Properties

Label 5824.2.a.bs.1.1
Level $5824$
Weight $2$
Character 5824.1
Self dual yes
Analytic conductor $46.505$
Analytic rank $1$
Dimension $3$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [5824,2,Mod(1,5824)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(5824, 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("5824.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 5824 = 2^{6} \cdot 7 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5824.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: \(46.5048741372\)
Analytic rank: \(1\)
Dimension: \(3\)
Coefficient field: 3.3.316.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 4x + 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 91)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-1.81361\) of defining polynomial
Character \(\chi\) \(=\) 5824.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-3.10278 q^{3} -2.81361 q^{5} +1.00000 q^{7} +6.62721 q^{9} +O(q^{10})\) \(q-3.10278 q^{3} -2.81361 q^{5} +1.00000 q^{7} +6.62721 q^{9} +3.10278 q^{11} -1.00000 q^{13} +8.72999 q^{15} -0.524438 q^{17} +0.813607 q^{19} -3.10278 q^{21} -7.33804 q^{23} +2.91638 q^{25} -11.2544 q^{27} -8.28917 q^{29} -1.39194 q^{31} -9.62721 q^{33} -2.81361 q^{35} +6.15165 q^{37} +3.10278 q^{39} -4.20555 q^{41} +6.75971 q^{43} -18.6464 q^{45} +5.97028 q^{47} +1.00000 q^{49} +1.62721 q^{51} +2.49472 q^{53} -8.72999 q^{55} -2.52444 q^{57} -4.47054 q^{59} +2.00000 q^{61} +6.62721 q^{63} +2.81361 q^{65} +10.0383 q^{67} +22.7683 q^{69} +8.72999 q^{71} -2.34307 q^{73} -9.04888 q^{75} +3.10278 q^{77} +13.5436 q^{79} +15.0383 q^{81} +16.4791 q^{83} +1.47556 q^{85} +25.7194 q^{87} -10.6464 q^{89} -1.00000 q^{91} +4.31889 q^{93} -2.28917 q^{95} -1.18639 q^{97} +20.5628 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 2 q^{3} - 2 q^{5} + 3 q^{7} + 7 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q - 2 q^{3} - 2 q^{5} + 3 q^{7} + 7 q^{9} + 2 q^{11} - 3 q^{13} + 6 q^{15} + 4 q^{17} - 4 q^{19} - 2 q^{21} - 10 q^{23} - 5 q^{25} - 8 q^{27} - 24 q^{29} + 4 q^{31} - 16 q^{33} - 2 q^{35} + 2 q^{39} + 2 q^{41} + 10 q^{43} - 22 q^{45} + 8 q^{47} + 3 q^{49} - 8 q^{51} - 8 q^{53} - 6 q^{55} - 2 q^{57} - 4 q^{59} + 6 q^{61} + 7 q^{63} + 2 q^{65} - 12 q^{67} + 6 q^{69} + 6 q^{71} - 10 q^{73} - 16 q^{75} + 2 q^{77} + 14 q^{79} + 3 q^{81} - 12 q^{83} + 10 q^{85} + 26 q^{87} + 2 q^{89} - 3 q^{91} + 22 q^{93} - 6 q^{95} - 10 q^{97} + 14 q^{99}+O(q^{100}) \) 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
\(3\) −3.10278 −1.79139 −0.895694 0.444671i \(-0.853321\pi\)
−0.895694 + 0.444671i \(0.853321\pi\)
\(4\) 0 0
\(5\) −2.81361 −1.25828 −0.629142 0.777291i \(-0.716593\pi\)
−0.629142 + 0.777291i \(0.716593\pi\)
\(6\) 0 0
\(7\) 1.00000 0.377964
\(8\) 0 0
\(9\) 6.62721 2.20907
\(10\) 0 0
\(11\) 3.10278 0.935522 0.467761 0.883855i \(-0.345061\pi\)
0.467761 + 0.883855i \(0.345061\pi\)
\(12\) 0 0
\(13\) −1.00000 −0.277350
\(14\) 0 0
\(15\) 8.72999 2.25407
\(16\) 0 0
\(17\) −0.524438 −0.127195 −0.0635974 0.997976i \(-0.520257\pi\)
−0.0635974 + 0.997976i \(0.520257\pi\)
\(18\) 0 0
\(19\) 0.813607 0.186654 0.0933271 0.995636i \(-0.470250\pi\)
0.0933271 + 0.995636i \(0.470250\pi\)
\(20\) 0 0
\(21\) −3.10278 −0.677081
\(22\) 0 0
\(23\) −7.33804 −1.53009 −0.765044 0.643978i \(-0.777283\pi\)
−0.765044 + 0.643978i \(0.777283\pi\)
\(24\) 0 0
\(25\) 2.91638 0.583276
\(26\) 0 0
\(27\) −11.2544 −2.16592
\(28\) 0 0
\(29\) −8.28917 −1.53926 −0.769630 0.638490i \(-0.779559\pi\)
−0.769630 + 0.638490i \(0.779559\pi\)
\(30\) 0 0
\(31\) −1.39194 −0.250000 −0.125000 0.992157i \(-0.539893\pi\)
−0.125000 + 0.992157i \(0.539893\pi\)
\(32\) 0 0
\(33\) −9.62721 −1.67588
\(34\) 0 0
\(35\) −2.81361 −0.475586
\(36\) 0 0
\(37\) 6.15165 1.01133 0.505663 0.862731i \(-0.331248\pi\)
0.505663 + 0.862731i \(0.331248\pi\)
\(38\) 0 0
\(39\) 3.10278 0.496842
\(40\) 0 0
\(41\) −4.20555 −0.656797 −0.328398 0.944539i \(-0.606509\pi\)
−0.328398 + 0.944539i \(0.606509\pi\)
\(42\) 0 0
\(43\) 6.75971 1.03085 0.515423 0.856936i \(-0.327635\pi\)
0.515423 + 0.856936i \(0.327635\pi\)
\(44\) 0 0
\(45\) −18.6464 −2.77964
\(46\) 0 0
\(47\) 5.97028 0.870855 0.435427 0.900224i \(-0.356597\pi\)
0.435427 + 0.900224i \(0.356597\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) 1.62721 0.227855
\(52\) 0 0
\(53\) 2.49472 0.342676 0.171338 0.985212i \(-0.445191\pi\)
0.171338 + 0.985212i \(0.445191\pi\)
\(54\) 0 0
\(55\) −8.72999 −1.17715
\(56\) 0 0
\(57\) −2.52444 −0.334370
\(58\) 0 0
\(59\) −4.47054 −0.582015 −0.291007 0.956721i \(-0.593990\pi\)
−0.291007 + 0.956721i \(0.593990\pi\)
\(60\) 0 0
\(61\) 2.00000 0.256074 0.128037 0.991769i \(-0.459132\pi\)
0.128037 + 0.991769i \(0.459132\pi\)
\(62\) 0 0
\(63\) 6.62721 0.834950
\(64\) 0 0
\(65\) 2.81361 0.348985
\(66\) 0 0
\(67\) 10.0383 1.22638 0.613188 0.789937i \(-0.289887\pi\)
0.613188 + 0.789937i \(0.289887\pi\)
\(68\) 0 0
\(69\) 22.7683 2.74098
\(70\) 0 0
\(71\) 8.72999 1.03606 0.518029 0.855363i \(-0.326666\pi\)
0.518029 + 0.855363i \(0.326666\pi\)
\(72\) 0 0
\(73\) −2.34307 −0.274235 −0.137118 0.990555i \(-0.543784\pi\)
−0.137118 + 0.990555i \(0.543784\pi\)
\(74\) 0 0
\(75\) −9.04888 −1.04487
\(76\) 0 0
\(77\) 3.10278 0.353594
\(78\) 0 0
\(79\) 13.5436 1.52377 0.761887 0.647710i \(-0.224273\pi\)
0.761887 + 0.647710i \(0.224273\pi\)
\(80\) 0 0
\(81\) 15.0383 1.67092
\(82\) 0 0
\(83\) 16.4791 1.80882 0.904410 0.426665i \(-0.140312\pi\)
0.904410 + 0.426665i \(0.140312\pi\)
\(84\) 0 0
\(85\) 1.47556 0.160047
\(86\) 0 0
\(87\) 25.7194 2.75741
\(88\) 0 0
\(89\) −10.6464 −1.12851 −0.564256 0.825600i \(-0.690837\pi\)
−0.564256 + 0.825600i \(0.690837\pi\)
\(90\) 0 0
\(91\) −1.00000 −0.104828
\(92\) 0 0
\(93\) 4.31889 0.447848
\(94\) 0 0
\(95\) −2.28917 −0.234864
\(96\) 0 0
\(97\) −1.18639 −0.120460 −0.0602300 0.998185i \(-0.519183\pi\)
−0.0602300 + 0.998185i \(0.519183\pi\)
\(98\) 0 0
\(99\) 20.5628 2.06663
\(100\) 0 0
\(101\) −13.1028 −1.30377 −0.651887 0.758316i \(-0.726023\pi\)
−0.651887 + 0.758316i \(0.726023\pi\)
\(102\) 0 0
\(103\) −4.41110 −0.434639 −0.217319 0.976101i \(-0.569731\pi\)
−0.217319 + 0.976101i \(0.569731\pi\)
\(104\) 0 0
\(105\) 8.72999 0.851960
\(106\) 0 0
\(107\) 0.578337 0.0559100 0.0279550 0.999609i \(-0.491100\pi\)
0.0279550 + 0.999609i \(0.491100\pi\)
\(108\) 0 0
\(109\) −5.57331 −0.533827 −0.266913 0.963721i \(-0.586004\pi\)
−0.266913 + 0.963721i \(0.586004\pi\)
\(110\) 0 0
\(111\) −19.0872 −1.81168
\(112\) 0 0
\(113\) 5.44584 0.512302 0.256151 0.966637i \(-0.417546\pi\)
0.256151 + 0.966637i \(0.417546\pi\)
\(114\) 0 0
\(115\) 20.6464 1.92528
\(116\) 0 0
\(117\) −6.62721 −0.612686
\(118\) 0 0
\(119\) −0.524438 −0.0480751
\(120\) 0 0
\(121\) −1.37279 −0.124799
\(122\) 0 0
\(123\) 13.0489 1.17658
\(124\) 0 0
\(125\) 5.86248 0.524356
\(126\) 0 0
\(127\) 12.8816 1.14306 0.571530 0.820581i \(-0.306350\pi\)
0.571530 + 0.820581i \(0.306350\pi\)
\(128\) 0 0
\(129\) −20.9739 −1.84664
\(130\) 0 0
\(131\) 9.04888 0.790604 0.395302 0.918551i \(-0.370640\pi\)
0.395302 + 0.918551i \(0.370640\pi\)
\(132\) 0 0
\(133\) 0.813607 0.0705486
\(134\) 0 0
\(135\) 31.6655 2.72533
\(136\) 0 0
\(137\) −6.25945 −0.534781 −0.267390 0.963588i \(-0.586161\pi\)
−0.267390 + 0.963588i \(0.586161\pi\)
\(138\) 0 0
\(139\) −11.5733 −0.981636 −0.490818 0.871262i \(-0.663302\pi\)
−0.490818 + 0.871262i \(0.663302\pi\)
\(140\) 0 0
\(141\) −18.5244 −1.56004
\(142\) 0 0
\(143\) −3.10278 −0.259467
\(144\) 0 0
\(145\) 23.3225 1.93682
\(146\) 0 0
\(147\) −3.10278 −0.255913
\(148\) 0 0
\(149\) −8.52444 −0.698349 −0.349175 0.937058i \(-0.613538\pi\)
−0.349175 + 0.937058i \(0.613538\pi\)
\(150\) 0 0
\(151\) 11.9844 0.975278 0.487639 0.873045i \(-0.337858\pi\)
0.487639 + 0.873045i \(0.337858\pi\)
\(152\) 0 0
\(153\) −3.47556 −0.280983
\(154\) 0 0
\(155\) 3.91638 0.314571
\(156\) 0 0
\(157\) 12.8277 1.02377 0.511883 0.859055i \(-0.328948\pi\)
0.511883 + 0.859055i \(0.328948\pi\)
\(158\) 0 0
\(159\) −7.74055 −0.613866
\(160\) 0 0
\(161\) −7.33804 −0.578319
\(162\) 0 0
\(163\) −13.4600 −1.05427 −0.527133 0.849783i \(-0.676733\pi\)
−0.527133 + 0.849783i \(0.676733\pi\)
\(164\) 0 0
\(165\) 27.0872 2.10873
\(166\) 0 0
\(167\) 2.02972 0.157064 0.0785322 0.996912i \(-0.474977\pi\)
0.0785322 + 0.996912i \(0.474977\pi\)
\(168\) 0 0
\(169\) 1.00000 0.0769231
\(170\) 0 0
\(171\) 5.39194 0.412332
\(172\) 0 0
\(173\) −20.2978 −1.54321 −0.771605 0.636102i \(-0.780546\pi\)
−0.771605 + 0.636102i \(0.780546\pi\)
\(174\) 0 0
\(175\) 2.91638 0.220458
\(176\) 0 0
\(177\) 13.8711 1.04261
\(178\) 0 0
\(179\) −11.0036 −0.822445 −0.411223 0.911535i \(-0.634898\pi\)
−0.411223 + 0.911535i \(0.634898\pi\)
\(180\) 0 0
\(181\) −0.691675 −0.0514118 −0.0257059 0.999670i \(-0.508183\pi\)
−0.0257059 + 0.999670i \(0.508183\pi\)
\(182\) 0 0
\(183\) −6.20555 −0.458727
\(184\) 0 0
\(185\) −17.3083 −1.27253
\(186\) 0 0
\(187\) −1.62721 −0.118994
\(188\) 0 0
\(189\) −11.2544 −0.818639
\(190\) 0 0
\(191\) −7.83276 −0.566759 −0.283379 0.959008i \(-0.591456\pi\)
−0.283379 + 0.959008i \(0.591456\pi\)
\(192\) 0 0
\(193\) −12.2056 −0.878575 −0.439287 0.898347i \(-0.644769\pi\)
−0.439287 + 0.898347i \(0.644769\pi\)
\(194\) 0 0
\(195\) −8.72999 −0.625167
\(196\) 0 0
\(197\) 18.8222 1.34103 0.670513 0.741898i \(-0.266074\pi\)
0.670513 + 0.741898i \(0.266074\pi\)
\(198\) 0 0
\(199\) −21.6116 −1.53201 −0.766004 0.642836i \(-0.777758\pi\)
−0.766004 + 0.642836i \(0.777758\pi\)
\(200\) 0 0
\(201\) −31.1466 −2.19691
\(202\) 0 0
\(203\) −8.28917 −0.581786
\(204\) 0 0
\(205\) 11.8328 0.826436
\(206\) 0 0
\(207\) −48.6308 −3.38007
\(208\) 0 0
\(209\) 2.52444 0.174619
\(210\) 0 0
\(211\) −17.3764 −1.19624 −0.598119 0.801407i \(-0.704085\pi\)
−0.598119 + 0.801407i \(0.704085\pi\)
\(212\) 0 0
\(213\) −27.0872 −1.85598
\(214\) 0 0
\(215\) −19.0192 −1.29710
\(216\) 0 0
\(217\) −1.39194 −0.0944913
\(218\) 0 0
\(219\) 7.27001 0.491262
\(220\) 0 0
\(221\) 0.524438 0.0352775
\(222\) 0 0
\(223\) 10.5486 0.706388 0.353194 0.935550i \(-0.385096\pi\)
0.353194 + 0.935550i \(0.385096\pi\)
\(224\) 0 0
\(225\) 19.3275 1.28850
\(226\) 0 0
\(227\) −6.95112 −0.461362 −0.230681 0.973029i \(-0.574095\pi\)
−0.230681 + 0.973029i \(0.574095\pi\)
\(228\) 0 0
\(229\) 21.0872 1.39348 0.696740 0.717323i \(-0.254633\pi\)
0.696740 + 0.717323i \(0.254633\pi\)
\(230\) 0 0
\(231\) −9.62721 −0.633424
\(232\) 0 0
\(233\) 6.08362 0.398551 0.199276 0.979943i \(-0.436141\pi\)
0.199276 + 0.979943i \(0.436141\pi\)
\(234\) 0 0
\(235\) −16.7980 −1.09578
\(236\) 0 0
\(237\) −42.0227 −2.72967
\(238\) 0 0
\(239\) 14.2056 0.918881 0.459440 0.888209i \(-0.348050\pi\)
0.459440 + 0.888209i \(0.348050\pi\)
\(240\) 0 0
\(241\) 8.44082 0.543721 0.271860 0.962337i \(-0.412361\pi\)
0.271860 + 0.962337i \(0.412361\pi\)
\(242\) 0 0
\(243\) −12.8972 −0.827357
\(244\) 0 0
\(245\) −2.81361 −0.179755
\(246\) 0 0
\(247\) −0.813607 −0.0517685
\(248\) 0 0
\(249\) −51.1310 −3.24030
\(250\) 0 0
\(251\) −23.9844 −1.51388 −0.756941 0.653483i \(-0.773307\pi\)
−0.756941 + 0.653483i \(0.773307\pi\)
\(252\) 0 0
\(253\) −22.7683 −1.43143
\(254\) 0 0
\(255\) −4.57834 −0.286707
\(256\) 0 0
\(257\) 15.6116 0.973827 0.486913 0.873450i \(-0.338123\pi\)
0.486913 + 0.873450i \(0.338123\pi\)
\(258\) 0 0
\(259\) 6.15165 0.382245
\(260\) 0 0
\(261\) −54.9341 −3.40033
\(262\) 0 0
\(263\) 15.1708 0.935472 0.467736 0.883868i \(-0.345070\pi\)
0.467736 + 0.883868i \(0.345070\pi\)
\(264\) 0 0
\(265\) −7.01916 −0.431183
\(266\) 0 0
\(267\) 33.0333 2.02160
\(268\) 0 0
\(269\) −23.2927 −1.42018 −0.710092 0.704109i \(-0.751347\pi\)
−0.710092 + 0.704109i \(0.751347\pi\)
\(270\) 0 0
\(271\) −12.7456 −0.774238 −0.387119 0.922030i \(-0.626530\pi\)
−0.387119 + 0.922030i \(0.626530\pi\)
\(272\) 0 0
\(273\) 3.10278 0.187788
\(274\) 0 0
\(275\) 9.04888 0.545668
\(276\) 0 0
\(277\) −8.12193 −0.488000 −0.244000 0.969775i \(-0.578460\pi\)
−0.244000 + 0.969775i \(0.578460\pi\)
\(278\) 0 0
\(279\) −9.22471 −0.552269
\(280\) 0 0
\(281\) 19.0333 1.13543 0.567715 0.823225i \(-0.307827\pi\)
0.567715 + 0.823225i \(0.307827\pi\)
\(282\) 0 0
\(283\) 11.1466 0.662598 0.331299 0.943526i \(-0.392513\pi\)
0.331299 + 0.943526i \(0.392513\pi\)
\(284\) 0 0
\(285\) 7.10278 0.420732
\(286\) 0 0
\(287\) −4.20555 −0.248246
\(288\) 0 0
\(289\) −16.7250 −0.983821
\(290\) 0 0
\(291\) 3.68111 0.215791
\(292\) 0 0
\(293\) 14.1758 0.828161 0.414080 0.910240i \(-0.364103\pi\)
0.414080 + 0.910240i \(0.364103\pi\)
\(294\) 0 0
\(295\) 12.5783 0.732339
\(296\) 0 0
\(297\) −34.9200 −2.02626
\(298\) 0 0
\(299\) 7.33804 0.424370
\(300\) 0 0
\(301\) 6.75971 0.389623
\(302\) 0 0
\(303\) 40.6550 2.33557
\(304\) 0 0
\(305\) −5.62721 −0.322213
\(306\) 0 0
\(307\) 13.5592 0.773863 0.386932 0.922108i \(-0.373535\pi\)
0.386932 + 0.922108i \(0.373535\pi\)
\(308\) 0 0
\(309\) 13.6867 0.778606
\(310\) 0 0
\(311\) −0.426686 −0.0241952 −0.0120976 0.999927i \(-0.503851\pi\)
−0.0120976 + 0.999927i \(0.503851\pi\)
\(312\) 0 0
\(313\) −18.1517 −1.02599 −0.512996 0.858391i \(-0.671464\pi\)
−0.512996 + 0.858391i \(0.671464\pi\)
\(314\) 0 0
\(315\) −18.6464 −1.05060
\(316\) 0 0
\(317\) −9.42166 −0.529173 −0.264587 0.964362i \(-0.585236\pi\)
−0.264587 + 0.964362i \(0.585236\pi\)
\(318\) 0 0
\(319\) −25.7194 −1.44001
\(320\) 0 0
\(321\) −1.79445 −0.100156
\(322\) 0 0
\(323\) −0.426686 −0.0237415
\(324\) 0 0
\(325\) −2.91638 −0.161772
\(326\) 0 0
\(327\) 17.2927 0.956291
\(328\) 0 0
\(329\) 5.97028 0.329152
\(330\) 0 0
\(331\) −17.4005 −0.956420 −0.478210 0.878246i \(-0.658714\pi\)
−0.478210 + 0.878246i \(0.658714\pi\)
\(332\) 0 0
\(333\) 40.7683 2.23409
\(334\) 0 0
\(335\) −28.2439 −1.54313
\(336\) 0 0
\(337\) 22.0524 1.20127 0.600637 0.799522i \(-0.294914\pi\)
0.600637 + 0.799522i \(0.294914\pi\)
\(338\) 0 0
\(339\) −16.8972 −0.917731
\(340\) 0 0
\(341\) −4.31889 −0.233881
\(342\) 0 0
\(343\) 1.00000 0.0539949
\(344\) 0 0
\(345\) −64.0610 −3.44893
\(346\) 0 0
\(347\) 25.3522 1.36098 0.680488 0.732759i \(-0.261768\pi\)
0.680488 + 0.732759i \(0.261768\pi\)
\(348\) 0 0
\(349\) 5.70529 0.305397 0.152699 0.988273i \(-0.451204\pi\)
0.152699 + 0.988273i \(0.451204\pi\)
\(350\) 0 0
\(351\) 11.2544 0.600717
\(352\) 0 0
\(353\) −28.6761 −1.52627 −0.763137 0.646237i \(-0.776342\pi\)
−0.763137 + 0.646237i \(0.776342\pi\)
\(354\) 0 0
\(355\) −24.5628 −1.30366
\(356\) 0 0
\(357\) 1.62721 0.0861212
\(358\) 0 0
\(359\) −11.0433 −0.582845 −0.291423 0.956594i \(-0.594129\pi\)
−0.291423 + 0.956594i \(0.594129\pi\)
\(360\) 0 0
\(361\) −18.3380 −0.965160
\(362\) 0 0
\(363\) 4.25945 0.223563
\(364\) 0 0
\(365\) 6.59247 0.345066
\(366\) 0 0
\(367\) −27.3466 −1.42748 −0.713741 0.700409i \(-0.753001\pi\)
−0.713741 + 0.700409i \(0.753001\pi\)
\(368\) 0 0
\(369\) −27.8711 −1.45091
\(370\) 0 0
\(371\) 2.49472 0.129519
\(372\) 0 0
\(373\) −16.1461 −0.836014 −0.418007 0.908444i \(-0.637271\pi\)
−0.418007 + 0.908444i \(0.637271\pi\)
\(374\) 0 0
\(375\) −18.1900 −0.939326
\(376\) 0 0
\(377\) 8.28917 0.426914
\(378\) 0 0
\(379\) 26.1305 1.34223 0.671117 0.741351i \(-0.265815\pi\)
0.671117 + 0.741351i \(0.265815\pi\)
\(380\) 0 0
\(381\) −39.9688 −2.04767
\(382\) 0 0
\(383\) 21.0489 1.07555 0.537774 0.843089i \(-0.319265\pi\)
0.537774 + 0.843089i \(0.319265\pi\)
\(384\) 0 0
\(385\) −8.72999 −0.444921
\(386\) 0 0
\(387\) 44.7980 2.27721
\(388\) 0 0
\(389\) 21.6061 1.09547 0.547736 0.836651i \(-0.315490\pi\)
0.547736 + 0.836651i \(0.315490\pi\)
\(390\) 0 0
\(391\) 3.84835 0.194619
\(392\) 0 0
\(393\) −28.0766 −1.41628
\(394\) 0 0
\(395\) −38.1063 −1.91734
\(396\) 0 0
\(397\) 27.6952 1.38998 0.694992 0.719017i \(-0.255408\pi\)
0.694992 + 0.719017i \(0.255408\pi\)
\(398\) 0 0
\(399\) −2.52444 −0.126380
\(400\) 0 0
\(401\) −2.57834 −0.128756 −0.0643780 0.997926i \(-0.520506\pi\)
−0.0643780 + 0.997926i \(0.520506\pi\)
\(402\) 0 0
\(403\) 1.39194 0.0693376
\(404\) 0 0
\(405\) −42.3119 −2.10250
\(406\) 0 0
\(407\) 19.0872 0.946117
\(408\) 0 0
\(409\) 15.1169 0.747483 0.373742 0.927533i \(-0.378075\pi\)
0.373742 + 0.927533i \(0.378075\pi\)
\(410\) 0 0
\(411\) 19.4217 0.958000
\(412\) 0 0
\(413\) −4.47054 −0.219981
\(414\) 0 0
\(415\) −46.3658 −2.27601
\(416\) 0 0
\(417\) 35.9094 1.75849
\(418\) 0 0
\(419\) −9.99446 −0.488261 −0.244131 0.969742i \(-0.578503\pi\)
−0.244131 + 0.969742i \(0.578503\pi\)
\(420\) 0 0
\(421\) 25.9250 1.26351 0.631753 0.775170i \(-0.282336\pi\)
0.631753 + 0.775170i \(0.282336\pi\)
\(422\) 0 0
\(423\) 39.5663 1.92378
\(424\) 0 0
\(425\) −1.52946 −0.0741898
\(426\) 0 0
\(427\) 2.00000 0.0967868
\(428\) 0 0
\(429\) 9.62721 0.464806
\(430\) 0 0
\(431\) −30.6761 −1.47762 −0.738808 0.673916i \(-0.764611\pi\)
−0.738808 + 0.673916i \(0.764611\pi\)
\(432\) 0 0
\(433\) −3.51941 −0.169132 −0.0845661 0.996418i \(-0.526950\pi\)
−0.0845661 + 0.996418i \(0.526950\pi\)
\(434\) 0 0
\(435\) −72.3643 −3.46960
\(436\) 0 0
\(437\) −5.97028 −0.285597
\(438\) 0 0
\(439\) −32.3517 −1.54406 −0.772030 0.635586i \(-0.780759\pi\)
−0.772030 + 0.635586i \(0.780759\pi\)
\(440\) 0 0
\(441\) 6.62721 0.315582
\(442\) 0 0
\(443\) 15.4458 0.733854 0.366927 0.930250i \(-0.380410\pi\)
0.366927 + 0.930250i \(0.380410\pi\)
\(444\) 0 0
\(445\) 29.9547 1.41999
\(446\) 0 0
\(447\) 26.4494 1.25101
\(448\) 0 0
\(449\) −14.4705 −0.682907 −0.341453 0.939899i \(-0.610919\pi\)
−0.341453 + 0.939899i \(0.610919\pi\)
\(450\) 0 0
\(451\) −13.0489 −0.614448
\(452\) 0 0
\(453\) −37.1849 −1.74710
\(454\) 0 0
\(455\) 2.81361 0.131904
\(456\) 0 0
\(457\) −34.6705 −1.62182 −0.810910 0.585171i \(-0.801027\pi\)
−0.810910 + 0.585171i \(0.801027\pi\)
\(458\) 0 0
\(459\) 5.90225 0.275493
\(460\) 0 0
\(461\) −12.5400 −0.584047 −0.292024 0.956411i \(-0.594329\pi\)
−0.292024 + 0.956411i \(0.594329\pi\)
\(462\) 0 0
\(463\) −12.1517 −0.564735 −0.282368 0.959306i \(-0.591120\pi\)
−0.282368 + 0.959306i \(0.591120\pi\)
\(464\) 0 0
\(465\) −12.1517 −0.563519
\(466\) 0 0
\(467\) −37.0333 −1.71370 −0.856848 0.515569i \(-0.827581\pi\)
−0.856848 + 0.515569i \(0.827581\pi\)
\(468\) 0 0
\(469\) 10.0383 0.463526
\(470\) 0 0
\(471\) −39.8016 −1.83396
\(472\) 0 0
\(473\) 20.9739 0.964379
\(474\) 0 0
\(475\) 2.37279 0.108871
\(476\) 0 0
\(477\) 16.5330 0.756996
\(478\) 0 0
\(479\) −12.0086 −0.548687 −0.274343 0.961632i \(-0.588460\pi\)
−0.274343 + 0.961632i \(0.588460\pi\)
\(480\) 0 0
\(481\) −6.15165 −0.280491
\(482\) 0 0
\(483\) 22.7683 1.03599
\(484\) 0 0
\(485\) 3.33804 0.151573
\(486\) 0 0
\(487\) −11.1184 −0.503821 −0.251911 0.967751i \(-0.581059\pi\)
−0.251911 + 0.967751i \(0.581059\pi\)
\(488\) 0 0
\(489\) 41.7633 1.88860
\(490\) 0 0
\(491\) 0.0594386 0.00268243 0.00134121 0.999999i \(-0.499573\pi\)
0.00134121 + 0.999999i \(0.499573\pi\)
\(492\) 0 0
\(493\) 4.34715 0.195786
\(494\) 0 0
\(495\) −57.8555 −2.60041
\(496\) 0 0
\(497\) 8.72999 0.391593
\(498\) 0 0
\(499\) 10.2978 0.460991 0.230496 0.973073i \(-0.425965\pi\)
0.230496 + 0.973073i \(0.425965\pi\)
\(500\) 0 0
\(501\) −6.29776 −0.281363
\(502\) 0 0
\(503\) 9.32391 0.415733 0.207866 0.978157i \(-0.433348\pi\)
0.207866 + 0.978157i \(0.433348\pi\)
\(504\) 0 0
\(505\) 36.8661 1.64052
\(506\) 0 0
\(507\) −3.10278 −0.137799
\(508\) 0 0
\(509\) −39.6952 −1.75946 −0.879730 0.475473i \(-0.842277\pi\)
−0.879730 + 0.475473i \(0.842277\pi\)
\(510\) 0 0
\(511\) −2.34307 −0.103651
\(512\) 0 0
\(513\) −9.15667 −0.404277
\(514\) 0 0
\(515\) 12.4111 0.546898
\(516\) 0 0
\(517\) 18.5244 0.814704
\(518\) 0 0
\(519\) 62.9794 2.76449
\(520\) 0 0
\(521\) 22.3627 0.979729 0.489865 0.871798i \(-0.337046\pi\)
0.489865 + 0.871798i \(0.337046\pi\)
\(522\) 0 0
\(523\) −20.6550 −0.903178 −0.451589 0.892226i \(-0.649143\pi\)
−0.451589 + 0.892226i \(0.649143\pi\)
\(524\) 0 0
\(525\) −9.04888 −0.394925
\(526\) 0 0
\(527\) 0.729988 0.0317988
\(528\) 0 0
\(529\) 30.8469 1.34117
\(530\) 0 0
\(531\) −29.6272 −1.28571
\(532\) 0 0
\(533\) 4.20555 0.182163
\(534\) 0 0
\(535\) −1.62721 −0.0703506
\(536\) 0 0
\(537\) 34.1416 1.47332
\(538\) 0 0
\(539\) 3.10278 0.133646
\(540\) 0 0
\(541\) −5.62167 −0.241695 −0.120847 0.992671i \(-0.538561\pi\)
−0.120847 + 0.992671i \(0.538561\pi\)
\(542\) 0 0
\(543\) 2.14611 0.0920985
\(544\) 0 0
\(545\) 15.6811 0.671705
\(546\) 0 0
\(547\) −10.3970 −0.444542 −0.222271 0.974985i \(-0.571347\pi\)
−0.222271 + 0.974985i \(0.571347\pi\)
\(548\) 0 0
\(549\) 13.2544 0.565685
\(550\) 0 0
\(551\) −6.74412 −0.287309
\(552\) 0 0
\(553\) 13.5436 0.575932
\(554\) 0 0
\(555\) 53.7038 2.27960
\(556\) 0 0
\(557\) −14.6550 −0.620951 −0.310475 0.950581i \(-0.600488\pi\)
−0.310475 + 0.950581i \(0.600488\pi\)
\(558\) 0 0
\(559\) −6.75971 −0.285905
\(560\) 0 0
\(561\) 5.04888 0.213164
\(562\) 0 0
\(563\) −24.7456 −1.04290 −0.521451 0.853281i \(-0.674609\pi\)
−0.521451 + 0.853281i \(0.674609\pi\)
\(564\) 0 0
\(565\) −15.3225 −0.644621
\(566\) 0 0
\(567\) 15.0383 0.631550
\(568\) 0 0
\(569\) −20.5330 −0.860789 −0.430395 0.902641i \(-0.641626\pi\)
−0.430395 + 0.902641i \(0.641626\pi\)
\(570\) 0 0
\(571\) −41.8953 −1.75326 −0.876631 0.481163i \(-0.840214\pi\)
−0.876631 + 0.481163i \(0.840214\pi\)
\(572\) 0 0
\(573\) 24.3033 1.01529
\(574\) 0 0
\(575\) −21.4005 −0.892464
\(576\) 0 0
\(577\) −20.1744 −0.839870 −0.419935 0.907554i \(-0.637947\pi\)
−0.419935 + 0.907554i \(0.637947\pi\)
\(578\) 0 0
\(579\) 37.8711 1.57387
\(580\) 0 0
\(581\) 16.4791 0.683670
\(582\) 0 0
\(583\) 7.74055 0.320581
\(584\) 0 0
\(585\) 18.6464 0.770933
\(586\) 0 0
\(587\) −18.7441 −0.773653 −0.386826 0.922153i \(-0.626429\pi\)
−0.386826 + 0.922153i \(0.626429\pi\)
\(588\) 0 0
\(589\) −1.13249 −0.0466636
\(590\) 0 0
\(591\) −58.4011 −2.40230
\(592\) 0 0
\(593\) −2.98084 −0.122409 −0.0612043 0.998125i \(-0.519494\pi\)
−0.0612043 + 0.998125i \(0.519494\pi\)
\(594\) 0 0
\(595\) 1.47556 0.0604921
\(596\) 0 0
\(597\) 67.0560 2.74442
\(598\) 0 0
\(599\) −7.47411 −0.305384 −0.152692 0.988274i \(-0.548794\pi\)
−0.152692 + 0.988274i \(0.548794\pi\)
\(600\) 0 0
\(601\) 21.4700 0.875780 0.437890 0.899028i \(-0.355726\pi\)
0.437890 + 0.899028i \(0.355726\pi\)
\(602\) 0 0
\(603\) 66.5260 2.70915
\(604\) 0 0
\(605\) 3.86248 0.157032
\(606\) 0 0
\(607\) 22.9044 0.929660 0.464830 0.885400i \(-0.346116\pi\)
0.464830 + 0.885400i \(0.346116\pi\)
\(608\) 0 0
\(609\) 25.7194 1.04220
\(610\) 0 0
\(611\) −5.97028 −0.241532
\(612\) 0 0
\(613\) 20.1461 0.813694 0.406847 0.913496i \(-0.366628\pi\)
0.406847 + 0.913496i \(0.366628\pi\)
\(614\) 0 0
\(615\) −36.7144 −1.48047
\(616\) 0 0
\(617\) −13.7844 −0.554939 −0.277470 0.960734i \(-0.589496\pi\)
−0.277470 + 0.960734i \(0.589496\pi\)
\(618\) 0 0
\(619\) −19.6655 −0.790424 −0.395212 0.918590i \(-0.629329\pi\)
−0.395212 + 0.918590i \(0.629329\pi\)
\(620\) 0 0
\(621\) 82.5855 3.31404
\(622\) 0 0
\(623\) −10.6464 −0.426538
\(624\) 0 0
\(625\) −31.0766 −1.24307
\(626\) 0 0
\(627\) −7.83276 −0.312810
\(628\) 0 0
\(629\) −3.22616 −0.128635
\(630\) 0 0
\(631\) 16.1672 0.643608 0.321804 0.946806i \(-0.395711\pi\)
0.321804 + 0.946806i \(0.395711\pi\)
\(632\) 0 0
\(633\) 53.9149 2.14293
\(634\) 0 0
\(635\) −36.2439 −1.43829
\(636\) 0 0
\(637\) −1.00000 −0.0396214
\(638\) 0 0
\(639\) 57.8555 2.28873
\(640\) 0 0
\(641\) −29.0036 −1.14557 −0.572786 0.819705i \(-0.694137\pi\)
−0.572786 + 0.819705i \(0.694137\pi\)
\(642\) 0 0
\(643\) −39.2233 −1.54681 −0.773407 0.633910i \(-0.781449\pi\)
−0.773407 + 0.633910i \(0.781449\pi\)
\(644\) 0 0
\(645\) 59.0122 2.32360
\(646\) 0 0
\(647\) 11.9844 0.471156 0.235578 0.971855i \(-0.424302\pi\)
0.235578 + 0.971855i \(0.424302\pi\)
\(648\) 0 0
\(649\) −13.8711 −0.544487
\(650\) 0 0
\(651\) 4.31889 0.169271
\(652\) 0 0
\(653\) −45.3311 −1.77394 −0.886971 0.461826i \(-0.847194\pi\)
−0.886971 + 0.461826i \(0.847194\pi\)
\(654\) 0 0
\(655\) −25.4600 −0.994804
\(656\) 0 0
\(657\) −15.5280 −0.605805
\(658\) 0 0
\(659\) 6.12193 0.238477 0.119238 0.992866i \(-0.461955\pi\)
0.119238 + 0.992866i \(0.461955\pi\)
\(660\) 0 0
\(661\) 27.5280 1.07072 0.535358 0.844625i \(-0.320177\pi\)
0.535358 + 0.844625i \(0.320177\pi\)
\(662\) 0 0
\(663\) −1.62721 −0.0631957
\(664\) 0 0
\(665\) −2.28917 −0.0887701
\(666\) 0 0
\(667\) 60.8263 2.35520
\(668\) 0 0
\(669\) −32.7300 −1.26541
\(670\) 0 0
\(671\) 6.20555 0.239563
\(672\) 0 0
\(673\) 27.9547 1.07757 0.538787 0.842442i \(-0.318883\pi\)
0.538787 + 0.842442i \(0.318883\pi\)
\(674\) 0 0
\(675\) −32.8222 −1.26333
\(676\) 0 0
\(677\) 12.6605 0.486583 0.243291 0.969953i \(-0.421773\pi\)
0.243291 + 0.969953i \(0.421773\pi\)
\(678\) 0 0
\(679\) −1.18639 −0.0455296
\(680\) 0 0
\(681\) 21.5678 0.826479
\(682\) 0 0
\(683\) −28.3033 −1.08300 −0.541498 0.840702i \(-0.682143\pi\)
−0.541498 + 0.840702i \(0.682143\pi\)
\(684\) 0 0
\(685\) 17.6116 0.672906
\(686\) 0 0
\(687\) −65.4288 −2.49626
\(688\) 0 0
\(689\) −2.49472 −0.0950412
\(690\) 0 0
\(691\) 12.2353 0.465452 0.232726 0.972542i \(-0.425236\pi\)
0.232726 + 0.972542i \(0.425236\pi\)
\(692\) 0 0
\(693\) 20.5628 0.781114
\(694\) 0 0
\(695\) 32.5628 1.23518
\(696\) 0 0
\(697\) 2.20555 0.0835412
\(698\) 0 0
\(699\) −18.8761 −0.713960
\(700\) 0 0
\(701\) −51.0419 −1.92783 −0.963913 0.266219i \(-0.914226\pi\)
−0.963913 + 0.266219i \(0.914226\pi\)
\(702\) 0 0
\(703\) 5.00502 0.188768
\(704\) 0 0
\(705\) 52.1205 1.96297
\(706\) 0 0
\(707\) −13.1028 −0.492781
\(708\) 0 0
\(709\) −42.5910 −1.59954 −0.799770 0.600307i \(-0.795045\pi\)
−0.799770 + 0.600307i \(0.795045\pi\)
\(710\) 0 0
\(711\) 89.7563 3.36612
\(712\) 0 0
\(713\) 10.2141 0.382523
\(714\) 0 0
\(715\) 8.72999 0.326483
\(716\) 0 0
\(717\) −44.0766 −1.64607
\(718\) 0 0
\(719\) 42.4933 1.58473 0.792366 0.610046i \(-0.208849\pi\)
0.792366 + 0.610046i \(0.208849\pi\)
\(720\) 0 0
\(721\) −4.41110 −0.164278
\(722\) 0 0
\(723\) −26.1900 −0.974015
\(724\) 0 0
\(725\) −24.1744 −0.897814
\(726\) 0 0
\(727\) 3.75614 0.139307 0.0696537 0.997571i \(-0.477811\pi\)
0.0696537 + 0.997571i \(0.477811\pi\)
\(728\) 0 0
\(729\) −5.09775 −0.188806
\(730\) 0 0
\(731\) −3.54505 −0.131118
\(732\) 0 0
\(733\) −45.7819 −1.69099 −0.845497 0.533980i \(-0.820696\pi\)
−0.845497 + 0.533980i \(0.820696\pi\)
\(734\) 0 0
\(735\) 8.72999 0.322010
\(736\) 0 0
\(737\) 31.1466 1.14730
\(738\) 0 0
\(739\) −14.0539 −0.516981 −0.258491 0.966014i \(-0.583225\pi\)
−0.258491 + 0.966014i \(0.583225\pi\)
\(740\) 0 0
\(741\) 2.52444 0.0927375
\(742\) 0 0
\(743\) −4.74557 −0.174098 −0.0870491 0.996204i \(-0.527744\pi\)
−0.0870491 + 0.996204i \(0.527744\pi\)
\(744\) 0 0
\(745\) 23.9844 0.878721
\(746\) 0 0
\(747\) 109.211 3.99581
\(748\) 0 0
\(749\) 0.578337 0.0211320
\(750\) 0 0
\(751\) −36.1008 −1.31734 −0.658669 0.752433i \(-0.728880\pi\)
−0.658669 + 0.752433i \(0.728880\pi\)
\(752\) 0 0
\(753\) 74.4182 2.71195
\(754\) 0 0
\(755\) −33.7194 −1.22718
\(756\) 0 0
\(757\) 1.03474 0.0376084 0.0188042 0.999823i \(-0.494014\pi\)
0.0188042 + 0.999823i \(0.494014\pi\)
\(758\) 0 0
\(759\) 70.6449 2.56425
\(760\) 0 0
\(761\) 29.8414 1.08175 0.540874 0.841104i \(-0.318094\pi\)
0.540874 + 0.841104i \(0.318094\pi\)
\(762\) 0 0
\(763\) −5.57331 −0.201768
\(764\) 0 0
\(765\) 9.77886 0.353556
\(766\) 0 0
\(767\) 4.47054 0.161422
\(768\) 0 0
\(769\) 23.6358 0.852329 0.426164 0.904646i \(-0.359864\pi\)
0.426164 + 0.904646i \(0.359864\pi\)
\(770\) 0 0
\(771\) −48.4394 −1.74450
\(772\) 0 0
\(773\) −13.0278 −0.468576 −0.234288 0.972167i \(-0.575276\pi\)
−0.234288 + 0.972167i \(0.575276\pi\)
\(774\) 0 0
\(775\) −4.05944 −0.145819
\(776\) 0 0
\(777\) −19.0872 −0.684749
\(778\) 0 0
\(779\) −3.42166 −0.122594
\(780\) 0 0
\(781\) 27.0872 0.969256
\(782\) 0 0
\(783\) 93.2898 3.33391
\(784\) 0 0
\(785\) −36.0922 −1.28819
\(786\) 0 0
\(787\) 46.2141 1.64736 0.823678 0.567058i \(-0.191918\pi\)
0.823678 + 0.567058i \(0.191918\pi\)
\(788\) 0 0
\(789\) −47.0716 −1.67579
\(790\) 0 0
\(791\) 5.44584 0.193632
\(792\) 0 0
\(793\) −2.00000 −0.0710221
\(794\) 0 0
\(795\) 21.7789 0.772417
\(796\) 0 0
\(797\) −53.1155 −1.88145 −0.940723 0.339176i \(-0.889852\pi\)
−0.940723 + 0.339176i \(0.889852\pi\)
\(798\) 0 0
\(799\) −3.13104 −0.110768
\(800\) 0 0
\(801\) −70.5558 −2.49297
\(802\) 0 0
\(803\) −7.27001 −0.256553
\(804\) 0 0
\(805\) 20.6464 0.727689
\(806\) 0 0
\(807\) 72.2721 2.54410
\(808\) 0 0
\(809\) −54.4635 −1.91484 −0.957418 0.288705i \(-0.906775\pi\)
−0.957418 + 0.288705i \(0.906775\pi\)
\(810\) 0 0
\(811\) 38.0978 1.33779 0.668897 0.743356i \(-0.266767\pi\)
0.668897 + 0.743356i \(0.266767\pi\)
\(812\) 0 0
\(813\) 39.5466 1.38696
\(814\) 0 0
\(815\) 37.8711 1.32657
\(816\) 0 0
\(817\) 5.49974 0.192412
\(818\) 0 0
\(819\) −6.62721 −0.231574
\(820\) 0 0
\(821\) 2.30330 0.0803858 0.0401929 0.999192i \(-0.487203\pi\)
0.0401929 + 0.999192i \(0.487203\pi\)
\(822\) 0 0
\(823\) −23.6172 −0.823243 −0.411621 0.911355i \(-0.635037\pi\)
−0.411621 + 0.911355i \(0.635037\pi\)
\(824\) 0 0
\(825\) −28.0766 −0.977503
\(826\) 0 0
\(827\) −48.1643 −1.67484 −0.837419 0.546562i \(-0.815936\pi\)
−0.837419 + 0.546562i \(0.815936\pi\)
\(828\) 0 0
\(829\) −13.0716 −0.453996 −0.226998 0.973895i \(-0.572891\pi\)
−0.226998 + 0.973895i \(0.572891\pi\)
\(830\) 0 0
\(831\) 25.2005 0.874197
\(832\) 0 0
\(833\) −0.524438 −0.0181707
\(834\) 0 0
\(835\) −5.71083 −0.197631
\(836\) 0 0
\(837\) 15.6655 0.541480
\(838\) 0 0
\(839\) 17.6756 0.610229 0.305114 0.952316i \(-0.401305\pi\)
0.305114 + 0.952316i \(0.401305\pi\)
\(840\) 0 0
\(841\) 39.7103 1.36932
\(842\) 0 0
\(843\) −59.0560 −2.03400
\(844\) 0 0
\(845\) −2.81361 −0.0967910
\(846\) 0 0
\(847\) −1.37279 −0.0471695
\(848\) 0 0
\(849\) −34.5855 −1.18697
\(850\) 0 0
\(851\) −45.1411 −1.54742
\(852\) 0 0
\(853\) 5.48970 0.187964 0.0939818 0.995574i \(-0.470040\pi\)
0.0939818 + 0.995574i \(0.470040\pi\)
\(854\) 0 0
\(855\) −15.1708 −0.518831
\(856\) 0 0
\(857\) −11.0489 −0.377422 −0.188711 0.982033i \(-0.560431\pi\)
−0.188711 + 0.982033i \(0.560431\pi\)
\(858\) 0 0
\(859\) 45.2616 1.54430 0.772152 0.635437i \(-0.219180\pi\)
0.772152 + 0.635437i \(0.219180\pi\)
\(860\) 0 0
\(861\) 13.0489 0.444705
\(862\) 0 0
\(863\) 5.90225 0.200915 0.100457 0.994941i \(-0.467969\pi\)
0.100457 + 0.994941i \(0.467969\pi\)
\(864\) 0 0
\(865\) 57.1099 1.94180
\(866\) 0 0
\(867\) 51.8938 1.76241
\(868\) 0 0
\(869\) 42.0227 1.42552
\(870\) 0 0
\(871\) −10.0383 −0.340135
\(872\) 0 0
\(873\) −7.86248 −0.266105
\(874\) 0 0
\(875\) 5.86248 0.198188
\(876\) 0 0
\(877\) 4.90727 0.165707 0.0828534 0.996562i \(-0.473597\pi\)
0.0828534 + 0.996562i \(0.473597\pi\)
\(878\) 0 0
\(879\) −43.9844 −1.48356
\(880\) 0 0
\(881\) 44.2822 1.49190 0.745952 0.665999i \(-0.231995\pi\)
0.745952 + 0.665999i \(0.231995\pi\)
\(882\) 0 0
\(883\) −58.8605 −1.98081 −0.990407 0.138181i \(-0.955874\pi\)
−0.990407 + 0.138181i \(0.955874\pi\)
\(884\) 0 0
\(885\) −39.0278 −1.31190
\(886\) 0 0
\(887\) 10.1289 0.340096 0.170048 0.985436i \(-0.445608\pi\)
0.170048 + 0.985436i \(0.445608\pi\)
\(888\) 0 0
\(889\) 12.8816 0.432036
\(890\) 0 0
\(891\) 46.6605 1.56319
\(892\) 0 0
\(893\) 4.85746 0.162549
\(894\) 0 0
\(895\) 30.9597 1.03487
\(896\) 0 0
\(897\) −22.7683 −0.760211
\(898\) 0 0
\(899\) 11.5381 0.384816
\(900\) 0 0
\(901\) −1.30833 −0.0435866
\(902\) 0 0
\(903\) −20.9739 −0.697966
\(904\) 0 0
\(905\) 1.94610 0.0646906
\(906\) 0 0
\(907\) 37.9547 1.26026 0.630132 0.776488i \(-0.283001\pi\)
0.630132 + 0.776488i \(0.283001\pi\)
\(908\) 0 0
\(909\) −86.8349 −2.88013
\(910\) 0 0
\(911\) −5.57477 −0.184700 −0.0923501 0.995727i \(-0.529438\pi\)
−0.0923501 + 0.995727i \(0.529438\pi\)
\(912\) 0 0
\(913\) 51.1310 1.69219
\(914\) 0 0
\(915\) 17.4600 0.577209
\(916\) 0 0
\(917\) 9.04888 0.298820
\(918\) 0 0
\(919\) −15.7844 −0.520679 −0.260340 0.965517i \(-0.583835\pi\)
−0.260340 + 0.965517i \(0.583835\pi\)
\(920\) 0 0
\(921\) −42.0711 −1.38629
\(922\) 0 0
\(923\) −8.72999 −0.287351
\(924\) 0 0
\(925\) 17.9406 0.589882
\(926\) 0 0
\(927\) −29.2333 −0.960148
\(928\) 0 0
\(929\) −45.2630 −1.48503 −0.742516 0.669829i \(-0.766368\pi\)
−0.742516 + 0.669829i \(0.766368\pi\)
\(930\) 0 0
\(931\) 0.813607 0.0266649
\(932\) 0 0
\(933\) 1.32391 0.0433429
\(934\) 0 0
\(935\) 4.57834 0.149728
\(936\) 0 0
\(937\) 53.6188 1.75165 0.875824 0.482630i \(-0.160318\pi\)
0.875824 + 0.482630i \(0.160318\pi\)
\(938\) 0 0
\(939\) 56.3205 1.83795
\(940\) 0 0
\(941\) −20.7753 −0.677255 −0.338628 0.940920i \(-0.609963\pi\)
−0.338628 + 0.940920i \(0.609963\pi\)
\(942\) 0 0
\(943\) 30.8605 1.00496
\(944\) 0 0
\(945\) 31.6655 1.03008
\(946\) 0 0
\(947\) −10.8605 −0.352919 −0.176460 0.984308i \(-0.556465\pi\)
−0.176460 + 0.984308i \(0.556465\pi\)
\(948\) 0 0
\(949\) 2.34307 0.0760592
\(950\) 0 0
\(951\) 29.2333 0.947955
\(952\) 0 0
\(953\) −25.7180 −0.833087 −0.416543 0.909116i \(-0.636759\pi\)
−0.416543 + 0.909116i \(0.636759\pi\)
\(954\) 0 0
\(955\) 22.0383 0.713143
\(956\) 0 0
\(957\) 79.8016 2.57962
\(958\) 0 0
\(959\) −6.25945 −0.202128
\(960\) 0 0
\(961\) −29.0625 −0.937500
\(962\) 0 0
\(963\) 3.83276 0.123509
\(964\) 0 0
\(965\) 34.3416 1.10550
\(966\) 0 0
\(967\) −33.5038 −1.07741 −0.538705 0.842494i \(-0.681086\pi\)
−0.538705 + 0.842494i \(0.681086\pi\)
\(968\) 0 0
\(969\) 1.32391 0.0425302
\(970\) 0 0
\(971\) 2.03831 0.0654126 0.0327063 0.999465i \(-0.489587\pi\)
0.0327063 + 0.999465i \(0.489587\pi\)
\(972\) 0 0
\(973\) −11.5733 −0.371023
\(974\) 0 0
\(975\) 9.04888 0.289796
\(976\) 0 0
\(977\) 15.1411 0.484406 0.242203 0.970226i \(-0.422130\pi\)
0.242203 + 0.970226i \(0.422130\pi\)
\(978\) 0 0
\(979\) −33.0333 −1.05575
\(980\) 0 0
\(981\) −36.9355 −1.17926
\(982\) 0 0
\(983\) −49.3124 −1.57282 −0.786411 0.617704i \(-0.788063\pi\)
−0.786411 + 0.617704i \(0.788063\pi\)
\(984\) 0 0
\(985\) −52.9583 −1.68739
\(986\) 0 0
\(987\) −18.5244 −0.589639
\(988\) 0 0
\(989\) −49.6030 −1.57728
\(990\) 0 0
\(991\) −5.43171 −0.172544 −0.0862720 0.996272i \(-0.527495\pi\)
−0.0862720 + 0.996272i \(0.527495\pi\)
\(992\) 0 0
\(993\) 53.9900 1.71332
\(994\) 0 0
\(995\) 60.8066 1.92770
\(996\) 0 0
\(997\) 53.6061 1.69772 0.848861 0.528616i \(-0.177289\pi\)
0.848861 + 0.528616i \(0.177289\pi\)
\(998\) 0 0
\(999\) −69.2333 −2.19044
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 5824.2.a.bs.1.1 3
4.3 odd 2 5824.2.a.by.1.3 3
8.3 odd 2 91.2.a.d.1.1 3
8.5 even 2 1456.2.a.t.1.3 3
24.11 even 2 819.2.a.i.1.3 3
40.19 odd 2 2275.2.a.m.1.3 3
56.3 even 6 637.2.e.i.79.3 6
56.11 odd 6 637.2.e.j.79.3 6
56.19 even 6 637.2.e.i.508.3 6
56.27 even 2 637.2.a.j.1.1 3
56.51 odd 6 637.2.e.j.508.3 6
104.51 odd 2 1183.2.a.i.1.3 3
104.83 even 4 1183.2.c.f.337.5 6
104.99 even 4 1183.2.c.f.337.2 6
168.83 odd 2 5733.2.a.x.1.3 3
728.363 even 2 8281.2.a.bg.1.3 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
91.2.a.d.1.1 3 8.3 odd 2
637.2.a.j.1.1 3 56.27 even 2
637.2.e.i.79.3 6 56.3 even 6
637.2.e.i.508.3 6 56.19 even 6
637.2.e.j.79.3 6 56.11 odd 6
637.2.e.j.508.3 6 56.51 odd 6
819.2.a.i.1.3 3 24.11 even 2
1183.2.a.i.1.3 3 104.51 odd 2
1183.2.c.f.337.2 6 104.99 even 4
1183.2.c.f.337.5 6 104.83 even 4
1456.2.a.t.1.3 3 8.5 even 2
2275.2.a.m.1.3 3 40.19 odd 2
5733.2.a.x.1.3 3 168.83 odd 2
5824.2.a.bs.1.1 3 1.1 even 1 trivial
5824.2.a.by.1.3 3 4.3 odd 2
8281.2.a.bg.1.3 3 728.363 even 2