Properties

Label 8046.2.a.k.1.3
Level $8046$
Weight $2$
Character 8046.1
Self dual yes
Analytic conductor $64.248$
Analytic rank $0$
Dimension $12$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [8046,2,Mod(1,8046)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8046, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("8046.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 8046 = 2 \cdot 3^{3} \cdot 149 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8046.a (trivial)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(64.2476334663\)
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: \( x^{12} - 3 x^{11} - 31 x^{10} + 82 x^{9} + 334 x^{8} - 684 x^{7} - 1561 x^{6} + 1551 x^{5} + 3573 x^{4} + 345 x^{3} - 1607 x^{2} - 594 x - 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(-1.28984\) of defining polynomial
Character \(\chi\) \(=\) 8046.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.00000 q^{2} +1.00000 q^{4} -1.28984 q^{5} +3.20787 q^{7} -1.00000 q^{8} +O(q^{10})\) \(q-1.00000 q^{2} +1.00000 q^{4} -1.28984 q^{5} +3.20787 q^{7} -1.00000 q^{8} +1.28984 q^{10} +3.26304 q^{11} -3.85048 q^{13} -3.20787 q^{14} +1.00000 q^{16} +5.41773 q^{17} -4.39994 q^{19} -1.28984 q^{20} -3.26304 q^{22} -0.544759 q^{23} -3.33630 q^{25} +3.85048 q^{26} +3.20787 q^{28} +5.75789 q^{29} +6.11583 q^{31} -1.00000 q^{32} -5.41773 q^{34} -4.13765 q^{35} -4.67003 q^{37} +4.39994 q^{38} +1.28984 q^{40} -9.17084 q^{41} +5.38737 q^{43} +3.26304 q^{44} +0.544759 q^{46} +12.5824 q^{47} +3.29040 q^{49} +3.33630 q^{50} -3.85048 q^{52} +10.7984 q^{53} -4.20882 q^{55} -3.20787 q^{56} -5.75789 q^{58} -3.15039 q^{59} -13.9696 q^{61} -6.11583 q^{62} +1.00000 q^{64} +4.96652 q^{65} -9.06275 q^{67} +5.41773 q^{68} +4.13765 q^{70} +5.97564 q^{71} +5.92590 q^{73} +4.67003 q^{74} -4.39994 q^{76} +10.4674 q^{77} +6.21363 q^{79} -1.28984 q^{80} +9.17084 q^{82} +0.323107 q^{83} -6.98803 q^{85} -5.38737 q^{86} -3.26304 q^{88} -4.91999 q^{89} -12.3518 q^{91} -0.544759 q^{92} -12.5824 q^{94} +5.67524 q^{95} +16.0890 q^{97} -3.29040 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 12 q^{2} + 12 q^{4} + 3 q^{5} - 6 q^{7} - 12 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 12 q^{2} + 12 q^{4} + 3 q^{5} - 6 q^{7} - 12 q^{8} - 3 q^{10} + 14 q^{11} - 3 q^{13} + 6 q^{14} + 12 q^{16} + 8 q^{17} - 4 q^{19} + 3 q^{20} - 14 q^{22} + 13 q^{23} + 11 q^{25} + 3 q^{26} - 6 q^{28} + 23 q^{29} - 14 q^{31} - 12 q^{32} - 8 q^{34} + 32 q^{35} - 19 q^{37} + 4 q^{38} - 3 q^{40} + 30 q^{41} - 15 q^{43} + 14 q^{44} - 13 q^{46} - q^{47} + 14 q^{49} - 11 q^{50} - 3 q^{52} + 16 q^{53} - 7 q^{55} + 6 q^{56} - 23 q^{58} + 26 q^{59} - 16 q^{61} + 14 q^{62} + 12 q^{64} + 8 q^{65} - 39 q^{67} + 8 q^{68} - 32 q^{70} + 15 q^{71} - 2 q^{73} + 19 q^{74} - 4 q^{76} + 34 q^{77} - 13 q^{79} + 3 q^{80} - 30 q^{82} + 6 q^{83} - 11 q^{85} + 15 q^{86} - 14 q^{88} + 18 q^{89} - 35 q^{91} + 13 q^{92} + q^{94} + 51 q^{95} + 19 q^{97} - 14 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.00000 −0.707107
\(3\) 0 0
\(4\) 1.00000 0.500000
\(5\) −1.28984 −0.576836 −0.288418 0.957505i \(-0.593129\pi\)
−0.288418 + 0.957505i \(0.593129\pi\)
\(6\) 0 0
\(7\) 3.20787 1.21246 0.606230 0.795290i \(-0.292681\pi\)
0.606230 + 0.795290i \(0.292681\pi\)
\(8\) −1.00000 −0.353553
\(9\) 0 0
\(10\) 1.28984 0.407885
\(11\) 3.26304 0.983844 0.491922 0.870639i \(-0.336294\pi\)
0.491922 + 0.870639i \(0.336294\pi\)
\(12\) 0 0
\(13\) −3.85048 −1.06793 −0.533966 0.845506i \(-0.679299\pi\)
−0.533966 + 0.845506i \(0.679299\pi\)
\(14\) −3.20787 −0.857338
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) 5.41773 1.31399 0.656996 0.753894i \(-0.271827\pi\)
0.656996 + 0.753894i \(0.271827\pi\)
\(18\) 0 0
\(19\) −4.39994 −1.00941 −0.504707 0.863290i \(-0.668400\pi\)
−0.504707 + 0.863290i \(0.668400\pi\)
\(20\) −1.28984 −0.288418
\(21\) 0 0
\(22\) −3.26304 −0.695683
\(23\) −0.544759 −0.113590 −0.0567951 0.998386i \(-0.518088\pi\)
−0.0567951 + 0.998386i \(0.518088\pi\)
\(24\) 0 0
\(25\) −3.33630 −0.667260
\(26\) 3.85048 0.755141
\(27\) 0 0
\(28\) 3.20787 0.606230
\(29\) 5.75789 1.06921 0.534606 0.845101i \(-0.320460\pi\)
0.534606 + 0.845101i \(0.320460\pi\)
\(30\) 0 0
\(31\) 6.11583 1.09844 0.549218 0.835679i \(-0.314926\pi\)
0.549218 + 0.835679i \(0.314926\pi\)
\(32\) −1.00000 −0.176777
\(33\) 0 0
\(34\) −5.41773 −0.929133
\(35\) −4.13765 −0.699390
\(36\) 0 0
\(37\) −4.67003 −0.767749 −0.383874 0.923385i \(-0.625410\pi\)
−0.383874 + 0.923385i \(0.625410\pi\)
\(38\) 4.39994 0.713764
\(39\) 0 0
\(40\) 1.28984 0.203942
\(41\) −9.17084 −1.43224 −0.716122 0.697975i \(-0.754085\pi\)
−0.716122 + 0.697975i \(0.754085\pi\)
\(42\) 0 0
\(43\) 5.38737 0.821566 0.410783 0.911733i \(-0.365255\pi\)
0.410783 + 0.911733i \(0.365255\pi\)
\(44\) 3.26304 0.491922
\(45\) 0 0
\(46\) 0.544759 0.0803203
\(47\) 12.5824 1.83534 0.917668 0.397348i \(-0.130070\pi\)
0.917668 + 0.397348i \(0.130070\pi\)
\(48\) 0 0
\(49\) 3.29040 0.470058
\(50\) 3.33630 0.471824
\(51\) 0 0
\(52\) −3.85048 −0.533966
\(53\) 10.7984 1.48328 0.741639 0.670799i \(-0.234049\pi\)
0.741639 + 0.670799i \(0.234049\pi\)
\(54\) 0 0
\(55\) −4.20882 −0.567517
\(56\) −3.20787 −0.428669
\(57\) 0 0
\(58\) −5.75789 −0.756048
\(59\) −3.15039 −0.410145 −0.205073 0.978747i \(-0.565743\pi\)
−0.205073 + 0.978747i \(0.565743\pi\)
\(60\) 0 0
\(61\) −13.9696 −1.78863 −0.894313 0.447442i \(-0.852335\pi\)
−0.894313 + 0.447442i \(0.852335\pi\)
\(62\) −6.11583 −0.776711
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) 4.96652 0.616021
\(66\) 0 0
\(67\) −9.06275 −1.10719 −0.553595 0.832786i \(-0.686745\pi\)
−0.553595 + 0.832786i \(0.686745\pi\)
\(68\) 5.41773 0.656996
\(69\) 0 0
\(70\) 4.13765 0.494544
\(71\) 5.97564 0.709178 0.354589 0.935022i \(-0.384621\pi\)
0.354589 + 0.935022i \(0.384621\pi\)
\(72\) 0 0
\(73\) 5.92590 0.693574 0.346787 0.937944i \(-0.387273\pi\)
0.346787 + 0.937944i \(0.387273\pi\)
\(74\) 4.67003 0.542881
\(75\) 0 0
\(76\) −4.39994 −0.504707
\(77\) 10.4674 1.19287
\(78\) 0 0
\(79\) 6.21363 0.699088 0.349544 0.936920i \(-0.386337\pi\)
0.349544 + 0.936920i \(0.386337\pi\)
\(80\) −1.28984 −0.144209
\(81\) 0 0
\(82\) 9.17084 1.01275
\(83\) 0.323107 0.0354656 0.0177328 0.999843i \(-0.494355\pi\)
0.0177328 + 0.999843i \(0.494355\pi\)
\(84\) 0 0
\(85\) −6.98803 −0.757958
\(86\) −5.38737 −0.580935
\(87\) 0 0
\(88\) −3.26304 −0.347841
\(89\) −4.91999 −0.521518 −0.260759 0.965404i \(-0.583973\pi\)
−0.260759 + 0.965404i \(0.583973\pi\)
\(90\) 0 0
\(91\) −12.3518 −1.29482
\(92\) −0.544759 −0.0567951
\(93\) 0 0
\(94\) −12.5824 −1.29778
\(95\) 5.67524 0.582267
\(96\) 0 0
\(97\) 16.0890 1.63359 0.816796 0.576927i \(-0.195748\pi\)
0.816796 + 0.576927i \(0.195748\pi\)
\(98\) −3.29040 −0.332381
\(99\) 0 0
\(100\) −3.33630 −0.333630
\(101\) 11.4226 1.13659 0.568294 0.822825i \(-0.307603\pi\)
0.568294 + 0.822825i \(0.307603\pi\)
\(102\) 0 0
\(103\) 3.09261 0.304723 0.152362 0.988325i \(-0.451312\pi\)
0.152362 + 0.988325i \(0.451312\pi\)
\(104\) 3.85048 0.377571
\(105\) 0 0
\(106\) −10.7984 −1.04884
\(107\) 5.60796 0.542142 0.271071 0.962559i \(-0.412622\pi\)
0.271071 + 0.962559i \(0.412622\pi\)
\(108\) 0 0
\(109\) −3.52073 −0.337225 −0.168612 0.985682i \(-0.553929\pi\)
−0.168612 + 0.985682i \(0.553929\pi\)
\(110\) 4.20882 0.401295
\(111\) 0 0
\(112\) 3.20787 0.303115
\(113\) 6.78189 0.637987 0.318993 0.947757i \(-0.396655\pi\)
0.318993 + 0.947757i \(0.396655\pi\)
\(114\) 0 0
\(115\) 0.702655 0.0655229
\(116\) 5.75789 0.534606
\(117\) 0 0
\(118\) 3.15039 0.290017
\(119\) 17.3794 1.59316
\(120\) 0 0
\(121\) −0.352555 −0.0320504
\(122\) 13.9696 1.26475
\(123\) 0 0
\(124\) 6.11583 0.549218
\(125\) 10.7525 0.961736
\(126\) 0 0
\(127\) 15.6435 1.38814 0.694070 0.719908i \(-0.255816\pi\)
0.694070 + 0.719908i \(0.255816\pi\)
\(128\) −1.00000 −0.0883883
\(129\) 0 0
\(130\) −4.96652 −0.435593
\(131\) 6.07893 0.531119 0.265559 0.964094i \(-0.414443\pi\)
0.265559 + 0.964094i \(0.414443\pi\)
\(132\) 0 0
\(133\) −14.1144 −1.22387
\(134\) 9.06275 0.782902
\(135\) 0 0
\(136\) −5.41773 −0.464567
\(137\) −16.0391 −1.37032 −0.685158 0.728395i \(-0.740267\pi\)
−0.685158 + 0.728395i \(0.740267\pi\)
\(138\) 0 0
\(139\) −10.6387 −0.902362 −0.451181 0.892432i \(-0.648997\pi\)
−0.451181 + 0.892432i \(0.648997\pi\)
\(140\) −4.13765 −0.349695
\(141\) 0 0
\(142\) −5.97564 −0.501465
\(143\) −12.5643 −1.05068
\(144\) 0 0
\(145\) −7.42678 −0.616761
\(146\) −5.92590 −0.490431
\(147\) 0 0
\(148\) −4.67003 −0.383874
\(149\) −1.00000 −0.0819232
\(150\) 0 0
\(151\) 6.15112 0.500571 0.250285 0.968172i \(-0.419476\pi\)
0.250285 + 0.968172i \(0.419476\pi\)
\(152\) 4.39994 0.356882
\(153\) 0 0
\(154\) −10.4674 −0.843487
\(155\) −7.88847 −0.633617
\(156\) 0 0
\(157\) −16.4628 −1.31387 −0.656937 0.753945i \(-0.728148\pi\)
−0.656937 + 0.753945i \(0.728148\pi\)
\(158\) −6.21363 −0.494330
\(159\) 0 0
\(160\) 1.28984 0.101971
\(161\) −1.74751 −0.137723
\(162\) 0 0
\(163\) 8.94721 0.700800 0.350400 0.936600i \(-0.386046\pi\)
0.350400 + 0.936600i \(0.386046\pi\)
\(164\) −9.17084 −0.716122
\(165\) 0 0
\(166\) −0.323107 −0.0250780
\(167\) 0.565153 0.0437329 0.0218664 0.999761i \(-0.493039\pi\)
0.0218664 + 0.999761i \(0.493039\pi\)
\(168\) 0 0
\(169\) 1.82620 0.140477
\(170\) 6.98803 0.535958
\(171\) 0 0
\(172\) 5.38737 0.410783
\(173\) 6.52301 0.495935 0.247968 0.968768i \(-0.420237\pi\)
0.247968 + 0.968768i \(0.420237\pi\)
\(174\) 0 0
\(175\) −10.7024 −0.809026
\(176\) 3.26304 0.245961
\(177\) 0 0
\(178\) 4.91999 0.368769
\(179\) −17.3901 −1.29979 −0.649897 0.760022i \(-0.725188\pi\)
−0.649897 + 0.760022i \(0.725188\pi\)
\(180\) 0 0
\(181\) −11.5623 −0.859421 −0.429711 0.902967i \(-0.641384\pi\)
−0.429711 + 0.902967i \(0.641384\pi\)
\(182\) 12.3518 0.915578
\(183\) 0 0
\(184\) 0.544759 0.0401602
\(185\) 6.02362 0.442865
\(186\) 0 0
\(187\) 17.6783 1.29276
\(188\) 12.5824 0.917668
\(189\) 0 0
\(190\) −5.67524 −0.411725
\(191\) −1.42498 −0.103108 −0.0515539 0.998670i \(-0.516417\pi\)
−0.0515539 + 0.998670i \(0.516417\pi\)
\(192\) 0 0
\(193\) −4.56918 −0.328897 −0.164449 0.986386i \(-0.552584\pi\)
−0.164449 + 0.986386i \(0.552584\pi\)
\(194\) −16.0890 −1.15512
\(195\) 0 0
\(196\) 3.29040 0.235029
\(197\) −4.57007 −0.325604 −0.162802 0.986659i \(-0.552053\pi\)
−0.162802 + 0.986659i \(0.552053\pi\)
\(198\) 0 0
\(199\) 25.4721 1.80567 0.902833 0.429991i \(-0.141483\pi\)
0.902833 + 0.429991i \(0.141483\pi\)
\(200\) 3.33630 0.235912
\(201\) 0 0
\(202\) −11.4226 −0.803690
\(203\) 18.4705 1.29638
\(204\) 0 0
\(205\) 11.8290 0.826170
\(206\) −3.09261 −0.215472
\(207\) 0 0
\(208\) −3.85048 −0.266983
\(209\) −14.3572 −0.993107
\(210\) 0 0
\(211\) 25.7505 1.77274 0.886369 0.462979i \(-0.153220\pi\)
0.886369 + 0.462979i \(0.153220\pi\)
\(212\) 10.7984 0.741639
\(213\) 0 0
\(214\) −5.60796 −0.383352
\(215\) −6.94887 −0.473909
\(216\) 0 0
\(217\) 19.6188 1.33181
\(218\) 3.52073 0.238454
\(219\) 0 0
\(220\) −4.20882 −0.283758
\(221\) −20.8609 −1.40325
\(222\) 0 0
\(223\) 12.3557 0.827398 0.413699 0.910414i \(-0.364237\pi\)
0.413699 + 0.910414i \(0.364237\pi\)
\(224\) −3.20787 −0.214335
\(225\) 0 0
\(226\) −6.78189 −0.451125
\(227\) −8.06407 −0.535231 −0.267615 0.963526i \(-0.586236\pi\)
−0.267615 + 0.963526i \(0.586236\pi\)
\(228\) 0 0
\(229\) 21.0700 1.39235 0.696173 0.717874i \(-0.254885\pi\)
0.696173 + 0.717874i \(0.254885\pi\)
\(230\) −0.702655 −0.0463317
\(231\) 0 0
\(232\) −5.75789 −0.378024
\(233\) −15.8020 −1.03522 −0.517612 0.855615i \(-0.673179\pi\)
−0.517612 + 0.855615i \(0.673179\pi\)
\(234\) 0 0
\(235\) −16.2294 −1.05869
\(236\) −3.15039 −0.205073
\(237\) 0 0
\(238\) −17.3794 −1.12654
\(239\) 4.53148 0.293117 0.146559 0.989202i \(-0.453180\pi\)
0.146559 + 0.989202i \(0.453180\pi\)
\(240\) 0 0
\(241\) 8.04690 0.518347 0.259173 0.965831i \(-0.416550\pi\)
0.259173 + 0.965831i \(0.416550\pi\)
\(242\) 0.352555 0.0226631
\(243\) 0 0
\(244\) −13.9696 −0.894313
\(245\) −4.24411 −0.271146
\(246\) 0 0
\(247\) 16.9419 1.07799
\(248\) −6.11583 −0.388356
\(249\) 0 0
\(250\) −10.7525 −0.680050
\(251\) −1.18604 −0.0748623 −0.0374312 0.999299i \(-0.511917\pi\)
−0.0374312 + 0.999299i \(0.511917\pi\)
\(252\) 0 0
\(253\) −1.77757 −0.111755
\(254\) −15.6435 −0.981563
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) −20.9570 −1.30726 −0.653632 0.756813i \(-0.726756\pi\)
−0.653632 + 0.756813i \(0.726756\pi\)
\(258\) 0 0
\(259\) −14.9808 −0.930864
\(260\) 4.96652 0.308011
\(261\) 0 0
\(262\) −6.07893 −0.375558
\(263\) 27.9860 1.72569 0.862844 0.505470i \(-0.168681\pi\)
0.862844 + 0.505470i \(0.168681\pi\)
\(264\) 0 0
\(265\) −13.9283 −0.855609
\(266\) 14.1144 0.865410
\(267\) 0 0
\(268\) −9.06275 −0.553595
\(269\) 16.7221 1.01956 0.509782 0.860304i \(-0.329726\pi\)
0.509782 + 0.860304i \(0.329726\pi\)
\(270\) 0 0
\(271\) −25.1645 −1.52863 −0.764316 0.644842i \(-0.776923\pi\)
−0.764316 + 0.644842i \(0.776923\pi\)
\(272\) 5.41773 0.328498
\(273\) 0 0
\(274\) 16.0391 0.968959
\(275\) −10.8865 −0.656480
\(276\) 0 0
\(277\) −31.5995 −1.89863 −0.949315 0.314328i \(-0.898221\pi\)
−0.949315 + 0.314328i \(0.898221\pi\)
\(278\) 10.6387 0.638066
\(279\) 0 0
\(280\) 4.13765 0.247272
\(281\) 11.3652 0.677989 0.338995 0.940788i \(-0.389913\pi\)
0.338995 + 0.940788i \(0.389913\pi\)
\(282\) 0 0
\(283\) 0.901374 0.0535811 0.0267906 0.999641i \(-0.491471\pi\)
0.0267906 + 0.999641i \(0.491471\pi\)
\(284\) 5.97564 0.354589
\(285\) 0 0
\(286\) 12.5643 0.742942
\(287\) −29.4188 −1.73654
\(288\) 0 0
\(289\) 12.3518 0.726577
\(290\) 7.42678 0.436116
\(291\) 0 0
\(292\) 5.92590 0.346787
\(293\) 14.2554 0.832808 0.416404 0.909180i \(-0.363290\pi\)
0.416404 + 0.909180i \(0.363290\pi\)
\(294\) 0 0
\(295\) 4.06351 0.236587
\(296\) 4.67003 0.271440
\(297\) 0 0
\(298\) 1.00000 0.0579284
\(299\) 2.09758 0.121306
\(300\) 0 0
\(301\) 17.2820 0.996115
\(302\) −6.15112 −0.353957
\(303\) 0 0
\(304\) −4.39994 −0.252354
\(305\) 18.0186 1.03174
\(306\) 0 0
\(307\) −25.2188 −1.43931 −0.719656 0.694331i \(-0.755701\pi\)
−0.719656 + 0.694331i \(0.755701\pi\)
\(308\) 10.4674 0.596436
\(309\) 0 0
\(310\) 7.88847 0.448035
\(311\) −1.85525 −0.105202 −0.0526008 0.998616i \(-0.516751\pi\)
−0.0526008 + 0.998616i \(0.516751\pi\)
\(312\) 0 0
\(313\) 7.47705 0.422628 0.211314 0.977418i \(-0.432226\pi\)
0.211314 + 0.977418i \(0.432226\pi\)
\(314\) 16.4628 0.929049
\(315\) 0 0
\(316\) 6.21363 0.349544
\(317\) −30.7715 −1.72830 −0.864150 0.503234i \(-0.832143\pi\)
−0.864150 + 0.503234i \(0.832143\pi\)
\(318\) 0 0
\(319\) 18.7882 1.05194
\(320\) −1.28984 −0.0721045
\(321\) 0 0
\(322\) 1.74751 0.0973852
\(323\) −23.8377 −1.32636
\(324\) 0 0
\(325\) 12.8464 0.712588
\(326\) −8.94721 −0.495540
\(327\) 0 0
\(328\) 9.17084 0.506375
\(329\) 40.3627 2.22527
\(330\) 0 0
\(331\) 24.9574 1.37178 0.685891 0.727704i \(-0.259413\pi\)
0.685891 + 0.727704i \(0.259413\pi\)
\(332\) 0.323107 0.0177328
\(333\) 0 0
\(334\) −0.565153 −0.0309238
\(335\) 11.6895 0.638668
\(336\) 0 0
\(337\) 14.0767 0.766805 0.383402 0.923581i \(-0.374752\pi\)
0.383402 + 0.923581i \(0.374752\pi\)
\(338\) −1.82620 −0.0993324
\(339\) 0 0
\(340\) −6.98803 −0.378979
\(341\) 19.9562 1.08069
\(342\) 0 0
\(343\) −11.8999 −0.642533
\(344\) −5.38737 −0.290467
\(345\) 0 0
\(346\) −6.52301 −0.350679
\(347\) 0.244603 0.0131310 0.00656549 0.999978i \(-0.497910\pi\)
0.00656549 + 0.999978i \(0.497910\pi\)
\(348\) 0 0
\(349\) −28.0705 −1.50258 −0.751289 0.659973i \(-0.770568\pi\)
−0.751289 + 0.659973i \(0.770568\pi\)
\(350\) 10.7024 0.572068
\(351\) 0 0
\(352\) −3.26304 −0.173921
\(353\) −13.8962 −0.739618 −0.369809 0.929108i \(-0.620577\pi\)
−0.369809 + 0.929108i \(0.620577\pi\)
\(354\) 0 0
\(355\) −7.70765 −0.409079
\(356\) −4.91999 −0.260759
\(357\) 0 0
\(358\) 17.3901 0.919093
\(359\) 28.8648 1.52342 0.761712 0.647915i \(-0.224359\pi\)
0.761712 + 0.647915i \(0.224359\pi\)
\(360\) 0 0
\(361\) 0.359449 0.0189184
\(362\) 11.5623 0.607703
\(363\) 0 0
\(364\) −12.3518 −0.647412
\(365\) −7.64349 −0.400078
\(366\) 0 0
\(367\) −3.27469 −0.170937 −0.0854686 0.996341i \(-0.527239\pi\)
−0.0854686 + 0.996341i \(0.527239\pi\)
\(368\) −0.544759 −0.0283975
\(369\) 0 0
\(370\) −6.02362 −0.313153
\(371\) 34.6399 1.79842
\(372\) 0 0
\(373\) 14.5287 0.752267 0.376134 0.926565i \(-0.377253\pi\)
0.376134 + 0.926565i \(0.377253\pi\)
\(374\) −17.6783 −0.914122
\(375\) 0 0
\(376\) −12.5824 −0.648889
\(377\) −22.1706 −1.14185
\(378\) 0 0
\(379\) 2.92973 0.150490 0.0752449 0.997165i \(-0.476026\pi\)
0.0752449 + 0.997165i \(0.476026\pi\)
\(380\) 5.67524 0.291133
\(381\) 0 0
\(382\) 1.42498 0.0729082
\(383\) 6.14540 0.314015 0.157008 0.987597i \(-0.449815\pi\)
0.157008 + 0.987597i \(0.449815\pi\)
\(384\) 0 0
\(385\) −13.5013 −0.688091
\(386\) 4.56918 0.232565
\(387\) 0 0
\(388\) 16.0890 0.816796
\(389\) −18.2567 −0.925650 −0.462825 0.886450i \(-0.653164\pi\)
−0.462825 + 0.886450i \(0.653164\pi\)
\(390\) 0 0
\(391\) −2.95136 −0.149257
\(392\) −3.29040 −0.166191
\(393\) 0 0
\(394\) 4.57007 0.230237
\(395\) −8.01461 −0.403259
\(396\) 0 0
\(397\) 18.5588 0.931438 0.465719 0.884933i \(-0.345796\pi\)
0.465719 + 0.884933i \(0.345796\pi\)
\(398\) −25.4721 −1.27680
\(399\) 0 0
\(400\) −3.33630 −0.166815
\(401\) 22.3568 1.11645 0.558223 0.829691i \(-0.311483\pi\)
0.558223 + 0.829691i \(0.311483\pi\)
\(402\) 0 0
\(403\) −23.5489 −1.17305
\(404\) 11.4226 0.568294
\(405\) 0 0
\(406\) −18.4705 −0.916677
\(407\) −15.2385 −0.755345
\(408\) 0 0
\(409\) 6.42743 0.317816 0.158908 0.987293i \(-0.449203\pi\)
0.158908 + 0.987293i \(0.449203\pi\)
\(410\) −11.8290 −0.584191
\(411\) 0 0
\(412\) 3.09261 0.152362
\(413\) −10.1060 −0.497285
\(414\) 0 0
\(415\) −0.416758 −0.0204578
\(416\) 3.85048 0.188785
\(417\) 0 0
\(418\) 14.3572 0.702233
\(419\) 31.2114 1.52477 0.762387 0.647121i \(-0.224027\pi\)
0.762387 + 0.647121i \(0.224027\pi\)
\(420\) 0 0
\(421\) 16.1359 0.786414 0.393207 0.919450i \(-0.371366\pi\)
0.393207 + 0.919450i \(0.371366\pi\)
\(422\) −25.7505 −1.25352
\(423\) 0 0
\(424\) −10.7984 −0.524418
\(425\) −18.0752 −0.876775
\(426\) 0 0
\(427\) −44.8127 −2.16864
\(428\) 5.60796 0.271071
\(429\) 0 0
\(430\) 6.94887 0.335104
\(431\) −5.79223 −0.279002 −0.139501 0.990222i \(-0.544550\pi\)
−0.139501 + 0.990222i \(0.544550\pi\)
\(432\) 0 0
\(433\) −1.81723 −0.0873305 −0.0436653 0.999046i \(-0.513904\pi\)
−0.0436653 + 0.999046i \(0.513904\pi\)
\(434\) −19.6188 −0.941731
\(435\) 0 0
\(436\) −3.52073 −0.168612
\(437\) 2.39691 0.114660
\(438\) 0 0
\(439\) 21.0055 1.00254 0.501268 0.865292i \(-0.332867\pi\)
0.501268 + 0.865292i \(0.332867\pi\)
\(440\) 4.20882 0.200647
\(441\) 0 0
\(442\) 20.8609 0.992250
\(443\) 37.1284 1.76402 0.882011 0.471229i \(-0.156189\pi\)
0.882011 + 0.471229i \(0.156189\pi\)
\(444\) 0 0
\(445\) 6.34602 0.300830
\(446\) −12.3557 −0.585058
\(447\) 0 0
\(448\) 3.20787 0.151557
\(449\) 26.8067 1.26509 0.632543 0.774525i \(-0.282011\pi\)
0.632543 + 0.774525i \(0.282011\pi\)
\(450\) 0 0
\(451\) −29.9248 −1.40911
\(452\) 6.78189 0.318993
\(453\) 0 0
\(454\) 8.06407 0.378465
\(455\) 15.9319 0.746901
\(456\) 0 0
\(457\) 18.8431 0.881443 0.440721 0.897644i \(-0.354723\pi\)
0.440721 + 0.897644i \(0.354723\pi\)
\(458\) −21.0700 −0.984537
\(459\) 0 0
\(460\) 0.702655 0.0327614
\(461\) 0.774819 0.0360869 0.0180435 0.999837i \(-0.494256\pi\)
0.0180435 + 0.999837i \(0.494256\pi\)
\(462\) 0 0
\(463\) 1.87804 0.0872799 0.0436400 0.999047i \(-0.486105\pi\)
0.0436400 + 0.999047i \(0.486105\pi\)
\(464\) 5.75789 0.267303
\(465\) 0 0
\(466\) 15.8020 0.732014
\(467\) 12.1725 0.563276 0.281638 0.959521i \(-0.409122\pi\)
0.281638 + 0.959521i \(0.409122\pi\)
\(468\) 0 0
\(469\) −29.0721 −1.34242
\(470\) 16.2294 0.748605
\(471\) 0 0
\(472\) 3.15039 0.145008
\(473\) 17.5792 0.808293
\(474\) 0 0
\(475\) 14.6795 0.673542
\(476\) 17.3794 0.796581
\(477\) 0 0
\(478\) −4.53148 −0.207265
\(479\) 17.2910 0.790048 0.395024 0.918671i \(-0.370736\pi\)
0.395024 + 0.918671i \(0.370736\pi\)
\(480\) 0 0
\(481\) 17.9819 0.819903
\(482\) −8.04690 −0.366526
\(483\) 0 0
\(484\) −0.352555 −0.0160252
\(485\) −20.7523 −0.942315
\(486\) 0 0
\(487\) −17.7705 −0.805257 −0.402628 0.915364i \(-0.631903\pi\)
−0.402628 + 0.915364i \(0.631903\pi\)
\(488\) 13.9696 0.632375
\(489\) 0 0
\(490\) 4.24411 0.191729
\(491\) 21.3837 0.965033 0.482517 0.875887i \(-0.339723\pi\)
0.482517 + 0.875887i \(0.339723\pi\)
\(492\) 0 0
\(493\) 31.1947 1.40494
\(494\) −16.9419 −0.762251
\(495\) 0 0
\(496\) 6.11583 0.274609
\(497\) 19.1691 0.859850
\(498\) 0 0
\(499\) 35.9564 1.60963 0.804814 0.593528i \(-0.202265\pi\)
0.804814 + 0.593528i \(0.202265\pi\)
\(500\) 10.7525 0.480868
\(501\) 0 0
\(502\) 1.18604 0.0529356
\(503\) 4.49120 0.200253 0.100126 0.994975i \(-0.468075\pi\)
0.100126 + 0.994975i \(0.468075\pi\)
\(504\) 0 0
\(505\) −14.7333 −0.655625
\(506\) 1.77757 0.0790227
\(507\) 0 0
\(508\) 15.6435 0.694070
\(509\) 35.6812 1.58154 0.790770 0.612113i \(-0.209680\pi\)
0.790770 + 0.612113i \(0.209680\pi\)
\(510\) 0 0
\(511\) 19.0095 0.840930
\(512\) −1.00000 −0.0441942
\(513\) 0 0
\(514\) 20.9570 0.924375
\(515\) −3.98898 −0.175775
\(516\) 0 0
\(517\) 41.0570 1.80568
\(518\) 14.9808 0.658221
\(519\) 0 0
\(520\) −4.96652 −0.217796
\(521\) −26.0613 −1.14177 −0.570883 0.821031i \(-0.693399\pi\)
−0.570883 + 0.821031i \(0.693399\pi\)
\(522\) 0 0
\(523\) 42.9756 1.87919 0.939596 0.342287i \(-0.111201\pi\)
0.939596 + 0.342287i \(0.111201\pi\)
\(524\) 6.07893 0.265559
\(525\) 0 0
\(526\) −27.9860 −1.22025
\(527\) 33.1339 1.44334
\(528\) 0 0
\(529\) −22.7032 −0.987097
\(530\) 13.9283 0.605007
\(531\) 0 0
\(532\) −14.1144 −0.611937
\(533\) 35.3122 1.52954
\(534\) 0 0
\(535\) −7.23339 −0.312727
\(536\) 9.06275 0.391451
\(537\) 0 0
\(538\) −16.7221 −0.720940
\(539\) 10.7367 0.462464
\(540\) 0 0
\(541\) −20.5023 −0.881462 −0.440731 0.897639i \(-0.645281\pi\)
−0.440731 + 0.897639i \(0.645281\pi\)
\(542\) 25.1645 1.08091
\(543\) 0 0
\(544\) −5.41773 −0.232283
\(545\) 4.54119 0.194523
\(546\) 0 0
\(547\) 42.7695 1.82869 0.914345 0.404935i \(-0.132706\pi\)
0.914345 + 0.404935i \(0.132706\pi\)
\(548\) −16.0391 −0.685158
\(549\) 0 0
\(550\) 10.8865 0.464202
\(551\) −25.3343 −1.07928
\(552\) 0 0
\(553\) 19.9325 0.847616
\(554\) 31.5995 1.34253
\(555\) 0 0
\(556\) −10.6387 −0.451181
\(557\) −41.5046 −1.75861 −0.879303 0.476263i \(-0.841991\pi\)
−0.879303 + 0.476263i \(0.841991\pi\)
\(558\) 0 0
\(559\) −20.7440 −0.877376
\(560\) −4.13765 −0.174848
\(561\) 0 0
\(562\) −11.3652 −0.479411
\(563\) −11.8254 −0.498382 −0.249191 0.968454i \(-0.580165\pi\)
−0.249191 + 0.968454i \(0.580165\pi\)
\(564\) 0 0
\(565\) −8.74758 −0.368014
\(566\) −0.901374 −0.0378876
\(567\) 0 0
\(568\) −5.97564 −0.250732
\(569\) 4.19085 0.175690 0.0878448 0.996134i \(-0.472002\pi\)
0.0878448 + 0.996134i \(0.472002\pi\)
\(570\) 0 0
\(571\) −13.5467 −0.566912 −0.283456 0.958985i \(-0.591481\pi\)
−0.283456 + 0.958985i \(0.591481\pi\)
\(572\) −12.5643 −0.525339
\(573\) 0 0
\(574\) 29.4188 1.22792
\(575\) 1.81748 0.0757942
\(576\) 0 0
\(577\) −30.9881 −1.29005 −0.645025 0.764162i \(-0.723153\pi\)
−0.645025 + 0.764162i \(0.723153\pi\)
\(578\) −12.3518 −0.513768
\(579\) 0 0
\(580\) −7.42678 −0.308380
\(581\) 1.03648 0.0430006
\(582\) 0 0
\(583\) 35.2357 1.45932
\(584\) −5.92590 −0.245215
\(585\) 0 0
\(586\) −14.2554 −0.588884
\(587\) −11.8328 −0.488391 −0.244196 0.969726i \(-0.578524\pi\)
−0.244196 + 0.969726i \(0.578524\pi\)
\(588\) 0 0
\(589\) −26.9093 −1.10878
\(590\) −4.06351 −0.167292
\(591\) 0 0
\(592\) −4.67003 −0.191937
\(593\) 26.1799 1.07508 0.537539 0.843239i \(-0.319354\pi\)
0.537539 + 0.843239i \(0.319354\pi\)
\(594\) 0 0
\(595\) −22.4167 −0.918994
\(596\) −1.00000 −0.0409616
\(597\) 0 0
\(598\) −2.09758 −0.0857766
\(599\) 28.2031 1.15235 0.576174 0.817327i \(-0.304545\pi\)
0.576174 + 0.817327i \(0.304545\pi\)
\(600\) 0 0
\(601\) 7.54747 0.307868 0.153934 0.988081i \(-0.450806\pi\)
0.153934 + 0.988081i \(0.450806\pi\)
\(602\) −17.2820 −0.704360
\(603\) 0 0
\(604\) 6.15112 0.250285
\(605\) 0.454740 0.0184878
\(606\) 0 0
\(607\) 13.6119 0.552490 0.276245 0.961087i \(-0.410910\pi\)
0.276245 + 0.961087i \(0.410910\pi\)
\(608\) 4.39994 0.178441
\(609\) 0 0
\(610\) −18.0186 −0.729553
\(611\) −48.4484 −1.96001
\(612\) 0 0
\(613\) −12.2099 −0.493155 −0.246578 0.969123i \(-0.579306\pi\)
−0.246578 + 0.969123i \(0.579306\pi\)
\(614\) 25.2188 1.01775
\(615\) 0 0
\(616\) −10.4674 −0.421744
\(617\) 17.9612 0.723090 0.361545 0.932355i \(-0.382249\pi\)
0.361545 + 0.932355i \(0.382249\pi\)
\(618\) 0 0
\(619\) −8.78344 −0.353036 −0.176518 0.984297i \(-0.556483\pi\)
−0.176518 + 0.984297i \(0.556483\pi\)
\(620\) −7.88847 −0.316809
\(621\) 0 0
\(622\) 1.85525 0.0743888
\(623\) −15.7827 −0.632319
\(624\) 0 0
\(625\) 2.81241 0.112496
\(626\) −7.47705 −0.298843
\(627\) 0 0
\(628\) −16.4628 −0.656937
\(629\) −25.3010 −1.00882
\(630\) 0 0
\(631\) −38.6453 −1.53845 −0.769223 0.638980i \(-0.779357\pi\)
−0.769223 + 0.638980i \(0.779357\pi\)
\(632\) −6.21363 −0.247165
\(633\) 0 0
\(634\) 30.7715 1.22209
\(635\) −20.1777 −0.800729
\(636\) 0 0
\(637\) −12.6696 −0.501989
\(638\) −18.7882 −0.743833
\(639\) 0 0
\(640\) 1.28984 0.0509856
\(641\) 6.71131 0.265081 0.132540 0.991178i \(-0.457687\pi\)
0.132540 + 0.991178i \(0.457687\pi\)
\(642\) 0 0
\(643\) −35.6962 −1.40772 −0.703859 0.710339i \(-0.748542\pi\)
−0.703859 + 0.710339i \(0.748542\pi\)
\(644\) −1.74751 −0.0688617
\(645\) 0 0
\(646\) 23.8377 0.937881
\(647\) 19.7034 0.774621 0.387311 0.921949i \(-0.373404\pi\)
0.387311 + 0.921949i \(0.373404\pi\)
\(648\) 0 0
\(649\) −10.2798 −0.403519
\(650\) −12.8464 −0.503876
\(651\) 0 0
\(652\) 8.94721 0.350400
\(653\) 1.14695 0.0448836 0.0224418 0.999748i \(-0.492856\pi\)
0.0224418 + 0.999748i \(0.492856\pi\)
\(654\) 0 0
\(655\) −7.84088 −0.306368
\(656\) −9.17084 −0.358061
\(657\) 0 0
\(658\) −40.3627 −1.57350
\(659\) 41.4619 1.61513 0.807563 0.589781i \(-0.200786\pi\)
0.807563 + 0.589781i \(0.200786\pi\)
\(660\) 0 0
\(661\) −8.79426 −0.342057 −0.171028 0.985266i \(-0.554709\pi\)
−0.171028 + 0.985266i \(0.554709\pi\)
\(662\) −24.9574 −0.969997
\(663\) 0 0
\(664\) −0.323107 −0.0125390
\(665\) 18.2054 0.705975
\(666\) 0 0
\(667\) −3.13666 −0.121452
\(668\) 0.565153 0.0218664
\(669\) 0 0
\(670\) −11.6895 −0.451606
\(671\) −45.5835 −1.75973
\(672\) 0 0
\(673\) 7.42888 0.286362 0.143181 0.989697i \(-0.454267\pi\)
0.143181 + 0.989697i \(0.454267\pi\)
\(674\) −14.0767 −0.542213
\(675\) 0 0
\(676\) 1.82620 0.0702386
\(677\) 2.76346 0.106209 0.0531043 0.998589i \(-0.483088\pi\)
0.0531043 + 0.998589i \(0.483088\pi\)
\(678\) 0 0
\(679\) 51.6114 1.98066
\(680\) 6.98803 0.267979
\(681\) 0 0
\(682\) −19.9562 −0.764163
\(683\) 49.1142 1.87930 0.939652 0.342133i \(-0.111149\pi\)
0.939652 + 0.342133i \(0.111149\pi\)
\(684\) 0 0
\(685\) 20.6880 0.790447
\(686\) 11.8999 0.454340
\(687\) 0 0
\(688\) 5.38737 0.205391
\(689\) −41.5792 −1.58404
\(690\) 0 0
\(691\) −11.1678 −0.424843 −0.212421 0.977178i \(-0.568135\pi\)
−0.212421 + 0.977178i \(0.568135\pi\)
\(692\) 6.52301 0.247968
\(693\) 0 0
\(694\) −0.244603 −0.00928500
\(695\) 13.7223 0.520515
\(696\) 0 0
\(697\) −49.6852 −1.88196
\(698\) 28.0705 1.06248
\(699\) 0 0
\(700\) −10.7024 −0.404513
\(701\) 9.04525 0.341635 0.170817 0.985303i \(-0.445359\pi\)
0.170817 + 0.985303i \(0.445359\pi\)
\(702\) 0 0
\(703\) 20.5479 0.774977
\(704\) 3.26304 0.122981
\(705\) 0 0
\(706\) 13.8962 0.522989
\(707\) 36.6421 1.37807
\(708\) 0 0
\(709\) 37.8659 1.42208 0.711042 0.703150i \(-0.248224\pi\)
0.711042 + 0.703150i \(0.248224\pi\)
\(710\) 7.70765 0.289263
\(711\) 0 0
\(712\) 4.91999 0.184384
\(713\) −3.33165 −0.124771
\(714\) 0 0
\(715\) 16.2060 0.606069
\(716\) −17.3901 −0.649897
\(717\) 0 0
\(718\) −28.8648 −1.07722
\(719\) −36.3141 −1.35429 −0.677143 0.735852i \(-0.736782\pi\)
−0.677143 + 0.735852i \(0.736782\pi\)
\(720\) 0 0
\(721\) 9.92066 0.369465
\(722\) −0.359449 −0.0133773
\(723\) 0 0
\(724\) −11.5623 −0.429711
\(725\) −19.2100 −0.713443
\(726\) 0 0
\(727\) −9.32073 −0.345687 −0.172843 0.984949i \(-0.555295\pi\)
−0.172843 + 0.984949i \(0.555295\pi\)
\(728\) 12.3518 0.457789
\(729\) 0 0
\(730\) 7.64349 0.282898
\(731\) 29.1873 1.07953
\(732\) 0 0
\(733\) −19.6052 −0.724134 −0.362067 0.932152i \(-0.617929\pi\)
−0.362067 + 0.932152i \(0.617929\pi\)
\(734\) 3.27469 0.120871
\(735\) 0 0
\(736\) 0.544759 0.0200801
\(737\) −29.5721 −1.08930
\(738\) 0 0
\(739\) 19.7365 0.726020 0.363010 0.931785i \(-0.381749\pi\)
0.363010 + 0.931785i \(0.381749\pi\)
\(740\) 6.02362 0.221433
\(741\) 0 0
\(742\) −34.6399 −1.27167
\(743\) 4.87522 0.178854 0.0894272 0.995993i \(-0.471496\pi\)
0.0894272 + 0.995993i \(0.471496\pi\)
\(744\) 0 0
\(745\) 1.28984 0.0472562
\(746\) −14.5287 −0.531933
\(747\) 0 0
\(748\) 17.6783 0.646382
\(749\) 17.9896 0.657325
\(750\) 0 0
\(751\) −35.8776 −1.30919 −0.654596 0.755979i \(-0.727161\pi\)
−0.654596 + 0.755979i \(0.727161\pi\)
\(752\) 12.5824 0.458834
\(753\) 0 0
\(754\) 22.1706 0.807407
\(755\) −7.93399 −0.288747
\(756\) 0 0
\(757\) 38.5845 1.40238 0.701189 0.712975i \(-0.252653\pi\)
0.701189 + 0.712975i \(0.252653\pi\)
\(758\) −2.92973 −0.106412
\(759\) 0 0
\(760\) −5.67524 −0.205862
\(761\) 8.51702 0.308742 0.154371 0.988013i \(-0.450665\pi\)
0.154371 + 0.988013i \(0.450665\pi\)
\(762\) 0 0
\(763\) −11.2940 −0.408871
\(764\) −1.42498 −0.0515539
\(765\) 0 0
\(766\) −6.14540 −0.222042
\(767\) 12.1305 0.438007
\(768\) 0 0
\(769\) 28.2406 1.01838 0.509191 0.860654i \(-0.329945\pi\)
0.509191 + 0.860654i \(0.329945\pi\)
\(770\) 13.5013 0.486554
\(771\) 0 0
\(772\) −4.56918 −0.164449
\(773\) −53.0040 −1.90642 −0.953210 0.302308i \(-0.902243\pi\)
−0.953210 + 0.302308i \(0.902243\pi\)
\(774\) 0 0
\(775\) −20.4042 −0.732942
\(776\) −16.0890 −0.577562
\(777\) 0 0
\(778\) 18.2567 0.654533
\(779\) 40.3511 1.44573
\(780\) 0 0
\(781\) 19.4988 0.697721
\(782\) 2.95136 0.105540
\(783\) 0 0
\(784\) 3.29040 0.117514
\(785\) 21.2344 0.757890
\(786\) 0 0
\(787\) −5.39006 −0.192135 −0.0960674 0.995375i \(-0.530626\pi\)
−0.0960674 + 0.995375i \(0.530626\pi\)
\(788\) −4.57007 −0.162802
\(789\) 0 0
\(790\) 8.01461 0.285147
\(791\) 21.7554 0.773533
\(792\) 0 0
\(793\) 53.7897 1.91013
\(794\) −18.5588 −0.658626
\(795\) 0 0
\(796\) 25.4721 0.902833
\(797\) −5.93084 −0.210081 −0.105041 0.994468i \(-0.533497\pi\)
−0.105041 + 0.994468i \(0.533497\pi\)
\(798\) 0 0
\(799\) 68.1682 2.41162
\(800\) 3.33630 0.117956
\(801\) 0 0
\(802\) −22.3568 −0.789447
\(803\) 19.3365 0.682369
\(804\) 0 0
\(805\) 2.25402 0.0794438
\(806\) 23.5489 0.829474
\(807\) 0 0
\(808\) −11.4226 −0.401845
\(809\) 44.5550 1.56647 0.783236 0.621725i \(-0.213568\pi\)
0.783236 + 0.621725i \(0.213568\pi\)
\(810\) 0 0
\(811\) 26.8980 0.944518 0.472259 0.881460i \(-0.343439\pi\)
0.472259 + 0.881460i \(0.343439\pi\)
\(812\) 18.4705 0.648189
\(813\) 0 0
\(814\) 15.2385 0.534110
\(815\) −11.5405 −0.404247
\(816\) 0 0
\(817\) −23.7041 −0.829301
\(818\) −6.42743 −0.224730
\(819\) 0 0
\(820\) 11.8290 0.413085
\(821\) −17.4081 −0.607547 −0.303773 0.952744i \(-0.598247\pi\)
−0.303773 + 0.952744i \(0.598247\pi\)
\(822\) 0 0
\(823\) −21.0176 −0.732628 −0.366314 0.930491i \(-0.619380\pi\)
−0.366314 + 0.930491i \(0.619380\pi\)
\(824\) −3.09261 −0.107736
\(825\) 0 0
\(826\) 10.1060 0.351633
\(827\) 6.58525 0.228992 0.114496 0.993424i \(-0.463475\pi\)
0.114496 + 0.993424i \(0.463475\pi\)
\(828\) 0 0
\(829\) −8.68704 −0.301713 −0.150857 0.988556i \(-0.548203\pi\)
−0.150857 + 0.988556i \(0.548203\pi\)
\(830\) 0.416758 0.0144659
\(831\) 0 0
\(832\) −3.85048 −0.133491
\(833\) 17.8265 0.617653
\(834\) 0 0
\(835\) −0.728960 −0.0252267
\(836\) −14.3572 −0.496554
\(837\) 0 0
\(838\) −31.2114 −1.07818
\(839\) 41.3804 1.42861 0.714305 0.699834i \(-0.246743\pi\)
0.714305 + 0.699834i \(0.246743\pi\)
\(840\) 0 0
\(841\) 4.15328 0.143216
\(842\) −16.1359 −0.556079
\(843\) 0 0
\(844\) 25.7505 0.886369
\(845\) −2.35552 −0.0810323
\(846\) 0 0
\(847\) −1.13095 −0.0388598
\(848\) 10.7984 0.370820
\(849\) 0 0
\(850\) 18.0752 0.619974
\(851\) 2.54404 0.0872087
\(852\) 0 0
\(853\) 31.8146 1.08931 0.544656 0.838660i \(-0.316660\pi\)
0.544656 + 0.838660i \(0.316660\pi\)
\(854\) 44.8127 1.53346
\(855\) 0 0
\(856\) −5.60796 −0.191676
\(857\) −42.9083 −1.46572 −0.732860 0.680379i \(-0.761815\pi\)
−0.732860 + 0.680379i \(0.761815\pi\)
\(858\) 0 0
\(859\) 37.2701 1.27164 0.635820 0.771838i \(-0.280662\pi\)
0.635820 + 0.771838i \(0.280662\pi\)
\(860\) −6.94887 −0.236954
\(861\) 0 0
\(862\) 5.79223 0.197284
\(863\) 8.20579 0.279328 0.139664 0.990199i \(-0.455398\pi\)
0.139664 + 0.990199i \(0.455398\pi\)
\(864\) 0 0
\(865\) −8.41367 −0.286073
\(866\) 1.81723 0.0617520
\(867\) 0 0
\(868\) 19.6188 0.665904
\(869\) 20.2753 0.687794
\(870\) 0 0
\(871\) 34.8959 1.18240
\(872\) 3.52073 0.119227
\(873\) 0 0
\(874\) −2.39691 −0.0810766
\(875\) 34.4927 1.16607
\(876\) 0 0
\(877\) −48.5804 −1.64044 −0.820222 0.572045i \(-0.806150\pi\)
−0.820222 + 0.572045i \(0.806150\pi\)
\(878\) −21.0055 −0.708900
\(879\) 0 0
\(880\) −4.20882 −0.141879
\(881\) −36.4874 −1.22929 −0.614645 0.788804i \(-0.710701\pi\)
−0.614645 + 0.788804i \(0.710701\pi\)
\(882\) 0 0
\(883\) −26.4845 −0.891273 −0.445637 0.895214i \(-0.647023\pi\)
−0.445637 + 0.895214i \(0.647023\pi\)
\(884\) −20.8609 −0.701627
\(885\) 0 0
\(886\) −37.1284 −1.24735
\(887\) 52.1259 1.75022 0.875108 0.483928i \(-0.160790\pi\)
0.875108 + 0.483928i \(0.160790\pi\)
\(888\) 0 0
\(889\) 50.1824 1.68306
\(890\) −6.34602 −0.212719
\(891\) 0 0
\(892\) 12.3557 0.413699
\(893\) −55.3619 −1.85262
\(894\) 0 0
\(895\) 22.4305 0.749768
\(896\) −3.20787 −0.107167
\(897\) 0 0
\(898\) −26.8067 −0.894551
\(899\) 35.2143 1.17446
\(900\) 0 0
\(901\) 58.5030 1.94902
\(902\) 29.9248 0.996388
\(903\) 0 0
\(904\) −6.78189 −0.225562
\(905\) 14.9136 0.495745
\(906\) 0 0
\(907\) 11.3037 0.375332 0.187666 0.982233i \(-0.439908\pi\)
0.187666 + 0.982233i \(0.439908\pi\)
\(908\) −8.06407 −0.267615
\(909\) 0 0
\(910\) −15.9319 −0.528139
\(911\) −42.3928 −1.40453 −0.702267 0.711913i \(-0.747829\pi\)
−0.702267 + 0.711913i \(0.747829\pi\)
\(912\) 0 0
\(913\) 1.05431 0.0348926
\(914\) −18.8431 −0.623274
\(915\) 0 0
\(916\) 21.0700 0.696173
\(917\) 19.5004 0.643960
\(918\) 0 0
\(919\) −10.6300 −0.350651 −0.175326 0.984510i \(-0.556098\pi\)
−0.175326 + 0.984510i \(0.556098\pi\)
\(920\) −0.702655 −0.0231658
\(921\) 0 0
\(922\) −0.774819 −0.0255173
\(923\) −23.0091 −0.757354
\(924\) 0 0
\(925\) 15.5806 0.512288
\(926\) −1.87804 −0.0617162
\(927\) 0 0
\(928\) −5.75789 −0.189012
\(929\) −4.08682 −0.134084 −0.0670421 0.997750i \(-0.521356\pi\)
−0.0670421 + 0.997750i \(0.521356\pi\)
\(930\) 0 0
\(931\) −14.4776 −0.474483
\(932\) −15.8020 −0.517612
\(933\) 0 0
\(934\) −12.1725 −0.398296
\(935\) −22.8022 −0.745713
\(936\) 0 0
\(937\) −36.3617 −1.18788 −0.593942 0.804508i \(-0.702429\pi\)
−0.593942 + 0.804508i \(0.702429\pi\)
\(938\) 29.0721 0.949237
\(939\) 0 0
\(940\) −16.2294 −0.529344
\(941\) −55.3089 −1.80302 −0.901509 0.432761i \(-0.857540\pi\)
−0.901509 + 0.432761i \(0.857540\pi\)
\(942\) 0 0
\(943\) 4.99590 0.162689
\(944\) −3.15039 −0.102536
\(945\) 0 0
\(946\) −17.5792 −0.571549
\(947\) 34.5292 1.12205 0.561025 0.827799i \(-0.310407\pi\)
0.561025 + 0.827799i \(0.310407\pi\)
\(948\) 0 0
\(949\) −22.8176 −0.740689
\(950\) −14.6795 −0.476266
\(951\) 0 0
\(952\) −17.3794 −0.563268
\(953\) 59.7894 1.93677 0.968385 0.249462i \(-0.0802538\pi\)
0.968385 + 0.249462i \(0.0802538\pi\)
\(954\) 0 0
\(955\) 1.83800 0.0594763
\(956\) 4.53148 0.146559
\(957\) 0 0
\(958\) −17.2910 −0.558648
\(959\) −51.4514 −1.66145
\(960\) 0 0
\(961\) 6.40336 0.206560
\(962\) −17.9819 −0.579759
\(963\) 0 0
\(964\) 8.04690 0.259173
\(965\) 5.89354 0.189720
\(966\) 0 0
\(967\) −0.362252 −0.0116492 −0.00582461 0.999983i \(-0.501854\pi\)
−0.00582461 + 0.999983i \(0.501854\pi\)
\(968\) 0.352555 0.0113315
\(969\) 0 0
\(970\) 20.7523 0.666317
\(971\) −1.80116 −0.0578021 −0.0289011 0.999582i \(-0.509201\pi\)
−0.0289011 + 0.999582i \(0.509201\pi\)
\(972\) 0 0
\(973\) −34.1275 −1.09408
\(974\) 17.7705 0.569402
\(975\) 0 0
\(976\) −13.9696 −0.447157
\(977\) 6.43661 0.205925 0.102963 0.994685i \(-0.467168\pi\)
0.102963 + 0.994685i \(0.467168\pi\)
\(978\) 0 0
\(979\) −16.0541 −0.513092
\(980\) −4.24411 −0.135573
\(981\) 0 0
\(982\) −21.3837 −0.682382
\(983\) −51.5087 −1.64287 −0.821437 0.570299i \(-0.806827\pi\)
−0.821437 + 0.570299i \(0.806827\pi\)
\(984\) 0 0
\(985\) 5.89468 0.187820
\(986\) −31.1947 −0.993441
\(987\) 0 0
\(988\) 16.9419 0.538993
\(989\) −2.93482 −0.0933218
\(990\) 0 0
\(991\) 13.4457 0.427118 0.213559 0.976930i \(-0.431494\pi\)
0.213559 + 0.976930i \(0.431494\pi\)
\(992\) −6.11583 −0.194178
\(993\) 0 0
\(994\) −19.1691 −0.608006
\(995\) −32.8550 −1.04157
\(996\) 0 0
\(997\) 3.77372 0.119515 0.0597575 0.998213i \(-0.480967\pi\)
0.0597575 + 0.998213i \(0.480967\pi\)
\(998\) −35.9564 −1.13818
\(999\) 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 8046.2.a.k.1.3 12
3.2 odd 2 8046.2.a.n.1.10 yes 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
8046.2.a.k.1.3 12 1.1 even 1 trivial
8046.2.a.n.1.10 yes 12 3.2 odd 2