Properties

Label 9680.2.a.cr.1.2
Level $9680$
Weight $2$
Character 9680.1
Self dual yes
Analytic conductor $77.295$
Analytic rank $1$
Dimension $4$
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] = [9680,2,Mod(1,9680)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(9680, 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("9680.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 9680 = 2^{4} \cdot 5 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 9680.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: \(77.2951891566\)
Analytic rank: \(1\)
Dimension: \(4\)
Coefficient field: 4.4.4752.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 2x^{3} - 3x^{2} + 4x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 4840)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-1.49551\) of defining polynomial
Character \(\chi\) \(=\) 9680.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-0.0947876 q^{3} -1.00000 q^{5} -0.826838 q^{7} -2.99102 q^{9} +O(q^{10})\) \(q-0.0947876 q^{3} -1.00000 q^{5} -0.826838 q^{7} -2.99102 q^{9} +2.99102 q^{13} +0.0947876 q^{15} +0.662661 q^{19} +0.0783740 q^{21} -0.473086 q^{23} +1.00000 q^{25} +0.567874 q^{27} -4.00000 q^{29} +4.64469 q^{31} +0.826838 q^{35} +5.98203 q^{37} -0.283511 q^{39} +0.394712 q^{41} -6.96562 q^{43} +2.99102 q^{45} +3.88724 q^{47} -6.31634 q^{49} -0.801440 q^{53} -0.0628121 q^{57} +1.71649 q^{59} -2.58429 q^{61} +2.47309 q^{63} -2.99102 q^{65} -5.69767 q^{67} +0.0448427 q^{69} +5.59086 q^{71} -8.00000 q^{73} -0.0947876 q^{75} +11.6357 q^{79} +8.91922 q^{81} +5.90606 q^{83} +0.379150 q^{87} +4.32532 q^{89} -2.47309 q^{91} -0.440259 q^{93} -0.662661 q^{95} +6.50591 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 2 q^{3} - 4 q^{5} + 6 q^{7} + 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 2 q^{3} - 4 q^{5} + 6 q^{7} + 4 q^{9} - 4 q^{13} - 2 q^{15} + 12 q^{21} - 4 q^{23} + 4 q^{25} + 2 q^{27} - 16 q^{29} - 16 q^{31} - 6 q^{35} - 8 q^{37} - 8 q^{39} - 8 q^{41} - 10 q^{43} - 4 q^{45} - 14 q^{47} - 4 q^{49} + 8 q^{53} - 12 q^{57} + 4 q^{61} + 12 q^{63} + 4 q^{65} + 2 q^{67} - 20 q^{69} - 8 q^{71} - 32 q^{73} + 2 q^{75} - 4 q^{79} - 8 q^{81} + 12 q^{83} - 8 q^{87} + 12 q^{89} - 12 q^{91} - 32 q^{93}+O(q^{100}) \) Copy content Toggle raw display

Coefficient data

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



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) −0.0947876 −0.0547256 −0.0273628 0.999626i \(-0.508711\pi\)
−0.0273628 + 0.999626i \(0.508711\pi\)
\(4\) 0 0
\(5\) −1.00000 −0.447214
\(6\) 0 0
\(7\) −0.826838 −0.312516 −0.156258 0.987716i \(-0.549943\pi\)
−0.156258 + 0.987716i \(0.549943\pi\)
\(8\) 0 0
\(9\) −2.99102 −0.997005
\(10\) 0 0
\(11\) 0 0
\(12\) 0 0
\(13\) 2.99102 0.829558 0.414779 0.909922i \(-0.363859\pi\)
0.414779 + 0.909922i \(0.363859\pi\)
\(14\) 0 0
\(15\) 0.0947876 0.0244740
\(16\) 0 0
\(17\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(18\) 0 0
\(19\) 0.662661 0.152025 0.0760125 0.997107i \(-0.475781\pi\)
0.0760125 + 0.997107i \(0.475781\pi\)
\(20\) 0 0
\(21\) 0.0783740 0.0171026
\(22\) 0 0
\(23\) −0.473086 −0.0986453 −0.0493227 0.998783i \(-0.515706\pi\)
−0.0493227 + 0.998783i \(0.515706\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) 0.567874 0.109287
\(28\) 0 0
\(29\) −4.00000 −0.742781 −0.371391 0.928477i \(-0.621119\pi\)
−0.371391 + 0.928477i \(0.621119\pi\)
\(30\) 0 0
\(31\) 4.64469 0.834211 0.417106 0.908858i \(-0.363044\pi\)
0.417106 + 0.908858i \(0.363044\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0.826838 0.139761
\(36\) 0 0
\(37\) 5.98203 0.983440 0.491720 0.870753i \(-0.336368\pi\)
0.491720 + 0.870753i \(0.336368\pi\)
\(38\) 0 0
\(39\) −0.283511 −0.0453981
\(40\) 0 0
\(41\) 0.394712 0.0616437 0.0308219 0.999525i \(-0.490188\pi\)
0.0308219 + 0.999525i \(0.490188\pi\)
\(42\) 0 0
\(43\) −6.96562 −1.06225 −0.531123 0.847295i \(-0.678230\pi\)
−0.531123 + 0.847295i \(0.678230\pi\)
\(44\) 0 0
\(45\) 2.99102 0.445874
\(46\) 0 0
\(47\) 3.88724 0.567013 0.283506 0.958970i \(-0.408502\pi\)
0.283506 + 0.958970i \(0.408502\pi\)
\(48\) 0 0
\(49\) −6.31634 −0.902334
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −0.801440 −0.110086 −0.0550431 0.998484i \(-0.517530\pi\)
−0.0550431 + 0.998484i \(0.517530\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) −0.0628121 −0.00831966
\(58\) 0 0
\(59\) 1.71649 0.223468 0.111734 0.993738i \(-0.464360\pi\)
0.111734 + 0.993738i \(0.464360\pi\)
\(60\) 0 0
\(61\) −2.58429 −0.330884 −0.165442 0.986220i \(-0.552905\pi\)
−0.165442 + 0.986220i \(0.552905\pi\)
\(62\) 0 0
\(63\) 2.47309 0.311580
\(64\) 0 0
\(65\) −2.99102 −0.370990
\(66\) 0 0
\(67\) −5.69767 −0.696081 −0.348040 0.937480i \(-0.613153\pi\)
−0.348040 + 0.937480i \(0.613153\pi\)
\(68\) 0 0
\(69\) 0.0448427 0.00539843
\(70\) 0 0
\(71\) 5.59086 0.663514 0.331757 0.943365i \(-0.392359\pi\)
0.331757 + 0.943365i \(0.392359\pi\)
\(72\) 0 0
\(73\) −8.00000 −0.936329 −0.468165 0.883641i \(-0.655085\pi\)
−0.468165 + 0.883641i \(0.655085\pi\)
\(74\) 0 0
\(75\) −0.0947876 −0.0109451
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 11.6357 1.30912 0.654560 0.756010i \(-0.272854\pi\)
0.654560 + 0.756010i \(0.272854\pi\)
\(80\) 0 0
\(81\) 8.91922 0.991024
\(82\) 0 0
\(83\) 5.90606 0.648275 0.324137 0.946010i \(-0.394926\pi\)
0.324137 + 0.946010i \(0.394926\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 0.379150 0.0406492
\(88\) 0 0
\(89\) 4.32532 0.458483 0.229242 0.973370i \(-0.426375\pi\)
0.229242 + 0.973370i \(0.426375\pi\)
\(90\) 0 0
\(91\) −2.47309 −0.259250
\(92\) 0 0
\(93\) −0.440259 −0.0456527
\(94\) 0 0
\(95\) −0.662661 −0.0679876
\(96\) 0 0
\(97\) 6.50591 0.660575 0.330288 0.943880i \(-0.392854\pi\)
0.330288 + 0.943880i \(0.392854\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 3.54248 0.352489 0.176245 0.984346i \(-0.443605\pi\)
0.176245 + 0.984346i \(0.443605\pi\)
\(102\) 0 0
\(103\) −16.4371 −1.61960 −0.809800 0.586706i \(-0.800424\pi\)
−0.809800 + 0.586706i \(0.800424\pi\)
\(104\) 0 0
\(105\) −0.0783740 −0.00764852
\(106\) 0 0
\(107\) 17.4387 1.68586 0.842932 0.538021i \(-0.180828\pi\)
0.842932 + 0.538021i \(0.180828\pi\)
\(108\) 0 0
\(109\) −3.24100 −0.310431 −0.155216 0.987881i \(-0.549607\pi\)
−0.155216 + 0.987881i \(0.549607\pi\)
\(110\) 0 0
\(111\) −0.567022 −0.0538194
\(112\) 0 0
\(113\) −12.3612 −1.16284 −0.581421 0.813603i \(-0.697503\pi\)
−0.581421 + 0.813603i \(0.697503\pi\)
\(114\) 0 0
\(115\) 0.473086 0.0441155
\(116\) 0 0
\(117\) −8.94617 −0.827074
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) 0 0
\(122\) 0 0
\(123\) −0.0374138 −0.00337349
\(124\) 0 0
\(125\) −1.00000 −0.0894427
\(126\) 0 0
\(127\) −2.73290 −0.242506 −0.121253 0.992622i \(-0.538691\pi\)
−0.121253 + 0.992622i \(0.538691\pi\)
\(128\) 0 0
\(129\) 0.660254 0.0581321
\(130\) 0 0
\(131\) −9.57290 −0.836388 −0.418194 0.908358i \(-0.637337\pi\)
−0.418194 + 0.908358i \(0.637337\pi\)
\(132\) 0 0
\(133\) −0.547914 −0.0475102
\(134\) 0 0
\(135\) −0.567874 −0.0488748
\(136\) 0 0
\(137\) −6.12676 −0.523445 −0.261722 0.965143i \(-0.584290\pi\)
−0.261722 + 0.965143i \(0.584290\pi\)
\(138\) 0 0
\(139\) 20.9551 1.77739 0.888693 0.458502i \(-0.151614\pi\)
0.888693 + 0.458502i \(0.151614\pi\)
\(140\) 0 0
\(141\) −0.368462 −0.0310301
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) 4.00000 0.332182
\(146\) 0 0
\(147\) 0.598710 0.0493808
\(148\) 0 0
\(149\) −11.9096 −0.975673 −0.487837 0.872935i \(-0.662214\pi\)
−0.487837 + 0.872935i \(0.662214\pi\)
\(150\) 0 0
\(151\) 16.2984 1.32634 0.663171 0.748468i \(-0.269210\pi\)
0.663171 + 0.748468i \(0.269210\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) −4.64469 −0.373071
\(156\) 0 0
\(157\) −11.5849 −0.924577 −0.462288 0.886730i \(-0.652971\pi\)
−0.462288 + 0.886730i \(0.652971\pi\)
\(158\) 0 0
\(159\) 0.0759666 0.00602454
\(160\) 0 0
\(161\) 0.391166 0.0308282
\(162\) 0 0
\(163\) 1.63967 0.128429 0.0642145 0.997936i \(-0.479546\pi\)
0.0642145 + 0.997936i \(0.479546\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 11.7550 0.909632 0.454816 0.890585i \(-0.349705\pi\)
0.454816 + 0.890585i \(0.349705\pi\)
\(168\) 0 0
\(169\) −4.05383 −0.311833
\(170\) 0 0
\(171\) −1.98203 −0.151570
\(172\) 0 0
\(173\) −14.4240 −1.09664 −0.548318 0.836270i \(-0.684732\pi\)
−0.548318 + 0.836270i \(0.684732\pi\)
\(174\) 0 0
\(175\) −0.826838 −0.0625031
\(176\) 0 0
\(177\) −0.162702 −0.0122294
\(178\) 0 0
\(179\) −6.86425 −0.513058 −0.256529 0.966536i \(-0.582579\pi\)
−0.256529 + 0.966536i \(0.582579\pi\)
\(180\) 0 0
\(181\) −23.9742 −1.78199 −0.890994 0.454016i \(-0.849991\pi\)
−0.890994 + 0.454016i \(0.849991\pi\)
\(182\) 0 0
\(183\) 0.244958 0.0181078
\(184\) 0 0
\(185\) −5.98203 −0.439808
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) −0.469540 −0.0341540
\(190\) 0 0
\(191\) −12.4551 −0.901221 −0.450610 0.892721i \(-0.648794\pi\)
−0.450610 + 0.892721i \(0.648794\pi\)
\(192\) 0 0
\(193\) −5.19261 −0.373772 −0.186886 0.982382i \(-0.559839\pi\)
−0.186886 + 0.982382i \(0.559839\pi\)
\(194\) 0 0
\(195\) 0.283511 0.0203027
\(196\) 0 0
\(197\) −8.68783 −0.618982 −0.309491 0.950902i \(-0.600159\pi\)
−0.309491 + 0.950902i \(0.600159\pi\)
\(198\) 0 0
\(199\) −13.5149 −0.958046 −0.479023 0.877802i \(-0.659009\pi\)
−0.479023 + 0.877802i \(0.659009\pi\)
\(200\) 0 0
\(201\) 0.540068 0.0380935
\(202\) 0 0
\(203\) 3.30735 0.232131
\(204\) 0 0
\(205\) −0.394712 −0.0275679
\(206\) 0 0
\(207\) 1.41501 0.0983499
\(208\) 0 0
\(209\) 0 0
\(210\) 0 0
\(211\) −18.0028 −1.23937 −0.619683 0.784852i \(-0.712739\pi\)
−0.619683 + 0.784852i \(0.712739\pi\)
\(212\) 0 0
\(213\) −0.529945 −0.0363112
\(214\) 0 0
\(215\) 6.96562 0.475051
\(216\) 0 0
\(217\) −3.84041 −0.260704
\(218\) 0 0
\(219\) 0.758301 0.0512412
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) −14.2784 −0.956153 −0.478076 0.878318i \(-0.658666\pi\)
−0.478076 + 0.878318i \(0.658666\pi\)
\(224\) 0 0
\(225\) −2.99102 −0.199401
\(226\) 0 0
\(227\) 13.3627 0.886916 0.443458 0.896295i \(-0.353752\pi\)
0.443458 + 0.896295i \(0.353752\pi\)
\(228\) 0 0
\(229\) 1.46713 0.0969508 0.0484754 0.998824i \(-0.484564\pi\)
0.0484754 + 0.998824i \(0.484564\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 12.0269 0.787907 0.393953 0.919130i \(-0.371107\pi\)
0.393953 + 0.919130i \(0.371107\pi\)
\(234\) 0 0
\(235\) −3.88724 −0.253576
\(236\) 0 0
\(237\) −1.10292 −0.0716424
\(238\) 0 0
\(239\) 27.0238 1.74803 0.874014 0.485902i \(-0.161509\pi\)
0.874014 + 0.485902i \(0.161509\pi\)
\(240\) 0 0
\(241\) −2.94377 −0.189625 −0.0948123 0.995495i \(-0.530225\pi\)
−0.0948123 + 0.995495i \(0.530225\pi\)
\(242\) 0 0
\(243\) −2.54905 −0.163522
\(244\) 0 0
\(245\) 6.31634 0.403536
\(246\) 0 0
\(247\) 1.98203 0.126114
\(248\) 0 0
\(249\) −0.559822 −0.0354773
\(250\) 0 0
\(251\) 24.8982 1.57156 0.785781 0.618505i \(-0.212261\pi\)
0.785781 + 0.618505i \(0.212261\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 13.4761 0.840617 0.420309 0.907381i \(-0.361922\pi\)
0.420309 + 0.907381i \(0.361922\pi\)
\(258\) 0 0
\(259\) −4.94617 −0.307340
\(260\) 0 0
\(261\) 11.9641 0.740557
\(262\) 0 0
\(263\) −4.83427 −0.298094 −0.149047 0.988830i \(-0.547621\pi\)
−0.149047 + 0.988830i \(0.547621\pi\)
\(264\) 0 0
\(265\) 0.801440 0.0492321
\(266\) 0 0
\(267\) −0.409987 −0.0250908
\(268\) 0 0
\(269\) −18.8373 −1.14853 −0.574265 0.818669i \(-0.694712\pi\)
−0.574265 + 0.818669i \(0.694712\pi\)
\(270\) 0 0
\(271\) −16.9102 −1.02722 −0.513612 0.858023i \(-0.671693\pi\)
−0.513612 + 0.858023i \(0.671693\pi\)
\(272\) 0 0
\(273\) 0.234418 0.0141876
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) −3.87919 −0.233078 −0.116539 0.993186i \(-0.537180\pi\)
−0.116539 + 0.993186i \(0.537180\pi\)
\(278\) 0 0
\(279\) −13.8923 −0.831713
\(280\) 0 0
\(281\) −17.6029 −1.05010 −0.525050 0.851071i \(-0.675953\pi\)
−0.525050 + 0.851071i \(0.675953\pi\)
\(282\) 0 0
\(283\) 0.781996 0.0464848 0.0232424 0.999730i \(-0.492601\pi\)
0.0232424 + 0.999730i \(0.492601\pi\)
\(284\) 0 0
\(285\) 0.0628121 0.00372067
\(286\) 0 0
\(287\) −0.326363 −0.0192646
\(288\) 0 0
\(289\) −17.0000 −1.00000
\(290\) 0 0
\(291\) −0.616680 −0.0361504
\(292\) 0 0
\(293\) −8.18362 −0.478092 −0.239046 0.971008i \(-0.576835\pi\)
−0.239046 + 0.971008i \(0.576835\pi\)
\(294\) 0 0
\(295\) −1.71649 −0.0999378
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) −1.41501 −0.0818320
\(300\) 0 0
\(301\) 5.75944 0.331969
\(302\) 0 0
\(303\) −0.335783 −0.0192902
\(304\) 0 0
\(305\) 2.58429 0.147976
\(306\) 0 0
\(307\) −27.4490 −1.56660 −0.783298 0.621647i \(-0.786464\pi\)
−0.783298 + 0.621647i \(0.786464\pi\)
\(308\) 0 0
\(309\) 1.55804 0.0886337
\(310\) 0 0
\(311\) −26.6907 −1.51349 −0.756745 0.653711i \(-0.773211\pi\)
−0.756745 + 0.653711i \(0.773211\pi\)
\(312\) 0 0
\(313\) −19.7296 −1.11519 −0.557593 0.830115i \(-0.688275\pi\)
−0.557593 + 0.830115i \(0.688275\pi\)
\(314\) 0 0
\(315\) −2.47309 −0.139343
\(316\) 0 0
\(317\) 9.24170 0.519066 0.259533 0.965734i \(-0.416431\pi\)
0.259533 + 0.965734i \(0.416431\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0 0
\(321\) −1.65297 −0.0922599
\(322\) 0 0
\(323\) 0 0
\(324\) 0 0
\(325\) 2.99102 0.165912
\(326\) 0 0
\(327\) 0.307206 0.0169885
\(328\) 0 0
\(329\) −3.21412 −0.177200
\(330\) 0 0
\(331\) −19.3833 −1.06540 −0.532702 0.846303i \(-0.678823\pi\)
−0.532702 + 0.846303i \(0.678823\pi\)
\(332\) 0 0
\(333\) −17.8923 −0.980494
\(334\) 0 0
\(335\) 5.69767 0.311297
\(336\) 0 0
\(337\) −29.9192 −1.62980 −0.814902 0.579599i \(-0.803209\pi\)
−0.814902 + 0.579599i \(0.803209\pi\)
\(338\) 0 0
\(339\) 1.17169 0.0636373
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) 11.0105 0.594509
\(344\) 0 0
\(345\) −0.0448427 −0.00241425
\(346\) 0 0
\(347\) 18.7892 1.00866 0.504328 0.863512i \(-0.331740\pi\)
0.504328 + 0.863512i \(0.331740\pi\)
\(348\) 0 0
\(349\) −2.74033 −0.146687 −0.0733433 0.997307i \(-0.523367\pi\)
−0.0733433 + 0.997307i \(0.523367\pi\)
\(350\) 0 0
\(351\) 1.69852 0.0906603
\(352\) 0 0
\(353\) 0.631538 0.0336134 0.0168067 0.999859i \(-0.494650\pi\)
0.0168067 + 0.999859i \(0.494650\pi\)
\(354\) 0 0
\(355\) −5.59086 −0.296732
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 0.0299849 0.00158254 0.000791272 1.00000i \(-0.499748\pi\)
0.000791272 1.00000i \(0.499748\pi\)
\(360\) 0 0
\(361\) −18.5609 −0.976888
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 8.00000 0.418739
\(366\) 0 0
\(367\) 5.90521 0.308250 0.154125 0.988051i \(-0.450744\pi\)
0.154125 + 0.988051i \(0.450744\pi\)
\(368\) 0 0
\(369\) −1.18059 −0.0614591
\(370\) 0 0
\(371\) 0.662661 0.0344037
\(372\) 0 0
\(373\) −5.38332 −0.278738 −0.139369 0.990241i \(-0.544507\pi\)
−0.139369 + 0.990241i \(0.544507\pi\)
\(374\) 0 0
\(375\) 0.0947876 0.00489481
\(376\) 0 0
\(377\) −11.9641 −0.616181
\(378\) 0 0
\(379\) −17.8940 −0.919156 −0.459578 0.888138i \(-0.651999\pi\)
−0.459578 + 0.888138i \(0.651999\pi\)
\(380\) 0 0
\(381\) 0.259045 0.0132713
\(382\) 0 0
\(383\) −11.8045 −0.603180 −0.301590 0.953438i \(-0.597517\pi\)
−0.301590 + 0.953438i \(0.597517\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 20.8343 1.05907
\(388\) 0 0
\(389\) 12.5418 0.635893 0.317947 0.948109i \(-0.397007\pi\)
0.317947 + 0.948109i \(0.397007\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0 0
\(393\) 0.907392 0.0457719
\(394\) 0 0
\(395\) −11.6357 −0.585456
\(396\) 0 0
\(397\) −18.1077 −0.908797 −0.454399 0.890798i \(-0.650146\pi\)
−0.454399 + 0.890798i \(0.650146\pi\)
\(398\) 0 0
\(399\) 0.0519354 0.00260002
\(400\) 0 0
\(401\) −24.2058 −1.20878 −0.604389 0.796689i \(-0.706583\pi\)
−0.604389 + 0.796689i \(0.706583\pi\)
\(402\) 0 0
\(403\) 13.8923 0.692027
\(404\) 0 0
\(405\) −8.91922 −0.443200
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) −4.82902 −0.238780 −0.119390 0.992847i \(-0.538094\pi\)
−0.119390 + 0.992847i \(0.538094\pi\)
\(410\) 0 0
\(411\) 0.580741 0.0286458
\(412\) 0 0
\(413\) −1.41926 −0.0698372
\(414\) 0 0
\(415\) −5.90606 −0.289917
\(416\) 0 0
\(417\) −1.98628 −0.0972686
\(418\) 0 0
\(419\) 16.7327 0.817445 0.408722 0.912659i \(-0.365974\pi\)
0.408722 + 0.912659i \(0.365974\pi\)
\(420\) 0 0
\(421\) 14.1447 0.689368 0.344684 0.938719i \(-0.387986\pi\)
0.344684 + 0.938719i \(0.387986\pi\)
\(422\) 0 0
\(423\) −11.6268 −0.565315
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 2.13679 0.103406
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) −11.7973 −0.568255 −0.284127 0.958787i \(-0.591704\pi\)
−0.284127 + 0.958787i \(0.591704\pi\)
\(432\) 0 0
\(433\) 0.717816 0.0344961 0.0172480 0.999851i \(-0.494510\pi\)
0.0172480 + 0.999851i \(0.494510\pi\)
\(434\) 0 0
\(435\) −0.379150 −0.0181789
\(436\) 0 0
\(437\) −0.313496 −0.0149966
\(438\) 0 0
\(439\) −13.0359 −0.622168 −0.311084 0.950382i \(-0.600692\pi\)
−0.311084 + 0.950382i \(0.600692\pi\)
\(440\) 0 0
\(441\) 18.8923 0.899632
\(442\) 0 0
\(443\) −12.3783 −0.588111 −0.294055 0.955788i \(-0.595005\pi\)
−0.294055 + 0.955788i \(0.595005\pi\)
\(444\) 0 0
\(445\) −4.32532 −0.205040
\(446\) 0 0
\(447\) 1.12888 0.0533943
\(448\) 0 0
\(449\) 23.1788 1.09388 0.546938 0.837173i \(-0.315793\pi\)
0.546938 + 0.837173i \(0.315793\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 0 0
\(453\) −1.54488 −0.0725849
\(454\) 0 0
\(455\) 2.47309 0.115940
\(456\) 0 0
\(457\) 22.8102 1.06702 0.533509 0.845794i \(-0.320873\pi\)
0.533509 + 0.845794i \(0.320873\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) −16.4527 −0.766279 −0.383140 0.923690i \(-0.625157\pi\)
−0.383140 + 0.923690i \(0.625157\pi\)
\(462\) 0 0
\(463\) −36.1708 −1.68100 −0.840499 0.541813i \(-0.817738\pi\)
−0.840499 + 0.541813i \(0.817738\pi\)
\(464\) 0 0
\(465\) 0.440259 0.0204165
\(466\) 0 0
\(467\) −23.9332 −1.10750 −0.553749 0.832684i \(-0.686803\pi\)
−0.553749 + 0.832684i \(0.686803\pi\)
\(468\) 0 0
\(469\) 4.71105 0.217536
\(470\) 0 0
\(471\) 1.09811 0.0505980
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) 0.662661 0.0304050
\(476\) 0 0
\(477\) 2.39712 0.109757
\(478\) 0 0
\(479\) −15.5101 −0.708674 −0.354337 0.935118i \(-0.615293\pi\)
−0.354337 + 0.935118i \(0.615293\pi\)
\(480\) 0 0
\(481\) 17.8923 0.815821
\(482\) 0 0
\(483\) −0.0370777 −0.00168709
\(484\) 0 0
\(485\) −6.50591 −0.295418
\(486\) 0 0
\(487\) 41.3114 1.87200 0.936000 0.352000i \(-0.114498\pi\)
0.936000 + 0.352000i \(0.114498\pi\)
\(488\) 0 0
\(489\) −0.155420 −0.00702835
\(490\) 0 0
\(491\) −3.79727 −0.171368 −0.0856842 0.996322i \(-0.527308\pi\)
−0.0856842 + 0.996322i \(0.527308\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −4.62274 −0.207358
\(498\) 0 0
\(499\) −3.62255 −0.162168 −0.0810839 0.996707i \(-0.525838\pi\)
−0.0810839 + 0.996707i \(0.525838\pi\)
\(500\) 0 0
\(501\) −1.11423 −0.0497802
\(502\) 0 0
\(503\) 16.6641 0.743017 0.371509 0.928430i \(-0.378841\pi\)
0.371509 + 0.928430i \(0.378841\pi\)
\(504\) 0 0
\(505\) −3.54248 −0.157638
\(506\) 0 0
\(507\) 0.384253 0.0170653
\(508\) 0 0
\(509\) −34.4969 −1.52905 −0.764525 0.644594i \(-0.777026\pi\)
−0.764525 + 0.644594i \(0.777026\pi\)
\(510\) 0 0
\(511\) 6.61471 0.292617
\(512\) 0 0
\(513\) 0.376308 0.0166144
\(514\) 0 0
\(515\) 16.4371 0.724307
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) 1.36722 0.0600141
\(520\) 0 0
\(521\) −9.44480 −0.413784 −0.206892 0.978364i \(-0.566335\pi\)
−0.206892 + 0.978364i \(0.566335\pi\)
\(522\) 0 0
\(523\) −8.38946 −0.366846 −0.183423 0.983034i \(-0.558718\pi\)
−0.183423 + 0.983034i \(0.558718\pi\)
\(524\) 0 0
\(525\) 0.0783740 0.00342052
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) −22.7762 −0.990269
\(530\) 0 0
\(531\) −5.13404 −0.222799
\(532\) 0 0
\(533\) 1.18059 0.0511371
\(534\) 0 0
\(535\) −17.4387 −0.753941
\(536\) 0 0
\(537\) 0.650646 0.0280774
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) 25.8725 1.11235 0.556174 0.831066i \(-0.312269\pi\)
0.556174 + 0.831066i \(0.312269\pi\)
\(542\) 0 0
\(543\) 2.27246 0.0975204
\(544\) 0 0
\(545\) 3.24100 0.138829
\(546\) 0 0
\(547\) 26.7804 1.14505 0.572523 0.819889i \(-0.305965\pi\)
0.572523 + 0.819889i \(0.305965\pi\)
\(548\) 0 0
\(549\) 7.72964 0.329893
\(550\) 0 0
\(551\) −2.65065 −0.112921
\(552\) 0 0
\(553\) −9.62085 −0.409120
\(554\) 0 0
\(555\) 0.567022 0.0240688
\(556\) 0 0
\(557\) −35.5490 −1.50626 −0.753129 0.657873i \(-0.771456\pi\)
−0.753129 + 0.657873i \(0.771456\pi\)
\(558\) 0 0
\(559\) −20.8343 −0.881196
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 25.1893 1.06160 0.530801 0.847497i \(-0.321891\pi\)
0.530801 + 0.847497i \(0.321891\pi\)
\(564\) 0 0
\(565\) 12.3612 0.520039
\(566\) 0 0
\(567\) −7.37475 −0.309710
\(568\) 0 0
\(569\) −7.98444 −0.334725 −0.167363 0.985895i \(-0.553525\pi\)
−0.167363 + 0.985895i \(0.553525\pi\)
\(570\) 0 0
\(571\) −19.8892 −0.832339 −0.416169 0.909287i \(-0.636628\pi\)
−0.416169 + 0.909287i \(0.636628\pi\)
\(572\) 0 0
\(573\) 1.18059 0.0493199
\(574\) 0 0
\(575\) −0.473086 −0.0197291
\(576\) 0 0
\(577\) 16.3444 0.680424 0.340212 0.940349i \(-0.389501\pi\)
0.340212 + 0.940349i \(0.389501\pi\)
\(578\) 0 0
\(579\) 0.492195 0.0204549
\(580\) 0 0
\(581\) −4.88336 −0.202596
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) 8.94617 0.369879
\(586\) 0 0
\(587\) 37.4781 1.54689 0.773444 0.633865i \(-0.218533\pi\)
0.773444 + 0.633865i \(0.218533\pi\)
\(588\) 0 0
\(589\) 3.07786 0.126821
\(590\) 0 0
\(591\) 0.823499 0.0338742
\(592\) 0 0
\(593\) −28.0028 −1.14994 −0.574969 0.818175i \(-0.694986\pi\)
−0.574969 + 0.818175i \(0.694986\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 1.28104 0.0524297
\(598\) 0 0
\(599\) −27.8522 −1.13801 −0.569004 0.822335i \(-0.692671\pi\)
−0.569004 + 0.822335i \(0.692671\pi\)
\(600\) 0 0
\(601\) 26.9282 1.09842 0.549212 0.835683i \(-0.314928\pi\)
0.549212 + 0.835683i \(0.314928\pi\)
\(602\) 0 0
\(603\) 17.0418 0.693996
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) 15.3833 0.624390 0.312195 0.950018i \(-0.398936\pi\)
0.312195 + 0.950018i \(0.398936\pi\)
\(608\) 0 0
\(609\) −0.313496 −0.0127035
\(610\) 0 0
\(611\) 11.6268 0.470370
\(612\) 0 0
\(613\) 29.0967 1.17520 0.587602 0.809150i \(-0.300072\pi\)
0.587602 + 0.809150i \(0.300072\pi\)
\(614\) 0 0
\(615\) 0.0374138 0.00150867
\(616\) 0 0
\(617\) −17.9832 −0.723975 −0.361988 0.932183i \(-0.617902\pi\)
−0.361988 + 0.932183i \(0.617902\pi\)
\(618\) 0 0
\(619\) 1.85641 0.0746153 0.0373076 0.999304i \(-0.488122\pi\)
0.0373076 + 0.999304i \(0.488122\pi\)
\(620\) 0 0
\(621\) −0.268653 −0.0107807
\(622\) 0 0
\(623\) −3.57634 −0.143283
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 0 0
\(630\) 0 0
\(631\) 2.61471 0.104090 0.0520449 0.998645i \(-0.483426\pi\)
0.0520449 + 0.998645i \(0.483426\pi\)
\(632\) 0 0
\(633\) 1.70645 0.0678251
\(634\) 0 0
\(635\) 2.73290 0.108452
\(636\) 0 0
\(637\) −18.8923 −0.748539
\(638\) 0 0
\(639\) −16.7224 −0.661526
\(640\) 0 0
\(641\) 31.3157 1.23690 0.618448 0.785826i \(-0.287762\pi\)
0.618448 + 0.785826i \(0.287762\pi\)
\(642\) 0 0
\(643\) −2.67383 −0.105445 −0.0527227 0.998609i \(-0.516790\pi\)
−0.0527227 + 0.998609i \(0.516790\pi\)
\(644\) 0 0
\(645\) −0.660254 −0.0259975
\(646\) 0 0
\(647\) −41.4002 −1.62761 −0.813804 0.581139i \(-0.802607\pi\)
−0.813804 + 0.581139i \(0.802607\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) 0.364023 0.0142672
\(652\) 0 0
\(653\) −26.2535 −1.02738 −0.513690 0.857976i \(-0.671722\pi\)
−0.513690 + 0.857976i \(0.671722\pi\)
\(654\) 0 0
\(655\) 9.57290 0.374044
\(656\) 0 0
\(657\) 23.9281 0.933525
\(658\) 0 0
\(659\) 7.50781 0.292463 0.146231 0.989250i \(-0.453286\pi\)
0.146231 + 0.989250i \(0.453286\pi\)
\(660\) 0 0
\(661\) 10.4161 0.405141 0.202571 0.979268i \(-0.435070\pi\)
0.202571 + 0.979268i \(0.435070\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 0.547914 0.0212472
\(666\) 0 0
\(667\) 1.89235 0.0732719
\(668\) 0 0
\(669\) 1.35342 0.0523261
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) −21.8881 −0.843724 −0.421862 0.906660i \(-0.638623\pi\)
−0.421862 + 0.906660i \(0.638623\pi\)
\(674\) 0 0
\(675\) 0.567874 0.0218575
\(676\) 0 0
\(677\) 18.7666 0.721261 0.360630 0.932709i \(-0.382562\pi\)
0.360630 + 0.932709i \(0.382562\pi\)
\(678\) 0 0
\(679\) −5.37934 −0.206440
\(680\) 0 0
\(681\) −1.26662 −0.0485370
\(682\) 0 0
\(683\) 44.8734 1.71703 0.858517 0.512785i \(-0.171386\pi\)
0.858517 + 0.512785i \(0.171386\pi\)
\(684\) 0 0
\(685\) 6.12676 0.234092
\(686\) 0 0
\(687\) −0.139066 −0.00530570
\(688\) 0 0
\(689\) −2.39712 −0.0913230
\(690\) 0 0
\(691\) 36.2936 1.38067 0.690336 0.723489i \(-0.257463\pi\)
0.690336 + 0.723489i \(0.257463\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −20.9551 −0.794871
\(696\) 0 0
\(697\) 0 0
\(698\) 0 0
\(699\) −1.14000 −0.0431187
\(700\) 0 0
\(701\) 23.5490 0.889432 0.444716 0.895672i \(-0.353305\pi\)
0.444716 + 0.895672i \(0.353305\pi\)
\(702\) 0 0
\(703\) 3.96406 0.149507
\(704\) 0 0
\(705\) 0.368462 0.0138771
\(706\) 0 0
\(707\) −2.92905 −0.110158
\(708\) 0 0
\(709\) 10.9024 0.409448 0.204724 0.978820i \(-0.434370\pi\)
0.204724 + 0.978820i \(0.434370\pi\)
\(710\) 0 0
\(711\) −34.8026 −1.30520
\(712\) 0 0
\(713\) −2.19734 −0.0822910
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) −2.56152 −0.0956619
\(718\) 0 0
\(719\) −18.9419 −0.706414 −0.353207 0.935545i \(-0.614909\pi\)
−0.353207 + 0.935545i \(0.614909\pi\)
\(720\) 0 0
\(721\) 13.5909 0.506150
\(722\) 0 0
\(723\) 0.279032 0.0103773
\(724\) 0 0
\(725\) −4.00000 −0.148556
\(726\) 0 0
\(727\) −29.3739 −1.08942 −0.544708 0.838626i \(-0.683360\pi\)
−0.544708 + 0.838626i \(0.683360\pi\)
\(728\) 0 0
\(729\) −26.5160 −0.982075
\(730\) 0 0
\(731\) 0 0
\(732\) 0 0
\(733\) 25.2654 0.933197 0.466599 0.884469i \(-0.345479\pi\)
0.466599 + 0.884469i \(0.345479\pi\)
\(734\) 0 0
\(735\) −0.598710 −0.0220838
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) −13.5161 −0.497199 −0.248600 0.968606i \(-0.579970\pi\)
−0.248600 + 0.968606i \(0.579970\pi\)
\(740\) 0 0
\(741\) −0.187872 −0.00690165
\(742\) 0 0
\(743\) −12.1594 −0.446084 −0.223042 0.974809i \(-0.571599\pi\)
−0.223042 + 0.974809i \(0.571599\pi\)
\(744\) 0 0
\(745\) 11.9096 0.436334
\(746\) 0 0
\(747\) −17.6651 −0.646333
\(748\) 0 0
\(749\) −14.4190 −0.526858
\(750\) 0 0
\(751\) −37.6069 −1.37229 −0.686147 0.727463i \(-0.740699\pi\)
−0.686147 + 0.727463i \(0.740699\pi\)
\(752\) 0 0
\(753\) −2.36004 −0.0860047
\(754\) 0 0
\(755\) −16.2984 −0.593158
\(756\) 0 0
\(757\) 32.3886 1.17718 0.588592 0.808430i \(-0.299682\pi\)
0.588592 + 0.808430i \(0.299682\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 52.9640 1.91994 0.959971 0.280098i \(-0.0903670\pi\)
0.959971 + 0.280098i \(0.0903670\pi\)
\(762\) 0 0
\(763\) 2.67978 0.0970145
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 5.13404 0.185380
\(768\) 0 0
\(769\) 10.6387 0.383643 0.191821 0.981430i \(-0.438561\pi\)
0.191821 + 0.981430i \(0.438561\pi\)
\(770\) 0 0
\(771\) −1.27737 −0.0460033
\(772\) 0 0
\(773\) 29.7953 1.07166 0.535831 0.844325i \(-0.319998\pi\)
0.535831 + 0.844325i \(0.319998\pi\)
\(774\) 0 0
\(775\) 4.64469 0.166842
\(776\) 0 0
\(777\) 0.468836 0.0168194
\(778\) 0 0
\(779\) 0.261561 0.00937138
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) −2.27150 −0.0811766
\(784\) 0 0
\(785\) 11.5849 0.413483
\(786\) 0 0
\(787\) 15.5486 0.554249 0.277125 0.960834i \(-0.410619\pi\)
0.277125 + 0.960834i \(0.410619\pi\)
\(788\) 0 0
\(789\) 0.458229 0.0163134
\(790\) 0 0
\(791\) 10.2207 0.363406
\(792\) 0 0
\(793\) −7.72964 −0.274488
\(794\) 0 0
\(795\) −0.0759666 −0.00269426
\(796\) 0 0
\(797\) 22.6399 0.801946 0.400973 0.916090i \(-0.368672\pi\)
0.400973 + 0.916090i \(0.368672\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 0 0
\(801\) −12.9371 −0.457110
\(802\) 0 0
\(803\) 0 0
\(804\) 0 0
\(805\) −0.391166 −0.0137868
\(806\) 0 0
\(807\) 1.78554 0.0628541
\(808\) 0 0
\(809\) −29.1638 −1.02534 −0.512672 0.858585i \(-0.671344\pi\)
−0.512672 + 0.858585i \(0.671344\pi\)
\(810\) 0 0
\(811\) −50.2445 −1.76432 −0.882161 0.470948i \(-0.843912\pi\)
−0.882161 + 0.470948i \(0.843912\pi\)
\(812\) 0 0
\(813\) 1.60288 0.0562155
\(814\) 0 0
\(815\) −1.63967 −0.0574352
\(816\) 0 0
\(817\) −4.61585 −0.161488
\(818\) 0 0
\(819\) 7.39704 0.258473
\(820\) 0 0
\(821\) 54.7708 1.91151 0.955756 0.294160i \(-0.0950399\pi\)
0.955756 + 0.294160i \(0.0950399\pi\)
\(822\) 0 0
\(823\) −24.4919 −0.853734 −0.426867 0.904314i \(-0.640383\pi\)
−0.426867 + 0.904314i \(0.640383\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 44.3069 1.54070 0.770352 0.637619i \(-0.220081\pi\)
0.770352 + 0.637619i \(0.220081\pi\)
\(828\) 0 0
\(829\) −50.9052 −1.76801 −0.884006 0.467476i \(-0.845163\pi\)
−0.884006 + 0.467476i \(0.845163\pi\)
\(830\) 0 0
\(831\) 0.367699 0.0127553
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) −11.7550 −0.406800
\(836\) 0 0
\(837\) 2.63760 0.0911688
\(838\) 0 0
\(839\) −43.3354 −1.49610 −0.748051 0.663641i \(-0.769010\pi\)
−0.748051 + 0.663641i \(0.769010\pi\)
\(840\) 0 0
\(841\) −13.0000 −0.448276
\(842\) 0 0
\(843\) 1.66853 0.0574674
\(844\) 0 0
\(845\) 4.05383 0.139456
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) −0.0741235 −0.00254391
\(850\) 0 0
\(851\) −2.83002 −0.0970117
\(852\) 0 0
\(853\) −11.1195 −0.380724 −0.190362 0.981714i \(-0.560966\pi\)
−0.190362 + 0.981714i \(0.560966\pi\)
\(854\) 0 0
\(855\) 1.98203 0.0677840
\(856\) 0 0
\(857\) −29.0430 −0.992088 −0.496044 0.868297i \(-0.665215\pi\)
−0.496044 + 0.868297i \(0.665215\pi\)
\(858\) 0 0
\(859\) 39.4993 1.34770 0.673850 0.738869i \(-0.264640\pi\)
0.673850 + 0.738869i \(0.264640\pi\)
\(860\) 0 0
\(861\) 0.0309352 0.00105427
\(862\) 0 0
\(863\) −31.1810 −1.06141 −0.530707 0.847556i \(-0.678073\pi\)
−0.530707 + 0.847556i \(0.678073\pi\)
\(864\) 0 0
\(865\) 14.4240 0.490430
\(866\) 0 0
\(867\) 1.61139 0.0547256
\(868\) 0 0
\(869\) 0 0
\(870\) 0 0
\(871\) −17.0418 −0.577440
\(872\) 0 0
\(873\) −19.4593 −0.658597
\(874\) 0 0
\(875\) 0.826838 0.0279522
\(876\) 0 0
\(877\) 5.04011 0.170192 0.0850962 0.996373i \(-0.472880\pi\)
0.0850962 + 0.996373i \(0.472880\pi\)
\(878\) 0 0
\(879\) 0.775706 0.0261639
\(880\) 0 0
\(881\) −19.1788 −0.646150 −0.323075 0.946373i \(-0.604717\pi\)
−0.323075 + 0.946373i \(0.604717\pi\)
\(882\) 0 0
\(883\) 7.36535 0.247864 0.123932 0.992291i \(-0.460450\pi\)
0.123932 + 0.992291i \(0.460450\pi\)
\(884\) 0 0
\(885\) 0.162702 0.00546916
\(886\) 0 0
\(887\) −34.1574 −1.14689 −0.573446 0.819243i \(-0.694394\pi\)
−0.573446 + 0.819243i \(0.694394\pi\)
\(888\) 0 0
\(889\) 2.25967 0.0757869
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 2.57593 0.0862001
\(894\) 0 0
\(895\) 6.86425 0.229447
\(896\) 0 0
\(897\) 0.134125 0.00447831
\(898\) 0 0
\(899\) −18.5788 −0.619637
\(900\) 0 0
\(901\) 0 0
\(902\) 0 0
\(903\) −0.545923 −0.0181672
\(904\) 0 0
\(905\) 23.9742 0.796929
\(906\) 0 0
\(907\) −32.5599 −1.08114 −0.540568 0.841301i \(-0.681790\pi\)
−0.540568 + 0.841301i \(0.681790\pi\)
\(908\) 0 0
\(909\) −10.5956 −0.351434
\(910\) 0 0
\(911\) −9.47896 −0.314052 −0.157026 0.987594i \(-0.550191\pi\)
−0.157026 + 0.987594i \(0.550191\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) 0 0
\(915\) −0.244958 −0.00809807
\(916\) 0 0
\(917\) 7.91524 0.261384
\(918\) 0 0
\(919\) −33.8624 −1.11702 −0.558508 0.829499i \(-0.688626\pi\)
−0.558508 + 0.829499i \(0.688626\pi\)
\(920\) 0 0
\(921\) 2.60182 0.0857330
\(922\) 0 0
\(923\) 16.7224 0.550423
\(924\) 0 0
\(925\) 5.98203 0.196688
\(926\) 0 0
\(927\) 49.1638 1.61475
\(928\) 0 0
\(929\) 24.9163 0.817477 0.408739 0.912652i \(-0.365969\pi\)
0.408739 + 0.912652i \(0.365969\pi\)
\(930\) 0 0
\(931\) −4.18559 −0.137177
\(932\) 0 0
\(933\) 2.52994 0.0828267
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) −14.0132 −0.457790 −0.228895 0.973451i \(-0.573511\pi\)
−0.228895 + 0.973451i \(0.573511\pi\)
\(938\) 0 0
\(939\) 1.87013 0.0610292
\(940\) 0 0
\(941\) −33.6453 −1.09681 −0.548403 0.836214i \(-0.684764\pi\)
−0.548403 + 0.836214i \(0.684764\pi\)
\(942\) 0 0
\(943\) −0.186733 −0.00608086
\(944\) 0 0
\(945\) 0.469540 0.0152741
\(946\) 0 0
\(947\) 55.9621 1.81852 0.909262 0.416225i \(-0.136647\pi\)
0.909262 + 0.416225i \(0.136647\pi\)
\(948\) 0 0
\(949\) −23.9281 −0.776740
\(950\) 0 0
\(951\) −0.875998 −0.0284062
\(952\) 0 0
\(953\) 14.9282 0.483572 0.241786 0.970330i \(-0.422267\pi\)
0.241786 + 0.970330i \(0.422267\pi\)
\(954\) 0 0
\(955\) 12.4551 0.403038
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 5.06584 0.163585
\(960\) 0 0
\(961\) −9.42684 −0.304091
\(962\) 0 0
\(963\) −52.1594 −1.68081
\(964\) 0 0
\(965\) 5.19261 0.167156
\(966\) 0 0
\(967\) 31.5986 1.01614 0.508072 0.861315i \(-0.330358\pi\)
0.508072 + 0.861315i \(0.330358\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) −56.5871 −1.81597 −0.907983 0.419006i \(-0.862379\pi\)
−0.907983 + 0.419006i \(0.862379\pi\)
\(972\) 0 0
\(973\) −17.3265 −0.555461
\(974\) 0 0
\(975\) −0.283511 −0.00907962
\(976\) 0 0
\(977\) 41.6948 1.33394 0.666968 0.745086i \(-0.267592\pi\)
0.666968 + 0.745086i \(0.267592\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 0 0
\(981\) 9.69387 0.309501
\(982\) 0 0
\(983\) 24.2687 0.774050 0.387025 0.922069i \(-0.373503\pi\)
0.387025 + 0.922069i \(0.373503\pi\)
\(984\) 0 0
\(985\) 8.68783 0.276817
\(986\) 0 0
\(987\) 0.304659 0.00969740
\(988\) 0 0
\(989\) 3.29534 0.104786
\(990\) 0 0
\(991\) −12.6670 −0.402381 −0.201191 0.979552i \(-0.564481\pi\)
−0.201191 + 0.979552i \(0.564481\pi\)
\(992\) 0 0
\(993\) 1.83730 0.0583049
\(994\) 0 0
\(995\) 13.5149 0.428451
\(996\) 0 0
\(997\) −19.3965 −0.614293 −0.307146 0.951662i \(-0.599374\pi\)
−0.307146 + 0.951662i \(0.599374\pi\)
\(998\) 0 0
\(999\) 3.39704 0.107478
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 9680.2.a.cr.1.2 4
4.3 odd 2 4840.2.a.w.1.3 4
11.10 odd 2 9680.2.a.cq.1.2 4
44.43 even 2 4840.2.a.x.1.3 yes 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
4840.2.a.w.1.3 4 4.3 odd 2
4840.2.a.x.1.3 yes 4 44.43 even 2
9680.2.a.cq.1.2 4 11.10 odd 2
9680.2.a.cr.1.2 4 1.1 even 1 trivial