Properties

Label 1152.4.d.h.577.2
Level $1152$
Weight $4$
Character 1152.577
Analytic conductor $67.970$
Analytic rank $0$
Dimension $2$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1152,4,Mod(577,1152)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1152, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1152.577");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1152 = 2^{7} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1152.d (of order \(2\), degree \(1\), minimal)

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: no
Analytic conductor: \(67.9702003266\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(i)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 128)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 577.2
Root \(1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 1152.577
Dual form 1152.4.d.h.577.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+12.0000i q^{5} +32.0000 q^{7} +O(q^{10})\) \(q+12.0000i q^{5} +32.0000 q^{7} +8.00000i q^{11} -20.0000i q^{13} +98.0000 q^{17} +88.0000i q^{19} +32.0000 q^{23} -19.0000 q^{25} -172.000i q^{29} +256.000 q^{31} +384.000i q^{35} -92.0000i q^{37} +102.000 q^{41} -296.000i q^{43} -320.000 q^{47} +681.000 q^{49} +76.0000i q^{53} -96.0000 q^{55} -408.000i q^{59} +636.000i q^{61} +240.000 q^{65} -552.000i q^{67} -416.000 q^{71} -138.000 q^{73} +256.000i q^{77} +64.0000 q^{79} +392.000i q^{83} +1176.00i q^{85} -582.000 q^{89} -640.000i q^{91} -1056.00 q^{95} +238.000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 64 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 64 q^{7} + 196 q^{17} + 64 q^{23} - 38 q^{25} + 512 q^{31} + 204 q^{41} - 640 q^{47} + 1362 q^{49} - 192 q^{55} + 480 q^{65} - 832 q^{71} - 276 q^{73} + 128 q^{79} - 1164 q^{89} - 2112 q^{95} + 476 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1152\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(641\) \(901\)
\(\chi(n)\) \(1\) \(1\) \(-1\)

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
\(5\) 12.0000i 1.07331i 0.843801 + 0.536656i \(0.180313\pi\)
−0.843801 + 0.536656i \(0.819687\pi\)
\(6\) 0 0
\(7\) 32.0000 1.72784 0.863919 0.503631i \(-0.168003\pi\)
0.863919 + 0.503631i \(0.168003\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 8.00000i 0.219281i 0.993971 + 0.109640i \(0.0349700\pi\)
−0.993971 + 0.109640i \(0.965030\pi\)
\(12\) 0 0
\(13\) − 20.0000i − 0.426692i −0.976977 0.213346i \(-0.931564\pi\)
0.976977 0.213346i \(-0.0684362\pi\)
\(14\) 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
\(19\) 88.0000i 1.06256i 0.847197 + 0.531279i \(0.178288\pi\)
−0.847197 + 0.531279i \(0.821712\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 32.0000 0.290107 0.145054 0.989424i \(-0.453665\pi\)
0.145054 + 0.989424i \(0.453665\pi\)
\(24\) 0 0
\(25\) −19.0000 −0.152000
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) − 172.000i − 1.10137i −0.834715 0.550683i \(-0.814367\pi\)
0.834715 0.550683i \(-0.185633\pi\)
\(30\) 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
\(35\) 384.000i 1.85451i
\(36\) 0 0
\(37\) − 92.0000i − 0.408776i −0.978890 0.204388i \(-0.934480\pi\)
0.978890 0.204388i \(-0.0655204\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 102.000 0.388530 0.194265 0.980949i \(-0.437768\pi\)
0.194265 + 0.980949i \(0.437768\pi\)
\(42\) 0 0
\(43\) − 296.000i − 1.04976i −0.851177 0.524879i \(-0.824111\pi\)
0.851177 0.524879i \(-0.175889\pi\)
\(44\) 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
\(53\) 76.0000i 0.196970i 0.995139 + 0.0984849i \(0.0313996\pi\)
−0.995139 + 0.0984849i \(0.968600\pi\)
\(54\) 0 0
\(55\) −96.0000 −0.235357
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) − 408.000i − 0.900289i −0.892956 0.450145i \(-0.851372\pi\)
0.892956 0.450145i \(-0.148628\pi\)
\(60\) 0 0
\(61\) 636.000i 1.33494i 0.744636 + 0.667471i \(0.232623\pi\)
−0.744636 + 0.667471i \(0.767377\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 240.000 0.457974
\(66\) 0 0
\(67\) − 552.000i − 1.00653i −0.864132 0.503265i \(-0.832132\pi\)
0.864132 0.503265i \(-0.167868\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −416.000 −0.695354 −0.347677 0.937614i \(-0.613029\pi\)
−0.347677 + 0.937614i \(0.613029\pi\)
\(72\) 0 0
\(73\) −138.000 −0.221256 −0.110628 0.993862i \(-0.535286\pi\)
−0.110628 + 0.993862i \(0.535286\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 256.000i 0.378882i
\(78\) 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
\(83\) 392.000i 0.518405i 0.965823 + 0.259202i \(0.0834597\pi\)
−0.965823 + 0.259202i \(0.916540\pi\)
\(84\) 0 0
\(85\) 1176.00i 1.50065i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −582.000 −0.693167 −0.346584 0.938019i \(-0.612658\pi\)
−0.346584 + 0.938019i \(0.612658\pi\)
\(90\) 0 0
\(91\) − 640.000i − 0.737255i
\(92\) 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
\(101\) 1468.00i 1.44625i 0.690716 + 0.723126i \(0.257295\pi\)
−0.690716 + 0.723126i \(0.742705\pi\)
\(102\) 0 0
\(103\) −992.000 −0.948977 −0.474489 0.880262i \(-0.657367\pi\)
−0.474489 + 0.880262i \(0.657367\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 584.000i 0.527639i 0.964572 + 0.263820i \(0.0849824\pi\)
−0.964572 + 0.263820i \(0.915018\pi\)
\(108\) 0 0
\(109\) − 740.000i − 0.650267i −0.945668 0.325134i \(-0.894591\pi\)
0.945668 0.325134i \(-0.105409\pi\)
\(110\) 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
\(115\) 384.000i 0.311376i
\(116\) 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
\(125\) 1272.00i 0.910169i
\(126\) 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
\(131\) − 2328.00i − 1.55266i −0.630327 0.776329i \(-0.717079\pi\)
0.630327 0.776329i \(-0.282921\pi\)
\(132\) 0 0
\(133\) 2816.00i 1.83593i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 1734.00 1.08135 0.540677 0.841230i \(-0.318168\pi\)
0.540677 + 0.841230i \(0.318168\pi\)
\(138\) 0 0
\(139\) 3032.00i 1.85015i 0.379784 + 0.925075i \(0.375998\pi\)
−0.379784 + 0.925075i \(0.624002\pi\)
\(140\) 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
\(149\) 1788.00i 0.983078i 0.870855 + 0.491539i \(0.163566\pi\)
−0.870855 + 0.491539i \(0.836434\pi\)
\(150\) 0 0
\(151\) 480.000 0.258688 0.129344 0.991600i \(-0.458713\pi\)
0.129344 + 0.991600i \(0.458713\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 3072.00i 1.59193i
\(156\) 0 0
\(157\) 2300.00i 1.16917i 0.811332 + 0.584586i \(0.198743\pi\)
−0.811332 + 0.584586i \(0.801257\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 1024.00 0.501258
\(162\) 0 0
\(163\) 1592.00i 0.765000i 0.923955 + 0.382500i \(0.124937\pi\)
−0.923955 + 0.382500i \(0.875063\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −2208.00 −1.02311 −0.511557 0.859249i \(-0.670931\pi\)
−0.511557 + 0.859249i \(0.670931\pi\)
\(168\) 0 0
\(169\) 1797.00 0.817934
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) − 3948.00i − 1.73503i −0.497408 0.867517i \(-0.665715\pi\)
0.497408 0.867517i \(-0.334285\pi\)
\(174\) 0 0
\(175\) −608.000 −0.262631
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) − 2104.00i − 0.878549i −0.898353 0.439275i \(-0.855235\pi\)
0.898353 0.439275i \(-0.144765\pi\)
\(180\) 0 0
\(181\) 1412.00i 0.579852i 0.957049 + 0.289926i \(0.0936306\pi\)
−0.957049 + 0.289926i \(0.906369\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 1104.00 0.438744
\(186\) 0 0
\(187\) 784.000i 0.306587i
\(188\) 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
\(197\) 684.000i 0.247376i 0.992321 + 0.123688i \(0.0394721\pi\)
−0.992321 + 0.123688i \(0.960528\pi\)
\(198\) 0 0
\(199\) −4064.00 −1.44769 −0.723843 0.689965i \(-0.757626\pi\)
−0.723843 + 0.689965i \(0.757626\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) − 5504.00i − 1.90298i
\(204\) 0 0
\(205\) 1224.00i 0.417014i
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) −704.000 −0.232999
\(210\) 0 0
\(211\) − 2120.00i − 0.691691i −0.938291 0.345846i \(-0.887592\pi\)
0.938291 0.345846i \(-0.112408\pi\)
\(212\) 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
\(221\) − 1960.00i − 0.596579i
\(222\) 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
\(227\) 3848.00i 1.12511i 0.826759 + 0.562557i \(0.190182\pi\)
−0.826759 + 0.562557i \(0.809818\pi\)
\(228\) 0 0
\(229\) − 652.000i − 0.188146i −0.995565 0.0940729i \(-0.970011\pi\)
0.995565 0.0940729i \(-0.0299887\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 3050.00 0.857563 0.428781 0.903408i \(-0.358943\pi\)
0.428781 + 0.903408i \(0.358943\pi\)
\(234\) 0 0
\(235\) − 3840.00i − 1.06593i
\(236\) 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
\(245\) 8172.00i 2.13098i
\(246\) 0 0
\(247\) 1760.00 0.453385
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) − 792.000i − 0.199166i −0.995029 0.0995829i \(-0.968249\pi\)
0.995029 0.0995829i \(-0.0317508\pi\)
\(252\) 0 0
\(253\) 256.000i 0.0636149i
\(254\) 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
\(259\) − 2944.00i − 0.706298i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) −3488.00 −0.817792 −0.408896 0.912581i \(-0.634086\pi\)
−0.408896 + 0.912581i \(0.634086\pi\)
\(264\) 0 0
\(265\) −912.000 −0.211410
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) − 4764.00i − 1.07980i −0.841729 0.539900i \(-0.818462\pi\)
0.841729 0.539900i \(-0.181538\pi\)
\(270\) 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
\(275\) − 152.000i − 0.0333307i
\(276\) 0 0
\(277\) 8596.00i 1.86456i 0.361735 + 0.932281i \(0.382184\pi\)
−0.361735 + 0.932281i \(0.617816\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 2874.00 0.610137 0.305068 0.952330i \(-0.401321\pi\)
0.305068 + 0.952330i \(0.401321\pi\)
\(282\) 0 0
\(283\) − 2888.00i − 0.606621i −0.952892 0.303311i \(-0.901908\pi\)
0.952892 0.303311i \(-0.0980920\pi\)
\(284\) 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
\(293\) 6540.00i 1.30400i 0.758221 + 0.651998i \(0.226069\pi\)
−0.758221 + 0.651998i \(0.773931\pi\)
\(294\) 0 0
\(295\) 4896.00 0.966292
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) − 640.000i − 0.123786i
\(300\) 0 0
\(301\) − 9472.00i − 1.81381i
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) −7632.00 −1.43281
\(306\) 0 0
\(307\) 10584.0i 1.96762i 0.179202 + 0.983812i \(0.442649\pi\)
−0.179202 + 0.983812i \(0.557351\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) −6368.00 −1.16108 −0.580540 0.814231i \(-0.697159\pi\)
−0.580540 + 0.814231i \(0.697159\pi\)
\(312\) 0 0
\(313\) −8758.00 −1.58157 −0.790785 0.612094i \(-0.790327\pi\)
−0.790785 + 0.612094i \(0.790327\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) − 716.000i − 0.126860i −0.997986 0.0634299i \(-0.979796\pi\)
0.997986 0.0634299i \(-0.0202039\pi\)
\(318\) 0 0
\(319\) 1376.00 0.241508
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 8624.00i 1.48561i
\(324\) 0 0
\(325\) 380.000i 0.0648573i
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) −10240.0 −1.71596
\(330\) 0 0
\(331\) 4408.00i 0.731981i 0.930619 + 0.365990i \(0.119270\pi\)
−0.930619 + 0.365990i \(0.880730\pi\)
\(332\) 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
\(341\) 2048.00i 0.325236i
\(342\) 0 0
\(343\) 10816.0 1.70265
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 5160.00i 0.798280i 0.916890 + 0.399140i \(0.130691\pi\)
−0.916890 + 0.399140i \(0.869309\pi\)
\(348\) 0 0
\(349\) 4876.00i 0.747869i 0.927455 + 0.373935i \(0.121992\pi\)
−0.927455 + 0.373935i \(0.878008\pi\)
\(350\) 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
\(355\) − 4992.00i − 0.746332i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −4128.00 −0.606873 −0.303437 0.952852i \(-0.598134\pi\)
−0.303437 + 0.952852i \(0.598134\pi\)
\(360\) 0 0
\(361\) −885.000 −0.129028
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) − 1656.00i − 0.237477i
\(366\) 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
\(371\) 2432.00i 0.340332i
\(372\) 0 0
\(373\) 4180.00i 0.580247i 0.956989 + 0.290124i \(0.0936964\pi\)
−0.956989 + 0.290124i \(0.906304\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −3440.00 −0.469944
\(378\) 0 0
\(379\) − 13736.0i − 1.86166i −0.365446 0.930832i \(-0.619084\pi\)
0.365446 0.930832i \(-0.380916\pi\)
\(380\) 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
\(389\) − 1732.00i − 0.225748i −0.993609 0.112874i \(-0.963994\pi\)
0.993609 0.112874i \(-0.0360056\pi\)
\(390\) 0 0
\(391\) 3136.00 0.405612
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 768.000i 0.0978285i
\(396\) 0 0
\(397\) − 10436.0i − 1.31931i −0.751567 0.659657i \(-0.770701\pi\)
0.751567 0.659657i \(-0.229299\pi\)
\(398\) 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
\(403\) − 5120.00i − 0.632867i
\(404\) 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 \(-0.598018\pi\)
−0.303088 + 0.952962i \(0.598018\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) − 13056.0i − 1.55555i
\(414\) 0 0
\(415\) −4704.00 −0.556410
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) − 8024.00i − 0.935556i −0.883846 0.467778i \(-0.845055\pi\)
0.883846 0.467778i \(-0.154945\pi\)
\(420\) 0 0
\(421\) − 2348.00i − 0.271816i −0.990721 0.135908i \(-0.956605\pi\)
0.990721 0.135908i \(-0.0433952\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) −1862.00 −0.212518
\(426\) 0 0
\(427\) 20352.0i 2.30656i
\(428\) 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
\(437\) 2816.00i 0.308255i
\(438\) 0 0
\(439\) −14112.0 −1.53423 −0.767117 0.641507i \(-0.778310\pi\)
−0.767117 + 0.641507i \(0.778310\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 4488.00i 0.481335i 0.970608 + 0.240667i \(0.0773663\pi\)
−0.970608 + 0.240667i \(0.922634\pi\)
\(444\) 0 0
\(445\) − 6984.00i − 0.743985i
\(446\) 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
\(451\) 816.000i 0.0851972i
\(452\) 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 \(-0.597538\pi\)
−0.301652 + 0.953418i \(0.597538\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) − 7068.00i − 0.714077i −0.934090 0.357039i \(-0.883786\pi\)
0.934090 0.357039i \(-0.116214\pi\)
\(462\) 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
\(467\) − 13080.0i − 1.29608i −0.761606 0.648041i \(-0.775589\pi\)
0.761606 0.648041i \(-0.224411\pi\)
\(468\) 0 0
\(469\) − 17664.0i − 1.73912i
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 2368.00 0.230192
\(474\) 0 0
\(475\) − 1672.00i − 0.161509i
\(476\) 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
\(485\) 2856.00i 0.267390i
\(486\) 0 0
\(487\) 1696.00 0.157809 0.0789046 0.996882i \(-0.474858\pi\)
0.0789046 + 0.996882i \(0.474858\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) − 3096.00i − 0.284563i −0.989826 0.142282i \(-0.954556\pi\)
0.989826 0.142282i \(-0.0454439\pi\)
\(492\) 0 0
\(493\) − 16856.0i − 1.53987i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −13312.0 −1.20146
\(498\) 0 0
\(499\) − 19208.0i − 1.72318i −0.507603 0.861591i \(-0.669468\pi\)
0.507603 0.861591i \(-0.330532\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) −16224.0 −1.43816 −0.719078 0.694929i \(-0.755436\pi\)
−0.719078 + 0.694929i \(0.755436\pi\)
\(504\) 0 0
\(505\) −17616.0 −1.55228
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) − 11292.0i − 0.983318i −0.870788 0.491659i \(-0.836391\pi\)
0.870788 0.491659i \(-0.163609\pi\)
\(510\) 0 0
\(511\) −4416.00 −0.382294
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) − 11904.0i − 1.01855i
\(516\) 0 0
\(517\) − 2560.00i − 0.217773i
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) −5178.00 −0.435417 −0.217709 0.976014i \(-0.569858\pi\)
−0.217709 + 0.976014i \(0.569858\pi\)
\(522\) 0 0
\(523\) − 6856.00i − 0.573216i −0.958048 0.286608i \(-0.907472\pi\)
0.958048 0.286608i \(-0.0925277\pi\)
\(524\) 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
\(533\) − 2040.00i − 0.165783i
\(534\) 0 0
\(535\) −7008.00 −0.566322
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 5448.00i 0.435365i
\(540\) 0 0
\(541\) − 13732.0i − 1.09128i −0.838018 0.545642i \(-0.816286\pi\)
0.838018 0.545642i \(-0.183714\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 8880.00 0.697940
\(546\) 0 0
\(547\) 10968.0i 0.857327i 0.903464 + 0.428663i \(0.141015\pi\)
−0.903464 + 0.428663i \(0.858985\pi\)
\(548\) 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
\(557\) − 25612.0i − 1.94832i −0.225855 0.974161i \(-0.572518\pi\)
0.225855 0.974161i \(-0.427482\pi\)
\(558\) 0 0
\(559\) −5920.00 −0.447924
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 9768.00i 0.731212i 0.930770 + 0.365606i \(0.119138\pi\)
−0.930770 + 0.365606i \(0.880862\pi\)
\(564\) 0 0
\(565\) 3624.00i 0.269846i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 22838.0 1.68263 0.841317 0.540542i \(-0.181781\pi\)
0.841317 + 0.540542i \(0.181781\pi\)
\(570\) 0 0
\(571\) 9208.00i 0.674856i 0.941351 + 0.337428i \(0.109557\pi\)
−0.941351 + 0.337428i \(0.890443\pi\)
\(572\) 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
\(581\) 12544.0i 0.895719i
\(582\) 0 0
\(583\) −608.000 −0.0431917
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) − 18040.0i − 1.26847i −0.773141 0.634234i \(-0.781316\pi\)
0.773141 0.634234i \(-0.218684\pi\)
\(588\) 0 0
\(589\) 22528.0i 1.57598i
\(590\) 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
\(595\) 37632.0i 2.59288i
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 18336.0 1.25073 0.625366 0.780331i \(-0.284950\pi\)
0.625366 + 0.780331i \(0.284950\pi\)
\(600\) 0 0
\(601\) 9286.00 0.630256 0.315128 0.949049i \(-0.397953\pi\)
0.315128 + 0.949049i \(0.397953\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 15204.0i 1.02170i
\(606\) 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
\(611\) 6400.00i 0.423758i
\(612\) 0 0
\(613\) − 5868.00i − 0.386633i −0.981136 0.193317i \(-0.938076\pi\)
0.981136 0.193317i \(-0.0619245\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −19286.0 −1.25839 −0.629194 0.777248i \(-0.716615\pi\)
−0.629194 + 0.777248i \(0.716615\pi\)
\(618\) 0 0
\(619\) 5240.00i 0.340248i 0.985423 + 0.170124i \(0.0544168\pi\)
−0.985423 + 0.170124i \(0.945583\pi\)
\(620\) 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
\(629\) − 9016.00i − 0.571529i
\(630\) 0 0
\(631\) −15520.0 −0.979147 −0.489573 0.871962i \(-0.662847\pi\)
−0.489573 + 0.871962i \(0.662847\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 19968.0i 1.24788i
\(636\) 0 0
\(637\) − 13620.0i − 0.847165i
\(638\) 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
\(643\) − 8232.00i − 0.504881i −0.967612 0.252440i \(-0.918767\pi\)
0.967612 0.252440i \(-0.0812332\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 24672.0 1.49916 0.749580 0.661914i \(-0.230256\pi\)
0.749580 + 0.661914i \(0.230256\pi\)
\(648\) 0 0
\(649\) 3264.00 0.197416
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 22052.0i 1.32153i 0.750591 + 0.660767i \(0.229769\pi\)
−0.750591 + 0.660767i \(0.770231\pi\)
\(654\) 0 0
\(655\) 27936.0 1.66649
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) − 12024.0i − 0.710757i −0.934723 0.355378i \(-0.884352\pi\)
0.934723 0.355378i \(-0.115648\pi\)
\(660\) 0 0
\(661\) − 19100.0i − 1.12391i −0.827168 0.561955i \(-0.810050\pi\)
0.827168 0.561955i \(-0.189950\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) −33792.0 −1.97052
\(666\) 0 0
\(667\) − 5504.00i − 0.319514i
\(668\) 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
\(677\) − 19684.0i − 1.11746i −0.829351 0.558728i \(-0.811290\pi\)
0.829351 0.558728i \(-0.188710\pi\)
\(678\) 0 0
\(679\) 7616.00 0.430450
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) − 19864.0i − 1.11285i −0.830899 0.556424i \(-0.812173\pi\)
0.830899 0.556424i \(-0.187827\pi\)
\(684\) 0 0
\(685\) 20808.0i 1.16063i
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 1520.00 0.0840456
\(690\) 0 0
\(691\) 3256.00i 0.179253i 0.995975 + 0.0896267i \(0.0285674\pi\)
−0.995975 + 0.0896267i \(0.971433\pi\)
\(692\) 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
\(701\) − 2876.00i − 0.154957i −0.996994 0.0774786i \(-0.975313\pi\)
0.996994 0.0774786i \(-0.0246870\pi\)
\(702\) 0 0
\(703\) 8096.00 0.434348
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 46976.0i 2.49889i
\(708\) 0 0
\(709\) 7300.00i 0.386682i 0.981132 + 0.193341i \(0.0619323\pi\)
−0.981132 + 0.193341i \(0.938068\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 8192.00 0.430284
\(714\) 0 0
\(715\) 1920.00i 0.100425i
\(716\) 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
\(725\) 3268.00i 0.167408i
\(726\) 0 0
\(727\) 8800.00 0.448933 0.224466 0.974482i \(-0.427936\pi\)
0.224466 + 0.974482i \(0.427936\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) − 29008.0i − 1.46771i
\(732\) 0 0
\(733\) − 21076.0i − 1.06202i −0.847366 0.531009i \(-0.821813\pi\)
0.847366 0.531009i \(-0.178187\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 4416.00 0.220713
\(738\) 0 0
\(739\) − 19336.0i − 0.962498i −0.876584 0.481249i \(-0.840183\pi\)
0.876584 0.481249i \(-0.159817\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 13664.0 0.674675 0.337338 0.941384i \(-0.390474\pi\)
0.337338 + 0.941384i \(0.390474\pi\)
\(744\) 0 0
\(745\) −21456.0 −1.05515
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 18688.0i 0.911675i
\(750\) 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
\(755\) 5760.00i 0.277653i
\(756\) 0 0
\(757\) 20004.0i 0.960446i 0.877146 + 0.480223i \(0.159444\pi\)
−0.877146 + 0.480223i \(0.840556\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 31478.0 1.49944 0.749722 0.661753i \(-0.230187\pi\)
0.749722 + 0.661753i \(0.230187\pi\)
\(762\) 0 0
\(763\) − 23680.0i − 1.12356i
\(764\) 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
\(773\) − 9604.00i − 0.446872i −0.974719 0.223436i \(-0.928273\pi\)
0.974719 0.223436i \(-0.0717273\pi\)
\(774\) 0 0
\(775\) −4864.00 −0.225445
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 8976.00i 0.412835i
\(780\) 0 0
\(781\) − 3328.00i − 0.152478i
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) −27600.0 −1.25489
\(786\) 0 0
\(787\) − 3144.00i − 0.142403i −0.997462 0.0712017i \(-0.977317\pi\)
0.997462 0.0712017i \(-0.0226834\pi\)
\(788\) 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
\(797\) 22084.0i 0.981500i 0.871300 + 0.490750i \(0.163277\pi\)
−0.871300 + 0.490750i \(0.836723\pi\)
\(798\) 0 0
\(799\) −31360.0 −1.38853
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) − 1104.00i − 0.0485172i
\(804\) 0 0
\(805\) 12288.0i 0.538006i
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 14950.0 0.649708 0.324854 0.945764i \(-0.394685\pi\)
0.324854 + 0.945764i \(0.394685\pi\)
\(810\) 0 0
\(811\) − 23432.0i − 1.01456i −0.861781 0.507280i \(-0.830651\pi\)
0.861781 0.507280i \(-0.169349\pi\)
\(812\) 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
\(821\) − 1044.00i − 0.0443798i −0.999754 0.0221899i \(-0.992936\pi\)
0.999754 0.0221899i \(-0.00706385\pi\)
\(822\) 0 0
\(823\) −18208.0 −0.771192 −0.385596 0.922668i \(-0.626004\pi\)
−0.385596 + 0.922668i \(0.626004\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 12488.0i 0.525091i 0.964920 + 0.262546i \(0.0845620\pi\)
−0.964920 + 0.262546i \(0.915438\pi\)
\(828\) 0 0
\(829\) 30172.0i 1.26407i 0.774938 + 0.632037i \(0.217781\pi\)
−0.774938 + 0.632037i \(0.782219\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 66738.0 2.77591
\(834\) 0 0
\(835\) − 26496.0i − 1.09812i
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) −32544.0 −1.33915 −0.669573 0.742746i \(-0.733523\pi\)
−0.669573 + 0.742746i \(0.733523\pi\)
\(840\) 0 0
\(841\) −5195.00 −0.213006
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 21564.0i 0.877898i
\(846\) 0 0
\(847\) 40544.0 1.64476
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) − 2944.00i − 0.118589i
\(852\) 0 0
\(853\) − 26156.0i − 1.04990i −0.851133 0.524950i \(-0.824084\pi\)
0.851133 0.524950i \(-0.175916\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 18646.0 0.743215 0.371607 0.928390i \(-0.378807\pi\)
0.371607 + 0.928390i \(0.378807\pi\)
\(858\) 0 0
\(859\) − 5800.00i − 0.230377i −0.993344 0.115188i \(-0.963253\pi\)
0.993344 0.115188i \(-0.0367471\pi\)
\(860\) 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
\(869\) 512.000i 0.0199867i
\(870\) 0 0
\(871\) −11040.0 −0.429479
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 40704.0i 1.57262i
\(876\) 0 0
\(877\) 3004.00i 0.115665i 0.998326 + 0.0578323i \(0.0184189\pi\)
−0.998326 + 0.0578323i \(0.981581\pi\)
\(878\) 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
\(883\) − 27880.0i − 1.06256i −0.847198 0.531278i \(-0.821712\pi\)
0.847198 0.531278i \(-0.178288\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) −7392.00 −0.279819 −0.139909 0.990164i \(-0.544681\pi\)
−0.139909 + 0.990164i \(0.544681\pi\)
\(888\) 0 0
\(889\) 53248.0 2.00886
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) − 28160.0i − 1.05525i
\(894\) 0 0
\(895\) 25248.0 0.942958
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) − 44032.0i − 1.63354i
\(900\) 0 0
\(901\) 7448.00i 0.275393i
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) −16944.0 −0.622362
\(906\) 0 0
\(907\) 29080.0i 1.06459i 0.846558 + 0.532296i \(0.178671\pi\)
−0.846558 + 0.532296i \(0.821329\pi\)
\(908\) 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
\(917\) − 74496.0i − 2.68274i
\(918\) 0 0
\(919\) 19680.0 0.706402 0.353201 0.935547i \(-0.385093\pi\)
0.353201 + 0.935547i \(0.385093\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 8320.00i 0.296702i
\(924\) 0 0
\(925\) 1748.00i 0.0621339i
\(926\) 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
\(931\) 59928.0i 2.10962i
\(932\) 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 \(-0.576248\pi\)
−0.237257 + 0.971447i \(0.576248\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 30692.0i 1.06326i 0.846976 + 0.531632i \(0.178421\pi\)
−0.846976 + 0.531632i \(0.821579\pi\)
\(942\) 0 0
\(943\) 3264.00 0.112715
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) − 4824.00i − 0.165532i −0.996569 0.0827661i \(-0.973625\pi\)
0.996569 0.0827661i \(-0.0263754\pi\)
\(948\) 0 0
\(949\) 2760.00i 0.0944082i
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) −22986.0 −0.781311 −0.390656 0.920537i \(-0.627752\pi\)
−0.390656 + 0.920537i \(0.627752\pi\)
\(954\) 0 0
\(955\) 44544.0i 1.50933i
\(956\) 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
\(965\) 19368.0i 0.646091i
\(966\) 0 0
\(967\) 17184.0 0.571458 0.285729 0.958310i \(-0.407764\pi\)
0.285729 + 0.958310i \(0.407764\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 2920.00i 0.0965059i 0.998835 + 0.0482530i \(0.0153654\pi\)
−0.998835 + 0.0482530i \(0.984635\pi\)
\(972\) 0 0
\(973\) 97024.0i 3.19676i
\(974\) 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
\(979\) − 4656.00i − 0.151998i
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) 44192.0 1.43388 0.716941 0.697134i \(-0.245542\pi\)
0.716941 + 0.697134i \(0.245542\pi\)
\(984\) 0 0
\(985\) −8208.00 −0.265511
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) − 9472.00i − 0.304542i
\(990\) 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
\(995\) − 48768.0i − 1.55382i
\(996\) 0 0
\(997\) − 11612.0i − 0.368862i −0.982845 0.184431i \(-0.940956\pi\)
0.982845 0.184431i \(-0.0590443\pi\)
\(998\) 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 1152.4.d.h.577.2 2
3.2 odd 2 128.4.b.d.65.1 yes 2
4.3 odd 2 1152.4.d.a.577.2 2
8.3 odd 2 1152.4.d.a.577.1 2
8.5 even 2 inner 1152.4.d.h.577.1 2
12.11 even 2 128.4.b.a.65.2 yes 2
16.3 odd 4 2304.4.a.n.1.1 1
16.5 even 4 2304.4.a.c.1.1 1
16.11 odd 4 2304.4.a.d.1.1 1
16.13 even 4 2304.4.a.m.1.1 1
24.5 odd 2 128.4.b.d.65.2 yes 2
24.11 even 2 128.4.b.a.65.1 2
48.5 odd 4 256.4.a.c.1.1 1
48.11 even 4 256.4.a.g.1.1 1
48.29 odd 4 256.4.a.f.1.1 1
48.35 even 4 256.4.a.b.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
128.4.b.a.65.1 2 24.11 even 2
128.4.b.a.65.2 yes 2 12.11 even 2
128.4.b.d.65.1 yes 2 3.2 odd 2
128.4.b.d.65.2 yes 2 24.5 odd 2
256.4.a.b.1.1 1 48.35 even 4
256.4.a.c.1.1 1 48.5 odd 4
256.4.a.f.1.1 1 48.29 odd 4
256.4.a.g.1.1 1 48.11 even 4
1152.4.d.a.577.1 2 8.3 odd 2
1152.4.d.a.577.2 2 4.3 odd 2
1152.4.d.h.577.1 2 8.5 even 2 inner
1152.4.d.h.577.2 2 1.1 even 1 trivial
2304.4.a.c.1.1 1 16.5 even 4
2304.4.a.d.1.1 1 16.11 odd 4
2304.4.a.m.1.1 1 16.13 even 4
2304.4.a.n.1.1 1 16.3 odd 4