Properties

Label 384.4.c.a.383.6
Level $384$
Weight $4$
Character 384.383
Analytic conductor $22.657$
Analytic rank $0$
Dimension $12$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [384,4,Mod(383,384)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(384, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0, 1]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("384.383");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 384 = 2^{7} \cdot 3 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 384.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: \(22.6567334422\)
Analytic rank: \(0\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} + 39x^{10} + 549x^{8} + 3500x^{6} + 10236x^{4} + 11952x^{2} + 4624 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{30}\cdot 3^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 383.6
Root \(-1.08600i\) of defining polynomial
Character \(\chi\) \(=\) 384.383
Dual form 384.4.c.a.383.5

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-3.04120 + 4.21320i) q^{3} -9.33303i q^{5} -36.3792i q^{7} +(-8.50216 - 25.6264i) q^{9} +O(q^{10})\) \(q+(-3.04120 + 4.21320i) q^{3} -9.33303i q^{5} -36.3792i q^{7} +(-8.50216 - 25.6264i) q^{9} -48.4408 q^{11} +25.8988 q^{13} +(39.3219 + 28.3836i) q^{15} +74.2219i q^{17} +82.9826i q^{19} +(153.273 + 110.637i) q^{21} -179.084 q^{23} +37.8946 q^{25} +(133.826 + 42.1138i) q^{27} +122.309i q^{29} +64.1893i q^{31} +(147.318 - 204.091i) q^{33} -339.528 q^{35} +5.01487 q^{37} +(-78.7635 + 109.117i) q^{39} -325.933i q^{41} +321.853i q^{43} +(-239.172 + 79.3509i) q^{45} +95.9780 q^{47} -980.446 q^{49} +(-312.712 - 225.724i) q^{51} +185.069i q^{53} +452.099i q^{55} +(-349.623 - 252.367i) q^{57} -226.316 q^{59} +198.851 q^{61} +(-932.268 + 309.302i) q^{63} -241.714i q^{65} -23.9524i q^{67} +(544.632 - 754.518i) q^{69} -399.741 q^{71} +669.631 q^{73} +(-115.245 + 159.658i) q^{75} +1762.24i q^{77} +229.453i q^{79} +(-584.427 + 435.760i) q^{81} +321.150 q^{83} +692.715 q^{85} +(-515.313 - 371.967i) q^{87} -131.446i q^{89} -942.177i q^{91} +(-270.443 - 195.213i) q^{93} +774.479 q^{95} +136.371 q^{97} +(411.851 + 1241.36i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 10 q^{3}+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 10 q^{3} - 36 q^{11} - 84 q^{15} + 136 q^{21} + 120 q^{23} - 300 q^{25} + 266 q^{27} - 116 q^{33} - 432 q^{35} + 528 q^{37} - 620 q^{39} + 440 q^{45} + 1248 q^{47} - 948 q^{49} + 1072 q^{51} - 172 q^{57} - 2508 q^{59} + 624 q^{61} - 2744 q^{63} - 24 q^{69} + 2040 q^{71} - 216 q^{73} + 3894 q^{75} - 1076 q^{81} - 4572 q^{83} + 480 q^{85} - 4156 q^{87} - 112 q^{93} + 5448 q^{95} - 48 q^{97} + 6044 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/384\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(133\) \(257\)
\(\chi(n)\) \(-1\) \(1\) \(-1\)

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
\(3\) −3.04120 + 4.21320i −0.585280 + 0.810831i
\(4\) 0 0
\(5\) 9.33303i 0.834771i −0.908729 0.417386i \(-0.862946\pi\)
0.908729 0.417386i \(-0.137054\pi\)
\(6\) 0 0
\(7\) 36.3792i 1.96429i −0.188120 0.982146i \(-0.560240\pi\)
0.188120 0.982146i \(-0.439760\pi\)
\(8\) 0 0
\(9\) −8.50216 25.6264i −0.314895 0.949127i
\(10\) 0 0
\(11\) −48.4408 −1.32777 −0.663884 0.747836i \(-0.731093\pi\)
−0.663884 + 0.747836i \(0.731093\pi\)
\(12\) 0 0
\(13\) 25.8988 0.552541 0.276270 0.961080i \(-0.410901\pi\)
0.276270 + 0.961080i \(0.410901\pi\)
\(14\) 0 0
\(15\) 39.3219 + 28.3836i 0.676859 + 0.488575i
\(16\) 0 0
\(17\) 74.2219i 1.05891i 0.848338 + 0.529455i \(0.177603\pi\)
−0.848338 + 0.529455i \(0.822397\pi\)
\(18\) 0 0
\(19\) 82.9826i 1.00197i 0.865455 + 0.500987i \(0.167030\pi\)
−0.865455 + 0.500987i \(0.832970\pi\)
\(20\) 0 0
\(21\) 153.273 + 110.637i 1.59271 + 1.14966i
\(22\) 0 0
\(23\) −179.084 −1.62355 −0.811775 0.583971i \(-0.801498\pi\)
−0.811775 + 0.583971i \(0.801498\pi\)
\(24\) 0 0
\(25\) 37.8946 0.303157
\(26\) 0 0
\(27\) 133.826 + 42.1138i 0.953883 + 0.300178i
\(28\) 0 0
\(29\) 122.309i 0.783180i 0.920140 + 0.391590i \(0.128075\pi\)
−0.920140 + 0.391590i \(0.871925\pi\)
\(30\) 0 0
\(31\) 64.1893i 0.371895i 0.982560 + 0.185947i \(0.0595354\pi\)
−0.982560 + 0.185947i \(0.940465\pi\)
\(32\) 0 0
\(33\) 147.318 204.091i 0.777116 1.07660i
\(34\) 0 0
\(35\) −339.528 −1.63973
\(36\) 0 0
\(37\) 5.01487 0.0222822 0.0111411 0.999938i \(-0.496454\pi\)
0.0111411 + 0.999938i \(0.496454\pi\)
\(38\) 0 0
\(39\) −78.7635 + 109.117i −0.323391 + 0.448017i
\(40\) 0 0
\(41\) 325.933i 1.24152i −0.784003 0.620758i \(-0.786825\pi\)
0.784003 0.620758i \(-0.213175\pi\)
\(42\) 0 0
\(43\) 321.853i 1.14144i 0.821143 + 0.570722i \(0.193337\pi\)
−0.821143 + 0.570722i \(0.806663\pi\)
\(44\) 0 0
\(45\) −239.172 + 79.3509i −0.792304 + 0.262865i
\(46\) 0 0
\(47\) 95.9780 0.297869 0.148934 0.988847i \(-0.452416\pi\)
0.148934 + 0.988847i \(0.452416\pi\)
\(48\) 0 0
\(49\) −980.446 −2.85844
\(50\) 0 0
\(51\) −312.712 225.724i −0.858597 0.619758i
\(52\) 0 0
\(53\) 185.069i 0.479645i 0.970817 + 0.239822i \(0.0770892\pi\)
−0.970817 + 0.239822i \(0.922911\pi\)
\(54\) 0 0
\(55\) 452.099i 1.10838i
\(56\) 0 0
\(57\) −349.623 252.367i −0.812432 0.586436i
\(58\) 0 0
\(59\) −226.316 −0.499386 −0.249693 0.968325i \(-0.580330\pi\)
−0.249693 + 0.968325i \(0.580330\pi\)
\(60\) 0 0
\(61\) 198.851 0.417382 0.208691 0.977982i \(-0.433080\pi\)
0.208691 + 0.977982i \(0.433080\pi\)
\(62\) 0 0
\(63\) −932.268 + 309.302i −1.86436 + 0.618545i
\(64\) 0 0
\(65\) 241.714i 0.461245i
\(66\) 0 0
\(67\) 23.9524i 0.0436753i −0.999762 0.0218377i \(-0.993048\pi\)
0.999762 0.0218377i \(-0.00695170\pi\)
\(68\) 0 0
\(69\) 544.632 754.518i 0.950231 1.31642i
\(70\) 0 0
\(71\) −399.741 −0.668176 −0.334088 0.942542i \(-0.608428\pi\)
−0.334088 + 0.942542i \(0.608428\pi\)
\(72\) 0 0
\(73\) 669.631 1.07362 0.536811 0.843703i \(-0.319629\pi\)
0.536811 + 0.843703i \(0.319629\pi\)
\(74\) 0 0
\(75\) −115.245 + 159.658i −0.177432 + 0.245809i
\(76\) 0 0
\(77\) 1762.24i 2.60812i
\(78\) 0 0
\(79\) 229.453i 0.326778i 0.986562 + 0.163389i \(0.0522426\pi\)
−0.986562 + 0.163389i \(0.947757\pi\)
\(80\) 0 0
\(81\) −584.427 + 435.760i −0.801683 + 0.597750i
\(82\) 0 0
\(83\) 321.150 0.424709 0.212354 0.977193i \(-0.431887\pi\)
0.212354 + 0.977193i \(0.431887\pi\)
\(84\) 0 0
\(85\) 692.715 0.883947
\(86\) 0 0
\(87\) −515.313 371.967i −0.635027 0.458380i
\(88\) 0 0
\(89\) 131.446i 0.156554i −0.996932 0.0782768i \(-0.975058\pi\)
0.996932 0.0782768i \(-0.0249418\pi\)
\(90\) 0 0
\(91\) 942.177i 1.08535i
\(92\) 0 0
\(93\) −270.443 195.213i −0.301544 0.217663i
\(94\) 0 0
\(95\) 774.479 0.836420
\(96\) 0 0
\(97\) 136.371 0.142746 0.0713732 0.997450i \(-0.477262\pi\)
0.0713732 + 0.997450i \(0.477262\pi\)
\(98\) 0 0
\(99\) 411.851 + 1241.36i 0.418107 + 1.26022i
\(100\) 0 0
\(101\) 411.095i 0.405005i 0.979282 + 0.202502i \(0.0649074\pi\)
−0.979282 + 0.202502i \(0.935093\pi\)
\(102\) 0 0
\(103\) 137.507i 0.131543i −0.997835 0.0657716i \(-0.979049\pi\)
0.997835 0.0657716i \(-0.0209509\pi\)
\(104\) 0 0
\(105\) 1032.57 1430.50i 0.959704 1.32955i
\(106\) 0 0
\(107\) −2113.08 −1.90915 −0.954577 0.297964i \(-0.903692\pi\)
−0.954577 + 0.297964i \(0.903692\pi\)
\(108\) 0 0
\(109\) −1310.07 −1.15121 −0.575606 0.817727i \(-0.695234\pi\)
−0.575606 + 0.817727i \(0.695234\pi\)
\(110\) 0 0
\(111\) −15.2512 + 21.1287i −0.0130413 + 0.0180671i
\(112\) 0 0
\(113\) 1403.62i 1.16851i 0.811571 + 0.584254i \(0.198613\pi\)
−0.811571 + 0.584254i \(0.801387\pi\)
\(114\) 0 0
\(115\) 1671.40i 1.35529i
\(116\) 0 0
\(117\) −220.196 663.693i −0.173992 0.524431i
\(118\) 0 0
\(119\) 2700.13 2.08001
\(120\) 0 0
\(121\) 1015.51 0.762967
\(122\) 0 0
\(123\) 1373.22 + 991.227i 1.00666 + 0.726634i
\(124\) 0 0
\(125\) 1520.30i 1.08784i
\(126\) 0 0
\(127\) 1583.69i 1.10653i −0.833004 0.553267i \(-0.813381\pi\)
0.833004 0.553267i \(-0.186619\pi\)
\(128\) 0 0
\(129\) −1356.03 978.820i −0.925519 0.668065i
\(130\) 0 0
\(131\) −160.326 −0.106929 −0.0534646 0.998570i \(-0.517026\pi\)
−0.0534646 + 0.998570i \(0.517026\pi\)
\(132\) 0 0
\(133\) 3018.84 1.96817
\(134\) 0 0
\(135\) 393.050 1249.00i 0.250580 0.796274i
\(136\) 0 0
\(137\) 1487.47i 0.927614i 0.885936 + 0.463807i \(0.153517\pi\)
−0.885936 + 0.463807i \(0.846483\pi\)
\(138\) 0 0
\(139\) 58.7660i 0.0358595i −0.999839 0.0179298i \(-0.994292\pi\)
0.999839 0.0179298i \(-0.00570752\pi\)
\(140\) 0 0
\(141\) −291.889 + 404.375i −0.174337 + 0.241521i
\(142\) 0 0
\(143\) −1254.56 −0.733646
\(144\) 0 0
\(145\) 1141.51 0.653776
\(146\) 0 0
\(147\) 2981.74 4130.82i 1.67299 2.31771i
\(148\) 0 0
\(149\) 940.146i 0.516911i −0.966023 0.258456i \(-0.916786\pi\)
0.966023 0.258456i \(-0.0832136\pi\)
\(150\) 0 0
\(151\) 1800.43i 0.970308i 0.874429 + 0.485154i \(0.161237\pi\)
−0.874429 + 0.485154i \(0.838763\pi\)
\(152\) 0 0
\(153\) 1902.04 631.046i 1.00504 0.333445i
\(154\) 0 0
\(155\) 599.081 0.310447
\(156\) 0 0
\(157\) −3024.10 −1.53726 −0.768629 0.639695i \(-0.779061\pi\)
−0.768629 + 0.639695i \(0.779061\pi\)
\(158\) 0 0
\(159\) −779.733 562.832i −0.388911 0.280726i
\(160\) 0 0
\(161\) 6514.94i 3.18913i
\(162\) 0 0
\(163\) 3124.46i 1.50139i −0.660650 0.750694i \(-0.729719\pi\)
0.660650 0.750694i \(-0.270281\pi\)
\(164\) 0 0
\(165\) −1904.79 1374.93i −0.898711 0.648714i
\(166\) 0 0
\(167\) 371.384 0.172087 0.0860435 0.996291i \(-0.472578\pi\)
0.0860435 + 0.996291i \(0.472578\pi\)
\(168\) 0 0
\(169\) −1526.25 −0.694699
\(170\) 0 0
\(171\) 2126.55 705.531i 0.951001 0.315517i
\(172\) 0 0
\(173\) 2931.24i 1.28820i 0.764943 + 0.644099i \(0.222767\pi\)
−0.764943 + 0.644099i \(0.777233\pi\)
\(174\) 0 0
\(175\) 1378.58i 0.595489i
\(176\) 0 0
\(177\) 688.272 953.513i 0.292281 0.404918i
\(178\) 0 0
\(179\) −2834.46 −1.18356 −0.591781 0.806098i \(-0.701575\pi\)
−0.591781 + 0.806098i \(0.701575\pi\)
\(180\) 0 0
\(181\) −1819.98 −0.747392 −0.373696 0.927551i \(-0.621910\pi\)
−0.373696 + 0.927551i \(0.621910\pi\)
\(182\) 0 0
\(183\) −604.747 + 837.800i −0.244285 + 0.338426i
\(184\) 0 0
\(185\) 46.8039i 0.0186005i
\(186\) 0 0
\(187\) 3595.37i 1.40599i
\(188\) 0 0
\(189\) 1532.07 4868.49i 0.589638 1.87370i
\(190\) 0 0
\(191\) 4627.06 1.75289 0.876446 0.481500i \(-0.159908\pi\)
0.876446 + 0.481500i \(0.159908\pi\)
\(192\) 0 0
\(193\) −2770.29 −1.03321 −0.516606 0.856223i \(-0.672805\pi\)
−0.516606 + 0.856223i \(0.672805\pi\)
\(194\) 0 0
\(195\) 1018.39 + 735.102i 0.373992 + 0.269958i
\(196\) 0 0
\(197\) 3880.01i 1.40324i −0.712550 0.701622i \(-0.752460\pi\)
0.712550 0.701622i \(-0.247540\pi\)
\(198\) 0 0
\(199\) 2015.00i 0.717785i −0.933379 0.358893i \(-0.883154\pi\)
0.933379 0.358893i \(-0.116846\pi\)
\(200\) 0 0
\(201\) 100.916 + 72.8440i 0.0354133 + 0.0255623i
\(202\) 0 0
\(203\) 4449.51 1.53839
\(204\) 0 0
\(205\) −3041.94 −1.03638
\(206\) 0 0
\(207\) 1522.60 + 4589.29i 0.511247 + 1.54095i
\(208\) 0 0
\(209\) 4019.74i 1.33039i
\(210\) 0 0
\(211\) 5611.42i 1.83084i 0.402505 + 0.915418i \(0.368140\pi\)
−0.402505 + 0.915418i \(0.631860\pi\)
\(212\) 0 0
\(213\) 1215.69 1684.19i 0.391070 0.541778i
\(214\) 0 0
\(215\) 3003.86 0.952845
\(216\) 0 0
\(217\) 2335.16 0.730510
\(218\) 0 0
\(219\) −2036.48 + 2821.29i −0.628369 + 0.870526i
\(220\) 0 0
\(221\) 1922.26i 0.585091i
\(222\) 0 0
\(223\) 2025.65i 0.608285i −0.952627 0.304142i \(-0.901630\pi\)
0.952627 0.304142i \(-0.0983699\pi\)
\(224\) 0 0
\(225\) −322.186 971.103i −0.0954625 0.287734i
\(226\) 0 0
\(227\) 404.049 0.118140 0.0590698 0.998254i \(-0.481187\pi\)
0.0590698 + 0.998254i \(0.481187\pi\)
\(228\) 0 0
\(229\) −4735.42 −1.36649 −0.683243 0.730191i \(-0.739431\pi\)
−0.683243 + 0.730191i \(0.739431\pi\)
\(230\) 0 0
\(231\) −7424.66 5359.32i −2.11475 1.52648i
\(232\) 0 0
\(233\) 3703.79i 1.04139i 0.853744 + 0.520694i \(0.174327\pi\)
−0.853744 + 0.520694i \(0.825673\pi\)
\(234\) 0 0
\(235\) 895.765i 0.248652i
\(236\) 0 0
\(237\) −966.733 697.814i −0.264962 0.191257i
\(238\) 0 0
\(239\) 6398.83 1.73182 0.865912 0.500197i \(-0.166739\pi\)
0.865912 + 0.500197i \(0.166739\pi\)
\(240\) 0 0
\(241\) 3766.48 1.00672 0.503362 0.864076i \(-0.332096\pi\)
0.503362 + 0.864076i \(0.332096\pi\)
\(242\) 0 0
\(243\) −58.5842 3787.54i −0.0154658 0.999880i
\(244\) 0 0
\(245\) 9150.53i 2.38615i
\(246\) 0 0
\(247\) 2149.15i 0.553632i
\(248\) 0 0
\(249\) −976.683 + 1353.07i −0.248573 + 0.344367i
\(250\) 0 0
\(251\) −1979.71 −0.497841 −0.248921 0.968524i \(-0.580076\pi\)
−0.248921 + 0.968524i \(0.580076\pi\)
\(252\) 0 0
\(253\) 8674.98 2.15570
\(254\) 0 0
\(255\) −2106.69 + 2918.55i −0.517356 + 0.716732i
\(256\) 0 0
\(257\) 1081.54i 0.262508i −0.991349 0.131254i \(-0.958100\pi\)
0.991349 0.131254i \(-0.0419004\pi\)
\(258\) 0 0
\(259\) 182.437i 0.0437687i
\(260\) 0 0
\(261\) 3134.34 1039.89i 0.743337 0.246619i
\(262\) 0 0
\(263\) −1121.20 −0.262875 −0.131438 0.991324i \(-0.541959\pi\)
−0.131438 + 0.991324i \(0.541959\pi\)
\(264\) 0 0
\(265\) 1727.25 0.400394
\(266\) 0 0
\(267\) 553.810 + 399.755i 0.126939 + 0.0916277i
\(268\) 0 0
\(269\) 1844.86i 0.418153i −0.977899 0.209076i \(-0.932954\pi\)
0.977899 0.209076i \(-0.0670457\pi\)
\(270\) 0 0
\(271\) 938.970i 0.210474i −0.994447 0.105237i \(-0.966440\pi\)
0.994447 0.105237i \(-0.0335601\pi\)
\(272\) 0 0
\(273\) 3969.58 + 2865.35i 0.880037 + 0.635234i
\(274\) 0 0
\(275\) −1835.64 −0.402522
\(276\) 0 0
\(277\) 1100.93 0.238803 0.119401 0.992846i \(-0.461902\pi\)
0.119401 + 0.992846i \(0.461902\pi\)
\(278\) 0 0
\(279\) 1644.94 545.748i 0.352975 0.117108i
\(280\) 0 0
\(281\) 748.208i 0.158841i 0.996841 + 0.0794205i \(0.0253070\pi\)
−0.996841 + 0.0794205i \(0.974693\pi\)
\(282\) 0 0
\(283\) 2150.15i 0.451636i 0.974170 + 0.225818i \(0.0725054\pi\)
−0.974170 + 0.225818i \(0.927495\pi\)
\(284\) 0 0
\(285\) −2355.35 + 3263.04i −0.489540 + 0.678195i
\(286\) 0 0
\(287\) −11857.2 −2.43870
\(288\) 0 0
\(289\) −595.891 −0.121289
\(290\) 0 0
\(291\) −414.733 + 574.559i −0.0835466 + 0.115743i
\(292\) 0 0
\(293\) 5252.83i 1.04735i 0.851918 + 0.523675i \(0.175439\pi\)
−0.851918 + 0.523675i \(0.824561\pi\)
\(294\) 0 0
\(295\) 2112.21i 0.416873i
\(296\) 0 0
\(297\) −6482.64 2040.03i −1.26654 0.398567i
\(298\) 0 0
\(299\) −4638.06 −0.897077
\(300\) 0 0
\(301\) 11708.8 2.24213
\(302\) 0 0
\(303\) −1732.03 1250.22i −0.328390 0.237041i
\(304\) 0 0
\(305\) 1855.88i 0.348418i
\(306\) 0 0
\(307\) 5337.06i 0.992190i 0.868268 + 0.496095i \(0.165233\pi\)
−0.868268 + 0.496095i \(0.834767\pi\)
\(308\) 0 0
\(309\) 579.344 + 418.186i 0.106659 + 0.0769896i
\(310\) 0 0
\(311\) 1034.85 0.188685 0.0943424 0.995540i \(-0.469925\pi\)
0.0943424 + 0.995540i \(0.469925\pi\)
\(312\) 0 0
\(313\) −9570.33 −1.72826 −0.864132 0.503265i \(-0.832132\pi\)
−0.864132 + 0.503265i \(0.832132\pi\)
\(314\) 0 0
\(315\) 2886.72 + 8700.89i 0.516344 + 1.55632i
\(316\) 0 0
\(317\) 3593.05i 0.636612i −0.947988 0.318306i \(-0.896886\pi\)
0.947988 0.318306i \(-0.103114\pi\)
\(318\) 0 0
\(319\) 5924.75i 1.03988i
\(320\) 0 0
\(321\) 6426.32 8902.85i 1.11739 1.54800i
\(322\) 0 0
\(323\) −6159.13 −1.06100
\(324\) 0 0
\(325\) 981.425 0.167507
\(326\) 0 0
\(327\) 3984.19 5519.60i 0.673781 0.933438i
\(328\) 0 0
\(329\) 3491.60i 0.585101i
\(330\) 0 0
\(331\) 4119.28i 0.684037i −0.939693 0.342018i \(-0.888890\pi\)
0.939693 0.342018i \(-0.111110\pi\)
\(332\) 0 0
\(333\) −42.6372 128.513i −0.00701654 0.0211486i
\(334\) 0 0
\(335\) −223.548 −0.0364589
\(336\) 0 0
\(337\) −6488.93 −1.04889 −0.524443 0.851446i \(-0.675726\pi\)
−0.524443 + 0.851446i \(0.675726\pi\)
\(338\) 0 0
\(339\) −5913.73 4268.69i −0.947462 0.683904i
\(340\) 0 0
\(341\) 3109.38i 0.493790i
\(342\) 0 0
\(343\) 23189.8i 3.65052i
\(344\) 0 0
\(345\) −7041.94 5083.06i −1.09891 0.793225i
\(346\) 0 0
\(347\) −7085.97 −1.09624 −0.548119 0.836400i \(-0.684656\pi\)
−0.548119 + 0.836400i \(0.684656\pi\)
\(348\) 0 0
\(349\) −6075.79 −0.931891 −0.465945 0.884814i \(-0.654286\pi\)
−0.465945 + 0.884814i \(0.654286\pi\)
\(350\) 0 0
\(351\) 3465.93 + 1090.70i 0.527059 + 0.165861i
\(352\) 0 0
\(353\) 4451.86i 0.671242i −0.941997 0.335621i \(-0.891054\pi\)
0.941997 0.335621i \(-0.108946\pi\)
\(354\) 0 0
\(355\) 3730.79i 0.557774i
\(356\) 0 0
\(357\) −8211.66 + 11376.2i −1.21739 + 1.68653i
\(358\) 0 0
\(359\) −10771.6 −1.58357 −0.791785 0.610800i \(-0.790848\pi\)
−0.791785 + 0.610800i \(0.790848\pi\)
\(360\) 0 0
\(361\) −27.1131 −0.00395293
\(362\) 0 0
\(363\) −3088.37 + 4278.54i −0.446549 + 0.618637i
\(364\) 0 0
\(365\) 6249.68i 0.896228i
\(366\) 0 0
\(367\) 6671.95i 0.948973i 0.880263 + 0.474486i \(0.157366\pi\)
−0.880263 + 0.474486i \(0.842634\pi\)
\(368\) 0 0
\(369\) −8352.48 + 2771.13i −1.17835 + 0.390947i
\(370\) 0 0
\(371\) 6732.66 0.942162
\(372\) 0 0
\(373\) −12618.5 −1.75164 −0.875819 0.482641i \(-0.839678\pi\)
−0.875819 + 0.482641i \(0.839678\pi\)
\(374\) 0 0
\(375\) 6405.33 + 4623.54i 0.882053 + 0.636690i
\(376\) 0 0
\(377\) 3167.66i 0.432739i
\(378\) 0 0
\(379\) 7915.38i 1.07279i −0.843968 0.536393i \(-0.819787\pi\)
0.843968 0.536393i \(-0.180213\pi\)
\(380\) 0 0
\(381\) 6672.40 + 4816.32i 0.897212 + 0.647632i
\(382\) 0 0
\(383\) −10820.5 −1.44360 −0.721802 0.692099i \(-0.756686\pi\)
−0.721802 + 0.692099i \(0.756686\pi\)
\(384\) 0 0
\(385\) 16447.0 2.17719
\(386\) 0 0
\(387\) 8247.94 2736.45i 1.08338 0.359435i
\(388\) 0 0
\(389\) 7990.57i 1.04149i 0.853714 + 0.520743i \(0.174345\pi\)
−0.853714 + 0.520743i \(0.825655\pi\)
\(390\) 0 0
\(391\) 13292.0i 1.71919i
\(392\) 0 0
\(393\) 487.583 675.485i 0.0625835 0.0867016i
\(394\) 0 0
\(395\) 2141.49 0.272785
\(396\) 0 0
\(397\) 5526.21 0.698620 0.349310 0.937007i \(-0.386416\pi\)
0.349310 + 0.937007i \(0.386416\pi\)
\(398\) 0 0
\(399\) −9180.91 + 12719.0i −1.15193 + 1.59585i
\(400\) 0 0
\(401\) 7844.55i 0.976903i −0.872591 0.488451i \(-0.837562\pi\)
0.872591 0.488451i \(-0.162438\pi\)
\(402\) 0 0
\(403\) 1662.43i 0.205487i
\(404\) 0 0
\(405\) 4066.96 + 5454.47i 0.498985 + 0.669222i
\(406\) 0 0
\(407\) −242.924 −0.0295855
\(408\) 0 0
\(409\) −4407.65 −0.532871 −0.266435 0.963853i \(-0.585846\pi\)
−0.266435 + 0.963853i \(0.585846\pi\)
\(410\) 0 0
\(411\) −6267.01 4523.70i −0.752138 0.542914i
\(412\) 0 0
\(413\) 8233.18i 0.980940i
\(414\) 0 0
\(415\) 2997.30i 0.354534i
\(416\) 0 0
\(417\) 247.593 + 178.720i 0.0290760 + 0.0209878i
\(418\) 0 0
\(419\) 12032.8 1.40296 0.701481 0.712688i \(-0.252523\pi\)
0.701481 + 0.712688i \(0.252523\pi\)
\(420\) 0 0
\(421\) 10320.3 1.19473 0.597365 0.801969i \(-0.296214\pi\)
0.597365 + 0.801969i \(0.296214\pi\)
\(422\) 0 0
\(423\) −816.021 2459.57i −0.0937973 0.282715i
\(424\) 0 0
\(425\) 2812.61i 0.321016i
\(426\) 0 0
\(427\) 7234.05i 0.819860i
\(428\) 0 0
\(429\) 3815.36 5285.70i 0.429388 0.594863i
\(430\) 0 0
\(431\) 8068.52 0.901733 0.450866 0.892591i \(-0.351115\pi\)
0.450866 + 0.892591i \(0.351115\pi\)
\(432\) 0 0
\(433\) −3236.75 −0.359234 −0.179617 0.983737i \(-0.557486\pi\)
−0.179617 + 0.983737i \(0.557486\pi\)
\(434\) 0 0
\(435\) −3471.58 + 4809.43i −0.382642 + 0.530102i
\(436\) 0 0
\(437\) 14860.9i 1.62676i
\(438\) 0 0
\(439\) 2207.71i 0.240019i 0.992773 + 0.120009i \(0.0382924\pi\)
−0.992773 + 0.120009i \(0.961708\pi\)
\(440\) 0 0
\(441\) 8335.91 + 25125.3i 0.900109 + 2.71302i
\(442\) 0 0
\(443\) −1058.18 −0.113489 −0.0567447 0.998389i \(-0.518072\pi\)
−0.0567447 + 0.998389i \(0.518072\pi\)
\(444\) 0 0
\(445\) −1226.79 −0.130686
\(446\) 0 0
\(447\) 3961.03 + 2859.18i 0.419128 + 0.302538i
\(448\) 0 0
\(449\) 10836.4i 1.13898i −0.822000 0.569488i \(-0.807142\pi\)
0.822000 0.569488i \(-0.192858\pi\)
\(450\) 0 0
\(451\) 15788.4i 1.64844i
\(452\) 0 0
\(453\) −7585.56 5475.46i −0.786756 0.567902i
\(454\) 0 0
\(455\) −8793.36 −0.906020
\(456\) 0 0
\(457\) −9827.12 −1.00589 −0.502947 0.864317i \(-0.667751\pi\)
−0.502947 + 0.864317i \(0.667751\pi\)
\(458\) 0 0
\(459\) −3125.77 + 9932.83i −0.317861 + 1.01008i
\(460\) 0 0
\(461\) 9311.74i 0.940761i −0.882464 0.470381i \(-0.844117\pi\)
0.882464 0.470381i \(-0.155883\pi\)
\(462\) 0 0
\(463\) 7601.37i 0.762993i −0.924370 0.381497i \(-0.875409\pi\)
0.924370 0.381497i \(-0.124591\pi\)
\(464\) 0 0
\(465\) −1821.93 + 2524.05i −0.181699 + 0.251720i
\(466\) 0 0
\(467\) 4446.20 0.440569 0.220284 0.975436i \(-0.429301\pi\)
0.220284 + 0.975436i \(0.429301\pi\)
\(468\) 0 0
\(469\) −871.368 −0.0857911
\(470\) 0 0
\(471\) 9196.90 12741.1i 0.899726 1.24646i
\(472\) 0 0
\(473\) 15590.8i 1.51557i
\(474\) 0 0
\(475\) 3144.59i 0.303756i
\(476\) 0 0
\(477\) 4742.65 1573.49i 0.455244 0.151038i
\(478\) 0 0
\(479\) 12285.9 1.17194 0.585970 0.810333i \(-0.300714\pi\)
0.585970 + 0.810333i \(0.300714\pi\)
\(480\) 0 0
\(481\) 129.879 0.0123118
\(482\) 0 0
\(483\) −27448.8 19813.3i −2.58584 1.86653i
\(484\) 0 0
\(485\) 1272.76i 0.119161i
\(486\) 0 0
\(487\) 8464.78i 0.787630i −0.919190 0.393815i \(-0.871155\pi\)
0.919190 0.393815i \(-0.128845\pi\)
\(488\) 0 0
\(489\) 13164.0 + 9502.11i 1.21737 + 0.878732i
\(490\) 0 0
\(491\) 6356.37 0.584234 0.292117 0.956383i \(-0.405640\pi\)
0.292117 + 0.956383i \(0.405640\pi\)
\(492\) 0 0
\(493\) −9078.02 −0.829317
\(494\) 0 0
\(495\) 11585.7 3843.82i 1.05200 0.349024i
\(496\) 0 0
\(497\) 14542.2i 1.31249i
\(498\) 0 0
\(499\) 11856.6i 1.06368i −0.846845 0.531840i \(-0.821501\pi\)
0.846845 0.531840i \(-0.178499\pi\)
\(500\) 0 0
\(501\) −1129.45 + 1564.72i −0.100719 + 0.139534i
\(502\) 0 0
\(503\) 17032.5 1.50982 0.754912 0.655826i \(-0.227679\pi\)
0.754912 + 0.655826i \(0.227679\pi\)
\(504\) 0 0
\(505\) 3836.76 0.338086
\(506\) 0 0
\(507\) 4641.65 6430.41i 0.406593 0.563283i
\(508\) 0 0
\(509\) 1563.18i 0.136123i 0.997681 + 0.0680617i \(0.0216815\pi\)
−0.997681 + 0.0680617i \(0.978319\pi\)
\(510\) 0 0
\(511\) 24360.6i 2.10891i
\(512\) 0 0
\(513\) −3494.72 + 11105.2i −0.300771 + 0.955767i
\(514\) 0 0
\(515\) −1283.36 −0.109809
\(516\) 0 0
\(517\) −4649.25 −0.395501
\(518\) 0 0
\(519\) −12349.9 8914.50i −1.04451 0.753956i
\(520\) 0 0
\(521\) 9096.23i 0.764900i −0.923976 0.382450i \(-0.875080\pi\)
0.923976 0.382450i \(-0.124920\pi\)
\(522\) 0 0
\(523\) 10108.4i 0.845143i 0.906330 + 0.422572i \(0.138873\pi\)
−0.906330 + 0.422572i \(0.861127\pi\)
\(524\) 0 0
\(525\) 5808.22 + 4192.53i 0.482841 + 0.348528i
\(526\) 0 0
\(527\) −4764.25 −0.393803
\(528\) 0 0
\(529\) 19904.2 1.63591
\(530\) 0 0
\(531\) 1924.17 + 5799.66i 0.157254 + 0.473980i
\(532\) 0 0
\(533\) 8441.26i 0.685988i
\(534\) 0 0
\(535\) 19721.5i 1.59371i
\(536\) 0 0
\(537\) 8620.18 11942.2i 0.692716 0.959670i
\(538\) 0 0
\(539\) 47493.6 3.79535
\(540\) 0 0
\(541\) −10097.6 −0.802462 −0.401231 0.915977i \(-0.631417\pi\)
−0.401231 + 0.915977i \(0.631417\pi\)
\(542\) 0 0
\(543\) 5534.93 7667.94i 0.437434 0.606009i
\(544\) 0 0
\(545\) 12226.9i 0.960998i
\(546\) 0 0
\(547\) 2064.83i 0.161400i 0.996738 + 0.0806998i \(0.0257155\pi\)
−0.996738 + 0.0806998i \(0.974284\pi\)
\(548\) 0 0
\(549\) −1690.66 5095.84i −0.131431 0.396148i
\(550\) 0 0
\(551\) −10149.5 −0.784727
\(552\) 0 0
\(553\) 8347.32 0.641888
\(554\) 0 0
\(555\) 197.194 + 142.340i 0.0150819 + 0.0108865i
\(556\) 0 0
\(557\) 12125.7i 0.922409i −0.887294 0.461204i \(-0.847417\pi\)
0.887294 0.461204i \(-0.152583\pi\)
\(558\) 0 0
\(559\) 8335.60i 0.630695i
\(560\) 0 0
\(561\) 15148.0 + 10934.2i 1.14002 + 0.822895i
\(562\) 0 0
\(563\) 7278.58 0.544859 0.272429 0.962176i \(-0.412173\pi\)
0.272429 + 0.962176i \(0.412173\pi\)
\(564\) 0 0
\(565\) 13100.0 0.975436
\(566\) 0 0
\(567\) 15852.6 + 21261.0i 1.17416 + 1.57474i
\(568\) 0 0
\(569\) 17428.1i 1.28405i 0.766685 + 0.642024i \(0.221905\pi\)
−0.766685 + 0.642024i \(0.778095\pi\)
\(570\) 0 0
\(571\) 2423.73i 0.177635i 0.996048 + 0.0888177i \(0.0283089\pi\)
−0.996048 + 0.0888177i \(0.971691\pi\)
\(572\) 0 0
\(573\) −14071.8 + 19494.7i −1.02593 + 1.42130i
\(574\) 0 0
\(575\) −6786.33 −0.492190
\(576\) 0 0
\(577\) −17471.1 −1.26054 −0.630268 0.776377i \(-0.717055\pi\)
−0.630268 + 0.776377i \(0.717055\pi\)
\(578\) 0 0
\(579\) 8425.03 11671.8i 0.604719 0.837761i
\(580\) 0 0
\(581\) 11683.2i 0.834252i
\(582\) 0 0
\(583\) 8964.88i 0.636857i
\(584\) 0 0
\(585\) −6194.27 + 2055.09i −0.437780 + 0.145244i
\(586\) 0 0
\(587\) −7273.06 −0.511399 −0.255700 0.966756i \(-0.582306\pi\)
−0.255700 + 0.966756i \(0.582306\pi\)
\(588\) 0 0
\(589\) −5326.60 −0.372629
\(590\) 0 0
\(591\) 16347.3 + 11799.9i 1.13779 + 0.821290i
\(592\) 0 0
\(593\) 19957.9i 1.38208i −0.722818 0.691038i \(-0.757154\pi\)
0.722818 0.691038i \(-0.242846\pi\)
\(594\) 0 0
\(595\) 25200.4i 1.73633i
\(596\) 0 0
\(597\) 8489.59 + 6128.01i 0.582003 + 0.420105i
\(598\) 0 0
\(599\) −6101.48 −0.416193 −0.208097 0.978108i \(-0.566727\pi\)
−0.208097 + 0.978108i \(0.566727\pi\)
\(600\) 0 0
\(601\) 16659.6 1.13071 0.565357 0.824846i \(-0.308738\pi\)
0.565357 + 0.824846i \(0.308738\pi\)
\(602\) 0 0
\(603\) −613.813 + 203.647i −0.0414534 + 0.0137531i
\(604\) 0 0
\(605\) 9477.77i 0.636903i
\(606\) 0 0
\(607\) 13728.6i 0.917999i 0.888437 + 0.458999i \(0.151792\pi\)
−0.888437 + 0.458999i \(0.848208\pi\)
\(608\) 0 0
\(609\) −13531.9 + 18746.7i −0.900392 + 1.24738i
\(610\) 0 0
\(611\) 2485.71 0.164585
\(612\) 0 0
\(613\) 7514.10 0.495093 0.247546 0.968876i \(-0.420376\pi\)
0.247546 + 0.968876i \(0.420376\pi\)
\(614\) 0 0
\(615\) 9251.15 12816.3i 0.606573 0.840330i
\(616\) 0 0
\(617\) 8052.78i 0.525434i 0.964873 + 0.262717i \(0.0846186\pi\)
−0.964873 + 0.262717i \(0.915381\pi\)
\(618\) 0 0
\(619\) 25194.4i 1.63594i −0.575261 0.817970i \(-0.695100\pi\)
0.575261 0.817970i \(-0.304900\pi\)
\(620\) 0 0
\(621\) −23966.1 7541.92i −1.54868 0.487354i
\(622\) 0 0
\(623\) −4781.91 −0.307517
\(624\) 0 0
\(625\) −9452.17 −0.604939
\(626\) 0 0
\(627\) 16936.0 + 12224.9i 1.07872 + 0.778650i
\(628\) 0 0
\(629\) 372.213i 0.0235948i
\(630\) 0 0
\(631\) 14945.1i 0.942876i −0.881899 0.471438i \(-0.843735\pi\)
0.881899 0.471438i \(-0.156265\pi\)
\(632\) 0 0
\(633\) −23642.1 17065.5i −1.48450 1.07155i
\(634\) 0 0
\(635\) −14780.6 −0.923702
\(636\) 0 0
\(637\) −25392.4 −1.57941
\(638\) 0 0
\(639\) 3398.66 + 10243.9i 0.210405 + 0.634184i
\(640\) 0 0
\(641\) 13909.9i 0.857109i 0.903516 + 0.428555i \(0.140977\pi\)
−0.903516 + 0.428555i \(0.859023\pi\)
\(642\) 0 0
\(643\) 13338.7i 0.818084i 0.912516 + 0.409042i \(0.134137\pi\)
−0.912516 + 0.409042i \(0.865863\pi\)
\(644\) 0 0
\(645\) −9135.36 + 12655.9i −0.557681 + 0.772597i
\(646\) 0 0
\(647\) −719.190 −0.0437006 −0.0218503 0.999761i \(-0.506956\pi\)
−0.0218503 + 0.999761i \(0.506956\pi\)
\(648\) 0 0
\(649\) 10962.9 0.663069
\(650\) 0 0
\(651\) −7101.69 + 9838.49i −0.427553 + 0.592321i
\(652\) 0 0
\(653\) 19019.4i 1.13980i −0.821714 0.569899i \(-0.806982\pi\)
0.821714 0.569899i \(-0.193018\pi\)
\(654\) 0 0
\(655\) 1496.32i 0.0892615i
\(656\) 0 0
\(657\) −5693.31 17160.2i −0.338078 1.01900i
\(658\) 0 0
\(659\) −22748.7 −1.34471 −0.672354 0.740230i \(-0.734717\pi\)
−0.672354 + 0.740230i \(0.734717\pi\)
\(660\) 0 0
\(661\) −23275.3 −1.36960 −0.684798 0.728733i \(-0.740110\pi\)
−0.684798 + 0.728733i \(0.740110\pi\)
\(662\) 0 0
\(663\) −8098.86 5845.98i −0.474410 0.342442i
\(664\) 0 0
\(665\) 28174.9i 1.64297i
\(666\) 0 0
\(667\) 21903.6i 1.27153i
\(668\) 0 0
\(669\) 8534.47 + 6160.41i 0.493216 + 0.356017i
\(670\) 0 0
\(671\) −9632.51 −0.554186
\(672\) 0 0
\(673\) −28195.5 −1.61495 −0.807473 0.589905i \(-0.799165\pi\)
−0.807473 + 0.589905i \(0.799165\pi\)
\(674\) 0 0
\(675\) 5071.29 + 1595.89i 0.289176 + 0.0910011i
\(676\) 0 0
\(677\) 32369.7i 1.83762i −0.394701 0.918809i \(-0.629152\pi\)
0.394701 0.918809i \(-0.370848\pi\)
\(678\) 0 0
\(679\) 4961.07i 0.280396i
\(680\) 0 0
\(681\) −1228.80 + 1702.34i −0.0691447 + 0.0957913i
\(682\) 0 0
\(683\) 1227.33 0.0687592 0.0343796 0.999409i \(-0.489054\pi\)
0.0343796 + 0.999409i \(0.489054\pi\)
\(684\) 0 0
\(685\) 13882.6 0.774345
\(686\) 0 0
\(687\) 14401.4 19951.3i 0.799776 1.10799i
\(688\) 0 0
\(689\) 4793.06i 0.265023i
\(690\) 0 0
\(691\) 3563.29i 0.196170i 0.995178 + 0.0980852i \(0.0312718\pi\)
−0.995178 + 0.0980852i \(0.968728\pi\)
\(692\) 0 0
\(693\) 45159.8 14982.8i 2.47544 0.821284i
\(694\) 0 0
\(695\) −548.465 −0.0299345
\(696\) 0 0
\(697\) 24191.3 1.31465
\(698\) 0 0
\(699\) −15604.8 11264.0i −0.844389 0.609503i
\(700\) 0 0
\(701\) 11942.4i 0.643448i 0.946834 + 0.321724i \(0.104262\pi\)
−0.946834 + 0.321724i \(0.895738\pi\)
\(702\) 0 0
\(703\) 416.147i 0.0223262i
\(704\) 0 0
\(705\) 3774.04 + 2724.21i 0.201615 + 0.145531i
\(706\) 0 0
\(707\) 14955.3 0.795547
\(708\) 0 0
\(709\) 22547.9 1.19436 0.597182 0.802105i \(-0.296287\pi\)
0.597182 + 0.802105i \(0.296287\pi\)
\(710\) 0 0
\(711\) 5880.06 1950.85i 0.310154 0.102901i
\(712\) 0 0
\(713\) 11495.3i 0.603790i
\(714\) 0 0
\(715\) 11708.8i 0.612426i
\(716\) 0 0
\(717\) −19460.1 + 26959.6i −1.01360 + 1.40422i
\(718\) 0 0
\(719\) −2034.54 −0.105529 −0.0527647 0.998607i \(-0.516803\pi\)
−0.0527647 + 0.998607i \(0.516803\pi\)
\(720\) 0 0
\(721\) −5002.39 −0.258389
\(722\) 0 0
\(723\) −11454.6 + 15869.0i −0.589215 + 0.816283i
\(724\) 0 0
\(725\) 4634.86i 0.237427i
\(726\) 0 0
\(727\) 26487.5i 1.35126i 0.737240 + 0.675631i \(0.236129\pi\)
−0.737240 + 0.675631i \(0.763871\pi\)
\(728\) 0 0
\(729\) 16135.9 + 11271.9i 0.819786 + 0.572670i
\(730\) 0 0
\(731\) −23888.5 −1.20869
\(732\) 0 0
\(733\) 38967.7 1.96358 0.981791 0.189964i \(-0.0608370\pi\)
0.981791 + 0.189964i \(0.0608370\pi\)
\(734\) 0 0
\(735\) −38553.0 27828.6i −1.93476 1.39656i
\(736\) 0 0
\(737\) 1160.27i 0.0579907i
\(738\) 0 0
\(739\) 14312.2i 0.712425i 0.934405 + 0.356212i \(0.115932\pi\)
−0.934405 + 0.356212i \(0.884068\pi\)
\(740\) 0 0
\(741\) −9054.80 6536.00i −0.448902 0.324030i
\(742\) 0 0
\(743\) −12790.0 −0.631520 −0.315760 0.948839i \(-0.602259\pi\)
−0.315760 + 0.948839i \(0.602259\pi\)
\(744\) 0 0
\(745\) −8774.41 −0.431503
\(746\) 0 0
\(747\) −2730.47 8229.93i −0.133739 0.403102i
\(748\) 0 0
\(749\) 76872.3i 3.75014i
\(750\) 0 0
\(751\) 19441.6i 0.944653i 0.881424 + 0.472327i \(0.156586\pi\)
−0.881424 + 0.472327i \(0.843414\pi\)
\(752\) 0 0
\(753\) 6020.70 8340.92i 0.291376 0.403665i
\(754\) 0 0
\(755\) 16803.4 0.809985
\(756\) 0 0
\(757\) −13901.0 −0.667424 −0.333712 0.942675i \(-0.608301\pi\)
−0.333712 + 0.942675i \(0.608301\pi\)
\(758\) 0 0
\(759\) −26382.4 + 36549.4i −1.26169 + 1.74791i
\(760\) 0 0
\(761\) 8818.58i 0.420070i 0.977694 + 0.210035i \(0.0673578\pi\)
−0.977694 + 0.210035i \(0.932642\pi\)
\(762\) 0 0
\(763\) 47659.3i 2.26132i
\(764\) 0 0
\(765\) −5889.57 17751.8i −0.278350 0.838978i
\(766\) 0 0
\(767\) −5861.30 −0.275931
\(768\) 0 0
\(769\) 1323.39 0.0620582 0.0310291 0.999518i \(-0.490122\pi\)
0.0310291 + 0.999518i \(0.490122\pi\)
\(770\) 0 0
\(771\) 4556.75 + 3289.18i 0.212850 + 0.153641i
\(772\) 0 0
\(773\) 34327.1i 1.59723i 0.601840 + 0.798617i \(0.294435\pi\)
−0.601840 + 0.798617i \(0.705565\pi\)
\(774\) 0 0
\(775\) 2432.43i 0.112743i
\(776\) 0 0
\(777\) 768.644 + 554.828i 0.0354890 + 0.0256169i
\(778\) 0 0
\(779\) 27046.7 1.24397
\(780\) 0 0
\(781\) 19363.8 0.887183
\(782\) 0 0
\(783\) −5150.91 + 16368.2i −0.235094 + 0.747063i
\(784\) 0 0
\(785\) 28224.0i 1.28326i
\(786\) 0 0
\(787\) 15375.3i 0.696405i −0.937419 0.348203i \(-0.886792\pi\)
0.937419 0.348203i \(-0.113208\pi\)
\(788\) 0 0
\(789\) 3409.80 4723.84i 0.153856 0.213147i
\(790\) 0 0
\(791\) 51062.5 2.29529
\(792\) 0 0
\(793\) 5150.00 0.230620
\(794\) 0 0
\(795\) −5252.93 + 7277.27i −0.234342 + 0.324652i
\(796\) 0 0
\(797\) 3347.25i 0.148765i −0.997230 0.0743826i \(-0.976301\pi\)
0.997230 0.0743826i \(-0.0236986\pi\)
\(798\) 0 0
\(799\) 7123.67i 0.315416i
\(800\) 0 0
\(801\) −3368.50 + 1117.58i −0.148589 + 0.0492979i
\(802\) 0 0
\(803\) −32437.4 −1.42552
\(804\) 0 0
\(805\) 60804.1 2.66219
\(806\) 0 0
\(807\) 7772.76 + 5610.59i 0.339051 + 0.244736i
\(808\) 0 0
\(809\) 39577.3i 1.71998i −0.510309 0.859991i \(-0.670469\pi\)
0.510309 0.859991i \(-0.329531\pi\)
\(810\) 0 0
\(811\) 12614.4i 0.546180i 0.961989 + 0.273090i \(0.0880457\pi\)
−0.961989 + 0.273090i \(0.911954\pi\)
\(812\) 0 0
\(813\) 3956.07 + 2855.60i 0.170659 + 0.123186i
\(814\) 0 0
\(815\) −29160.6 −1.25332
\(816\) 0 0
\(817\) −26708.2 −1.14370
\(818\) 0 0
\(819\) −24144.6 + 8010.54i −1.03014 + 0.341772i
\(820\) 0 0
\(821\) 18243.6i 0.775526i 0.921759 + 0.387763i \(0.126752\pi\)
−0.921759 + 0.387763i \(0.873248\pi\)
\(822\) 0 0
\(823\) 18350.6i 0.777230i −0.921400 0.388615i \(-0.872954\pi\)
0.921400 0.388615i \(-0.127046\pi\)
\(824\) 0 0
\(825\) 5582.57 7733.94i 0.235588 0.326377i
\(826\) 0 0
\(827\) −15785.4 −0.663739 −0.331869 0.943325i \(-0.607679\pi\)
−0.331869 + 0.943325i \(0.607679\pi\)
\(828\) 0 0
\(829\) −36102.8 −1.51255 −0.756274 0.654255i \(-0.772982\pi\)
−0.756274 + 0.654255i \(0.772982\pi\)
\(830\) 0 0
\(831\) −3348.15 + 4638.43i −0.139766 + 0.193629i
\(832\) 0 0
\(833\) 72770.6i 3.02683i
\(834\) 0 0
\(835\) 3466.13i 0.143653i
\(836\) 0 0
\(837\) −2703.26 + 8590.21i −0.111635 + 0.354744i
\(838\) 0 0
\(839\) 18961.8 0.780255 0.390127 0.920761i \(-0.372431\pi\)
0.390127 + 0.920761i \(0.372431\pi\)
\(840\) 0 0
\(841\) 9429.48 0.386628
\(842\) 0 0
\(843\) −3152.35 2275.45i −0.128793 0.0929665i
\(844\) 0 0
\(845\) 14244.6i 0.579914i
\(846\) 0 0
\(847\) 36943.4i 1.49869i
\(848\) 0 0
\(849\) −9059.00 6539.03i −0.366201 0.264333i
\(850\) 0 0
\(851\) −898.084 −0.0361762
\(852\) 0 0
\(853\) −27761.8 −1.11435 −0.557177 0.830394i \(-0.688116\pi\)
−0.557177 + 0.830394i \(0.688116\pi\)
\(854\) 0 0
\(855\) −6584.74 19847.1i −0.263384 0.793868i
\(856\) 0 0
\(857\) 2082.01i 0.0829874i 0.999139 + 0.0414937i \(0.0132117\pi\)
−0.999139 + 0.0414937i \(0.986788\pi\)
\(858\) 0 0
\(859\) 34995.2i 1.39001i 0.719004 + 0.695005i \(0.244598\pi\)
−0.719004 + 0.695005i \(0.755402\pi\)
\(860\) 0 0
\(861\) 36060.1 49956.6i 1.42732 1.97737i
\(862\) 0 0
\(863\) 6045.46 0.238459 0.119229 0.992867i \(-0.461958\pi\)
0.119229 + 0.992867i \(0.461958\pi\)
\(864\) 0 0
\(865\) 27357.3 1.07535
\(866\) 0 0
\(867\) 1812.23 2510.61i 0.0709878 0.0983446i
\(868\) 0 0
\(869\) 11114.9i 0.433886i
\(870\) 0 0
\(871\) 620.337i 0.0241324i
\(872\) 0 0
\(873\) −1159.45 3494.70i −0.0449501 0.135484i
\(874\) 0 0
\(875\) −55307.3 −2.13683
\(876\) 0 0
\(877\) −30101.8 −1.15903 −0.579514 0.814963i \(-0.696757\pi\)
−0.579514 + 0.814963i \(0.696757\pi\)
\(878\) 0 0
\(879\) −22131.2 15974.9i −0.849224 0.612993i
\(880\) 0 0
\(881\) 16655.3i 0.636926i 0.947935 + 0.318463i \(0.103167\pi\)
−0.947935 + 0.318463i \(0.896833\pi\)
\(882\) 0 0
\(883\) 24940.5i 0.950528i 0.879843 + 0.475264i \(0.157647\pi\)
−0.879843 + 0.475264i \(0.842353\pi\)
\(884\) 0 0
\(885\) −8899.16 6423.66i −0.338014 0.243987i
\(886\) 0 0
\(887\) −32143.9 −1.21678 −0.608391 0.793637i \(-0.708185\pi\)
−0.608391 + 0.793637i \(0.708185\pi\)
\(888\) 0 0
\(889\) −57613.3 −2.17355
\(890\) 0 0
\(891\) 28310.1 21108.5i 1.06445 0.793673i
\(892\) 0 0
\(893\) 7964.51i 0.298457i
\(894\) 0 0
\(895\) 26454.1i 0.988004i
\(896\) 0 0
\(897\) 14105.3 19541.1i 0.525041 0.727378i
\(898\) 0 0
\(899\) −7850.94 −0.291261
\(900\) 0 0
\(901\) −13736.2 −0.507900
\(902\) 0 0
\(903\) −35608.7 + 49331.3i −1.31227 + 1.81799i
\(904\) 0 0
\(905\) 16985.9i 0.623902i
\(906\) 0 0
\(907\) 3376.58i 0.123613i 0.998088 + 0.0618067i \(0.0196862\pi\)
−0.998088 + 0.0618067i \(0.980314\pi\)
\(908\) 0 0
\(909\) 10534.9 3495.19i 0.384401 0.127534i
\(910\) 0 0
\(911\) 45248.4 1.64561 0.822803 0.568326i \(-0.192409\pi\)
0.822803 + 0.568326i \(0.192409\pi\)
\(912\) 0 0
\(913\) −15556.8 −0.563914
\(914\) 0 0
\(915\) 7819.21 + 5644.12i 0.282508 + 0.203922i
\(916\) 0 0
\(917\) 5832.52i 0.210040i
\(918\) 0 0
\(919\) 18545.2i 0.665668i −0.942985 0.332834i \(-0.891995\pi\)
0.942985 0.332834i \(-0.108005\pi\)
\(920\) 0 0
\(921\) −22486.1 16231.1i −0.804498 0.580709i
\(922\) 0 0
\(923\) −10352.8 −0.369195
\(924\) 0 0
\(925\) 190.037 0.00675499
\(926\) 0 0
\(927\) −3523.81 + 1169.11i −0.124851 + 0.0414223i
\(928\) 0 0
\(929\) 33976.7i 1.19993i 0.800024 + 0.599967i \(0.204820\pi\)
−0.800024 + 0.599967i \(0.795180\pi\)
\(930\) 0 0
\(931\) 81360.0i 2.86409i
\(932\) 0 0
\(933\) −3147.19 + 4360.03i −0.110433 + 0.152992i
\(934\) 0 0
\(935\) −33555.7 −1.17368
\(936\) 0 0
\(937\) −22803.1 −0.795031 −0.397516 0.917595i \(-0.630128\pi\)
−0.397516 + 0.917595i \(0.630128\pi\)
\(938\) 0 0
\(939\) 29105.3 40321.7i 1.01152 1.40133i
\(940\) 0 0
\(941\) 24174.7i 0.837484i −0.908105 0.418742i \(-0.862471\pi\)
0.908105 0.418742i \(-0.137529\pi\)
\(942\) 0 0
\(943\) 58369.4i 2.01566i
\(944\) 0 0
\(945\) −45437.7 14298.8i −1.56412 0.492213i
\(946\) 0 0
\(947\) −8044.99 −0.276058 −0.138029 0.990428i \(-0.544077\pi\)
−0.138029 + 0.990428i \(0.544077\pi\)
\(948\) 0 0
\(949\) 17342.6 0.593220
\(950\) 0 0
\(951\) 15138.3 + 10927.2i 0.516185 + 0.372596i
\(952\) 0 0
\(953\) 39730.8i 1.35048i 0.737598 + 0.675240i \(0.235960\pi\)
−0.737598 + 0.675240i \(0.764040\pi\)
\(954\) 0 0
\(955\) 43184.5i 1.46326i
\(956\) 0 0
\(957\) 24962.2 + 18018.4i 0.843169 + 0.608622i
\(958\) 0 0
\(959\) 54112.9 1.82210
\(960\) 0 0
\(961\) 25670.7 0.861694
\(962\) 0 0
\(963\) 17965.8 + 54150.8i 0.601183 + 1.81203i
\(964\) 0 0
\(965\) 25855.2i 0.862496i
\(966\) 0 0
\(967\) 17685.2i 0.588126i −0.955786 0.294063i \(-0.904992\pi\)
0.955786 0.294063i \(-0.0950076\pi\)
\(968\) 0 0
\(969\) 18731.2 25949.7i 0.620982 0.860292i
\(970\) 0 0
\(971\) 44010.1 1.45453 0.727267 0.686355i \(-0.240790\pi\)
0.727267 + 0.686355i \(0.240790\pi\)
\(972\) 0 0
\(973\) −2137.86 −0.0704385
\(974\) 0 0
\(975\) −2984.71 + 4134.94i −0.0980382 + 0.135820i
\(976\) 0 0
\(977\) 33518.4i 1.09759i 0.835956 + 0.548796i \(0.184914\pi\)
−0.835956 + 0.548796i \(0.815086\pi\)
\(978\) 0 0
\(979\) 6367.36i 0.207867i
\(980\) 0 0
\(981\) 11138.4 + 33572.4i 0.362511 + 1.09265i
\(982\) 0 0
\(983\) −43740.3 −1.41923 −0.709613 0.704592i \(-0.751130\pi\)
−0.709613 + 0.704592i \(0.751130\pi\)
\(984\) 0 0
\(985\) −36212.2 −1.17139
\(986\) 0 0
\(987\) 14710.8 + 10618.7i 0.474418 + 0.342448i
\(988\) 0 0
\(989\) 57638.8i 1.85319i
\(990\) 0 0
\(991\) 10468.7i 0.335568i 0.985824 + 0.167784i \(0.0536611\pi\)
−0.985824 + 0.167784i \(0.946339\pi\)
\(992\) 0 0
\(993\) 17355.4 + 12527.6i 0.554638 + 0.400353i
\(994\) 0 0
\(995\) −18806.0 −0.599186
\(996\) 0 0
\(997\) −9561.56 −0.303729 −0.151864 0.988401i \(-0.548528\pi\)
−0.151864 + 0.988401i \(0.548528\pi\)
\(998\) 0 0
\(999\) 671.121 + 211.195i 0.0212546 + 0.00668862i
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 384.4.c.a.383.6 yes 12
3.2 odd 2 384.4.c.d.383.8 yes 12
4.3 odd 2 384.4.c.d.383.7 yes 12
8.3 odd 2 384.4.c.b.383.6 yes 12
8.5 even 2 384.4.c.c.383.7 yes 12
12.11 even 2 inner 384.4.c.a.383.5 12
16.3 odd 4 768.4.f.g.383.10 12
16.5 even 4 768.4.f.h.383.10 12
16.11 odd 4 768.4.f.f.383.3 12
16.13 even 4 768.4.f.e.383.3 12
24.5 odd 2 384.4.c.b.383.5 yes 12
24.11 even 2 384.4.c.c.383.8 yes 12
48.5 odd 4 768.4.f.g.383.9 12
48.11 even 4 768.4.f.e.383.4 12
48.29 odd 4 768.4.f.f.383.4 12
48.35 even 4 768.4.f.h.383.9 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
384.4.c.a.383.5 12 12.11 even 2 inner
384.4.c.a.383.6 yes 12 1.1 even 1 trivial
384.4.c.b.383.5 yes 12 24.5 odd 2
384.4.c.b.383.6 yes 12 8.3 odd 2
384.4.c.c.383.7 yes 12 8.5 even 2
384.4.c.c.383.8 yes 12 24.11 even 2
384.4.c.d.383.7 yes 12 4.3 odd 2
384.4.c.d.383.8 yes 12 3.2 odd 2
768.4.f.e.383.3 12 16.13 even 4
768.4.f.e.383.4 12 48.11 even 4
768.4.f.f.383.3 12 16.11 odd 4
768.4.f.f.383.4 12 48.29 odd 4
768.4.f.g.383.9 12 48.5 odd 4
768.4.f.g.383.10 12 16.3 odd 4
768.4.f.h.383.9 12 48.35 even 4
768.4.f.h.383.10 12 16.5 even 4