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Properties

Label 2304.4.a.c.1.1
Level $2304$
Weight $4$
Character 2304.1
Self dual yes
Analytic conductor $135.940$
Analytic rank $1$
Dimension $1$
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] = [2304,4,Mod(1,2304)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2304, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 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("2304.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2304 = 2^{8} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 2304.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: \(135.940400653\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 128)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 2304.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-12.0000 q^{5} -32.0000 q^{7} +O(q^{10})\) \(q-12.0000 q^{5} -32.0000 q^{7} -8.00000 q^{11} -20.0000 q^{13} +98.0000 q^{17} +88.0000 q^{19} -32.0000 q^{23} +19.0000 q^{25} -172.000 q^{29} +256.000 q^{31} +384.000 q^{35} +92.0000 q^{37} -102.000 q^{41} +296.000 q^{43} -320.000 q^{47} +681.000 q^{49} -76.0000 q^{53} +96.0000 q^{55} +408.000 q^{59} +636.000 q^{61} +240.000 q^{65} -552.000 q^{67} +416.000 q^{71} +138.000 q^{73} +256.000 q^{77} +64.0000 q^{79} +392.000 q^{83} -1176.00 q^{85} +582.000 q^{89} +640.000 q^{91} -1056.00 q^{95} +238.000 q^{97} +O(q^{100})\)

Display 100 coefficients 10 coefficients

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
\(5\) −12.0000 −1.07331 −0.536656 − 0.843801i \(-0.680313\pi\)
−0.536656 + 0.843801i \(0.680313\pi\)
\(6\) 0 0
\(7\) −32.0000 −1.72784 −0.863919 − 0.503631i \(-0.831997\pi\)
−0.863919 + 0.503631i \(0.831997\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) −8.00000 −0.219281 −0.109640 − 0.993971i \(-0.534970\pi\)
−0.109640 + 0.993971i \(0.534970\pi\)
\(12\) 0 0
\(13\) −20.0000 −0.426692 −0.213346 − 0.976977i \(-0.568436\pi\)
−0.213346 + 0.976977i \(0.568436\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 98.0000 1.39815 0.699073 − 0.715050i \(-0.253596\pi\)
0.699073 + 0.715050i \(0.253596\pi\)
\(18\) 0 0
\(19\) 88.0000 1.06256 0.531279 − 0.847197i \(-0.321712\pi\)
0.531279 + 0.847197i \(0.321712\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −32.0000 −0.290107 −0.145054 − 0.989424i \(-0.546335\pi\)
−0.145054 + 0.989424i \(0.546335\pi\)
\(24\) 0 0
\(25\) 19.0000 0.152000
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −172.000 −1.10137 −0.550683 − 0.834715i \(-0.685633\pi\)
−0.550683 + 0.834715i \(0.685633\pi\)
\(30\) 0 0
\(31\) 256.000 1.48319 0.741596 − 0.670847i \(-0.234069\pi\)
0.741596 + 0.670847i \(0.234069\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 384.000 1.85451
\(36\) 0 0
\(37\) 92.0000 0.408776 0.204388 − 0.978890i \(-0.434480\pi\)
0.204388 + 0.978890i \(0.434480\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −102.000 −0.388530 −0.194265 − 0.980949i \(-0.562232\pi\)
−0.194265 + 0.980949i \(0.562232\pi\)
\(42\) 0 0
\(43\) 296.000 1.04976 0.524879 − 0.851177i \(-0.324111\pi\)
0.524879 + 0.851177i \(0.324111\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −320.000 −0.993123 −0.496562 − 0.868001i \(-0.665404\pi\)
−0.496562 + 0.868001i \(0.665404\pi\)
\(48\) 0 0
\(49\) 681.000 1.98542
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −76.0000 −0.196970 −0.0984849 − 0.995139i \(-0.531400\pi\)
−0.0984849 + 0.995139i \(0.531400\pi\)
\(54\) 0 0
\(55\) 96.0000 0.235357
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 408.000 0.900289 0.450145 − 0.892956i \(-0.351372\pi\)
0.450145 + 0.892956i \(0.351372\pi\)
\(60\) 0 0
\(61\) 636.000 1.33494 0.667471 − 0.744636i \(-0.267377\pi\)
0.667471 + 0.744636i \(0.267377\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 240.000 0.457974
\(66\) 0 0
\(67\) −552.000 −1.00653 −0.503265 − 0.864132i \(-0.667868\pi\)
−0.503265 + 0.864132i \(0.667868\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 416.000 0.695354 0.347677 − 0.937614i \(-0.386971\pi\)
0.347677 + 0.937614i \(0.386971\pi\)
\(72\) 0 0
\(73\) 138.000 0.221256 0.110628 − 0.993862i \(-0.464714\pi\)
0.110628 + 0.993862i \(0.464714\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 256.000 0.378882
\(78\) 0 0
\(79\) 64.0000 0.0911464 0.0455732 − 0.998961i \(-0.485489\pi\)
0.0455732 + 0.998961i \(0.485489\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 392.000 0.518405 0.259202 − 0.965823i \(-0.416540\pi\)
0.259202 + 0.965823i \(0.416540\pi\)
\(84\) 0 0
\(85\) −1176.00 −1.50065
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 582.000 0.693167 0.346584 − 0.938019i \(-0.387342\pi\)
0.346584 + 0.938019i \(0.387342\pi\)
\(90\) 0 0
\(91\) 640.000 0.737255
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −1056.00 −1.14046
\(96\) 0 0
\(97\) 238.000 0.249126 0.124563 − 0.992212i \(-0.460247\pi\)
0.124563 + 0.992212i \(0.460247\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) −1468.00 −1.44625 −0.723126 − 0.690716i \(-0.757295\pi\)
−0.723126 + 0.690716i \(0.757295\pi\)
\(102\) 0 0
\(103\) 992.000 0.948977 0.474489 − 0.880262i \(-0.342633\pi\)
0.474489 + 0.880262i \(0.342633\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −584.000 −0.527639 −0.263820 − 0.964572i \(-0.584982\pi\)
−0.263820 + 0.964572i \(0.584982\pi\)
\(108\) 0 0
\(109\) −740.000 −0.650267 −0.325134 − 0.945668i \(-0.605409\pi\)
−0.325134 + 0.945668i \(0.605409\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 302.000 0.251414 0.125707 − 0.992067i \(-0.459880\pi\)
0.125707 + 0.992067i \(0.459880\pi\)
\(114\) 0 0
\(115\) 384.000 0.311376
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) −3136.00 −2.41577
\(120\) 0 0
\(121\) −1267.00 −0.951916
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 1272.00 0.910169
\(126\) 0 0
\(127\) 1664.00 1.16265 0.581323 − 0.813673i \(-0.302535\pi\)
0.581323 + 0.813673i \(0.302535\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) −2328.00 −1.55266 −0.776329 − 0.630327i \(-0.782921\pi\)
−0.776329 + 0.630327i \(0.782921\pi\)
\(132\) 0 0
\(133\) −2816.00 −1.83593
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −1734.00 −1.08135 −0.540677 − 0.841230i \(-0.681832\pi\)
−0.540677 + 0.841230i \(0.681832\pi\)
\(138\) 0 0
\(139\) −3032.00 −1.85015 −0.925075 − 0.379784i \(-0.875998\pi\)
−0.925075 + 0.379784i \(0.875998\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 160.000 0.0935655
\(144\) 0 0
\(145\) 2064.00 1.18211
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) −1788.00 −0.983078 −0.491539 − 0.870855i \(-0.663566\pi\)
−0.491539 + 0.870855i \(0.663566\pi\)
\(150\) 0 0
\(151\) −480.000 −0.258688 −0.129344 − 0.991600i \(-0.541287\pi\)
−0.129344 + 0.991600i \(0.541287\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) −3072.00 −1.59193
\(156\) 0 0
\(157\) 2300.00 1.16917 0.584586 − 0.811332i \(-0.301257\pi\)
0.584586 + 0.811332i \(0.301257\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 1024.00 0.501258
\(162\) 0 0
\(163\) 1592.00 0.765000 0.382500 − 0.923955i \(-0.375063\pi\)
0.382500 + 0.923955i \(0.375063\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 2208.00 1.02311 0.511557 − 0.859249i \(-0.329069\pi\)
0.511557 + 0.859249i \(0.329069\pi\)
\(168\) 0 0
\(169\) −1797.00 −0.817934
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) −3948.00 −1.73503 −0.867517 − 0.497408i \(-0.834285\pi\)
−0.867517 + 0.497408i \(0.834285\pi\)
\(174\) 0 0
\(175\) −608.000 −0.262631
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) −2104.00 −0.878549 −0.439275 − 0.898353i \(-0.644765\pi\)
−0.439275 + 0.898353i \(0.644765\pi\)
\(180\) 0 0
\(181\) −1412.00 −0.579852 −0.289926 − 0.957049i \(-0.593631\pi\)
−0.289926 + 0.957049i \(0.593631\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) −1104.00 −0.438744
\(186\) 0 0
\(187\) −784.000 −0.306587
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 3712.00 1.40624 0.703118 − 0.711074i \(-0.251791\pi\)
0.703118 + 0.711074i \(0.251791\pi\)
\(192\) 0 0
\(193\) 1614.00 0.601960 0.300980 − 0.953630i \(-0.402686\pi\)
0.300980 + 0.953630i \(0.402686\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −684.000 −0.247376 −0.123688 − 0.992321i \(-0.539472\pi\)
−0.123688 + 0.992321i \(0.539472\pi\)
\(198\) 0 0
\(199\) 4064.00 1.44769 0.723843 − 0.689965i \(-0.242374\pi\)
0.723843 + 0.689965i \(0.242374\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 5504.00 1.90298
\(204\) 0 0
\(205\) 1224.00 0.417014
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) −704.000 −0.232999
\(210\) 0 0
\(211\) −2120.00 −0.691691 −0.345846 − 0.938291i \(-0.612408\pi\)
−0.345846 + 0.938291i \(0.612408\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) −3552.00 −1.12672
\(216\) 0 0
\(217\) −8192.00 −2.56272
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) −1960.00 −0.596579
\(222\) 0 0
\(223\) −2816.00 −0.845620 −0.422810 − 0.906218i \(-0.638956\pi\)
−0.422810 + 0.906218i \(0.638956\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 3848.00 1.12511 0.562557 − 0.826759i \(-0.309818\pi\)
0.562557 + 0.826759i \(0.309818\pi\)
\(228\) 0 0
\(229\) 652.000 0.188146 0.0940729 − 0.995565i \(-0.470011\pi\)
0.0940729 + 0.995565i \(0.470011\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −3050.00 −0.857563 −0.428781 − 0.903408i \(-0.641057\pi\)
−0.428781 + 0.903408i \(0.641057\pi\)
\(234\) 0 0
\(235\) 3840.00 1.06593
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) −6336.00 −1.71482 −0.857410 − 0.514635i \(-0.827928\pi\)
−0.857410 + 0.514635i \(0.827928\pi\)
\(240\) 0 0
\(241\) −4610.00 −1.23218 −0.616092 − 0.787674i \(-0.711285\pi\)
−0.616092 + 0.787674i \(0.711285\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) −8172.00 −2.13098
\(246\) 0 0
\(247\) −1760.00 −0.453385
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 792.000 0.199166 0.0995829 − 0.995029i \(-0.468249\pi\)
0.0995829 + 0.995029i \(0.468249\pi\)
\(252\) 0 0
\(253\) 256.000 0.0636149
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 5374.00 1.30436 0.652181 − 0.758063i \(-0.273854\pi\)
0.652181 + 0.758063i \(0.273854\pi\)
\(258\) 0 0
\(259\) −2944.00 −0.706298
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 3488.00 0.817792 0.408896 − 0.912581i \(-0.365914\pi\)
0.408896 + 0.912581i \(0.365914\pi\)
\(264\) 0 0
\(265\) 912.000 0.211410
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) −4764.00 −1.07980 −0.539900 − 0.841729i \(-0.681538\pi\)
−0.539900 + 0.841729i \(0.681538\pi\)
\(270\) 0 0
\(271\) −1344.00 −0.301263 −0.150631 − 0.988590i \(-0.548131\pi\)
−0.150631 + 0.988590i \(0.548131\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −152.000 −0.0333307
\(276\) 0 0
\(277\) −8596.00 −1.86456 −0.932281 − 0.361735i \(-0.882184\pi\)
−0.932281 + 0.361735i \(0.882184\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) −2874.00 −0.610137 −0.305068 − 0.952330i \(-0.598679\pi\)
−0.305068 + 0.952330i \(0.598679\pi\)
\(282\) 0 0
\(283\) 2888.00 0.606621 0.303311 − 0.952892i \(-0.401908\pi\)
0.303311 + 0.952892i \(0.401908\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 3264.00 0.671316
\(288\) 0 0
\(289\) 4691.00 0.954814
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) −6540.00 −1.30400 −0.651998 − 0.758221i \(-0.726069\pi\)
−0.651998 + 0.758221i \(0.726069\pi\)
\(294\) 0 0
\(295\) −4896.00 −0.966292
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 640.000 0.123786
\(300\) 0 0
\(301\) −9472.00 −1.81381
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) −7632.00 −1.43281
\(306\) 0 0
\(307\) 10584.0 1.96762 0.983812 − 0.179202i \(-0.0573515\pi\)
0.983812 + 0.179202i \(0.0573515\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 6368.00 1.16108 0.580540 − 0.814231i \(-0.302841\pi\)
0.580540 + 0.814231i \(0.302841\pi\)
\(312\) 0 0
\(313\) 8758.00 1.58157 0.790785 − 0.612094i \(-0.209673\pi\)
0.790785 + 0.612094i \(0.209673\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −716.000 −0.126860 −0.0634299 − 0.997986i \(-0.520204\pi\)
−0.0634299 + 0.997986i \(0.520204\pi\)
\(318\) 0 0
\(319\) 1376.00 0.241508
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 8624.00 1.48561
\(324\) 0 0
\(325\) −380.000 −0.0648573
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 10240.0 1.71596
\(330\) 0 0
\(331\) −4408.00 −0.731981 −0.365990 − 0.930619i \(-0.619270\pi\)
−0.365990 + 0.930619i \(0.619270\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 6624.00 1.08032
\(336\) 0 0
\(337\) 1202.00 0.194294 0.0971471 − 0.995270i \(-0.469028\pi\)
0.0971471 + 0.995270i \(0.469028\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) −2048.00 −0.325236
\(342\) 0 0
\(343\) −10816.0 −1.70265
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −5160.00 −0.798280 −0.399140 − 0.916890i \(-0.630691\pi\)
−0.399140 + 0.916890i \(0.630691\pi\)
\(348\) 0 0
\(349\) 4876.00 0.747869 0.373935 − 0.927455i \(-0.378008\pi\)
0.373935 + 0.927455i \(0.378008\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) −4834.00 −0.728861 −0.364430 − 0.931231i \(-0.618736\pi\)
−0.364430 + 0.931231i \(0.618736\pi\)
\(354\) 0 0
\(355\) −4992.00 −0.746332
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 4128.00 0.606873 0.303437 − 0.952852i \(-0.401866\pi\)
0.303437 + 0.952852i \(0.401866\pi\)
\(360\) 0 0
\(361\) 885.000 0.129028
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) −1656.00 −0.237477
\(366\) 0 0
\(367\) 4416.00 0.628102 0.314051 − 0.949406i \(-0.398314\pi\)
0.314051 + 0.949406i \(0.398314\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 2432.00 0.340332
\(372\) 0 0
\(373\) −4180.00 −0.580247 −0.290124 − 0.956989i \(-0.593696\pi\)
−0.290124 + 0.956989i \(0.593696\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 3440.00 0.469944
\(378\) 0 0
\(379\) 13736.0 1.86166 0.930832 − 0.365446i \(-0.119084\pi\)
0.930832 + 0.365446i \(0.119084\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 512.000 0.0683080 0.0341540 − 0.999417i \(-0.489126\pi\)
0.0341540 + 0.999417i \(0.489126\pi\)
\(384\) 0 0
\(385\) −3072.00 −0.406659
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 1732.00 0.225748 0.112874 − 0.993609i \(-0.463994\pi\)
0.112874 + 0.993609i \(0.463994\pi\)
\(390\) 0 0
\(391\) −3136.00 −0.405612
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) −768.000 −0.0978285
\(396\) 0 0
\(397\) −10436.0 −1.31931 −0.659657 − 0.751567i \(-0.729299\pi\)
−0.659657 + 0.751567i \(0.729299\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 12130.0 1.51058 0.755291 − 0.655390i \(-0.227496\pi\)
0.755291 + 0.655390i \(0.227496\pi\)
\(402\) 0 0
\(403\) −5120.00 −0.632867
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) −736.000 −0.0896368
\(408\) 0 0
\(409\) 5014.00 0.606177 0.303088 − 0.952962i \(-0.401982\pi\)
0.303088 + 0.952962i \(0.401982\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) −13056.0 −1.55555
\(414\) 0 0
\(415\) −4704.00 −0.556410
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) −8024.00 −0.935556 −0.467778 − 0.883846i \(-0.654945\pi\)
−0.467778 + 0.883846i \(0.654945\pi\)
\(420\) 0 0
\(421\) 2348.00 0.271816 0.135908 − 0.990721i \(-0.456605\pi\)
0.135908 + 0.990721i \(0.456605\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 1862.00 0.212518
\(426\) 0 0
\(427\) −20352.0 −2.30656
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) −1728.00 −0.193120 −0.0965601 − 0.995327i \(-0.530784\pi\)
−0.0965601 + 0.995327i \(0.530784\pi\)
\(432\) 0 0
\(433\) 62.0000 0.00688113 0.00344057 − 0.999994i \(-0.498905\pi\)
0.00344057 + 0.999994i \(0.498905\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −2816.00 −0.308255
\(438\) 0 0
\(439\) 14112.0 1.53423 0.767117 − 0.641507i \(-0.221690\pi\)
0.767117 + 0.641507i \(0.221690\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −4488.00 −0.481335 −0.240667 − 0.970608i \(-0.577366\pi\)
−0.240667 + 0.970608i \(0.577366\pi\)
\(444\) 0 0
\(445\) −6984.00 −0.743985
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 2482.00 0.260875 0.130437 − 0.991457i \(-0.458362\pi\)
0.130437 + 0.991457i \(0.458362\pi\)
\(450\) 0 0
\(451\) 816.000 0.0851972
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) −7680.00 −0.791305
\(456\) 0 0
\(457\) 5894.00 0.603303 0.301652 − 0.953418i \(-0.402462\pi\)
0.301652 + 0.953418i \(0.402462\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) −7068.00 −0.714077 −0.357039 − 0.934090i \(-0.616214\pi\)
−0.357039 + 0.934090i \(0.616214\pi\)
\(462\) 0 0
\(463\) −7616.00 −0.764461 −0.382231 − 0.924067i \(-0.624844\pi\)
−0.382231 + 0.924067i \(0.624844\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −13080.0 −1.29608 −0.648041 − 0.761606i \(-0.724411\pi\)
−0.648041 + 0.761606i \(0.724411\pi\)
\(468\) 0 0
\(469\) 17664.0 1.73912
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) −2368.00 −0.230192
\(474\) 0 0
\(475\) 1672.00 0.161509
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) −13568.0 −1.29423 −0.647117 − 0.762391i \(-0.724025\pi\)
−0.647117 + 0.762391i \(0.724025\pi\)
\(480\) 0 0
\(481\) −1840.00 −0.174422
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) −2856.00 −0.267390
\(486\) 0 0
\(487\) −1696.00 −0.157809 −0.0789046 − 0.996882i \(-0.525142\pi\)
−0.0789046 + 0.996882i \(0.525142\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 3096.00 0.284563 0.142282 − 0.989826i \(-0.454556\pi\)
0.142282 + 0.989826i \(0.454556\pi\)
\(492\) 0 0
\(493\) −16856.0 −1.53987
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −13312.0 −1.20146
\(498\) 0 0
\(499\) −19208.0 −1.72318 −0.861591 − 0.507603i \(-0.830532\pi\)
−0.861591 + 0.507603i \(0.830532\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 16224.0 1.43816 0.719078 − 0.694929i \(-0.244564\pi\)
0.719078 + 0.694929i \(0.244564\pi\)
\(504\) 0 0
\(505\) 17616.0 1.55228
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) −11292.0 −0.983318 −0.491659 − 0.870788i \(-0.663609\pi\)
−0.491659 + 0.870788i \(0.663609\pi\)
\(510\) 0 0
\(511\) −4416.00 −0.382294
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) −11904.0 −1.01855
\(516\) 0 0
\(517\) 2560.00 0.217773
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 5178.00 0.435417 0.217709 − 0.976014i \(-0.430142\pi\)
0.217709 + 0.976014i \(0.430142\pi\)
\(522\) 0 0
\(523\) 6856.00 0.573216 0.286608 − 0.958048i \(-0.407472\pi\)
0.286608 + 0.958048i \(0.407472\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 25088.0 2.07372
\(528\) 0 0
\(529\) −11143.0 −0.915838
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 2040.00 0.165783
\(534\) 0 0
\(535\) 7008.00 0.566322
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) −5448.00 −0.435365
\(540\) 0 0
\(541\) −13732.0 −1.09128 −0.545642 − 0.838018i \(-0.683714\pi\)
−0.545642 + 0.838018i \(0.683714\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 8880.00 0.697940
\(546\) 0 0
\(547\) 10968.0 0.857327 0.428663 − 0.903464i \(-0.358985\pi\)
0.428663 + 0.903464i \(0.358985\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) −15136.0 −1.17026
\(552\) 0 0
\(553\) −2048.00 −0.157486
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −25612.0 −1.94832 −0.974161 − 0.225855i \(-0.927482\pi\)
−0.974161 + 0.225855i \(0.927482\pi\)
\(558\) 0 0
\(559\) −5920.00 −0.447924
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 9768.00 0.731212 0.365606 − 0.930770i \(-0.380862\pi\)
0.365606 + 0.930770i \(0.380862\pi\)
\(564\) 0 0
\(565\) −3624.00 −0.269846
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −22838.0 −1.68263 −0.841317 − 0.540542i \(-0.818219\pi\)
−0.841317 + 0.540542i \(0.818219\pi\)
\(570\) 0 0
\(571\) −9208.00 −0.674856 −0.337428 − 0.941351i \(-0.609557\pi\)
−0.337428 + 0.941351i \(0.609557\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) −608.000 −0.0440963
\(576\) 0 0
\(577\) −10878.0 −0.784848 −0.392424 − 0.919785i \(-0.628363\pi\)
−0.392424 + 0.919785i \(0.628363\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) −12544.0 −0.895719
\(582\) 0 0
\(583\) 608.000 0.0431917
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 18040.0 1.26847 0.634234 − 0.773141i \(-0.281316\pi\)
0.634234 + 0.773141i \(0.281316\pi\)
\(588\) 0 0
\(589\) 22528.0 1.57598
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) −26994.0 −1.86933 −0.934663 − 0.355534i \(-0.884299\pi\)
−0.934663 + 0.355534i \(0.884299\pi\)
\(594\) 0 0
\(595\) 37632.0 2.59288
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) −18336.0 −1.25073 −0.625366 − 0.780331i \(-0.715050\pi\)
−0.625366 + 0.780331i \(0.715050\pi\)
\(600\) 0 0
\(601\) −9286.00 −0.630256 −0.315128 − 0.949049i \(-0.602047\pi\)
−0.315128 + 0.949049i \(0.602047\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 15204.0 1.02170
\(606\) 0 0
\(607\) 17536.0 1.17259 0.586297 − 0.810096i \(-0.300585\pi\)
0.586297 + 0.810096i \(0.300585\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 6400.00 0.423758
\(612\) 0 0
\(613\) 5868.00 0.386633 0.193317 − 0.981136i \(-0.438076\pi\)
0.193317 + 0.981136i \(0.438076\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 19286.0 1.25839 0.629194 − 0.777248i \(-0.283385\pi\)
0.629194 + 0.777248i \(0.283385\pi\)
\(618\) 0 0
\(619\) −5240.00 −0.340248 −0.170124 − 0.985423i \(-0.554417\pi\)
−0.170124 + 0.985423i \(0.554417\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) −18624.0 −1.19768
\(624\) 0 0
\(625\) −17639.0 −1.12890
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 9016.00 0.571529
\(630\) 0 0
\(631\) 15520.0 0.979147 0.489573 − 0.871962i \(-0.337153\pi\)
0.489573 + 0.871962i \(0.337153\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) −19968.0 −1.24788
\(636\) 0 0
\(637\) −13620.0 −0.847165
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) −654.000 −0.0402987 −0.0201493 − 0.999797i \(-0.506414\pi\)
−0.0201493 + 0.999797i \(0.506414\pi\)
\(642\) 0 0
\(643\) −8232.00 −0.504881 −0.252440 − 0.967612i \(-0.581233\pi\)
−0.252440 + 0.967612i \(0.581233\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −24672.0 −1.49916 −0.749580 − 0.661914i \(-0.769744\pi\)
−0.749580 + 0.661914i \(0.769744\pi\)
\(648\) 0 0
\(649\) −3264.00 −0.197416
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 22052.0 1.32153 0.660767 − 0.750591i \(-0.270231\pi\)
0.660767 + 0.750591i \(0.270231\pi\)
\(654\) 0 0
\(655\) 27936.0 1.66649
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) −12024.0 −0.710757 −0.355378 − 0.934723i \(-0.615648\pi\)
−0.355378 + 0.934723i \(0.615648\pi\)
\(660\) 0 0
\(661\) 19100.0 1.12391 0.561955 − 0.827168i \(-0.310050\pi\)
0.561955 + 0.827168i \(0.310050\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 33792.0 1.97052
\(666\) 0 0
\(667\) 5504.00 0.319514
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) −5088.00 −0.292727
\(672\) 0 0
\(673\) 9902.00 0.567153 0.283577 − 0.958950i \(-0.408479\pi\)
0.283577 + 0.958950i \(0.408479\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 19684.0 1.11746 0.558728 − 0.829351i \(-0.311290\pi\)
0.558728 + 0.829351i \(0.311290\pi\)
\(678\) 0 0
\(679\) −7616.00 −0.430450
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 19864.0 1.11285 0.556424 − 0.830899i \(-0.312173\pi\)
0.556424 + 0.830899i \(0.312173\pi\)
\(684\) 0 0
\(685\) 20808.0 1.16063
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 1520.00 0.0840456
\(690\) 0 0
\(691\) 3256.00 0.179253 0.0896267 − 0.995975i \(-0.471433\pi\)
0.0896267 + 0.995975i \(0.471433\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 36384.0 1.98579
\(696\) 0 0
\(697\) −9996.00 −0.543222
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) −2876.00 −0.154957 −0.0774786 − 0.996994i \(-0.524687\pi\)
−0.0774786 + 0.996994i \(0.524687\pi\)
\(702\) 0 0
\(703\) 8096.00 0.434348
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 46976.0 2.49889
\(708\) 0 0
\(709\) −7300.00 −0.386682 −0.193341 − 0.981132i \(-0.561932\pi\)
−0.193341 + 0.981132i \(0.561932\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) −8192.00 −0.430284
\(714\) 0 0
\(715\) −1920.00 −0.100425
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) −2880.00 −0.149382 −0.0746912 − 0.997207i \(-0.523797\pi\)
−0.0746912 + 0.997207i \(0.523797\pi\)
\(720\) 0 0
\(721\) −31744.0 −1.63968
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) −3268.00 −0.167408
\(726\) 0 0
\(727\) −8800.00 −0.448933 −0.224466 − 0.974482i \(-0.572064\pi\)
−0.224466 + 0.974482i \(0.572064\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 29008.0 1.46771
\(732\) 0 0
\(733\) −21076.0 −1.06202 −0.531009 − 0.847366i \(-0.678187\pi\)
−0.531009 + 0.847366i \(0.678187\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 4416.00 0.220713
\(738\) 0 0
\(739\) −19336.0 −0.962498 −0.481249 − 0.876584i \(-0.659817\pi\)
−0.481249 + 0.876584i \(0.659817\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) −13664.0 −0.674675 −0.337338 − 0.941384i \(-0.609526\pi\)
−0.337338 + 0.941384i \(0.609526\pi\)
\(744\) 0 0
\(745\) 21456.0 1.05515
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 18688.0 0.911675
\(750\) 0 0
\(751\) −19520.0 −0.948462 −0.474231 − 0.880400i \(-0.657274\pi\)
−0.474231 + 0.880400i \(0.657274\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 5760.00 0.277653
\(756\) 0 0
\(757\) −20004.0 −0.960446 −0.480223 − 0.877146i \(-0.659444\pi\)
−0.480223 + 0.877146i \(0.659444\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −31478.0 −1.49944 −0.749722 − 0.661753i \(-0.769813\pi\)
−0.749722 + 0.661753i \(0.769813\pi\)
\(762\) 0 0
\(763\) 23680.0 1.12356
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) −8160.00 −0.384147
\(768\) 0 0
\(769\) 7054.00 0.330785 0.165393 − 0.986228i \(-0.447111\pi\)
0.165393 + 0.986228i \(0.447111\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 9604.00 0.446872 0.223436 − 0.974719i \(-0.428273\pi\)
0.223436 + 0.974719i \(0.428273\pi\)
\(774\) 0 0
\(775\) 4864.00 0.225445
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) −8976.00 −0.412835
\(780\) 0 0
\(781\) −3328.00 −0.152478
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) −27600.0 −1.25489
\(786\) 0 0
\(787\) −3144.00 −0.142403 −0.0712017 − 0.997462i \(-0.522683\pi\)
−0.0712017 + 0.997462i \(0.522683\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) −9664.00 −0.434402
\(792\) 0 0
\(793\) −12720.0 −0.569610
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 22084.0 0.981500 0.490750 − 0.871300i \(-0.336723\pi\)
0.490750 + 0.871300i \(0.336723\pi\)
\(798\) 0 0
\(799\) −31360.0 −1.38853
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) −1104.00 −0.0485172
\(804\) 0 0
\(805\) −12288.0 −0.538006
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) −14950.0 −0.649708 −0.324854 − 0.945764i \(-0.605315\pi\)
−0.324854 + 0.945764i \(0.605315\pi\)
\(810\) 0 0
\(811\) 23432.0 1.01456 0.507280 − 0.861781i \(-0.330651\pi\)
0.507280 + 0.861781i \(0.330651\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) −19104.0 −0.821085
\(816\) 0 0
\(817\) 26048.0 1.11543
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 1044.00 0.0443798 0.0221899 − 0.999754i \(-0.492936\pi\)
0.0221899 + 0.999754i \(0.492936\pi\)
\(822\) 0 0
\(823\) 18208.0 0.771192 0.385596 − 0.922668i \(-0.373996\pi\)
0.385596 + 0.922668i \(0.373996\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −12488.0 −0.525091 −0.262546 − 0.964920i \(-0.584562\pi\)
−0.262546 + 0.964920i \(0.584562\pi\)
\(828\) 0 0
\(829\) 30172.0 1.26407 0.632037 − 0.774938i \(-0.282219\pi\)
0.632037 + 0.774938i \(0.282219\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 66738.0 2.77591
\(834\) 0 0
\(835\) −26496.0 −1.09812
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 32544.0 1.33915 0.669573 − 0.742746i \(-0.266477\pi\)
0.669573 + 0.742746i \(0.266477\pi\)
\(840\) 0 0
\(841\) 5195.00 0.213006
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 21564.0 0.877898
\(846\) 0 0
\(847\) 40544.0 1.64476
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) −2944.00 −0.118589
\(852\) 0 0
\(853\) 26156.0 1.04990 0.524950 − 0.851133i \(-0.324084\pi\)
0.524950 + 0.851133i \(0.324084\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −18646.0 −0.743215 −0.371607 − 0.928390i \(-0.621193\pi\)
−0.371607 + 0.928390i \(0.621193\pi\)
\(858\) 0 0
\(859\) 5800.00 0.230377 0.115188 − 0.993344i \(-0.463253\pi\)
0.115188 + 0.993344i \(0.463253\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) −25088.0 −0.989578 −0.494789 − 0.869013i \(-0.664755\pi\)
−0.494789 + 0.869013i \(0.664755\pi\)
\(864\) 0 0
\(865\) 47376.0 1.86223
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) −512.000 −0.0199867
\(870\) 0 0
\(871\) 11040.0 0.429479
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) −40704.0 −1.57262
\(876\) 0 0
\(877\) 3004.00 0.115665 0.0578323 − 0.998326i \(-0.481581\pi\)
0.0578323 + 0.998326i \(0.481581\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) −43282.0 −1.65517 −0.827587 − 0.561338i \(-0.810287\pi\)
−0.827587 + 0.561338i \(0.810287\pi\)
\(882\) 0 0
\(883\) −27880.0 −1.06256 −0.531278 − 0.847198i \(-0.678288\pi\)
−0.531278 + 0.847198i \(0.678288\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 7392.00 0.279819 0.139909 − 0.990164i \(-0.455319\pi\)
0.139909 + 0.990164i \(0.455319\pi\)
\(888\) 0 0
\(889\) −53248.0 −2.00886
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) −28160.0 −1.05525
\(894\) 0 0
\(895\) 25248.0 0.942958
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) −44032.0 −1.63354
\(900\) 0 0
\(901\) −7448.00 −0.275393
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 16944.0 0.622362
\(906\) 0 0
\(907\) −29080.0 −1.06459 −0.532296 − 0.846558i \(-0.678671\pi\)
−0.532296 + 0.846558i \(0.678671\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 26688.0 0.970596 0.485298 − 0.874349i \(-0.338711\pi\)
0.485298 + 0.874349i \(0.338711\pi\)
\(912\) 0 0
\(913\) −3136.00 −0.113676
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 74496.0 2.68274
\(918\) 0 0
\(919\) −19680.0 −0.706402 −0.353201 − 0.935547i \(-0.614907\pi\)
−0.353201 + 0.935547i \(0.614907\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) −8320.00 −0.296702
\(924\) 0 0
\(925\) 1748.00 0.0621339
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 48466.0 1.71164 0.855822 − 0.517270i \(-0.173052\pi\)
0.855822 + 0.517270i \(0.173052\pi\)
\(930\) 0 0
\(931\) 59928.0 2.10962
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 9408.00 0.329064
\(936\) 0 0
\(937\) 13610.0 0.474514 0.237257 − 0.971447i \(-0.423752\pi\)
0.237257 + 0.971447i \(0.423752\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 30692.0 1.06326 0.531632 − 0.846976i \(-0.321579\pi\)
0.531632 + 0.846976i \(0.321579\pi\)
\(942\) 0 0
\(943\) 3264.00 0.112715
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −4824.00 −0.165532 −0.0827661 − 0.996569i \(-0.526375\pi\)
−0.0827661 + 0.996569i \(0.526375\pi\)
\(948\) 0 0
\(949\) −2760.00 −0.0944082
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 22986.0 0.781311 0.390656 − 0.920537i \(-0.372248\pi\)
0.390656 + 0.920537i \(0.372248\pi\)
\(954\) 0 0
\(955\) −44544.0 −1.50933
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 55488.0 1.86841
\(960\) 0 0
\(961\) 35745.0 1.19986
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) −19368.0 −0.646091
\(966\) 0 0
\(967\) −17184.0 −0.571458 −0.285729 − 0.958310i \(-0.592236\pi\)
−0.285729 + 0.958310i \(0.592236\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) −2920.00 −0.0965059 −0.0482530 − 0.998835i \(-0.515365\pi\)
−0.0482530 + 0.998835i \(0.515365\pi\)
\(972\) 0 0
\(973\) 97024.0 3.19676
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 27042.0 0.885517 0.442759 − 0.896641i \(-0.354000\pi\)
0.442759 + 0.896641i \(0.354000\pi\)
\(978\) 0 0
\(979\) −4656.00 −0.151998
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) −44192.0 −1.43388 −0.716941 − 0.697134i \(-0.754458\pi\)
−0.716941 + 0.697134i \(0.754458\pi\)
\(984\) 0 0
\(985\) 8208.00 0.265511
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) −9472.00 −0.304542
\(990\) 0 0
\(991\) −29824.0 −0.955995 −0.477997 − 0.878361i \(-0.658637\pi\)
−0.477997 + 0.878361i \(0.658637\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) −48768.0 −1.55382
\(996\) 0 0
\(997\) 11612.0 0.368862 0.184431 − 0.982845i \(-0.440956\pi\)
0.184431 + 0.982845i \(0.440956\pi\)
\(998\) 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 2304.4.a.c.1.1 1
3.2 odd 2 256.4.a.c.1.1 1
4.3 odd 2 2304.4.a.d.1.1 1
8.3 odd 2 2304.4.a.n.1.1 1
8.5 even 2 2304.4.a.m.1.1 1
12.11 even 2 256.4.a.g.1.1 1
16.3 odd 4 1152.4.d.a.577.2 2
16.5 even 4 1152.4.d.h.577.1 2
16.11 odd 4 1152.4.d.a.577.1 2
16.13 even 4 1152.4.d.h.577.2 2
24.5 odd 2 256.4.a.f.1.1 1
24.11 even 2 256.4.a.b.1.1 1
48.5 odd 4 128.4.b.d.65.2 yes 2
48.11 even 4 128.4.b.a.65.1 ✓ 2
48.29 odd 4 128.4.b.d.65.1 yes 2
48.35 even 4 128.4.b.a.65.2 yes 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
128.4.b.a.65.1 ✓ 2 48.11 even 4
128.4.b.a.65.2 yes 2 48.35 even 4
128.4.b.d.65.1 yes 2 48.29 odd 4
128.4.b.d.65.2 yes 2 48.5 odd 4
256.4.a.b.1.1 1 24.11 even 2
256.4.a.c.1.1 1 3.2 odd 2
256.4.a.f.1.1 1 24.5 odd 2
256.4.a.g.1.1 1 12.11 even 2
1152.4.d.a.577.1 2 16.11 odd 4
1152.4.d.a.577.2 2 16.3 odd 4
1152.4.d.h.577.1 2 16.5 even 4
1152.4.d.h.577.2 2 16.13 even 4
2304.4.a.c.1.1 1 1.1 even 1 trivial
2304.4.a.d.1.1 1 4.3 odd 2
2304.4.a.m.1.1 1 8.5 even 2
2304.4.a.n.1.1 1 8.3 odd 2