Properties

Label 4840.2.a.bd.1.5
Level $4840$
Weight $2$
Character 4840.1
Self dual yes
Analytic conductor $38.648$
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] = [4840,2,Mod(1,4840)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4840, 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("4840.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4840 = 2^{3} \cdot 5 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4840.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: \(38.6475945783\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: 6.6.22733568.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - 8x^{4} - 2x^{3} + 16x^{2} + 8x - 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.5
Root \(-2.08589\) of defining polynomial
Character \(\chi\) \(=\) 4840.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.35095 q^{3} +1.00000 q^{5} -3.00679 q^{7} -1.17493 q^{9} +O(q^{10})\) \(q+1.35095 q^{3} +1.00000 q^{5} -3.00679 q^{7} -1.17493 q^{9} +5.69877 q^{13} +1.35095 q^{15} +4.70191 q^{17} -3.69877 q^{19} -4.06204 q^{21} -6.83866 q^{23} +1.00000 q^{25} -5.64013 q^{27} +10.6956 q^{29} -3.50491 q^{31} -3.00679 q^{35} +5.79409 q^{37} +7.69877 q^{39} +8.39893 q^{41} +2.80716 q^{43} -1.17493 q^{45} +5.14504 q^{47} +2.04081 q^{49} +6.35205 q^{51} -12.2204 q^{53} -4.99686 q^{57} +3.30528 q^{59} +10.7990 q^{61} +3.53276 q^{63} +5.69877 q^{65} +11.5328 q^{67} -9.23871 q^{69} +2.05440 q^{71} +3.29589 q^{73} +1.35095 q^{75} -4.67984 q^{79} -4.09477 q^{81} -12.5247 q^{83} +4.70191 q^{85} +14.4493 q^{87} -4.32366 q^{89} -17.1350 q^{91} -4.73497 q^{93} -3.69877 q^{95} +2.94235 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 2 q^{3} + 6 q^{5} + 4 q^{7} + 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q - 2 q^{3} + 6 q^{5} + 4 q^{7} + 4 q^{9} - 2 q^{15} + 8 q^{17} + 12 q^{19} - 8 q^{21} - 8 q^{23} + 6 q^{25} - 14 q^{27} + 16 q^{29} - 4 q^{31} + 4 q^{35} + 8 q^{37} + 12 q^{39} + 32 q^{41} - 4 q^{43} + 4 q^{45} - 6 q^{47} + 16 q^{49} + 40 q^{51} + 8 q^{53} - 16 q^{57} + 4 q^{59} + 16 q^{61} + 28 q^{63} - 2 q^{67} + 8 q^{69} - 28 q^{71} + 16 q^{73} - 2 q^{75} - 10 q^{81} + 12 q^{83} + 8 q^{85} - 24 q^{87} + 18 q^{89} - 24 q^{91} + 20 q^{93} + 12 q^{95}+O(q^{100}) \) 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
\(3\) 1.35095 0.779973 0.389987 0.920821i \(-0.372480\pi\)
0.389987 + 0.920821i \(0.372480\pi\)
\(4\) 0 0
\(5\) 1.00000 0.447214
\(6\) 0 0
\(7\) −3.00679 −1.13646 −0.568231 0.822869i \(-0.692372\pi\)
−0.568231 + 0.822869i \(0.692372\pi\)
\(8\) 0 0
\(9\) −1.17493 −0.391642
\(10\) 0 0
\(11\) 0 0
\(12\) 0 0
\(13\) 5.69877 1.58055 0.790277 0.612750i \(-0.209937\pi\)
0.790277 + 0.612750i \(0.209937\pi\)
\(14\) 0 0
\(15\) 1.35095 0.348815
\(16\) 0 0
\(17\) 4.70191 1.14038 0.570190 0.821513i \(-0.306870\pi\)
0.570190 + 0.821513i \(0.306870\pi\)
\(18\) 0 0
\(19\) −3.69877 −0.848556 −0.424278 0.905532i \(-0.639472\pi\)
−0.424278 + 0.905532i \(0.639472\pi\)
\(20\) 0 0
\(21\) −4.06204 −0.886409
\(22\) 0 0
\(23\) −6.83866 −1.42596 −0.712980 0.701185i \(-0.752655\pi\)
−0.712980 + 0.701185i \(0.752655\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) −5.64013 −1.08544
\(28\) 0 0
\(29\) 10.6956 1.98613 0.993064 0.117573i \(-0.0375113\pi\)
0.993064 + 0.117573i \(0.0375113\pi\)
\(30\) 0 0
\(31\) −3.50491 −0.629501 −0.314750 0.949174i \(-0.601921\pi\)
−0.314750 + 0.949174i \(0.601921\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −3.00679 −0.508241
\(36\) 0 0
\(37\) 5.79409 0.952542 0.476271 0.879298i \(-0.341988\pi\)
0.476271 + 0.879298i \(0.341988\pi\)
\(38\) 0 0
\(39\) 7.69877 1.23279
\(40\) 0 0
\(41\) 8.39893 1.31169 0.655846 0.754895i \(-0.272312\pi\)
0.655846 + 0.754895i \(0.272312\pi\)
\(42\) 0 0
\(43\) 2.80716 0.428088 0.214044 0.976824i \(-0.431336\pi\)
0.214044 + 0.976824i \(0.431336\pi\)
\(44\) 0 0
\(45\) −1.17493 −0.175148
\(46\) 0 0
\(47\) 5.14504 0.750481 0.375241 0.926927i \(-0.377560\pi\)
0.375241 + 0.926927i \(0.377560\pi\)
\(48\) 0 0
\(49\) 2.04081 0.291544
\(50\) 0 0
\(51\) 6.35205 0.889466
\(52\) 0 0
\(53\) −12.2204 −1.67860 −0.839301 0.543667i \(-0.817035\pi\)
−0.839301 + 0.543667i \(0.817035\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) −4.99686 −0.661851
\(58\) 0 0
\(59\) 3.30528 0.430311 0.215155 0.976580i \(-0.430974\pi\)
0.215155 + 0.976580i \(0.430974\pi\)
\(60\) 0 0
\(61\) 10.7990 1.38267 0.691334 0.722536i \(-0.257024\pi\)
0.691334 + 0.722536i \(0.257024\pi\)
\(62\) 0 0
\(63\) 3.53276 0.445086
\(64\) 0 0
\(65\) 5.69877 0.706845
\(66\) 0 0
\(67\) 11.5328 1.40895 0.704475 0.709729i \(-0.251183\pi\)
0.704475 + 0.709729i \(0.251183\pi\)
\(68\) 0 0
\(69\) −9.23871 −1.11221
\(70\) 0 0
\(71\) 2.05440 0.243812 0.121906 0.992542i \(-0.461099\pi\)
0.121906 + 0.992542i \(0.461099\pi\)
\(72\) 0 0
\(73\) 3.29589 0.385755 0.192877 0.981223i \(-0.438218\pi\)
0.192877 + 0.981223i \(0.438218\pi\)
\(74\) 0 0
\(75\) 1.35095 0.155995
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) −4.67984 −0.526523 −0.263261 0.964725i \(-0.584798\pi\)
−0.263261 + 0.964725i \(0.584798\pi\)
\(80\) 0 0
\(81\) −4.09477 −0.454975
\(82\) 0 0
\(83\) −12.5247 −1.37477 −0.687385 0.726293i \(-0.741241\pi\)
−0.687385 + 0.726293i \(0.741241\pi\)
\(84\) 0 0
\(85\) 4.70191 0.509993
\(86\) 0 0
\(87\) 14.4493 1.54913
\(88\) 0 0
\(89\) −4.32366 −0.458307 −0.229153 0.973390i \(-0.573596\pi\)
−0.229153 + 0.973390i \(0.573596\pi\)
\(90\) 0 0
\(91\) −17.1350 −1.79624
\(92\) 0 0
\(93\) −4.73497 −0.490994
\(94\) 0 0
\(95\) −3.69877 −0.379486
\(96\) 0 0
\(97\) 2.94235 0.298751 0.149375 0.988781i \(-0.452274\pi\)
0.149375 + 0.988781i \(0.452274\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 2.69796 0.268457 0.134228 0.990950i \(-0.457144\pi\)
0.134228 + 0.990950i \(0.457144\pi\)
\(102\) 0 0
\(103\) 16.1167 1.58803 0.794014 0.607900i \(-0.207988\pi\)
0.794014 + 0.607900i \(0.207988\pi\)
\(104\) 0 0
\(105\) −4.06204 −0.396414
\(106\) 0 0
\(107\) 18.7756 1.81511 0.907554 0.419935i \(-0.137947\pi\)
0.907554 + 0.419935i \(0.137947\pi\)
\(108\) 0 0
\(109\) −14.8927 −1.42646 −0.713232 0.700928i \(-0.752769\pi\)
−0.713232 + 0.700928i \(0.752769\pi\)
\(110\) 0 0
\(111\) 7.82754 0.742957
\(112\) 0 0
\(113\) 5.05396 0.475437 0.237718 0.971334i \(-0.423600\pi\)
0.237718 + 0.971334i \(0.423600\pi\)
\(114\) 0 0
\(115\) −6.83866 −0.637709
\(116\) 0 0
\(117\) −6.69563 −0.619011
\(118\) 0 0
\(119\) −14.1377 −1.29600
\(120\) 0 0
\(121\) 0 0
\(122\) 0 0
\(123\) 11.3466 1.02308
\(124\) 0 0
\(125\) 1.00000 0.0894427
\(126\) 0 0
\(127\) 18.2618 1.62047 0.810236 0.586104i \(-0.199339\pi\)
0.810236 + 0.586104i \(0.199339\pi\)
\(128\) 0 0
\(129\) 3.79234 0.333897
\(130\) 0 0
\(131\) 9.29322 0.811953 0.405976 0.913884i \(-0.366932\pi\)
0.405976 + 0.913884i \(0.366932\pi\)
\(132\) 0 0
\(133\) 11.1214 0.964351
\(134\) 0 0
\(135\) −5.64013 −0.485425
\(136\) 0 0
\(137\) 5.14826 0.439846 0.219923 0.975517i \(-0.429419\pi\)
0.219923 + 0.975517i \(0.429419\pi\)
\(138\) 0 0
\(139\) −6.73066 −0.570887 −0.285444 0.958396i \(-0.592141\pi\)
−0.285444 + 0.958396i \(0.592141\pi\)
\(140\) 0 0
\(141\) 6.95071 0.585355
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) 10.6956 0.888224
\(146\) 0 0
\(147\) 2.75704 0.227397
\(148\) 0 0
\(149\) −5.37053 −0.439971 −0.219985 0.975503i \(-0.570601\pi\)
−0.219985 + 0.975503i \(0.570601\pi\)
\(150\) 0 0
\(151\) −1.94713 −0.158455 −0.0792277 0.996857i \(-0.525245\pi\)
−0.0792277 + 0.996857i \(0.525245\pi\)
\(152\) 0 0
\(153\) −5.52439 −0.446621
\(154\) 0 0
\(155\) −3.50491 −0.281521
\(156\) 0 0
\(157\) 7.29589 0.582276 0.291138 0.956681i \(-0.405966\pi\)
0.291138 + 0.956681i \(0.405966\pi\)
\(158\) 0 0
\(159\) −16.5092 −1.30926
\(160\) 0 0
\(161\) 20.5625 1.62055
\(162\) 0 0
\(163\) 3.96332 0.310431 0.155216 0.987881i \(-0.450393\pi\)
0.155216 + 0.987881i \(0.450393\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −7.41532 −0.573815 −0.286907 0.957958i \(-0.592627\pi\)
−0.286907 + 0.957958i \(0.592627\pi\)
\(168\) 0 0
\(169\) 19.4760 1.49815
\(170\) 0 0
\(171\) 4.34578 0.332330
\(172\) 0 0
\(173\) 8.59979 0.653830 0.326915 0.945054i \(-0.393991\pi\)
0.326915 + 0.945054i \(0.393991\pi\)
\(174\) 0 0
\(175\) −3.00679 −0.227292
\(176\) 0 0
\(177\) 4.46528 0.335631
\(178\) 0 0
\(179\) 1.13004 0.0844632 0.0422316 0.999108i \(-0.486553\pi\)
0.0422316 + 0.999108i \(0.486553\pi\)
\(180\) 0 0
\(181\) 9.98170 0.741934 0.370967 0.928646i \(-0.379026\pi\)
0.370967 + 0.928646i \(0.379026\pi\)
\(182\) 0 0
\(183\) 14.5889 1.07844
\(184\) 0 0
\(185\) 5.79409 0.425990
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) 16.9587 1.23356
\(190\) 0 0
\(191\) 18.4991 1.33855 0.669274 0.743016i \(-0.266605\pi\)
0.669274 + 0.743016i \(0.266605\pi\)
\(192\) 0 0
\(193\) 9.38556 0.675587 0.337794 0.941220i \(-0.390319\pi\)
0.337794 + 0.941220i \(0.390319\pi\)
\(194\) 0 0
\(195\) 7.69877 0.551320
\(196\) 0 0
\(197\) −13.4556 −0.958673 −0.479337 0.877631i \(-0.659123\pi\)
−0.479337 + 0.877631i \(0.659123\pi\)
\(198\) 0 0
\(199\) 23.5214 1.66739 0.833695 0.552225i \(-0.186221\pi\)
0.833695 + 0.552225i \(0.186221\pi\)
\(200\) 0 0
\(201\) 15.5802 1.09894
\(202\) 0 0
\(203\) −32.1596 −2.25716
\(204\) 0 0
\(205\) 8.39893 0.586606
\(206\) 0 0
\(207\) 8.03492 0.558466
\(208\) 0 0
\(209\) 0 0
\(210\) 0 0
\(211\) −5.65886 −0.389572 −0.194786 0.980846i \(-0.562401\pi\)
−0.194786 + 0.980846i \(0.562401\pi\)
\(212\) 0 0
\(213\) 2.77540 0.190167
\(214\) 0 0
\(215\) 2.80716 0.191447
\(216\) 0 0
\(217\) 10.5386 0.715403
\(218\) 0 0
\(219\) 4.45259 0.300878
\(220\) 0 0
\(221\) 26.7951 1.80243
\(222\) 0 0
\(223\) −18.6856 −1.25128 −0.625641 0.780112i \(-0.715162\pi\)
−0.625641 + 0.780112i \(0.715162\pi\)
\(224\) 0 0
\(225\) −1.17493 −0.0783284
\(226\) 0 0
\(227\) 11.5721 0.768065 0.384033 0.923319i \(-0.374535\pi\)
0.384033 + 0.923319i \(0.374535\pi\)
\(228\) 0 0
\(229\) −21.2785 −1.40612 −0.703062 0.711128i \(-0.748184\pi\)
−0.703062 + 0.711128i \(0.748184\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −14.1659 −0.928039 −0.464019 0.885825i \(-0.653593\pi\)
−0.464019 + 0.885825i \(0.653593\pi\)
\(234\) 0 0
\(235\) 5.14504 0.335626
\(236\) 0 0
\(237\) −6.32224 −0.410674
\(238\) 0 0
\(239\) −7.56145 −0.489110 −0.244555 0.969635i \(-0.578642\pi\)
−0.244555 + 0.969635i \(0.578642\pi\)
\(240\) 0 0
\(241\) −4.70376 −0.302996 −0.151498 0.988458i \(-0.548410\pi\)
−0.151498 + 0.988458i \(0.548410\pi\)
\(242\) 0 0
\(243\) 11.3885 0.730575
\(244\) 0 0
\(245\) 2.04081 0.130383
\(246\) 0 0
\(247\) −21.0784 −1.34119
\(248\) 0 0
\(249\) −16.9203 −1.07228
\(250\) 0 0
\(251\) 29.2749 1.84781 0.923907 0.382616i \(-0.124977\pi\)
0.923907 + 0.382616i \(0.124977\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 0 0
\(255\) 6.35205 0.397781
\(256\) 0 0
\(257\) 11.5154 0.718312 0.359156 0.933278i \(-0.383065\pi\)
0.359156 + 0.933278i \(0.383065\pi\)
\(258\) 0 0
\(259\) −17.4216 −1.08253
\(260\) 0 0
\(261\) −12.5666 −0.777851
\(262\) 0 0
\(263\) 4.69882 0.289742 0.144871 0.989451i \(-0.453723\pi\)
0.144871 + 0.989451i \(0.453723\pi\)
\(264\) 0 0
\(265\) −12.2204 −0.750694
\(266\) 0 0
\(267\) −5.84106 −0.357467
\(268\) 0 0
\(269\) −12.6594 −0.771855 −0.385927 0.922529i \(-0.626118\pi\)
−0.385927 + 0.922529i \(0.626118\pi\)
\(270\) 0 0
\(271\) 19.9465 1.21166 0.605832 0.795593i \(-0.292840\pi\)
0.605832 + 0.795593i \(0.292840\pi\)
\(272\) 0 0
\(273\) −23.1486 −1.40102
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) −22.6688 −1.36204 −0.681018 0.732267i \(-0.738462\pi\)
−0.681018 + 0.732267i \(0.738462\pi\)
\(278\) 0 0
\(279\) 4.11801 0.246539
\(280\) 0 0
\(281\) −1.59445 −0.0951171 −0.0475586 0.998868i \(-0.515144\pi\)
−0.0475586 + 0.998868i \(0.515144\pi\)
\(282\) 0 0
\(283\) −6.09579 −0.362357 −0.181178 0.983450i \(-0.557991\pi\)
−0.181178 + 0.983450i \(0.557991\pi\)
\(284\) 0 0
\(285\) −4.99686 −0.295989
\(286\) 0 0
\(287\) −25.2538 −1.49069
\(288\) 0 0
\(289\) 5.10792 0.300466
\(290\) 0 0
\(291\) 3.97498 0.233017
\(292\) 0 0
\(293\) −23.4163 −1.36800 −0.683999 0.729483i \(-0.739761\pi\)
−0.683999 + 0.729483i \(0.739761\pi\)
\(294\) 0 0
\(295\) 3.30528 0.192441
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) −38.9720 −2.25381
\(300\) 0 0
\(301\) −8.44055 −0.486505
\(302\) 0 0
\(303\) 3.64481 0.209389
\(304\) 0 0
\(305\) 10.7990 0.618348
\(306\) 0 0
\(307\) 2.62127 0.149604 0.0748019 0.997198i \(-0.476168\pi\)
0.0748019 + 0.997198i \(0.476168\pi\)
\(308\) 0 0
\(309\) 21.7729 1.23862
\(310\) 0 0
\(311\) −30.3266 −1.71967 −0.859833 0.510575i \(-0.829433\pi\)
−0.859833 + 0.510575i \(0.829433\pi\)
\(312\) 0 0
\(313\) −12.3246 −0.696629 −0.348314 0.937378i \(-0.613246\pi\)
−0.348314 + 0.937378i \(0.613246\pi\)
\(314\) 0 0
\(315\) 3.53276 0.199048
\(316\) 0 0
\(317\) −22.3636 −1.25606 −0.628032 0.778187i \(-0.716139\pi\)
−0.628032 + 0.778187i \(0.716139\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0 0
\(321\) 25.3650 1.41574
\(322\) 0 0
\(323\) −17.3913 −0.967676
\(324\) 0 0
\(325\) 5.69877 0.316111
\(326\) 0 0
\(327\) −20.1194 −1.11260
\(328\) 0 0
\(329\) −15.4701 −0.852893
\(330\) 0 0
\(331\) −28.0559 −1.54209 −0.771045 0.636781i \(-0.780265\pi\)
−0.771045 + 0.636781i \(0.780265\pi\)
\(332\) 0 0
\(333\) −6.80762 −0.373056
\(334\) 0 0
\(335\) 11.5328 0.630102
\(336\) 0 0
\(337\) 30.7483 1.67496 0.837482 0.546464i \(-0.184027\pi\)
0.837482 + 0.546464i \(0.184027\pi\)
\(338\) 0 0
\(339\) 6.82766 0.370828
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) 14.9113 0.805132
\(344\) 0 0
\(345\) −9.23871 −0.497396
\(346\) 0 0
\(347\) 21.4212 1.14995 0.574974 0.818172i \(-0.305012\pi\)
0.574974 + 0.818172i \(0.305012\pi\)
\(348\) 0 0
\(349\) 22.9134 1.22653 0.613264 0.789878i \(-0.289856\pi\)
0.613264 + 0.789878i \(0.289856\pi\)
\(350\) 0 0
\(351\) −32.1418 −1.71560
\(352\) 0 0
\(353\) −1.25101 −0.0665843 −0.0332922 0.999446i \(-0.510599\pi\)
−0.0332922 + 0.999446i \(0.510599\pi\)
\(354\) 0 0
\(355\) 2.05440 0.109036
\(356\) 0 0
\(357\) −19.0993 −1.01084
\(358\) 0 0
\(359\) 36.1951 1.91030 0.955152 0.296115i \(-0.0956910\pi\)
0.955152 + 0.296115i \(0.0956910\pi\)
\(360\) 0 0
\(361\) −5.31912 −0.279953
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 3.29589 0.172515
\(366\) 0 0
\(367\) −38.2633 −1.99733 −0.998664 0.0516727i \(-0.983545\pi\)
−0.998664 + 0.0516727i \(0.983545\pi\)
\(368\) 0 0
\(369\) −9.86811 −0.513714
\(370\) 0 0
\(371\) 36.7442 1.90767
\(372\) 0 0
\(373\) −26.8965 −1.39265 −0.696324 0.717728i \(-0.745182\pi\)
−0.696324 + 0.717728i \(0.745182\pi\)
\(374\) 0 0
\(375\) 1.35095 0.0697629
\(376\) 0 0
\(377\) 60.9519 3.13918
\(378\) 0 0
\(379\) 12.2189 0.627644 0.313822 0.949482i \(-0.398390\pi\)
0.313822 + 0.949482i \(0.398390\pi\)
\(380\) 0 0
\(381\) 24.6708 1.26392
\(382\) 0 0
\(383\) −5.43228 −0.277577 −0.138788 0.990322i \(-0.544321\pi\)
−0.138788 + 0.990322i \(0.544321\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) −3.29820 −0.167657
\(388\) 0 0
\(389\) −6.49665 −0.329393 −0.164697 0.986344i \(-0.552664\pi\)
−0.164697 + 0.986344i \(0.552664\pi\)
\(390\) 0 0
\(391\) −32.1547 −1.62614
\(392\) 0 0
\(393\) 12.5547 0.633301
\(394\) 0 0
\(395\) −4.67984 −0.235468
\(396\) 0 0
\(397\) −19.1186 −0.959536 −0.479768 0.877395i \(-0.659279\pi\)
−0.479768 + 0.877395i \(0.659279\pi\)
\(398\) 0 0
\(399\) 15.0245 0.752168
\(400\) 0 0
\(401\) −12.9874 −0.648562 −0.324281 0.945961i \(-0.605122\pi\)
−0.324281 + 0.945961i \(0.605122\pi\)
\(402\) 0 0
\(403\) −19.9737 −0.994960
\(404\) 0 0
\(405\) −4.09477 −0.203471
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) −2.03820 −0.100783 −0.0503913 0.998730i \(-0.516047\pi\)
−0.0503913 + 0.998730i \(0.516047\pi\)
\(410\) 0 0
\(411\) 6.95506 0.343068
\(412\) 0 0
\(413\) −9.93829 −0.489031
\(414\) 0 0
\(415\) −12.5247 −0.614816
\(416\) 0 0
\(417\) −9.09281 −0.445277
\(418\) 0 0
\(419\) 21.8002 1.06501 0.532504 0.846427i \(-0.321251\pi\)
0.532504 + 0.846427i \(0.321251\pi\)
\(420\) 0 0
\(421\) −16.3277 −0.795763 −0.397882 0.917437i \(-0.630254\pi\)
−0.397882 + 0.917437i \(0.630254\pi\)
\(422\) 0 0
\(423\) −6.04504 −0.293920
\(424\) 0 0
\(425\) 4.70191 0.228076
\(426\) 0 0
\(427\) −32.4703 −1.57135
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 0.582599 0.0280628 0.0140314 0.999902i \(-0.495534\pi\)
0.0140314 + 0.999902i \(0.495534\pi\)
\(432\) 0 0
\(433\) 20.0423 0.963170 0.481585 0.876399i \(-0.340061\pi\)
0.481585 + 0.876399i \(0.340061\pi\)
\(434\) 0 0
\(435\) 14.4493 0.692791
\(436\) 0 0
\(437\) 25.2946 1.21001
\(438\) 0 0
\(439\) 15.9036 0.759038 0.379519 0.925184i \(-0.376089\pi\)
0.379519 + 0.925184i \(0.376089\pi\)
\(440\) 0 0
\(441\) −2.39780 −0.114181
\(442\) 0 0
\(443\) −12.1974 −0.579514 −0.289757 0.957100i \(-0.593575\pi\)
−0.289757 + 0.957100i \(0.593575\pi\)
\(444\) 0 0
\(445\) −4.32366 −0.204961
\(446\) 0 0
\(447\) −7.25533 −0.343165
\(448\) 0 0
\(449\) 15.5212 0.732489 0.366245 0.930519i \(-0.380643\pi\)
0.366245 + 0.930519i \(0.380643\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 0 0
\(453\) −2.63048 −0.123591
\(454\) 0 0
\(455\) −17.1350 −0.803302
\(456\) 0 0
\(457\) 20.1078 0.940604 0.470302 0.882506i \(-0.344145\pi\)
0.470302 + 0.882506i \(0.344145\pi\)
\(458\) 0 0
\(459\) −26.5194 −1.23782
\(460\) 0 0
\(461\) 13.2592 0.617541 0.308770 0.951137i \(-0.400083\pi\)
0.308770 + 0.951137i \(0.400083\pi\)
\(462\) 0 0
\(463\) −8.73477 −0.405939 −0.202970 0.979185i \(-0.565059\pi\)
−0.202970 + 0.979185i \(0.565059\pi\)
\(464\) 0 0
\(465\) −4.73497 −0.219579
\(466\) 0 0
\(467\) −36.0072 −1.66621 −0.833107 0.553111i \(-0.813440\pi\)
−0.833107 + 0.553111i \(0.813440\pi\)
\(468\) 0 0
\(469\) −34.6766 −1.60122
\(470\) 0 0
\(471\) 9.85641 0.454159
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) −3.69877 −0.169711
\(476\) 0 0
\(477\) 14.3581 0.657411
\(478\) 0 0
\(479\) 12.5656 0.574139 0.287069 0.957910i \(-0.407319\pi\)
0.287069 + 0.957910i \(0.407319\pi\)
\(480\) 0 0
\(481\) 33.0192 1.50554
\(482\) 0 0
\(483\) 27.7789 1.26398
\(484\) 0 0
\(485\) 2.94235 0.133605
\(486\) 0 0
\(487\) 29.8003 1.35038 0.675189 0.737644i \(-0.264062\pi\)
0.675189 + 0.737644i \(0.264062\pi\)
\(488\) 0 0
\(489\) 5.35426 0.242128
\(490\) 0 0
\(491\) 8.62233 0.389120 0.194560 0.980891i \(-0.437672\pi\)
0.194560 + 0.980891i \(0.437672\pi\)
\(492\) 0 0
\(493\) 50.2898 2.26494
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −6.17716 −0.277083
\(498\) 0 0
\(499\) 33.2734 1.48952 0.744761 0.667332i \(-0.232564\pi\)
0.744761 + 0.667332i \(0.232564\pi\)
\(500\) 0 0
\(501\) −10.0178 −0.447560
\(502\) 0 0
\(503\) −9.56522 −0.426492 −0.213246 0.976999i \(-0.568404\pi\)
−0.213246 + 0.976999i \(0.568404\pi\)
\(504\) 0 0
\(505\) 2.69796 0.120057
\(506\) 0 0
\(507\) 26.3111 1.16852
\(508\) 0 0
\(509\) −21.4598 −0.951187 −0.475593 0.879665i \(-0.657767\pi\)
−0.475593 + 0.879665i \(0.657767\pi\)
\(510\) 0 0
\(511\) −9.91007 −0.438396
\(512\) 0 0
\(513\) 20.8615 0.921059
\(514\) 0 0
\(515\) 16.1167 0.710187
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) 11.6179 0.509970
\(520\) 0 0
\(521\) 12.3687 0.541881 0.270941 0.962596i \(-0.412665\pi\)
0.270941 + 0.962596i \(0.412665\pi\)
\(522\) 0 0
\(523\) −13.9337 −0.609280 −0.304640 0.952468i \(-0.598536\pi\)
−0.304640 + 0.952468i \(0.598536\pi\)
\(524\) 0 0
\(525\) −4.06204 −0.177282
\(526\) 0 0
\(527\) −16.4798 −0.717870
\(528\) 0 0
\(529\) 23.7673 1.03336
\(530\) 0 0
\(531\) −3.88346 −0.168528
\(532\) 0 0
\(533\) 47.8635 2.07320
\(534\) 0 0
\(535\) 18.7756 0.811741
\(536\) 0 0
\(537\) 1.52663 0.0658790
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) 38.0571 1.63620 0.818100 0.575076i \(-0.195027\pi\)
0.818100 + 0.575076i \(0.195027\pi\)
\(542\) 0 0
\(543\) 13.4848 0.578688
\(544\) 0 0
\(545\) −14.8927 −0.637934
\(546\) 0 0
\(547\) −33.5537 −1.43465 −0.717325 0.696738i \(-0.754634\pi\)
−0.717325 + 0.696738i \(0.754634\pi\)
\(548\) 0 0
\(549\) −12.6880 −0.541510
\(550\) 0 0
\(551\) −39.5607 −1.68534
\(552\) 0 0
\(553\) 14.0713 0.598373
\(554\) 0 0
\(555\) 7.82754 0.332261
\(556\) 0 0
\(557\) −45.9285 −1.94605 −0.973025 0.230698i \(-0.925899\pi\)
−0.973025 + 0.230698i \(0.925899\pi\)
\(558\) 0 0
\(559\) 15.9973 0.676616
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) −19.1695 −0.807898 −0.403949 0.914781i \(-0.632363\pi\)
−0.403949 + 0.914781i \(0.632363\pi\)
\(564\) 0 0
\(565\) 5.05396 0.212622
\(566\) 0 0
\(567\) 12.3121 0.517061
\(568\) 0 0
\(569\) −16.7995 −0.704271 −0.352136 0.935949i \(-0.614544\pi\)
−0.352136 + 0.935949i \(0.614544\pi\)
\(570\) 0 0
\(571\) 33.5389 1.40356 0.701780 0.712394i \(-0.252389\pi\)
0.701780 + 0.712394i \(0.252389\pi\)
\(572\) 0 0
\(573\) 24.9914 1.04403
\(574\) 0 0
\(575\) −6.83866 −0.285192
\(576\) 0 0
\(577\) 18.7364 0.780008 0.390004 0.920813i \(-0.372474\pi\)
0.390004 + 0.920813i \(0.372474\pi\)
\(578\) 0 0
\(579\) 12.6795 0.526940
\(580\) 0 0
\(581\) 37.6593 1.56237
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) −6.69563 −0.276830
\(586\) 0 0
\(587\) 6.76096 0.279055 0.139527 0.990218i \(-0.455442\pi\)
0.139527 + 0.990218i \(0.455442\pi\)
\(588\) 0 0
\(589\) 12.9639 0.534167
\(590\) 0 0
\(591\) −18.1779 −0.747739
\(592\) 0 0
\(593\) −15.3645 −0.630946 −0.315473 0.948935i \(-0.602163\pi\)
−0.315473 + 0.948935i \(0.602163\pi\)
\(594\) 0 0
\(595\) −14.1377 −0.579588
\(596\) 0 0
\(597\) 31.7764 1.30052
\(598\) 0 0
\(599\) −4.76071 −0.194517 −0.0972587 0.995259i \(-0.531007\pi\)
−0.0972587 + 0.995259i \(0.531007\pi\)
\(600\) 0 0
\(601\) −2.84016 −0.115853 −0.0579263 0.998321i \(-0.518449\pi\)
−0.0579263 + 0.998321i \(0.518449\pi\)
\(602\) 0 0
\(603\) −13.5501 −0.551804
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) −22.3497 −0.907148 −0.453574 0.891219i \(-0.649851\pi\)
−0.453574 + 0.891219i \(0.649851\pi\)
\(608\) 0 0
\(609\) −43.4461 −1.76052
\(610\) 0 0
\(611\) 29.3204 1.18618
\(612\) 0 0
\(613\) −19.2315 −0.776754 −0.388377 0.921501i \(-0.626964\pi\)
−0.388377 + 0.921501i \(0.626964\pi\)
\(614\) 0 0
\(615\) 11.3466 0.457537
\(616\) 0 0
\(617\) −14.8714 −0.598700 −0.299350 0.954143i \(-0.596770\pi\)
−0.299350 + 0.954143i \(0.596770\pi\)
\(618\) 0 0
\(619\) −34.7781 −1.39785 −0.698925 0.715195i \(-0.746338\pi\)
−0.698925 + 0.715195i \(0.746338\pi\)
\(620\) 0 0
\(621\) 38.5709 1.54780
\(622\) 0 0
\(623\) 13.0003 0.520848
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 27.2433 1.08626
\(630\) 0 0
\(631\) −19.2646 −0.766913 −0.383456 0.923559i \(-0.625266\pi\)
−0.383456 + 0.923559i \(0.625266\pi\)
\(632\) 0 0
\(633\) −7.64486 −0.303856
\(634\) 0 0
\(635\) 18.2618 0.724697
\(636\) 0 0
\(637\) 11.6301 0.460802
\(638\) 0 0
\(639\) −2.41377 −0.0954872
\(640\) 0 0
\(641\) −41.6297 −1.64428 −0.822138 0.569289i \(-0.807219\pi\)
−0.822138 + 0.569289i \(0.807219\pi\)
\(642\) 0 0
\(643\) 12.9034 0.508862 0.254431 0.967091i \(-0.418112\pi\)
0.254431 + 0.967091i \(0.418112\pi\)
\(644\) 0 0
\(645\) 3.79234 0.149323
\(646\) 0 0
\(647\) −8.73477 −0.343399 −0.171700 0.985149i \(-0.554926\pi\)
−0.171700 + 0.985149i \(0.554926\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) 14.2371 0.557995
\(652\) 0 0
\(653\) −41.5489 −1.62594 −0.812968 0.582308i \(-0.802150\pi\)
−0.812968 + 0.582308i \(0.802150\pi\)
\(654\) 0 0
\(655\) 9.29322 0.363116
\(656\) 0 0
\(657\) −3.87243 −0.151078
\(658\) 0 0
\(659\) 34.8683 1.35827 0.679137 0.734011i \(-0.262354\pi\)
0.679137 + 0.734011i \(0.262354\pi\)
\(660\) 0 0
\(661\) −3.05492 −0.118823 −0.0594113 0.998234i \(-0.518922\pi\)
−0.0594113 + 0.998234i \(0.518922\pi\)
\(662\) 0 0
\(663\) 36.1989 1.40585
\(664\) 0 0
\(665\) 11.1214 0.431271
\(666\) 0 0
\(667\) −73.1438 −2.83214
\(668\) 0 0
\(669\) −25.2434 −0.975966
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) −6.56543 −0.253079 −0.126539 0.991962i \(-0.540387\pi\)
−0.126539 + 0.991962i \(0.540387\pi\)
\(674\) 0 0
\(675\) −5.64013 −0.217089
\(676\) 0 0
\(677\) 8.92088 0.342857 0.171429 0.985197i \(-0.445162\pi\)
0.171429 + 0.985197i \(0.445162\pi\)
\(678\) 0 0
\(679\) −8.84705 −0.339519
\(680\) 0 0
\(681\) 15.6333 0.599070
\(682\) 0 0
\(683\) −9.77717 −0.374113 −0.187057 0.982349i \(-0.559895\pi\)
−0.187057 + 0.982349i \(0.559895\pi\)
\(684\) 0 0
\(685\) 5.14826 0.196705
\(686\) 0 0
\(687\) −28.7463 −1.09674
\(688\) 0 0
\(689\) −69.6413 −2.65312
\(690\) 0 0
\(691\) −6.19944 −0.235838 −0.117919 0.993023i \(-0.537622\pi\)
−0.117919 + 0.993023i \(0.537622\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −6.73066 −0.255309
\(696\) 0 0
\(697\) 39.4910 1.49583
\(698\) 0 0
\(699\) −19.1375 −0.723845
\(700\) 0 0
\(701\) 18.7398 0.707791 0.353896 0.935285i \(-0.384857\pi\)
0.353896 + 0.935285i \(0.384857\pi\)
\(702\) 0 0
\(703\) −21.4310 −0.808285
\(704\) 0 0
\(705\) 6.95071 0.261779
\(706\) 0 0
\(707\) −8.11220 −0.305091
\(708\) 0 0
\(709\) −48.8505 −1.83462 −0.917309 0.398176i \(-0.869643\pi\)
−0.917309 + 0.398176i \(0.869643\pi\)
\(710\) 0 0
\(711\) 5.49846 0.206208
\(712\) 0 0
\(713\) 23.9689 0.897643
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) −10.2152 −0.381492
\(718\) 0 0
\(719\) −10.4572 −0.389988 −0.194994 0.980804i \(-0.562469\pi\)
−0.194994 + 0.980804i \(0.562469\pi\)
\(720\) 0 0
\(721\) −48.4597 −1.80473
\(722\) 0 0
\(723\) −6.35456 −0.236329
\(724\) 0 0
\(725\) 10.6956 0.397226
\(726\) 0 0
\(727\) −32.6965 −1.21265 −0.606323 0.795219i \(-0.707356\pi\)
−0.606323 + 0.795219i \(0.707356\pi\)
\(728\) 0 0
\(729\) 27.6697 1.02480
\(730\) 0 0
\(731\) 13.1990 0.488183
\(732\) 0 0
\(733\) 4.32167 0.159624 0.0798122 0.996810i \(-0.474568\pi\)
0.0798122 + 0.996810i \(0.474568\pi\)
\(734\) 0 0
\(735\) 2.75704 0.101695
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) −8.71832 −0.320709 −0.160354 0.987060i \(-0.551264\pi\)
−0.160354 + 0.987060i \(0.551264\pi\)
\(740\) 0 0
\(741\) −28.4760 −1.04609
\(742\) 0 0
\(743\) 32.8875 1.20653 0.603263 0.797542i \(-0.293867\pi\)
0.603263 + 0.797542i \(0.293867\pi\)
\(744\) 0 0
\(745\) −5.37053 −0.196761
\(746\) 0 0
\(747\) 14.7157 0.538418
\(748\) 0 0
\(749\) −56.4544 −2.06280
\(750\) 0 0
\(751\) −43.9740 −1.60464 −0.802318 0.596897i \(-0.796400\pi\)
−0.802318 + 0.596897i \(0.796400\pi\)
\(752\) 0 0
\(753\) 39.5490 1.44125
\(754\) 0 0
\(755\) −1.94713 −0.0708634
\(756\) 0 0
\(757\) −21.9285 −0.797004 −0.398502 0.917168i \(-0.630470\pi\)
−0.398502 + 0.917168i \(0.630470\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 24.3364 0.882195 0.441097 0.897459i \(-0.354589\pi\)
0.441097 + 0.897459i \(0.354589\pi\)
\(762\) 0 0
\(763\) 44.7793 1.62112
\(764\) 0 0
\(765\) −5.52439 −0.199735
\(766\) 0 0
\(767\) 18.8360 0.680129
\(768\) 0 0
\(769\) 28.4041 1.02428 0.512138 0.858903i \(-0.328854\pi\)
0.512138 + 0.858903i \(0.328854\pi\)
\(770\) 0 0
\(771\) 15.5568 0.560264
\(772\) 0 0
\(773\) −6.15809 −0.221491 −0.110746 0.993849i \(-0.535324\pi\)
−0.110746 + 0.993849i \(0.535324\pi\)
\(774\) 0 0
\(775\) −3.50491 −0.125900
\(776\) 0 0
\(777\) −23.5358 −0.844342
\(778\) 0 0
\(779\) −31.0657 −1.11304
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) −60.3247 −2.15583
\(784\) 0 0
\(785\) 7.29589 0.260402
\(786\) 0 0
\(787\) −14.4250 −0.514195 −0.257098 0.966385i \(-0.582766\pi\)
−0.257098 + 0.966385i \(0.582766\pi\)
\(788\) 0 0
\(789\) 6.34789 0.225991
\(790\) 0 0
\(791\) −15.1962 −0.540315
\(792\) 0 0
\(793\) 61.5409 2.18538
\(794\) 0 0
\(795\) −16.5092 −0.585521
\(796\) 0 0
\(797\) 13.7588 0.487362 0.243681 0.969855i \(-0.421645\pi\)
0.243681 + 0.969855i \(0.421645\pi\)
\(798\) 0 0
\(799\) 24.1915 0.855834
\(800\) 0 0
\(801\) 5.07998 0.179492
\(802\) 0 0
\(803\) 0 0
\(804\) 0 0
\(805\) 20.5625 0.724731
\(806\) 0 0
\(807\) −17.1022 −0.602026
\(808\) 0 0
\(809\) −9.68668 −0.340566 −0.170283 0.985395i \(-0.554468\pi\)
−0.170283 + 0.985395i \(0.554468\pi\)
\(810\) 0 0
\(811\) 6.12820 0.215190 0.107595 0.994195i \(-0.465685\pi\)
0.107595 + 0.994195i \(0.465685\pi\)
\(812\) 0 0
\(813\) 26.9468 0.945065
\(814\) 0 0
\(815\) 3.96332 0.138829
\(816\) 0 0
\(817\) −10.3830 −0.363256
\(818\) 0 0
\(819\) 20.1324 0.703482
\(820\) 0 0
\(821\) −23.8825 −0.833505 −0.416753 0.909020i \(-0.636832\pi\)
−0.416753 + 0.909020i \(0.636832\pi\)
\(822\) 0 0
\(823\) −34.2553 −1.19406 −0.597032 0.802217i \(-0.703654\pi\)
−0.597032 + 0.802217i \(0.703654\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −6.94363 −0.241454 −0.120727 0.992686i \(-0.538523\pi\)
−0.120727 + 0.992686i \(0.538523\pi\)
\(828\) 0 0
\(829\) 30.1998 1.04888 0.524442 0.851446i \(-0.324274\pi\)
0.524442 + 0.851446i \(0.324274\pi\)
\(830\) 0 0
\(831\) −30.6245 −1.06235
\(832\) 0 0
\(833\) 9.59570 0.332471
\(834\) 0 0
\(835\) −7.41532 −0.256618
\(836\) 0 0
\(837\) 19.7682 0.683288
\(838\) 0 0
\(839\) 50.0106 1.72656 0.863280 0.504726i \(-0.168407\pi\)
0.863280 + 0.504726i \(0.168407\pi\)
\(840\) 0 0
\(841\) 85.3965 2.94471
\(842\) 0 0
\(843\) −2.15403 −0.0741888
\(844\) 0 0
\(845\) 19.4760 0.669993
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) −8.23512 −0.282629
\(850\) 0 0
\(851\) −39.6238 −1.35829
\(852\) 0 0
\(853\) 40.9697 1.40278 0.701388 0.712780i \(-0.252564\pi\)
0.701388 + 0.712780i \(0.252564\pi\)
\(854\) 0 0
\(855\) 4.34578 0.148622
\(856\) 0 0
\(857\) 45.8893 1.56755 0.783774 0.621046i \(-0.213292\pi\)
0.783774 + 0.621046i \(0.213292\pi\)
\(858\) 0 0
\(859\) −5.42402 −0.185065 −0.0925327 0.995710i \(-0.529496\pi\)
−0.0925327 + 0.995710i \(0.529496\pi\)
\(860\) 0 0
\(861\) −34.1168 −1.16270
\(862\) 0 0
\(863\) 24.7119 0.841202 0.420601 0.907246i \(-0.361819\pi\)
0.420601 + 0.907246i \(0.361819\pi\)
\(864\) 0 0
\(865\) 8.59979 0.292402
\(866\) 0 0
\(867\) 6.90056 0.234355
\(868\) 0 0
\(869\) 0 0
\(870\) 0 0
\(871\) 65.7225 2.22692
\(872\) 0 0
\(873\) −3.45705 −0.117003
\(874\) 0 0
\(875\) −3.00679 −0.101648
\(876\) 0 0
\(877\) −33.0006 −1.11435 −0.557175 0.830395i \(-0.688115\pi\)
−0.557175 + 0.830395i \(0.688115\pi\)
\(878\) 0 0
\(879\) −31.6344 −1.06700
\(880\) 0 0
\(881\) −24.5289 −0.826401 −0.413200 0.910640i \(-0.635589\pi\)
−0.413200 + 0.910640i \(0.635589\pi\)
\(882\) 0 0
\(883\) 24.7421 0.832639 0.416320 0.909218i \(-0.363320\pi\)
0.416320 + 0.909218i \(0.363320\pi\)
\(884\) 0 0
\(885\) 4.46528 0.150099
\(886\) 0 0
\(887\) −41.1605 −1.38203 −0.691017 0.722838i \(-0.742837\pi\)
−0.691017 + 0.722838i \(0.742837\pi\)
\(888\) 0 0
\(889\) −54.9095 −1.84160
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) −19.0303 −0.636825
\(894\) 0 0
\(895\) 1.13004 0.0377731
\(896\) 0 0
\(897\) −52.6493 −1.75791
\(898\) 0 0
\(899\) −37.4872 −1.25027
\(900\) 0 0
\(901\) −57.4592 −1.91424
\(902\) 0 0
\(903\) −11.4028 −0.379461
\(904\) 0 0
\(905\) 9.98170 0.331803
\(906\) 0 0
\(907\) 27.4276 0.910717 0.455358 0.890308i \(-0.349511\pi\)
0.455358 + 0.890308i \(0.349511\pi\)
\(908\) 0 0
\(909\) −3.16990 −0.105139
\(910\) 0 0
\(911\) −3.80200 −0.125966 −0.0629830 0.998015i \(-0.520061\pi\)
−0.0629830 + 0.998015i \(0.520061\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) 0 0
\(915\) 14.5889 0.482294
\(916\) 0 0
\(917\) −27.9428 −0.922753
\(918\) 0 0
\(919\) −27.4379 −0.905093 −0.452546 0.891741i \(-0.649484\pi\)
−0.452546 + 0.891741i \(0.649484\pi\)
\(920\) 0 0
\(921\) 3.54121 0.116687
\(922\) 0 0
\(923\) 11.7075 0.385359
\(924\) 0 0
\(925\) 5.79409 0.190508
\(926\) 0 0
\(927\) −18.9359 −0.621938
\(928\) 0 0
\(929\) 5.96209 0.195610 0.0978048 0.995206i \(-0.468818\pi\)
0.0978048 + 0.995206i \(0.468818\pi\)
\(930\) 0 0
\(931\) −7.54849 −0.247392
\(932\) 0 0
\(933\) −40.9699 −1.34129
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) −23.5196 −0.768353 −0.384176 0.923260i \(-0.625515\pi\)
−0.384176 + 0.923260i \(0.625515\pi\)
\(938\) 0 0
\(939\) −16.6500 −0.543352
\(940\) 0 0
\(941\) 45.3577 1.47862 0.739309 0.673366i \(-0.235152\pi\)
0.739309 + 0.673366i \(0.235152\pi\)
\(942\) 0 0
\(943\) −57.4374 −1.87042
\(944\) 0 0
\(945\) 16.9587 0.551667
\(946\) 0 0
\(947\) 52.0311 1.69078 0.845391 0.534148i \(-0.179367\pi\)
0.845391 + 0.534148i \(0.179367\pi\)
\(948\) 0 0
\(949\) 18.7825 0.609706
\(950\) 0 0
\(951\) −30.2122 −0.979697
\(952\) 0 0
\(953\) 26.5884 0.861284 0.430642 0.902523i \(-0.358287\pi\)
0.430642 + 0.902523i \(0.358287\pi\)
\(954\) 0 0
\(955\) 18.4991 0.598617
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) −15.4798 −0.499868
\(960\) 0 0
\(961\) −18.7156 −0.603729
\(962\) 0 0
\(963\) −22.0600 −0.710872
\(964\) 0 0
\(965\) 9.38556 0.302132
\(966\) 0 0
\(967\) −14.5187 −0.466890 −0.233445 0.972370i \(-0.575000\pi\)
−0.233445 + 0.972370i \(0.575000\pi\)
\(968\) 0 0
\(969\) −23.4948 −0.754761
\(970\) 0 0
\(971\) 55.0093 1.76533 0.882667 0.469999i \(-0.155746\pi\)
0.882667 + 0.469999i \(0.155746\pi\)
\(972\) 0 0
\(973\) 20.2377 0.648791
\(974\) 0 0
\(975\) 7.69877 0.246558
\(976\) 0 0
\(977\) 37.7764 1.20857 0.604287 0.796767i \(-0.293458\pi\)
0.604287 + 0.796767i \(0.293458\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 0 0
\(981\) 17.4978 0.558663
\(982\) 0 0
\(983\) −20.0610 −0.639845 −0.319922 0.947444i \(-0.603657\pi\)
−0.319922 + 0.947444i \(0.603657\pi\)
\(984\) 0 0
\(985\) −13.4556 −0.428732
\(986\) 0 0
\(987\) −20.8994 −0.665234
\(988\) 0 0
\(989\) −19.1972 −0.610436
\(990\) 0 0
\(991\) −56.3026 −1.78851 −0.894255 0.447557i \(-0.852294\pi\)
−0.894255 + 0.447557i \(0.852294\pi\)
\(992\) 0 0
\(993\) −37.9021 −1.20279
\(994\) 0 0
\(995\) 23.5214 0.745680
\(996\) 0 0
\(997\) 27.9469 0.885087 0.442544 0.896747i \(-0.354076\pi\)
0.442544 + 0.896747i \(0.354076\pi\)
\(998\) 0 0
\(999\) −32.6794 −1.03393
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 4840.2.a.bd.1.5 yes 6
4.3 odd 2 9680.2.a.da.1.2 6
11.10 odd 2 4840.2.a.bc.1.5 6
44.43 even 2 9680.2.a.db.1.2 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
4840.2.a.bc.1.5 6 11.10 odd 2
4840.2.a.bd.1.5 yes 6 1.1 even 1 trivial
9680.2.a.da.1.2 6 4.3 odd 2
9680.2.a.db.1.2 6 44.43 even 2