.418724 0.908113i \(-0.362477\pi\)
0.418724 + 0.908113i \(0.362477\pi\)
\(840\) 0 0
\(841\) −10308.5 −0.422669
\(842\) 0 0
\(843\) −55568.2 −2.27031
\(844\) 0 0
\(845\) −29580.1 −1.20425
\(846\) 0 0
\(847\) 14704.9 0.596535
\(848\) 0 0
\(849\) 10975.9 0.443691
\(850\) 0 0
\(851\) 52660.1 2.12123
\(852\) 0 0
\(853\) −18949.3 −0.760625 −0.380313 0.924858i \(-0.624184\pi\)
−0.380313 + 0.924858i \(0.624184\pi\)
\(854\) 0 0
\(855\) 19502.7 0.780093
\(856\) 0 0
\(857\) −41367.8 −1.64889 −0.824444 0.565943i \(-0.808512\pi\)
−0.824444 + 0.565943i \(0.808512\pi\)
\(858\) 0 0
\(859\) 17909.6 0.711370 0.355685 0.934606i \(-0.384248\pi\)
0.355685 + 0.934606i \(0.384248\pi\)
\(860\) 0 0
\(861\) −5750.62 −0.227620
\(862\) 0 0
\(863\) −28855.9 −1.13820 −0.569099 0.822269i \(-0.692708\pi\)
−0.569099 + 0.822269i \(0.692708\pi\)
\(864\) 0 0
\(865\) −10474.0 −0.411708
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) −743.407 −0.0290200
\(870\) 0 0
\(871\) −548.389 −0.0213335
\(872\) 0 0
\(873\) 86222.9 3.34273
\(874\) 0 0
\(875\) 14605.4 0.564290
\(876\) 0 0
\(877\) 36116.6 1.39062 0.695309 0.718711i \(-0.255268\pi\)
0.695309 + 0.718711i \(0.255268\pi\)
\(878\) 0 0
\(879\) −31843.8 −1.22192
\(880\) 0 0
\(881\) 31133.0 1.19058 0.595289 0.803512i \(-0.297038\pi\)
0.595289 + 0.803512i \(0.297038\pi\)
\(882\) 0 0
\(883\) −44665.2 −1.70227 −0.851135 0.524946i \(-0.824085\pi\)
−0.851135 + 0.524946i \(0.824085\pi\)
\(884\) 0 0
\(885\) −118867. −4.51488
\(886\) 0 0
\(887\) −36950.4 −1.39873 −0.699365 0.714765i \(-0.746534\pi\)
−0.699365 + 0.714765i \(0.746534\pi\)
\(888\) 0 0
\(889\) 9610.50 0.362571
\(890\) 0 0
\(891\) −126766. −4.76635
\(892\) 0 0
\(893\) 8987.57 0.336795
\(894\) 0 0
\(895\) −5410.09 −0.202055
\(896\) 0 0
\(897\) 128519. 4.78385
\(898\) 0 0
\(899\) −495.633 −0.0183874
\(900\) 0 0
\(901\) 0 0
\(902\) 0 0
\(903\) 1731.47 0.0638092
\(904\) 0 0
\(905\) 27275.9 1.00186
\(906\) 0 0
\(907\) −1462.19 −0.0535296 −0.0267648 0.999642i \(-0.508521\pi\)
−0.0267648 + 0.999642i \(0.508521\pi\)
\(908\) 0 0
\(909\) −47863.0 −1.74644
\(910\) 0 0
\(911\) 993.504 0.0361320 0.0180660 0.999837i \(-0.494249\pi\)
0.0180660 + 0.999837i \(0.494249\pi\)
\(912\) 0 0
\(913\) −57559.8 −2.08648
\(914\) 0 0
\(915\) 23855.1 0.861887
\(916\) 0 0
\(917\) 11662.1 0.419976
\(918\) 0 0
\(919\) −12596.0 −0.452125 −0.226063 0.974113i \(-0.572585\pi\)
−0.226063 + 0.974113i \(0.572585\pi\)
\(920\) 0 0
\(921\) −26394.5 −0.944332
\(922\) 0 0
\(923\) 3359.95 0.119820
\(924\) 0 0
\(925\) 18087.2 0.642922
\(926\) 0 0
\(927\) −12383.1 −0.438742
\(928\) 0 0
\(929\) 26013.8 0.918713 0.459357 0.888252i \(-0.348080\pi\)
0.459357 + 0.888252i \(0.348080\pi\)
\(930\) 0 0
\(931\) 287.395 0.0101171
\(932\) 0 0
\(933\) −40034.8 −1.40480
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 22507.8 0.784737 0.392368 0.919808i \(-0.371656\pi\)
0.392368 + 0.919808i \(0.371656\pi\)
\(938\) 0 0
\(939\) 35211.1 1.22372
\(940\) 0 0
\(941\) −35241.7 −1.22088 −0.610439 0.792063i \(-0.709007\pi\)
−0.610439 + 0.792063i \(0.709007\pi\)
\(942\) 0 0
\(943\) −6144.07 −0.212172
\(944\) 0 0
\(945\) −118171. −4.06782
\(946\) 0 0
\(947\) −35820.8 −1.22917 −0.614583 0.788852i \(-0.710676\pi\)
−0.614583 + 0.788852i \(0.710676\pi\)
\(948\) 0 0
\(949\) 62326.8 2.13194
\(950\) 0 0
\(951\) 16573.0 0.565106
\(952\) 0 0
\(953\) 25732.2 0.874658 0.437329 0.899302i \(-0.355925\pi\)
0.437329 + 0.899302i \(0.355925\pi\)
\(954\) 0 0
\(955\) −5850.73 −0.198246
\(956\) 0 0
\(957\) 55181.8 1.86392
\(958\) 0 0
\(959\) 43904.4 1.47836
\(960\) 0 0
\(961\) −29773.6 −0.999414
\(962\) 0 0
\(963\) −40016.7 −1.33907
\(964\) 0 0
\(965\) −24796.0 −0.827161
\(966\) 0 0
\(967\) −34792.6 −1.15704 −0.578518 0.815669i \(-0.696369\pi\)
−0.578518 + 0.815669i \(0.696369\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) −19826.3 −0.655260 −0.327630 0.944806i \(-0.606250\pi\)
−0.327630 + 0.944806i \(0.606250\pi\)
\(972\) 0 0
\(973\) −40611.7 −1.33808
\(974\) 0 0
\(975\) 44142.4 1.44994
\(976\) 0 0
\(977\) 4133.51 0.135356 0.0676779 0.997707i \(-0.478441\pi\)
0.0676779 + 0.997707i \(0.478441\pi\)
\(978\) 0 0
\(979\) 63172.2 2.06230
\(980\) 0 0
\(981\) 15740.3 0.512283
\(982\) 0 0
\(983\) 41660.6 1.35175 0.675873 0.737018i \(-0.263767\pi\)
0.675873 + 0.737018i \(0.263767\pi\)
\(984\) 0 0
\(985\) −35736.2 −1.15599
\(986\) 0 0
\(987\) −85800.0 −2.76702
\(988\) 0 0
\(989\) 1849.93 0.0594787
\(990\) 0 0
\(991\) −2903.34 −0.0930652 −0.0465326 0.998917i \(-0.514817\pi\)
−0.0465326 + 0.998917i \(0.514817\pi\)
\(992\) 0 0
\(993\) 43206.6 1.38079
\(994\) 0 0
\(995\) 16639.5 0.530159
\(996\) 0 0
\(997\) −43692.3 −1.38791 −0.693957 0.720017i \(-0.744134\pi\)
−0.693957 + 0.720017i \(0.744134\pi\)
\(998\) 0 0
\(999\) 127685. 4.04382
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 2312.4.a.q.1.18 yes 18
17.16 even 2 2312.4.a.n.1.1 18
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2312.4.a.n.1.1 18 17.16 even 2
2312.4.a.q.1.18 yes 18 1.1 even 1 trivial