Properties

Label 1840.4.a.ba.1.7
Level $1840$
Weight $4$
Character 1840.1
Self dual yes
Analytic conductor $108.564$
Analytic rank $0$
Dimension $10$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1840,4,Mod(1,1840)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1840, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0, 0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1840.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1840 = 2^{4} \cdot 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1840.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: \(108.563514411\)
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: \( x^{10} - 5 x^{9} - 192 x^{8} + 762 x^{7} + 12246 x^{6} - 33828 x^{5} - 298243 x^{4} + 383603 x^{3} + 2423016 x^{2} + 864576 x + 57408 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{7} \)
Twist minimal: no (minimal twist has level 920)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.7
Root \(-2.78233\) of defining polynomial
Character \(\chi\) \(=\) 1840.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+2.78233 q^{3} +5.00000 q^{5} +11.8146 q^{7} -19.2586 q^{9} +O(q^{10})\) \(q+2.78233 q^{3} +5.00000 q^{5} +11.8146 q^{7} -19.2586 q^{9} -0.796382 q^{11} +13.0701 q^{13} +13.9116 q^{15} -99.3670 q^{17} -54.5930 q^{19} +32.8722 q^{21} +23.0000 q^{23} +25.0000 q^{25} -128.707 q^{27} -38.4110 q^{29} +219.099 q^{31} -2.21580 q^{33} +59.0731 q^{35} +381.602 q^{37} +36.3653 q^{39} +266.522 q^{41} -41.4155 q^{43} -96.2932 q^{45} +314.164 q^{47} -203.415 q^{49} -276.472 q^{51} +532.322 q^{53} -3.98191 q^{55} -151.896 q^{57} +433.619 q^{59} +438.950 q^{61} -227.534 q^{63} +65.3504 q^{65} +61.2709 q^{67} +63.9936 q^{69} +652.909 q^{71} +373.885 q^{73} +69.5582 q^{75} -9.40896 q^{77} -716.640 q^{79} +161.879 q^{81} -1241.01 q^{83} -496.835 q^{85} -106.872 q^{87} -476.408 q^{89} +154.418 q^{91} +609.607 q^{93} -272.965 q^{95} +701.769 q^{97} +15.3372 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q - 5 q^{3} + 50 q^{5} - 14 q^{7} + 139 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 10 q - 5 q^{3} + 50 q^{5} - 14 q^{7} + 139 q^{9} + 56 q^{11} + 49 q^{13} - 25 q^{15} + 240 q^{17} - 88 q^{19} + 346 q^{21} + 230 q^{23} + 250 q^{25} - 449 q^{27} + 319 q^{29} + 109 q^{31} + 504 q^{33} - 70 q^{35} + 580 q^{37} - 107 q^{39} + 259 q^{41} - 330 q^{43} + 695 q^{45} - 227 q^{47} + 630 q^{49} + 192 q^{51} - 186 q^{53} + 280 q^{55} + 1708 q^{57} - 262 q^{59} + 1000 q^{61} - 722 q^{63} + 245 q^{65} - 354 q^{67} - 115 q^{69} - 599 q^{71} + 355 q^{73} - 125 q^{75} + 1776 q^{77} + 1068 q^{79} + 3490 q^{81} - 754 q^{83} + 1200 q^{85} - 2675 q^{87} + 1740 q^{89} - 690 q^{91} + 1669 q^{93} - 440 q^{95} + 2592 q^{97} - 916 q^{99}+O(q^{100}) \) 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
\(3\) 2.78233 0.535459 0.267730 0.963494i \(-0.413727\pi\)
0.267730 + 0.963494i \(0.413727\pi\)
\(4\) 0 0
\(5\) 5.00000 0.447214
\(6\) 0 0
\(7\) 11.8146 0.637930 0.318965 0.947767i \(-0.396665\pi\)
0.318965 + 0.947767i \(0.396665\pi\)
\(8\) 0 0
\(9\) −19.2586 −0.713283
\(10\) 0 0
\(11\) −0.796382 −0.0218289 −0.0109145 0.999940i \(-0.503474\pi\)
−0.0109145 + 0.999940i \(0.503474\pi\)
\(12\) 0 0
\(13\) 13.0701 0.278845 0.139423 0.990233i \(-0.455475\pi\)
0.139423 + 0.990233i \(0.455475\pi\)
\(14\) 0 0
\(15\) 13.9116 0.239465
\(16\) 0 0
\(17\) −99.3670 −1.41765 −0.708825 0.705385i \(-0.750774\pi\)
−0.708825 + 0.705385i \(0.750774\pi\)
\(18\) 0 0
\(19\) −54.5930 −0.659184 −0.329592 0.944123i \(-0.606911\pi\)
−0.329592 + 0.944123i \(0.606911\pi\)
\(20\) 0 0
\(21\) 32.8722 0.341586
\(22\) 0 0
\(23\) 23.0000 0.208514
\(24\) 0 0
\(25\) 25.0000 0.200000
\(26\) 0 0
\(27\) −128.707 −0.917394
\(28\) 0 0
\(29\) −38.4110 −0.245956 −0.122978 0.992409i \(-0.539245\pi\)
−0.122978 + 0.992409i \(0.539245\pi\)
\(30\) 0 0
\(31\) 219.099 1.26940 0.634700 0.772758i \(-0.281123\pi\)
0.634700 + 0.772758i \(0.281123\pi\)
\(32\) 0 0
\(33\) −2.21580 −0.0116885
\(34\) 0 0
\(35\) 59.0731 0.285291
\(36\) 0 0
\(37\) 381.602 1.69554 0.847769 0.530365i \(-0.177945\pi\)
0.847769 + 0.530365i \(0.177945\pi\)
\(38\) 0 0
\(39\) 36.3653 0.149310
\(40\) 0 0
\(41\) 266.522 1.01521 0.507607 0.861589i \(-0.330530\pi\)
0.507607 + 0.861589i \(0.330530\pi\)
\(42\) 0 0
\(43\) −41.4155 −0.146879 −0.0734395 0.997300i \(-0.523398\pi\)
−0.0734395 + 0.997300i \(0.523398\pi\)
\(44\) 0 0
\(45\) −96.2932 −0.318990
\(46\) 0 0
\(47\) 314.164 0.975012 0.487506 0.873120i \(-0.337907\pi\)
0.487506 + 0.873120i \(0.337907\pi\)
\(48\) 0 0
\(49\) −203.415 −0.593045
\(50\) 0 0
\(51\) −276.472 −0.759094
\(52\) 0 0
\(53\) 532.322 1.37962 0.689812 0.723989i \(-0.257693\pi\)
0.689812 + 0.723989i \(0.257693\pi\)
\(54\) 0 0
\(55\) −3.98191 −0.00976220
\(56\) 0 0
\(57\) −151.896 −0.352966
\(58\) 0 0
\(59\) 433.619 0.956820 0.478410 0.878137i \(-0.341213\pi\)
0.478410 + 0.878137i \(0.341213\pi\)
\(60\) 0 0
\(61\) 438.950 0.921340 0.460670 0.887571i \(-0.347609\pi\)
0.460670 + 0.887571i \(0.347609\pi\)
\(62\) 0 0
\(63\) −227.534 −0.455025
\(64\) 0 0
\(65\) 65.3504 0.124703
\(66\) 0 0
\(67\) 61.2709 0.111723 0.0558614 0.998439i \(-0.482210\pi\)
0.0558614 + 0.998439i \(0.482210\pi\)
\(68\) 0 0
\(69\) 63.9936 0.111651
\(70\) 0 0
\(71\) 652.909 1.09135 0.545677 0.837996i \(-0.316273\pi\)
0.545677 + 0.837996i \(0.316273\pi\)
\(72\) 0 0
\(73\) 373.885 0.599452 0.299726 0.954025i \(-0.403105\pi\)
0.299726 + 0.954025i \(0.403105\pi\)
\(74\) 0 0
\(75\) 69.5582 0.107092
\(76\) 0 0
\(77\) −9.40896 −0.0139253
\(78\) 0 0
\(79\) −716.640 −1.02061 −0.510306 0.859993i \(-0.670468\pi\)
−0.510306 + 0.859993i \(0.670468\pi\)
\(80\) 0 0
\(81\) 161.879 0.222056
\(82\) 0 0
\(83\) −1241.01 −1.64119 −0.820597 0.571508i \(-0.806359\pi\)
−0.820597 + 0.571508i \(0.806359\pi\)
\(84\) 0 0
\(85\) −496.835 −0.633992
\(86\) 0 0
\(87\) −106.872 −0.131700
\(88\) 0 0
\(89\) −476.408 −0.567406 −0.283703 0.958912i \(-0.591563\pi\)
−0.283703 + 0.958912i \(0.591563\pi\)
\(90\) 0 0
\(91\) 154.418 0.177884
\(92\) 0 0
\(93\) 609.607 0.679713
\(94\) 0 0
\(95\) −272.965 −0.294796
\(96\) 0 0
\(97\) 701.769 0.734576 0.367288 0.930107i \(-0.380286\pi\)
0.367288 + 0.930107i \(0.380286\pi\)
\(98\) 0 0
\(99\) 15.3372 0.0155702
\(100\) 0 0
\(101\) 1226.44 1.20827 0.604134 0.796883i \(-0.293519\pi\)
0.604134 + 0.796883i \(0.293519\pi\)
\(102\) 0 0
\(103\) 904.021 0.864814 0.432407 0.901679i \(-0.357664\pi\)
0.432407 + 0.901679i \(0.357664\pi\)
\(104\) 0 0
\(105\) 164.361 0.152762
\(106\) 0 0
\(107\) 1888.98 1.70668 0.853339 0.521357i \(-0.174574\pi\)
0.853339 + 0.521357i \(0.174574\pi\)
\(108\) 0 0
\(109\) 1887.37 1.65851 0.829255 0.558870i \(-0.188765\pi\)
0.829255 + 0.558870i \(0.188765\pi\)
\(110\) 0 0
\(111\) 1061.74 0.907892
\(112\) 0 0
\(113\) −534.175 −0.444699 −0.222349 0.974967i \(-0.571373\pi\)
−0.222349 + 0.974967i \(0.571373\pi\)
\(114\) 0 0
\(115\) 115.000 0.0932505
\(116\) 0 0
\(117\) −251.712 −0.198896
\(118\) 0 0
\(119\) −1173.98 −0.904361
\(120\) 0 0
\(121\) −1330.37 −0.999523
\(122\) 0 0
\(123\) 741.552 0.543606
\(124\) 0 0
\(125\) 125.000 0.0894427
\(126\) 0 0
\(127\) −1754.75 −1.22605 −0.613027 0.790062i \(-0.710048\pi\)
−0.613027 + 0.790062i \(0.710048\pi\)
\(128\) 0 0
\(129\) −115.231 −0.0786478
\(130\) 0 0
\(131\) −983.194 −0.655741 −0.327870 0.944723i \(-0.606331\pi\)
−0.327870 + 0.944723i \(0.606331\pi\)
\(132\) 0 0
\(133\) −644.996 −0.420513
\(134\) 0 0
\(135\) −643.534 −0.410271
\(136\) 0 0
\(137\) 3174.46 1.97965 0.989826 0.142286i \(-0.0454451\pi\)
0.989826 + 0.142286i \(0.0454451\pi\)
\(138\) 0 0
\(139\) −2106.78 −1.28557 −0.642787 0.766045i \(-0.722222\pi\)
−0.642787 + 0.766045i \(0.722222\pi\)
\(140\) 0 0
\(141\) 874.108 0.522079
\(142\) 0 0
\(143\) −10.4088 −0.00608690
\(144\) 0 0
\(145\) −192.055 −0.109995
\(146\) 0 0
\(147\) −565.966 −0.317552
\(148\) 0 0
\(149\) 3228.82 1.77527 0.887635 0.460548i \(-0.152347\pi\)
0.887635 + 0.460548i \(0.152347\pi\)
\(150\) 0 0
\(151\) −1843.21 −0.993368 −0.496684 0.867931i \(-0.665449\pi\)
−0.496684 + 0.867931i \(0.665449\pi\)
\(152\) 0 0
\(153\) 1913.67 1.01119
\(154\) 0 0
\(155\) 1095.50 0.567693
\(156\) 0 0
\(157\) 1104.59 0.561503 0.280752 0.959780i \(-0.409416\pi\)
0.280752 + 0.959780i \(0.409416\pi\)
\(158\) 0 0
\(159\) 1481.09 0.738732
\(160\) 0 0
\(161\) 271.736 0.133018
\(162\) 0 0
\(163\) −1912.97 −0.919234 −0.459617 0.888117i \(-0.652013\pi\)
−0.459617 + 0.888117i \(0.652013\pi\)
\(164\) 0 0
\(165\) −11.0790 −0.00522726
\(166\) 0 0
\(167\) −257.835 −0.119472 −0.0597362 0.998214i \(-0.519026\pi\)
−0.0597362 + 0.998214i \(0.519026\pi\)
\(168\) 0 0
\(169\) −2026.17 −0.922245
\(170\) 0 0
\(171\) 1051.39 0.470185
\(172\) 0 0
\(173\) 3749.16 1.64765 0.823824 0.566846i \(-0.191836\pi\)
0.823824 + 0.566846i \(0.191836\pi\)
\(174\) 0 0
\(175\) 295.366 0.127586
\(176\) 0 0
\(177\) 1206.47 0.512338
\(178\) 0 0
\(179\) −2040.75 −0.852137 −0.426068 0.904691i \(-0.640102\pi\)
−0.426068 + 0.904691i \(0.640102\pi\)
\(180\) 0 0
\(181\) −3926.66 −1.61252 −0.806261 0.591561i \(-0.798512\pi\)
−0.806261 + 0.591561i \(0.798512\pi\)
\(182\) 0 0
\(183\) 1221.30 0.493340
\(184\) 0 0
\(185\) 1908.01 0.758268
\(186\) 0 0
\(187\) 79.1341 0.0309458
\(188\) 0 0
\(189\) −1520.62 −0.585233
\(190\) 0 0
\(191\) −2334.49 −0.884386 −0.442193 0.896920i \(-0.645799\pi\)
−0.442193 + 0.896920i \(0.645799\pi\)
\(192\) 0 0
\(193\) 4396.43 1.63970 0.819849 0.572580i \(-0.194057\pi\)
0.819849 + 0.572580i \(0.194057\pi\)
\(194\) 0 0
\(195\) 181.826 0.0667737
\(196\) 0 0
\(197\) −2513.52 −0.909039 −0.454519 0.890737i \(-0.650189\pi\)
−0.454519 + 0.890737i \(0.650189\pi\)
\(198\) 0 0
\(199\) 1817.29 0.647360 0.323680 0.946167i \(-0.395080\pi\)
0.323680 + 0.946167i \(0.395080\pi\)
\(200\) 0 0
\(201\) 170.476 0.0598230
\(202\) 0 0
\(203\) −453.811 −0.156903
\(204\) 0 0
\(205\) 1332.61 0.454017
\(206\) 0 0
\(207\) −442.949 −0.148730
\(208\) 0 0
\(209\) 43.4769 0.0143893
\(210\) 0 0
\(211\) 3223.78 1.05182 0.525911 0.850540i \(-0.323725\pi\)
0.525911 + 0.850540i \(0.323725\pi\)
\(212\) 0 0
\(213\) 1816.61 0.584375
\(214\) 0 0
\(215\) −207.077 −0.0656863
\(216\) 0 0
\(217\) 2588.58 0.809789
\(218\) 0 0
\(219\) 1040.27 0.320982
\(220\) 0 0
\(221\) −1298.74 −0.395305
\(222\) 0 0
\(223\) 1399.93 0.420387 0.210194 0.977660i \(-0.432591\pi\)
0.210194 + 0.977660i \(0.432591\pi\)
\(224\) 0 0
\(225\) −481.466 −0.142657
\(226\) 0 0
\(227\) −4637.45 −1.35594 −0.677970 0.735089i \(-0.737140\pi\)
−0.677970 + 0.735089i \(0.737140\pi\)
\(228\) 0 0
\(229\) −2676.10 −0.772235 −0.386117 0.922450i \(-0.626184\pi\)
−0.386117 + 0.922450i \(0.626184\pi\)
\(230\) 0 0
\(231\) −26.1788 −0.00745645
\(232\) 0 0
\(233\) 119.364 0.0335615 0.0167807 0.999859i \(-0.494658\pi\)
0.0167807 + 0.999859i \(0.494658\pi\)
\(234\) 0 0
\(235\) 1570.82 0.436039
\(236\) 0 0
\(237\) −1993.93 −0.546496
\(238\) 0 0
\(239\) 1128.32 0.305376 0.152688 0.988274i \(-0.451207\pi\)
0.152688 + 0.988274i \(0.451207\pi\)
\(240\) 0 0
\(241\) −1945.38 −0.519972 −0.259986 0.965612i \(-0.583718\pi\)
−0.259986 + 0.965612i \(0.583718\pi\)
\(242\) 0 0
\(243\) 3925.48 1.03630
\(244\) 0 0
\(245\) −1017.07 −0.265218
\(246\) 0 0
\(247\) −713.536 −0.183810
\(248\) 0 0
\(249\) −3452.91 −0.878792
\(250\) 0 0
\(251\) 3354.62 0.843593 0.421796 0.906691i \(-0.361400\pi\)
0.421796 + 0.906691i \(0.361400\pi\)
\(252\) 0 0
\(253\) −18.3168 −0.00455165
\(254\) 0 0
\(255\) −1382.36 −0.339477
\(256\) 0 0
\(257\) −3182.83 −0.772527 −0.386263 0.922388i \(-0.626234\pi\)
−0.386263 + 0.922388i \(0.626234\pi\)
\(258\) 0 0
\(259\) 4508.48 1.08163
\(260\) 0 0
\(261\) 739.743 0.175437
\(262\) 0 0
\(263\) 216.610 0.0507861 0.0253930 0.999678i \(-0.491916\pi\)
0.0253930 + 0.999678i \(0.491916\pi\)
\(264\) 0 0
\(265\) 2661.61 0.616986
\(266\) 0 0
\(267\) −1325.52 −0.303823
\(268\) 0 0
\(269\) 2941.31 0.666672 0.333336 0.942808i \(-0.391826\pi\)
0.333336 + 0.942808i \(0.391826\pi\)
\(270\) 0 0
\(271\) 5696.98 1.27700 0.638500 0.769622i \(-0.279555\pi\)
0.638500 + 0.769622i \(0.279555\pi\)
\(272\) 0 0
\(273\) 429.642 0.0952496
\(274\) 0 0
\(275\) −19.9096 −0.00436579
\(276\) 0 0
\(277\) 7180.55 1.55754 0.778768 0.627312i \(-0.215845\pi\)
0.778768 + 0.627312i \(0.215845\pi\)
\(278\) 0 0
\(279\) −4219.56 −0.905442
\(280\) 0 0
\(281\) 676.626 0.143644 0.0718222 0.997417i \(-0.477119\pi\)
0.0718222 + 0.997417i \(0.477119\pi\)
\(282\) 0 0
\(283\) −3513.90 −0.738092 −0.369046 0.929411i \(-0.620315\pi\)
−0.369046 + 0.929411i \(0.620315\pi\)
\(284\) 0 0
\(285\) −759.478 −0.157851
\(286\) 0 0
\(287\) 3148.86 0.647635
\(288\) 0 0
\(289\) 4960.80 1.00973
\(290\) 0 0
\(291\) 1952.55 0.393336
\(292\) 0 0
\(293\) 5774.30 1.15132 0.575662 0.817688i \(-0.304744\pi\)
0.575662 + 0.817688i \(0.304744\pi\)
\(294\) 0 0
\(295\) 2168.10 0.427903
\(296\) 0 0
\(297\) 102.500 0.0200257
\(298\) 0 0
\(299\) 300.612 0.0581433
\(300\) 0 0
\(301\) −489.308 −0.0936985
\(302\) 0 0
\(303\) 3412.35 0.646978
\(304\) 0 0
\(305\) 2194.75 0.412036
\(306\) 0 0
\(307\) 9182.94 1.70716 0.853580 0.520962i \(-0.174427\pi\)
0.853580 + 0.520962i \(0.174427\pi\)
\(308\) 0 0
\(309\) 2515.28 0.463073
\(310\) 0 0
\(311\) 5497.37 1.00234 0.501169 0.865349i \(-0.332903\pi\)
0.501169 + 0.865349i \(0.332903\pi\)
\(312\) 0 0
\(313\) −8163.92 −1.47429 −0.737144 0.675736i \(-0.763826\pi\)
−0.737144 + 0.675736i \(0.763826\pi\)
\(314\) 0 0
\(315\) −1137.67 −0.203493
\(316\) 0 0
\(317\) −7450.10 −1.32000 −0.659999 0.751267i \(-0.729443\pi\)
−0.659999 + 0.751267i \(0.729443\pi\)
\(318\) 0 0
\(319\) 30.5898 0.00536897
\(320\) 0 0
\(321\) 5255.76 0.913857
\(322\) 0 0
\(323\) 5424.74 0.934492
\(324\) 0 0
\(325\) 326.752 0.0557691
\(326\) 0 0
\(327\) 5251.30 0.888065
\(328\) 0 0
\(329\) 3711.73 0.621989
\(330\) 0 0
\(331\) 413.619 0.0686845 0.0343422 0.999410i \(-0.489066\pi\)
0.0343422 + 0.999410i \(0.489066\pi\)
\(332\) 0 0
\(333\) −7349.13 −1.20940
\(334\) 0 0
\(335\) 306.354 0.0499640
\(336\) 0 0
\(337\) 5197.04 0.840061 0.420031 0.907510i \(-0.362019\pi\)
0.420031 + 0.907510i \(0.362019\pi\)
\(338\) 0 0
\(339\) −1486.25 −0.238118
\(340\) 0 0
\(341\) −174.487 −0.0277097
\(342\) 0 0
\(343\) −6455.68 −1.01625
\(344\) 0 0
\(345\) 319.968 0.0499318
\(346\) 0 0
\(347\) 2307.38 0.356964 0.178482 0.983943i \(-0.442881\pi\)
0.178482 + 0.983943i \(0.442881\pi\)
\(348\) 0 0
\(349\) −6718.97 −1.03054 −0.515270 0.857028i \(-0.672308\pi\)
−0.515270 + 0.857028i \(0.672308\pi\)
\(350\) 0 0
\(351\) −1682.21 −0.255811
\(352\) 0 0
\(353\) 3970.93 0.598728 0.299364 0.954139i \(-0.403225\pi\)
0.299364 + 0.954139i \(0.403225\pi\)
\(354\) 0 0
\(355\) 3264.55 0.488068
\(356\) 0 0
\(357\) −3266.41 −0.484248
\(358\) 0 0
\(359\) 6042.26 0.888296 0.444148 0.895953i \(-0.353506\pi\)
0.444148 + 0.895953i \(0.353506\pi\)
\(360\) 0 0
\(361\) −3878.60 −0.565477
\(362\) 0 0
\(363\) −3701.52 −0.535204
\(364\) 0 0
\(365\) 1869.43 0.268083
\(366\) 0 0
\(367\) −1233.72 −0.175476 −0.0877381 0.996144i \(-0.527964\pi\)
−0.0877381 + 0.996144i \(0.527964\pi\)
\(368\) 0 0
\(369\) −5132.86 −0.724135
\(370\) 0 0
\(371\) 6289.18 0.880103
\(372\) 0 0
\(373\) 372.550 0.0517156 0.0258578 0.999666i \(-0.491768\pi\)
0.0258578 + 0.999666i \(0.491768\pi\)
\(374\) 0 0
\(375\) 347.791 0.0478929
\(376\) 0 0
\(377\) −502.035 −0.0685838
\(378\) 0 0
\(379\) −1488.11 −0.201686 −0.100843 0.994902i \(-0.532154\pi\)
−0.100843 + 0.994902i \(0.532154\pi\)
\(380\) 0 0
\(381\) −4882.29 −0.656503
\(382\) 0 0
\(383\) −6945.88 −0.926678 −0.463339 0.886181i \(-0.653349\pi\)
−0.463339 + 0.886181i \(0.653349\pi\)
\(384\) 0 0
\(385\) −47.0448 −0.00622760
\(386\) 0 0
\(387\) 797.606 0.104766
\(388\) 0 0
\(389\) 988.407 0.128828 0.0644141 0.997923i \(-0.479482\pi\)
0.0644141 + 0.997923i \(0.479482\pi\)
\(390\) 0 0
\(391\) −2285.44 −0.295600
\(392\) 0 0
\(393\) −2735.57 −0.351123
\(394\) 0 0
\(395\) −3583.20 −0.456431
\(396\) 0 0
\(397\) 7028.59 0.888552 0.444276 0.895890i \(-0.353461\pi\)
0.444276 + 0.895890i \(0.353461\pi\)
\(398\) 0 0
\(399\) −1794.59 −0.225168
\(400\) 0 0
\(401\) −8768.39 −1.09195 −0.545976 0.837801i \(-0.683841\pi\)
−0.545976 + 0.837801i \(0.683841\pi\)
\(402\) 0 0
\(403\) 2863.65 0.353967
\(404\) 0 0
\(405\) 809.395 0.0993065
\(406\) 0 0
\(407\) −303.901 −0.0370118
\(408\) 0 0
\(409\) 10656.9 1.28838 0.644191 0.764865i \(-0.277194\pi\)
0.644191 + 0.764865i \(0.277194\pi\)
\(410\) 0 0
\(411\) 8832.39 1.06002
\(412\) 0 0
\(413\) 5123.05 0.610384
\(414\) 0 0
\(415\) −6205.07 −0.733964
\(416\) 0 0
\(417\) −5861.76 −0.688373
\(418\) 0 0
\(419\) −13788.2 −1.60764 −0.803818 0.594875i \(-0.797202\pi\)
−0.803818 + 0.594875i \(0.797202\pi\)
\(420\) 0 0
\(421\) −1826.43 −0.211436 −0.105718 0.994396i \(-0.533714\pi\)
−0.105718 + 0.994396i \(0.533714\pi\)
\(422\) 0 0
\(423\) −6050.38 −0.695460
\(424\) 0 0
\(425\) −2484.17 −0.283530
\(426\) 0 0
\(427\) 5186.03 0.587750
\(428\) 0 0
\(429\) −28.9607 −0.00325929
\(430\) 0 0
\(431\) 9130.86 1.02046 0.510230 0.860038i \(-0.329560\pi\)
0.510230 + 0.860038i \(0.329560\pi\)
\(432\) 0 0
\(433\) 10619.7 1.17864 0.589318 0.807901i \(-0.299396\pi\)
0.589318 + 0.807901i \(0.299396\pi\)
\(434\) 0 0
\(435\) −534.360 −0.0588979
\(436\) 0 0
\(437\) −1255.64 −0.137449
\(438\) 0 0
\(439\) −17399.5 −1.89165 −0.945826 0.324675i \(-0.894745\pi\)
−0.945826 + 0.324675i \(0.894745\pi\)
\(440\) 0 0
\(441\) 3917.49 0.423009
\(442\) 0 0
\(443\) 1827.41 0.195988 0.0979942 0.995187i \(-0.468757\pi\)
0.0979942 + 0.995187i \(0.468757\pi\)
\(444\) 0 0
\(445\) −2382.04 −0.253752
\(446\) 0 0
\(447\) 8983.64 0.950585
\(448\) 0 0
\(449\) −11699.1 −1.22966 −0.614828 0.788661i \(-0.710775\pi\)
−0.614828 + 0.788661i \(0.710775\pi\)
\(450\) 0 0
\(451\) −212.254 −0.0221610
\(452\) 0 0
\(453\) −5128.43 −0.531908
\(454\) 0 0
\(455\) 772.091 0.0795521
\(456\) 0 0
\(457\) −7472.63 −0.764890 −0.382445 0.923978i \(-0.624918\pi\)
−0.382445 + 0.923978i \(0.624918\pi\)
\(458\) 0 0
\(459\) 12789.2 1.30054
\(460\) 0 0
\(461\) −10572.2 −1.06811 −0.534053 0.845451i \(-0.679332\pi\)
−0.534053 + 0.845451i \(0.679332\pi\)
\(462\) 0 0
\(463\) −14833.5 −1.48892 −0.744461 0.667666i \(-0.767293\pi\)
−0.744461 + 0.667666i \(0.767293\pi\)
\(464\) 0 0
\(465\) 3048.03 0.303977
\(466\) 0 0
\(467\) 10795.7 1.06974 0.534868 0.844936i \(-0.320361\pi\)
0.534868 + 0.844936i \(0.320361\pi\)
\(468\) 0 0
\(469\) 723.892 0.0712713
\(470\) 0 0
\(471\) 3073.34 0.300662
\(472\) 0 0
\(473\) 32.9825 0.00320621
\(474\) 0 0
\(475\) −1364.83 −0.131837
\(476\) 0 0
\(477\) −10251.8 −0.984062
\(478\) 0 0
\(479\) 18591.9 1.77346 0.886730 0.462287i \(-0.152971\pi\)
0.886730 + 0.462287i \(0.152971\pi\)
\(480\) 0 0
\(481\) 4987.57 0.472793
\(482\) 0 0
\(483\) 756.060 0.0712255
\(484\) 0 0
\(485\) 3508.85 0.328512
\(486\) 0 0
\(487\) −2239.82 −0.208410 −0.104205 0.994556i \(-0.533230\pi\)
−0.104205 + 0.994556i \(0.533230\pi\)
\(488\) 0 0
\(489\) −5322.50 −0.492212
\(490\) 0 0
\(491\) 20077.9 1.84543 0.922714 0.385486i \(-0.125966\pi\)
0.922714 + 0.385486i \(0.125966\pi\)
\(492\) 0 0
\(493\) 3816.78 0.348680
\(494\) 0 0
\(495\) 76.6862 0.00696321
\(496\) 0 0
\(497\) 7713.88 0.696207
\(498\) 0 0
\(499\) 6629.39 0.594734 0.297367 0.954763i \(-0.403892\pi\)
0.297367 + 0.954763i \(0.403892\pi\)
\(500\) 0 0
\(501\) −717.383 −0.0639727
\(502\) 0 0
\(503\) 17560.4 1.55662 0.778308 0.627882i \(-0.216078\pi\)
0.778308 + 0.627882i \(0.216078\pi\)
\(504\) 0 0
\(505\) 6132.19 0.540354
\(506\) 0 0
\(507\) −5637.48 −0.493825
\(508\) 0 0
\(509\) 754.641 0.0657149 0.0328574 0.999460i \(-0.489539\pi\)
0.0328574 + 0.999460i \(0.489539\pi\)
\(510\) 0 0
\(511\) 4417.32 0.382408
\(512\) 0 0
\(513\) 7026.49 0.604731
\(514\) 0 0
\(515\) 4520.10 0.386756
\(516\) 0 0
\(517\) −250.195 −0.0212835
\(518\) 0 0
\(519\) 10431.4 0.882249
\(520\) 0 0
\(521\) 9578.48 0.805452 0.402726 0.915320i \(-0.368063\pi\)
0.402726 + 0.915320i \(0.368063\pi\)
\(522\) 0 0
\(523\) −17200.1 −1.43806 −0.719031 0.694978i \(-0.755414\pi\)
−0.719031 + 0.694978i \(0.755414\pi\)
\(524\) 0 0
\(525\) 821.804 0.0683171
\(526\) 0 0
\(527\) −21771.3 −1.79957
\(528\) 0 0
\(529\) 529.000 0.0434783
\(530\) 0 0
\(531\) −8350.92 −0.682484
\(532\) 0 0
\(533\) 3483.47 0.283088
\(534\) 0 0
\(535\) 9444.89 0.763249
\(536\) 0 0
\(537\) −5678.02 −0.456285
\(538\) 0 0
\(539\) 161.996 0.0129455
\(540\) 0 0
\(541\) 13456.9 1.06942 0.534712 0.845034i \(-0.320420\pi\)
0.534712 + 0.845034i \(0.320420\pi\)
\(542\) 0 0
\(543\) −10925.3 −0.863440
\(544\) 0 0
\(545\) 9436.87 0.741709
\(546\) 0 0
\(547\) 20923.9 1.63554 0.817771 0.575543i \(-0.195209\pi\)
0.817771 + 0.575543i \(0.195209\pi\)
\(548\) 0 0
\(549\) −8453.58 −0.657176
\(550\) 0 0
\(551\) 2096.97 0.162131
\(552\) 0 0
\(553\) −8466.84 −0.651079
\(554\) 0 0
\(555\) 5308.71 0.406022
\(556\) 0 0
\(557\) −7522.86 −0.572269 −0.286135 0.958189i \(-0.592370\pi\)
−0.286135 + 0.958189i \(0.592370\pi\)
\(558\) 0 0
\(559\) −541.304 −0.0409566
\(560\) 0 0
\(561\) 220.177 0.0165702
\(562\) 0 0
\(563\) −18991.9 −1.42169 −0.710845 0.703349i \(-0.751687\pi\)
−0.710845 + 0.703349i \(0.751687\pi\)
\(564\) 0 0
\(565\) −2670.88 −0.198875
\(566\) 0 0
\(567\) 1912.54 0.141656
\(568\) 0 0
\(569\) 5469.43 0.402971 0.201485 0.979492i \(-0.435423\pi\)
0.201485 + 0.979492i \(0.435423\pi\)
\(570\) 0 0
\(571\) 6785.97 0.497345 0.248673 0.968588i \(-0.420006\pi\)
0.248673 + 0.968588i \(0.420006\pi\)
\(572\) 0 0
\(573\) −6495.32 −0.473553
\(574\) 0 0
\(575\) 575.000 0.0417029
\(576\) 0 0
\(577\) −10572.1 −0.762778 −0.381389 0.924415i \(-0.624554\pi\)
−0.381389 + 0.924415i \(0.624554\pi\)
\(578\) 0 0
\(579\) 12232.3 0.877992
\(580\) 0 0
\(581\) −14662.1 −1.04697
\(582\) 0 0
\(583\) −423.932 −0.0301157
\(584\) 0 0
\(585\) −1258.56 −0.0889489
\(586\) 0 0
\(587\) −7392.04 −0.519765 −0.259883 0.965640i \(-0.583684\pi\)
−0.259883 + 0.965640i \(0.583684\pi\)
\(588\) 0 0
\(589\) −11961.3 −0.836769
\(590\) 0 0
\(591\) −6993.43 −0.486753
\(592\) 0 0
\(593\) 9214.68 0.638114 0.319057 0.947735i \(-0.396634\pi\)
0.319057 + 0.947735i \(0.396634\pi\)
\(594\) 0 0
\(595\) −5869.92 −0.404442
\(596\) 0 0
\(597\) 5056.31 0.346635
\(598\) 0 0
\(599\) 7519.28 0.512904 0.256452 0.966557i \(-0.417446\pi\)
0.256452 + 0.966557i \(0.417446\pi\)
\(600\) 0 0
\(601\) 19066.1 1.29405 0.647025 0.762469i \(-0.276013\pi\)
0.647025 + 0.762469i \(0.276013\pi\)
\(602\) 0 0
\(603\) −1179.99 −0.0796900
\(604\) 0 0
\(605\) −6651.83 −0.447000
\(606\) 0 0
\(607\) 16134.2 1.07886 0.539428 0.842032i \(-0.318640\pi\)
0.539428 + 0.842032i \(0.318640\pi\)
\(608\) 0 0
\(609\) −1262.65 −0.0840152
\(610\) 0 0
\(611\) 4106.16 0.271878
\(612\) 0 0
\(613\) 4349.49 0.286581 0.143291 0.989681i \(-0.454232\pi\)
0.143291 + 0.989681i \(0.454232\pi\)
\(614\) 0 0
\(615\) 3707.76 0.243108
\(616\) 0 0
\(617\) 18710.3 1.22082 0.610412 0.792084i \(-0.291004\pi\)
0.610412 + 0.792084i \(0.291004\pi\)
\(618\) 0 0
\(619\) −18079.4 −1.17395 −0.586973 0.809606i \(-0.699681\pi\)
−0.586973 + 0.809606i \(0.699681\pi\)
\(620\) 0 0
\(621\) −2960.26 −0.191290
\(622\) 0 0
\(623\) −5628.58 −0.361965
\(624\) 0 0
\(625\) 625.000 0.0400000
\(626\) 0 0
\(627\) 120.967 0.00770488
\(628\) 0 0
\(629\) −37918.6 −2.40368
\(630\) 0 0
\(631\) −1360.87 −0.0858565 −0.0429282 0.999078i \(-0.513669\pi\)
−0.0429282 + 0.999078i \(0.513669\pi\)
\(632\) 0 0
\(633\) 8969.62 0.563208
\(634\) 0 0
\(635\) −8773.75 −0.548308
\(636\) 0 0
\(637\) −2658.65 −0.165368
\(638\) 0 0
\(639\) −12574.1 −0.778444
\(640\) 0 0
\(641\) 10888.4 0.670927 0.335464 0.942053i \(-0.391107\pi\)
0.335464 + 0.942053i \(0.391107\pi\)
\(642\) 0 0
\(643\) −16758.6 −1.02783 −0.513915 0.857841i \(-0.671806\pi\)
−0.513915 + 0.857841i \(0.671806\pi\)
\(644\) 0 0
\(645\) −576.157 −0.0351724
\(646\) 0 0
\(647\) −415.017 −0.0252180 −0.0126090 0.999921i \(-0.504014\pi\)
−0.0126090 + 0.999921i \(0.504014\pi\)
\(648\) 0 0
\(649\) −345.327 −0.0208864
\(650\) 0 0
\(651\) 7202.28 0.433609
\(652\) 0 0
\(653\) −25781.0 −1.54501 −0.772503 0.635011i \(-0.780995\pi\)
−0.772503 + 0.635011i \(0.780995\pi\)
\(654\) 0 0
\(655\) −4915.97 −0.293256
\(656\) 0 0
\(657\) −7200.53 −0.427579
\(658\) 0 0
\(659\) 8667.93 0.512374 0.256187 0.966627i \(-0.417534\pi\)
0.256187 + 0.966627i \(0.417534\pi\)
\(660\) 0 0
\(661\) −10039.2 −0.590740 −0.295370 0.955383i \(-0.595443\pi\)
−0.295370 + 0.955383i \(0.595443\pi\)
\(662\) 0 0
\(663\) −3613.51 −0.211670
\(664\) 0 0
\(665\) −3224.98 −0.188059
\(666\) 0 0
\(667\) −883.452 −0.0512855
\(668\) 0 0
\(669\) 3895.07 0.225100
\(670\) 0 0
\(671\) −349.572 −0.0201119
\(672\) 0 0
\(673\) 24776.4 1.41911 0.709554 0.704651i \(-0.248896\pi\)
0.709554 + 0.704651i \(0.248896\pi\)
\(674\) 0 0
\(675\) −3217.67 −0.183479
\(676\) 0 0
\(677\) −5458.64 −0.309885 −0.154943 0.987923i \(-0.549519\pi\)
−0.154943 + 0.987923i \(0.549519\pi\)
\(678\) 0 0
\(679\) 8291.14 0.468608
\(680\) 0 0
\(681\) −12902.9 −0.726051
\(682\) 0 0
\(683\) −22846.9 −1.27996 −0.639981 0.768391i \(-0.721058\pi\)
−0.639981 + 0.768391i \(0.721058\pi\)
\(684\) 0 0
\(685\) 15872.3 0.885327
\(686\) 0 0
\(687\) −7445.80 −0.413500
\(688\) 0 0
\(689\) 6957.50 0.384702
\(690\) 0 0
\(691\) 9932.95 0.546841 0.273421 0.961895i \(-0.411845\pi\)
0.273421 + 0.961895i \(0.411845\pi\)
\(692\) 0 0
\(693\) 181.204 0.00993270
\(694\) 0 0
\(695\) −10533.9 −0.574926
\(696\) 0 0
\(697\) −26483.5 −1.43922
\(698\) 0 0
\(699\) 332.111 0.0179708
\(700\) 0 0
\(701\) −3272.30 −0.176310 −0.0881549 0.996107i \(-0.528097\pi\)
−0.0881549 + 0.996107i \(0.528097\pi\)
\(702\) 0 0
\(703\) −20832.8 −1.11767
\(704\) 0 0
\(705\) 4370.54 0.233481
\(706\) 0 0
\(707\) 14489.9 0.770790
\(708\) 0 0
\(709\) 11979.8 0.634573 0.317287 0.948330i \(-0.397228\pi\)
0.317287 + 0.948330i \(0.397228\pi\)
\(710\) 0 0
\(711\) 13801.5 0.727985
\(712\) 0 0
\(713\) 5039.29 0.264688
\(714\) 0 0
\(715\) −52.0439 −0.00272214
\(716\) 0 0
\(717\) 3139.35 0.163516
\(718\) 0 0
\(719\) −33489.7 −1.73707 −0.868537 0.495625i \(-0.834939\pi\)
−0.868537 + 0.495625i \(0.834939\pi\)
\(720\) 0 0
\(721\) 10680.7 0.551690
\(722\) 0 0
\(723\) −5412.70 −0.278424
\(724\) 0 0
\(725\) −960.274 −0.0491913
\(726\) 0 0
\(727\) −26092.7 −1.33112 −0.665559 0.746345i \(-0.731807\pi\)
−0.665559 + 0.746345i \(0.731807\pi\)
\(728\) 0 0
\(729\) 6551.25 0.332838
\(730\) 0 0
\(731\) 4115.33 0.208223
\(732\) 0 0
\(733\) 1802.12 0.0908088 0.0454044 0.998969i \(-0.485542\pi\)
0.0454044 + 0.998969i \(0.485542\pi\)
\(734\) 0 0
\(735\) −2829.83 −0.142013
\(736\) 0 0
\(737\) −48.7950 −0.00243879
\(738\) 0 0
\(739\) −17571.0 −0.874639 −0.437319 0.899306i \(-0.644072\pi\)
−0.437319 + 0.899306i \(0.644072\pi\)
\(740\) 0 0
\(741\) −1985.29 −0.0984230
\(742\) 0 0
\(743\) −9110.13 −0.449823 −0.224911 0.974379i \(-0.572209\pi\)
−0.224911 + 0.974379i \(0.572209\pi\)
\(744\) 0 0
\(745\) 16144.1 0.793925
\(746\) 0 0
\(747\) 23900.3 1.17064
\(748\) 0 0
\(749\) 22317.6 1.08874
\(750\) 0 0
\(751\) −5447.61 −0.264695 −0.132348 0.991203i \(-0.542252\pi\)
−0.132348 + 0.991203i \(0.542252\pi\)
\(752\) 0 0
\(753\) 9333.66 0.451710
\(754\) 0 0
\(755\) −9216.07 −0.444248
\(756\) 0 0
\(757\) 13631.5 0.654486 0.327243 0.944940i \(-0.393880\pi\)
0.327243 + 0.944940i \(0.393880\pi\)
\(758\) 0 0
\(759\) −50.9633 −0.00243722
\(760\) 0 0
\(761\) −5851.77 −0.278747 −0.139374 0.990240i \(-0.544509\pi\)
−0.139374 + 0.990240i \(0.544509\pi\)
\(762\) 0 0
\(763\) 22298.6 1.05801
\(764\) 0 0
\(765\) 9568.37 0.452216
\(766\) 0 0
\(767\) 5667.44 0.266805
\(768\) 0 0
\(769\) −36459.9 −1.70972 −0.854862 0.518856i \(-0.826358\pi\)
−0.854862 + 0.518856i \(0.826358\pi\)
\(770\) 0 0
\(771\) −8855.68 −0.413657
\(772\) 0 0
\(773\) −28659.7 −1.33353 −0.666765 0.745268i \(-0.732321\pi\)
−0.666765 + 0.745268i \(0.732321\pi\)
\(774\) 0 0
\(775\) 5477.49 0.253880
\(776\) 0 0
\(777\) 12544.1 0.579171
\(778\) 0 0
\(779\) −14550.2 −0.669213
\(780\) 0 0
\(781\) −519.965 −0.0238231
\(782\) 0 0
\(783\) 4943.75 0.225639
\(784\) 0 0
\(785\) 5522.96 0.251112
\(786\) 0 0
\(787\) −43165.1 −1.95511 −0.977553 0.210691i \(-0.932429\pi\)
−0.977553 + 0.210691i \(0.932429\pi\)
\(788\) 0 0
\(789\) 602.680 0.0271939
\(790\) 0 0
\(791\) −6311.08 −0.283687
\(792\) 0 0
\(793\) 5737.11 0.256911
\(794\) 0 0
\(795\) 7405.47 0.330371
\(796\) 0 0
\(797\) −32530.8 −1.44580 −0.722898 0.690955i \(-0.757190\pi\)
−0.722898 + 0.690955i \(0.757190\pi\)
\(798\) 0 0
\(799\) −31217.6 −1.38222
\(800\) 0 0
\(801\) 9174.97 0.404721
\(802\) 0 0
\(803\) −297.756 −0.0130854
\(804\) 0 0
\(805\) 1358.68 0.0594873
\(806\) 0 0
\(807\) 8183.69 0.356976
\(808\) 0 0
\(809\) −27629.4 −1.20074 −0.600369 0.799723i \(-0.704980\pi\)
−0.600369 + 0.799723i \(0.704980\pi\)
\(810\) 0 0
\(811\) −26826.1 −1.16152 −0.580758 0.814076i \(-0.697244\pi\)
−0.580758 + 0.814076i \(0.697244\pi\)
\(812\) 0 0
\(813\) 15850.9 0.683782
\(814\) 0 0
\(815\) −9564.83 −0.411094
\(816\) 0 0
\(817\) 2260.99 0.0968203
\(818\) 0 0
\(819\) −2973.89 −0.126882
\(820\) 0 0
\(821\) 647.832 0.0275390 0.0137695 0.999905i \(-0.495617\pi\)
0.0137695 + 0.999905i \(0.495617\pi\)
\(822\) 0 0
\(823\) 8631.34 0.365577 0.182788 0.983152i \(-0.441488\pi\)
0.182788 + 0.983152i \(0.441488\pi\)
\(824\) 0 0
\(825\) −55.3949 −0.00233770
\(826\) 0 0
\(827\) −32459.8 −1.36486 −0.682430 0.730951i \(-0.739077\pi\)
−0.682430 + 0.730951i \(0.739077\pi\)
\(828\) 0 0
\(829\) 7056.18 0.295623 0.147811 0.989016i \(-0.452777\pi\)
0.147811 + 0.989016i \(0.452777\pi\)
\(830\) 0 0
\(831\) 19978.7 0.833998
\(832\) 0 0
\(833\) 20212.7 0.840730
\(834\) 0 0
\(835\) −1289.18 −0.0534297
\(836\) 0 0
\(837\) −28199.6 −1.16454
\(838\) 0 0
\(839\) 23143.8 0.952341 0.476171 0.879353i \(-0.342024\pi\)
0.476171 + 0.879353i \(0.342024\pi\)
\(840\) 0 0
\(841\) −22913.6 −0.939505
\(842\) 0 0
\(843\) 1882.59 0.0769158
\(844\) 0 0
\(845\) −10130.9 −0.412441
\(846\) 0 0
\(847\) −15717.8 −0.637626
\(848\) 0 0
\(849\) −9776.84 −0.395218
\(850\) 0 0
\(851\) 8776.84 0.353544
\(852\) 0 0
\(853\) 27135.9 1.08923 0.544616 0.838686i \(-0.316676\pi\)
0.544616 + 0.838686i \(0.316676\pi\)
\(854\) 0 0
\(855\) 5256.94 0.210273
\(856\) 0 0
\(857\) −45885.3 −1.82895 −0.914475 0.404642i \(-0.867396\pi\)
−0.914475 + 0.404642i \(0.867396\pi\)
\(858\) 0 0
\(859\) −39582.8 −1.57223 −0.786116 0.618079i \(-0.787911\pi\)
−0.786116 + 0.618079i \(0.787911\pi\)
\(860\) 0 0
\(861\) 8761.16 0.346782
\(862\) 0 0
\(863\) −27037.8 −1.06648 −0.533242 0.845963i \(-0.679027\pi\)
−0.533242 + 0.845963i \(0.679027\pi\)
\(864\) 0 0
\(865\) 18745.8 0.736851
\(866\) 0 0
\(867\) 13802.6 0.540669
\(868\) 0 0
\(869\) 570.719 0.0222789
\(870\) 0 0
\(871\) 800.816 0.0311534
\(872\) 0 0
\(873\) −13515.1 −0.523961
\(874\) 0 0
\(875\) 1476.83 0.0570582
\(876\) 0 0
\(877\) 17048.2 0.656416 0.328208 0.944605i \(-0.393555\pi\)
0.328208 + 0.944605i \(0.393555\pi\)
\(878\) 0 0
\(879\) 16066.0 0.616488
\(880\) 0 0
\(881\) 41694.6 1.59447 0.797235 0.603669i \(-0.206295\pi\)
0.797235 + 0.603669i \(0.206295\pi\)
\(882\) 0 0
\(883\) −43429.5 −1.65517 −0.827587 0.561338i \(-0.810287\pi\)
−0.827587 + 0.561338i \(0.810287\pi\)
\(884\) 0 0
\(885\) 6032.35 0.229125
\(886\) 0 0
\(887\) −25491.5 −0.964960 −0.482480 0.875907i \(-0.660264\pi\)
−0.482480 + 0.875907i \(0.660264\pi\)
\(888\) 0 0
\(889\) −20731.7 −0.782137
\(890\) 0 0
\(891\) −128.917 −0.00484725
\(892\) 0 0
\(893\) −17151.2 −0.642712
\(894\) 0 0
\(895\) −10203.7 −0.381087
\(896\) 0 0
\(897\) 836.402 0.0311334
\(898\) 0 0
\(899\) −8415.82 −0.312217
\(900\) 0 0
\(901\) −52895.2 −1.95582
\(902\) 0 0
\(903\) −1361.42 −0.0501718
\(904\) 0 0
\(905\) −19633.3 −0.721141
\(906\) 0 0
\(907\) −27372.8 −1.00209 −0.501046 0.865420i \(-0.667051\pi\)
−0.501046 + 0.865420i \(0.667051\pi\)
\(908\) 0 0
\(909\) −23619.5 −0.861837
\(910\) 0 0
\(911\) 36324.8 1.32107 0.660535 0.750796i \(-0.270330\pi\)
0.660535 + 0.750796i \(0.270330\pi\)
\(912\) 0 0
\(913\) 988.322 0.0358255
\(914\) 0 0
\(915\) 6106.51 0.220628
\(916\) 0 0
\(917\) −11616.1 −0.418317
\(918\) 0 0
\(919\) 19730.6 0.708217 0.354108 0.935204i \(-0.384784\pi\)
0.354108 + 0.935204i \(0.384784\pi\)
\(920\) 0 0
\(921\) 25550.0 0.914115
\(922\) 0 0
\(923\) 8533.58 0.304319
\(924\) 0 0
\(925\) 9540.04 0.339108
\(926\) 0 0
\(927\) −17410.2 −0.616857
\(928\) 0 0
\(929\) 41756.0 1.47467 0.737336 0.675526i \(-0.236083\pi\)
0.737336 + 0.675526i \(0.236083\pi\)
\(930\) 0 0
\(931\) 11105.0 0.390926
\(932\) 0 0
\(933\) 15295.5 0.536712
\(934\) 0 0
\(935\) 395.670 0.0138394
\(936\) 0 0
\(937\) 9708.13 0.338475 0.169237 0.985575i \(-0.445870\pi\)
0.169237 + 0.985575i \(0.445870\pi\)
\(938\) 0 0
\(939\) −22714.7 −0.789421
\(940\) 0 0
\(941\) −4191.60 −0.145210 −0.0726049 0.997361i \(-0.523131\pi\)
−0.0726049 + 0.997361i \(0.523131\pi\)
\(942\) 0 0
\(943\) 6130.01 0.211687
\(944\) 0 0
\(945\) −7603.11 −0.261724
\(946\) 0 0
\(947\) 5517.16 0.189318 0.0946588 0.995510i \(-0.469824\pi\)
0.0946588 + 0.995510i \(0.469824\pi\)
\(948\) 0 0
\(949\) 4886.72 0.167154
\(950\) 0 0
\(951\) −20728.6 −0.706805
\(952\) 0 0
\(953\) 42788.0 1.45440 0.727198 0.686427i \(-0.240822\pi\)
0.727198 + 0.686427i \(0.240822\pi\)
\(954\) 0 0
\(955\) −11672.5 −0.395510
\(956\) 0 0
\(957\) 85.1109 0.00287486
\(958\) 0 0
\(959\) 37505.0 1.26288
\(960\) 0 0
\(961\) 18213.6 0.611379
\(962\) 0 0
\(963\) −36379.2 −1.21734
\(964\) 0 0
\(965\) 21982.1 0.733295
\(966\) 0 0
\(967\) 8776.98 0.291881 0.145940 0.989293i \(-0.453379\pi\)
0.145940 + 0.989293i \(0.453379\pi\)
\(968\) 0 0
\(969\) 15093.4 0.500382
\(970\) 0 0
\(971\) −23660.2 −0.781968 −0.390984 0.920397i \(-0.627865\pi\)
−0.390984 + 0.920397i \(0.627865\pi\)
\(972\) 0 0
\(973\) −24890.8 −0.820106
\(974\) 0 0
\(975\) 909.132 0.0298621
\(976\) 0 0
\(977\) −9077.16 −0.297241 −0.148620 0.988894i \(-0.547483\pi\)
−0.148620 + 0.988894i \(0.547483\pi\)
\(978\) 0 0
\(979\) 379.403 0.0123859
\(980\) 0 0
\(981\) −36348.3 −1.18299
\(982\) 0 0
\(983\) 56743.7 1.84114 0.920571 0.390576i \(-0.127724\pi\)
0.920571 + 0.390576i \(0.127724\pi\)
\(984\) 0 0
\(985\) −12567.6 −0.406534
\(986\) 0 0
\(987\) 10327.3 0.333050
\(988\) 0 0
\(989\) −952.556 −0.0306264
\(990\) 0 0
\(991\) −20911.3 −0.670302 −0.335151 0.942165i \(-0.608787\pi\)
−0.335151 + 0.942165i \(0.608787\pi\)
\(992\) 0 0
\(993\) 1150.82 0.0367778
\(994\) 0 0
\(995\) 9086.47 0.289508
\(996\) 0 0
\(997\) −34257.9 −1.08822 −0.544112 0.839013i \(-0.683133\pi\)
−0.544112 + 0.839013i \(0.683133\pi\)
\(998\) 0 0
\(999\) −49114.7 −1.55548
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 1840.4.a.ba.1.7 10
4.3 odd 2 920.4.a.h.1.4 10
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
920.4.a.h.1.4 10 4.3 odd 2
1840.4.a.ba.1.7 10 1.1 even 1 trivial