Properties

Label 9800.2.a.cx.1.1
Level $9800$
Weight $2$
Character 9800.1
Self dual yes
Analytic conductor $78.253$
Analytic rank $0$
Dimension $6$
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] = [9800,2,Mod(1,9800)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(9800, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("9800.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 9800 = 2^{3} \cdot 5^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 9800.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: \(78.2533939809\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: 6.6.239575536.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - x^{5} - 12x^{4} + 8x^{3} + 35x^{2} - 15x - 8 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 280)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-2.54431\) of defining polynomial
Character \(\chi\) \(=\) 9800.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-2.54431 q^{3} +3.47349 q^{9} +O(q^{10})\) \(q-2.54431 q^{3} +3.47349 q^{9} -3.91228 q^{11} +3.47349 q^{13} -5.08262 q^{17} -5.25188 q^{19} -6.01780 q^{23} -1.20471 q^{27} -10.2660 q^{29} +2.04350 q^{31} +9.95405 q^{33} +0.405001 q^{37} -8.83763 q^{39} -2.57534 q^{41} -6.15769 q^{43} -0.480556 q^{47} +12.9317 q^{51} +0.466428 q^{53} +13.3624 q^{57} -10.3113 q^{59} +8.08088 q^{61} +4.15169 q^{67} +15.3111 q^{69} -4.28658 q^{71} +4.81483 q^{73} -12.6639 q^{79} -7.35533 q^{81} +8.42280 q^{83} +26.1199 q^{87} +2.38595 q^{89} -5.19930 q^{93} +1.32080 q^{97} -13.5893 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + q^{3} + 7 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q + q^{3} + 7 q^{9} + q^{11} + 7 q^{13} + 4 q^{17} - 5 q^{19} - 6 q^{23} + 7 q^{27} - 3 q^{29} - 2 q^{31} + 16 q^{33} - 9 q^{37} + 10 q^{39} + 6 q^{41} + 3 q^{43} + 27 q^{47} + 5 q^{53} + 26 q^{57} - 24 q^{59} + 9 q^{61} - 17 q^{67} + 15 q^{69} + 4 q^{71} + 18 q^{73} - 22 q^{79} - 6 q^{81} + 9 q^{83} + 39 q^{87} + 15 q^{89} - 10 q^{93} + 12 q^{97} + 11 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Coefficient data

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



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) −2.54431 −1.46896 −0.734478 0.678633i \(-0.762573\pi\)
−0.734478 + 0.678633i \(0.762573\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) 0 0
\(8\) 0 0
\(9\) 3.47349 1.15783
\(10\) 0 0
\(11\) −3.91228 −1.17960 −0.589799 0.807550i \(-0.700793\pi\)
−0.589799 + 0.807550i \(0.700793\pi\)
\(12\) 0 0
\(13\) 3.47349 0.963373 0.481687 0.876344i \(-0.340024\pi\)
0.481687 + 0.876344i \(0.340024\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −5.08262 −1.23272 −0.616358 0.787466i \(-0.711393\pi\)
−0.616358 + 0.787466i \(0.711393\pi\)
\(18\) 0 0
\(19\) −5.25188 −1.20486 −0.602432 0.798170i \(-0.705802\pi\)
−0.602432 + 0.798170i \(0.705802\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −6.01780 −1.25480 −0.627399 0.778698i \(-0.715880\pi\)
−0.627399 + 0.778698i \(0.715880\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) −1.20471 −0.231846
\(28\) 0 0
\(29\) −10.2660 −1.90635 −0.953175 0.302419i \(-0.902206\pi\)
−0.953175 + 0.302419i \(0.902206\pi\)
\(30\) 0 0
\(31\) 2.04350 0.367024 0.183512 0.983017i \(-0.441253\pi\)
0.183512 + 0.983017i \(0.441253\pi\)
\(32\) 0 0
\(33\) 9.95405 1.73278
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 0.405001 0.0665818 0.0332909 0.999446i \(-0.489401\pi\)
0.0332909 + 0.999446i \(0.489401\pi\)
\(38\) 0 0
\(39\) −8.83763 −1.41515
\(40\) 0 0
\(41\) −2.57534 −0.402200 −0.201100 0.979571i \(-0.564452\pi\)
−0.201100 + 0.979571i \(0.564452\pi\)
\(42\) 0 0
\(43\) −6.15769 −0.939038 −0.469519 0.882922i \(-0.655573\pi\)
−0.469519 + 0.882922i \(0.655573\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −0.480556 −0.0700963 −0.0350481 0.999386i \(-0.511158\pi\)
−0.0350481 + 0.999386i \(0.511158\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 0 0
\(51\) 12.9317 1.81081
\(52\) 0 0
\(53\) 0.466428 0.0640688 0.0320344 0.999487i \(-0.489801\pi\)
0.0320344 + 0.999487i \(0.489801\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 13.3624 1.76989
\(58\) 0 0
\(59\) −10.3113 −1.34242 −0.671208 0.741269i \(-0.734224\pi\)
−0.671208 + 0.741269i \(0.734224\pi\)
\(60\) 0 0
\(61\) 8.08088 1.03465 0.517325 0.855789i \(-0.326928\pi\)
0.517325 + 0.855789i \(0.326928\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 4.15169 0.507210 0.253605 0.967308i \(-0.418384\pi\)
0.253605 + 0.967308i \(0.418384\pi\)
\(68\) 0 0
\(69\) 15.3111 1.84324
\(70\) 0 0
\(71\) −4.28658 −0.508724 −0.254362 0.967109i \(-0.581865\pi\)
−0.254362 + 0.967109i \(0.581865\pi\)
\(72\) 0 0
\(73\) 4.81483 0.563533 0.281767 0.959483i \(-0.409080\pi\)
0.281767 + 0.959483i \(0.409080\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) −12.6639 −1.42481 −0.712403 0.701771i \(-0.752393\pi\)
−0.712403 + 0.701771i \(0.752393\pi\)
\(80\) 0 0
\(81\) −7.35533 −0.817259
\(82\) 0 0
\(83\) 8.42280 0.924522 0.462261 0.886744i \(-0.347038\pi\)
0.462261 + 0.886744i \(0.347038\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 26.1199 2.80034
\(88\) 0 0
\(89\) 2.38595 0.252910 0.126455 0.991972i \(-0.459640\pi\)
0.126455 + 0.991972i \(0.459640\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) −5.19930 −0.539142
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 1.32080 0.134107 0.0670537 0.997749i \(-0.478640\pi\)
0.0670537 + 0.997749i \(0.478640\pi\)
\(98\) 0 0
\(99\) −13.5893 −1.36577
\(100\) 0 0
\(101\) 7.75166 0.771319 0.385660 0.922641i \(-0.373974\pi\)
0.385660 + 0.922641i \(0.373974\pi\)
\(102\) 0 0
\(103\) 2.20364 0.217131 0.108565 0.994089i \(-0.465374\pi\)
0.108565 + 0.994089i \(0.465374\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −10.8235 −1.04635 −0.523176 0.852225i \(-0.675253\pi\)
−0.523176 + 0.852225i \(0.675253\pi\)
\(108\) 0 0
\(109\) −11.0433 −1.05776 −0.528880 0.848697i \(-0.677388\pi\)
−0.528880 + 0.848697i \(0.677388\pi\)
\(110\) 0 0
\(111\) −1.03045 −0.0978057
\(112\) 0 0
\(113\) 7.29167 0.685942 0.342971 0.939346i \(-0.388567\pi\)
0.342971 + 0.939346i \(0.388567\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 12.0651 1.11542
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) 4.30597 0.391452
\(122\) 0 0
\(123\) 6.55244 0.590814
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) 7.66305 0.679986 0.339993 0.940428i \(-0.389575\pi\)
0.339993 + 0.940428i \(0.389575\pi\)
\(128\) 0 0
\(129\) 15.6670 1.37941
\(130\) 0 0
\(131\) −17.2119 −1.50381 −0.751906 0.659270i \(-0.770865\pi\)
−0.751906 + 0.659270i \(0.770865\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −5.26072 −0.449454 −0.224727 0.974422i \(-0.572149\pi\)
−0.224727 + 0.974422i \(0.572149\pi\)
\(138\) 0 0
\(139\) −15.2148 −1.29050 −0.645250 0.763971i \(-0.723247\pi\)
−0.645250 + 0.763971i \(0.723247\pi\)
\(140\) 0 0
\(141\) 1.22268 0.102968
\(142\) 0 0
\(143\) −13.5893 −1.13639
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) −1.01721 −0.0833335 −0.0416667 0.999132i \(-0.513267\pi\)
−0.0416667 + 0.999132i \(0.513267\pi\)
\(150\) 0 0
\(151\) 13.2583 1.07894 0.539472 0.842004i \(-0.318624\pi\)
0.539472 + 0.842004i \(0.318624\pi\)
\(152\) 0 0
\(153\) −17.6544 −1.42728
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 17.7722 1.41838 0.709189 0.705019i \(-0.249061\pi\)
0.709189 + 0.705019i \(0.249061\pi\)
\(158\) 0 0
\(159\) −1.18674 −0.0941143
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) 2.91952 0.228675 0.114337 0.993442i \(-0.463526\pi\)
0.114337 + 0.993442i \(0.463526\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −9.21710 −0.713240 −0.356620 0.934249i \(-0.616071\pi\)
−0.356620 + 0.934249i \(0.616071\pi\)
\(168\) 0 0
\(169\) −0.934854 −0.0719118
\(170\) 0 0
\(171\) −18.2424 −1.39503
\(172\) 0 0
\(173\) −21.6803 −1.64832 −0.824162 0.566354i \(-0.808353\pi\)
−0.824162 + 0.566354i \(0.808353\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 26.2351 1.97195
\(178\) 0 0
\(179\) −24.9965 −1.86833 −0.934164 0.356843i \(-0.883853\pi\)
−0.934164 + 0.356843i \(0.883853\pi\)
\(180\) 0 0
\(181\) −4.17900 −0.310623 −0.155311 0.987866i \(-0.549638\pi\)
−0.155311 + 0.987866i \(0.549638\pi\)
\(182\) 0 0
\(183\) −20.5602 −1.51986
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 19.8846 1.45411
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −8.97169 −0.649169 −0.324585 0.945857i \(-0.605224\pi\)
−0.324585 + 0.945857i \(0.605224\pi\)
\(192\) 0 0
\(193\) −18.2218 −1.31164 −0.655818 0.754919i \(-0.727676\pi\)
−0.655818 + 0.754919i \(0.727676\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 23.5632 1.67881 0.839403 0.543509i \(-0.182905\pi\)
0.839403 + 0.543509i \(0.182905\pi\)
\(198\) 0 0
\(199\) 20.7163 1.46854 0.734271 0.678856i \(-0.237524\pi\)
0.734271 + 0.678856i \(0.237524\pi\)
\(200\) 0 0
\(201\) −10.5632 −0.745069
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) −20.9028 −1.45284
\(208\) 0 0
\(209\) 20.5469 1.42126
\(210\) 0 0
\(211\) 7.69865 0.529997 0.264998 0.964249i \(-0.414629\pi\)
0.264998 + 0.964249i \(0.414629\pi\)
\(212\) 0 0
\(213\) 10.9064 0.747292
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) −12.2504 −0.827805
\(220\) 0 0
\(221\) −17.6544 −1.18757
\(222\) 0 0
\(223\) 12.7136 0.851366 0.425683 0.904872i \(-0.360034\pi\)
0.425683 + 0.904872i \(0.360034\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) −10.7229 −0.711706 −0.355853 0.934542i \(-0.615810\pi\)
−0.355853 + 0.934542i \(0.615810\pi\)
\(228\) 0 0
\(229\) −24.7892 −1.63812 −0.819059 0.573709i \(-0.805504\pi\)
−0.819059 + 0.573709i \(0.805504\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 22.7544 1.49069 0.745345 0.666679i \(-0.232285\pi\)
0.745345 + 0.666679i \(0.232285\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 32.2210 2.09298
\(238\) 0 0
\(239\) −1.49071 −0.0964258 −0.0482129 0.998837i \(-0.515353\pi\)
−0.0482129 + 0.998837i \(0.515353\pi\)
\(240\) 0 0
\(241\) 26.1677 1.68561 0.842806 0.538217i \(-0.180902\pi\)
0.842806 + 0.538217i \(0.180902\pi\)
\(242\) 0 0
\(243\) 22.3283 1.43236
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) −18.2424 −1.16073
\(248\) 0 0
\(249\) −21.4302 −1.35808
\(250\) 0 0
\(251\) 4.90277 0.309460 0.154730 0.987957i \(-0.450549\pi\)
0.154730 + 0.987957i \(0.450549\pi\)
\(252\) 0 0
\(253\) 23.5433 1.48016
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 12.3399 0.769739 0.384870 0.922971i \(-0.374246\pi\)
0.384870 + 0.922971i \(0.374246\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) −35.6589 −2.20723
\(262\) 0 0
\(263\) −20.1247 −1.24094 −0.620471 0.784229i \(-0.713059\pi\)
−0.620471 + 0.784229i \(0.713059\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) −6.07059 −0.371514
\(268\) 0 0
\(269\) −14.9394 −0.910874 −0.455437 0.890268i \(-0.650517\pi\)
−0.455437 + 0.890268i \(0.650517\pi\)
\(270\) 0 0
\(271\) −5.80932 −0.352891 −0.176446 0.984310i \(-0.556460\pi\)
−0.176446 + 0.984310i \(0.556460\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) −25.9390 −1.55853 −0.779263 0.626697i \(-0.784406\pi\)
−0.779263 + 0.626697i \(0.784406\pi\)
\(278\) 0 0
\(279\) 7.09809 0.424952
\(280\) 0 0
\(281\) −14.8840 −0.887904 −0.443952 0.896051i \(-0.646424\pi\)
−0.443952 + 0.896051i \(0.646424\pi\)
\(282\) 0 0
\(283\) 9.71540 0.577520 0.288760 0.957401i \(-0.406757\pi\)
0.288760 + 0.957401i \(0.406757\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) 8.83301 0.519589
\(290\) 0 0
\(291\) −3.36053 −0.196998
\(292\) 0 0
\(293\) −8.08596 −0.472387 −0.236194 0.971706i \(-0.575900\pi\)
−0.236194 + 0.971706i \(0.575900\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 4.71316 0.273485
\(298\) 0 0
\(299\) −20.9028 −1.20884
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) −19.7226 −1.13303
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) 8.81809 0.503275 0.251637 0.967822i \(-0.419031\pi\)
0.251637 + 0.967822i \(0.419031\pi\)
\(308\) 0 0
\(309\) −5.60673 −0.318956
\(310\) 0 0
\(311\) 9.66439 0.548017 0.274009 0.961727i \(-0.411650\pi\)
0.274009 + 0.961727i \(0.411650\pi\)
\(312\) 0 0
\(313\) 27.1864 1.53667 0.768334 0.640049i \(-0.221086\pi\)
0.768334 + 0.640049i \(0.221086\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −12.1583 −0.682876 −0.341438 0.939904i \(-0.610914\pi\)
−0.341438 + 0.939904i \(0.610914\pi\)
\(318\) 0 0
\(319\) 40.1635 2.24873
\(320\) 0 0
\(321\) 27.5384 1.53704
\(322\) 0 0
\(323\) 26.6933 1.48526
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) 28.0976 1.55380
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) −3.42945 −0.188500 −0.0942499 0.995549i \(-0.530045\pi\)
−0.0942499 + 0.995549i \(0.530045\pi\)
\(332\) 0 0
\(333\) 1.40677 0.0770904
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 1.39770 0.0761375 0.0380687 0.999275i \(-0.487879\pi\)
0.0380687 + 0.999275i \(0.487879\pi\)
\(338\) 0 0
\(339\) −18.5522 −1.00762
\(340\) 0 0
\(341\) −7.99477 −0.432941
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −3.46302 −0.185904 −0.0929522 0.995671i \(-0.529630\pi\)
−0.0929522 + 0.995671i \(0.529630\pi\)
\(348\) 0 0
\(349\) −19.2360 −1.02968 −0.514838 0.857287i \(-0.672148\pi\)
−0.514838 + 0.857287i \(0.672148\pi\)
\(350\) 0 0
\(351\) −4.18454 −0.223354
\(352\) 0 0
\(353\) 17.2651 0.918931 0.459465 0.888196i \(-0.348041\pi\)
0.459465 + 0.888196i \(0.348041\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 0.867633 0.0457919 0.0228960 0.999738i \(-0.492711\pi\)
0.0228960 + 0.999738i \(0.492711\pi\)
\(360\) 0 0
\(361\) 8.58226 0.451698
\(362\) 0 0
\(363\) −10.9557 −0.575025
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) −16.4984 −0.861210 −0.430605 0.902541i \(-0.641700\pi\)
−0.430605 + 0.902541i \(0.641700\pi\)
\(368\) 0 0
\(369\) −8.94541 −0.465679
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) −20.3871 −1.05560 −0.527801 0.849368i \(-0.676983\pi\)
−0.527801 + 0.849368i \(0.676983\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −35.6589 −1.83653
\(378\) 0 0
\(379\) 13.1652 0.676251 0.338126 0.941101i \(-0.390207\pi\)
0.338126 + 0.941101i \(0.390207\pi\)
\(380\) 0 0
\(381\) −19.4971 −0.998869
\(382\) 0 0
\(383\) 5.19162 0.265280 0.132640 0.991164i \(-0.457655\pi\)
0.132640 + 0.991164i \(0.457655\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) −21.3887 −1.08725
\(388\) 0 0
\(389\) 31.1419 1.57896 0.789478 0.613779i \(-0.210351\pi\)
0.789478 + 0.613779i \(0.210351\pi\)
\(390\) 0 0
\(391\) 30.5862 1.54681
\(392\) 0 0
\(393\) 43.7924 2.20903
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) −15.1065 −0.758174 −0.379087 0.925361i \(-0.623762\pi\)
−0.379087 + 0.925361i \(0.623762\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 16.2471 0.811340 0.405670 0.914020i \(-0.367038\pi\)
0.405670 + 0.914020i \(0.367038\pi\)
\(402\) 0 0
\(403\) 7.09809 0.353581
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) −1.58448 −0.0785397
\(408\) 0 0
\(409\) 33.3206 1.64760 0.823798 0.566883i \(-0.191851\pi\)
0.823798 + 0.566883i \(0.191851\pi\)
\(410\) 0 0
\(411\) 13.3849 0.660228
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 38.7110 1.89569
\(418\) 0 0
\(419\) 11.3290 0.553457 0.276729 0.960948i \(-0.410750\pi\)
0.276729 + 0.960948i \(0.410750\pi\)
\(420\) 0 0
\(421\) 12.5047 0.609440 0.304720 0.952442i \(-0.401437\pi\)
0.304720 + 0.952442i \(0.401437\pi\)
\(422\) 0 0
\(423\) −1.66921 −0.0811596
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) 34.5753 1.66931
\(430\) 0 0
\(431\) −5.14713 −0.247928 −0.123964 0.992287i \(-0.539561\pi\)
−0.123964 + 0.992287i \(0.539561\pi\)
\(432\) 0 0
\(433\) 1.88420 0.0905491 0.0452745 0.998975i \(-0.485584\pi\)
0.0452745 + 0.998975i \(0.485584\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 31.6048 1.51186
\(438\) 0 0
\(439\) 10.5904 0.505454 0.252727 0.967538i \(-0.418673\pi\)
0.252727 + 0.967538i \(0.418673\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 24.3510 1.15695 0.578476 0.815700i \(-0.303648\pi\)
0.578476 + 0.815700i \(0.303648\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 2.58811 0.122413
\(448\) 0 0
\(449\) 20.2506 0.955684 0.477842 0.878446i \(-0.341419\pi\)
0.477842 + 0.878446i \(0.341419\pi\)
\(450\) 0 0
\(451\) 10.0754 0.474434
\(452\) 0 0
\(453\) −33.7331 −1.58492
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −19.1864 −0.897503 −0.448752 0.893657i \(-0.648131\pi\)
−0.448752 + 0.893657i \(0.648131\pi\)
\(458\) 0 0
\(459\) 6.12307 0.285801
\(460\) 0 0
\(461\) −13.4096 −0.624548 −0.312274 0.949992i \(-0.601091\pi\)
−0.312274 + 0.949992i \(0.601091\pi\)
\(462\) 0 0
\(463\) −17.7864 −0.826604 −0.413302 0.910594i \(-0.635625\pi\)
−0.413302 + 0.910594i \(0.635625\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 15.2100 0.703835 0.351918 0.936031i \(-0.385530\pi\)
0.351918 + 0.936031i \(0.385530\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) −45.2180 −2.08353
\(472\) 0 0
\(473\) 24.0906 1.10769
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 1.62013 0.0741809
\(478\) 0 0
\(479\) −12.5684 −0.574266 −0.287133 0.957891i \(-0.592702\pi\)
−0.287133 + 0.957891i \(0.592702\pi\)
\(480\) 0 0
\(481\) 1.40677 0.0641431
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) 13.7901 0.624888 0.312444 0.949936i \(-0.398852\pi\)
0.312444 + 0.949936i \(0.398852\pi\)
\(488\) 0 0
\(489\) −7.42816 −0.335913
\(490\) 0 0
\(491\) 27.4423 1.23845 0.619226 0.785213i \(-0.287447\pi\)
0.619226 + 0.785213i \(0.287447\pi\)
\(492\) 0 0
\(493\) 52.1782 2.34999
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) −19.7379 −0.883590 −0.441795 0.897116i \(-0.645658\pi\)
−0.441795 + 0.897116i \(0.645658\pi\)
\(500\) 0 0
\(501\) 23.4511 1.04772
\(502\) 0 0
\(503\) 5.00417 0.223125 0.111562 0.993757i \(-0.464415\pi\)
0.111562 + 0.993757i \(0.464415\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 2.37855 0.105635
\(508\) 0 0
\(509\) 22.7592 1.00878 0.504392 0.863475i \(-0.331717\pi\)
0.504392 + 0.863475i \(0.331717\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) 6.32699 0.279343
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 1.88007 0.0826854
\(518\) 0 0
\(519\) 55.1614 2.42131
\(520\) 0 0
\(521\) −17.5371 −0.768315 −0.384158 0.923267i \(-0.625508\pi\)
−0.384158 + 0.923267i \(0.625508\pi\)
\(522\) 0 0
\(523\) −13.7901 −0.602998 −0.301499 0.953467i \(-0.597487\pi\)
−0.301499 + 0.953467i \(0.597487\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −10.3863 −0.452436
\(528\) 0 0
\(529\) 13.2139 0.574517
\(530\) 0 0
\(531\) −35.8162 −1.55429
\(532\) 0 0
\(533\) −8.94541 −0.387469
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 63.5988 2.74449
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) 46.0001 1.97770 0.988849 0.148922i \(-0.0475805\pi\)
0.988849 + 0.148922i \(0.0475805\pi\)
\(542\) 0 0
\(543\) 10.6327 0.456291
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 26.4411 1.13054 0.565270 0.824906i \(-0.308772\pi\)
0.565270 + 0.824906i \(0.308772\pi\)
\(548\) 0 0
\(549\) 28.0689 1.19795
\(550\) 0 0
\(551\) 53.9159 2.29689
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −31.7219 −1.34410 −0.672050 0.740506i \(-0.734586\pi\)
−0.672050 + 0.740506i \(0.734586\pi\)
\(558\) 0 0
\(559\) −21.3887 −0.904644
\(560\) 0 0
\(561\) −50.5926 −2.13602
\(562\) 0 0
\(563\) −20.7249 −0.873448 −0.436724 0.899595i \(-0.643861\pi\)
−0.436724 + 0.899595i \(0.643861\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 42.5454 1.78360 0.891799 0.452432i \(-0.149444\pi\)
0.891799 + 0.452432i \(0.149444\pi\)
\(570\) 0 0
\(571\) 30.1062 1.25990 0.629952 0.776634i \(-0.283074\pi\)
0.629952 + 0.776634i \(0.283074\pi\)
\(572\) 0 0
\(573\) 22.8267 0.953601
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) −10.2683 −0.427473 −0.213736 0.976891i \(-0.568563\pi\)
−0.213736 + 0.976891i \(0.568563\pi\)
\(578\) 0 0
\(579\) 46.3619 1.92674
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) −1.82480 −0.0755755
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 41.5546 1.71514 0.857570 0.514367i \(-0.171973\pi\)
0.857570 + 0.514367i \(0.171973\pi\)
\(588\) 0 0
\(589\) −10.7322 −0.442214
\(590\) 0 0
\(591\) −59.9519 −2.46609
\(592\) 0 0
\(593\) 23.9303 0.982698 0.491349 0.870963i \(-0.336504\pi\)
0.491349 + 0.870963i \(0.336504\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) −52.7087 −2.15722
\(598\) 0 0
\(599\) 9.08267 0.371108 0.185554 0.982634i \(-0.440592\pi\)
0.185554 + 0.982634i \(0.440592\pi\)
\(600\) 0 0
\(601\) −29.8162 −1.21623 −0.608114 0.793850i \(-0.708074\pi\)
−0.608114 + 0.793850i \(0.708074\pi\)
\(602\) 0 0
\(603\) 14.4209 0.587263
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) 14.5480 0.590486 0.295243 0.955422i \(-0.404599\pi\)
0.295243 + 0.955422i \(0.404599\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) −1.66921 −0.0675289
\(612\) 0 0
\(613\) 10.3228 0.416934 0.208467 0.978029i \(-0.433153\pi\)
0.208467 + 0.978029i \(0.433153\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 13.0906 0.527008 0.263504 0.964658i \(-0.415122\pi\)
0.263504 + 0.964658i \(0.415122\pi\)
\(618\) 0 0
\(619\) −32.5588 −1.30865 −0.654325 0.756213i \(-0.727047\pi\)
−0.654325 + 0.756213i \(0.727047\pi\)
\(620\) 0 0
\(621\) 7.24969 0.290920
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) −52.2775 −2.08776
\(628\) 0 0
\(629\) −2.05847 −0.0820764
\(630\) 0 0
\(631\) −13.2537 −0.527620 −0.263810 0.964575i \(-0.584979\pi\)
−0.263810 + 0.964575i \(0.584979\pi\)
\(632\) 0 0
\(633\) −19.5877 −0.778542
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) −14.8894 −0.589016
\(640\) 0 0
\(641\) 0.703073 0.0277697 0.0138848 0.999904i \(-0.495580\pi\)
0.0138848 + 0.999904i \(0.495580\pi\)
\(642\) 0 0
\(643\) 4.64163 0.183048 0.0915240 0.995803i \(-0.470826\pi\)
0.0915240 + 0.995803i \(0.470826\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 12.0354 0.473161 0.236580 0.971612i \(-0.423973\pi\)
0.236580 + 0.971612i \(0.423973\pi\)
\(648\) 0 0
\(649\) 40.3407 1.58351
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 18.2464 0.714036 0.357018 0.934098i \(-0.383794\pi\)
0.357018 + 0.934098i \(0.383794\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 16.7243 0.652476
\(658\) 0 0
\(659\) 29.4231 1.14616 0.573081 0.819499i \(-0.305748\pi\)
0.573081 + 0.819499i \(0.305748\pi\)
\(660\) 0 0
\(661\) −17.2957 −0.672723 −0.336361 0.941733i \(-0.609196\pi\)
−0.336361 + 0.941733i \(0.609196\pi\)
\(662\) 0 0
\(663\) 44.9183 1.74448
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 61.7788 2.39208
\(668\) 0 0
\(669\) −32.3473 −1.25062
\(670\) 0 0
\(671\) −31.6147 −1.22047
\(672\) 0 0
\(673\) −28.7525 −1.10833 −0.554164 0.832407i \(-0.686962\pi\)
−0.554164 + 0.832407i \(0.686962\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) −5.48594 −0.210842 −0.105421 0.994428i \(-0.533619\pi\)
−0.105421 + 0.994428i \(0.533619\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 27.2824 1.04546
\(682\) 0 0
\(683\) −11.3768 −0.435319 −0.217660 0.976025i \(-0.569842\pi\)
−0.217660 + 0.976025i \(0.569842\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) 63.0714 2.40632
\(688\) 0 0
\(689\) 1.62013 0.0617222
\(690\) 0 0
\(691\) −45.3261 −1.72429 −0.862144 0.506664i \(-0.830879\pi\)
−0.862144 + 0.506664i \(0.830879\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 13.0894 0.495798
\(698\) 0 0
\(699\) −57.8941 −2.18976
\(700\) 0 0
\(701\) 4.87634 0.184177 0.0920884 0.995751i \(-0.470646\pi\)
0.0920884 + 0.995751i \(0.470646\pi\)
\(702\) 0 0
\(703\) −2.12702 −0.0802220
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) −24.8887 −0.934713 −0.467357 0.884069i \(-0.654794\pi\)
−0.467357 + 0.884069i \(0.654794\pi\)
\(710\) 0 0
\(711\) −43.9881 −1.64968
\(712\) 0 0
\(713\) −12.2974 −0.460541
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 3.79281 0.141645
\(718\) 0 0
\(719\) −5.43573 −0.202719 −0.101359 0.994850i \(-0.532319\pi\)
−0.101359 + 0.994850i \(0.532319\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) −66.5787 −2.47609
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) 17.1112 0.634621 0.317310 0.948322i \(-0.397220\pi\)
0.317310 + 0.948322i \(0.397220\pi\)
\(728\) 0 0
\(729\) −34.7441 −1.28682
\(730\) 0 0
\(731\) 31.2972 1.15757
\(732\) 0 0
\(733\) 7.85744 0.290221 0.145111 0.989415i \(-0.453646\pi\)
0.145111 + 0.989415i \(0.453646\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −16.2426 −0.598304
\(738\) 0 0
\(739\) 51.0109 1.87647 0.938234 0.346002i \(-0.112461\pi\)
0.938234 + 0.346002i \(0.112461\pi\)
\(740\) 0 0
\(741\) 46.4142 1.70507
\(742\) 0 0
\(743\) 6.82733 0.250470 0.125235 0.992127i \(-0.460031\pi\)
0.125235 + 0.992127i \(0.460031\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 29.2565 1.07044
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) 32.3570 1.18072 0.590361 0.807139i \(-0.298985\pi\)
0.590361 + 0.807139i \(0.298985\pi\)
\(752\) 0 0
\(753\) −12.4742 −0.454583
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) −29.1180 −1.05831 −0.529156 0.848524i \(-0.677491\pi\)
−0.529156 + 0.848524i \(0.677491\pi\)
\(758\) 0 0
\(759\) −59.9014 −2.17428
\(760\) 0 0
\(761\) 28.1730 1.02127 0.510636 0.859797i \(-0.329410\pi\)
0.510636 + 0.859797i \(0.329410\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) −35.8162 −1.29325
\(768\) 0 0
\(769\) 16.5017 0.595065 0.297532 0.954712i \(-0.403836\pi\)
0.297532 + 0.954712i \(0.403836\pi\)
\(770\) 0 0
\(771\) −31.3964 −1.13071
\(772\) 0 0
\(773\) −18.4891 −0.665005 −0.332503 0.943102i \(-0.607893\pi\)
−0.332503 + 0.943102i \(0.607893\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 13.5254 0.484596
\(780\) 0 0
\(781\) 16.7703 0.600089
\(782\) 0 0
\(783\) 12.3675 0.441980
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) 15.7443 0.561224 0.280612 0.959821i \(-0.409463\pi\)
0.280612 + 0.959821i \(0.409463\pi\)
\(788\) 0 0
\(789\) 51.2034 1.82289
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 28.0689 0.996755
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −29.9307 −1.06020 −0.530100 0.847935i \(-0.677846\pi\)
−0.530100 + 0.847935i \(0.677846\pi\)
\(798\) 0 0
\(799\) 2.44248 0.0864088
\(800\) 0 0
\(801\) 8.28758 0.292827
\(802\) 0 0
\(803\) −18.8370 −0.664743
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 38.0105 1.33803
\(808\) 0 0
\(809\) 5.16658 0.181647 0.0908236 0.995867i \(-0.471050\pi\)
0.0908236 + 0.995867i \(0.471050\pi\)
\(810\) 0 0
\(811\) −38.4698 −1.35086 −0.675430 0.737425i \(-0.736042\pi\)
−0.675430 + 0.737425i \(0.736042\pi\)
\(812\) 0 0
\(813\) 14.7807 0.518381
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 32.3394 1.13141
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −3.69551 −0.128974 −0.0644872 0.997919i \(-0.520541\pi\)
−0.0644872 + 0.997919i \(0.520541\pi\)
\(822\) 0 0
\(823\) −42.4030 −1.47807 −0.739037 0.673665i \(-0.764719\pi\)
−0.739037 + 0.673665i \(0.764719\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −40.0950 −1.39424 −0.697120 0.716954i \(-0.745536\pi\)
−0.697120 + 0.716954i \(0.745536\pi\)
\(828\) 0 0
\(829\) 53.8264 1.86947 0.934734 0.355347i \(-0.115637\pi\)
0.934734 + 0.355347i \(0.115637\pi\)
\(830\) 0 0
\(831\) 65.9968 2.28941
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) −2.46183 −0.0850931
\(838\) 0 0
\(839\) −25.1002 −0.866556 −0.433278 0.901260i \(-0.642643\pi\)
−0.433278 + 0.901260i \(0.642643\pi\)
\(840\) 0 0
\(841\) 76.3910 2.63417
\(842\) 0 0
\(843\) 37.8694 1.30429
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) −24.7189 −0.848352
\(850\) 0 0
\(851\) −2.43721 −0.0835466
\(852\) 0 0
\(853\) 26.7320 0.915286 0.457643 0.889136i \(-0.348694\pi\)
0.457643 + 0.889136i \(0.348694\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 46.5891 1.59145 0.795726 0.605657i \(-0.207089\pi\)
0.795726 + 0.605657i \(0.207089\pi\)
\(858\) 0 0
\(859\) −18.6535 −0.636448 −0.318224 0.948016i \(-0.603086\pi\)
−0.318224 + 0.948016i \(0.603086\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) −17.8239 −0.606733 −0.303367 0.952874i \(-0.598111\pi\)
−0.303367 + 0.952874i \(0.598111\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) −22.4739 −0.763253
\(868\) 0 0
\(869\) 49.5450 1.68070
\(870\) 0 0
\(871\) 14.4209 0.488633
\(872\) 0 0
\(873\) 4.58781 0.155274
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 26.7793 0.904273 0.452136 0.891949i \(-0.350662\pi\)
0.452136 + 0.891949i \(0.350662\pi\)
\(878\) 0 0
\(879\) 20.5732 0.693916
\(880\) 0 0
\(881\) 17.5843 0.592429 0.296215 0.955121i \(-0.404276\pi\)
0.296215 + 0.955121i \(0.404276\pi\)
\(882\) 0 0
\(883\) −18.9638 −0.638182 −0.319091 0.947724i \(-0.603378\pi\)
−0.319091 + 0.947724i \(0.603378\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) −11.1654 −0.374897 −0.187448 0.982274i \(-0.560022\pi\)
−0.187448 + 0.982274i \(0.560022\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) 28.7761 0.964037
\(892\) 0 0
\(893\) 2.52382 0.0844565
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 53.1830 1.77573
\(898\) 0 0
\(899\) −20.9786 −0.699676
\(900\) 0 0
\(901\) −2.37068 −0.0789787
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) 38.1647 1.26724 0.633619 0.773645i \(-0.281569\pi\)
0.633619 + 0.773645i \(0.281569\pi\)
\(908\) 0 0
\(909\) 26.9253 0.893057
\(910\) 0 0
\(911\) 47.0251 1.55801 0.779006 0.627016i \(-0.215724\pi\)
0.779006 + 0.627016i \(0.215724\pi\)
\(912\) 0 0
\(913\) −32.9524 −1.09056
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) 25.3879 0.837469 0.418734 0.908109i \(-0.362474\pi\)
0.418734 + 0.908109i \(0.362474\pi\)
\(920\) 0 0
\(921\) −22.4359 −0.739288
\(922\) 0 0
\(923\) −14.8894 −0.490091
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) 7.65432 0.251401
\(928\) 0 0
\(929\) 7.64709 0.250893 0.125446 0.992100i \(-0.459964\pi\)
0.125446 + 0.992100i \(0.459964\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 0 0
\(933\) −24.5892 −0.805013
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 1.40923 0.0460375 0.0230188 0.999735i \(-0.492672\pi\)
0.0230188 + 0.999735i \(0.492672\pi\)
\(938\) 0 0
\(939\) −69.1706 −2.25730
\(940\) 0 0
\(941\) 52.4101 1.70852 0.854260 0.519845i \(-0.174010\pi\)
0.854260 + 0.519845i \(0.174010\pi\)
\(942\) 0 0
\(943\) 15.4978 0.504679
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −32.7229 −1.06335 −0.531675 0.846948i \(-0.678437\pi\)
−0.531675 + 0.846948i \(0.678437\pi\)
\(948\) 0 0
\(949\) 16.7243 0.542893
\(950\) 0 0
\(951\) 30.9344 1.00312
\(952\) 0 0
\(953\) −41.7747 −1.35322 −0.676608 0.736344i \(-0.736551\pi\)
−0.676608 + 0.736344i \(0.736551\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) −102.188 −3.30328
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) −26.8241 −0.865293
\(962\) 0 0
\(963\) −37.5955 −1.21150
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) −6.47375 −0.208182 −0.104091 0.994568i \(-0.533193\pi\)
−0.104091 + 0.994568i \(0.533193\pi\)
\(968\) 0 0
\(969\) −67.9159 −2.18177
\(970\) 0 0
\(971\) −33.1743 −1.06462 −0.532308 0.846551i \(-0.678675\pi\)
−0.532308 + 0.846551i \(0.678675\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 40.0711 1.28199 0.640994 0.767546i \(-0.278522\pi\)
0.640994 + 0.767546i \(0.278522\pi\)
\(978\) 0 0
\(979\) −9.33452 −0.298333
\(980\) 0 0
\(981\) −38.3589 −1.22471
\(982\) 0 0
\(983\) 0.852116 0.0271783 0.0135891 0.999908i \(-0.495674\pi\)
0.0135891 + 0.999908i \(0.495674\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 37.0557 1.17830
\(990\) 0 0
\(991\) −5.02760 −0.159707 −0.0798535 0.996807i \(-0.525445\pi\)
−0.0798535 + 0.996807i \(0.525445\pi\)
\(992\) 0 0
\(993\) 8.72558 0.276898
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) 27.0291 0.856021 0.428011 0.903774i \(-0.359215\pi\)
0.428011 + 0.903774i \(0.359215\pi\)
\(998\) 0 0
\(999\) −0.487908 −0.0154367
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 9800.2.a.cx.1.1 6
5.2 odd 4 1960.2.g.f.1569.11 12
5.3 odd 4 1960.2.g.f.1569.2 12
5.4 even 2 9800.2.a.cv.1.6 6
7.2 even 3 1400.2.q.n.1201.6 12
7.4 even 3 1400.2.q.n.401.6 12
7.6 odd 2 9800.2.a.cw.1.6 6
35.2 odd 12 280.2.bg.a.249.2 yes 24
35.4 even 6 1400.2.q.o.401.1 12
35.9 even 6 1400.2.q.o.1201.1 12
35.13 even 4 1960.2.g.e.1569.11 12
35.18 odd 12 280.2.bg.a.9.2 24
35.23 odd 12 280.2.bg.a.249.11 yes 24
35.27 even 4 1960.2.g.e.1569.2 12
35.32 odd 12 280.2.bg.a.9.11 yes 24
35.34 odd 2 9800.2.a.cy.1.1 6
140.23 even 12 560.2.bw.f.529.2 24
140.67 even 12 560.2.bw.f.289.2 24
140.107 even 12 560.2.bw.f.529.11 24
140.123 even 12 560.2.bw.f.289.11 24
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
280.2.bg.a.9.2 24 35.18 odd 12
280.2.bg.a.9.11 yes 24 35.32 odd 12
280.2.bg.a.249.2 yes 24 35.2 odd 12
280.2.bg.a.249.11 yes 24 35.23 odd 12
560.2.bw.f.289.2 24 140.67 even 12
560.2.bw.f.289.11 24 140.123 even 12
560.2.bw.f.529.2 24 140.23 even 12
560.2.bw.f.529.11 24 140.107 even 12
1400.2.q.n.401.6 12 7.4 even 3
1400.2.q.n.1201.6 12 7.2 even 3
1400.2.q.o.401.1 12 35.4 even 6
1400.2.q.o.1201.1 12 35.9 even 6
1960.2.g.e.1569.2 12 35.27 even 4
1960.2.g.e.1569.11 12 35.13 even 4
1960.2.g.f.1569.2 12 5.3 odd 4
1960.2.g.f.1569.11 12 5.2 odd 4
9800.2.a.cv.1.6 6 5.4 even 2
9800.2.a.cw.1.6 6 7.6 odd 2
9800.2.a.cx.1.1 6 1.1 even 1 trivial
9800.2.a.cy.1.1 6 35.34 odd 2