Properties

Label 8036.2.a.l.1.5
Level $8036$
Weight $2$
Character 8036.1
Self dual yes
Analytic conductor $64.168$
Analytic rank $1$
Dimension $5$
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] = [8036,2,Mod(1,8036)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8036, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("8036.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 8036 = 2^{2} \cdot 7^{2} \cdot 41 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8036.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: \(64.1677830643\)
Analytic rank: \(1\)
Dimension: \(5\)
Coefficient field: 5.5.470117.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{5} - 2x^{4} - 6x^{3} + 8x^{2} + 7x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 1148)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.5
Root \(2.78088\) of defining polynomial
Character \(\chi\) \(=\) 8036.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+2.78088 q^{3} +0.592821 q^{5} +4.73330 q^{9} +O(q^{10})\) \(q+2.78088 q^{3} +0.592821 q^{5} +4.73330 q^{9} -2.21318 q^{11} -5.00816 q^{13} +1.64856 q^{15} -4.32612 q^{17} +2.38471 q^{19} +1.67368 q^{23} -4.64856 q^{25} +4.82010 q^{27} -1.27244 q^{29} -2.49766 q^{31} -6.15458 q^{33} -4.24526 q^{37} -13.9271 q^{39} -1.00000 q^{41} -8.15665 q^{43} +2.80600 q^{45} -1.65958 q^{47} -12.0304 q^{51} -7.60486 q^{53} -1.31202 q^{55} +6.63161 q^{57} +6.12638 q^{59} -5.08395 q^{61} -2.96894 q^{65} -14.0092 q^{67} +4.65431 q^{69} -2.62939 q^{71} +16.2577 q^{73} -12.9271 q^{75} +4.77561 q^{79} -0.795771 q^{81} -11.0170 q^{83} -2.56461 q^{85} -3.53851 q^{87} -6.37164 q^{89} -6.94569 q^{93} +1.41371 q^{95} +11.9608 q^{97} -10.4756 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5 q + 2 q^{3} + q^{5} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 5 q + 2 q^{3} + q^{5} + q^{9} - 2 q^{11} + q^{13} - 9 q^{15} + 3 q^{17} + 4 q^{19} - 8 q^{23} - 6 q^{25} + 8 q^{27} - 9 q^{29} + 11 q^{31} - 5 q^{33} - 11 q^{37} - 17 q^{39} - 5 q^{41} - 27 q^{43} + 3 q^{45} + 3 q^{47} - 3 q^{51} - 19 q^{53} + 13 q^{55} - 11 q^{57} + 15 q^{59} + 7 q^{65} - 21 q^{67} - 14 q^{69} - 16 q^{71} + 10 q^{73} - 12 q^{75} - 14 q^{79} - 7 q^{81} + 2 q^{83} - 21 q^{85} + 36 q^{87} - 6 q^{89} + 17 q^{93} + 9 q^{95} - 20 q^{97} - 17 q^{99}+O(q^{100}) \) 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
\(3\) 2.78088 1.60554 0.802771 0.596287i \(-0.203358\pi\)
0.802771 + 0.596287i \(0.203358\pi\)
\(4\) 0 0
\(5\) 0.592821 0.265117 0.132559 0.991175i \(-0.457681\pi\)
0.132559 + 0.991175i \(0.457681\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) 0 0
\(9\) 4.73330 1.57777
\(10\) 0 0
\(11\) −2.21318 −0.667298 −0.333649 0.942697i \(-0.608280\pi\)
−0.333649 + 0.942697i \(0.608280\pi\)
\(12\) 0 0
\(13\) −5.00816 −1.38901 −0.694507 0.719486i \(-0.744378\pi\)
−0.694507 + 0.719486i \(0.744378\pi\)
\(14\) 0 0
\(15\) 1.64856 0.425657
\(16\) 0 0
\(17\) −4.32612 −1.04924 −0.524619 0.851337i \(-0.675792\pi\)
−0.524619 + 0.851337i \(0.675792\pi\)
\(18\) 0 0
\(19\) 2.38471 0.547091 0.273546 0.961859i \(-0.411804\pi\)
0.273546 + 0.961859i \(0.411804\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 1.67368 0.348986 0.174493 0.984658i \(-0.444171\pi\)
0.174493 + 0.984658i \(0.444171\pi\)
\(24\) 0 0
\(25\) −4.64856 −0.929713
\(26\) 0 0
\(27\) 4.82010 0.927629
\(28\) 0 0
\(29\) −1.27244 −0.236287 −0.118143 0.992997i \(-0.537694\pi\)
−0.118143 + 0.992997i \(0.537694\pi\)
\(30\) 0 0
\(31\) −2.49766 −0.448593 −0.224296 0.974521i \(-0.572008\pi\)
−0.224296 + 0.974521i \(0.572008\pi\)
\(32\) 0 0
\(33\) −6.15458 −1.07138
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −4.24526 −0.697917 −0.348958 0.937138i \(-0.613465\pi\)
−0.348958 + 0.937138i \(0.613465\pi\)
\(38\) 0 0
\(39\) −13.9271 −2.23012
\(40\) 0 0
\(41\) −1.00000 −0.156174
\(42\) 0 0
\(43\) −8.15665 −1.24388 −0.621939 0.783066i \(-0.713655\pi\)
−0.621939 + 0.783066i \(0.713655\pi\)
\(44\) 0 0
\(45\) 2.80600 0.418293
\(46\) 0 0
\(47\) −1.65958 −0.242074 −0.121037 0.992648i \(-0.538622\pi\)
−0.121037 + 0.992648i \(0.538622\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 0 0
\(51\) −12.0304 −1.68460
\(52\) 0 0
\(53\) −7.60486 −1.04461 −0.522304 0.852759i \(-0.674927\pi\)
−0.522304 + 0.852759i \(0.674927\pi\)
\(54\) 0 0
\(55\) −1.31202 −0.176912
\(56\) 0 0
\(57\) 6.63161 0.878378
\(58\) 0 0
\(59\) 6.12638 0.797586 0.398793 0.917041i \(-0.369429\pi\)
0.398793 + 0.917041i \(0.369429\pi\)
\(60\) 0 0
\(61\) −5.08395 −0.650933 −0.325467 0.945554i \(-0.605521\pi\)
−0.325467 + 0.945554i \(0.605521\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −2.96894 −0.368252
\(66\) 0 0
\(67\) −14.0092 −1.71149 −0.855747 0.517394i \(-0.826902\pi\)
−0.855747 + 0.517394i \(0.826902\pi\)
\(68\) 0 0
\(69\) 4.65431 0.560313
\(70\) 0 0
\(71\) −2.62939 −0.312051 −0.156026 0.987753i \(-0.549868\pi\)
−0.156026 + 0.987753i \(0.549868\pi\)
\(72\) 0 0
\(73\) 16.2577 1.90282 0.951410 0.307926i \(-0.0996348\pi\)
0.951410 + 0.307926i \(0.0996348\pi\)
\(74\) 0 0
\(75\) −12.9271 −1.49269
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 4.77561 0.537298 0.268649 0.963238i \(-0.413423\pi\)
0.268649 + 0.963238i \(0.413423\pi\)
\(80\) 0 0
\(81\) −0.795771 −0.0884190
\(82\) 0 0
\(83\) −11.0170 −1.20927 −0.604636 0.796502i \(-0.706681\pi\)
−0.604636 + 0.796502i \(0.706681\pi\)
\(84\) 0 0
\(85\) −2.56461 −0.278171
\(86\) 0 0
\(87\) −3.53851 −0.379368
\(88\) 0 0
\(89\) −6.37164 −0.675392 −0.337696 0.941255i \(-0.609648\pi\)
−0.337696 + 0.941255i \(0.609648\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) −6.94569 −0.720235
\(94\) 0 0
\(95\) 1.41371 0.145043
\(96\) 0 0
\(97\) 11.9608 1.21444 0.607219 0.794535i \(-0.292285\pi\)
0.607219 + 0.794535i \(0.292285\pi\)
\(98\) 0 0
\(99\) −10.4756 −1.05284
\(100\) 0 0
\(101\) 18.3506 1.82595 0.912977 0.408011i \(-0.133778\pi\)
0.912977 + 0.408011i \(0.133778\pi\)
\(102\) 0 0
\(103\) 16.0362 1.58009 0.790045 0.613048i \(-0.210057\pi\)
0.790045 + 0.613048i \(0.210057\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −10.5723 −1.02207 −0.511034 0.859561i \(-0.670737\pi\)
−0.511034 + 0.859561i \(0.670737\pi\)
\(108\) 0 0
\(109\) 7.49481 0.717872 0.358936 0.933362i \(-0.383140\pi\)
0.358936 + 0.933362i \(0.383140\pi\)
\(110\) 0 0
\(111\) −11.8056 −1.12053
\(112\) 0 0
\(113\) 12.3907 1.16562 0.582809 0.812609i \(-0.301953\pi\)
0.582809 + 0.812609i \(0.301953\pi\)
\(114\) 0 0
\(115\) 0.992192 0.0925224
\(116\) 0 0
\(117\) −23.7051 −2.19154
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) −6.10185 −0.554713
\(122\) 0 0
\(123\) −2.78088 −0.250744
\(124\) 0 0
\(125\) −5.71987 −0.511601
\(126\) 0 0
\(127\) −6.60383 −0.585995 −0.292998 0.956113i \(-0.594653\pi\)
−0.292998 + 0.956113i \(0.594653\pi\)
\(128\) 0 0
\(129\) −22.6827 −1.99710
\(130\) 0 0
\(131\) −11.3120 −0.988336 −0.494168 0.869367i \(-0.664527\pi\)
−0.494168 + 0.869367i \(0.664527\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) 2.85746 0.245931
\(136\) 0 0
\(137\) −3.12052 −0.266604 −0.133302 0.991075i \(-0.542558\pi\)
−0.133302 + 0.991075i \(0.542558\pi\)
\(138\) 0 0
\(139\) −2.85393 −0.242067 −0.121034 0.992648i \(-0.538621\pi\)
−0.121034 + 0.992648i \(0.538621\pi\)
\(140\) 0 0
\(141\) −4.61509 −0.388660
\(142\) 0 0
\(143\) 11.0840 0.926886
\(144\) 0 0
\(145\) −0.754330 −0.0626437
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 0.0502334 0.00411528 0.00205764 0.999998i \(-0.499345\pi\)
0.00205764 + 0.999998i \(0.499345\pi\)
\(150\) 0 0
\(151\) 13.6578 1.11145 0.555726 0.831366i \(-0.312440\pi\)
0.555726 + 0.831366i \(0.312440\pi\)
\(152\) 0 0
\(153\) −20.4768 −1.65545
\(154\) 0 0
\(155\) −1.48066 −0.118930
\(156\) 0 0
\(157\) 5.22014 0.416613 0.208306 0.978064i \(-0.433205\pi\)
0.208306 + 0.978064i \(0.433205\pi\)
\(158\) 0 0
\(159\) −21.1482 −1.67716
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) −9.97366 −0.781197 −0.390599 0.920561i \(-0.627732\pi\)
−0.390599 + 0.920561i \(0.627732\pi\)
\(164\) 0 0
\(165\) −3.64856 −0.284040
\(166\) 0 0
\(167\) −12.9041 −0.998545 −0.499273 0.866445i \(-0.666399\pi\)
−0.499273 + 0.866445i \(0.666399\pi\)
\(168\) 0 0
\(169\) 12.0817 0.929360
\(170\) 0 0
\(171\) 11.2876 0.863182
\(172\) 0 0
\(173\) 0.327578 0.0249053 0.0124526 0.999922i \(-0.496036\pi\)
0.0124526 + 0.999922i \(0.496036\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 17.0367 1.28056
\(178\) 0 0
\(179\) −4.70431 −0.351616 −0.175808 0.984424i \(-0.556254\pi\)
−0.175808 + 0.984424i \(0.556254\pi\)
\(180\) 0 0
\(181\) 0.993451 0.0738426 0.0369213 0.999318i \(-0.488245\pi\)
0.0369213 + 0.999318i \(0.488245\pi\)
\(182\) 0 0
\(183\) −14.1379 −1.04510
\(184\) 0 0
\(185\) −2.51668 −0.185030
\(186\) 0 0
\(187\) 9.57447 0.700155
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −14.5734 −1.05449 −0.527247 0.849712i \(-0.676776\pi\)
−0.527247 + 0.849712i \(0.676776\pi\)
\(192\) 0 0
\(193\) 10.7510 0.773875 0.386938 0.922106i \(-0.373533\pi\)
0.386938 + 0.922106i \(0.373533\pi\)
\(194\) 0 0
\(195\) −8.25627 −0.591244
\(196\) 0 0
\(197\) −16.9790 −1.20970 −0.604851 0.796339i \(-0.706767\pi\)
−0.604851 + 0.796339i \(0.706767\pi\)
\(198\) 0 0
\(199\) −10.9681 −0.777506 −0.388753 0.921342i \(-0.627094\pi\)
−0.388753 + 0.921342i \(0.627094\pi\)
\(200\) 0 0
\(201\) −38.9579 −2.74788
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) −0.592821 −0.0414044
\(206\) 0 0
\(207\) 7.92203 0.550619
\(208\) 0 0
\(209\) −5.27780 −0.365073
\(210\) 0 0
\(211\) −8.54912 −0.588545 −0.294273 0.955721i \(-0.595077\pi\)
−0.294273 + 0.955721i \(0.595077\pi\)
\(212\) 0 0
\(213\) −7.31202 −0.501011
\(214\) 0 0
\(215\) −4.83543 −0.329774
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) 45.2108 3.05506
\(220\) 0 0
\(221\) 21.6659 1.45741
\(222\) 0 0
\(223\) 6.25505 0.418869 0.209435 0.977823i \(-0.432838\pi\)
0.209435 + 0.977823i \(0.432838\pi\)
\(224\) 0 0
\(225\) −22.0030 −1.46687
\(226\) 0 0
\(227\) 19.4817 1.29305 0.646523 0.762895i \(-0.276222\pi\)
0.646523 + 0.762895i \(0.276222\pi\)
\(228\) 0 0
\(229\) −25.8356 −1.70727 −0.853633 0.520875i \(-0.825606\pi\)
−0.853633 + 0.520875i \(0.825606\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 15.3055 1.00270 0.501348 0.865246i \(-0.332838\pi\)
0.501348 + 0.865246i \(0.332838\pi\)
\(234\) 0 0
\(235\) −0.983831 −0.0641781
\(236\) 0 0
\(237\) 13.2804 0.862655
\(238\) 0 0
\(239\) 15.2062 0.983607 0.491804 0.870706i \(-0.336338\pi\)
0.491804 + 0.870706i \(0.336338\pi\)
\(240\) 0 0
\(241\) −11.4744 −0.739128 −0.369564 0.929205i \(-0.620493\pi\)
−0.369564 + 0.929205i \(0.620493\pi\)
\(242\) 0 0
\(243\) −16.6732 −1.06959
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) −11.9430 −0.759917
\(248\) 0 0
\(249\) −30.6370 −1.94154
\(250\) 0 0
\(251\) 8.21160 0.518312 0.259156 0.965835i \(-0.416556\pi\)
0.259156 + 0.965835i \(0.416556\pi\)
\(252\) 0 0
\(253\) −3.70415 −0.232878
\(254\) 0 0
\(255\) −7.13189 −0.446616
\(256\) 0 0
\(257\) −14.7648 −0.921001 −0.460500 0.887659i \(-0.652330\pi\)
−0.460500 + 0.887659i \(0.652330\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) −6.02285 −0.372805
\(262\) 0 0
\(263\) 16.4721 1.01571 0.507857 0.861441i \(-0.330438\pi\)
0.507857 + 0.861441i \(0.330438\pi\)
\(264\) 0 0
\(265\) −4.50832 −0.276944
\(266\) 0 0
\(267\) −17.7188 −1.08437
\(268\) 0 0
\(269\) 10.4540 0.637389 0.318694 0.947858i \(-0.396756\pi\)
0.318694 + 0.947858i \(0.396756\pi\)
\(270\) 0 0
\(271\) 29.4555 1.78930 0.894648 0.446771i \(-0.147426\pi\)
0.894648 + 0.446771i \(0.147426\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 10.2881 0.620396
\(276\) 0 0
\(277\) −14.5127 −0.871985 −0.435993 0.899950i \(-0.643603\pi\)
−0.435993 + 0.899950i \(0.643603\pi\)
\(278\) 0 0
\(279\) −11.8222 −0.707775
\(280\) 0 0
\(281\) 11.7868 0.703143 0.351571 0.936161i \(-0.385647\pi\)
0.351571 + 0.936161i \(0.385647\pi\)
\(282\) 0 0
\(283\) 7.22160 0.429280 0.214640 0.976693i \(-0.431142\pi\)
0.214640 + 0.976693i \(0.431142\pi\)
\(284\) 0 0
\(285\) 3.93135 0.232873
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) 1.71532 0.100901
\(290\) 0 0
\(291\) 33.2616 1.94983
\(292\) 0 0
\(293\) 27.0496 1.58026 0.790129 0.612941i \(-0.210014\pi\)
0.790129 + 0.612941i \(0.210014\pi\)
\(294\) 0 0
\(295\) 3.63184 0.211454
\(296\) 0 0
\(297\) −10.6677 −0.619005
\(298\) 0 0
\(299\) −8.38206 −0.484747
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) 51.0309 2.93165
\(304\) 0 0
\(305\) −3.01387 −0.172574
\(306\) 0 0
\(307\) 4.45012 0.253982 0.126991 0.991904i \(-0.459468\pi\)
0.126991 + 0.991904i \(0.459468\pi\)
\(308\) 0 0
\(309\) 44.5947 2.53690
\(310\) 0 0
\(311\) −8.67308 −0.491805 −0.245903 0.969295i \(-0.579084\pi\)
−0.245903 + 0.969295i \(0.579084\pi\)
\(312\) 0 0
\(313\) −3.28722 −0.185805 −0.0929023 0.995675i \(-0.529614\pi\)
−0.0929023 + 0.995675i \(0.529614\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 2.27116 0.127561 0.0637806 0.997964i \(-0.479684\pi\)
0.0637806 + 0.997964i \(0.479684\pi\)
\(318\) 0 0
\(319\) 2.81614 0.157674
\(320\) 0 0
\(321\) −29.4004 −1.64097
\(322\) 0 0
\(323\) −10.3166 −0.574029
\(324\) 0 0
\(325\) 23.2808 1.29138
\(326\) 0 0
\(327\) 20.8422 1.15257
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) −5.94977 −0.327029 −0.163514 0.986541i \(-0.552283\pi\)
−0.163514 + 0.986541i \(0.552283\pi\)
\(332\) 0 0
\(333\) −20.0941 −1.10115
\(334\) 0 0
\(335\) −8.30494 −0.453747
\(336\) 0 0
\(337\) −26.3739 −1.43668 −0.718338 0.695695i \(-0.755097\pi\)
−0.718338 + 0.695695i \(0.755097\pi\)
\(338\) 0 0
\(339\) 34.4571 1.87145
\(340\) 0 0
\(341\) 5.52776 0.299345
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) 2.75917 0.148549
\(346\) 0 0
\(347\) −8.52887 −0.457854 −0.228927 0.973444i \(-0.573522\pi\)
−0.228927 + 0.973444i \(0.573522\pi\)
\(348\) 0 0
\(349\) 9.92722 0.531392 0.265696 0.964057i \(-0.414398\pi\)
0.265696 + 0.964057i \(0.414398\pi\)
\(350\) 0 0
\(351\) −24.1398 −1.28849
\(352\) 0 0
\(353\) 29.9113 1.59202 0.796010 0.605284i \(-0.206940\pi\)
0.796010 + 0.605284i \(0.206940\pi\)
\(354\) 0 0
\(355\) −1.55876 −0.0827302
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −1.51710 −0.0800692 −0.0400346 0.999198i \(-0.512747\pi\)
−0.0400346 + 0.999198i \(0.512747\pi\)
\(360\) 0 0
\(361\) −13.3131 −0.700691
\(362\) 0 0
\(363\) −16.9685 −0.890616
\(364\) 0 0
\(365\) 9.63790 0.504471
\(366\) 0 0
\(367\) 11.5791 0.604424 0.302212 0.953241i \(-0.402275\pi\)
0.302212 + 0.953241i \(0.402275\pi\)
\(368\) 0 0
\(369\) −4.73330 −0.246406
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) 6.21111 0.321599 0.160800 0.986987i \(-0.448593\pi\)
0.160800 + 0.986987i \(0.448593\pi\)
\(374\) 0 0
\(375\) −15.9063 −0.821396
\(376\) 0 0
\(377\) 6.37260 0.328206
\(378\) 0 0
\(379\) −21.1641 −1.08713 −0.543564 0.839367i \(-0.682926\pi\)
−0.543564 + 0.839367i \(0.682926\pi\)
\(380\) 0 0
\(381\) −18.3645 −0.940841
\(382\) 0 0
\(383\) 2.18271 0.111531 0.0557655 0.998444i \(-0.482240\pi\)
0.0557655 + 0.998444i \(0.482240\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) −38.6079 −1.96255
\(388\) 0 0
\(389\) −27.1345 −1.37578 −0.687888 0.725817i \(-0.741462\pi\)
−0.687888 + 0.725817i \(0.741462\pi\)
\(390\) 0 0
\(391\) −7.24054 −0.366170
\(392\) 0 0
\(393\) −31.4574 −1.58681
\(394\) 0 0
\(395\) 2.83108 0.142447
\(396\) 0 0
\(397\) 16.6475 0.835514 0.417757 0.908559i \(-0.362816\pi\)
0.417757 + 0.908559i \(0.362816\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −12.8927 −0.643829 −0.321914 0.946769i \(-0.604326\pi\)
−0.321914 + 0.946769i \(0.604326\pi\)
\(402\) 0 0
\(403\) 12.5087 0.623102
\(404\) 0 0
\(405\) −0.471749 −0.0234414
\(406\) 0 0
\(407\) 9.39551 0.465718
\(408\) 0 0
\(409\) −6.53906 −0.323336 −0.161668 0.986845i \(-0.551687\pi\)
−0.161668 + 0.986845i \(0.551687\pi\)
\(410\) 0 0
\(411\) −8.67779 −0.428044
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) −6.53110 −0.320599
\(416\) 0 0
\(417\) −7.93645 −0.388650
\(418\) 0 0
\(419\) −5.74856 −0.280836 −0.140418 0.990092i \(-0.544845\pi\)
−0.140418 + 0.990092i \(0.544845\pi\)
\(420\) 0 0
\(421\) −6.88749 −0.335676 −0.167838 0.985815i \(-0.553679\pi\)
−0.167838 + 0.985815i \(0.553679\pi\)
\(422\) 0 0
\(423\) −7.85528 −0.381937
\(424\) 0 0
\(425\) 20.1102 0.975490
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) 30.8231 1.48816
\(430\) 0 0
\(431\) 35.7117 1.72017 0.860086 0.510148i \(-0.170410\pi\)
0.860086 + 0.510148i \(0.170410\pi\)
\(432\) 0 0
\(433\) 19.5183 0.937988 0.468994 0.883201i \(-0.344617\pi\)
0.468994 + 0.883201i \(0.344617\pi\)
\(434\) 0 0
\(435\) −2.09770 −0.100577
\(436\) 0 0
\(437\) 3.99125 0.190927
\(438\) 0 0
\(439\) 1.16465 0.0555859 0.0277930 0.999614i \(-0.491152\pi\)
0.0277930 + 0.999614i \(0.491152\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 17.7880 0.845134 0.422567 0.906332i \(-0.361129\pi\)
0.422567 + 0.906332i \(0.361129\pi\)
\(444\) 0 0
\(445\) −3.77724 −0.179058
\(446\) 0 0
\(447\) 0.139693 0.00660726
\(448\) 0 0
\(449\) −24.6447 −1.16305 −0.581527 0.813527i \(-0.697545\pi\)
−0.581527 + 0.813527i \(0.697545\pi\)
\(450\) 0 0
\(451\) 2.21318 0.104214
\(452\) 0 0
\(453\) 37.9806 1.78448
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −5.39264 −0.252257 −0.126128 0.992014i \(-0.540255\pi\)
−0.126128 + 0.992014i \(0.540255\pi\)
\(458\) 0 0
\(459\) −20.8523 −0.973304
\(460\) 0 0
\(461\) −32.1343 −1.49665 −0.748323 0.663335i \(-0.769141\pi\)
−0.748323 + 0.663335i \(0.769141\pi\)
\(462\) 0 0
\(463\) −6.34660 −0.294951 −0.147476 0.989066i \(-0.547115\pi\)
−0.147476 + 0.989066i \(0.547115\pi\)
\(464\) 0 0
\(465\) −4.11755 −0.190947
\(466\) 0 0
\(467\) 5.03372 0.232933 0.116466 0.993195i \(-0.462843\pi\)
0.116466 + 0.993195i \(0.462843\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 14.5166 0.668890
\(472\) 0 0
\(473\) 18.0521 0.830037
\(474\) 0 0
\(475\) −11.0855 −0.508638
\(476\) 0 0
\(477\) −35.9961 −1.64815
\(478\) 0 0
\(479\) −35.6160 −1.62734 −0.813668 0.581329i \(-0.802533\pi\)
−0.813668 + 0.581329i \(0.802533\pi\)
\(480\) 0 0
\(481\) 21.2610 0.969416
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 7.09062 0.321969
\(486\) 0 0
\(487\) 11.4713 0.519815 0.259908 0.965633i \(-0.416308\pi\)
0.259908 + 0.965633i \(0.416308\pi\)
\(488\) 0 0
\(489\) −27.7356 −1.25425
\(490\) 0 0
\(491\) 42.0829 1.89917 0.949587 0.313504i \(-0.101503\pi\)
0.949587 + 0.313504i \(0.101503\pi\)
\(492\) 0 0
\(493\) 5.50474 0.247921
\(494\) 0 0
\(495\) −6.21017 −0.279126
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) −42.8986 −1.92040 −0.960202 0.279306i \(-0.909896\pi\)
−0.960202 + 0.279306i \(0.909896\pi\)
\(500\) 0 0
\(501\) −35.8846 −1.60321
\(502\) 0 0
\(503\) −25.1067 −1.11945 −0.559727 0.828677i \(-0.689094\pi\)
−0.559727 + 0.828677i \(0.689094\pi\)
\(504\) 0 0
\(505\) 10.8786 0.484092
\(506\) 0 0
\(507\) 33.5977 1.49213
\(508\) 0 0
\(509\) 11.0937 0.491721 0.245861 0.969305i \(-0.420929\pi\)
0.245861 + 0.969305i \(0.420929\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) 11.4946 0.507498
\(514\) 0 0
\(515\) 9.50657 0.418910
\(516\) 0 0
\(517\) 3.67294 0.161536
\(518\) 0 0
\(519\) 0.910955 0.0399865
\(520\) 0 0
\(521\) −39.1255 −1.71412 −0.857060 0.515216i \(-0.827712\pi\)
−0.857060 + 0.515216i \(0.827712\pi\)
\(522\) 0 0
\(523\) −22.7377 −0.994250 −0.497125 0.867679i \(-0.665611\pi\)
−0.497125 + 0.867679i \(0.665611\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 10.8052 0.470681
\(528\) 0 0
\(529\) −20.1988 −0.878208
\(530\) 0 0
\(531\) 28.9980 1.25840
\(532\) 0 0
\(533\) 5.00816 0.216928
\(534\) 0 0
\(535\) −6.26750 −0.270968
\(536\) 0 0
\(537\) −13.0821 −0.564535
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) 8.82913 0.379594 0.189797 0.981823i \(-0.439217\pi\)
0.189797 + 0.981823i \(0.439217\pi\)
\(542\) 0 0
\(543\) 2.76267 0.118557
\(544\) 0 0
\(545\) 4.44308 0.190320
\(546\) 0 0
\(547\) −33.3632 −1.42651 −0.713253 0.700907i \(-0.752779\pi\)
−0.713253 + 0.700907i \(0.752779\pi\)
\(548\) 0 0
\(549\) −24.0639 −1.02702
\(550\) 0 0
\(551\) −3.03441 −0.129270
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) −6.99858 −0.297073
\(556\) 0 0
\(557\) 16.8730 0.714933 0.357467 0.933926i \(-0.383641\pi\)
0.357467 + 0.933926i \(0.383641\pi\)
\(558\) 0 0
\(559\) 40.8498 1.72776
\(560\) 0 0
\(561\) 26.6255 1.12413
\(562\) 0 0
\(563\) 23.6024 0.994723 0.497362 0.867543i \(-0.334302\pi\)
0.497362 + 0.867543i \(0.334302\pi\)
\(564\) 0 0
\(565\) 7.34546 0.309026
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 17.4601 0.731964 0.365982 0.930622i \(-0.380733\pi\)
0.365982 + 0.930622i \(0.380733\pi\)
\(570\) 0 0
\(571\) 25.4898 1.06672 0.533358 0.845890i \(-0.320930\pi\)
0.533358 + 0.845890i \(0.320930\pi\)
\(572\) 0 0
\(573\) −40.5269 −1.69304
\(574\) 0 0
\(575\) −7.78021 −0.324457
\(576\) 0 0
\(577\) 15.6392 0.651067 0.325533 0.945531i \(-0.394456\pi\)
0.325533 + 0.945531i \(0.394456\pi\)
\(578\) 0 0
\(579\) 29.8973 1.24249
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 16.8309 0.697065
\(584\) 0 0
\(585\) −14.0529 −0.581016
\(586\) 0 0
\(587\) −44.5500 −1.83878 −0.919388 0.393351i \(-0.871316\pi\)
−0.919388 + 0.393351i \(0.871316\pi\)
\(588\) 0 0
\(589\) −5.95620 −0.245421
\(590\) 0 0
\(591\) −47.2165 −1.94223
\(592\) 0 0
\(593\) 35.3296 1.45081 0.725407 0.688320i \(-0.241651\pi\)
0.725407 + 0.688320i \(0.241651\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) −30.5009 −1.24832
\(598\) 0 0
\(599\) 7.87532 0.321777 0.160888 0.986973i \(-0.448564\pi\)
0.160888 + 0.986973i \(0.448564\pi\)
\(600\) 0 0
\(601\) −2.51170 −0.102454 −0.0512272 0.998687i \(-0.516313\pi\)
−0.0512272 + 0.998687i \(0.516313\pi\)
\(602\) 0 0
\(603\) −66.3097 −2.70034
\(604\) 0 0
\(605\) −3.61730 −0.147064
\(606\) 0 0
\(607\) 13.6137 0.552563 0.276281 0.961077i \(-0.410898\pi\)
0.276281 + 0.961077i \(0.410898\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 8.31143 0.336244
\(612\) 0 0
\(613\) −25.1790 −1.01697 −0.508486 0.861070i \(-0.669795\pi\)
−0.508486 + 0.861070i \(0.669795\pi\)
\(614\) 0 0
\(615\) −1.64856 −0.0664765
\(616\) 0 0
\(617\) −0.129305 −0.00520562 −0.00260281 0.999997i \(-0.500829\pi\)
−0.00260281 + 0.999997i \(0.500829\pi\)
\(618\) 0 0
\(619\) −33.9871 −1.36606 −0.683029 0.730392i \(-0.739338\pi\)
−0.683029 + 0.730392i \(0.739338\pi\)
\(620\) 0 0
\(621\) 8.06731 0.323730
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) 19.8520 0.794079
\(626\) 0 0
\(627\) −14.6769 −0.586140
\(628\) 0 0
\(629\) 18.3655 0.732281
\(630\) 0 0
\(631\) −9.20584 −0.366479 −0.183239 0.983068i \(-0.558658\pi\)
−0.183239 + 0.983068i \(0.558658\pi\)
\(632\) 0 0
\(633\) −23.7741 −0.944935
\(634\) 0 0
\(635\) −3.91489 −0.155358
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) −12.4457 −0.492344
\(640\) 0 0
\(641\) 19.1014 0.754462 0.377231 0.926119i \(-0.376876\pi\)
0.377231 + 0.926119i \(0.376876\pi\)
\(642\) 0 0
\(643\) −37.6674 −1.48546 −0.742728 0.669593i \(-0.766469\pi\)
−0.742728 + 0.669593i \(0.766469\pi\)
\(644\) 0 0
\(645\) −13.4468 −0.529465
\(646\) 0 0
\(647\) 43.4110 1.70666 0.853331 0.521369i \(-0.174578\pi\)
0.853331 + 0.521369i \(0.174578\pi\)
\(648\) 0 0
\(649\) −13.5588 −0.532228
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −11.9271 −0.466744 −0.233372 0.972387i \(-0.574976\pi\)
−0.233372 + 0.972387i \(0.574976\pi\)
\(654\) 0 0
\(655\) −6.70600 −0.262025
\(656\) 0 0
\(657\) 76.9526 3.00221
\(658\) 0 0
\(659\) −20.7327 −0.807632 −0.403816 0.914840i \(-0.632316\pi\)
−0.403816 + 0.914840i \(0.632316\pi\)
\(660\) 0 0
\(661\) 24.4832 0.952288 0.476144 0.879367i \(-0.342034\pi\)
0.476144 + 0.879367i \(0.342034\pi\)
\(662\) 0 0
\(663\) 60.2503 2.33993
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) −2.12966 −0.0824609
\(668\) 0 0
\(669\) 17.3946 0.672512
\(670\) 0 0
\(671\) 11.2517 0.434366
\(672\) 0 0
\(673\) 29.1195 1.12247 0.561237 0.827655i \(-0.310325\pi\)
0.561237 + 0.827655i \(0.310325\pi\)
\(674\) 0 0
\(675\) −22.4065 −0.862428
\(676\) 0 0
\(677\) 27.4949 1.05671 0.528357 0.849022i \(-0.322808\pi\)
0.528357 + 0.849022i \(0.322808\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 54.1763 2.07604
\(682\) 0 0
\(683\) 43.3060 1.65706 0.828529 0.559946i \(-0.189178\pi\)
0.828529 + 0.559946i \(0.189178\pi\)
\(684\) 0 0
\(685\) −1.84991 −0.0706813
\(686\) 0 0
\(687\) −71.8458 −2.74109
\(688\) 0 0
\(689\) 38.0864 1.45097
\(690\) 0 0
\(691\) 2.51216 0.0955669 0.0477835 0.998858i \(-0.484784\pi\)
0.0477835 + 0.998858i \(0.484784\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −1.69187 −0.0641763
\(696\) 0 0
\(697\) 4.32612 0.163864
\(698\) 0 0
\(699\) 42.5627 1.60987
\(700\) 0 0
\(701\) −9.69744 −0.366267 −0.183134 0.983088i \(-0.558624\pi\)
−0.183134 + 0.983088i \(0.558624\pi\)
\(702\) 0 0
\(703\) −10.1237 −0.381824
\(704\) 0 0
\(705\) −2.73592 −0.103041
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) 26.7425 1.00433 0.502167 0.864770i \(-0.332536\pi\)
0.502167 + 0.864770i \(0.332536\pi\)
\(710\) 0 0
\(711\) 22.6044 0.847731
\(712\) 0 0
\(713\) −4.18028 −0.156553
\(714\) 0 0
\(715\) 6.57079 0.245734
\(716\) 0 0
\(717\) 42.2866 1.57922
\(718\) 0 0
\(719\) −12.6483 −0.471701 −0.235850 0.971789i \(-0.575788\pi\)
−0.235850 + 0.971789i \(0.575788\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) −31.9088 −1.18670
\(724\) 0 0
\(725\) 5.91503 0.219679
\(726\) 0 0
\(727\) 34.5407 1.28105 0.640523 0.767939i \(-0.278718\pi\)
0.640523 + 0.767939i \(0.278718\pi\)
\(728\) 0 0
\(729\) −43.9790 −1.62885
\(730\) 0 0
\(731\) 35.2866 1.30512
\(732\) 0 0
\(733\) −13.1244 −0.484759 −0.242379 0.970182i \(-0.577928\pi\)
−0.242379 + 0.970182i \(0.577928\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 31.0048 1.14208
\(738\) 0 0
\(739\) 52.7457 1.94028 0.970142 0.242539i \(-0.0779802\pi\)
0.970142 + 0.242539i \(0.0779802\pi\)
\(740\) 0 0
\(741\) −33.2122 −1.22008
\(742\) 0 0
\(743\) −8.10219 −0.297241 −0.148620 0.988894i \(-0.547483\pi\)
−0.148620 + 0.988894i \(0.547483\pi\)
\(744\) 0 0
\(745\) 0.0297794 0.00109103
\(746\) 0 0
\(747\) −52.1467 −1.90795
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) −53.2853 −1.94441 −0.972204 0.234137i \(-0.924774\pi\)
−0.972204 + 0.234137i \(0.924774\pi\)
\(752\) 0 0
\(753\) 22.8355 0.832172
\(754\) 0 0
\(755\) 8.09660 0.294665
\(756\) 0 0
\(757\) 24.0792 0.875174 0.437587 0.899176i \(-0.355833\pi\)
0.437587 + 0.899176i \(0.355833\pi\)
\(758\) 0 0
\(759\) −10.3008 −0.373896
\(760\) 0 0
\(761\) −29.2174 −1.05913 −0.529565 0.848269i \(-0.677645\pi\)
−0.529565 + 0.848269i \(0.677645\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) −12.1391 −0.438890
\(766\) 0 0
\(767\) −30.6819 −1.10786
\(768\) 0 0
\(769\) 41.5367 1.49785 0.748926 0.662653i \(-0.230570\pi\)
0.748926 + 0.662653i \(0.230570\pi\)
\(770\) 0 0
\(771\) −41.0591 −1.47871
\(772\) 0 0
\(773\) −36.4847 −1.31226 −0.656131 0.754647i \(-0.727808\pi\)
−0.656131 + 0.754647i \(0.727808\pi\)
\(774\) 0 0
\(775\) 11.6105 0.417062
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) −2.38471 −0.0854413
\(780\) 0 0
\(781\) 5.81930 0.208231
\(782\) 0 0
\(783\) −6.13330 −0.219186
\(784\) 0 0
\(785\) 3.09461 0.110451
\(786\) 0 0
\(787\) −32.1940 −1.14759 −0.573797 0.818998i \(-0.694530\pi\)
−0.573797 + 0.818998i \(0.694530\pi\)
\(788\) 0 0
\(789\) 45.8070 1.63077
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 25.4612 0.904155
\(794\) 0 0
\(795\) −12.5371 −0.444645
\(796\) 0 0
\(797\) 32.2909 1.14380 0.571901 0.820323i \(-0.306206\pi\)
0.571901 + 0.820323i \(0.306206\pi\)
\(798\) 0 0
\(799\) 7.17953 0.253994
\(800\) 0 0
\(801\) −30.1589 −1.06561
\(802\) 0 0
\(803\) −35.9812 −1.26975
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 29.0712 1.02335
\(808\) 0 0
\(809\) −20.3729 −0.716275 −0.358137 0.933669i \(-0.616588\pi\)
−0.358137 + 0.933669i \(0.616588\pi\)
\(810\) 0 0
\(811\) −18.1851 −0.638563 −0.319282 0.947660i \(-0.603442\pi\)
−0.319282 + 0.947660i \(0.603442\pi\)
\(812\) 0 0
\(813\) 81.9124 2.87279
\(814\) 0 0
\(815\) −5.91259 −0.207109
\(816\) 0 0
\(817\) −19.4513 −0.680514
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −45.3250 −1.58185 −0.790927 0.611911i \(-0.790401\pi\)
−0.790927 + 0.611911i \(0.790401\pi\)
\(822\) 0 0
\(823\) 35.9058 1.25160 0.625798 0.779985i \(-0.284773\pi\)
0.625798 + 0.779985i \(0.284773\pi\)
\(824\) 0 0
\(825\) 28.6100 0.996071
\(826\) 0 0
\(827\) −40.1662 −1.39671 −0.698357 0.715749i \(-0.746085\pi\)
−0.698357 + 0.715749i \(0.746085\pi\)
\(828\) 0 0
\(829\) −5.09462 −0.176943 −0.0884717 0.996079i \(-0.528198\pi\)
−0.0884717 + 0.996079i \(0.528198\pi\)
\(830\) 0 0
\(831\) −40.3582 −1.40001
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) −7.64979 −0.264732
\(836\) 0 0
\(837\) −12.0390 −0.416128
\(838\) 0 0
\(839\) −18.0941 −0.624677 −0.312338 0.949971i \(-0.601112\pi\)
−0.312338 + 0.949971i \(0.601112\pi\)
\(840\) 0 0
\(841\) −27.3809 −0.944169
\(842\) 0 0
\(843\) 32.7778 1.12893
\(844\) 0 0
\(845\) 7.16227 0.246390
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) 20.0824 0.689227
\(850\) 0 0
\(851\) −7.10521 −0.243563
\(852\) 0 0
\(853\) −45.5790 −1.56060 −0.780298 0.625408i \(-0.784933\pi\)
−0.780298 + 0.625408i \(0.784933\pi\)
\(854\) 0 0
\(855\) 6.69151 0.228845
\(856\) 0 0
\(857\) −53.4497 −1.82581 −0.912904 0.408174i \(-0.866166\pi\)
−0.912904 + 0.408174i \(0.866166\pi\)
\(858\) 0 0
\(859\) 3.28760 0.112171 0.0560857 0.998426i \(-0.482138\pi\)
0.0560857 + 0.998426i \(0.482138\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) −1.87675 −0.0638853 −0.0319426 0.999490i \(-0.510169\pi\)
−0.0319426 + 0.999490i \(0.510169\pi\)
\(864\) 0 0
\(865\) 0.194195 0.00660283
\(866\) 0 0
\(867\) 4.77010 0.162001
\(868\) 0 0
\(869\) −10.5693 −0.358538
\(870\) 0 0
\(871\) 70.1603 2.37729
\(872\) 0 0
\(873\) 56.6141 1.91610
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) −5.69488 −0.192302 −0.0961512 0.995367i \(-0.530653\pi\)
−0.0961512 + 0.995367i \(0.530653\pi\)
\(878\) 0 0
\(879\) 75.2218 2.53717
\(880\) 0 0
\(881\) 0.468017 0.0157679 0.00788395 0.999969i \(-0.497490\pi\)
0.00788395 + 0.999969i \(0.497490\pi\)
\(882\) 0 0
\(883\) 44.3412 1.49220 0.746100 0.665834i \(-0.231924\pi\)
0.746100 + 0.665834i \(0.231924\pi\)
\(884\) 0 0
\(885\) 10.0997 0.339498
\(886\) 0 0
\(887\) 52.6344 1.76729 0.883645 0.468158i \(-0.155082\pi\)
0.883645 + 0.468158i \(0.155082\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) 1.76118 0.0590018
\(892\) 0 0
\(893\) −3.95762 −0.132437
\(894\) 0 0
\(895\) −2.78881 −0.0932196
\(896\) 0 0
\(897\) −23.3095 −0.778282
\(898\) 0 0
\(899\) 3.17813 0.105996
\(900\) 0 0
\(901\) 32.8995 1.09604
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 0.588938 0.0195770
\(906\) 0 0
\(907\) −7.81824 −0.259600 −0.129800 0.991540i \(-0.541434\pi\)
−0.129800 + 0.991540i \(0.541434\pi\)
\(908\) 0 0
\(909\) 86.8589 2.88093
\(910\) 0 0
\(911\) 5.22863 0.173232 0.0866162 0.996242i \(-0.472395\pi\)
0.0866162 + 0.996242i \(0.472395\pi\)
\(912\) 0 0
\(913\) 24.3826 0.806945
\(914\) 0 0
\(915\) −8.38122 −0.277074
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) −18.0843 −0.596547 −0.298273 0.954480i \(-0.596411\pi\)
−0.298273 + 0.954480i \(0.596411\pi\)
\(920\) 0 0
\(921\) 12.3753 0.407779
\(922\) 0 0
\(923\) 13.1684 0.433443
\(924\) 0 0
\(925\) 19.7344 0.648862
\(926\) 0 0
\(927\) 75.9040 2.49301
\(928\) 0 0
\(929\) −40.3901 −1.32515 −0.662577 0.748994i \(-0.730537\pi\)
−0.662577 + 0.748994i \(0.730537\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 0 0
\(933\) −24.1188 −0.789614
\(934\) 0 0
\(935\) 5.67594 0.185623
\(936\) 0 0
\(937\) −1.45124 −0.0474099 −0.0237049 0.999719i \(-0.507546\pi\)
−0.0237049 + 0.999719i \(0.507546\pi\)
\(938\) 0 0
\(939\) −9.14136 −0.298317
\(940\) 0 0
\(941\) −1.03771 −0.0338282 −0.0169141 0.999857i \(-0.505384\pi\)
−0.0169141 + 0.999857i \(0.505384\pi\)
\(942\) 0 0
\(943\) −1.67368 −0.0545025
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 51.0133 1.65771 0.828854 0.559465i \(-0.188993\pi\)
0.828854 + 0.559465i \(0.188993\pi\)
\(948\) 0 0
\(949\) −81.4212 −2.64305
\(950\) 0 0
\(951\) 6.31584 0.204805
\(952\) 0 0
\(953\) 9.39299 0.304269 0.152134 0.988360i \(-0.451385\pi\)
0.152134 + 0.988360i \(0.451385\pi\)
\(954\) 0 0
\(955\) −8.63941 −0.279565
\(956\) 0 0
\(957\) 7.83135 0.253152
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) −24.7617 −0.798765
\(962\) 0 0
\(963\) −50.0421 −1.61258
\(964\) 0 0
\(965\) 6.37343 0.205168
\(966\) 0 0
\(967\) 22.1155 0.711187 0.355594 0.934641i \(-0.384279\pi\)
0.355594 + 0.934641i \(0.384279\pi\)
\(968\) 0 0
\(969\) −28.6891 −0.921628
\(970\) 0 0
\(971\) −29.0253 −0.931468 −0.465734 0.884925i \(-0.654210\pi\)
−0.465734 + 0.884925i \(0.654210\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) 64.7410 2.07337
\(976\) 0 0
\(977\) 19.4228 0.621391 0.310695 0.950510i \(-0.399438\pi\)
0.310695 + 0.950510i \(0.399438\pi\)
\(978\) 0 0
\(979\) 14.1016 0.450688
\(980\) 0 0
\(981\) 35.4752 1.13264
\(982\) 0 0
\(983\) −39.4311 −1.25766 −0.628828 0.777545i \(-0.716465\pi\)
−0.628828 + 0.777545i \(0.716465\pi\)
\(984\) 0 0
\(985\) −10.0655 −0.320713
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) −13.6516 −0.434096
\(990\) 0 0
\(991\) −24.3793 −0.774435 −0.387218 0.921988i \(-0.626564\pi\)
−0.387218 + 0.921988i \(0.626564\pi\)
\(992\) 0 0
\(993\) −16.5456 −0.525059
\(994\) 0 0
\(995\) −6.50210 −0.206130
\(996\) 0 0
\(997\) 46.3658 1.46842 0.734210 0.678923i \(-0.237553\pi\)
0.734210 + 0.678923i \(0.237553\pi\)
\(998\) 0 0
\(999\) −20.4626 −0.647408
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 8036.2.a.l.1.5 5
7.6 odd 2 1148.2.a.c.1.1 5
28.27 even 2 4592.2.a.be.1.5 5
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1148.2.a.c.1.1 5 7.6 odd 2
4592.2.a.be.1.5 5 28.27 even 2
8036.2.a.l.1.5 5 1.1 even 1 trivial