Properties

Label 8008.2.a.z.1.9
Level $8008$
Weight $2$
Character 8008.1
Self dual yes
Analytic conductor $63.944$
Analytic rank $0$
Dimension $15$
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] = [8008,2,Mod(1,8008)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8008, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0, 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("8008.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 8008 = 2^{3} \cdot 7 \cdot 11 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8008.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: \(63.9442019386\)
Analytic rank: \(0\)
Dimension: \(15\)
Coefficient field: \(\mathbb{Q}[x]/(x^{15} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{15} - x^{14} - 35 x^{13} + 32 x^{12} + 477 x^{11} - 392 x^{10} - 3236 x^{9} + 2330 x^{8} + 11690 x^{7} - 7119 x^{6} - 22246 x^{5} + 11137 x^{4} + 20034 x^{3} - 8392 x^{2} + \cdots + 2560 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{6} \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.9
Root \(-0.804723\) of defining polynomial
Character \(\chi\) \(=\) 8008.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+0.804723 q^{3} -0.337817 q^{5} -1.00000 q^{7} -2.35242 q^{9} +O(q^{10})\) \(q+0.804723 q^{3} -0.337817 q^{5} -1.00000 q^{7} -2.35242 q^{9} -1.00000 q^{11} +1.00000 q^{13} -0.271849 q^{15} -1.72499 q^{17} +3.84801 q^{19} -0.804723 q^{21} -1.18228 q^{23} -4.88588 q^{25} -4.30721 q^{27} +8.19613 q^{29} +4.39204 q^{31} -0.804723 q^{33} +0.337817 q^{35} -8.55083 q^{37} +0.804723 q^{39} +3.35008 q^{41} +7.49568 q^{43} +0.794689 q^{45} -4.05444 q^{47} +1.00000 q^{49} -1.38814 q^{51} -8.73577 q^{53} +0.337817 q^{55} +3.09658 q^{57} +3.60840 q^{59} +3.64244 q^{61} +2.35242 q^{63} -0.337817 q^{65} -5.01726 q^{67} -0.951407 q^{69} -0.791482 q^{71} +8.54964 q^{73} -3.93178 q^{75} +1.00000 q^{77} -15.5648 q^{79} +3.59115 q^{81} +11.6191 q^{83} +0.582732 q^{85} +6.59561 q^{87} -4.70415 q^{89} -1.00000 q^{91} +3.53438 q^{93} -1.29992 q^{95} -9.95235 q^{97} +2.35242 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 15 q - q^{3} + 4 q^{5} - 15 q^{7} + 26 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 15 q - q^{3} + 4 q^{5} - 15 q^{7} + 26 q^{9} - 15 q^{11} + 15 q^{13} - 6 q^{15} + 8 q^{17} - 17 q^{19} + q^{21} + 7 q^{23} + 33 q^{25} - 4 q^{27} + 14 q^{29} - 4 q^{31} + q^{33} - 4 q^{35} + 3 q^{37} - q^{39} - 13 q^{43} + 20 q^{45} + 6 q^{47} + 15 q^{49} + 8 q^{51} + 38 q^{53} - 4 q^{55} + 24 q^{57} - 18 q^{59} + 23 q^{61} - 26 q^{63} + 4 q^{65} - 8 q^{67} + 43 q^{69} - 12 q^{71} + 11 q^{73} + 12 q^{75} + 15 q^{77} - q^{79} + 51 q^{81} - 16 q^{83} + 13 q^{85} - 25 q^{87} + 28 q^{89} - 15 q^{91} - 14 q^{93} + 49 q^{95} + 30 q^{97} - 26 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Coefficient data

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



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 0.804723 0.464607 0.232303 0.972643i \(-0.425374\pi\)
0.232303 + 0.972643i \(0.425374\pi\)
\(4\) 0 0
\(5\) −0.337817 −0.151076 −0.0755382 0.997143i \(-0.524067\pi\)
−0.0755382 + 0.997143i \(0.524067\pi\)
\(6\) 0 0
\(7\) −1.00000 −0.377964
\(8\) 0 0
\(9\) −2.35242 −0.784141
\(10\) 0 0
\(11\) −1.00000 −0.301511
\(12\) 0 0
\(13\) 1.00000 0.277350
\(14\) 0 0
\(15\) −0.271849 −0.0701912
\(16\) 0 0
\(17\) −1.72499 −0.418372 −0.209186 0.977876i \(-0.567081\pi\)
−0.209186 + 0.977876i \(0.567081\pi\)
\(18\) 0 0
\(19\) 3.84801 0.882794 0.441397 0.897312i \(-0.354483\pi\)
0.441397 + 0.897312i \(0.354483\pi\)
\(20\) 0 0
\(21\) −0.804723 −0.175605
\(22\) 0 0
\(23\) −1.18228 −0.246522 −0.123261 0.992374i \(-0.539335\pi\)
−0.123261 + 0.992374i \(0.539335\pi\)
\(24\) 0 0
\(25\) −4.88588 −0.977176
\(26\) 0 0
\(27\) −4.30721 −0.828924
\(28\) 0 0
\(29\) 8.19613 1.52198 0.760991 0.648762i \(-0.224713\pi\)
0.760991 + 0.648762i \(0.224713\pi\)
\(30\) 0 0
\(31\) 4.39204 0.788834 0.394417 0.918932i \(-0.370947\pi\)
0.394417 + 0.918932i \(0.370947\pi\)
\(32\) 0 0
\(33\) −0.804723 −0.140084
\(34\) 0 0
\(35\) 0.337817 0.0571015
\(36\) 0 0
\(37\) −8.55083 −1.40575 −0.702874 0.711315i \(-0.748100\pi\)
−0.702874 + 0.711315i \(0.748100\pi\)
\(38\) 0 0
\(39\) 0.804723 0.128859
\(40\) 0 0
\(41\) 3.35008 0.523195 0.261598 0.965177i \(-0.415751\pi\)
0.261598 + 0.965177i \(0.415751\pi\)
\(42\) 0 0
\(43\) 7.49568 1.14308 0.571541 0.820574i \(-0.306346\pi\)
0.571541 + 0.820574i \(0.306346\pi\)
\(44\) 0 0
\(45\) 0.794689 0.118465
\(46\) 0 0
\(47\) −4.05444 −0.591401 −0.295701 0.955281i \(-0.595553\pi\)
−0.295701 + 0.955281i \(0.595553\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) −1.38814 −0.194378
\(52\) 0 0
\(53\) −8.73577 −1.19995 −0.599975 0.800019i \(-0.704823\pi\)
−0.599975 + 0.800019i \(0.704823\pi\)
\(54\) 0 0
\(55\) 0.337817 0.0455513
\(56\) 0 0
\(57\) 3.09658 0.410152
\(58\) 0 0
\(59\) 3.60840 0.469774 0.234887 0.972023i \(-0.424528\pi\)
0.234887 + 0.972023i \(0.424528\pi\)
\(60\) 0 0
\(61\) 3.64244 0.466367 0.233183 0.972433i \(-0.425086\pi\)
0.233183 + 0.972433i \(0.425086\pi\)
\(62\) 0 0
\(63\) 2.35242 0.296377
\(64\) 0 0
\(65\) −0.337817 −0.0419011
\(66\) 0 0
\(67\) −5.01726 −0.612956 −0.306478 0.951878i \(-0.599151\pi\)
−0.306478 + 0.951878i \(0.599151\pi\)
\(68\) 0 0
\(69\) −0.951407 −0.114536
\(70\) 0 0
\(71\) −0.791482 −0.0939316 −0.0469658 0.998896i \(-0.514955\pi\)
−0.0469658 + 0.998896i \(0.514955\pi\)
\(72\) 0 0
\(73\) 8.54964 1.00066 0.500330 0.865835i \(-0.333212\pi\)
0.500330 + 0.865835i \(0.333212\pi\)
\(74\) 0 0
\(75\) −3.93178 −0.454003
\(76\) 0 0
\(77\) 1.00000 0.113961
\(78\) 0 0
\(79\) −15.5648 −1.75118 −0.875589 0.483057i \(-0.839526\pi\)
−0.875589 + 0.483057i \(0.839526\pi\)
\(80\) 0 0
\(81\) 3.59115 0.399017
\(82\) 0 0
\(83\) 11.6191 1.27536 0.637680 0.770301i \(-0.279894\pi\)
0.637680 + 0.770301i \(0.279894\pi\)
\(84\) 0 0
\(85\) 0.582732 0.0632061
\(86\) 0 0
\(87\) 6.59561 0.707124
\(88\) 0 0
\(89\) −4.70415 −0.498639 −0.249319 0.968421i \(-0.580207\pi\)
−0.249319 + 0.968421i \(0.580207\pi\)
\(90\) 0 0
\(91\) −1.00000 −0.104828
\(92\) 0 0
\(93\) 3.53438 0.366498
\(94\) 0 0
\(95\) −1.29992 −0.133369
\(96\) 0 0
\(97\) −9.95235 −1.01051 −0.505254 0.862971i \(-0.668601\pi\)
−0.505254 + 0.862971i \(0.668601\pi\)
\(98\) 0 0
\(99\) 2.35242 0.236427
\(100\) 0 0
\(101\) 18.2805 1.81898 0.909488 0.415730i \(-0.136474\pi\)
0.909488 + 0.415730i \(0.136474\pi\)
\(102\) 0 0
\(103\) 12.6336 1.24483 0.622413 0.782689i \(-0.286153\pi\)
0.622413 + 0.782689i \(0.286153\pi\)
\(104\) 0 0
\(105\) 0.271849 0.0265298
\(106\) 0 0
\(107\) 1.18691 0.114743 0.0573715 0.998353i \(-0.481728\pi\)
0.0573715 + 0.998353i \(0.481728\pi\)
\(108\) 0 0
\(109\) 14.4650 1.38550 0.692749 0.721178i \(-0.256399\pi\)
0.692749 + 0.721178i \(0.256399\pi\)
\(110\) 0 0
\(111\) −6.88105 −0.653120
\(112\) 0 0
\(113\) −11.5775 −1.08912 −0.544558 0.838723i \(-0.683303\pi\)
−0.544558 + 0.838723i \(0.683303\pi\)
\(114\) 0 0
\(115\) 0.399394 0.0372437
\(116\) 0 0
\(117\) −2.35242 −0.217481
\(118\) 0 0
\(119\) 1.72499 0.158130
\(120\) 0 0
\(121\) 1.00000 0.0909091
\(122\) 0 0
\(123\) 2.69589 0.243080
\(124\) 0 0
\(125\) 3.33962 0.298705
\(126\) 0 0
\(127\) 19.9915 1.77396 0.886981 0.461806i \(-0.152798\pi\)
0.886981 + 0.461806i \(0.152798\pi\)
\(128\) 0 0
\(129\) 6.03195 0.531083
\(130\) 0 0
\(131\) −4.62283 −0.403898 −0.201949 0.979396i \(-0.564728\pi\)
−0.201949 + 0.979396i \(0.564728\pi\)
\(132\) 0 0
\(133\) −3.84801 −0.333665
\(134\) 0 0
\(135\) 1.45505 0.125231
\(136\) 0 0
\(137\) 17.8908 1.52852 0.764258 0.644911i \(-0.223106\pi\)
0.764258 + 0.644911i \(0.223106\pi\)
\(138\) 0 0
\(139\) −8.09613 −0.686705 −0.343352 0.939207i \(-0.611562\pi\)
−0.343352 + 0.939207i \(0.611562\pi\)
\(140\) 0 0
\(141\) −3.26270 −0.274769
\(142\) 0 0
\(143\) −1.00000 −0.0836242
\(144\) 0 0
\(145\) −2.76879 −0.229936
\(146\) 0 0
\(147\) 0.804723 0.0663724
\(148\) 0 0
\(149\) 3.23448 0.264979 0.132489 0.991184i \(-0.457703\pi\)
0.132489 + 0.991184i \(0.457703\pi\)
\(150\) 0 0
\(151\) −21.5150 −1.75087 −0.875433 0.483339i \(-0.839424\pi\)
−0.875433 + 0.483339i \(0.839424\pi\)
\(152\) 0 0
\(153\) 4.05791 0.328062
\(154\) 0 0
\(155\) −1.48371 −0.119174
\(156\) 0 0
\(157\) 15.1699 1.21069 0.605344 0.795964i \(-0.293036\pi\)
0.605344 + 0.795964i \(0.293036\pi\)
\(158\) 0 0
\(159\) −7.02987 −0.557505
\(160\) 0 0
\(161\) 1.18228 0.0931767
\(162\) 0 0
\(163\) 16.7280 1.31024 0.655120 0.755525i \(-0.272618\pi\)
0.655120 + 0.755525i \(0.272618\pi\)
\(164\) 0 0
\(165\) 0.271849 0.0211634
\(166\) 0 0
\(167\) 18.6673 1.44452 0.722259 0.691622i \(-0.243104\pi\)
0.722259 + 0.691622i \(0.243104\pi\)
\(168\) 0 0
\(169\) 1.00000 0.0769231
\(170\) 0 0
\(171\) −9.05214 −0.692234
\(172\) 0 0
\(173\) 2.84617 0.216391 0.108195 0.994130i \(-0.465493\pi\)
0.108195 + 0.994130i \(0.465493\pi\)
\(174\) 0 0
\(175\) 4.88588 0.369338
\(176\) 0 0
\(177\) 2.90376 0.218260
\(178\) 0 0
\(179\) 14.8283 1.10832 0.554160 0.832410i \(-0.313039\pi\)
0.554160 + 0.832410i \(0.313039\pi\)
\(180\) 0 0
\(181\) −19.2463 −1.43056 −0.715281 0.698837i \(-0.753701\pi\)
−0.715281 + 0.698837i \(0.753701\pi\)
\(182\) 0 0
\(183\) 2.93115 0.216677
\(184\) 0 0
\(185\) 2.88862 0.212375
\(186\) 0 0
\(187\) 1.72499 0.126144
\(188\) 0 0
\(189\) 4.30721 0.313304
\(190\) 0 0
\(191\) 4.53657 0.328254 0.164127 0.986439i \(-0.447519\pi\)
0.164127 + 0.986439i \(0.447519\pi\)
\(192\) 0 0
\(193\) 6.96448 0.501314 0.250657 0.968076i \(-0.419353\pi\)
0.250657 + 0.968076i \(0.419353\pi\)
\(194\) 0 0
\(195\) −0.271849 −0.0194675
\(196\) 0 0
\(197\) 3.05837 0.217900 0.108950 0.994047i \(-0.465251\pi\)
0.108950 + 0.994047i \(0.465251\pi\)
\(198\) 0 0
\(199\) −11.1399 −0.789688 −0.394844 0.918748i \(-0.629201\pi\)
−0.394844 + 0.918748i \(0.629201\pi\)
\(200\) 0 0
\(201\) −4.03750 −0.284783
\(202\) 0 0
\(203\) −8.19613 −0.575255
\(204\) 0 0
\(205\) −1.13172 −0.0790425
\(206\) 0 0
\(207\) 2.78122 0.193308
\(208\) 0 0
\(209\) −3.84801 −0.266172
\(210\) 0 0
\(211\) 14.6682 1.00980 0.504900 0.863178i \(-0.331529\pi\)
0.504900 + 0.863178i \(0.331529\pi\)
\(212\) 0 0
\(213\) −0.636924 −0.0436413
\(214\) 0 0
\(215\) −2.53217 −0.172693
\(216\) 0 0
\(217\) −4.39204 −0.298151
\(218\) 0 0
\(219\) 6.88009 0.464913
\(220\) 0 0
\(221\) −1.72499 −0.116035
\(222\) 0 0
\(223\) 20.4786 1.37135 0.685674 0.727909i \(-0.259507\pi\)
0.685674 + 0.727909i \(0.259507\pi\)
\(224\) 0 0
\(225\) 11.4936 0.766243
\(226\) 0 0
\(227\) −9.36703 −0.621712 −0.310856 0.950457i \(-0.600616\pi\)
−0.310856 + 0.950457i \(0.600616\pi\)
\(228\) 0 0
\(229\) 25.0926 1.65816 0.829081 0.559128i \(-0.188864\pi\)
0.829081 + 0.559128i \(0.188864\pi\)
\(230\) 0 0
\(231\) 0.804723 0.0529469
\(232\) 0 0
\(233\) 23.9120 1.56653 0.783263 0.621691i \(-0.213554\pi\)
0.783263 + 0.621691i \(0.213554\pi\)
\(234\) 0 0
\(235\) 1.36966 0.0893468
\(236\) 0 0
\(237\) −12.5254 −0.813609
\(238\) 0 0
\(239\) −15.6714 −1.01370 −0.506850 0.862034i \(-0.669190\pi\)
−0.506850 + 0.862034i \(0.669190\pi\)
\(240\) 0 0
\(241\) −10.8205 −0.697008 −0.348504 0.937307i \(-0.613310\pi\)
−0.348504 + 0.937307i \(0.613310\pi\)
\(242\) 0 0
\(243\) 15.8115 1.01431
\(244\) 0 0
\(245\) −0.337817 −0.0215824
\(246\) 0 0
\(247\) 3.84801 0.244843
\(248\) 0 0
\(249\) 9.35014 0.592541
\(250\) 0 0
\(251\) 25.5868 1.61503 0.807513 0.589850i \(-0.200813\pi\)
0.807513 + 0.589850i \(0.200813\pi\)
\(252\) 0 0
\(253\) 1.18228 0.0743293
\(254\) 0 0
\(255\) 0.468937 0.0293660
\(256\) 0 0
\(257\) −0.0997874 −0.00622457 −0.00311228 0.999995i \(-0.500991\pi\)
−0.00311228 + 0.999995i \(0.500991\pi\)
\(258\) 0 0
\(259\) 8.55083 0.531323
\(260\) 0 0
\(261\) −19.2808 −1.19345
\(262\) 0 0
\(263\) −22.4546 −1.38461 −0.692305 0.721605i \(-0.743405\pi\)
−0.692305 + 0.721605i \(0.743405\pi\)
\(264\) 0 0
\(265\) 2.95109 0.181284
\(266\) 0 0
\(267\) −3.78554 −0.231671
\(268\) 0 0
\(269\) 31.5223 1.92195 0.960973 0.276642i \(-0.0892217\pi\)
0.960973 + 0.276642i \(0.0892217\pi\)
\(270\) 0 0
\(271\) −3.29902 −0.200401 −0.100201 0.994967i \(-0.531948\pi\)
−0.100201 + 0.994967i \(0.531948\pi\)
\(272\) 0 0
\(273\) −0.804723 −0.0487040
\(274\) 0 0
\(275\) 4.88588 0.294630
\(276\) 0 0
\(277\) −14.3345 −0.861279 −0.430640 0.902524i \(-0.641712\pi\)
−0.430640 + 0.902524i \(0.641712\pi\)
\(278\) 0 0
\(279\) −10.3319 −0.618557
\(280\) 0 0
\(281\) 4.75341 0.283565 0.141782 0.989898i \(-0.454717\pi\)
0.141782 + 0.989898i \(0.454717\pi\)
\(282\) 0 0
\(283\) 16.7122 0.993437 0.496719 0.867912i \(-0.334538\pi\)
0.496719 + 0.867912i \(0.334538\pi\)
\(284\) 0 0
\(285\) −1.04608 −0.0619643
\(286\) 0 0
\(287\) −3.35008 −0.197749
\(288\) 0 0
\(289\) −14.0244 −0.824965
\(290\) 0 0
\(291\) −8.00888 −0.469489
\(292\) 0 0
\(293\) 22.9887 1.34301 0.671507 0.740998i \(-0.265647\pi\)
0.671507 + 0.740998i \(0.265647\pi\)
\(294\) 0 0
\(295\) −1.21898 −0.0709718
\(296\) 0 0
\(297\) 4.30721 0.249930
\(298\) 0 0
\(299\) −1.18228 −0.0683730
\(300\) 0 0
\(301\) −7.49568 −0.432044
\(302\) 0 0
\(303\) 14.7107 0.845109
\(304\) 0 0
\(305\) −1.23048 −0.0704570
\(306\) 0 0
\(307\) 2.25287 0.128578 0.0642890 0.997931i \(-0.479522\pi\)
0.0642890 + 0.997931i \(0.479522\pi\)
\(308\) 0 0
\(309\) 10.1665 0.578354
\(310\) 0 0
\(311\) −19.2444 −1.09125 −0.545625 0.838029i \(-0.683708\pi\)
−0.545625 + 0.838029i \(0.683708\pi\)
\(312\) 0 0
\(313\) 24.1366 1.36428 0.682140 0.731222i \(-0.261050\pi\)
0.682140 + 0.731222i \(0.261050\pi\)
\(314\) 0 0
\(315\) −0.794689 −0.0447756
\(316\) 0 0
\(317\) 5.52023 0.310047 0.155024 0.987911i \(-0.450455\pi\)
0.155024 + 0.987911i \(0.450455\pi\)
\(318\) 0 0
\(319\) −8.19613 −0.458895
\(320\) 0 0
\(321\) 0.955135 0.0533104
\(322\) 0 0
\(323\) −6.63778 −0.369336
\(324\) 0 0
\(325\) −4.88588 −0.271020
\(326\) 0 0
\(327\) 11.6403 0.643712
\(328\) 0 0
\(329\) 4.05444 0.223529
\(330\) 0 0
\(331\) −7.60003 −0.417735 −0.208868 0.977944i \(-0.566978\pi\)
−0.208868 + 0.977944i \(0.566978\pi\)
\(332\) 0 0
\(333\) 20.1152 1.10230
\(334\) 0 0
\(335\) 1.69492 0.0926032
\(336\) 0 0
\(337\) −4.11912 −0.224383 −0.112191 0.993687i \(-0.535787\pi\)
−0.112191 + 0.993687i \(0.535787\pi\)
\(338\) 0 0
\(339\) −9.31665 −0.506011
\(340\) 0 0
\(341\) −4.39204 −0.237842
\(342\) 0 0
\(343\) −1.00000 −0.0539949
\(344\) 0 0
\(345\) 0.321402 0.0173037
\(346\) 0 0
\(347\) 5.22235 0.280350 0.140175 0.990127i \(-0.455233\pi\)
0.140175 + 0.990127i \(0.455233\pi\)
\(348\) 0 0
\(349\) 16.9918 0.909549 0.454774 0.890607i \(-0.349720\pi\)
0.454774 + 0.890607i \(0.349720\pi\)
\(350\) 0 0
\(351\) −4.30721 −0.229902
\(352\) 0 0
\(353\) 20.5693 1.09480 0.547398 0.836872i \(-0.315618\pi\)
0.547398 + 0.836872i \(0.315618\pi\)
\(354\) 0 0
\(355\) 0.267376 0.0141909
\(356\) 0 0
\(357\) 1.38814 0.0734681
\(358\) 0 0
\(359\) −1.77260 −0.0935541 −0.0467771 0.998905i \(-0.514895\pi\)
−0.0467771 + 0.998905i \(0.514895\pi\)
\(360\) 0 0
\(361\) −4.19282 −0.220675
\(362\) 0 0
\(363\) 0.804723 0.0422370
\(364\) 0 0
\(365\) −2.88821 −0.151176
\(366\) 0 0
\(367\) −9.53966 −0.497966 −0.248983 0.968508i \(-0.580096\pi\)
−0.248983 + 0.968508i \(0.580096\pi\)
\(368\) 0 0
\(369\) −7.88081 −0.410259
\(370\) 0 0
\(371\) 8.73577 0.453539
\(372\) 0 0
\(373\) 28.8531 1.49396 0.746978 0.664849i \(-0.231504\pi\)
0.746978 + 0.664849i \(0.231504\pi\)
\(374\) 0 0
\(375\) 2.68747 0.138780
\(376\) 0 0
\(377\) 8.19613 0.422122
\(378\) 0 0
\(379\) 6.74582 0.346509 0.173255 0.984877i \(-0.444572\pi\)
0.173255 + 0.984877i \(0.444572\pi\)
\(380\) 0 0
\(381\) 16.0876 0.824195
\(382\) 0 0
\(383\) 8.38307 0.428355 0.214177 0.976795i \(-0.431293\pi\)
0.214177 + 0.976795i \(0.431293\pi\)
\(384\) 0 0
\(385\) −0.337817 −0.0172168
\(386\) 0 0
\(387\) −17.6330 −0.896336
\(388\) 0 0
\(389\) 7.44907 0.377683 0.188841 0.982008i \(-0.439527\pi\)
0.188841 + 0.982008i \(0.439527\pi\)
\(390\) 0 0
\(391\) 2.03942 0.103138
\(392\) 0 0
\(393\) −3.72009 −0.187654
\(394\) 0 0
\(395\) 5.25806 0.264562
\(396\) 0 0
\(397\) −32.2952 −1.62085 −0.810425 0.585843i \(-0.800763\pi\)
−0.810425 + 0.585843i \(0.800763\pi\)
\(398\) 0 0
\(399\) −3.09658 −0.155023
\(400\) 0 0
\(401\) −19.8338 −0.990454 −0.495227 0.868764i \(-0.664915\pi\)
−0.495227 + 0.868764i \(0.664915\pi\)
\(402\) 0 0
\(403\) 4.39204 0.218783
\(404\) 0 0
\(405\) −1.21315 −0.0602820
\(406\) 0 0
\(407\) 8.55083 0.423849
\(408\) 0 0
\(409\) −17.3553 −0.858163 −0.429082 0.903266i \(-0.641163\pi\)
−0.429082 + 0.903266i \(0.641163\pi\)
\(410\) 0 0
\(411\) 14.3971 0.710159
\(412\) 0 0
\(413\) −3.60840 −0.177558
\(414\) 0 0
\(415\) −3.92513 −0.192677
\(416\) 0 0
\(417\) −6.51514 −0.319048
\(418\) 0 0
\(419\) −28.5545 −1.39498 −0.697490 0.716595i \(-0.745700\pi\)
−0.697490 + 0.716595i \(0.745700\pi\)
\(420\) 0 0
\(421\) 7.37781 0.359572 0.179786 0.983706i \(-0.442459\pi\)
0.179786 + 0.983706i \(0.442459\pi\)
\(422\) 0 0
\(423\) 9.53776 0.463742
\(424\) 0 0
\(425\) 8.42810 0.408823
\(426\) 0 0
\(427\) −3.64244 −0.176270
\(428\) 0 0
\(429\) −0.804723 −0.0388524
\(430\) 0 0
\(431\) −8.08144 −0.389269 −0.194635 0.980876i \(-0.562352\pi\)
−0.194635 + 0.980876i \(0.562352\pi\)
\(432\) 0 0
\(433\) 4.91199 0.236055 0.118028 0.993010i \(-0.462343\pi\)
0.118028 + 0.993010i \(0.462343\pi\)
\(434\) 0 0
\(435\) −2.22811 −0.106830
\(436\) 0 0
\(437\) −4.54942 −0.217628
\(438\) 0 0
\(439\) −9.96827 −0.475760 −0.237880 0.971295i \(-0.576452\pi\)
−0.237880 + 0.971295i \(0.576452\pi\)
\(440\) 0 0
\(441\) −2.35242 −0.112020
\(442\) 0 0
\(443\) 9.86795 0.468841 0.234420 0.972135i \(-0.424681\pi\)
0.234420 + 0.972135i \(0.424681\pi\)
\(444\) 0 0
\(445\) 1.58914 0.0753326
\(446\) 0 0
\(447\) 2.60286 0.123111
\(448\) 0 0
\(449\) −14.5207 −0.685276 −0.342638 0.939467i \(-0.611320\pi\)
−0.342638 + 0.939467i \(0.611320\pi\)
\(450\) 0 0
\(451\) −3.35008 −0.157749
\(452\) 0 0
\(453\) −17.3136 −0.813465
\(454\) 0 0
\(455\) 0.337817 0.0158371
\(456\) 0 0
\(457\) −6.82950 −0.319471 −0.159735 0.987160i \(-0.551064\pi\)
−0.159735 + 0.987160i \(0.551064\pi\)
\(458\) 0 0
\(459\) 7.42991 0.346798
\(460\) 0 0
\(461\) 16.0736 0.748623 0.374312 0.927303i \(-0.377879\pi\)
0.374312 + 0.927303i \(0.377879\pi\)
\(462\) 0 0
\(463\) 18.5734 0.863177 0.431589 0.902071i \(-0.357953\pi\)
0.431589 + 0.902071i \(0.357953\pi\)
\(464\) 0 0
\(465\) −1.19397 −0.0553692
\(466\) 0 0
\(467\) 15.2575 0.706034 0.353017 0.935617i \(-0.385156\pi\)
0.353017 + 0.935617i \(0.385156\pi\)
\(468\) 0 0
\(469\) 5.01726 0.231676
\(470\) 0 0
\(471\) 12.2075 0.562494
\(472\) 0 0
\(473\) −7.49568 −0.344652
\(474\) 0 0
\(475\) −18.8009 −0.862645
\(476\) 0 0
\(477\) 20.5502 0.940930
\(478\) 0 0
\(479\) −19.8332 −0.906200 −0.453100 0.891460i \(-0.649682\pi\)
−0.453100 + 0.891460i \(0.649682\pi\)
\(480\) 0 0
\(481\) −8.55083 −0.389884
\(482\) 0 0
\(483\) 0.951407 0.0432905
\(484\) 0 0
\(485\) 3.36207 0.152664
\(486\) 0 0
\(487\) 32.1839 1.45839 0.729196 0.684305i \(-0.239894\pi\)
0.729196 + 0.684305i \(0.239894\pi\)
\(488\) 0 0
\(489\) 13.4614 0.608746
\(490\) 0 0
\(491\) −7.29478 −0.329209 −0.164604 0.986360i \(-0.552635\pi\)
−0.164604 + 0.986360i \(0.552635\pi\)
\(492\) 0 0
\(493\) −14.1383 −0.636755
\(494\) 0 0
\(495\) −0.794689 −0.0357186
\(496\) 0 0
\(497\) 0.791482 0.0355028
\(498\) 0 0
\(499\) −37.7508 −1.68996 −0.844979 0.534799i \(-0.820387\pi\)
−0.844979 + 0.534799i \(0.820387\pi\)
\(500\) 0 0
\(501\) 15.0220 0.671133
\(502\) 0 0
\(503\) 18.7322 0.835228 0.417614 0.908625i \(-0.362866\pi\)
0.417614 + 0.908625i \(0.362866\pi\)
\(504\) 0 0
\(505\) −6.17546 −0.274804
\(506\) 0 0
\(507\) 0.804723 0.0357390
\(508\) 0 0
\(509\) −14.5377 −0.644370 −0.322185 0.946677i \(-0.604417\pi\)
−0.322185 + 0.946677i \(0.604417\pi\)
\(510\) 0 0
\(511\) −8.54964 −0.378214
\(512\) 0 0
\(513\) −16.5742 −0.731769
\(514\) 0 0
\(515\) −4.26785 −0.188064
\(516\) 0 0
\(517\) 4.05444 0.178314
\(518\) 0 0
\(519\) 2.29038 0.100537
\(520\) 0 0
\(521\) 20.4815 0.897312 0.448656 0.893705i \(-0.351903\pi\)
0.448656 + 0.893705i \(0.351903\pi\)
\(522\) 0 0
\(523\) −31.0237 −1.35657 −0.678285 0.734799i \(-0.737277\pi\)
−0.678285 + 0.734799i \(0.737277\pi\)
\(524\) 0 0
\(525\) 3.93178 0.171597
\(526\) 0 0
\(527\) −7.57623 −0.330026
\(528\) 0 0
\(529\) −21.6022 −0.939227
\(530\) 0 0
\(531\) −8.48848 −0.368369
\(532\) 0 0
\(533\) 3.35008 0.145108
\(534\) 0 0
\(535\) −0.400959 −0.0173350
\(536\) 0 0
\(537\) 11.9327 0.514933
\(538\) 0 0
\(539\) −1.00000 −0.0430730
\(540\) 0 0
\(541\) −18.2602 −0.785066 −0.392533 0.919738i \(-0.628401\pi\)
−0.392533 + 0.919738i \(0.628401\pi\)
\(542\) 0 0
\(543\) −15.4879 −0.664649
\(544\) 0 0
\(545\) −4.88654 −0.209316
\(546\) 0 0
\(547\) −23.3950 −1.00030 −0.500149 0.865940i \(-0.666721\pi\)
−0.500149 + 0.865940i \(0.666721\pi\)
\(548\) 0 0
\(549\) −8.56856 −0.365697
\(550\) 0 0
\(551\) 31.5388 1.34360
\(552\) 0 0
\(553\) 15.5648 0.661883
\(554\) 0 0
\(555\) 2.32454 0.0986710
\(556\) 0 0
\(557\) 19.2014 0.813589 0.406794 0.913520i \(-0.366647\pi\)
0.406794 + 0.913520i \(0.366647\pi\)
\(558\) 0 0
\(559\) 7.49568 0.317034
\(560\) 0 0
\(561\) 1.38814 0.0586073
\(562\) 0 0
\(563\) −19.0333 −0.802158 −0.401079 0.916043i \(-0.631365\pi\)
−0.401079 + 0.916043i \(0.631365\pi\)
\(564\) 0 0
\(565\) 3.91107 0.164540
\(566\) 0 0
\(567\) −3.59115 −0.150814
\(568\) 0 0
\(569\) −27.6935 −1.16097 −0.580487 0.814270i \(-0.697138\pi\)
−0.580487 + 0.814270i \(0.697138\pi\)
\(570\) 0 0
\(571\) −7.55319 −0.316091 −0.158046 0.987432i \(-0.550519\pi\)
−0.158046 + 0.987432i \(0.550519\pi\)
\(572\) 0 0
\(573\) 3.65068 0.152509
\(574\) 0 0
\(575\) 5.77648 0.240896
\(576\) 0 0
\(577\) 13.1096 0.545762 0.272881 0.962048i \(-0.412024\pi\)
0.272881 + 0.962048i \(0.412024\pi\)
\(578\) 0 0
\(579\) 5.60447 0.232914
\(580\) 0 0
\(581\) −11.6191 −0.482041
\(582\) 0 0
\(583\) 8.73577 0.361799
\(584\) 0 0
\(585\) 0.794689 0.0328563
\(586\) 0 0
\(587\) −38.7363 −1.59882 −0.799409 0.600787i \(-0.794854\pi\)
−0.799409 + 0.600787i \(0.794854\pi\)
\(588\) 0 0
\(589\) 16.9006 0.696378
\(590\) 0 0
\(591\) 2.46114 0.101238
\(592\) 0 0
\(593\) −8.05735 −0.330876 −0.165438 0.986220i \(-0.552904\pi\)
−0.165438 + 0.986220i \(0.552904\pi\)
\(594\) 0 0
\(595\) −0.582732 −0.0238897
\(596\) 0 0
\(597\) −8.96454 −0.366894
\(598\) 0 0
\(599\) 41.2329 1.68473 0.842366 0.538906i \(-0.181162\pi\)
0.842366 + 0.538906i \(0.181162\pi\)
\(600\) 0 0
\(601\) 23.8925 0.974595 0.487297 0.873236i \(-0.337983\pi\)
0.487297 + 0.873236i \(0.337983\pi\)
\(602\) 0 0
\(603\) 11.8027 0.480643
\(604\) 0 0
\(605\) −0.337817 −0.0137342
\(606\) 0 0
\(607\) −16.3096 −0.661986 −0.330993 0.943633i \(-0.607384\pi\)
−0.330993 + 0.943633i \(0.607384\pi\)
\(608\) 0 0
\(609\) −6.59561 −0.267268
\(610\) 0 0
\(611\) −4.05444 −0.164025
\(612\) 0 0
\(613\) −5.34724 −0.215973 −0.107986 0.994152i \(-0.534440\pi\)
−0.107986 + 0.994152i \(0.534440\pi\)
\(614\) 0 0
\(615\) −0.910718 −0.0367237
\(616\) 0 0
\(617\) 25.1242 1.01146 0.505731 0.862691i \(-0.331223\pi\)
0.505731 + 0.862691i \(0.331223\pi\)
\(618\) 0 0
\(619\) −5.58680 −0.224553 −0.112276 0.993677i \(-0.535814\pi\)
−0.112276 + 0.993677i \(0.535814\pi\)
\(620\) 0 0
\(621\) 5.09233 0.204348
\(622\) 0 0
\(623\) 4.70415 0.188468
\(624\) 0 0
\(625\) 23.3012 0.932049
\(626\) 0 0
\(627\) −3.09658 −0.123666
\(628\) 0 0
\(629\) 14.7501 0.588125
\(630\) 0 0
\(631\) 25.4265 1.01221 0.506106 0.862471i \(-0.331085\pi\)
0.506106 + 0.862471i \(0.331085\pi\)
\(632\) 0 0
\(633\) 11.8038 0.469160
\(634\) 0 0
\(635\) −6.75348 −0.268004
\(636\) 0 0
\(637\) 1.00000 0.0396214
\(638\) 0 0
\(639\) 1.86190 0.0736556
\(640\) 0 0
\(641\) −7.11538 −0.281041 −0.140520 0.990078i \(-0.544878\pi\)
−0.140520 + 0.990078i \(0.544878\pi\)
\(642\) 0 0
\(643\) −12.4604 −0.491390 −0.245695 0.969347i \(-0.579016\pi\)
−0.245695 + 0.969347i \(0.579016\pi\)
\(644\) 0 0
\(645\) −2.03770 −0.0802342
\(646\) 0 0
\(647\) 2.36127 0.0928313 0.0464156 0.998922i \(-0.485220\pi\)
0.0464156 + 0.998922i \(0.485220\pi\)
\(648\) 0 0
\(649\) −3.60840 −0.141642
\(650\) 0 0
\(651\) −3.53438 −0.138523
\(652\) 0 0
\(653\) −27.8552 −1.09006 −0.545029 0.838417i \(-0.683481\pi\)
−0.545029 + 0.838417i \(0.683481\pi\)
\(654\) 0 0
\(655\) 1.56167 0.0610195
\(656\) 0 0
\(657\) −20.1123 −0.784657
\(658\) 0 0
\(659\) 24.4355 0.951874 0.475937 0.879479i \(-0.342109\pi\)
0.475937 + 0.879479i \(0.342109\pi\)
\(660\) 0 0
\(661\) 11.8633 0.461429 0.230715 0.973021i \(-0.425894\pi\)
0.230715 + 0.973021i \(0.425894\pi\)
\(662\) 0 0
\(663\) −1.38814 −0.0539109
\(664\) 0 0
\(665\) 1.29992 0.0504089
\(666\) 0 0
\(667\) −9.69012 −0.375203
\(668\) 0 0
\(669\) 16.4796 0.637138
\(670\) 0 0
\(671\) −3.64244 −0.140615
\(672\) 0 0
\(673\) 8.56662 0.330219 0.165109 0.986275i \(-0.447202\pi\)
0.165109 + 0.986275i \(0.447202\pi\)
\(674\) 0 0
\(675\) 21.0445 0.810004
\(676\) 0 0
\(677\) 44.4161 1.70705 0.853524 0.521054i \(-0.174461\pi\)
0.853524 + 0.521054i \(0.174461\pi\)
\(678\) 0 0
\(679\) 9.95235 0.381936
\(680\) 0 0
\(681\) −7.53786 −0.288851
\(682\) 0 0
\(683\) −31.8614 −1.21914 −0.609572 0.792731i \(-0.708659\pi\)
−0.609572 + 0.792731i \(0.708659\pi\)
\(684\) 0 0
\(685\) −6.04382 −0.230923
\(686\) 0 0
\(687\) 20.1925 0.770394
\(688\) 0 0
\(689\) −8.73577 −0.332806
\(690\) 0 0
\(691\) −36.1899 −1.37673 −0.688365 0.725365i \(-0.741671\pi\)
−0.688365 + 0.725365i \(0.741671\pi\)
\(692\) 0 0
\(693\) −2.35242 −0.0893611
\(694\) 0 0
\(695\) 2.73501 0.103745
\(696\) 0 0
\(697\) −5.77887 −0.218890
\(698\) 0 0
\(699\) 19.2425 0.727818
\(700\) 0 0
\(701\) 27.7478 1.04802 0.524010 0.851712i \(-0.324436\pi\)
0.524010 + 0.851712i \(0.324436\pi\)
\(702\) 0 0
\(703\) −32.9037 −1.24099
\(704\) 0 0
\(705\) 1.10220 0.0415111
\(706\) 0 0
\(707\) −18.2805 −0.687508
\(708\) 0 0
\(709\) 24.7540 0.929656 0.464828 0.885401i \(-0.346116\pi\)
0.464828 + 0.885401i \(0.346116\pi\)
\(710\) 0 0
\(711\) 36.6150 1.37317
\(712\) 0 0
\(713\) −5.19262 −0.194465
\(714\) 0 0
\(715\) 0.337817 0.0126336
\(716\) 0 0
\(717\) −12.6112 −0.470972
\(718\) 0 0
\(719\) −13.3846 −0.499160 −0.249580 0.968354i \(-0.580292\pi\)
−0.249580 + 0.968354i \(0.580292\pi\)
\(720\) 0 0
\(721\) −12.6336 −0.470500
\(722\) 0 0
\(723\) −8.70748 −0.323835
\(724\) 0 0
\(725\) −40.0453 −1.48725
\(726\) 0 0
\(727\) 1.06117 0.0393565 0.0196783 0.999806i \(-0.493736\pi\)
0.0196783 + 0.999806i \(0.493736\pi\)
\(728\) 0 0
\(729\) 1.95044 0.0722384
\(730\) 0 0
\(731\) −12.9300 −0.478233
\(732\) 0 0
\(733\) 23.7890 0.878665 0.439333 0.898324i \(-0.355215\pi\)
0.439333 + 0.898324i \(0.355215\pi\)
\(734\) 0 0
\(735\) −0.271849 −0.0100273
\(736\) 0 0
\(737\) 5.01726 0.184813
\(738\) 0 0
\(739\) −17.7619 −0.653383 −0.326691 0.945131i \(-0.605934\pi\)
−0.326691 + 0.945131i \(0.605934\pi\)
\(740\) 0 0
\(741\) 3.09658 0.113756
\(742\) 0 0
\(743\) −20.9218 −0.767548 −0.383774 0.923427i \(-0.625376\pi\)
−0.383774 + 0.923427i \(0.625376\pi\)
\(744\) 0 0
\(745\) −1.09266 −0.0400321
\(746\) 0 0
\(747\) −27.3330 −1.00006
\(748\) 0 0
\(749\) −1.18691 −0.0433688
\(750\) 0 0
\(751\) 20.8653 0.761386 0.380693 0.924701i \(-0.375685\pi\)
0.380693 + 0.924701i \(0.375685\pi\)
\(752\) 0 0
\(753\) 20.5903 0.750352
\(754\) 0 0
\(755\) 7.26814 0.264515
\(756\) 0 0
\(757\) 27.0153 0.981887 0.490944 0.871191i \(-0.336652\pi\)
0.490944 + 0.871191i \(0.336652\pi\)
\(758\) 0 0
\(759\) 0.951407 0.0345339
\(760\) 0 0
\(761\) 19.7456 0.715776 0.357888 0.933765i \(-0.383497\pi\)
0.357888 + 0.933765i \(0.383497\pi\)
\(762\) 0 0
\(763\) −14.4650 −0.523669
\(764\) 0 0
\(765\) −1.37083 −0.0495625
\(766\) 0 0
\(767\) 3.60840 0.130292
\(768\) 0 0
\(769\) −16.3583 −0.589895 −0.294947 0.955514i \(-0.595302\pi\)
−0.294947 + 0.955514i \(0.595302\pi\)
\(770\) 0 0
\(771\) −0.0803012 −0.00289198
\(772\) 0 0
\(773\) 12.7453 0.458417 0.229208 0.973377i \(-0.426386\pi\)
0.229208 + 0.973377i \(0.426386\pi\)
\(774\) 0 0
\(775\) −21.4590 −0.770830
\(776\) 0 0
\(777\) 6.88105 0.246856
\(778\) 0 0
\(779\) 12.8912 0.461874
\(780\) 0 0
\(781\) 0.791482 0.0283215
\(782\) 0 0
\(783\) −35.3025 −1.26161
\(784\) 0 0
\(785\) −5.12464 −0.182906
\(786\) 0 0
\(787\) 16.8399 0.600279 0.300139 0.953895i \(-0.402967\pi\)
0.300139 + 0.953895i \(0.402967\pi\)
\(788\) 0 0
\(789\) −18.0697 −0.643299
\(790\) 0 0
\(791\) 11.5775 0.411647
\(792\) 0 0
\(793\) 3.64244 0.129347
\(794\) 0 0
\(795\) 2.37481 0.0842259
\(796\) 0 0
\(797\) 10.1295 0.358804 0.179402 0.983776i \(-0.442584\pi\)
0.179402 + 0.983776i \(0.442584\pi\)
\(798\) 0 0
\(799\) 6.99388 0.247426
\(800\) 0 0
\(801\) 11.0661 0.391003
\(802\) 0 0
\(803\) −8.54964 −0.301710
\(804\) 0 0
\(805\) −0.399394 −0.0140768
\(806\) 0 0
\(807\) 25.3667 0.892949
\(808\) 0 0
\(809\) 11.6959 0.411204 0.205602 0.978636i \(-0.434085\pi\)
0.205602 + 0.978636i \(0.434085\pi\)
\(810\) 0 0
\(811\) −41.4726 −1.45630 −0.728150 0.685417i \(-0.759620\pi\)
−0.728150 + 0.685417i \(0.759620\pi\)
\(812\) 0 0
\(813\) −2.65480 −0.0931077
\(814\) 0 0
\(815\) −5.65101 −0.197946
\(816\) 0 0
\(817\) 28.8435 1.00911
\(818\) 0 0
\(819\) 2.35242 0.0822003
\(820\) 0 0
\(821\) 26.6978 0.931759 0.465880 0.884848i \(-0.345738\pi\)
0.465880 + 0.884848i \(0.345738\pi\)
\(822\) 0 0
\(823\) −26.3736 −0.919327 −0.459664 0.888093i \(-0.652030\pi\)
−0.459664 + 0.888093i \(0.652030\pi\)
\(824\) 0 0
\(825\) 3.93178 0.136887
\(826\) 0 0
\(827\) 22.0398 0.766399 0.383200 0.923666i \(-0.374822\pi\)
0.383200 + 0.923666i \(0.374822\pi\)
\(828\) 0 0
\(829\) −8.08626 −0.280848 −0.140424 0.990091i \(-0.544846\pi\)
−0.140424 + 0.990091i \(0.544846\pi\)
\(830\) 0 0
\(831\) −11.5353 −0.400156
\(832\) 0 0
\(833\) −1.72499 −0.0597674
\(834\) 0 0
\(835\) −6.30613 −0.218233
\(836\) 0 0
\(837\) −18.9175 −0.653883
\(838\) 0 0
\(839\) 44.2263 1.52686 0.763431 0.645889i \(-0.223513\pi\)
0.763431 + 0.645889i \(0.223513\pi\)
\(840\) 0 0
\(841\) 38.1765 1.31643
\(842\) 0 0
\(843\) 3.82518 0.131746
\(844\) 0 0
\(845\) −0.337817 −0.0116213
\(846\) 0 0
\(847\) −1.00000 −0.0343604
\(848\) 0 0
\(849\) 13.4487 0.461558
\(850\) 0 0
\(851\) 10.1095 0.346548
\(852\) 0 0
\(853\) 30.0636 1.02936 0.514679 0.857383i \(-0.327911\pi\)
0.514679 + 0.857383i \(0.327911\pi\)
\(854\) 0 0
\(855\) 3.05797 0.104580
\(856\) 0 0
\(857\) −7.85196 −0.268218 −0.134109 0.990967i \(-0.542817\pi\)
−0.134109 + 0.990967i \(0.542817\pi\)
\(858\) 0 0
\(859\) −42.7699 −1.45929 −0.729645 0.683826i \(-0.760315\pi\)
−0.729645 + 0.683826i \(0.760315\pi\)
\(860\) 0 0
\(861\) −2.69589 −0.0918756
\(862\) 0 0
\(863\) −1.33604 −0.0454793 −0.0227396 0.999741i \(-0.507239\pi\)
−0.0227396 + 0.999741i \(0.507239\pi\)
\(864\) 0 0
\(865\) −0.961487 −0.0326915
\(866\) 0 0
\(867\) −11.2858 −0.383284
\(868\) 0 0
\(869\) 15.5648 0.528000
\(870\) 0 0
\(871\) −5.01726 −0.170003
\(872\) 0 0
\(873\) 23.4121 0.792380
\(874\) 0 0
\(875\) −3.33962 −0.112900
\(876\) 0 0
\(877\) −30.0933 −1.01618 −0.508089 0.861305i \(-0.669648\pi\)
−0.508089 + 0.861305i \(0.669648\pi\)
\(878\) 0 0
\(879\) 18.4995 0.623974
\(880\) 0 0
\(881\) 13.2296 0.445717 0.222858 0.974851i \(-0.428461\pi\)
0.222858 + 0.974851i \(0.428461\pi\)
\(882\) 0 0
\(883\) 28.3466 0.953940 0.476970 0.878920i \(-0.341735\pi\)
0.476970 + 0.878920i \(0.341735\pi\)
\(884\) 0 0
\(885\) −0.980941 −0.0329740
\(886\) 0 0
\(887\) −12.2119 −0.410035 −0.205017 0.978758i \(-0.565725\pi\)
−0.205017 + 0.978758i \(0.565725\pi\)
\(888\) 0 0
\(889\) −19.9915 −0.670495
\(890\) 0 0
\(891\) −3.59115 −0.120308
\(892\) 0 0
\(893\) −15.6015 −0.522085
\(894\) 0 0
\(895\) −5.00926 −0.167441
\(896\) 0 0
\(897\) −0.951407 −0.0317666
\(898\) 0 0
\(899\) 35.9977 1.20059
\(900\) 0 0
\(901\) 15.0691 0.502025
\(902\) 0 0
\(903\) −6.03195 −0.200731
\(904\) 0 0
\(905\) 6.50172 0.216124
\(906\) 0 0
\(907\) −6.23015 −0.206869 −0.103434 0.994636i \(-0.532983\pi\)
−0.103434 + 0.994636i \(0.532983\pi\)
\(908\) 0 0
\(909\) −43.0034 −1.42633
\(910\) 0 0
\(911\) 28.1522 0.932725 0.466362 0.884594i \(-0.345564\pi\)
0.466362 + 0.884594i \(0.345564\pi\)
\(912\) 0 0
\(913\) −11.6191 −0.384535
\(914\) 0 0
\(915\) −0.990195 −0.0327348
\(916\) 0 0
\(917\) 4.62283 0.152659
\(918\) 0 0
\(919\) −9.87149 −0.325631 −0.162815 0.986657i \(-0.552057\pi\)
−0.162815 + 0.986657i \(0.552057\pi\)
\(920\) 0 0
\(921\) 1.81293 0.0597382
\(922\) 0 0
\(923\) −0.791482 −0.0260519
\(924\) 0 0
\(925\) 41.7783 1.37366
\(926\) 0 0
\(927\) −29.7195 −0.976118
\(928\) 0 0
\(929\) −27.9140 −0.915827 −0.457914 0.888997i \(-0.651403\pi\)
−0.457914 + 0.888997i \(0.651403\pi\)
\(930\) 0 0
\(931\) 3.84801 0.126113
\(932\) 0 0
\(933\) −15.4864 −0.507002
\(934\) 0 0
\(935\) −0.582732 −0.0190574
\(936\) 0 0
\(937\) −11.9572 −0.390624 −0.195312 0.980741i \(-0.562572\pi\)
−0.195312 + 0.980741i \(0.562572\pi\)
\(938\) 0 0
\(939\) 19.4232 0.633853
\(940\) 0 0
\(941\) −3.77722 −0.123134 −0.0615669 0.998103i \(-0.519610\pi\)
−0.0615669 + 0.998103i \(0.519610\pi\)
\(942\) 0 0
\(943\) −3.96074 −0.128979
\(944\) 0 0
\(945\) −1.45505 −0.0473328
\(946\) 0 0
\(947\) −44.1233 −1.43382 −0.716908 0.697168i \(-0.754443\pi\)
−0.716908 + 0.697168i \(0.754443\pi\)
\(948\) 0 0
\(949\) 8.54964 0.277533
\(950\) 0 0
\(951\) 4.44226 0.144050
\(952\) 0 0
\(953\) 5.94441 0.192558 0.0962791 0.995354i \(-0.469306\pi\)
0.0962791 + 0.995354i \(0.469306\pi\)
\(954\) 0 0
\(955\) −1.53253 −0.0495915
\(956\) 0 0
\(957\) −6.59561 −0.213206
\(958\) 0 0
\(959\) −17.8908 −0.577724
\(960\) 0 0
\(961\) −11.7100 −0.377741
\(962\) 0 0
\(963\) −2.79212 −0.0899747
\(964\) 0 0
\(965\) −2.35272 −0.0757368
\(966\) 0 0
\(967\) −47.4748 −1.52669 −0.763343 0.645993i \(-0.776444\pi\)
−0.763343 + 0.645993i \(0.776444\pi\)
\(968\) 0 0
\(969\) −5.34157 −0.171596
\(970\) 0 0
\(971\) 57.7191 1.85230 0.926148 0.377161i \(-0.123100\pi\)
0.926148 + 0.377161i \(0.123100\pi\)
\(972\) 0 0
\(973\) 8.09613 0.259550
\(974\) 0 0
\(975\) −3.93178 −0.125918
\(976\) 0 0
\(977\) 22.9982 0.735776 0.367888 0.929870i \(-0.380081\pi\)
0.367888 + 0.929870i \(0.380081\pi\)
\(978\) 0 0
\(979\) 4.70415 0.150345
\(980\) 0 0
\(981\) −34.0279 −1.08643
\(982\) 0 0
\(983\) 16.3095 0.520194 0.260097 0.965583i \(-0.416246\pi\)
0.260097 + 0.965583i \(0.416246\pi\)
\(984\) 0 0
\(985\) −1.03317 −0.0329196
\(986\) 0 0
\(987\) 3.26270 0.103853
\(988\) 0 0
\(989\) −8.86200 −0.281795
\(990\) 0 0
\(991\) −37.1657 −1.18061 −0.590304 0.807181i \(-0.700992\pi\)
−0.590304 + 0.807181i \(0.700992\pi\)
\(992\) 0 0
\(993\) −6.11591 −0.194083
\(994\) 0 0
\(995\) 3.76326 0.119303
\(996\) 0 0
\(997\) 30.2441 0.957840 0.478920 0.877859i \(-0.341028\pi\)
0.478920 + 0.877859i \(0.341028\pi\)
\(998\) 0 0
\(999\) 36.8303 1.16526
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 8008.2.a.z.1.9 15
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
8008.2.a.z.1.9 15 1.1 even 1 trivial