Properties

Label 324.4.a.c.1.1
Level $324$
Weight $4$
Character 324.1
Self dual yes
Analytic conductor $19.117$
Analytic rank $1$
Dimension $3$
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] = [324,4,Mod(1,324)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(324, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("324.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 324 = 2^{2} \cdot 3^{4} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 324.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: \(19.1166188419\)
Analytic rank: \(1\)
Dimension: \(3\)
Coefficient field: 3.3.1509.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 7x + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2\cdot 3^{2} \)
Twist minimal: no (minimal twist has level 36)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-2.47735\) of defining polynomial
Character \(\chi\) \(=\) 324.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-13.8439 q^{5} +30.7080 q^{7} +O(q^{10})\) \(q-13.8439 q^{5} +30.7080 q^{7} -43.9046 q^{11} -12.2371 q^{13} +76.0286 q^{17} -44.1789 q^{19} -78.6271 q^{23} +66.6530 q^{25} -92.7806 q^{29} -143.066 q^{31} -425.118 q^{35} -32.4741 q^{37} -335.555 q^{41} -498.331 q^{43} -281.765 q^{47} +599.981 q^{49} -628.565 q^{53} +607.810 q^{55} +504.654 q^{59} +371.901 q^{61} +169.408 q^{65} -162.661 q^{67} +433.512 q^{71} -629.645 q^{73} -1348.22 q^{77} +172.730 q^{79} -174.858 q^{83} -1052.53 q^{85} +336.716 q^{89} -375.775 q^{91} +611.608 q^{95} +84.3187 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 6 q^{5} + 6 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 3 q - 6 q^{5} + 6 q^{7} - 51 q^{11} - 12 q^{13} - 111 q^{17} + 15 q^{19} - 210 q^{23} + 3 q^{25} - 456 q^{29} - 48 q^{31} - 552 q^{35} - 48 q^{37} - 897 q^{41} - 129 q^{43} - 522 q^{47} + 225 q^{49} - 1104 q^{53} - 108 q^{55} - 453 q^{59} + 402 q^{61} - 1110 q^{65} + 213 q^{67} + 60 q^{71} + 375 q^{73} - 1128 q^{77} - 552 q^{79} + 612 q^{83} - 1188 q^{85} - 462 q^{89} - 132 q^{91} + 2184 q^{95} - 93 q^{97}+O(q^{100}) \) 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 0
\(4\) 0 0
\(5\) −13.8439 −1.23823 −0.619117 0.785299i \(-0.712509\pi\)
−0.619117 + 0.785299i \(0.712509\pi\)
\(6\) 0 0
\(7\) 30.7080 1.65808 0.829038 0.559192i \(-0.188889\pi\)
0.829038 + 0.559192i \(0.188889\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) −43.9046 −1.20343 −0.601715 0.798711i \(-0.705516\pi\)
−0.601715 + 0.798711i \(0.705516\pi\)
\(12\) 0 0
\(13\) −12.2371 −0.261073 −0.130536 0.991444i \(-0.541670\pi\)
−0.130536 + 0.991444i \(0.541670\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 76.0286 1.08468 0.542342 0.840158i \(-0.317538\pi\)
0.542342 + 0.840158i \(0.317538\pi\)
\(18\) 0 0
\(19\) −44.1789 −0.533439 −0.266720 0.963774i \(-0.585940\pi\)
−0.266720 + 0.963774i \(0.585940\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −78.6271 −0.712821 −0.356410 0.934330i \(-0.615999\pi\)
−0.356410 + 0.934330i \(0.615999\pi\)
\(24\) 0 0
\(25\) 66.6530 0.533224
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −92.7806 −0.594101 −0.297050 0.954862i \(-0.596003\pi\)
−0.297050 + 0.954862i \(0.596003\pi\)
\(30\) 0 0
\(31\) −143.066 −0.828884 −0.414442 0.910076i \(-0.636023\pi\)
−0.414442 + 0.910076i \(0.636023\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −425.118 −2.05309
\(36\) 0 0
\(37\) −32.4741 −0.144289 −0.0721447 0.997394i \(-0.522984\pi\)
−0.0721447 + 0.997394i \(0.522984\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −335.555 −1.27817 −0.639084 0.769137i \(-0.720686\pi\)
−0.639084 + 0.769137i \(0.720686\pi\)
\(42\) 0 0
\(43\) −498.331 −1.76732 −0.883660 0.468129i \(-0.844928\pi\)
−0.883660 + 0.468129i \(0.844928\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −281.765 −0.874461 −0.437230 0.899350i \(-0.644041\pi\)
−0.437230 + 0.899350i \(0.644041\pi\)
\(48\) 0 0
\(49\) 599.981 1.74922
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −628.565 −1.62906 −0.814529 0.580123i \(-0.803005\pi\)
−0.814529 + 0.580123i \(0.803005\pi\)
\(54\) 0 0
\(55\) 607.810 1.49013
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 504.654 1.11356 0.556782 0.830658i \(-0.312036\pi\)
0.556782 + 0.830658i \(0.312036\pi\)
\(60\) 0 0
\(61\) 371.901 0.780607 0.390304 0.920686i \(-0.372370\pi\)
0.390304 + 0.920686i \(0.372370\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 169.408 0.323269
\(66\) 0 0
\(67\) −162.661 −0.296600 −0.148300 0.988942i \(-0.547380\pi\)
−0.148300 + 0.988942i \(0.547380\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 433.512 0.724626 0.362313 0.932056i \(-0.381987\pi\)
0.362313 + 0.932056i \(0.381987\pi\)
\(72\) 0 0
\(73\) −629.645 −1.00951 −0.504756 0.863262i \(-0.668418\pi\)
−0.504756 + 0.863262i \(0.668418\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −1348.22 −1.99538
\(78\) 0 0
\(79\) 172.730 0.245995 0.122998 0.992407i \(-0.460749\pi\)
0.122998 + 0.992407i \(0.460749\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) −174.858 −0.231242 −0.115621 0.993293i \(-0.536886\pi\)
−0.115621 + 0.993293i \(0.536886\pi\)
\(84\) 0 0
\(85\) −1052.53 −1.34309
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 336.716 0.401032 0.200516 0.979690i \(-0.435738\pi\)
0.200516 + 0.979690i \(0.435738\pi\)
\(90\) 0 0
\(91\) −375.775 −0.432879
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 611.608 0.660523
\(96\) 0 0
\(97\) 84.3187 0.0882605 0.0441303 0.999026i \(-0.485948\pi\)
0.0441303 + 0.999026i \(0.485948\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) −1749.80 −1.72388 −0.861941 0.507008i \(-0.830751\pi\)
−0.861941 + 0.507008i \(0.830751\pi\)
\(102\) 0 0
\(103\) 111.396 0.106564 0.0532822 0.998579i \(-0.483032\pi\)
0.0532822 + 0.998579i \(0.483032\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 895.520 0.809095 0.404548 0.914517i \(-0.367429\pi\)
0.404548 + 0.914517i \(0.367429\pi\)
\(108\) 0 0
\(109\) −716.957 −0.630019 −0.315009 0.949089i \(-0.602008\pi\)
−0.315009 + 0.949089i \(0.602008\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 231.051 0.192349 0.0961746 0.995364i \(-0.469339\pi\)
0.0961746 + 0.995364i \(0.469339\pi\)
\(114\) 0 0
\(115\) 1088.50 0.882639
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 2334.68 1.79849
\(120\) 0 0
\(121\) 596.612 0.448244
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 807.748 0.577978
\(126\) 0 0
\(127\) 1715.22 1.19843 0.599217 0.800586i \(-0.295478\pi\)
0.599217 + 0.800586i \(0.295478\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 1332.87 0.888957 0.444479 0.895789i \(-0.353389\pi\)
0.444479 + 0.895789i \(0.353389\pi\)
\(132\) 0 0
\(133\) −1356.65 −0.884483
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 2518.98 1.57089 0.785443 0.618935i \(-0.212435\pi\)
0.785443 + 0.618935i \(0.212435\pi\)
\(138\) 0 0
\(139\) 622.883 0.380088 0.190044 0.981776i \(-0.439137\pi\)
0.190044 + 0.981776i \(0.439137\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 537.263 0.314183
\(144\) 0 0
\(145\) 1284.44 0.735636
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 2408.95 1.32449 0.662243 0.749289i \(-0.269604\pi\)
0.662243 + 0.749289i \(0.269604\pi\)
\(150\) 0 0
\(151\) 67.6436 0.0364553 0.0182277 0.999834i \(-0.494198\pi\)
0.0182277 + 0.999834i \(0.494198\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 1980.59 1.02635
\(156\) 0 0
\(157\) −4.39352 −0.00223338 −0.00111669 0.999999i \(-0.500355\pi\)
−0.00111669 + 0.999999i \(0.500355\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) −2414.48 −1.18191
\(162\) 0 0
\(163\) −2863.88 −1.37617 −0.688087 0.725628i \(-0.741549\pi\)
−0.688087 + 0.725628i \(0.741549\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 429.514 0.199023 0.0995114 0.995036i \(-0.468272\pi\)
0.0995114 + 0.995036i \(0.468272\pi\)
\(168\) 0 0
\(169\) −2047.25 −0.931841
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) −2574.29 −1.13133 −0.565663 0.824637i \(-0.691380\pi\)
−0.565663 + 0.824637i \(0.691380\pi\)
\(174\) 0 0
\(175\) 2046.78 0.884126
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 807.448 0.337159 0.168580 0.985688i \(-0.446082\pi\)
0.168580 + 0.985688i \(0.446082\pi\)
\(180\) 0 0
\(181\) 4296.57 1.76443 0.882215 0.470847i \(-0.156052\pi\)
0.882215 + 0.470847i \(0.156052\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 449.568 0.178664
\(186\) 0 0
\(187\) −3338.00 −1.30534
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −306.295 −0.116035 −0.0580176 0.998316i \(-0.518478\pi\)
−0.0580176 + 0.998316i \(0.518478\pi\)
\(192\) 0 0
\(193\) −1712.35 −0.638642 −0.319321 0.947647i \(-0.603455\pi\)
−0.319321 + 0.947647i \(0.603455\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −263.636 −0.0953466 −0.0476733 0.998863i \(-0.515181\pi\)
−0.0476733 + 0.998863i \(0.515181\pi\)
\(198\) 0 0
\(199\) −3835.87 −1.36642 −0.683211 0.730221i \(-0.739417\pi\)
−0.683211 + 0.730221i \(0.739417\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) −2849.11 −0.985064
\(204\) 0 0
\(205\) 4645.38 1.58267
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 1939.66 0.641957
\(210\) 0 0
\(211\) −4069.32 −1.32770 −0.663848 0.747867i \(-0.731078\pi\)
−0.663848 + 0.747867i \(0.731078\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 6898.84 2.18836
\(216\) 0 0
\(217\) −4393.27 −1.37435
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) −930.365 −0.283182
\(222\) 0 0
\(223\) 6156.14 1.84864 0.924318 0.381622i \(-0.124635\pi\)
0.924318 + 0.381622i \(0.124635\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 24.2614 0.00709378 0.00354689 0.999994i \(-0.498871\pi\)
0.00354689 + 0.999994i \(0.498871\pi\)
\(228\) 0 0
\(229\) −632.419 −0.182495 −0.0912476 0.995828i \(-0.529085\pi\)
−0.0912476 + 0.995828i \(0.529085\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −3163.59 −0.889502 −0.444751 0.895654i \(-0.646708\pi\)
−0.444751 + 0.895654i \(0.646708\pi\)
\(234\) 0 0
\(235\) 3900.72 1.08279
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 2221.15 0.601147 0.300573 0.953759i \(-0.402822\pi\)
0.300573 + 0.953759i \(0.402822\pi\)
\(240\) 0 0
\(241\) −601.459 −0.160761 −0.0803805 0.996764i \(-0.525614\pi\)
−0.0803805 + 0.996764i \(0.525614\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) −8306.07 −2.16594
\(246\) 0 0
\(247\) 540.620 0.139266
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 1350.71 0.339665 0.169833 0.985473i \(-0.445677\pi\)
0.169833 + 0.985473i \(0.445677\pi\)
\(252\) 0 0
\(253\) 3452.09 0.857830
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −7043.91 −1.70968 −0.854838 0.518894i \(-0.826344\pi\)
−0.854838 + 0.518894i \(0.826344\pi\)
\(258\) 0 0
\(259\) −997.215 −0.239243
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 2324.26 0.544944 0.272472 0.962164i \(-0.412159\pi\)
0.272472 + 0.962164i \(0.412159\pi\)
\(264\) 0 0
\(265\) 8701.78 2.01716
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) −3675.87 −0.833167 −0.416584 0.909097i \(-0.636773\pi\)
−0.416584 + 0.909097i \(0.636773\pi\)
\(270\) 0 0
\(271\) −1881.48 −0.421741 −0.210871 0.977514i \(-0.567630\pi\)
−0.210871 + 0.977514i \(0.567630\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −2926.37 −0.641698
\(276\) 0 0
\(277\) 1464.02 0.317560 0.158780 0.987314i \(-0.449244\pi\)
0.158780 + 0.987314i \(0.449244\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) −8232.56 −1.74773 −0.873867 0.486165i \(-0.838395\pi\)
−0.873867 + 0.486165i \(0.838395\pi\)
\(282\) 0 0
\(283\) 1688.73 0.354715 0.177358 0.984146i \(-0.443245\pi\)
0.177358 + 0.984146i \(0.443245\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) −10304.2 −2.11930
\(288\) 0 0
\(289\) 867.343 0.176540
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 511.732 0.102033 0.0510166 0.998698i \(-0.483754\pi\)
0.0510166 + 0.998698i \(0.483754\pi\)
\(294\) 0 0
\(295\) −6986.37 −1.37885
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 962.163 0.186098
\(300\) 0 0
\(301\) −15302.7 −2.93035
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) −5148.55 −0.966575
\(306\) 0 0
\(307\) −3171.98 −0.589690 −0.294845 0.955545i \(-0.595268\pi\)
−0.294845 + 0.955545i \(0.595268\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 5060.78 0.922735 0.461367 0.887209i \(-0.347359\pi\)
0.461367 + 0.887209i \(0.347359\pi\)
\(312\) 0 0
\(313\) −9465.84 −1.70940 −0.854698 0.519125i \(-0.826258\pi\)
−0.854698 + 0.519125i \(0.826258\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 6417.13 1.13698 0.568489 0.822691i \(-0.307528\pi\)
0.568489 + 0.822691i \(0.307528\pi\)
\(318\) 0 0
\(319\) 4073.49 0.714959
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) −3358.86 −0.578613
\(324\) 0 0
\(325\) −815.637 −0.139210
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) −8652.44 −1.44992
\(330\) 0 0
\(331\) 6664.74 1.10673 0.553364 0.832940i \(-0.313344\pi\)
0.553364 + 0.832940i \(0.313344\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 2251.86 0.367260
\(336\) 0 0
\(337\) 10488.1 1.69532 0.847660 0.530540i \(-0.178011\pi\)
0.847660 + 0.530540i \(0.178011\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 6281.25 0.997503
\(342\) 0 0
\(343\) 7891.37 1.24226
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 2148.96 0.332455 0.166228 0.986087i \(-0.446841\pi\)
0.166228 + 0.986087i \(0.446841\pi\)
\(348\) 0 0
\(349\) −2777.06 −0.425939 −0.212969 0.977059i \(-0.568313\pi\)
−0.212969 + 0.977059i \(0.568313\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 7797.44 1.17568 0.587841 0.808976i \(-0.299978\pi\)
0.587841 + 0.808976i \(0.299978\pi\)
\(354\) 0 0
\(355\) −6001.49 −0.897257
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 2309.31 0.339500 0.169750 0.985487i \(-0.445704\pi\)
0.169750 + 0.985487i \(0.445704\pi\)
\(360\) 0 0
\(361\) −4907.22 −0.715443
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 8716.73 1.25001
\(366\) 0 0
\(367\) −6211.32 −0.883455 −0.441728 0.897149i \(-0.645634\pi\)
−0.441728 + 0.897149i \(0.645634\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) −19302.0 −2.70110
\(372\) 0 0
\(373\) 8699.22 1.20758 0.603792 0.797142i \(-0.293656\pi\)
0.603792 + 0.797142i \(0.293656\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 1135.36 0.155104
\(378\) 0 0
\(379\) 5954.77 0.807061 0.403531 0.914966i \(-0.367783\pi\)
0.403531 + 0.914966i \(0.367783\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 4256.70 0.567904 0.283952 0.958839i \(-0.408354\pi\)
0.283952 + 0.958839i \(0.408354\pi\)
\(384\) 0 0
\(385\) 18664.6 2.47075
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 1602.04 0.208809 0.104404 0.994535i \(-0.466706\pi\)
0.104404 + 0.994535i \(0.466706\pi\)
\(390\) 0 0
\(391\) −5977.90 −0.773186
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) −2391.25 −0.304600
\(396\) 0 0
\(397\) −7987.87 −1.00982 −0.504912 0.863171i \(-0.668475\pi\)
−0.504912 + 0.863171i \(0.668475\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −7069.49 −0.880383 −0.440191 0.897904i \(-0.645089\pi\)
−0.440191 + 0.897904i \(0.645089\pi\)
\(402\) 0 0
\(403\) 1750.70 0.216399
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 1425.76 0.173642
\(408\) 0 0
\(409\) −11864.8 −1.43441 −0.717207 0.696861i \(-0.754580\pi\)
−0.717207 + 0.696861i \(0.754580\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 15496.9 1.84637
\(414\) 0 0
\(415\) 2420.71 0.286332
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) −12342.6 −1.43908 −0.719541 0.694450i \(-0.755648\pi\)
−0.719541 + 0.694450i \(0.755648\pi\)
\(420\) 0 0
\(421\) 5465.86 0.632755 0.316377 0.948633i \(-0.397533\pi\)
0.316377 + 0.948633i \(0.397533\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 5067.54 0.578380
\(426\) 0 0
\(427\) 11420.3 1.29431
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 8997.67 1.00557 0.502787 0.864410i \(-0.332308\pi\)
0.502787 + 0.864410i \(0.332308\pi\)
\(432\) 0 0
\(433\) −10967.4 −1.21723 −0.608615 0.793466i \(-0.708275\pi\)
−0.608615 + 0.793466i \(0.708275\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 3473.66 0.380246
\(438\) 0 0
\(439\) 7127.02 0.774838 0.387419 0.921904i \(-0.373367\pi\)
0.387419 + 0.921904i \(0.373367\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 8868.55 0.951146 0.475573 0.879676i \(-0.342241\pi\)
0.475573 + 0.879676i \(0.342241\pi\)
\(444\) 0 0
\(445\) −4661.46 −0.496571
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 9835.05 1.03373 0.516865 0.856067i \(-0.327099\pi\)
0.516865 + 0.856067i \(0.327099\pi\)
\(450\) 0 0
\(451\) 14732.4 1.53819
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 5202.19 0.536005
\(456\) 0 0
\(457\) −8886.49 −0.909612 −0.454806 0.890591i \(-0.650291\pi\)
−0.454806 + 0.890591i \(0.650291\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 1865.79 0.188500 0.0942499 0.995549i \(-0.469955\pi\)
0.0942499 + 0.995549i \(0.469955\pi\)
\(462\) 0 0
\(463\) −13623.4 −1.36746 −0.683729 0.729736i \(-0.739643\pi\)
−0.683729 + 0.729736i \(0.739643\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −9328.97 −0.924397 −0.462199 0.886776i \(-0.652939\pi\)
−0.462199 + 0.886776i \(0.652939\pi\)
\(468\) 0 0
\(469\) −4994.99 −0.491785
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 21879.0 2.12685
\(474\) 0 0
\(475\) −2944.66 −0.284443
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 2660.30 0.253762 0.126881 0.991918i \(-0.459503\pi\)
0.126881 + 0.991918i \(0.459503\pi\)
\(480\) 0 0
\(481\) 397.387 0.0376701
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) −1167.30 −0.109287
\(486\) 0 0
\(487\) −20450.7 −1.90289 −0.951447 0.307813i \(-0.900403\pi\)
−0.951447 + 0.307813i \(0.900403\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 20654.9 1.89846 0.949228 0.314588i \(-0.101866\pi\)
0.949228 + 0.314588i \(0.101866\pi\)
\(492\) 0 0
\(493\) −7053.98 −0.644412
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 13312.3 1.20149
\(498\) 0 0
\(499\) 17466.6 1.56696 0.783480 0.621417i \(-0.213443\pi\)
0.783480 + 0.621417i \(0.213443\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 19951.2 1.76855 0.884275 0.466966i \(-0.154653\pi\)
0.884275 + 0.466966i \(0.154653\pi\)
\(504\) 0 0
\(505\) 24224.1 2.13457
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) −7277.77 −0.633755 −0.316878 0.948466i \(-0.602634\pi\)
−0.316878 + 0.948466i \(0.602634\pi\)
\(510\) 0 0
\(511\) −19335.1 −1.67385
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) −1542.15 −0.131952
\(516\) 0 0
\(517\) 12370.8 1.05235
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) −11664.5 −0.980866 −0.490433 0.871479i \(-0.663161\pi\)
−0.490433 + 0.871479i \(0.663161\pi\)
\(522\) 0 0
\(523\) −8925.51 −0.746243 −0.373122 0.927782i \(-0.621713\pi\)
−0.373122 + 0.927782i \(0.621713\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −10877.1 −0.899077
\(528\) 0 0
\(529\) −5984.78 −0.491887
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 4106.20 0.333695
\(534\) 0 0
\(535\) −12397.5 −1.00185
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) −26341.9 −2.10506
\(540\) 0 0
\(541\) 12334.1 0.980193 0.490097 0.871668i \(-0.336962\pi\)
0.490097 + 0.871668i \(0.336962\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 9925.47 0.780111
\(546\) 0 0
\(547\) 4569.63 0.357190 0.178595 0.983923i \(-0.442845\pi\)
0.178595 + 0.983923i \(0.442845\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 4098.95 0.316917
\(552\) 0 0
\(553\) 5304.19 0.407879
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −8154.40 −0.620310 −0.310155 0.950686i \(-0.600381\pi\)
−0.310155 + 0.950686i \(0.600381\pi\)
\(558\) 0 0
\(559\) 6098.10 0.461399
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) −16173.6 −1.21072 −0.605362 0.795950i \(-0.706972\pi\)
−0.605362 + 0.795950i \(0.706972\pi\)
\(564\) 0 0
\(565\) −3198.65 −0.238174
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −9097.53 −0.670278 −0.335139 0.942169i \(-0.608783\pi\)
−0.335139 + 0.942169i \(0.608783\pi\)
\(570\) 0 0
\(571\) −3295.09 −0.241498 −0.120749 0.992683i \(-0.538530\pi\)
−0.120749 + 0.992683i \(0.538530\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) −5240.73 −0.380093
\(576\) 0 0
\(577\) 15544.7 1.12155 0.560775 0.827968i \(-0.310503\pi\)
0.560775 + 0.827968i \(0.310503\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) −5369.52 −0.383417
\(582\) 0 0
\(583\) 27596.9 1.96046
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −4688.66 −0.329680 −0.164840 0.986320i \(-0.552711\pi\)
−0.164840 + 0.986320i \(0.552711\pi\)
\(588\) 0 0
\(589\) 6320.50 0.442159
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 16637.6 1.15215 0.576075 0.817397i \(-0.304584\pi\)
0.576075 + 0.817397i \(0.304584\pi\)
\(594\) 0 0
\(595\) −32321.1 −2.22695
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) −5867.46 −0.400230 −0.200115 0.979772i \(-0.564132\pi\)
−0.200115 + 0.979772i \(0.564132\pi\)
\(600\) 0 0
\(601\) 20675.8 1.40330 0.701649 0.712522i \(-0.252447\pi\)
0.701649 + 0.712522i \(0.252447\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) −8259.43 −0.555031
\(606\) 0 0
\(607\) 12607.7 0.843051 0.421525 0.906817i \(-0.361495\pi\)
0.421525 + 0.906817i \(0.361495\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 3447.97 0.228298
\(612\) 0 0
\(613\) 12469.7 0.821608 0.410804 0.911724i \(-0.365248\pi\)
0.410804 + 0.911724i \(0.365248\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 18587.0 1.21278 0.606388 0.795169i \(-0.292618\pi\)
0.606388 + 0.795169i \(0.292618\pi\)
\(618\) 0 0
\(619\) −11945.9 −0.775682 −0.387841 0.921726i \(-0.626779\pi\)
−0.387841 + 0.921726i \(0.626779\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 10339.9 0.664941
\(624\) 0 0
\(625\) −19514.0 −1.24890
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) −2468.96 −0.156509
\(630\) 0 0
\(631\) 4285.35 0.270360 0.135180 0.990821i \(-0.456839\pi\)
0.135180 + 0.990821i \(0.456839\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) −23745.3 −1.48394
\(636\) 0 0
\(637\) −7342.00 −0.456673
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) −21758.2 −1.34071 −0.670357 0.742039i \(-0.733859\pi\)
−0.670357 + 0.742039i \(0.733859\pi\)
\(642\) 0 0
\(643\) −7343.09 −0.450363 −0.225181 0.974317i \(-0.572297\pi\)
−0.225181 + 0.974317i \(0.572297\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 9483.24 0.576236 0.288118 0.957595i \(-0.406970\pi\)
0.288118 + 0.957595i \(0.406970\pi\)
\(648\) 0 0
\(649\) −22156.6 −1.34010
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 4784.18 0.286707 0.143353 0.989672i \(-0.454211\pi\)
0.143353 + 0.989672i \(0.454211\pi\)
\(654\) 0 0
\(655\) −18452.1 −1.10074
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) −32067.9 −1.89558 −0.947791 0.318892i \(-0.896689\pi\)
−0.947791 + 0.318892i \(0.896689\pi\)
\(660\) 0 0
\(661\) −7525.28 −0.442813 −0.221407 0.975182i \(-0.571065\pi\)
−0.221407 + 0.975182i \(0.571065\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 18781.3 1.09520
\(666\) 0 0
\(667\) 7295.07 0.423487
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) −16328.2 −0.939406
\(672\) 0 0
\(673\) −7341.95 −0.420522 −0.210261 0.977645i \(-0.567431\pi\)
−0.210261 + 0.977645i \(0.567431\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 9521.03 0.540507 0.270253 0.962789i \(-0.412893\pi\)
0.270253 + 0.962789i \(0.412893\pi\)
\(678\) 0 0
\(679\) 2589.26 0.146343
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) −1475.06 −0.0826377 −0.0413188 0.999146i \(-0.513156\pi\)
−0.0413188 + 0.999146i \(0.513156\pi\)
\(684\) 0 0
\(685\) −34872.5 −1.94512
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 7691.79 0.425303
\(690\) 0 0
\(691\) 34160.7 1.88066 0.940329 0.340267i \(-0.110518\pi\)
0.940329 + 0.340267i \(0.110518\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −8623.12 −0.470638
\(696\) 0 0
\(697\) −25511.8 −1.38641
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) −29427.2 −1.58552 −0.792759 0.609535i \(-0.791356\pi\)
−0.792759 + 0.609535i \(0.791356\pi\)
\(702\) 0 0
\(703\) 1434.67 0.0769696
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) −53733.0 −2.85833
\(708\) 0 0
\(709\) 19692.9 1.04313 0.521566 0.853211i \(-0.325348\pi\)
0.521566 + 0.853211i \(0.325348\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 11248.8 0.590845
\(714\) 0 0
\(715\) −7437.80 −0.389032
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) −33746.9 −1.75041 −0.875206 0.483751i \(-0.839274\pi\)
−0.875206 + 0.483751i \(0.839274\pi\)
\(720\) 0 0
\(721\) 3420.74 0.176692
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) −6184.11 −0.316789
\(726\) 0 0
\(727\) 20760.8 1.05912 0.529558 0.848274i \(-0.322358\pi\)
0.529558 + 0.848274i \(0.322358\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) −37887.4 −1.91698
\(732\) 0 0
\(733\) 14244.0 0.717757 0.358878 0.933384i \(-0.383159\pi\)
0.358878 + 0.933384i \(0.383159\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 7141.55 0.356937
\(738\) 0 0
\(739\) 27490.2 1.36839 0.684196 0.729298i \(-0.260153\pi\)
0.684196 + 0.729298i \(0.260153\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 36468.1 1.80065 0.900326 0.435217i \(-0.143328\pi\)
0.900326 + 0.435217i \(0.143328\pi\)
\(744\) 0 0
\(745\) −33349.2 −1.64003
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 27499.6 1.34154
\(750\) 0 0
\(751\) 38683.1 1.87958 0.939791 0.341751i \(-0.111020\pi\)
0.939791 + 0.341751i \(0.111020\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) −936.449 −0.0451402
\(756\) 0 0
\(757\) 37768.4 1.81337 0.906683 0.421813i \(-0.138606\pi\)
0.906683 + 0.421813i \(0.138606\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −20813.2 −0.991429 −0.495714 0.868486i \(-0.665094\pi\)
−0.495714 + 0.868486i \(0.665094\pi\)
\(762\) 0 0
\(763\) −22016.3 −1.04462
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) −6175.47 −0.290722
\(768\) 0 0
\(769\) 6121.70 0.287067 0.143533 0.989645i \(-0.454154\pi\)
0.143533 + 0.989645i \(0.454154\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) −34251.2 −1.59370 −0.796850 0.604177i \(-0.793502\pi\)
−0.796850 + 0.604177i \(0.793502\pi\)
\(774\) 0 0
\(775\) −9535.78 −0.441981
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 14824.5 0.681825
\(780\) 0 0
\(781\) −19033.2 −0.872037
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 60.8234 0.00276545
\(786\) 0 0
\(787\) −9122.68 −0.413200 −0.206600 0.978425i \(-0.566240\pi\)
−0.206600 + 0.978425i \(0.566240\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 7095.12 0.318930
\(792\) 0 0
\(793\) −4550.97 −0.203795
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −10012.2 −0.444983 −0.222492 0.974935i \(-0.571419\pi\)
−0.222492 + 0.974935i \(0.571419\pi\)
\(798\) 0 0
\(799\) −21422.2 −0.948514
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 27644.3 1.21488
\(804\) 0 0
\(805\) 33425.8 1.46348
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 22375.5 0.972413 0.486207 0.873844i \(-0.338380\pi\)
0.486207 + 0.873844i \(0.338380\pi\)
\(810\) 0 0
\(811\) 970.867 0.0420367 0.0210183 0.999779i \(-0.493309\pi\)
0.0210183 + 0.999779i \(0.493309\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 39647.2 1.70403
\(816\) 0 0
\(817\) 22015.7 0.942758
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 3334.40 0.141743 0.0708717 0.997485i \(-0.477422\pi\)
0.0708717 + 0.997485i \(0.477422\pi\)
\(822\) 0 0
\(823\) 3714.29 0.157317 0.0786585 0.996902i \(-0.474936\pi\)
0.0786585 + 0.996902i \(0.474936\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −30744.8 −1.29275 −0.646374 0.763020i \(-0.723716\pi\)
−0.646374 + 0.763020i \(0.723716\pi\)
\(828\) 0 0
\(829\) −29789.6 −1.24805 −0.624026 0.781404i \(-0.714504\pi\)
−0.624026 + 0.781404i \(0.714504\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 45615.7 1.89735
\(834\) 0 0
\(835\) −5946.14 −0.246437
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 32304.0 1.32927 0.664635 0.747168i \(-0.268587\pi\)
0.664635 + 0.747168i \(0.268587\pi\)
\(840\) 0 0
\(841\) −15780.8 −0.647044
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 28341.9 1.15384
\(846\) 0 0
\(847\) 18320.8 0.743222
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 2553.34 0.102853
\(852\) 0 0
\(853\) −17420.6 −0.699260 −0.349630 0.936888i \(-0.613693\pi\)
−0.349630 + 0.936888i \(0.613693\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −34483.8 −1.37450 −0.687248 0.726423i \(-0.741182\pi\)
−0.687248 + 0.726423i \(0.741182\pi\)
\(858\) 0 0
\(859\) −15174.4 −0.602730 −0.301365 0.953509i \(-0.597442\pi\)
−0.301365 + 0.953509i \(0.597442\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 19314.7 0.761852 0.380926 0.924605i \(-0.375605\pi\)
0.380926 + 0.924605i \(0.375605\pi\)
\(864\) 0 0
\(865\) 35638.1 1.40085
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) −7583.63 −0.296038
\(870\) 0 0
\(871\) 1990.49 0.0774341
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 24804.3 0.958331
\(876\) 0 0
\(877\) −13744.8 −0.529222 −0.264611 0.964355i \(-0.585244\pi\)
−0.264611 + 0.964355i \(0.585244\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) −13269.0 −0.507429 −0.253714 0.967279i \(-0.581652\pi\)
−0.253714 + 0.967279i \(0.581652\pi\)
\(882\) 0 0
\(883\) −5785.26 −0.220487 −0.110243 0.993905i \(-0.535163\pi\)
−0.110243 + 0.993905i \(0.535163\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 1338.82 0.0506802 0.0253401 0.999679i \(-0.491933\pi\)
0.0253401 + 0.999679i \(0.491933\pi\)
\(888\) 0 0
\(889\) 52671.0 1.98710
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 12448.1 0.466471
\(894\) 0 0
\(895\) −11178.2 −0.417482
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 13273.7 0.492440
\(900\) 0 0
\(901\) −47788.9 −1.76701
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) −59481.3 −2.18478
\(906\) 0 0
\(907\) −13647.5 −0.499622 −0.249811 0.968295i \(-0.580368\pi\)
−0.249811 + 0.968295i \(0.580368\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 39201.3 1.42568 0.712842 0.701325i \(-0.247408\pi\)
0.712842 + 0.701325i \(0.247408\pi\)
\(912\) 0 0
\(913\) 7677.05 0.278284
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 40929.8 1.47396
\(918\) 0 0
\(919\) −4733.10 −0.169892 −0.0849459 0.996386i \(-0.527072\pi\)
−0.0849459 + 0.996386i \(0.527072\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) −5304.91 −0.189180
\(924\) 0 0
\(925\) −2164.50 −0.0769386
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) −29332.8 −1.03593 −0.517964 0.855402i \(-0.673310\pi\)
−0.517964 + 0.855402i \(0.673310\pi\)
\(930\) 0 0
\(931\) −26506.5 −0.933100
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 46210.9 1.61632
\(936\) 0 0
\(937\) −39028.8 −1.36074 −0.680371 0.732867i \(-0.738182\pi\)
−0.680371 + 0.732867i \(0.738182\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) −26632.5 −0.922629 −0.461314 0.887237i \(-0.652622\pi\)
−0.461314 + 0.887237i \(0.652622\pi\)
\(942\) 0 0
\(943\) 26383.7 0.911105
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −51212.3 −1.75731 −0.878657 0.477454i \(-0.841560\pi\)
−0.878657 + 0.477454i \(0.841560\pi\)
\(948\) 0 0
\(949\) 7705.00 0.263556
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) −6350.46 −0.215857 −0.107928 0.994159i \(-0.534422\pi\)
−0.107928 + 0.994159i \(0.534422\pi\)
\(954\) 0 0
\(955\) 4240.31 0.143679
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 77352.9 2.60465
\(960\) 0 0
\(961\) −9323.16 −0.312952
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 23705.6 0.790788
\(966\) 0 0
\(967\) −39546.1 −1.31512 −0.657558 0.753404i \(-0.728411\pi\)
−0.657558 + 0.753404i \(0.728411\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) −21043.4 −0.695483 −0.347742 0.937590i \(-0.613051\pi\)
−0.347742 + 0.937590i \(0.613051\pi\)
\(972\) 0 0
\(973\) 19127.5 0.630215
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 28818.1 0.943677 0.471838 0.881685i \(-0.343591\pi\)
0.471838 + 0.881685i \(0.343591\pi\)
\(978\) 0 0
\(979\) −14783.4 −0.482614
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) −29223.7 −0.948210 −0.474105 0.880468i \(-0.657228\pi\)
−0.474105 + 0.880468i \(0.657228\pi\)
\(984\) 0 0
\(985\) 3649.75 0.118061
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 39182.3 1.25978
\(990\) 0 0
\(991\) 1667.46 0.0534496 0.0267248 0.999643i \(-0.491492\pi\)
0.0267248 + 0.999643i \(0.491492\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 53103.4 1.69195
\(996\) 0 0
\(997\) 33972.0 1.07914 0.539571 0.841940i \(-0.318587\pi\)
0.539571 + 0.841940i \(0.318587\pi\)
\(998\) 0 0
\(999\) 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 324.4.a.c.1.1 3
3.2 odd 2 324.4.a.d.1.3 3
4.3 odd 2 1296.4.a.v.1.1 3
9.2 odd 6 108.4.e.a.37.1 6
9.4 even 3 36.4.e.a.25.1 yes 6
9.5 odd 6 108.4.e.a.73.1 6
9.7 even 3 36.4.e.a.13.1 6
12.11 even 2 1296.4.a.w.1.3 3
36.7 odd 6 144.4.i.d.49.3 6
36.11 even 6 432.4.i.d.145.1 6
36.23 even 6 432.4.i.d.289.1 6
36.31 odd 6 144.4.i.d.97.3 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
36.4.e.a.13.1 6 9.7 even 3
36.4.e.a.25.1 yes 6 9.4 even 3
108.4.e.a.37.1 6 9.2 odd 6
108.4.e.a.73.1 6 9.5 odd 6
144.4.i.d.49.3 6 36.7 odd 6
144.4.i.d.97.3 6 36.31 odd 6
324.4.a.c.1.1 3 1.1 even 1 trivial
324.4.a.d.1.3 3 3.2 odd 2
432.4.i.d.145.1 6 36.11 even 6
432.4.i.d.289.1 6 36.23 even 6
1296.4.a.v.1.1 3 4.3 odd 2
1296.4.a.w.1.3 3 12.11 even 2