Properties

Label 136.4.b.b.33.8
Level $136$
Weight $4$
Character 136.33
Analytic conductor $8.024$
Analytic rank $0$
Dimension $8$
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] = [136,4,Mod(33,136)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(136, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 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("136.33");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 136 = 2^{3} \cdot 17 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 136.b (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: \(8.02425976078\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 95x^{6} + 756x^{4} + 1780x^{2} + 1152 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{13} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 33.8
Root \(2.20783i\) of defining polynomial
Character \(\chi\) \(=\) 136.33
Dual form 136.4.b.b.33.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+9.26065i q^{3} -16.4090i q^{5} -34.5010i q^{7} -58.7597 q^{9} +O(q^{10})\) \(q+9.26065i q^{3} -16.4090i q^{5} -34.5010i q^{7} -58.7597 q^{9} -7.42843i q^{11} -42.0242 q^{13} +151.958 q^{15} +(-26.0870 - 65.0574i) q^{17} +59.9095 q^{19} +319.502 q^{21} -49.4728i q^{23} -144.255 q^{25} -294.116i q^{27} +259.369i q^{29} +92.2215i q^{31} +68.7922 q^{33} -566.126 q^{35} -207.564i q^{37} -389.172i q^{39} -176.800i q^{41} -19.0809 q^{43} +964.188i q^{45} +80.1389 q^{47} -847.318 q^{49} +(602.474 - 241.583i) q^{51} +319.869 q^{53} -121.893 q^{55} +554.801i q^{57} +11.0809 q^{59} -712.648i q^{61} +2027.27i q^{63} +689.575i q^{65} +484.980 q^{67} +458.150 q^{69} +443.968i q^{71} -337.468i q^{73} -1335.89i q^{75} -256.288 q^{77} -840.905i q^{79} +1137.19 q^{81} +456.004 q^{83} +(-1067.53 + 428.062i) q^{85} -2401.93 q^{87} -1205.43 q^{89} +1449.88i q^{91} -854.032 q^{93} -983.055i q^{95} +638.484i q^{97} +436.493i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 132 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 132 q^{9} + 44 q^{13} + 24 q^{15} + 28 q^{17} + 48 q^{19} + 308 q^{21} - 520 q^{25} + 812 q^{33} - 1064 q^{35} + 8 q^{43} + 312 q^{47} - 1124 q^{49} + 408 q^{51} + 472 q^{53} + 1416 q^{55} - 72 q^{59} - 624 q^{67} - 180 q^{69} + 1660 q^{77} + 3156 q^{81} + 2472 q^{83} - 2160 q^{85} - 6664 q^{87} + 68 q^{89} - 4036 q^{93}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/136\mathbb{Z}\right)^\times\).

\(n\) \(69\) \(103\) \(105\)
\(\chi(n)\) \(1\) \(1\) \(-1\)

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
\(3\) 9.26065i 1.78221i 0.453794 + 0.891107i \(0.350070\pi\)
−0.453794 + 0.891107i \(0.649930\pi\)
\(4\) 0 0
\(5\) 16.4090i 1.46766i −0.679331 0.733832i \(-0.737730\pi\)
0.679331 0.733832i \(-0.262270\pi\)
\(6\) 0 0
\(7\) 34.5010i 1.86288i −0.363898 0.931439i \(-0.618554\pi\)
0.363898 0.931439i \(-0.381446\pi\)
\(8\) 0 0
\(9\) −58.7597 −2.17629
\(10\) 0 0
\(11\) 7.42843i 0.203614i −0.994804 0.101807i \(-0.967538\pi\)
0.994804 0.101807i \(-0.0324625\pi\)
\(12\) 0 0
\(13\) −42.0242 −0.896571 −0.448285 0.893890i \(-0.647965\pi\)
−0.448285 + 0.893890i \(0.647965\pi\)
\(14\) 0 0
\(15\) 151.958 2.61569
\(16\) 0 0
\(17\) −26.0870 65.0574i −0.372178 0.928161i
\(18\) 0 0
\(19\) 59.9095 0.723378 0.361689 0.932299i \(-0.382200\pi\)
0.361689 + 0.932299i \(0.382200\pi\)
\(20\) 0 0
\(21\) 319.502 3.32005
\(22\) 0 0
\(23\) 49.4728i 0.448513i −0.974530 0.224256i \(-0.928005\pi\)
0.974530 0.224256i \(-0.0719953\pi\)
\(24\) 0 0
\(25\) −144.255 −1.15404
\(26\) 0 0
\(27\) 294.116i 2.09639i
\(28\) 0 0
\(29\) 259.369i 1.66081i 0.557157 + 0.830407i \(0.311892\pi\)
−0.557157 + 0.830407i \(0.688108\pi\)
\(30\) 0 0
\(31\) 92.2215i 0.534306i 0.963654 + 0.267153i \(0.0860829\pi\)
−0.963654 + 0.267153i \(0.913917\pi\)
\(32\) 0 0
\(33\) 68.7922 0.362884
\(34\) 0 0
\(35\) −566.126 −2.73408
\(36\) 0 0
\(37\) 207.564i 0.922252i −0.887335 0.461126i \(-0.847446\pi\)
0.887335 0.461126i \(-0.152554\pi\)
\(38\) 0 0
\(39\) 389.172i 1.59788i
\(40\) 0 0
\(41\) 176.800i 0.673454i −0.941602 0.336727i \(-0.890680\pi\)
0.941602 0.336727i \(-0.109320\pi\)
\(42\) 0 0
\(43\) −19.0809 −0.0676700 −0.0338350 0.999427i \(-0.510772\pi\)
−0.0338350 + 0.999427i \(0.510772\pi\)
\(44\) 0 0
\(45\) 964.188i 3.19406i
\(46\) 0 0
\(47\) 80.1389 0.248712 0.124356 0.992238i \(-0.460314\pi\)
0.124356 + 0.992238i \(0.460314\pi\)
\(48\) 0 0
\(49\) −847.318 −2.47031
\(50\) 0 0
\(51\) 602.474 241.583i 1.65418 0.663301i
\(52\) 0 0
\(53\) 319.869 0.829006 0.414503 0.910048i \(-0.363955\pi\)
0.414503 + 0.910048i \(0.363955\pi\)
\(54\) 0 0
\(55\) −121.893 −0.298837
\(56\) 0 0
\(57\) 554.801i 1.28921i
\(58\) 0 0
\(59\) 11.0809 0.0244510 0.0122255 0.999925i \(-0.496108\pi\)
0.0122255 + 0.999925i \(0.496108\pi\)
\(60\) 0 0
\(61\) 712.648i 1.49582i −0.663798 0.747912i \(-0.731057\pi\)
0.663798 0.747912i \(-0.268943\pi\)
\(62\) 0 0
\(63\) 2027.27i 4.05415i
\(64\) 0 0
\(65\) 689.575i 1.31587i
\(66\) 0 0
\(67\) 484.980 0.884325 0.442163 0.896935i \(-0.354211\pi\)
0.442163 + 0.896935i \(0.354211\pi\)
\(68\) 0 0
\(69\) 458.150 0.799345
\(70\) 0 0
\(71\) 443.968i 0.742103i 0.928612 + 0.371051i \(0.121003\pi\)
−0.928612 + 0.371051i \(0.878997\pi\)
\(72\) 0 0
\(73\) 337.468i 0.541063i −0.962711 0.270531i \(-0.912801\pi\)
0.962711 0.270531i \(-0.0871994\pi\)
\(74\) 0 0
\(75\) 1335.89i 2.05674i
\(76\) 0 0
\(77\) −256.288 −0.379309
\(78\) 0 0
\(79\) 840.905i 1.19759i −0.800904 0.598793i \(-0.795647\pi\)
0.800904 0.598793i \(-0.204353\pi\)
\(80\) 0 0
\(81\) 1137.19 1.55993
\(82\) 0 0
\(83\) 456.004 0.603047 0.301524 0.953459i \(-0.402505\pi\)
0.301524 + 0.953459i \(0.402505\pi\)
\(84\) 0 0
\(85\) −1067.53 + 428.062i −1.36223 + 0.546233i
\(86\) 0 0
\(87\) −2401.93 −2.95993
\(88\) 0 0
\(89\) −1205.43 −1.43568 −0.717841 0.696207i \(-0.754870\pi\)
−0.717841 + 0.696207i \(0.754870\pi\)
\(90\) 0 0
\(91\) 1449.88i 1.67020i
\(92\) 0 0
\(93\) −854.032 −0.952247
\(94\) 0 0
\(95\) 983.055i 1.06168i
\(96\) 0 0
\(97\) 638.484i 0.668332i 0.942514 + 0.334166i \(0.108455\pi\)
−0.942514 + 0.334166i \(0.891545\pi\)
\(98\) 0 0
\(99\) 436.493i 0.443123i
\(100\) 0 0
\(101\) 739.395 0.728441 0.364220 0.931313i \(-0.381335\pi\)
0.364220 + 0.931313i \(0.381335\pi\)
\(102\) 0 0
\(103\) 1588.54 1.51965 0.759824 0.650129i \(-0.225285\pi\)
0.759824 + 0.650129i \(0.225285\pi\)
\(104\) 0 0
\(105\) 5242.70i 4.87271i
\(106\) 0 0
\(107\) 202.298i 0.182775i −0.995815 0.0913875i \(-0.970870\pi\)
0.995815 0.0913875i \(-0.0291302\pi\)
\(108\) 0 0
\(109\) 1244.71i 1.09378i −0.837204 0.546890i \(-0.815812\pi\)
0.837204 0.546890i \(-0.184188\pi\)
\(110\) 0 0
\(111\) 1922.18 1.64365
\(112\) 0 0
\(113\) 1492.28i 1.24232i 0.783685 + 0.621159i \(0.213338\pi\)
−0.783685 + 0.621159i \(0.786662\pi\)
\(114\) 0 0
\(115\) −811.798 −0.658266
\(116\) 0 0
\(117\) 2469.33 1.95119
\(118\) 0 0
\(119\) −2244.54 + 900.028i −1.72905 + 0.693323i
\(120\) 0 0
\(121\) 1275.82 0.958541
\(122\) 0 0
\(123\) 1637.29 1.20024
\(124\) 0 0
\(125\) 315.954i 0.226078i
\(126\) 0 0
\(127\) 2054.49 1.43548 0.717741 0.696311i \(-0.245176\pi\)
0.717741 + 0.696311i \(0.245176\pi\)
\(128\) 0 0
\(129\) 176.702i 0.120602i
\(130\) 0 0
\(131\) 1892.72i 1.26235i 0.775640 + 0.631175i \(0.217427\pi\)
−0.775640 + 0.631175i \(0.782573\pi\)
\(132\) 0 0
\(133\) 2066.94i 1.34757i
\(134\) 0 0
\(135\) −4826.14 −3.07680
\(136\) 0 0
\(137\) −848.618 −0.529214 −0.264607 0.964356i \(-0.585242\pi\)
−0.264607 + 0.964356i \(0.585242\pi\)
\(138\) 0 0
\(139\) 20.6414i 0.0125955i 0.999980 + 0.00629777i \(0.00200466\pi\)
−0.999980 + 0.00629777i \(0.997995\pi\)
\(140\) 0 0
\(141\) 742.139i 0.443258i
\(142\) 0 0
\(143\) 312.174i 0.182555i
\(144\) 0 0
\(145\) 4255.98 2.43752
\(146\) 0 0
\(147\) 7846.71i 4.40263i
\(148\) 0 0
\(149\) −1202.59 −0.661206 −0.330603 0.943770i \(-0.607252\pi\)
−0.330603 + 0.943770i \(0.607252\pi\)
\(150\) 0 0
\(151\) −2080.77 −1.12140 −0.560698 0.828020i \(-0.689467\pi\)
−0.560698 + 0.828020i \(0.689467\pi\)
\(152\) 0 0
\(153\) 1532.87 + 3822.76i 0.809966 + 2.01994i
\(154\) 0 0
\(155\) 1513.26 0.784182
\(156\) 0 0
\(157\) 1153.74 0.586486 0.293243 0.956038i \(-0.405266\pi\)
0.293243 + 0.956038i \(0.405266\pi\)
\(158\) 0 0
\(159\) 2962.19i 1.47747i
\(160\) 0 0
\(161\) −1706.86 −0.835524
\(162\) 0 0
\(163\) 1461.35i 0.702221i −0.936334 0.351110i \(-0.885804\pi\)
0.936334 0.351110i \(-0.114196\pi\)
\(164\) 0 0
\(165\) 1128.81i 0.532592i
\(166\) 0 0
\(167\) 4157.98i 1.92667i 0.268296 + 0.963337i \(0.413539\pi\)
−0.268296 + 0.963337i \(0.586461\pi\)
\(168\) 0 0
\(169\) −430.965 −0.196160
\(170\) 0 0
\(171\) −3520.27 −1.57428
\(172\) 0 0
\(173\) 2504.08i 1.10047i −0.835010 0.550235i \(-0.814538\pi\)
0.835010 0.550235i \(-0.185462\pi\)
\(174\) 0 0
\(175\) 4976.94i 2.14983i
\(176\) 0 0
\(177\) 102.616i 0.0435769i
\(178\) 0 0
\(179\) 839.685 0.350620 0.175310 0.984513i \(-0.443907\pi\)
0.175310 + 0.984513i \(0.443907\pi\)
\(180\) 0 0
\(181\) 500.491i 0.205532i 0.994706 + 0.102766i \(0.0327692\pi\)
−0.994706 + 0.102766i \(0.967231\pi\)
\(182\) 0 0
\(183\) 6599.59 2.66588
\(184\) 0 0
\(185\) −3405.92 −1.35356
\(186\) 0 0
\(187\) −483.275 + 193.786i −0.188987 + 0.0757808i
\(188\) 0 0
\(189\) −10147.3 −3.90532
\(190\) 0 0
\(191\) 199.937 0.0757430 0.0378715 0.999283i \(-0.487942\pi\)
0.0378715 + 0.999283i \(0.487942\pi\)
\(192\) 0 0
\(193\) 2758.52i 1.02882i −0.857544 0.514411i \(-0.828011\pi\)
0.857544 0.514411i \(-0.171989\pi\)
\(194\) 0 0
\(195\) −6385.92 −2.34515
\(196\) 0 0
\(197\) 3379.87i 1.22236i 0.791490 + 0.611182i \(0.209306\pi\)
−0.791490 + 0.611182i \(0.790694\pi\)
\(198\) 0 0
\(199\) 2729.46i 0.972293i −0.873877 0.486147i \(-0.838402\pi\)
0.873877 0.486147i \(-0.161598\pi\)
\(200\) 0 0
\(201\) 4491.24i 1.57606i
\(202\) 0 0
\(203\) 8948.48 3.09389
\(204\) 0 0
\(205\) −2901.12 −0.988404
\(206\) 0 0
\(207\) 2907.01i 0.976092i
\(208\) 0 0
\(209\) 445.034i 0.147290i
\(210\) 0 0
\(211\) 2149.01i 0.701157i −0.936533 0.350579i \(-0.885985\pi\)
0.936533 0.350579i \(-0.114015\pi\)
\(212\) 0 0
\(213\) −4111.43 −1.32259
\(214\) 0 0
\(215\) 313.098i 0.0993168i
\(216\) 0 0
\(217\) 3181.73 0.995346
\(218\) 0 0
\(219\) 3125.17 0.964290
\(220\) 0 0
\(221\) 1096.29 + 2733.99i 0.333684 + 0.832162i
\(222\) 0 0
\(223\) −115.665 −0.0347332 −0.0173666 0.999849i \(-0.505528\pi\)
−0.0173666 + 0.999849i \(0.505528\pi\)
\(224\) 0 0
\(225\) 8476.38 2.51152
\(226\) 0 0
\(227\) 150.739i 0.0440743i 0.999757 + 0.0220372i \(0.00701521\pi\)
−0.999757 + 0.0220372i \(0.992985\pi\)
\(228\) 0 0
\(229\) −252.262 −0.0727946 −0.0363973 0.999337i \(-0.511588\pi\)
−0.0363973 + 0.999337i \(0.511588\pi\)
\(230\) 0 0
\(231\) 2373.40i 0.676009i
\(232\) 0 0
\(233\) 1025.45i 0.288323i 0.989554 + 0.144161i \(0.0460484\pi\)
−0.989554 + 0.144161i \(0.953952\pi\)
\(234\) 0 0
\(235\) 1315.00i 0.365026i
\(236\) 0 0
\(237\) 7787.33 2.13435
\(238\) 0 0
\(239\) 3803.45 1.02939 0.514696 0.857373i \(-0.327905\pi\)
0.514696 + 0.857373i \(0.327905\pi\)
\(240\) 0 0
\(241\) 2073.99i 0.554346i 0.960820 + 0.277173i \(0.0893975\pi\)
−0.960820 + 0.277173i \(0.910603\pi\)
\(242\) 0 0
\(243\) 2590.02i 0.683743i
\(244\) 0 0
\(245\) 13903.6i 3.62559i
\(246\) 0 0
\(247\) −2517.65 −0.648560
\(248\) 0 0
\(249\) 4222.89i 1.07476i
\(250\) 0 0
\(251\) −1242.05 −0.312341 −0.156170 0.987730i \(-0.549915\pi\)
−0.156170 + 0.987730i \(0.549915\pi\)
\(252\) 0 0
\(253\) −367.505 −0.0913236
\(254\) 0 0
\(255\) −3964.13 9885.99i −0.973504 2.42778i
\(256\) 0 0
\(257\) 1469.82 0.356751 0.178375 0.983963i \(-0.442916\pi\)
0.178375 + 0.983963i \(0.442916\pi\)
\(258\) 0 0
\(259\) −7161.16 −1.71804
\(260\) 0 0
\(261\) 15240.4i 3.61441i
\(262\) 0 0
\(263\) −4308.70 −1.01021 −0.505106 0.863057i \(-0.668547\pi\)
−0.505106 + 0.863057i \(0.668547\pi\)
\(264\) 0 0
\(265\) 5248.72i 1.21670i
\(266\) 0 0
\(267\) 11163.1i 2.55869i
\(268\) 0 0
\(269\) 3812.72i 0.864184i −0.901830 0.432092i \(-0.857776\pi\)
0.901830 0.432092i \(-0.142224\pi\)
\(270\) 0 0
\(271\) 2859.05 0.640868 0.320434 0.947271i \(-0.396171\pi\)
0.320434 + 0.947271i \(0.396171\pi\)
\(272\) 0 0
\(273\) −13426.8 −2.97666
\(274\) 0 0
\(275\) 1071.59i 0.234979i
\(276\) 0 0
\(277\) 5377.92i 1.16653i −0.812283 0.583263i \(-0.801776\pi\)
0.812283 0.583263i \(-0.198224\pi\)
\(278\) 0 0
\(279\) 5418.91i 1.16280i
\(280\) 0 0
\(281\) 1867.29 0.396418 0.198209 0.980160i \(-0.436488\pi\)
0.198209 + 0.980160i \(0.436488\pi\)
\(282\) 0 0
\(283\) 4296.76i 0.902529i 0.892390 + 0.451265i \(0.149027\pi\)
−0.892390 + 0.451265i \(0.850973\pi\)
\(284\) 0 0
\(285\) 9103.73 1.89213
\(286\) 0 0
\(287\) −6099.79 −1.25456
\(288\) 0 0
\(289\) −3551.94 + 3394.31i −0.722967 + 0.690883i
\(290\) 0 0
\(291\) −5912.78 −1.19111
\(292\) 0 0
\(293\) 8816.10 1.75782 0.878911 0.476985i \(-0.158270\pi\)
0.878911 + 0.476985i \(0.158270\pi\)
\(294\) 0 0
\(295\) 181.826i 0.0358859i
\(296\) 0 0
\(297\) −2184.82 −0.426855
\(298\) 0 0
\(299\) 2079.06i 0.402123i
\(300\) 0 0
\(301\) 658.309i 0.126061i
\(302\) 0 0
\(303\) 6847.28i 1.29824i
\(304\) 0 0
\(305\) −11693.8 −2.19537
\(306\) 0 0
\(307\) 6143.84 1.14217 0.571087 0.820890i \(-0.306522\pi\)
0.571087 + 0.820890i \(0.306522\pi\)
\(308\) 0 0
\(309\) 14710.9i 2.70834i
\(310\) 0 0
\(311\) 7409.80i 1.35103i −0.737345 0.675516i \(-0.763921\pi\)
0.737345 0.675516i \(-0.236079\pi\)
\(312\) 0 0
\(313\) 9455.98i 1.70761i −0.520589 0.853807i \(-0.674288\pi\)
0.520589 0.853807i \(-0.325712\pi\)
\(314\) 0 0
\(315\) 33265.4 5.95014
\(316\) 0 0
\(317\) 1045.23i 0.185191i −0.995704 0.0925957i \(-0.970484\pi\)
0.995704 0.0925957i \(-0.0295164\pi\)
\(318\) 0 0
\(319\) 1926.71 0.338165
\(320\) 0 0
\(321\) 1873.42 0.325744
\(322\) 0 0
\(323\) −1562.86 3897.56i −0.269226 0.671412i
\(324\) 0 0
\(325\) 6062.20 1.03468
\(326\) 0 0
\(327\) 11526.9 1.94935
\(328\) 0 0
\(329\) 2764.87i 0.463320i
\(330\) 0 0
\(331\) −6370.04 −1.05779 −0.528896 0.848687i \(-0.677394\pi\)
−0.528896 + 0.848687i \(0.677394\pi\)
\(332\) 0 0
\(333\) 12196.4i 2.00708i
\(334\) 0 0
\(335\) 7958.04i 1.29789i
\(336\) 0 0
\(337\) 9216.37i 1.48976i −0.667201 0.744878i \(-0.732508\pi\)
0.667201 0.744878i \(-0.267492\pi\)
\(338\) 0 0
\(339\) −13819.5 −2.21408
\(340\) 0 0
\(341\) 685.062 0.108792
\(342\) 0 0
\(343\) 17399.4i 2.73901i
\(344\) 0 0
\(345\) 7517.79i 1.17317i
\(346\) 0 0
\(347\) 7810.63i 1.20835i 0.796852 + 0.604174i \(0.206497\pi\)
−0.796852 + 0.604174i \(0.793503\pi\)
\(348\) 0 0
\(349\) 8377.96 1.28499 0.642495 0.766290i \(-0.277899\pi\)
0.642495 + 0.766290i \(0.277899\pi\)
\(350\) 0 0
\(351\) 12360.0i 1.87956i
\(352\) 0 0
\(353\) 500.551 0.0754721 0.0377360 0.999288i \(-0.487985\pi\)
0.0377360 + 0.999288i \(0.487985\pi\)
\(354\) 0 0
\(355\) 7285.06 1.08916
\(356\) 0 0
\(357\) −8334.84 20786.0i −1.23565 3.08154i
\(358\) 0 0
\(359\) −12643.0 −1.85870 −0.929351 0.369198i \(-0.879632\pi\)
−0.929351 + 0.369198i \(0.879632\pi\)
\(360\) 0 0
\(361\) −3269.85 −0.476724
\(362\) 0 0
\(363\) 11814.9i 1.70833i
\(364\) 0 0
\(365\) −5537.50 −0.794099
\(366\) 0 0
\(367\) 640.446i 0.0910927i 0.998962 + 0.0455464i \(0.0145029\pi\)
−0.998962 + 0.0455464i \(0.985497\pi\)
\(368\) 0 0
\(369\) 10388.7i 1.46563i
\(370\) 0 0
\(371\) 11035.8i 1.54434i
\(372\) 0 0
\(373\) −1138.01 −0.157972 −0.0789861 0.996876i \(-0.525168\pi\)
−0.0789861 + 0.996876i \(0.525168\pi\)
\(374\) 0 0
\(375\) −2925.94 −0.402920
\(376\) 0 0
\(377\) 10899.8i 1.48904i
\(378\) 0 0
\(379\) 12320.8i 1.66987i 0.550352 + 0.834933i \(0.314493\pi\)
−0.550352 + 0.834933i \(0.685507\pi\)
\(380\) 0 0
\(381\) 19025.9i 2.55833i
\(382\) 0 0
\(383\) −9934.77 −1.32544 −0.662719 0.748868i \(-0.730598\pi\)
−0.662719 + 0.748868i \(0.730598\pi\)
\(384\) 0 0
\(385\) 4205.43i 0.556698i
\(386\) 0 0
\(387\) 1121.19 0.147269
\(388\) 0 0
\(389\) 3323.46 0.433178 0.216589 0.976263i \(-0.430507\pi\)
0.216589 + 0.976263i \(0.430507\pi\)
\(390\) 0 0
\(391\) −3218.57 + 1290.60i −0.416292 + 0.166927i
\(392\) 0 0
\(393\) −17527.8 −2.24978
\(394\) 0 0
\(395\) −13798.4 −1.75765
\(396\) 0 0
\(397\) 3094.45i 0.391199i 0.980684 + 0.195599i \(0.0626652\pi\)
−0.980684 + 0.195599i \(0.937335\pi\)
\(398\) 0 0
\(399\) 19141.2 2.40165
\(400\) 0 0
\(401\) 6261.50i 0.779761i 0.920865 + 0.389881i \(0.127484\pi\)
−0.920865 + 0.389881i \(0.872516\pi\)
\(402\) 0 0
\(403\) 3875.54i 0.479043i
\(404\) 0 0
\(405\) 18660.2i 2.28946i
\(406\) 0 0
\(407\) −1541.88 −0.187784
\(408\) 0 0
\(409\) 13966.7 1.68853 0.844264 0.535927i \(-0.180038\pi\)
0.844264 + 0.535927i \(0.180038\pi\)
\(410\) 0 0
\(411\) 7858.76i 0.943172i
\(412\) 0 0
\(413\) 382.302i 0.0455492i
\(414\) 0 0
\(415\) 7482.56i 0.885071i
\(416\) 0 0
\(417\) −191.153 −0.0224479
\(418\) 0 0
\(419\) 6308.88i 0.735582i 0.929908 + 0.367791i \(0.119886\pi\)
−0.929908 + 0.367791i \(0.880114\pi\)
\(420\) 0 0
\(421\) −3142.63 −0.363806 −0.181903 0.983317i \(-0.558226\pi\)
−0.181903 + 0.983317i \(0.558226\pi\)
\(422\) 0 0
\(423\) −4708.94 −0.541268
\(424\) 0 0
\(425\) 3763.18 + 9384.85i 0.429508 + 1.07113i
\(426\) 0 0
\(427\) −24587.1 −2.78654
\(428\) 0 0
\(429\) −2890.94 −0.325351
\(430\) 0 0
\(431\) 6900.34i 0.771178i −0.922671 0.385589i \(-0.873998\pi\)
0.922671 0.385589i \(-0.126002\pi\)
\(432\) 0 0
\(433\) −15236.0 −1.69098 −0.845490 0.533991i \(-0.820692\pi\)
−0.845490 + 0.533991i \(0.820692\pi\)
\(434\) 0 0
\(435\) 39413.2i 4.34418i
\(436\) 0 0
\(437\) 2963.89i 0.324444i
\(438\) 0 0
\(439\) 9594.64i 1.04311i −0.853216 0.521557i \(-0.825351\pi\)
0.853216 0.521557i \(-0.174649\pi\)
\(440\) 0 0
\(441\) 49788.1 5.37611
\(442\) 0 0
\(443\) −3375.02 −0.361968 −0.180984 0.983486i \(-0.557928\pi\)
−0.180984 + 0.983486i \(0.557928\pi\)
\(444\) 0 0
\(445\) 19780.0i 2.10710i
\(446\) 0 0
\(447\) 11136.7i 1.17841i
\(448\) 0 0
\(449\) 2918.05i 0.306707i 0.988171 + 0.153354i \(0.0490073\pi\)
−0.988171 + 0.153354i \(0.950993\pi\)
\(450\) 0 0
\(451\) −1313.35 −0.137125
\(452\) 0 0
\(453\) 19269.3i 1.99857i
\(454\) 0 0
\(455\) 23791.0 2.45130
\(456\) 0 0
\(457\) −3829.50 −0.391983 −0.195992 0.980606i \(-0.562793\pi\)
−0.195992 + 0.980606i \(0.562793\pi\)
\(458\) 0 0
\(459\) −19134.4 + 7672.60i −1.94579 + 0.780232i
\(460\) 0 0
\(461\) 6988.84 0.706080 0.353040 0.935608i \(-0.385148\pi\)
0.353040 + 0.935608i \(0.385148\pi\)
\(462\) 0 0
\(463\) 8669.67 0.870224 0.435112 0.900376i \(-0.356709\pi\)
0.435112 + 0.900376i \(0.356709\pi\)
\(464\) 0 0
\(465\) 14013.8i 1.39758i
\(466\) 0 0
\(467\) 12728.0 1.26120 0.630600 0.776108i \(-0.282809\pi\)
0.630600 + 0.776108i \(0.282809\pi\)
\(468\) 0 0
\(469\) 16732.3i 1.64739i
\(470\) 0 0
\(471\) 10684.4i 1.04524i
\(472\) 0 0
\(473\) 141.741i 0.0137786i
\(474\) 0 0
\(475\) −8642.24 −0.834807
\(476\) 0 0
\(477\) −18795.4 −1.80415
\(478\) 0 0
\(479\) 13282.8i 1.26703i −0.773730 0.633516i \(-0.781611\pi\)
0.773730 0.633516i \(-0.218389\pi\)
\(480\) 0 0
\(481\) 8722.72i 0.826864i
\(482\) 0 0
\(483\) 15806.6i 1.48908i
\(484\) 0 0
\(485\) 10476.9 0.980888
\(486\) 0 0
\(487\) 5014.37i 0.466577i −0.972408 0.233288i \(-0.925051\pi\)
0.972408 0.233288i \(-0.0749486\pi\)
\(488\) 0 0
\(489\) 13533.1 1.25151
\(490\) 0 0
\(491\) 14518.4 1.33443 0.667217 0.744863i \(-0.267485\pi\)
0.667217 + 0.744863i \(0.267485\pi\)
\(492\) 0 0
\(493\) 16873.9 6766.16i 1.54150 0.618119i
\(494\) 0 0
\(495\) 7162.40 0.650356
\(496\) 0 0
\(497\) 15317.3 1.38245
\(498\) 0 0
\(499\) 18158.2i 1.62901i 0.580158 + 0.814504i \(0.302991\pi\)
−0.580158 + 0.814504i \(0.697009\pi\)
\(500\) 0 0
\(501\) −38505.6 −3.43374
\(502\) 0 0
\(503\) 12331.2i 1.09309i −0.837431 0.546543i \(-0.815943\pi\)
0.837431 0.546543i \(-0.184057\pi\)
\(504\) 0 0
\(505\) 12132.7i 1.06911i
\(506\) 0 0
\(507\) 3991.01i 0.349600i
\(508\) 0 0
\(509\) −4051.78 −0.352833 −0.176417 0.984316i \(-0.556451\pi\)
−0.176417 + 0.984316i \(0.556451\pi\)
\(510\) 0 0
\(511\) −11643.0 −1.00793
\(512\) 0 0
\(513\) 17620.3i 1.51648i
\(514\) 0 0
\(515\) 26066.4i 2.23033i
\(516\) 0 0
\(517\) 595.307i 0.0506413i
\(518\) 0 0
\(519\) 23189.4 1.96127
\(520\) 0 0
\(521\) 6101.23i 0.513052i −0.966537 0.256526i \(-0.917422\pi\)
0.966537 0.256526i \(-0.0825779\pi\)
\(522\) 0 0
\(523\) −1233.70 −0.103147 −0.0515734 0.998669i \(-0.516424\pi\)
−0.0515734 + 0.998669i \(0.516424\pi\)
\(524\) 0 0
\(525\) −46089.7 −3.83146
\(526\) 0 0
\(527\) 5999.69 2405.78i 0.495922 0.198857i
\(528\) 0 0
\(529\) 9719.44 0.798836
\(530\) 0 0
\(531\) −651.110 −0.0532124
\(532\) 0 0
\(533\) 7429.90i 0.603799i
\(534\) 0 0
\(535\) −3319.51 −0.268252
\(536\) 0 0
\(537\) 7776.03i 0.624880i
\(538\) 0 0
\(539\) 6294.24i 0.502991i
\(540\) 0 0
\(541\) 8968.57i 0.712734i −0.934346 0.356367i \(-0.884015\pi\)
0.934346 0.356367i \(-0.115985\pi\)
\(542\) 0 0
\(543\) −4634.87 −0.366301
\(544\) 0 0
\(545\) −20424.5 −1.60530
\(546\) 0 0
\(547\) 7187.71i 0.561836i 0.959732 + 0.280918i \(0.0906389\pi\)
−0.959732 + 0.280918i \(0.909361\pi\)
\(548\) 0 0
\(549\) 41875.0i 3.25534i
\(550\) 0 0
\(551\) 15538.7i 1.20140i
\(552\) 0 0
\(553\) −29012.0 −2.23095
\(554\) 0 0
\(555\) 31541.0i 2.41233i
\(556\) 0 0
\(557\) 12502.2 0.951047 0.475524 0.879703i \(-0.342259\pi\)
0.475524 + 0.879703i \(0.342259\pi\)
\(558\) 0 0
\(559\) 801.860 0.0606709
\(560\) 0 0
\(561\) −1794.58 4475.44i −0.135058 0.336815i
\(562\) 0 0
\(563\) 20391.1 1.52643 0.763216 0.646143i \(-0.223619\pi\)
0.763216 + 0.646143i \(0.223619\pi\)
\(564\) 0 0
\(565\) 24486.8 1.82331
\(566\) 0 0
\(567\) 39234.2i 2.90597i
\(568\) 0 0
\(569\) 3407.53 0.251057 0.125528 0.992090i \(-0.459937\pi\)
0.125528 + 0.992090i \(0.459937\pi\)
\(570\) 0 0
\(571\) 9682.41i 0.709626i −0.934937 0.354813i \(-0.884545\pi\)
0.934937 0.354813i \(-0.115455\pi\)
\(572\) 0 0
\(573\) 1851.54i 0.134990i
\(574\) 0 0
\(575\) 7136.69i 0.517601i
\(576\) 0 0
\(577\) −10357.7 −0.747309 −0.373654 0.927568i \(-0.621895\pi\)
−0.373654 + 0.927568i \(0.621895\pi\)
\(578\) 0 0
\(579\) 25545.7 1.83358
\(580\) 0 0
\(581\) 15732.6i 1.12340i
\(582\) 0 0
\(583\) 2376.12i 0.168798i
\(584\) 0 0
\(585\) 40519.2i 2.86370i
\(586\) 0 0
\(587\) −760.786 −0.0534940 −0.0267470 0.999642i \(-0.508515\pi\)
−0.0267470 + 0.999642i \(0.508515\pi\)
\(588\) 0 0
\(589\) 5524.95i 0.386505i
\(590\) 0 0
\(591\) −31299.8 −2.17851
\(592\) 0 0
\(593\) 23861.5 1.65241 0.826203 0.563373i \(-0.190497\pi\)
0.826203 + 0.563373i \(0.190497\pi\)
\(594\) 0 0
\(595\) 14768.5 + 36830.7i 1.01757 + 2.53767i
\(596\) 0 0
\(597\) 25276.6 1.73283
\(598\) 0 0
\(599\) 40.2985 0.00274884 0.00137442 0.999999i \(-0.499563\pi\)
0.00137442 + 0.999999i \(0.499563\pi\)
\(600\) 0 0
\(601\) 28351.4i 1.92426i 0.272594 + 0.962129i \(0.412118\pi\)
−0.272594 + 0.962129i \(0.587882\pi\)
\(602\) 0 0
\(603\) −28497.3 −1.92454
\(604\) 0 0
\(605\) 20934.9i 1.40682i
\(606\) 0 0
\(607\) 380.879i 0.0254686i −0.999919 0.0127343i \(-0.995946\pi\)
0.999919 0.0127343i \(-0.00405356\pi\)
\(608\) 0 0
\(609\) 82868.8i 5.51398i
\(610\) 0 0
\(611\) −3367.78 −0.222988
\(612\) 0 0
\(613\) −18449.5 −1.21561 −0.607804 0.794087i \(-0.707949\pi\)
−0.607804 + 0.794087i \(0.707949\pi\)
\(614\) 0 0
\(615\) 26866.2i 1.76155i
\(616\) 0 0
\(617\) 13692.2i 0.893397i −0.894685 0.446698i \(-0.852600\pi\)
0.894685 0.446698i \(-0.147400\pi\)
\(618\) 0 0
\(619\) 15000.2i 0.974004i 0.873401 + 0.487002i \(0.161910\pi\)
−0.873401 + 0.487002i \(0.838090\pi\)
\(620\) 0 0
\(621\) −14550.7 −0.940259
\(622\) 0 0
\(623\) 41588.7i 2.67450i
\(624\) 0 0
\(625\) −12847.4 −0.822232
\(626\) 0 0
\(627\) 4121.31 0.262503
\(628\) 0 0
\(629\) −13503.6 + 5414.73i −0.855998 + 0.343242i
\(630\) 0 0
\(631\) −25659.3 −1.61883 −0.809413 0.587240i \(-0.800215\pi\)
−0.809413 + 0.587240i \(0.800215\pi\)
\(632\) 0 0
\(633\) 19901.3 1.24961
\(634\) 0 0
\(635\) 33712.0i 2.10680i
\(636\) 0 0
\(637\) 35607.9 2.21481
\(638\) 0 0
\(639\) 26087.4i 1.61503i
\(640\) 0 0
\(641\) 6709.60i 0.413437i −0.978400 0.206719i \(-0.933722\pi\)
0.978400 0.206719i \(-0.0662785\pi\)
\(642\) 0 0
\(643\) 11976.4i 0.734528i 0.930117 + 0.367264i \(0.119705\pi\)
−0.930117 + 0.367264i \(0.880295\pi\)
\(644\) 0 0
\(645\) −2899.49 −0.177004
\(646\) 0 0
\(647\) 21265.3 1.29215 0.646077 0.763272i \(-0.276408\pi\)
0.646077 + 0.763272i \(0.276408\pi\)
\(648\) 0 0
\(649\) 82.3137i 0.00497857i
\(650\) 0 0
\(651\) 29464.9i 1.77392i
\(652\) 0 0
\(653\) 19671.9i 1.17890i 0.807806 + 0.589449i \(0.200655\pi\)
−0.807806 + 0.589449i \(0.799345\pi\)
\(654\) 0 0
\(655\) 31057.6 1.85271
\(656\) 0 0
\(657\) 19829.5i 1.17751i
\(658\) 0 0
\(659\) −14782.6 −0.873824 −0.436912 0.899504i \(-0.643928\pi\)
−0.436912 + 0.899504i \(0.643928\pi\)
\(660\) 0 0
\(661\) 27770.8 1.63413 0.817063 0.576548i \(-0.195601\pi\)
0.817063 + 0.576548i \(0.195601\pi\)
\(662\) 0 0
\(663\) −25318.5 + 10152.3i −1.48309 + 0.594697i
\(664\) 0 0
\(665\) −33916.4 −1.97777
\(666\) 0 0
\(667\) 12831.7 0.744896
\(668\) 0 0
\(669\) 1071.14i 0.0619021i
\(670\) 0 0
\(671\) −5293.86 −0.304571
\(672\) 0 0
\(673\) 21891.3i 1.25386i −0.779077 0.626929i \(-0.784312\pi\)
0.779077 0.626929i \(-0.215688\pi\)
\(674\) 0 0
\(675\) 42427.6i 2.41932i
\(676\) 0 0
\(677\) 1946.42i 0.110498i 0.998473 + 0.0552488i \(0.0175952\pi\)
−0.998473 + 0.0552488i \(0.982405\pi\)
\(678\) 0 0
\(679\) 22028.3 1.24502
\(680\) 0 0
\(681\) −1395.94 −0.0785498
\(682\) 0 0
\(683\) 10516.5i 0.589169i −0.955625 0.294584i \(-0.904819\pi\)
0.955625 0.294584i \(-0.0951812\pi\)
\(684\) 0 0
\(685\) 13925.0i 0.776709i
\(686\) 0 0
\(687\) 2336.12i 0.129736i
\(688\) 0 0
\(689\) −13442.2 −0.743263
\(690\) 0 0
\(691\) 26695.9i 1.46970i −0.678232 0.734848i \(-0.737253\pi\)
0.678232 0.734848i \(-0.262747\pi\)
\(692\) 0 0
\(693\) 15059.4 0.825484
\(694\) 0 0
\(695\) 338.704 0.0184860
\(696\) 0 0
\(697\) −11502.2 + 4612.20i −0.625074 + 0.250645i
\(698\) 0 0
\(699\) −9496.30 −0.513853
\(700\) 0 0
\(701\) −22317.0 −1.20243 −0.601214 0.799088i \(-0.705316\pi\)
−0.601214 + 0.799088i \(0.705316\pi\)
\(702\) 0 0
\(703\) 12435.1i 0.667137i
\(704\) 0 0
\(705\) 12177.7 0.650554
\(706\) 0 0
\(707\) 25509.8i 1.35700i
\(708\) 0 0
\(709\) 28249.7i 1.49639i 0.663478 + 0.748196i \(0.269080\pi\)
−0.663478 + 0.748196i \(0.730920\pi\)
\(710\) 0 0
\(711\) 49411.3i 2.60629i
\(712\) 0 0
\(713\) 4562.46 0.239643
\(714\) 0 0
\(715\) 5122.46 0.267929
\(716\) 0 0
\(717\) 35222.4i 1.83460i
\(718\) 0 0
\(719\) 30991.2i 1.60748i −0.594982 0.803739i \(-0.702841\pi\)
0.594982 0.803739i \(-0.297159\pi\)
\(720\) 0 0
\(721\) 54806.3i 2.83092i
\(722\) 0 0
\(723\) −19206.5 −0.987963
\(724\) 0 0
\(725\) 37415.2i 1.91664i
\(726\) 0 0
\(727\) 13328.0 0.679929 0.339965 0.940438i \(-0.389585\pi\)
0.339965 + 0.940438i \(0.389585\pi\)
\(728\) 0 0
\(729\) 6718.94 0.341358
\(730\) 0 0
\(731\) 497.764 + 1241.35i 0.0251853 + 0.0628086i
\(732\) 0 0
\(733\) −27976.7 −1.40974 −0.704872 0.709334i \(-0.748996\pi\)
−0.704872 + 0.709334i \(0.748996\pi\)
\(734\) 0 0
\(735\) −128757. −6.46158
\(736\) 0 0
\(737\) 3602.64i 0.180061i
\(738\) 0 0
\(739\) 33971.2 1.69100 0.845501 0.533974i \(-0.179302\pi\)
0.845501 + 0.533974i \(0.179302\pi\)
\(740\) 0 0
\(741\) 23315.1i 1.15587i
\(742\) 0 0
\(743\) 25922.4i 1.27995i 0.768397 + 0.639974i \(0.221055\pi\)
−0.768397 + 0.639974i \(0.778945\pi\)
\(744\) 0 0
\(745\) 19733.2i 0.970428i
\(746\) 0 0
\(747\) −26794.6 −1.31240
\(748\) 0 0
\(749\) −6979.49 −0.340488
\(750\) 0 0
\(751\) 18950.5i 0.920792i 0.887714 + 0.460396i \(0.152293\pi\)
−0.887714 + 0.460396i \(0.847707\pi\)
\(752\) 0 0
\(753\) 11502.2i 0.556658i
\(754\) 0 0
\(755\) 34143.4i 1.64583i
\(756\) 0 0
\(757\) −8184.07 −0.392940 −0.196470 0.980510i \(-0.562948\pi\)
−0.196470 + 0.980510i \(0.562948\pi\)
\(758\) 0 0
\(759\) 3403.34i 0.162758i
\(760\) 0 0
\(761\) −25803.9 −1.22916 −0.614579 0.788855i \(-0.710674\pi\)
−0.614579 + 0.788855i \(0.710674\pi\)
\(762\) 0 0
\(763\) −42943.9 −2.03758
\(764\) 0 0
\(765\) 62727.6 25152.8i 2.96460 1.18876i
\(766\) 0 0
\(767\) −465.666 −0.0219221
\(768\) 0 0
\(769\) 5222.53 0.244901 0.122451 0.992475i \(-0.460925\pi\)
0.122451 + 0.992475i \(0.460925\pi\)
\(770\) 0 0
\(771\) 13611.5i 0.635806i
\(772\) 0 0
\(773\) 32636.6 1.51857 0.759286 0.650757i \(-0.225548\pi\)
0.759286 + 0.650757i \(0.225548\pi\)
\(774\) 0 0
\(775\) 13303.4i 0.616610i
\(776\) 0 0
\(777\) 66317.0i 3.06192i
\(778\) 0 0
\(779\) 10592.0i 0.487162i
\(780\) 0 0
\(781\) 3297.99 0.151103
\(782\) 0 0
\(783\) 76284.5 3.48172
\(784\) 0 0
\(785\) 18931.7i 0.860764i
\(786\) 0 0
\(787\) 17014.8i 0.770664i 0.922778 + 0.385332i \(0.125913\pi\)
−0.922778 + 0.385332i \(0.874087\pi\)
\(788\) 0 0
\(789\) 39901.4i 1.80042i
\(790\) 0 0
\(791\) 51485.1 2.31429
\(792\) 0 0
\(793\) 29948.5i 1.34111i
\(794\) 0 0
\(795\) 48606.6 2.16842
\(796\) 0 0
\(797\) −20826.6 −0.925616 −0.462808 0.886458i \(-0.653158\pi\)
−0.462808 + 0.886458i \(0.653158\pi\)
\(798\) 0 0
\(799\) −2090.59 5213.63i −0.0925652 0.230845i
\(800\) 0 0
\(801\) 70831.0 3.12446
\(802\) 0 0
\(803\) −2506.86 −0.110168
\(804\) 0 0
\(805\) 28007.8i 1.22627i
\(806\) 0 0
\(807\) 35308.2 1.54016
\(808\) 0 0
\(809\) 8639.25i 0.375451i 0.982222 + 0.187726i \(0.0601115\pi\)
−0.982222 + 0.187726i \(0.939888\pi\)
\(810\) 0 0
\(811\) 22319.6i 0.966397i −0.875511 0.483199i \(-0.839475\pi\)
0.875511 0.483199i \(-0.160525\pi\)
\(812\) 0 0
\(813\) 26476.7i 1.14216i
\(814\) 0 0
\(815\) −23979.3 −1.03062
\(816\) 0 0
\(817\) −1143.13 −0.0489510
\(818\) 0 0
\(819\) 85194.4i 3.63484i
\(820\) 0 0
\(821\) 9361.94i 0.397971i −0.980002 0.198985i \(-0.936235\pi\)
0.980002 0.198985i \(-0.0637646\pi\)
\(822\) 0 0
\(823\) 19128.8i 0.810192i 0.914274 + 0.405096i \(0.132762\pi\)
−0.914274 + 0.405096i \(0.867238\pi\)
\(824\) 0 0
\(825\) −9923.61 −0.418783
\(826\) 0 0
\(827\) 40640.9i 1.70886i 0.519570 + 0.854428i \(0.326092\pi\)
−0.519570 + 0.854428i \(0.673908\pi\)
\(828\) 0 0
\(829\) −25220.3 −1.05662 −0.528310 0.849052i \(-0.677174\pi\)
−0.528310 + 0.849052i \(0.677174\pi\)
\(830\) 0 0
\(831\) 49803.0 2.07900
\(832\) 0 0
\(833\) 22104.0 + 55124.3i 0.919397 + 2.29285i
\(834\) 0 0
\(835\) 68228.3 2.82771
\(836\) 0 0
\(837\) 27123.8 1.12011
\(838\) 0 0
\(839\) 4544.50i 0.187001i 0.995619 + 0.0935004i \(0.0298056\pi\)
−0.995619 + 0.0935004i \(0.970194\pi\)
\(840\) 0 0
\(841\) −42883.3 −1.75830
\(842\) 0 0
\(843\) 17292.4i 0.706501i
\(844\) 0 0
\(845\) 7071.69i 0.287898i
\(846\) 0 0
\(847\) 44017.0i 1.78565i
\(848\) 0 0
\(849\) −39790.8 −1.60850
\(850\) 0 0
\(851\) −10268.8 −0.413641
\(852\) 0 0
\(853\) 22395.8i 0.898964i −0.893289 0.449482i \(-0.851609\pi\)
0.893289 0.449482i \(-0.148391\pi\)
\(854\) 0 0
\(855\) 57764.0i 2.31051i
\(856\) 0 0
\(857\) 5735.42i 0.228610i −0.993446 0.114305i \(-0.963536\pi\)
0.993446 0.114305i \(-0.0364640\pi\)
\(858\) 0 0
\(859\) −34718.6 −1.37903 −0.689513 0.724273i \(-0.742175\pi\)
−0.689513 + 0.724273i \(0.742175\pi\)
\(860\) 0 0
\(861\) 56488.0i 2.23590i
\(862\) 0 0
\(863\) −22239.9 −0.877237 −0.438619 0.898673i \(-0.644532\pi\)
−0.438619 + 0.898673i \(0.644532\pi\)
\(864\) 0 0
\(865\) −41089.4 −1.61512
\(866\) 0 0
\(867\) −31433.5 32893.2i −1.23130 1.28848i
\(868\) 0 0
\(869\) −6246.61 −0.243845
\(870\) 0 0
\(871\) −20380.9 −0.792860
\(872\) 0 0
\(873\) 37517.1i 1.45448i
\(874\) 0 0
\(875\) 10900.7 0.421156
\(876\) 0 0
\(877\) 10379.6i 0.399650i −0.979832 0.199825i \(-0.935963\pi\)
0.979832 0.199825i \(-0.0640373\pi\)
\(878\) 0 0
\(879\) 81642.9i 3.13282i
\(880\) 0 0
\(881\) 44934.3i 1.71836i −0.511672 0.859181i \(-0.670974\pi\)
0.511672 0.859181i \(-0.329026\pi\)
\(882\) 0 0
\(883\) −15846.0 −0.603918 −0.301959 0.953321i \(-0.597640\pi\)
−0.301959 + 0.953321i \(0.597640\pi\)
\(884\) 0 0
\(885\) 1683.83 0.0639563
\(886\) 0 0
\(887\) 30415.0i 1.15134i 0.817683 + 0.575669i \(0.195258\pi\)
−0.817683 + 0.575669i \(0.804742\pi\)
\(888\) 0 0
\(889\) 70881.8i 2.67413i
\(890\) 0 0
\(891\) 8447.56i 0.317625i
\(892\) 0 0
\(893\) 4801.08 0.179913
\(894\) 0 0
\(895\) 13778.4i 0.514593i
\(896\) 0 0
\(897\) −19253.4 −0.716670
\(898\) 0 0
\(899\) −23919.4 −0.887382
\(900\) 0 0
\(901\) −8344.42 20809.8i −0.308538 0.769451i
\(902\) 0 0
\(903\) −6096.38 −0.224667
\(904\) 0 0
\(905\) 8212.55 0.301651
\(906\) 0 0
\(907\) 7288.75i 0.266835i −0.991060 0.133417i \(-0.957405\pi\)
0.991060 0.133417i \(-0.0425951\pi\)
\(908\) 0 0
\(909\) −43446.6 −1.58530
\(910\) 0 0
\(911\) 10232.6i 0.372140i −0.982536 0.186070i \(-0.940425\pi\)
0.982536 0.186070i \(-0.0595752\pi\)
\(912\) 0 0
\(913\) 3387.39i 0.122789i
\(914\) 0 0
\(915\) 108293.i 3.91261i
\(916\) 0 0
\(917\) 65300.8 2.35160
\(918\) 0 0
\(919\) −26533.8 −0.952415 −0.476207 0.879333i \(-0.657989\pi\)
−0.476207 + 0.879333i \(0.657989\pi\)
\(920\) 0 0
\(921\) 56895.9i 2.03560i
\(922\) 0 0
\(923\) 18657.4i 0.665348i
\(924\) 0 0
\(925\) 29942.1i 1.06431i
\(926\) 0 0
\(927\) −93342.3 −3.30719
\(928\) 0 0
\(929\) 7037.69i 0.248546i 0.992248 + 0.124273i \(0.0396599\pi\)
−0.992248 + 0.124273i \(0.960340\pi\)
\(930\) 0 0
\(931\) −50762.4 −1.78697
\(932\) 0 0
\(933\) 68619.6 2.40783
\(934\) 0 0
\(935\) 3179.83 + 7930.05i 0.111221 + 0.277369i
\(936\) 0 0
\(937\) 16895.0 0.589047 0.294523 0.955644i \(-0.404839\pi\)
0.294523 + 0.955644i \(0.404839\pi\)
\(938\) 0 0
\(939\) 87568.5 3.04333
\(940\) 0 0
\(941\) 51103.6i 1.77038i 0.465229 + 0.885190i \(0.345972\pi\)
−0.465229 + 0.885190i \(0.654028\pi\)
\(942\) 0 0
\(943\) −8746.81 −0.302052
\(944\) 0 0
\(945\) 166507.i 5.73170i
\(946\) 0 0
\(947\) 22510.4i 0.772429i −0.922409 0.386215i \(-0.873782\pi\)
0.922409 0.386215i \(-0.126218\pi\)
\(948\) 0 0
\(949\) 14181.8i 0.485101i
\(950\) 0 0
\(951\) 9679.47 0.330051
\(952\) 0 0
\(953\) −4260.98 −0.144834 −0.0724169 0.997374i \(-0.523071\pi\)
−0.0724169 + 0.997374i \(0.523071\pi\)
\(954\) 0 0
\(955\) 3280.76i 0.111165i
\(956\) 0 0
\(957\) 17842.5i 0.602683i
\(958\) 0 0
\(959\) 29278.1i 0.985861i
\(960\) 0 0
\(961\) 21286.2 0.714517
\(962\) 0 0
\(963\) 11887.0i 0.397771i
\(964\) 0 0
\(965\) −45264.5 −1.50996
\(966\) 0 0
\(967\) 5646.95 0.187791 0.0938953 0.995582i \(-0.470068\pi\)
0.0938953 + 0.995582i \(0.470068\pi\)
\(968\) 0 0
\(969\) 36093.9 14473.1i 1.19660 0.479818i
\(970\) 0 0
\(971\) 3320.02 0.109726 0.0548632 0.998494i \(-0.482528\pi\)
0.0548632 + 0.998494i \(0.482528\pi\)
\(972\) 0 0
\(973\) 712.148 0.0234639
\(974\) 0 0
\(975\) 56140.0i 1.84402i
\(976\) 0 0
\(977\) −5821.02 −0.190615 −0.0953076 0.995448i \(-0.530383\pi\)
−0.0953076 + 0.995448i \(0.530383\pi\)
\(978\) 0 0
\(979\) 8954.49i 0.292326i
\(980\) 0 0
\(981\) 73139.1i 2.38038i
\(982\) 0 0
\(983\) 234.506i 0.00760894i −0.999993 0.00380447i \(-0.998789\pi\)
0.999993 0.00380447i \(-0.00121100\pi\)
\(984\) 0 0
\(985\) 55460.3 1.79402
\(986\) 0 0
\(987\) 25604.5 0.825735
\(988\) 0 0
\(989\) 943.985i 0.0303508i
\(990\) 0 0
\(991\) 16708.2i 0.535573i −0.963478 0.267787i \(-0.913708\pi\)
0.963478 0.267787i \(-0.0862922\pi\)
\(992\) 0 0
\(993\) 58990.8i 1.88521i
\(994\) 0 0
\(995\) −44787.7 −1.42700
\(996\) 0 0
\(997\) 10281.9i 0.326611i −0.986576 0.163305i \(-0.947784\pi\)
0.986576 0.163305i \(-0.0522156\pi\)
\(998\) 0 0
\(999\) −61047.8 −1.93340
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 136.4.b.b.33.8 yes 8
3.2 odd 2 1224.4.c.e.577.7 8
4.3 odd 2 272.4.b.f.33.1 8
17.4 even 4 2312.4.a.k.1.8 8
17.13 even 4 2312.4.a.k.1.1 8
17.16 even 2 inner 136.4.b.b.33.1 8
51.50 odd 2 1224.4.c.e.577.2 8
68.67 odd 2 272.4.b.f.33.8 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
136.4.b.b.33.1 8 17.16 even 2 inner
136.4.b.b.33.8 yes 8 1.1 even 1 trivial
272.4.b.f.33.1 8 4.3 odd 2
272.4.b.f.33.8 8 68.67 odd 2
1224.4.c.e.577.2 8 51.50 odd 2
1224.4.c.e.577.7 8 3.2 odd 2
2312.4.a.k.1.1 8 17.13 even 4
2312.4.a.k.1.8 8 17.4 even 4