Properties

Label 4140.3.c.a.4049.13
Level $4140$
Weight $3$
Character 4140.4049
Analytic conductor $112.807$
Analytic rank $0$
Dimension $88$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [4140,3,Mod(4049,4140)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4140, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 1, 0]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("4140.4049");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4140 = 2^{2} \cdot 3^{2} \cdot 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 4140.c (of order \(2\), degree \(1\), minimal)

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: no
Analytic conductor: \(112.806829445\)
Analytic rank: \(0\)
Dimension: \(88\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 4049.13
Character \(\chi\) \(=\) 4140.4049
Dual form 4140.3.c.a.4049.14

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-4.49878 - 2.18197i) q^{5} -10.0913i q^{7} +O(q^{10})\) \(q+(-4.49878 - 2.18197i) q^{5} -10.0913i q^{7} +0.456013i q^{11} +0.213952i q^{13} +5.67892 q^{17} +25.0820 q^{19} +4.79583 q^{23} +(15.4780 + 19.6324i) q^{25} -50.8783i q^{29} +14.3745 q^{31} +(-22.0189 + 45.3985i) q^{35} +64.5468i q^{37} -47.9986i q^{41} -0.886502i q^{43} +61.7325 q^{47} -52.8342 q^{49} +74.2027 q^{53} +(0.995007 - 2.05150i) q^{55} +20.8696i q^{59} +72.4855 q^{61} +(0.466837 - 0.962522i) q^{65} +47.2709i q^{67} +77.6702i q^{71} +76.1012i q^{73} +4.60176 q^{77} +123.462 q^{79} -116.582 q^{83} +(-25.5482 - 12.3912i) q^{85} +52.0421i q^{89} +2.15905 q^{91} +(-112.838 - 54.7282i) q^{95} -15.7460i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 88 q+O(q^{10}) \) Copy content Toggle raw display \( 88 q - 16 q^{19} - 48 q^{25} + 272 q^{31} - 600 q^{49} + 112 q^{55} + 448 q^{61} - 32 q^{79} - 264 q^{85} - 16 q^{91}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/4140\mathbb{Z}\right)^\times\).

\(n\) \(461\) \(1657\) \(2071\) \(3961\)
\(\chi(n)\) \(-1\) \(-1\) \(1\) \(1\)

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 0
\(4\) 0 0
\(5\) −4.49878 2.18197i −0.899756 0.436394i
\(6\) 0 0
\(7\) 10.0913i 1.44161i −0.693136 0.720807i \(-0.743772\pi\)
0.693136 0.720807i \(-0.256228\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 0.456013i 0.0414557i 0.999785 + 0.0207279i \(0.00659836\pi\)
−0.999785 + 0.0207279i \(0.993402\pi\)
\(12\) 0 0
\(13\) 0.213952i 0.0164578i 0.999966 + 0.00822892i \(0.00261938\pi\)
−0.999966 + 0.00822892i \(0.997381\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 5.67892 0.334054 0.167027 0.985952i \(-0.446583\pi\)
0.167027 + 0.985952i \(0.446583\pi\)
\(18\) 0 0
\(19\) 25.0820 1.32011 0.660053 0.751219i \(-0.270534\pi\)
0.660053 + 0.751219i \(0.270534\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 4.79583 0.208514
\(24\) 0 0
\(25\) 15.4780 + 19.6324i 0.619120 + 0.785296i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 50.8783i 1.75443i −0.480102 0.877213i \(-0.659400\pi\)
0.480102 0.877213i \(-0.340600\pi\)
\(30\) 0 0
\(31\) 14.3745 0.463695 0.231847 0.972752i \(-0.425523\pi\)
0.231847 + 0.972752i \(0.425523\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −22.0189 + 45.3985i −0.629112 + 1.29710i
\(36\) 0 0
\(37\) 64.5468i 1.74451i 0.489052 + 0.872255i \(0.337343\pi\)
−0.489052 + 0.872255i \(0.662657\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 47.9986i 1.17070i −0.810782 0.585348i \(-0.800958\pi\)
0.810782 0.585348i \(-0.199042\pi\)
\(42\) 0 0
\(43\) 0.886502i 0.0206163i −0.999947 0.0103082i \(-0.996719\pi\)
0.999947 0.0103082i \(-0.00328125\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 61.7325 1.31346 0.656729 0.754127i \(-0.271940\pi\)
0.656729 + 0.754127i \(0.271940\pi\)
\(48\) 0 0
\(49\) −52.8342 −1.07825
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 74.2027 1.40005 0.700025 0.714118i \(-0.253172\pi\)
0.700025 + 0.714118i \(0.253172\pi\)
\(54\) 0 0
\(55\) 0.995007 2.05150i 0.0180910 0.0373000i
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 20.8696i 0.353722i 0.984236 + 0.176861i \(0.0565944\pi\)
−0.984236 + 0.176861i \(0.943406\pi\)
\(60\) 0 0
\(61\) 72.4855 1.18829 0.594144 0.804359i \(-0.297491\pi\)
0.594144 + 0.804359i \(0.297491\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 0.466837 0.962522i 0.00718211 0.0148080i
\(66\) 0 0
\(67\) 47.2709i 0.705536i 0.935711 + 0.352768i \(0.114759\pi\)
−0.935711 + 0.352768i \(0.885241\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 77.6702i 1.09395i 0.837150 + 0.546973i \(0.184220\pi\)
−0.837150 + 0.546973i \(0.815780\pi\)
\(72\) 0 0
\(73\) 76.1012i 1.04248i 0.853410 + 0.521241i \(0.174531\pi\)
−0.853410 + 0.521241i \(0.825469\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 4.60176 0.0597631
\(78\) 0 0
\(79\) 123.462 1.56280 0.781402 0.624028i \(-0.214505\pi\)
0.781402 + 0.624028i \(0.214505\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) −116.582 −1.40460 −0.702302 0.711879i \(-0.747844\pi\)
−0.702302 + 0.711879i \(0.747844\pi\)
\(84\) 0 0
\(85\) −25.5482 12.3912i −0.300567 0.145779i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 52.0421i 0.584743i 0.956305 + 0.292371i \(0.0944443\pi\)
−0.956305 + 0.292371i \(0.905556\pi\)
\(90\) 0 0
\(91\) 2.15905 0.0237258
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −112.838 54.7282i −1.18777 0.576086i
\(96\) 0 0
\(97\) 15.7460i 0.162330i −0.996701 0.0811650i \(-0.974136\pi\)
0.996701 0.0811650i \(-0.0258641\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 46.8637i 0.463997i −0.972716 0.231999i \(-0.925474\pi\)
0.972716 0.231999i \(-0.0745265\pi\)
\(102\) 0 0
\(103\) 89.5615i 0.869529i 0.900544 + 0.434764i \(0.143168\pi\)
−0.900544 + 0.434764i \(0.856832\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 28.6251 0.267525 0.133762 0.991013i \(-0.457294\pi\)
0.133762 + 0.991013i \(0.457294\pi\)
\(108\) 0 0
\(109\) 188.237 1.72695 0.863474 0.504393i \(-0.168284\pi\)
0.863474 + 0.504393i \(0.168284\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) −63.0780 −0.558212 −0.279106 0.960260i \(-0.590038\pi\)
−0.279106 + 0.960260i \(0.590038\pi\)
\(114\) 0 0
\(115\) −21.5754 10.4644i −0.187612 0.0909945i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 57.3077i 0.481577i
\(120\) 0 0
\(121\) 120.792 0.998281
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) −26.7948 122.094i −0.214358 0.976755i
\(126\) 0 0
\(127\) 209.870i 1.65252i −0.563289 0.826260i \(-0.690464\pi\)
0.563289 0.826260i \(-0.309536\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 117.183i 0.894527i −0.894402 0.447264i \(-0.852399\pi\)
0.894402 0.447264i \(-0.147601\pi\)
\(132\) 0 0
\(133\) 253.110i 1.90308i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 63.8768 0.466254 0.233127 0.972446i \(-0.425104\pi\)
0.233127 + 0.972446i \(0.425104\pi\)
\(138\) 0 0
\(139\) 92.0960 0.662561 0.331281 0.943532i \(-0.392519\pi\)
0.331281 + 0.943532i \(0.392519\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) −0.0975649 −0.000682272
\(144\) 0 0
\(145\) −111.015 + 228.890i −0.765621 + 1.57855i
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 127.096i 0.852990i 0.904490 + 0.426495i \(0.140252\pi\)
−0.904490 + 0.426495i \(0.859748\pi\)
\(150\) 0 0
\(151\) −251.780 −1.66741 −0.833707 0.552207i \(-0.813786\pi\)
−0.833707 + 0.552207i \(0.813786\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) −64.6679 31.3648i −0.417212 0.202354i
\(156\) 0 0
\(157\) 195.073i 1.24251i −0.783610 0.621253i \(-0.786624\pi\)
0.783610 0.621253i \(-0.213376\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 48.3961i 0.300597i
\(162\) 0 0
\(163\) 128.920i 0.790922i 0.918483 + 0.395461i \(0.129415\pi\)
−0.918483 + 0.395461i \(0.870585\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −37.5842 −0.225055 −0.112528 0.993649i \(-0.535895\pi\)
−0.112528 + 0.993649i \(0.535895\pi\)
\(168\) 0 0
\(169\) 168.954 0.999729
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) −90.9797 −0.525894 −0.262947 0.964810i \(-0.584694\pi\)
−0.262947 + 0.964810i \(0.584694\pi\)
\(174\) 0 0
\(175\) 198.116 156.193i 1.13209 0.892532i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 19.5078i 0.108982i 0.998514 + 0.0544911i \(0.0173537\pi\)
−0.998514 + 0.0544911i \(0.982646\pi\)
\(180\) 0 0
\(181\) −174.603 −0.964656 −0.482328 0.875991i \(-0.660209\pi\)
−0.482328 + 0.875991i \(0.660209\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 140.839 290.382i 0.761294 1.56963i
\(186\) 0 0
\(187\) 2.58966i 0.0138485i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 98.4380i 0.515382i −0.966227 0.257691i \(-0.917038\pi\)
0.966227 0.257691i \(-0.0829617\pi\)
\(192\) 0 0
\(193\) 330.925i 1.71464i −0.514785 0.857319i \(-0.672128\pi\)
0.514785 0.857319i \(-0.327872\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −309.042 −1.56874 −0.784371 0.620292i \(-0.787014\pi\)
−0.784371 + 0.620292i \(0.787014\pi\)
\(198\) 0 0
\(199\) −12.2565 −0.0615906 −0.0307953 0.999526i \(-0.509804\pi\)
−0.0307953 + 0.999526i \(0.509804\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) −513.428 −2.52920
\(204\) 0 0
\(205\) −104.731 + 215.935i −0.510885 + 1.05334i
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 11.4377i 0.0547259i
\(210\) 0 0
\(211\) −30.4591 −0.144356 −0.0721780 0.997392i \(-0.522995\pi\)
−0.0721780 + 0.997392i \(0.522995\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) −1.93432 + 3.98817i −0.00899684 + 0.0185496i
\(216\) 0 0
\(217\) 145.058i 0.668469i
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 1.21502i 0.00549781i
\(222\) 0 0
\(223\) 34.0476i 0.152680i −0.997082 0.0763400i \(-0.975677\pi\)
0.997082 0.0763400i \(-0.0243234\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) −390.716 −1.72121 −0.860607 0.509270i \(-0.829916\pi\)
−0.860607 + 0.509270i \(0.829916\pi\)
\(228\) 0 0
\(229\) 60.7427 0.265252 0.132626 0.991166i \(-0.457659\pi\)
0.132626 + 0.991166i \(0.457659\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 170.929 0.733603 0.366801 0.930299i \(-0.380453\pi\)
0.366801 + 0.930299i \(0.380453\pi\)
\(234\) 0 0
\(235\) −277.721 134.699i −1.18179 0.573185i
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 11.1922i 0.0468293i −0.999726 0.0234147i \(-0.992546\pi\)
0.999726 0.0234147i \(-0.00745380\pi\)
\(240\) 0 0
\(241\) 83.6136 0.346944 0.173472 0.984839i \(-0.444501\pi\)
0.173472 + 0.984839i \(0.444501\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 237.689 + 115.283i 0.970161 + 0.470542i
\(246\) 0 0
\(247\) 5.36634i 0.0217261i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 420.150i 1.67391i −0.547275 0.836953i \(-0.684335\pi\)
0.547275 0.836953i \(-0.315665\pi\)
\(252\) 0 0
\(253\) 2.18696i 0.00864412i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 179.388 0.698009 0.349004 0.937121i \(-0.386520\pi\)
0.349004 + 0.937121i \(0.386520\pi\)
\(258\) 0 0
\(259\) 651.361 2.51491
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 441.954 1.68043 0.840217 0.542251i \(-0.182428\pi\)
0.840217 + 0.542251i \(0.182428\pi\)
\(264\) 0 0
\(265\) −333.821 161.908i −1.25970 0.610974i
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 27.3172i 0.101551i 0.998710 + 0.0507754i \(0.0161693\pi\)
−0.998710 + 0.0507754i \(0.983831\pi\)
\(270\) 0 0
\(271\) 291.313 1.07496 0.537478 0.843278i \(-0.319377\pi\)
0.537478 + 0.843278i \(0.319377\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −8.95263 + 7.05817i −0.0325550 + 0.0256661i
\(276\) 0 0
\(277\) 188.753i 0.681417i 0.940169 + 0.340709i \(0.110667\pi\)
−0.940169 + 0.340709i \(0.889333\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 472.884i 1.68286i −0.540364 0.841431i \(-0.681713\pi\)
0.540364 0.841431i \(-0.318287\pi\)
\(282\) 0 0
\(283\) 281.426i 0.994438i −0.867625 0.497219i \(-0.834355\pi\)
0.867625 0.497219i \(-0.165645\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) −484.368 −1.68769
\(288\) 0 0
\(289\) −256.750 −0.888408
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 388.192 1.32489 0.662443 0.749112i \(-0.269520\pi\)
0.662443 + 0.749112i \(0.269520\pi\)
\(294\) 0 0
\(295\) 45.5369 93.8878i 0.154362 0.318264i
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 1.02608i 0.00343170i
\(300\) 0 0
\(301\) −8.94595 −0.0297208
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) −326.096 158.161i −1.06917 0.518562i
\(306\) 0 0
\(307\) 213.835i 0.696531i −0.937396 0.348265i \(-0.886771\pi\)
0.937396 0.348265i \(-0.113229\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 331.277i 1.06520i 0.846368 + 0.532599i \(0.178785\pi\)
−0.846368 + 0.532599i \(0.821215\pi\)
\(312\) 0 0
\(313\) 392.652i 1.25448i −0.778826 0.627239i \(-0.784185\pi\)
0.778826 0.627239i \(-0.215815\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 83.6598 0.263911 0.131956 0.991256i \(-0.457874\pi\)
0.131956 + 0.991256i \(0.457874\pi\)
\(318\) 0 0
\(319\) 23.2012 0.0727310
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 142.439 0.440987
\(324\) 0 0
\(325\) −4.20039 + 3.31155i −0.0129243 + 0.0101894i
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 622.961i 1.89350i
\(330\) 0 0
\(331\) −113.701 −0.343507 −0.171753 0.985140i \(-0.554943\pi\)
−0.171753 + 0.985140i \(0.554943\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 103.144 212.661i 0.307892 0.634810i
\(336\) 0 0
\(337\) 278.872i 0.827513i 0.910387 + 0.413757i \(0.135784\pi\)
−0.910387 + 0.413757i \(0.864216\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 6.55498i 0.0192228i
\(342\) 0 0
\(343\) 38.6922i 0.112805i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −252.355 −0.727247 −0.363623 0.931546i \(-0.618460\pi\)
−0.363623 + 0.931546i \(0.618460\pi\)
\(348\) 0 0
\(349\) 333.461 0.955477 0.477738 0.878502i \(-0.341457\pi\)
0.477738 + 0.878502i \(0.341457\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) −221.720 −0.628102 −0.314051 0.949406i \(-0.601686\pi\)
−0.314051 + 0.949406i \(0.601686\pi\)
\(354\) 0 0
\(355\) 169.474 349.421i 0.477392 0.984284i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 66.9184i 0.186402i 0.995647 + 0.0932011i \(0.0297099\pi\)
−0.995647 + 0.0932011i \(0.970290\pi\)
\(360\) 0 0
\(361\) 268.107 0.742678
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 166.051 342.362i 0.454933 0.937979i
\(366\) 0 0
\(367\) 459.145i 1.25108i 0.780194 + 0.625538i \(0.215120\pi\)
−0.780194 + 0.625538i \(0.784880\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 748.801i 2.01833i
\(372\) 0 0
\(373\) 280.330i 0.751556i 0.926710 + 0.375778i \(0.122624\pi\)
−0.926710 + 0.375778i \(0.877376\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 10.8855 0.0288741
\(378\) 0 0
\(379\) 441.415 1.16468 0.582341 0.812944i \(-0.302137\pi\)
0.582341 + 0.812944i \(0.302137\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) −26.3805 −0.0688786 −0.0344393 0.999407i \(-0.510965\pi\)
−0.0344393 + 0.999407i \(0.510965\pi\)
\(384\) 0 0
\(385\) −20.7023 10.0409i −0.0537722 0.0260803i
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 525.157i 1.35002i −0.737810 0.675008i \(-0.764140\pi\)
0.737810 0.675008i \(-0.235860\pi\)
\(390\) 0 0
\(391\) 27.2352 0.0696551
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) −555.426 269.389i −1.40614 0.681999i
\(396\) 0 0
\(397\) 520.403i 1.31084i 0.755265 + 0.655419i \(0.227508\pi\)
−0.755265 + 0.655419i \(0.772492\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 182.030i 0.453939i 0.973902 + 0.226970i \(0.0728818\pi\)
−0.973902 + 0.226970i \(0.927118\pi\)
\(402\) 0 0
\(403\) 3.07546i 0.00763142i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) −29.4342 −0.0723199
\(408\) 0 0
\(409\) −563.216 −1.37706 −0.688529 0.725209i \(-0.741743\pi\)
−0.688529 + 0.725209i \(0.741743\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 210.602 0.509931
\(414\) 0 0
\(415\) 524.477 + 254.379i 1.26380 + 0.612961i
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 125.092i 0.298549i 0.988796 + 0.149274i \(0.0476937\pi\)
−0.988796 + 0.149274i \(0.952306\pi\)
\(420\) 0 0
\(421\) −83.9028 −0.199294 −0.0996470 0.995023i \(-0.531771\pi\)
−0.0996470 + 0.995023i \(0.531771\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 87.8984 + 111.491i 0.206820 + 0.262332i
\(426\) 0 0
\(427\) 731.473i 1.71305i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 796.050i 1.84698i −0.383619 0.923491i \(-0.625323\pi\)
0.383619 0.923491i \(-0.374677\pi\)
\(432\) 0 0
\(433\) 146.392i 0.338087i 0.985609 + 0.169044i \(0.0540678\pi\)
−0.985609 + 0.169044i \(0.945932\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 120.289 0.275261
\(438\) 0 0
\(439\) 248.094 0.565135 0.282568 0.959247i \(-0.408814\pi\)
0.282568 + 0.959247i \(0.408814\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 247.807 0.559383 0.279691 0.960090i \(-0.409768\pi\)
0.279691 + 0.960090i \(0.409768\pi\)
\(444\) 0 0
\(445\) 113.554 234.126i 0.255178 0.526126i
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 814.811i 1.81472i −0.420351 0.907362i \(-0.638093\pi\)
0.420351 0.907362i \(-0.361907\pi\)
\(450\) 0 0
\(451\) 21.8880 0.0485321
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) −9.71309 4.71099i −0.0213475 0.0103538i
\(456\) 0 0
\(457\) 225.952i 0.494425i −0.968961 0.247213i \(-0.920485\pi\)
0.968961 0.247213i \(-0.0795146\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 249.107i 0.540362i 0.962810 + 0.270181i \(0.0870836\pi\)
−0.962810 + 0.270181i \(0.912916\pi\)
\(462\) 0 0
\(463\) 78.6025i 0.169768i −0.996391 0.0848839i \(-0.972948\pi\)
0.996391 0.0848839i \(-0.0270519\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 506.114 1.08376 0.541878 0.840457i \(-0.317714\pi\)
0.541878 + 0.840457i \(0.317714\pi\)
\(468\) 0 0
\(469\) 477.025 1.01711
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 0.404256 0.000854665
\(474\) 0 0
\(475\) 388.219 + 492.420i 0.817304 + 1.03667i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 394.563i 0.823723i 0.911247 + 0.411861i \(0.135121\pi\)
−0.911247 + 0.411861i \(0.864879\pi\)
\(480\) 0 0
\(481\) −13.8099 −0.0287109
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) −34.3574 + 70.8378i −0.0708399 + 0.146057i
\(486\) 0 0
\(487\) 606.423i 1.24522i 0.782532 + 0.622610i \(0.213928\pi\)
−0.782532 + 0.622610i \(0.786072\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 329.600i 0.671283i 0.941990 + 0.335641i \(0.108953\pi\)
−0.941990 + 0.335641i \(0.891047\pi\)
\(492\) 0 0
\(493\) 288.934i 0.586074i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 783.793 1.57705
\(498\) 0 0
\(499\) 385.046 0.771635 0.385817 0.922575i \(-0.373920\pi\)
0.385817 + 0.922575i \(0.373920\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) −464.735 −0.923926 −0.461963 0.886899i \(-0.652855\pi\)
−0.461963 + 0.886899i \(0.652855\pi\)
\(504\) 0 0
\(505\) −102.255 + 210.829i −0.202486 + 0.417484i
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 944.209i 1.85503i 0.373788 + 0.927514i \(0.378059\pi\)
−0.373788 + 0.927514i \(0.621941\pi\)
\(510\) 0 0
\(511\) 767.959 1.50286
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 195.421 402.917i 0.379457 0.782363i
\(516\) 0 0
\(517\) 28.1508i 0.0544504i
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 702.408i 1.34819i 0.738644 + 0.674096i \(0.235467\pi\)
−0.738644 + 0.674096i \(0.764533\pi\)
\(522\) 0 0
\(523\) 487.025i 0.931214i −0.884992 0.465607i \(-0.845836\pi\)
0.884992 0.465607i \(-0.154164\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 81.6319 0.154899
\(528\) 0 0
\(529\) 23.0000 0.0434783
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 10.2694 0.0192671
\(534\) 0 0
\(535\) −128.778 62.4592i −0.240707 0.116746i
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 24.0931i 0.0446996i
\(540\) 0 0
\(541\) 142.175 0.262800 0.131400 0.991329i \(-0.458053\pi\)
0.131400 + 0.991329i \(0.458053\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) −846.838 410.728i −1.55383 0.753630i
\(546\) 0 0
\(547\) 1011.54i 1.84924i 0.380888 + 0.924621i \(0.375618\pi\)
−0.380888 + 0.924621i \(0.624382\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 1276.13i 2.31603i
\(552\) 0 0
\(553\) 1245.89i 2.25296i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −804.458 −1.44427 −0.722134 0.691753i \(-0.756839\pi\)
−0.722134 + 0.691753i \(0.756839\pi\)
\(558\) 0 0
\(559\) 0.189669 0.000339300
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) −692.845 −1.23063 −0.615316 0.788281i \(-0.710971\pi\)
−0.615316 + 0.788281i \(0.710971\pi\)
\(564\) 0 0
\(565\) 283.774 + 137.634i 0.502254 + 0.243600i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 299.787i 0.526866i 0.964678 + 0.263433i \(0.0848548\pi\)
−0.964678 + 0.263433i \(0.915145\pi\)
\(570\) 0 0
\(571\) 522.141 0.914432 0.457216 0.889356i \(-0.348847\pi\)
0.457216 + 0.889356i \(0.348847\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 74.2299 + 94.1537i 0.129096 + 0.163746i
\(576\) 0 0
\(577\) 72.0944i 0.124947i −0.998047 0.0624735i \(-0.980101\pi\)
0.998047 0.0624735i \(-0.0198989\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 1176.46i 2.02490i
\(582\) 0 0
\(583\) 33.8374i 0.0580401i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −579.228 −0.986760 −0.493380 0.869814i \(-0.664239\pi\)
−0.493380 + 0.869814i \(0.664239\pi\)
\(588\) 0 0
\(589\) 360.542 0.612126
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 15.3152 0.0258266 0.0129133 0.999917i \(-0.495889\pi\)
0.0129133 + 0.999917i \(0.495889\pi\)
\(594\) 0 0
\(595\) −125.044 + 257.815i −0.210157 + 0.433302i
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 656.299i 1.09566i −0.836591 0.547829i \(-0.815455\pi\)
0.836591 0.547829i \(-0.184545\pi\)
\(600\) 0 0
\(601\) 589.765 0.981307 0.490653 0.871355i \(-0.336758\pi\)
0.490653 + 0.871355i \(0.336758\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) −543.417 263.565i −0.898209 0.435644i
\(606\) 0 0
\(607\) 430.985i 0.710025i −0.934862 0.355012i \(-0.884477\pi\)
0.934862 0.355012i \(-0.115523\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 13.2078i 0.0216167i
\(612\) 0 0
\(613\) 231.974i 0.378423i 0.981936 + 0.189212i \(0.0605933\pi\)
−0.981936 + 0.189212i \(0.939407\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −268.727 −0.435538 −0.217769 0.976000i \(-0.569878\pi\)
−0.217769 + 0.976000i \(0.569878\pi\)
\(618\) 0 0
\(619\) −1156.45 −1.86826 −0.934129 0.356937i \(-0.883821\pi\)
−0.934129 + 0.356937i \(0.883821\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 525.172 0.842973
\(624\) 0 0
\(625\) −145.863 + 607.741i −0.233380 + 0.972386i
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 366.557i 0.582761i
\(630\) 0 0
\(631\) −872.840 −1.38326 −0.691632 0.722250i \(-0.743108\pi\)
−0.691632 + 0.722250i \(0.743108\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) −457.930 + 944.158i −0.721150 + 1.48686i
\(636\) 0 0
\(637\) 11.3040i 0.0177457i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 1077.19i 1.68049i −0.542207 0.840245i \(-0.682411\pi\)
0.542207 0.840245i \(-0.317589\pi\)
\(642\) 0 0
\(643\) 567.675i 0.882854i −0.897297 0.441427i \(-0.854472\pi\)
0.897297 0.441427i \(-0.145528\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 455.891 0.704623 0.352311 0.935883i \(-0.385396\pi\)
0.352311 + 0.935883i \(0.385396\pi\)
\(648\) 0 0
\(649\) −9.51682 −0.0146638
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −818.894 −1.25405 −0.627025 0.778999i \(-0.715728\pi\)
−0.627025 + 0.778999i \(0.715728\pi\)
\(654\) 0 0
\(655\) −255.690 + 527.180i −0.390366 + 0.804856i
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 43.6188i 0.0661893i 0.999452 + 0.0330947i \(0.0105363\pi\)
−0.999452 + 0.0330947i \(0.989464\pi\)
\(660\) 0 0
\(661\) −699.302 −1.05795 −0.528973 0.848639i \(-0.677423\pi\)
−0.528973 + 0.848639i \(0.677423\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) −552.278 + 1138.69i −0.830494 + 1.71231i
\(666\) 0 0
\(667\) 244.004i 0.365823i
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 33.0544i 0.0492613i
\(672\) 0 0
\(673\) 982.321i 1.45962i −0.683652 0.729808i \(-0.739610\pi\)
0.683652 0.729808i \(-0.260390\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) −611.511 −0.903266 −0.451633 0.892204i \(-0.649158\pi\)
−0.451633 + 0.892204i \(0.649158\pi\)
\(678\) 0 0
\(679\) −158.898 −0.234017
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 30.0480 0.0439942 0.0219971 0.999758i \(-0.492998\pi\)
0.0219971 + 0.999758i \(0.492998\pi\)
\(684\) 0 0
\(685\) −287.368 139.377i −0.419515 0.203471i
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 15.8758i 0.0230418i
\(690\) 0 0
\(691\) −740.707 −1.07194 −0.535968 0.844239i \(-0.680053\pi\)
−0.535968 + 0.844239i \(0.680053\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −414.320 200.951i −0.596143 0.289138i
\(696\) 0 0
\(697\) 272.580i 0.391076i
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 1033.99i 1.47502i −0.675336 0.737511i \(-0.736001\pi\)
0.675336 0.737511i \(-0.263999\pi\)
\(702\) 0 0
\(703\) 1618.96i 2.30294i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) −472.915 −0.668904
\(708\) 0 0
\(709\) 365.755 0.515874 0.257937 0.966162i \(-0.416957\pi\)
0.257937 + 0.966162i \(0.416957\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 68.9379 0.0966871
\(714\) 0 0
\(715\) 0.438923 + 0.212884i 0.000613878 + 0.000297739i
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 0.979382i 0.00136214i −1.00000 0.000681072i \(-0.999783\pi\)
1.00000 0.000681072i \(-0.000216792\pi\)
\(720\) 0 0
\(721\) 903.791 1.25352
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 998.864 787.495i 1.37774 1.08620i
\(726\) 0 0
\(727\) 193.163i 0.265699i −0.991136 0.132850i \(-0.957587\pi\)
0.991136 0.132850i \(-0.0424127\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 5.03438i 0.00688697i
\(732\) 0 0
\(733\) 609.311i 0.831257i 0.909534 + 0.415628i \(0.136438\pi\)
−0.909534 + 0.415628i \(0.863562\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −21.5561 −0.0292485
\(738\) 0 0
\(739\) 1143.30 1.54709 0.773547 0.633739i \(-0.218481\pi\)
0.773547 + 0.633739i \(0.218481\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 41.3555 0.0556602 0.0278301 0.999613i \(-0.491140\pi\)
0.0278301 + 0.999613i \(0.491140\pi\)
\(744\) 0 0
\(745\) 277.319 571.775i 0.372240 0.767483i
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 288.865i 0.385667i
\(750\) 0 0
\(751\) 1414.17 1.88305 0.941525 0.336943i \(-0.109393\pi\)
0.941525 + 0.336943i \(0.109393\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 1132.70 + 549.376i 1.50027 + 0.727650i
\(756\) 0 0
\(757\) 145.741i 0.192524i 0.995356 + 0.0962621i \(0.0306887\pi\)
−0.995356 + 0.0962621i \(0.969311\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 450.235i 0.591636i 0.955244 + 0.295818i \(0.0955923\pi\)
−0.955244 + 0.295818i \(0.904408\pi\)
\(762\) 0 0
\(763\) 1899.56i 2.48959i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) −4.46510 −0.00582151
\(768\) 0 0
\(769\) −204.046 −0.265339 −0.132669 0.991160i \(-0.542355\pi\)
−0.132669 + 0.991160i \(0.542355\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 1474.26 1.90720 0.953599 0.301080i \(-0.0973472\pi\)
0.953599 + 0.301080i \(0.0973472\pi\)
\(774\) 0 0
\(775\) 222.489 + 282.207i 0.287083 + 0.364138i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 1203.90i 1.54544i
\(780\) 0 0
\(781\) −35.4186 −0.0453503
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) −425.645 + 877.592i −0.542222 + 1.11795i
\(786\) 0 0
\(787\) 786.144i 0.998912i −0.866339 0.499456i \(-0.833533\pi\)
0.866339 0.499456i \(-0.166467\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 636.538i 0.804726i
\(792\) 0 0
\(793\) 15.5084i 0.0195566i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −1050.07 −1.31753 −0.658765 0.752349i \(-0.728921\pi\)
−0.658765 + 0.752349i \(0.728921\pi\)
\(798\) 0 0
\(799\) 350.574 0.438766
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) −34.7031 −0.0432168
\(804\) 0 0
\(805\) −105.599 + 217.724i −0.131179 + 0.270464i
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 1251.93i 1.54750i 0.633490 + 0.773751i \(0.281622\pi\)
−0.633490 + 0.773751i \(0.718378\pi\)
\(810\) 0 0
\(811\) 1068.89 1.31799 0.658997 0.752146i \(-0.270981\pi\)
0.658997 + 0.752146i \(0.270981\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 281.300 579.984i 0.345154 0.711636i
\(816\) 0 0
\(817\) 22.2352i 0.0272157i
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 1013.06i 1.23394i 0.786988 + 0.616968i \(0.211639\pi\)
−0.786988 + 0.616968i \(0.788361\pi\)
\(822\) 0 0
\(823\) 42.2166i 0.0512960i 0.999671 + 0.0256480i \(0.00816490\pi\)
−0.999671 + 0.0256480i \(0.991835\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −992.092 −1.19963 −0.599814 0.800140i \(-0.704759\pi\)
−0.599814 + 0.800140i \(0.704759\pi\)
\(828\) 0 0
\(829\) −1062.71 −1.28192 −0.640959 0.767575i \(-0.721463\pi\)
−0.640959 + 0.767575i \(0.721463\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) −300.041 −0.360194
\(834\) 0 0
\(835\) 169.083 + 82.0077i 0.202495 + 0.0982128i
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 1630.82i 1.94376i −0.235469 0.971882i \(-0.575663\pi\)
0.235469 0.971882i \(-0.424337\pi\)
\(840\) 0 0
\(841\) −1747.61 −2.07801
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) −760.088 368.653i −0.899512 0.436276i
\(846\) 0 0
\(847\) 1218.95i 1.43914i
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 309.556i 0.363755i
\(852\) 0 0
\(853\) 713.109i 0.836001i 0.908447 + 0.418000i \(0.137269\pi\)
−0.908447 + 0.418000i \(0.862731\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 680.264 0.793774 0.396887 0.917868i \(-0.370091\pi\)
0.396887 + 0.917868i \(0.370091\pi\)
\(858\) 0 0
\(859\) −170.477 −0.198459 −0.0992297 0.995065i \(-0.531638\pi\)
−0.0992297 + 0.995065i \(0.531638\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 1090.72 1.26387 0.631933 0.775023i \(-0.282262\pi\)
0.631933 + 0.775023i \(0.282262\pi\)
\(864\) 0 0
\(865\) 409.298 + 198.515i 0.473176 + 0.229497i
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 56.3001i 0.0647872i
\(870\) 0 0
\(871\) −10.1137 −0.0116116
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) −1232.09 + 270.394i −1.40810 + 0.309022i
\(876\) 0 0
\(877\) 1080.87i 1.23247i −0.787564 0.616233i \(-0.788658\pi\)
0.787564 0.616233i \(-0.211342\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 1632.76i 1.85330i 0.375923 + 0.926651i \(0.377326\pi\)
−0.375923 + 0.926651i \(0.622674\pi\)
\(882\) 0 0
\(883\) 429.629i 0.486557i −0.969957 0.243278i \(-0.921777\pi\)
0.969957 0.243278i \(-0.0782228\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 1537.25 1.73308 0.866542 0.499104i \(-0.166338\pi\)
0.866542 + 0.499104i \(0.166338\pi\)
\(888\) 0 0
\(889\) −2117.86 −2.38229
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 1548.38 1.73390
\(894\) 0 0
\(895\) 42.5655 87.7613i 0.0475592 0.0980573i
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 731.353i 0.813518i
\(900\) 0 0
\(901\) 421.391 0.467693
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 785.499 + 380.978i 0.867955 + 0.420970i
\(906\) 0 0
\(907\) 491.466i 0.541859i 0.962599 + 0.270930i \(0.0873311\pi\)
−0.962599 + 0.270930i \(0.912669\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 276.630i 0.303655i 0.988407 + 0.151827i \(0.0485158\pi\)
−0.988407 + 0.151827i \(0.951484\pi\)
\(912\) 0 0
\(913\) 53.1630i 0.0582289i
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) −1182.53 −1.28956
\(918\) 0 0
\(919\) −1723.53 −1.87544 −0.937722 0.347387i \(-0.887069\pi\)
−0.937722 + 0.347387i \(0.887069\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) −16.6177 −0.0180040
\(924\) 0 0
\(925\) −1267.21 + 999.056i −1.36996 + 1.08006i
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 435.765i 0.469069i −0.972108 0.234534i \(-0.924643\pi\)
0.972108 0.234534i \(-0.0753566\pi\)
\(930\) 0 0
\(931\) −1325.19 −1.42340
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 5.65057 11.6503i 0.00604339 0.0124602i
\(936\) 0 0
\(937\) 1439.57i 1.53636i 0.640237 + 0.768178i \(0.278836\pi\)
−0.640237 + 0.768178i \(0.721164\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 349.601i 0.371521i 0.982595 + 0.185761i \(0.0594749\pi\)
−0.982595 + 0.185761i \(0.940525\pi\)
\(942\) 0 0
\(943\) 230.193i 0.244107i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 1204.96 1.27240 0.636198 0.771526i \(-0.280506\pi\)
0.636198 + 0.771526i \(0.280506\pi\)
\(948\) 0 0
\(949\) −16.2820 −0.0171570
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) −169.630 −0.177995 −0.0889977 0.996032i \(-0.528366\pi\)
−0.0889977 + 0.996032i \(0.528366\pi\)
\(954\) 0 0
\(955\) −214.789 + 442.851i −0.224910 + 0.463718i
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 644.600i 0.672158i
\(960\) 0 0
\(961\) −754.373 −0.784987
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) −722.069 + 1488.76i −0.748258 + 1.54276i
\(966\) 0 0
\(967\) 62.3819i 0.0645108i 0.999480 + 0.0322554i \(0.0102690\pi\)
−0.999480 + 0.0322554i \(0.989731\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 934.246i 0.962148i −0.876680 0.481074i \(-0.840247\pi\)
0.876680 0.481074i \(-0.159753\pi\)
\(972\) 0 0
\(973\) 929.368i 0.955157i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −611.285 −0.625676 −0.312838 0.949807i \(-0.601280\pi\)
−0.312838 + 0.949807i \(0.601280\pi\)
\(978\) 0 0
\(979\) −23.7319 −0.0242409
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) −553.953 −0.563533 −0.281767 0.959483i \(-0.590920\pi\)
−0.281767 + 0.959483i \(0.590920\pi\)
\(984\) 0 0
\(985\) 1390.31 + 674.321i 1.41148 + 0.684590i
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 4.25151i 0.00429880i
\(990\) 0 0
\(991\) −1609.14 −1.62375 −0.811874 0.583832i \(-0.801553\pi\)
−0.811874 + 0.583832i \(0.801553\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 55.1394 + 26.7434i 0.0554165 + 0.0268778i
\(996\) 0 0
\(997\) 26.1069i 0.0261855i 0.999914 + 0.0130927i \(0.00416767\pi\)
−0.999914 + 0.0130927i \(0.995832\pi\)
\(998\) 0 0
\(999\) 0 0
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 4140.3.c.a.4049.13 88
3.2 odd 2 inner 4140.3.c.a.4049.76 yes 88
5.4 even 2 inner 4140.3.c.a.4049.75 yes 88
15.14 odd 2 inner 4140.3.c.a.4049.14 yes 88
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
4140.3.c.a.4049.13 88 1.1 even 1 trivial
4140.3.c.a.4049.14 yes 88 15.14 odd 2 inner
4140.3.c.a.4049.75 yes 88 5.4 even 2 inner
4140.3.c.a.4049.76 yes 88 3.2 odd 2 inner