Properties

Label 3344.2.a.u.1.5
Level $3344$
Weight $2$
Character 3344.1
Self dual yes
Analytic conductor $26.702$
Analytic rank $0$
Dimension $5$
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] = [3344,2,Mod(1,3344)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3344, 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("3344.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3344 = 2^{4} \cdot 11 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3344.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: \(26.7019744359\)
Analytic rank: \(0\)
Dimension: \(5\)
Coefficient field: 5.5.3979184.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{5} - 2x^{4} - 9x^{3} + 18x^{2} + 5x - 12 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 1672)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.5
Root \(0.915789\) of defining polynomial
Character \(\chi\) \(=\) 3344.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+2.85680 q^{3} +1.91579 q^{5} -1.08421 q^{7} +5.16133 q^{9} +O(q^{10})\) \(q+2.85680 q^{3} +1.91579 q^{5} -1.08421 q^{7} +5.16133 q^{9} +1.00000 q^{11} +5.31769 q^{13} +5.47303 q^{15} +2.00000 q^{17} -1.00000 q^{19} -3.09738 q^{21} +3.07712 q^{23} -1.32975 q^{25} +6.17450 q^{27} +1.29854 q^{29} -4.99189 q^{31} +2.85680 q^{33} -2.07712 q^{35} -3.24554 q^{37} +15.1916 q^{39} +2.99503 q^{41} +8.87494 q^{43} +9.88802 q^{45} -6.32266 q^{47} -5.82449 q^{49} +5.71361 q^{51} +2.00000 q^{53} +1.91579 q^{55} -2.85680 q^{57} -3.72669 q^{59} -14.3731 q^{61} -5.59597 q^{63} +10.1876 q^{65} -5.78576 q^{67} +8.79073 q^{69} -12.3102 q^{71} +12.9461 q^{73} -3.79884 q^{75} -1.08421 q^{77} -0.168422 q^{79} +2.15534 q^{81} -1.58998 q^{83} +3.83158 q^{85} +3.70966 q^{87} +6.32376 q^{89} -5.76550 q^{91} -14.2608 q^{93} -1.91579 q^{95} +14.5305 q^{97} +5.16133 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5 q - q^{3} + 7 q^{5} - 8 q^{7} + 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 5 q - q^{3} + 7 q^{5} - 8 q^{7} + 8 q^{9} + 5 q^{11} + 4 q^{13} - q^{15} + 10 q^{17} - 5 q^{19} + 2 q^{21} - 5 q^{23} + 6 q^{25} - 7 q^{27} + 16 q^{29} - q^{31} - q^{33} + 10 q^{35} - q^{37} + 4 q^{39} + 28 q^{41} - 4 q^{43} + 12 q^{45} + 4 q^{47} - q^{49} - 2 q^{51} + 10 q^{53} + 7 q^{55} + q^{57} + q^{59} - 16 q^{61} - 12 q^{63} + 24 q^{65} + 9 q^{67} - 7 q^{69} - 7 q^{71} + 8 q^{73} + 8 q^{75} - 8 q^{77} - 6 q^{79} + 5 q^{81} - 4 q^{83} + 14 q^{85} + 2 q^{87} + 31 q^{89} + 12 q^{91} - 7 q^{93} - 7 q^{95} + 13 q^{97} + 8 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Coefficient data

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



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 2.85680 1.64938 0.824688 0.565587i \(-0.191351\pi\)
0.824688 + 0.565587i \(0.191351\pi\)
\(4\) 0 0
\(5\) 1.91579 0.856767 0.428383 0.903597i \(-0.359083\pi\)
0.428383 + 0.903597i \(0.359083\pi\)
\(6\) 0 0
\(7\) −1.08421 −0.409793 −0.204897 0.978784i \(-0.565686\pi\)
−0.204897 + 0.978784i \(0.565686\pi\)
\(8\) 0 0
\(9\) 5.16133 1.72044
\(10\) 0 0
\(11\) 1.00000 0.301511
\(12\) 0 0
\(13\) 5.31769 1.47486 0.737431 0.675422i \(-0.236038\pi\)
0.737431 + 0.675422i \(0.236038\pi\)
\(14\) 0 0
\(15\) 5.47303 1.41313
\(16\) 0 0
\(17\) 2.00000 0.485071 0.242536 0.970143i \(-0.422021\pi\)
0.242536 + 0.970143i \(0.422021\pi\)
\(18\) 0 0
\(19\) −1.00000 −0.229416
\(20\) 0 0
\(21\) −3.09738 −0.675903
\(22\) 0 0
\(23\) 3.07712 0.641624 0.320812 0.947143i \(-0.396044\pi\)
0.320812 + 0.947143i \(0.396044\pi\)
\(24\) 0 0
\(25\) −1.32975 −0.265950
\(26\) 0 0
\(27\) 6.17450 1.18828
\(28\) 0 0
\(29\) 1.29854 0.241132 0.120566 0.992705i \(-0.461529\pi\)
0.120566 + 0.992705i \(0.461529\pi\)
\(30\) 0 0
\(31\) −4.99189 −0.896569 −0.448285 0.893891i \(-0.647965\pi\)
−0.448285 + 0.893891i \(0.647965\pi\)
\(32\) 0 0
\(33\) 2.85680 0.497306
\(34\) 0 0
\(35\) −2.07712 −0.351097
\(36\) 0 0
\(37\) −3.24554 −0.533564 −0.266782 0.963757i \(-0.585960\pi\)
−0.266782 + 0.963757i \(0.585960\pi\)
\(38\) 0 0
\(39\) 15.1916 2.43260
\(40\) 0 0
\(41\) 2.99503 0.467746 0.233873 0.972267i \(-0.424860\pi\)
0.233873 + 0.972267i \(0.424860\pi\)
\(42\) 0 0
\(43\) 8.87494 1.35342 0.676708 0.736252i \(-0.263406\pi\)
0.676708 + 0.736252i \(0.263406\pi\)
\(44\) 0 0
\(45\) 9.88802 1.47402
\(46\) 0 0
\(47\) −6.32266 −0.922255 −0.461127 0.887334i \(-0.652555\pi\)
−0.461127 + 0.887334i \(0.652555\pi\)
\(48\) 0 0
\(49\) −5.82449 −0.832070
\(50\) 0 0
\(51\) 5.71361 0.800065
\(52\) 0 0
\(53\) 2.00000 0.274721 0.137361 0.990521i \(-0.456138\pi\)
0.137361 + 0.990521i \(0.456138\pi\)
\(54\) 0 0
\(55\) 1.91579 0.258325
\(56\) 0 0
\(57\) −2.85680 −0.378393
\(58\) 0 0
\(59\) −3.72669 −0.485174 −0.242587 0.970130i \(-0.577996\pi\)
−0.242587 + 0.970130i \(0.577996\pi\)
\(60\) 0 0
\(61\) −14.3731 −1.84029 −0.920144 0.391579i \(-0.871929\pi\)
−0.920144 + 0.391579i \(0.871929\pi\)
\(62\) 0 0
\(63\) −5.59597 −0.705026
\(64\) 0 0
\(65\) 10.1876 1.26361
\(66\) 0 0
\(67\) −5.78576 −0.706843 −0.353422 0.935464i \(-0.614982\pi\)
−0.353422 + 0.935464i \(0.614982\pi\)
\(68\) 0 0
\(69\) 8.79073 1.05828
\(70\) 0 0
\(71\) −12.3102 −1.46095 −0.730475 0.682939i \(-0.760701\pi\)
−0.730475 + 0.682939i \(0.760701\pi\)
\(72\) 0 0
\(73\) 12.9461 1.51522 0.757611 0.652706i \(-0.226366\pi\)
0.757611 + 0.652706i \(0.226366\pi\)
\(74\) 0 0
\(75\) −3.79884 −0.438653
\(76\) 0 0
\(77\) −1.08421 −0.123557
\(78\) 0 0
\(79\) −0.168422 −0.0189489 −0.00947447 0.999955i \(-0.503016\pi\)
−0.00947447 + 0.999955i \(0.503016\pi\)
\(80\) 0 0
\(81\) 2.15534 0.239482
\(82\) 0 0
\(83\) −1.58998 −0.174523 −0.0872616 0.996185i \(-0.527812\pi\)
−0.0872616 + 0.996185i \(0.527812\pi\)
\(84\) 0 0
\(85\) 3.83158 0.415593
\(86\) 0 0
\(87\) 3.70966 0.397718
\(88\) 0 0
\(89\) 6.32376 0.670317 0.335159 0.942162i \(-0.391210\pi\)
0.335159 + 0.942162i \(0.391210\pi\)
\(90\) 0 0
\(91\) −5.76550 −0.604389
\(92\) 0 0
\(93\) −14.2608 −1.47878
\(94\) 0 0
\(95\) −1.91579 −0.196556
\(96\) 0 0
\(97\) 14.5305 1.47535 0.737674 0.675157i \(-0.235924\pi\)
0.737674 + 0.675157i \(0.235924\pi\)
\(98\) 0 0
\(99\) 5.16133 0.518733
\(100\) 0 0
\(101\) 11.1407 1.10854 0.554268 0.832338i \(-0.312998\pi\)
0.554268 + 0.832338i \(0.312998\pi\)
\(102\) 0 0
\(103\) −7.18057 −0.707522 −0.353761 0.935336i \(-0.615097\pi\)
−0.353761 + 0.935336i \(0.615097\pi\)
\(104\) 0 0
\(105\) −5.93392 −0.579092
\(106\) 0 0
\(107\) 2.49108 0.240822 0.120411 0.992724i \(-0.461579\pi\)
0.120411 + 0.992724i \(0.461579\pi\)
\(108\) 0 0
\(109\) 4.48115 0.429216 0.214608 0.976700i \(-0.431153\pi\)
0.214608 + 0.976700i \(0.431153\pi\)
\(110\) 0 0
\(111\) −9.27188 −0.880048
\(112\) 0 0
\(113\) 15.7510 1.48173 0.740864 0.671655i \(-0.234416\pi\)
0.740864 + 0.671655i \(0.234416\pi\)
\(114\) 0 0
\(115\) 5.89511 0.549722
\(116\) 0 0
\(117\) 27.4464 2.53742
\(118\) 0 0
\(119\) −2.16842 −0.198779
\(120\) 0 0
\(121\) 1.00000 0.0909091
\(122\) 0 0
\(123\) 8.55622 0.771489
\(124\) 0 0
\(125\) −12.1265 −1.08462
\(126\) 0 0
\(127\) 2.62341 0.232790 0.116395 0.993203i \(-0.462866\pi\)
0.116395 + 0.993203i \(0.462866\pi\)
\(128\) 0 0
\(129\) 25.3540 2.23229
\(130\) 0 0
\(131\) 11.4987 1.00464 0.502322 0.864681i \(-0.332479\pi\)
0.502322 + 0.864681i \(0.332479\pi\)
\(132\) 0 0
\(133\) 1.08421 0.0940130
\(134\) 0 0
\(135\) 11.8290 1.01808
\(136\) 0 0
\(137\) 0.161330 0.0137834 0.00689169 0.999976i \(-0.497806\pi\)
0.00689169 + 0.999976i \(0.497806\pi\)
\(138\) 0 0
\(139\) −14.3697 −1.21882 −0.609410 0.792855i \(-0.708594\pi\)
−0.609410 + 0.792855i \(0.708594\pi\)
\(140\) 0 0
\(141\) −18.0626 −1.52115
\(142\) 0 0
\(143\) 5.31769 0.444688
\(144\) 0 0
\(145\) 2.48772 0.206594
\(146\) 0 0
\(147\) −16.6394 −1.37240
\(148\) 0 0
\(149\) 3.09020 0.253159 0.126580 0.991956i \(-0.459600\pi\)
0.126580 + 0.991956i \(0.459600\pi\)
\(150\) 0 0
\(151\) 3.09020 0.251477 0.125738 0.992063i \(-0.459870\pi\)
0.125738 + 0.992063i \(0.459870\pi\)
\(152\) 0 0
\(153\) 10.3227 0.834538
\(154\) 0 0
\(155\) −9.56340 −0.768151
\(156\) 0 0
\(157\) 7.03232 0.561241 0.280620 0.959819i \(-0.409460\pi\)
0.280620 + 0.959819i \(0.409460\pi\)
\(158\) 0 0
\(159\) 5.71361 0.453119
\(160\) 0 0
\(161\) −3.33625 −0.262933
\(162\) 0 0
\(163\) −4.36318 −0.341751 −0.170875 0.985293i \(-0.554660\pi\)
−0.170875 + 0.985293i \(0.554660\pi\)
\(164\) 0 0
\(165\) 5.47303 0.426075
\(166\) 0 0
\(167\) −2.75131 −0.212903 −0.106451 0.994318i \(-0.533949\pi\)
−0.106451 + 0.994318i \(0.533949\pi\)
\(168\) 0 0
\(169\) 15.2779 1.17522
\(170\) 0 0
\(171\) −5.16133 −0.394697
\(172\) 0 0
\(173\) −9.20690 −0.699988 −0.349994 0.936752i \(-0.613816\pi\)
−0.349994 + 0.936752i \(0.613816\pi\)
\(174\) 0 0
\(175\) 1.44173 0.108985
\(176\) 0 0
\(177\) −10.6464 −0.800234
\(178\) 0 0
\(179\) 1.18647 0.0886810 0.0443405 0.999016i \(-0.485881\pi\)
0.0443405 + 0.999016i \(0.485881\pi\)
\(180\) 0 0
\(181\) −24.9593 −1.85521 −0.927606 0.373560i \(-0.878137\pi\)
−0.927606 + 0.373560i \(0.878137\pi\)
\(182\) 0 0
\(183\) −41.0612 −3.03533
\(184\) 0 0
\(185\) −6.21777 −0.457140
\(186\) 0 0
\(187\) 2.00000 0.146254
\(188\) 0 0
\(189\) −6.69446 −0.486950
\(190\) 0 0
\(191\) 10.3096 0.745975 0.372987 0.927836i \(-0.378333\pi\)
0.372987 + 0.927836i \(0.378333\pi\)
\(192\) 0 0
\(193\) 26.1125 1.87962 0.939808 0.341704i \(-0.111004\pi\)
0.939808 + 0.341704i \(0.111004\pi\)
\(194\) 0 0
\(195\) 29.1039 2.08417
\(196\) 0 0
\(197\) −10.0768 −0.717941 −0.358971 0.933349i \(-0.616872\pi\)
−0.358971 + 0.933349i \(0.616872\pi\)
\(198\) 0 0
\(199\) −10.0002 −0.708893 −0.354447 0.935076i \(-0.615331\pi\)
−0.354447 + 0.935076i \(0.615331\pi\)
\(200\) 0 0
\(201\) −16.5288 −1.16585
\(202\) 0 0
\(203\) −1.40789 −0.0988143
\(204\) 0 0
\(205\) 5.73785 0.400749
\(206\) 0 0
\(207\) 15.8820 1.10388
\(208\) 0 0
\(209\) −1.00000 −0.0691714
\(210\) 0 0
\(211\) 3.10438 0.213715 0.106857 0.994274i \(-0.465921\pi\)
0.106857 + 0.994274i \(0.465921\pi\)
\(212\) 0 0
\(213\) −35.1678 −2.40966
\(214\) 0 0
\(215\) 17.0025 1.15956
\(216\) 0 0
\(217\) 5.41226 0.367408
\(218\) 0 0
\(219\) 36.9844 2.49917
\(220\) 0 0
\(221\) 10.6354 0.715414
\(222\) 0 0
\(223\) 15.4097 1.03191 0.515955 0.856615i \(-0.327437\pi\)
0.515955 + 0.856615i \(0.327437\pi\)
\(224\) 0 0
\(225\) −6.86329 −0.457553
\(226\) 0 0
\(227\) −13.9183 −0.923790 −0.461895 0.886935i \(-0.652830\pi\)
−0.461895 + 0.886935i \(0.652830\pi\)
\(228\) 0 0
\(229\) −18.4742 −1.22081 −0.610406 0.792089i \(-0.708994\pi\)
−0.610406 + 0.792089i \(0.708994\pi\)
\(230\) 0 0
\(231\) −3.09738 −0.203793
\(232\) 0 0
\(233\) 28.7221 1.88165 0.940824 0.338896i \(-0.110053\pi\)
0.940824 + 0.338896i \(0.110053\pi\)
\(234\) 0 0
\(235\) −12.1129 −0.790157
\(236\) 0 0
\(237\) −0.481148 −0.0312540
\(238\) 0 0
\(239\) −18.9274 −1.22431 −0.612157 0.790736i \(-0.709698\pi\)
−0.612157 + 0.790736i \(0.709698\pi\)
\(240\) 0 0
\(241\) −23.5821 −1.51905 −0.759527 0.650476i \(-0.774570\pi\)
−0.759527 + 0.650476i \(0.774570\pi\)
\(242\) 0 0
\(243\) −12.3661 −0.793286
\(244\) 0 0
\(245\) −11.1585 −0.712890
\(246\) 0 0
\(247\) −5.31769 −0.338357
\(248\) 0 0
\(249\) −4.54226 −0.287854
\(250\) 0 0
\(251\) −14.7745 −0.932558 −0.466279 0.884638i \(-0.654406\pi\)
−0.466279 + 0.884638i \(0.654406\pi\)
\(252\) 0 0
\(253\) 3.07712 0.193457
\(254\) 0 0
\(255\) 10.9461 0.685469
\(256\) 0 0
\(257\) −8.53160 −0.532187 −0.266093 0.963947i \(-0.585733\pi\)
−0.266093 + 0.963947i \(0.585733\pi\)
\(258\) 0 0
\(259\) 3.51885 0.218651
\(260\) 0 0
\(261\) 6.70217 0.414854
\(262\) 0 0
\(263\) −23.8573 −1.47110 −0.735551 0.677469i \(-0.763077\pi\)
−0.735551 + 0.677469i \(0.763077\pi\)
\(264\) 0 0
\(265\) 3.83158 0.235372
\(266\) 0 0
\(267\) 18.0658 1.10561
\(268\) 0 0
\(269\) 14.7534 0.899528 0.449764 0.893147i \(-0.351508\pi\)
0.449764 + 0.893147i \(0.351508\pi\)
\(270\) 0 0
\(271\) −3.61665 −0.219696 −0.109848 0.993948i \(-0.535036\pi\)
−0.109848 + 0.993948i \(0.535036\pi\)
\(272\) 0 0
\(273\) −16.4709 −0.996865
\(274\) 0 0
\(275\) −1.32975 −0.0801871
\(276\) 0 0
\(277\) −0.871327 −0.0523529 −0.0261765 0.999657i \(-0.508333\pi\)
−0.0261765 + 0.999657i \(0.508333\pi\)
\(278\) 0 0
\(279\) −25.7648 −1.54250
\(280\) 0 0
\(281\) −25.5658 −1.52513 −0.762565 0.646912i \(-0.776060\pi\)
−0.762565 + 0.646912i \(0.776060\pi\)
\(282\) 0 0
\(283\) 21.5513 1.28109 0.640547 0.767919i \(-0.278708\pi\)
0.640547 + 0.767919i \(0.278708\pi\)
\(284\) 0 0
\(285\) −5.47303 −0.324195
\(286\) 0 0
\(287\) −3.24725 −0.191679
\(288\) 0 0
\(289\) −13.0000 −0.764706
\(290\) 0 0
\(291\) 41.5108 2.43341
\(292\) 0 0
\(293\) 2.07530 0.121240 0.0606201 0.998161i \(-0.480692\pi\)
0.0606201 + 0.998161i \(0.480692\pi\)
\(294\) 0 0
\(295\) −7.13955 −0.415681
\(296\) 0 0
\(297\) 6.17450 0.358281
\(298\) 0 0
\(299\) 16.3632 0.946307
\(300\) 0 0
\(301\) −9.62231 −0.554620
\(302\) 0 0
\(303\) 31.8267 1.82839
\(304\) 0 0
\(305\) −27.5359 −1.57670
\(306\) 0 0
\(307\) −9.47971 −0.541036 −0.270518 0.962715i \(-0.587195\pi\)
−0.270518 + 0.962715i \(0.587195\pi\)
\(308\) 0 0
\(309\) −20.5135 −1.16697
\(310\) 0 0
\(311\) −8.49108 −0.481485 −0.240743 0.970589i \(-0.577391\pi\)
−0.240743 + 0.970589i \(0.577391\pi\)
\(312\) 0 0
\(313\) −9.21577 −0.520906 −0.260453 0.965487i \(-0.583872\pi\)
−0.260453 + 0.965487i \(0.583872\pi\)
\(314\) 0 0
\(315\) −10.7207 −0.604043
\(316\) 0 0
\(317\) −28.2279 −1.58544 −0.792718 0.609589i \(-0.791335\pi\)
−0.792718 + 0.609589i \(0.791335\pi\)
\(318\) 0 0
\(319\) 1.29854 0.0727041
\(320\) 0 0
\(321\) 7.11654 0.397206
\(322\) 0 0
\(323\) −2.00000 −0.111283
\(324\) 0 0
\(325\) −7.07122 −0.392240
\(326\) 0 0
\(327\) 12.8018 0.707939
\(328\) 0 0
\(329\) 6.85510 0.377934
\(330\) 0 0
\(331\) 1.72813 0.0949867 0.0474933 0.998872i \(-0.484877\pi\)
0.0474933 + 0.998872i \(0.484877\pi\)
\(332\) 0 0
\(333\) −16.7513 −0.917966
\(334\) 0 0
\(335\) −11.0843 −0.605600
\(336\) 0 0
\(337\) 13.2502 0.721783 0.360892 0.932608i \(-0.382472\pi\)
0.360892 + 0.932608i \(0.382472\pi\)
\(338\) 0 0
\(339\) 44.9975 2.44393
\(340\) 0 0
\(341\) −4.99189 −0.270326
\(342\) 0 0
\(343\) 13.9044 0.750770
\(344\) 0 0
\(345\) 16.8412 0.906699
\(346\) 0 0
\(347\) −8.88126 −0.476771 −0.238386 0.971171i \(-0.576618\pi\)
−0.238386 + 0.971171i \(0.576618\pi\)
\(348\) 0 0
\(349\) −24.0464 −1.28717 −0.643586 0.765373i \(-0.722554\pi\)
−0.643586 + 0.765373i \(0.722554\pi\)
\(350\) 0 0
\(351\) 32.8341 1.75255
\(352\) 0 0
\(353\) 9.90522 0.527202 0.263601 0.964632i \(-0.415090\pi\)
0.263601 + 0.964632i \(0.415090\pi\)
\(354\) 0 0
\(355\) −23.5837 −1.25169
\(356\) 0 0
\(357\) −6.19476 −0.327861
\(358\) 0 0
\(359\) 12.9158 0.681669 0.340835 0.940123i \(-0.389290\pi\)
0.340835 + 0.940123i \(0.389290\pi\)
\(360\) 0 0
\(361\) 1.00000 0.0526316
\(362\) 0 0
\(363\) 2.85680 0.149943
\(364\) 0 0
\(365\) 24.8019 1.29819
\(366\) 0 0
\(367\) 30.7858 1.60701 0.803503 0.595300i \(-0.202967\pi\)
0.803503 + 0.595300i \(0.202967\pi\)
\(368\) 0 0
\(369\) 15.4584 0.804730
\(370\) 0 0
\(371\) −2.16842 −0.112579
\(372\) 0 0
\(373\) −22.4250 −1.16112 −0.580561 0.814216i \(-0.697167\pi\)
−0.580561 + 0.814216i \(0.697167\pi\)
\(374\) 0 0
\(375\) −34.6430 −1.78895
\(376\) 0 0
\(377\) 6.90522 0.355637
\(378\) 0 0
\(379\) −7.51924 −0.386237 −0.193119 0.981175i \(-0.561860\pi\)
−0.193119 + 0.981175i \(0.561860\pi\)
\(380\) 0 0
\(381\) 7.49456 0.383958
\(382\) 0 0
\(383\) 16.5491 0.845617 0.422809 0.906219i \(-0.361044\pi\)
0.422809 + 0.906219i \(0.361044\pi\)
\(384\) 0 0
\(385\) −2.07712 −0.105860
\(386\) 0 0
\(387\) 45.8065 2.32847
\(388\) 0 0
\(389\) 35.5840 1.80418 0.902089 0.431549i \(-0.142033\pi\)
0.902089 + 0.431549i \(0.142033\pi\)
\(390\) 0 0
\(391\) 6.15424 0.311233
\(392\) 0 0
\(393\) 32.8495 1.65704
\(394\) 0 0
\(395\) −0.322661 −0.0162348
\(396\) 0 0
\(397\) 12.3529 0.619976 0.309988 0.950740i \(-0.399675\pi\)
0.309988 + 0.950740i \(0.399675\pi\)
\(398\) 0 0
\(399\) 3.09738 0.155063
\(400\) 0 0
\(401\) −32.9049 −1.64319 −0.821596 0.570070i \(-0.806916\pi\)
−0.821596 + 0.570070i \(0.806916\pi\)
\(402\) 0 0
\(403\) −26.5453 −1.32232
\(404\) 0 0
\(405\) 4.12918 0.205180
\(406\) 0 0
\(407\) −3.24554 −0.160876
\(408\) 0 0
\(409\) −25.8018 −1.27582 −0.637908 0.770113i \(-0.720200\pi\)
−0.637908 + 0.770113i \(0.720200\pi\)
\(410\) 0 0
\(411\) 0.460889 0.0227340
\(412\) 0 0
\(413\) 4.04052 0.198821
\(414\) 0 0
\(415\) −3.04607 −0.149526
\(416\) 0 0
\(417\) −41.0513 −2.01029
\(418\) 0 0
\(419\) −21.2971 −1.04043 −0.520216 0.854035i \(-0.674149\pi\)
−0.520216 + 0.854035i \(0.674149\pi\)
\(420\) 0 0
\(421\) 29.4734 1.43645 0.718223 0.695813i \(-0.244956\pi\)
0.718223 + 0.695813i \(0.244956\pi\)
\(422\) 0 0
\(423\) −32.6333 −1.58669
\(424\) 0 0
\(425\) −2.65950 −0.129005
\(426\) 0 0
\(427\) 15.5835 0.754138
\(428\) 0 0
\(429\) 15.1916 0.733458
\(430\) 0 0
\(431\) 21.7861 1.04940 0.524701 0.851287i \(-0.324177\pi\)
0.524701 + 0.851287i \(0.324177\pi\)
\(432\) 0 0
\(433\) 21.3657 1.02677 0.513386 0.858158i \(-0.328391\pi\)
0.513386 + 0.858158i \(0.328391\pi\)
\(434\) 0 0
\(435\) 7.10693 0.340751
\(436\) 0 0
\(437\) −3.07712 −0.147199
\(438\) 0 0
\(439\) −8.81816 −0.420868 −0.210434 0.977608i \(-0.567488\pi\)
−0.210434 + 0.977608i \(0.567488\pi\)
\(440\) 0 0
\(441\) −30.0621 −1.43153
\(442\) 0 0
\(443\) 27.5521 1.30904 0.654521 0.756044i \(-0.272870\pi\)
0.654521 + 0.756044i \(0.272870\pi\)
\(444\) 0 0
\(445\) 12.1150 0.574306
\(446\) 0 0
\(447\) 8.82810 0.417555
\(448\) 0 0
\(449\) 8.16732 0.385440 0.192720 0.981254i \(-0.438269\pi\)
0.192720 + 0.981254i \(0.438269\pi\)
\(450\) 0 0
\(451\) 2.99503 0.141031
\(452\) 0 0
\(453\) 8.82810 0.414780
\(454\) 0 0
\(455\) −11.0455 −0.517820
\(456\) 0 0
\(457\) −4.97240 −0.232599 −0.116300 0.993214i \(-0.537103\pi\)
−0.116300 + 0.993214i \(0.537103\pi\)
\(458\) 0 0
\(459\) 12.3490 0.576402
\(460\) 0 0
\(461\) −15.9039 −0.740721 −0.370360 0.928888i \(-0.620766\pi\)
−0.370360 + 0.928888i \(0.620766\pi\)
\(462\) 0 0
\(463\) −4.24715 −0.197382 −0.0986908 0.995118i \(-0.531465\pi\)
−0.0986908 + 0.995118i \(0.531465\pi\)
\(464\) 0 0
\(465\) −27.3208 −1.26697
\(466\) 0 0
\(467\) −27.2216 −1.25967 −0.629833 0.776731i \(-0.716877\pi\)
−0.629833 + 0.776731i \(0.716877\pi\)
\(468\) 0 0
\(469\) 6.27299 0.289660
\(470\) 0 0
\(471\) 20.0900 0.925697
\(472\) 0 0
\(473\) 8.87494 0.408070
\(474\) 0 0
\(475\) 1.32975 0.0610132
\(476\) 0 0
\(477\) 10.3227 0.472642
\(478\) 0 0
\(479\) −9.08201 −0.414968 −0.207484 0.978238i \(-0.566527\pi\)
−0.207484 + 0.978238i \(0.566527\pi\)
\(480\) 0 0
\(481\) −17.2588 −0.786933
\(482\) 0 0
\(483\) −9.53100 −0.433676
\(484\) 0 0
\(485\) 27.8374 1.26403
\(486\) 0 0
\(487\) −17.6203 −0.798451 −0.399225 0.916853i \(-0.630721\pi\)
−0.399225 + 0.916853i \(0.630721\pi\)
\(488\) 0 0
\(489\) −12.4647 −0.563675
\(490\) 0 0
\(491\) −22.7082 −1.02481 −0.512404 0.858744i \(-0.671245\pi\)
−0.512404 + 0.858744i \(0.671245\pi\)
\(492\) 0 0
\(493\) 2.59707 0.116966
\(494\) 0 0
\(495\) 9.88802 0.444433
\(496\) 0 0
\(497\) 13.3468 0.598687
\(498\) 0 0
\(499\) 4.11592 0.184254 0.0921270 0.995747i \(-0.470633\pi\)
0.0921270 + 0.995747i \(0.470633\pi\)
\(500\) 0 0
\(501\) −7.85996 −0.351157
\(502\) 0 0
\(503\) −39.1716 −1.74658 −0.873288 0.487205i \(-0.838017\pi\)
−0.873288 + 0.487205i \(0.838017\pi\)
\(504\) 0 0
\(505\) 21.3431 0.949757
\(506\) 0 0
\(507\) 43.6459 1.93838
\(508\) 0 0
\(509\) −17.4018 −0.771322 −0.385661 0.922640i \(-0.626027\pi\)
−0.385661 + 0.922640i \(0.626027\pi\)
\(510\) 0 0
\(511\) −14.0363 −0.620928
\(512\) 0 0
\(513\) −6.17450 −0.272611
\(514\) 0 0
\(515\) −13.7565 −0.606182
\(516\) 0 0
\(517\) −6.32266 −0.278070
\(518\) 0 0
\(519\) −26.3023 −1.15454
\(520\) 0 0
\(521\) 20.2691 0.888004 0.444002 0.896026i \(-0.353558\pi\)
0.444002 + 0.896026i \(0.353558\pi\)
\(522\) 0 0
\(523\) 7.23875 0.316529 0.158264 0.987397i \(-0.449410\pi\)
0.158264 + 0.987397i \(0.449410\pi\)
\(524\) 0 0
\(525\) 4.11875 0.179757
\(526\) 0 0
\(527\) −9.98377 −0.434900
\(528\) 0 0
\(529\) −13.5313 −0.588319
\(530\) 0 0
\(531\) −19.2347 −0.834714
\(532\) 0 0
\(533\) 15.9267 0.689861
\(534\) 0 0
\(535\) 4.77239 0.206328
\(536\) 0 0
\(537\) 3.38951 0.146268
\(538\) 0 0
\(539\) −5.82449 −0.250878
\(540\) 0 0
\(541\) 38.1394 1.63974 0.819870 0.572549i \(-0.194046\pi\)
0.819870 + 0.572549i \(0.194046\pi\)
\(542\) 0 0
\(543\) −71.3039 −3.05994
\(544\) 0 0
\(545\) 8.58494 0.367738
\(546\) 0 0
\(547\) −1.50161 −0.0642044 −0.0321022 0.999485i \(-0.510220\pi\)
−0.0321022 + 0.999485i \(0.510220\pi\)
\(548\) 0 0
\(549\) −74.1844 −3.16611
\(550\) 0 0
\(551\) −1.29854 −0.0553195
\(552\) 0 0
\(553\) 0.182605 0.00776515
\(554\) 0 0
\(555\) −17.7630 −0.753996
\(556\) 0 0
\(557\) 26.0847 1.10524 0.552622 0.833432i \(-0.313627\pi\)
0.552622 + 0.833432i \(0.313627\pi\)
\(558\) 0 0
\(559\) 47.1942 1.99610
\(560\) 0 0
\(561\) 5.71361 0.241229
\(562\) 0 0
\(563\) −27.9759 −1.17904 −0.589521 0.807753i \(-0.700684\pi\)
−0.589521 + 0.807753i \(0.700684\pi\)
\(564\) 0 0
\(565\) 30.1756 1.26950
\(566\) 0 0
\(567\) −2.33684 −0.0981382
\(568\) 0 0
\(569\) 21.2097 0.889155 0.444578 0.895740i \(-0.353354\pi\)
0.444578 + 0.895740i \(0.353354\pi\)
\(570\) 0 0
\(571\) 19.1347 0.800761 0.400380 0.916349i \(-0.368878\pi\)
0.400380 + 0.916349i \(0.368878\pi\)
\(572\) 0 0
\(573\) 29.4525 1.23039
\(574\) 0 0
\(575\) −4.09181 −0.170640
\(576\) 0 0
\(577\) −45.9979 −1.91492 −0.957459 0.288568i \(-0.906821\pi\)
−0.957459 + 0.288568i \(0.906821\pi\)
\(578\) 0 0
\(579\) 74.5982 3.10019
\(580\) 0 0
\(581\) 1.72387 0.0715184
\(582\) 0 0
\(583\) 2.00000 0.0828315
\(584\) 0 0
\(585\) 52.5815 2.17398
\(586\) 0 0
\(587\) 46.3398 1.91265 0.956325 0.292306i \(-0.0944226\pi\)
0.956325 + 0.292306i \(0.0944226\pi\)
\(588\) 0 0
\(589\) 4.99189 0.205687
\(590\) 0 0
\(591\) −28.7874 −1.18416
\(592\) 0 0
\(593\) 17.8898 0.734644 0.367322 0.930094i \(-0.380275\pi\)
0.367322 + 0.930094i \(0.380275\pi\)
\(594\) 0 0
\(595\) −4.15424 −0.170307
\(596\) 0 0
\(597\) −28.5685 −1.16923
\(598\) 0 0
\(599\) 18.8508 0.770222 0.385111 0.922870i \(-0.374163\pi\)
0.385111 + 0.922870i \(0.374163\pi\)
\(600\) 0 0
\(601\) −9.16363 −0.373792 −0.186896 0.982380i \(-0.559843\pi\)
−0.186896 + 0.982380i \(0.559843\pi\)
\(602\) 0 0
\(603\) −29.8622 −1.21608
\(604\) 0 0
\(605\) 1.91579 0.0778879
\(606\) 0 0
\(607\) −1.11636 −0.0453118 −0.0226559 0.999743i \(-0.507212\pi\)
−0.0226559 + 0.999743i \(0.507212\pi\)
\(608\) 0 0
\(609\) −4.02206 −0.162982
\(610\) 0 0
\(611\) −33.6220 −1.36020
\(612\) 0 0
\(613\) 9.64609 0.389602 0.194801 0.980843i \(-0.437594\pi\)
0.194801 + 0.980843i \(0.437594\pi\)
\(614\) 0 0
\(615\) 16.3919 0.660986
\(616\) 0 0
\(617\) −38.4415 −1.54760 −0.773799 0.633431i \(-0.781646\pi\)
−0.773799 + 0.633431i \(0.781646\pi\)
\(618\) 0 0
\(619\) 39.2599 1.57799 0.788995 0.614400i \(-0.210602\pi\)
0.788995 + 0.614400i \(0.210602\pi\)
\(620\) 0 0
\(621\) 18.9997 0.762430
\(622\) 0 0
\(623\) −6.85629 −0.274692
\(624\) 0 0
\(625\) −16.5830 −0.663320
\(626\) 0 0
\(627\) −2.85680 −0.114090
\(628\) 0 0
\(629\) −6.49108 −0.258816
\(630\) 0 0
\(631\) −14.9952 −0.596951 −0.298476 0.954417i \(-0.596478\pi\)
−0.298476 + 0.954417i \(0.596478\pi\)
\(632\) 0 0
\(633\) 8.86862 0.352496
\(634\) 0 0
\(635\) 5.02590 0.199447
\(636\) 0 0
\(637\) −30.9728 −1.22719
\(638\) 0 0
\(639\) −63.5369 −2.51348
\(640\) 0 0
\(641\) 15.3586 0.606628 0.303314 0.952891i \(-0.401907\pi\)
0.303314 + 0.952891i \(0.401907\pi\)
\(642\) 0 0
\(643\) −36.0013 −1.41975 −0.709876 0.704327i \(-0.751249\pi\)
−0.709876 + 0.704327i \(0.751249\pi\)
\(644\) 0 0
\(645\) 48.5728 1.91255
\(646\) 0 0
\(647\) 5.30041 0.208381 0.104190 0.994557i \(-0.466775\pi\)
0.104190 + 0.994557i \(0.466775\pi\)
\(648\) 0 0
\(649\) −3.72669 −0.146285
\(650\) 0 0
\(651\) 15.4618 0.605994
\(652\) 0 0
\(653\) 25.0255 0.979325 0.489662 0.871912i \(-0.337120\pi\)
0.489662 + 0.871912i \(0.337120\pi\)
\(654\) 0 0
\(655\) 22.0290 0.860746
\(656\) 0 0
\(657\) 66.8189 2.60686
\(658\) 0 0
\(659\) 28.7896 1.12148 0.560742 0.827990i \(-0.310516\pi\)
0.560742 + 0.827990i \(0.310516\pi\)
\(660\) 0 0
\(661\) −15.8286 −0.615661 −0.307831 0.951441i \(-0.599603\pi\)
−0.307831 + 0.951441i \(0.599603\pi\)
\(662\) 0 0
\(663\) 30.3832 1.17999
\(664\) 0 0
\(665\) 2.07712 0.0805472
\(666\) 0 0
\(667\) 3.99575 0.154716
\(668\) 0 0
\(669\) 44.0225 1.70201
\(670\) 0 0
\(671\) −14.3731 −0.554868
\(672\) 0 0
\(673\) 19.5104 0.752071 0.376035 0.926605i \(-0.377287\pi\)
0.376035 + 0.926605i \(0.377287\pi\)
\(674\) 0 0
\(675\) −8.21055 −0.316024
\(676\) 0 0
\(677\) 19.3832 0.744957 0.372479 0.928041i \(-0.378508\pi\)
0.372479 + 0.928041i \(0.378508\pi\)
\(678\) 0 0
\(679\) −15.7541 −0.604588
\(680\) 0 0
\(681\) −39.7619 −1.52368
\(682\) 0 0
\(683\) −4.67657 −0.178944 −0.0894720 0.995989i \(-0.528518\pi\)
−0.0894720 + 0.995989i \(0.528518\pi\)
\(684\) 0 0
\(685\) 0.309075 0.0118092
\(686\) 0 0
\(687\) −52.7773 −2.01358
\(688\) 0 0
\(689\) 10.6354 0.405176
\(690\) 0 0
\(691\) −13.7096 −0.521539 −0.260769 0.965401i \(-0.583976\pi\)
−0.260769 + 0.965401i \(0.583976\pi\)
\(692\) 0 0
\(693\) −5.59597 −0.212573
\(694\) 0 0
\(695\) −27.5293 −1.04424
\(696\) 0 0
\(697\) 5.99007 0.226890
\(698\) 0 0
\(699\) 82.0534 3.10355
\(700\) 0 0
\(701\) −40.8786 −1.54396 −0.771981 0.635646i \(-0.780734\pi\)
−0.771981 + 0.635646i \(0.780734\pi\)
\(702\) 0 0
\(703\) 3.24554 0.122408
\(704\) 0 0
\(705\) −34.6041 −1.30327
\(706\) 0 0
\(707\) −12.0788 −0.454271
\(708\) 0 0
\(709\) −33.3293 −1.25171 −0.625855 0.779940i \(-0.715250\pi\)
−0.625855 + 0.779940i \(0.715250\pi\)
\(710\) 0 0
\(711\) −0.869281 −0.0326006
\(712\) 0 0
\(713\) −15.3606 −0.575260
\(714\) 0 0
\(715\) 10.1876 0.380994
\(716\) 0 0
\(717\) −54.0720 −2.01936
\(718\) 0 0
\(719\) 24.5647 0.916110 0.458055 0.888924i \(-0.348546\pi\)
0.458055 + 0.888924i \(0.348546\pi\)
\(720\) 0 0
\(721\) 7.78525 0.289938
\(722\) 0 0
\(723\) −67.3693 −2.50549
\(724\) 0 0
\(725\) −1.72673 −0.0641292
\(726\) 0 0
\(727\) −8.42755 −0.312560 −0.156280 0.987713i \(-0.549950\pi\)
−0.156280 + 0.987713i \(0.549950\pi\)
\(728\) 0 0
\(729\) −41.7936 −1.54791
\(730\) 0 0
\(731\) 17.7499 0.656503
\(732\) 0 0
\(733\) −11.3192 −0.418084 −0.209042 0.977907i \(-0.567034\pi\)
−0.209042 + 0.977907i \(0.567034\pi\)
\(734\) 0 0
\(735\) −31.8776 −1.17582
\(736\) 0 0
\(737\) −5.78576 −0.213121
\(738\) 0 0
\(739\) 47.8757 1.76114 0.880568 0.473919i \(-0.157161\pi\)
0.880568 + 0.473919i \(0.157161\pi\)
\(740\) 0 0
\(741\) −15.1916 −0.558078
\(742\) 0 0
\(743\) −49.9347 −1.83193 −0.915963 0.401262i \(-0.868572\pi\)
−0.915963 + 0.401262i \(0.868572\pi\)
\(744\) 0 0
\(745\) 5.92017 0.216898
\(746\) 0 0
\(747\) −8.20642 −0.300257
\(748\) 0 0
\(749\) −2.70086 −0.0986872
\(750\) 0 0
\(751\) 16.9473 0.618417 0.309209 0.950994i \(-0.399936\pi\)
0.309209 + 0.950994i \(0.399936\pi\)
\(752\) 0 0
\(753\) −42.2079 −1.53814
\(754\) 0 0
\(755\) 5.92017 0.215457
\(756\) 0 0
\(757\) 0.939402 0.0341431 0.0170716 0.999854i \(-0.494566\pi\)
0.0170716 + 0.999854i \(0.494566\pi\)
\(758\) 0 0
\(759\) 8.79073 0.319083
\(760\) 0 0
\(761\) 10.1954 0.369584 0.184792 0.982778i \(-0.440839\pi\)
0.184792 + 0.982778i \(0.440839\pi\)
\(762\) 0 0
\(763\) −4.85851 −0.175890
\(764\) 0 0
\(765\) 19.7760 0.715004
\(766\) 0 0
\(767\) −19.8174 −0.715565
\(768\) 0 0
\(769\) −2.22600 −0.0802718 −0.0401359 0.999194i \(-0.512779\pi\)
−0.0401359 + 0.999194i \(0.512779\pi\)
\(770\) 0 0
\(771\) −24.3731 −0.877776
\(772\) 0 0
\(773\) −24.2189 −0.871092 −0.435546 0.900166i \(-0.643445\pi\)
−0.435546 + 0.900166i \(0.643445\pi\)
\(774\) 0 0
\(775\) 6.63797 0.238443
\(776\) 0 0
\(777\) 10.0527 0.360638
\(778\) 0 0
\(779\) −2.99503 −0.107308
\(780\) 0 0
\(781\) −12.3102 −0.440493
\(782\) 0 0
\(783\) 8.01781 0.286533
\(784\) 0 0
\(785\) 13.4725 0.480852
\(786\) 0 0
\(787\) 36.7756 1.31091 0.655454 0.755235i \(-0.272477\pi\)
0.655454 + 0.755235i \(0.272477\pi\)
\(788\) 0 0
\(789\) −68.1556 −2.42640
\(790\) 0 0
\(791\) −17.0774 −0.607202
\(792\) 0 0
\(793\) −76.4318 −2.71417
\(794\) 0 0
\(795\) 10.9461 0.388217
\(796\) 0 0
\(797\) 42.1391 1.49264 0.746322 0.665585i \(-0.231818\pi\)
0.746322 + 0.665585i \(0.231818\pi\)
\(798\) 0 0
\(799\) −12.6453 −0.447359
\(800\) 0 0
\(801\) 32.6390 1.15324
\(802\) 0 0
\(803\) 12.9461 0.456857
\(804\) 0 0
\(805\) −6.39154 −0.225272
\(806\) 0 0
\(807\) 42.1475 1.48366
\(808\) 0 0
\(809\) −11.6056 −0.408030 −0.204015 0.978968i \(-0.565399\pi\)
−0.204015 + 0.978968i \(0.565399\pi\)
\(810\) 0 0
\(811\) 52.3945 1.83982 0.919910 0.392128i \(-0.128261\pi\)
0.919910 + 0.392128i \(0.128261\pi\)
\(812\) 0 0
\(813\) −10.3321 −0.362361
\(814\) 0 0
\(815\) −8.35893 −0.292801
\(816\) 0 0
\(817\) −8.87494 −0.310495
\(818\) 0 0
\(819\) −29.7577 −1.03982
\(820\) 0 0
\(821\) −49.2156 −1.71764 −0.858818 0.512281i \(-0.828801\pi\)
−0.858818 + 0.512281i \(0.828801\pi\)
\(822\) 0 0
\(823\) −20.6059 −0.718277 −0.359138 0.933284i \(-0.616929\pi\)
−0.359138 + 0.933284i \(0.616929\pi\)
\(824\) 0 0
\(825\) −3.79884 −0.132259
\(826\) 0 0
\(827\) −28.6595 −0.996589 −0.498295 0.867008i \(-0.666040\pi\)
−0.498295 + 0.867008i \(0.666040\pi\)
\(828\) 0 0
\(829\) −10.1774 −0.353476 −0.176738 0.984258i \(-0.556555\pi\)
−0.176738 + 0.984258i \(0.556555\pi\)
\(830\) 0 0
\(831\) −2.48921 −0.0863497
\(832\) 0 0
\(833\) −11.6490 −0.403613
\(834\) 0 0
\(835\) −5.27093 −0.182408
\(836\) 0 0
\(837\) −30.8224 −1.06538
\(838\) 0 0
\(839\) 18.4526 0.637055 0.318528 0.947914i \(-0.396812\pi\)
0.318528 + 0.947914i \(0.396812\pi\)
\(840\) 0 0
\(841\) −27.3138 −0.941855
\(842\) 0 0
\(843\) −73.0366 −2.51551
\(844\) 0 0
\(845\) 29.2692 1.00689
\(846\) 0 0
\(847\) −1.08421 −0.0372539
\(848\) 0 0
\(849\) 61.5680 2.11301
\(850\) 0 0
\(851\) −9.98692 −0.342347
\(852\) 0 0
\(853\) 31.8680 1.09114 0.545570 0.838065i \(-0.316313\pi\)
0.545570 + 0.838065i \(0.316313\pi\)
\(854\) 0 0
\(855\) −9.88802 −0.338163
\(856\) 0 0
\(857\) −2.31725 −0.0791559 −0.0395779 0.999216i \(-0.512601\pi\)
−0.0395779 + 0.999216i \(0.512601\pi\)
\(858\) 0 0
\(859\) 44.0900 1.50433 0.752166 0.658973i \(-0.229009\pi\)
0.752166 + 0.658973i \(0.229009\pi\)
\(860\) 0 0
\(861\) −9.27675 −0.316151
\(862\) 0 0
\(863\) 5.09373 0.173393 0.0866964 0.996235i \(-0.472369\pi\)
0.0866964 + 0.996235i \(0.472369\pi\)
\(864\) 0 0
\(865\) −17.6385 −0.599726
\(866\) 0 0
\(867\) −37.1385 −1.26129
\(868\) 0 0
\(869\) −0.168422 −0.00571332
\(870\) 0 0
\(871\) −30.7669 −1.04250
\(872\) 0 0
\(873\) 74.9967 2.53825
\(874\) 0 0
\(875\) 13.1477 0.444472
\(876\) 0 0
\(877\) 13.8573 0.467928 0.233964 0.972245i \(-0.424830\pi\)
0.233964 + 0.972245i \(0.424830\pi\)
\(878\) 0 0
\(879\) 5.92872 0.199971
\(880\) 0 0
\(881\) −32.6376 −1.09959 −0.549794 0.835300i \(-0.685294\pi\)
−0.549794 + 0.835300i \(0.685294\pi\)
\(882\) 0 0
\(883\) 31.3091 1.05363 0.526817 0.849979i \(-0.323385\pi\)
0.526817 + 0.849979i \(0.323385\pi\)
\(884\) 0 0
\(885\) −20.3963 −0.685614
\(886\) 0 0
\(887\) 35.6199 1.19600 0.598000 0.801496i \(-0.295962\pi\)
0.598000 + 0.801496i \(0.295962\pi\)
\(888\) 0 0
\(889\) −2.84433 −0.0953957
\(890\) 0 0
\(891\) 2.15534 0.0722066
\(892\) 0 0
\(893\) 6.32266 0.211580
\(894\) 0 0
\(895\) 2.27303 0.0759789
\(896\) 0 0
\(897\) 46.7464 1.56082
\(898\) 0 0
\(899\) −6.48214 −0.216192
\(900\) 0 0
\(901\) 4.00000 0.133259
\(902\) 0 0
\(903\) −27.4890 −0.914778
\(904\) 0 0
\(905\) −47.8168 −1.58948
\(906\) 0 0
\(907\) 41.1629 1.36679 0.683396 0.730048i \(-0.260502\pi\)
0.683396 + 0.730048i \(0.260502\pi\)
\(908\) 0 0
\(909\) 57.5006 1.90717
\(910\) 0 0
\(911\) 8.30560 0.275177 0.137588 0.990490i \(-0.456065\pi\)
0.137588 + 0.990490i \(0.456065\pi\)
\(912\) 0 0
\(913\) −1.58998 −0.0526207
\(914\) 0 0
\(915\) −78.6645 −2.60057
\(916\) 0 0
\(917\) −12.4670 −0.411696
\(918\) 0 0
\(919\) 17.5343 0.578402 0.289201 0.957268i \(-0.406610\pi\)
0.289201 + 0.957268i \(0.406610\pi\)
\(920\) 0 0
\(921\) −27.0817 −0.892372
\(922\) 0 0
\(923\) −65.4618 −2.15470
\(924\) 0 0
\(925\) 4.31577 0.141902
\(926\) 0 0
\(927\) −37.0613 −1.21725
\(928\) 0 0
\(929\) −17.8335 −0.585097 −0.292549 0.956251i \(-0.594503\pi\)
−0.292549 + 0.956251i \(0.594503\pi\)
\(930\) 0 0
\(931\) 5.82449 0.190890
\(932\) 0 0
\(933\) −24.2574 −0.794150
\(934\) 0 0
\(935\) 3.83158 0.125306
\(936\) 0 0
\(937\) 15.8865 0.518988 0.259494 0.965745i \(-0.416444\pi\)
0.259494 + 0.965745i \(0.416444\pi\)
\(938\) 0 0
\(939\) −26.3276 −0.859170
\(940\) 0 0
\(941\) −14.0364 −0.457575 −0.228787 0.973476i \(-0.573476\pi\)
−0.228787 + 0.973476i \(0.573476\pi\)
\(942\) 0 0
\(943\) 9.21607 0.300117
\(944\) 0 0
\(945\) −12.8252 −0.417203
\(946\) 0 0
\(947\) 30.1959 0.981234 0.490617 0.871375i \(-0.336771\pi\)
0.490617 + 0.871375i \(0.336771\pi\)
\(948\) 0 0
\(949\) 68.8432 2.23475
\(950\) 0 0
\(951\) −80.6415 −2.61498
\(952\) 0 0
\(953\) 8.71300 0.282242 0.141121 0.989992i \(-0.454929\pi\)
0.141121 + 0.989992i \(0.454929\pi\)
\(954\) 0 0
\(955\) 19.7510 0.639127
\(956\) 0 0
\(957\) 3.70966 0.119916
\(958\) 0 0
\(959\) −0.174916 −0.00564834
\(960\) 0 0
\(961\) −6.08108 −0.196164
\(962\) 0 0
\(963\) 12.8573 0.414321
\(964\) 0 0
\(965\) 50.0259 1.61039
\(966\) 0 0
\(967\) −43.1106 −1.38634 −0.693171 0.720773i \(-0.743787\pi\)
−0.693171 + 0.720773i \(0.743787\pi\)
\(968\) 0 0
\(969\) −5.71361 −0.183548
\(970\) 0 0
\(971\) −21.2824 −0.682985 −0.341493 0.939884i \(-0.610932\pi\)
−0.341493 + 0.939884i \(0.610932\pi\)
\(972\) 0 0
\(973\) 15.5798 0.499464
\(974\) 0 0
\(975\) −20.2011 −0.646952
\(976\) 0 0
\(977\) −13.2820 −0.424928 −0.212464 0.977169i \(-0.568149\pi\)
−0.212464 + 0.977169i \(0.568149\pi\)
\(978\) 0 0
\(979\) 6.32376 0.202108
\(980\) 0 0
\(981\) 23.1287 0.738442
\(982\) 0 0
\(983\) −59.4823 −1.89719 −0.948596 0.316491i \(-0.897495\pi\)
−0.948596 + 0.316491i \(0.897495\pi\)
\(984\) 0 0
\(985\) −19.3050 −0.615108
\(986\) 0 0
\(987\) 19.5837 0.623355
\(988\) 0 0
\(989\) 27.3092 0.868384
\(990\) 0 0
\(991\) −56.0005 −1.77891 −0.889457 0.457019i \(-0.848917\pi\)
−0.889457 + 0.457019i \(0.848917\pi\)
\(992\) 0 0
\(993\) 4.93693 0.156669
\(994\) 0 0
\(995\) −19.1582 −0.607356
\(996\) 0 0
\(997\) 16.3839 0.518883 0.259441 0.965759i \(-0.416462\pi\)
0.259441 + 0.965759i \(0.416462\pi\)
\(998\) 0 0
\(999\) −20.0396 −0.634025
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 3344.2.a.u.1.5 5
4.3 odd 2 1672.2.a.g.1.1 5
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1672.2.a.g.1.1 5 4.3 odd 2
3344.2.a.u.1.5 5 1.1 even 1 trivial