Properties

Label 6020.2.a.k.1.5
Level $6020$
Weight $2$
Character 6020.1
Self dual yes
Analytic conductor $48.070$
Analytic rank $0$
Dimension $13$
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] = [6020,2,Mod(1,6020)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(6020, 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("6020.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 6020 = 2^{2} \cdot 5 \cdot 7 \cdot 43 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6020.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: \(48.0699420168\)
Analytic rank: \(0\)
Dimension: \(13\)
Coefficient field: \(\mathbb{Q}[x]/(x^{13} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{13} - 27 x^{11} - 2 x^{10} + 268 x^{9} + 37 x^{8} - 1201 x^{7} - 189 x^{6} + 2384 x^{5} + 231 x^{4} - 1729 x^{3} + 20 x^{2} + 105 x - 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.5
Root \(-1.30831\) of defining polynomial
Character \(\chi\) \(=\) 6020.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.30831 q^{3} +1.00000 q^{5} -1.00000 q^{7} -1.28831 q^{9} +O(q^{10})\) \(q-1.30831 q^{3} +1.00000 q^{5} -1.00000 q^{7} -1.28831 q^{9} +1.23892 q^{11} +2.90271 q^{13} -1.30831 q^{15} -6.12861 q^{17} -5.73312 q^{19} +1.30831 q^{21} +1.92684 q^{23} +1.00000 q^{25} +5.61046 q^{27} +7.54393 q^{29} -9.21855 q^{31} -1.62090 q^{33} -1.00000 q^{35} -7.93752 q^{37} -3.79766 q^{39} +1.47495 q^{41} +1.00000 q^{43} -1.28831 q^{45} +5.53769 q^{47} +1.00000 q^{49} +8.01815 q^{51} +11.4850 q^{53} +1.23892 q^{55} +7.50073 q^{57} -4.80965 q^{59} +1.61188 q^{61} +1.28831 q^{63} +2.90271 q^{65} -4.62612 q^{67} -2.52091 q^{69} +14.5579 q^{71} -3.30509 q^{73} -1.30831 q^{75} -1.23892 q^{77} +2.77804 q^{79} -3.47531 q^{81} -8.41048 q^{83} -6.12861 q^{85} -9.86983 q^{87} +16.1801 q^{89} -2.90271 q^{91} +12.0608 q^{93} -5.73312 q^{95} +1.64353 q^{97} -1.59612 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 13 q + 13 q^{5} - 13 q^{7} + 15 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 13 q + 13 q^{5} - 13 q^{7} + 15 q^{9} + 11 q^{13} + 16 q^{17} + 3 q^{23} + 13 q^{25} + 6 q^{27} + 10 q^{29} - q^{31} + 14 q^{33} - 13 q^{35} + 16 q^{37} - 14 q^{39} + 23 q^{41} + 13 q^{43} + 15 q^{45} + 2 q^{47} + 13 q^{49} + 4 q^{51} + 20 q^{53} + 22 q^{57} + 2 q^{59} + 5 q^{61} - 15 q^{63} + 11 q^{65} + 19 q^{67} + 16 q^{69} + 4 q^{71} + 34 q^{73} - 15 q^{79} + 17 q^{81} + 27 q^{83} + 16 q^{85} - 5 q^{87} + 3 q^{89} - 11 q^{91} + 35 q^{93} + 45 q^{97} + q^{99}+O(q^{100}) \) 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
\(3\) −1.30831 −0.755356 −0.377678 0.925937i \(-0.623277\pi\)
−0.377678 + 0.925937i \(0.623277\pi\)
\(4\) 0 0
\(5\) 1.00000 0.447214
\(6\) 0 0
\(7\) −1.00000 −0.377964
\(8\) 0 0
\(9\) −1.28831 −0.429438
\(10\) 0 0
\(11\) 1.23892 0.373548 0.186774 0.982403i \(-0.440197\pi\)
0.186774 + 0.982403i \(0.440197\pi\)
\(12\) 0 0
\(13\) 2.90271 0.805067 0.402534 0.915405i \(-0.368130\pi\)
0.402534 + 0.915405i \(0.368130\pi\)
\(14\) 0 0
\(15\) −1.30831 −0.337805
\(16\) 0 0
\(17\) −6.12861 −1.48641 −0.743203 0.669065i \(-0.766695\pi\)
−0.743203 + 0.669065i \(0.766695\pi\)
\(18\) 0 0
\(19\) −5.73312 −1.31527 −0.657634 0.753337i \(-0.728443\pi\)
−0.657634 + 0.753337i \(0.728443\pi\)
\(20\) 0 0
\(21\) 1.30831 0.285498
\(22\) 0 0
\(23\) 1.92684 0.401774 0.200887 0.979614i \(-0.435618\pi\)
0.200887 + 0.979614i \(0.435618\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) 5.61046 1.07973
\(28\) 0 0
\(29\) 7.54393 1.40087 0.700436 0.713715i \(-0.252989\pi\)
0.700436 + 0.713715i \(0.252989\pi\)
\(30\) 0 0
\(31\) −9.21855 −1.65570 −0.827850 0.560949i \(-0.810436\pi\)
−0.827850 + 0.560949i \(0.810436\pi\)
\(32\) 0 0
\(33\) −1.62090 −0.282162
\(34\) 0 0
\(35\) −1.00000 −0.169031
\(36\) 0 0
\(37\) −7.93752 −1.30492 −0.652460 0.757823i \(-0.726263\pi\)
−0.652460 + 0.757823i \(0.726263\pi\)
\(38\) 0 0
\(39\) −3.79766 −0.608112
\(40\) 0 0
\(41\) 1.47495 0.230349 0.115174 0.993345i \(-0.463257\pi\)
0.115174 + 0.993345i \(0.463257\pi\)
\(42\) 0 0
\(43\) 1.00000 0.152499
\(44\) 0 0
\(45\) −1.28831 −0.192050
\(46\) 0 0
\(47\) 5.53769 0.807755 0.403878 0.914813i \(-0.367662\pi\)
0.403878 + 0.914813i \(0.367662\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) 8.01815 1.12277
\(52\) 0 0
\(53\) 11.4850 1.57758 0.788790 0.614662i \(-0.210708\pi\)
0.788790 + 0.614662i \(0.210708\pi\)
\(54\) 0 0
\(55\) 1.23892 0.167056
\(56\) 0 0
\(57\) 7.50073 0.993496
\(58\) 0 0
\(59\) −4.80965 −0.626163 −0.313081 0.949726i \(-0.601361\pi\)
−0.313081 + 0.949726i \(0.601361\pi\)
\(60\) 0 0
\(61\) 1.61188 0.206381 0.103190 0.994662i \(-0.467095\pi\)
0.103190 + 0.994662i \(0.467095\pi\)
\(62\) 0 0
\(63\) 1.28831 0.162312
\(64\) 0 0
\(65\) 2.90271 0.360037
\(66\) 0 0
\(67\) −4.62612 −0.565171 −0.282585 0.959242i \(-0.591192\pi\)
−0.282585 + 0.959242i \(0.591192\pi\)
\(68\) 0 0
\(69\) −2.52091 −0.303482
\(70\) 0 0
\(71\) 14.5579 1.72771 0.863853 0.503745i \(-0.168045\pi\)
0.863853 + 0.503745i \(0.168045\pi\)
\(72\) 0 0
\(73\) −3.30509 −0.386832 −0.193416 0.981117i \(-0.561957\pi\)
−0.193416 + 0.981117i \(0.561957\pi\)
\(74\) 0 0
\(75\) −1.30831 −0.151071
\(76\) 0 0
\(77\) −1.23892 −0.141188
\(78\) 0 0
\(79\) 2.77804 0.312554 0.156277 0.987713i \(-0.450051\pi\)
0.156277 + 0.987713i \(0.450051\pi\)
\(80\) 0 0
\(81\) −3.47531 −0.386145
\(82\) 0 0
\(83\) −8.41048 −0.923170 −0.461585 0.887096i \(-0.652719\pi\)
−0.461585 + 0.887096i \(0.652719\pi\)
\(84\) 0 0
\(85\) −6.12861 −0.664741
\(86\) 0 0
\(87\) −9.86983 −1.05816
\(88\) 0 0
\(89\) 16.1801 1.71509 0.857544 0.514410i \(-0.171989\pi\)
0.857544 + 0.514410i \(0.171989\pi\)
\(90\) 0 0
\(91\) −2.90271 −0.304287
\(92\) 0 0
\(93\) 12.0608 1.25064
\(94\) 0 0
\(95\) −5.73312 −0.588206
\(96\) 0 0
\(97\) 1.64353 0.166875 0.0834376 0.996513i \(-0.473410\pi\)
0.0834376 + 0.996513i \(0.473410\pi\)
\(98\) 0 0
\(99\) −1.59612 −0.160416
\(100\) 0 0
\(101\) −13.1247 −1.30596 −0.652980 0.757375i \(-0.726482\pi\)
−0.652980 + 0.757375i \(0.726482\pi\)
\(102\) 0 0
\(103\) 0.157689 0.0155375 0.00776876 0.999970i \(-0.497527\pi\)
0.00776876 + 0.999970i \(0.497527\pi\)
\(104\) 0 0
\(105\) 1.30831 0.127678
\(106\) 0 0
\(107\) −17.9459 −1.73489 −0.867447 0.497529i \(-0.834241\pi\)
−0.867447 + 0.497529i \(0.834241\pi\)
\(108\) 0 0
\(109\) 3.86657 0.370350 0.185175 0.982706i \(-0.440715\pi\)
0.185175 + 0.982706i \(0.440715\pi\)
\(110\) 0 0
\(111\) 10.3848 0.985679
\(112\) 0 0
\(113\) 1.45169 0.136563 0.0682815 0.997666i \(-0.478248\pi\)
0.0682815 + 0.997666i \(0.478248\pi\)
\(114\) 0 0
\(115\) 1.92684 0.179679
\(116\) 0 0
\(117\) −3.73960 −0.345726
\(118\) 0 0
\(119\) 6.12861 0.561809
\(120\) 0 0
\(121\) −9.46508 −0.860462
\(122\) 0 0
\(123\) −1.92970 −0.173995
\(124\) 0 0
\(125\) 1.00000 0.0894427
\(126\) 0 0
\(127\) 12.7997 1.13579 0.567896 0.823101i \(-0.307758\pi\)
0.567896 + 0.823101i \(0.307758\pi\)
\(128\) 0 0
\(129\) −1.30831 −0.115191
\(130\) 0 0
\(131\) 18.4073 1.60825 0.804126 0.594458i \(-0.202633\pi\)
0.804126 + 0.594458i \(0.202633\pi\)
\(132\) 0 0
\(133\) 5.73312 0.497125
\(134\) 0 0
\(135\) 5.61046 0.482872
\(136\) 0 0
\(137\) 12.9856 1.10943 0.554716 0.832040i \(-0.312827\pi\)
0.554716 + 0.832040i \(0.312827\pi\)
\(138\) 0 0
\(139\) −8.05439 −0.683164 −0.341582 0.939852i \(-0.610963\pi\)
−0.341582 + 0.939852i \(0.610963\pi\)
\(140\) 0 0
\(141\) −7.24504 −0.610142
\(142\) 0 0
\(143\) 3.59623 0.300732
\(144\) 0 0
\(145\) 7.54393 0.626489
\(146\) 0 0
\(147\) −1.30831 −0.107908
\(148\) 0 0
\(149\) −3.80585 −0.311787 −0.155894 0.987774i \(-0.549826\pi\)
−0.155894 + 0.987774i \(0.549826\pi\)
\(150\) 0 0
\(151\) −4.28473 −0.348686 −0.174343 0.984685i \(-0.555780\pi\)
−0.174343 + 0.984685i \(0.555780\pi\)
\(152\) 0 0
\(153\) 7.89558 0.638319
\(154\) 0 0
\(155\) −9.21855 −0.740452
\(156\) 0 0
\(157\) 16.4427 1.31227 0.656135 0.754643i \(-0.272190\pi\)
0.656135 + 0.754643i \(0.272190\pi\)
\(158\) 0 0
\(159\) −15.0259 −1.19163
\(160\) 0 0
\(161\) −1.92684 −0.151856
\(162\) 0 0
\(163\) 13.4106 1.05040 0.525198 0.850980i \(-0.323991\pi\)
0.525198 + 0.850980i \(0.323991\pi\)
\(164\) 0 0
\(165\) −1.62090 −0.126187
\(166\) 0 0
\(167\) 6.57973 0.509155 0.254577 0.967052i \(-0.418064\pi\)
0.254577 + 0.967052i \(0.418064\pi\)
\(168\) 0 0
\(169\) −4.57426 −0.351867
\(170\) 0 0
\(171\) 7.38606 0.564826
\(172\) 0 0
\(173\) 21.3221 1.62109 0.810546 0.585675i \(-0.199170\pi\)
0.810546 + 0.585675i \(0.199170\pi\)
\(174\) 0 0
\(175\) −1.00000 −0.0755929
\(176\) 0 0
\(177\) 6.29253 0.472975
\(178\) 0 0
\(179\) −6.54369 −0.489098 −0.244549 0.969637i \(-0.578640\pi\)
−0.244549 + 0.969637i \(0.578640\pi\)
\(180\) 0 0
\(181\) −1.70779 −0.126939 −0.0634694 0.997984i \(-0.520217\pi\)
−0.0634694 + 0.997984i \(0.520217\pi\)
\(182\) 0 0
\(183\) −2.10885 −0.155891
\(184\) 0 0
\(185\) −7.93752 −0.583578
\(186\) 0 0
\(187\) −7.59286 −0.555245
\(188\) 0 0
\(189\) −5.61046 −0.408101
\(190\) 0 0
\(191\) −20.0481 −1.45063 −0.725315 0.688417i \(-0.758306\pi\)
−0.725315 + 0.688417i \(0.758306\pi\)
\(192\) 0 0
\(193\) 22.5384 1.62235 0.811174 0.584806i \(-0.198829\pi\)
0.811174 + 0.584806i \(0.198829\pi\)
\(194\) 0 0
\(195\) −3.79766 −0.271956
\(196\) 0 0
\(197\) 14.1729 1.00978 0.504889 0.863184i \(-0.331533\pi\)
0.504889 + 0.863184i \(0.331533\pi\)
\(198\) 0 0
\(199\) 4.48412 0.317871 0.158935 0.987289i \(-0.449194\pi\)
0.158935 + 0.987289i \(0.449194\pi\)
\(200\) 0 0
\(201\) 6.05242 0.426905
\(202\) 0 0
\(203\) −7.54393 −0.529480
\(204\) 0 0
\(205\) 1.47495 0.103015
\(206\) 0 0
\(207\) −2.48237 −0.172537
\(208\) 0 0
\(209\) −7.10288 −0.491317
\(210\) 0 0
\(211\) −6.29006 −0.433026 −0.216513 0.976280i \(-0.569468\pi\)
−0.216513 + 0.976280i \(0.569468\pi\)
\(212\) 0 0
\(213\) −19.0463 −1.30503
\(214\) 0 0
\(215\) 1.00000 0.0681994
\(216\) 0 0
\(217\) 9.21855 0.625796
\(218\) 0 0
\(219\) 4.32410 0.292196
\(220\) 0 0
\(221\) −17.7896 −1.19666
\(222\) 0 0
\(223\) 23.6695 1.58503 0.792515 0.609852i \(-0.208771\pi\)
0.792515 + 0.609852i \(0.208771\pi\)
\(224\) 0 0
\(225\) −1.28831 −0.0858876
\(226\) 0 0
\(227\) −22.3568 −1.48387 −0.741936 0.670471i \(-0.766092\pi\)
−0.741936 + 0.670471i \(0.766092\pi\)
\(228\) 0 0
\(229\) −6.64565 −0.439157 −0.219579 0.975595i \(-0.570468\pi\)
−0.219579 + 0.975595i \(0.570468\pi\)
\(230\) 0 0
\(231\) 1.62090 0.106647
\(232\) 0 0
\(233\) 3.71170 0.243162 0.121581 0.992582i \(-0.461204\pi\)
0.121581 + 0.992582i \(0.461204\pi\)
\(234\) 0 0
\(235\) 5.53769 0.361239
\(236\) 0 0
\(237\) −3.63455 −0.236089
\(238\) 0 0
\(239\) 30.3199 1.96123 0.980615 0.195945i \(-0.0627773\pi\)
0.980615 + 0.195945i \(0.0627773\pi\)
\(240\) 0 0
\(241\) 0.743833 0.0479145 0.0239572 0.999713i \(-0.492373\pi\)
0.0239572 + 0.999713i \(0.492373\pi\)
\(242\) 0 0
\(243\) −12.2846 −0.788057
\(244\) 0 0
\(245\) 1.00000 0.0638877
\(246\) 0 0
\(247\) −16.6416 −1.05888
\(248\) 0 0
\(249\) 11.0035 0.697322
\(250\) 0 0
\(251\) −1.41950 −0.0895980 −0.0447990 0.998996i \(-0.514265\pi\)
−0.0447990 + 0.998996i \(0.514265\pi\)
\(252\) 0 0
\(253\) 2.38720 0.150082
\(254\) 0 0
\(255\) 8.01815 0.502116
\(256\) 0 0
\(257\) −15.6806 −0.978126 −0.489063 0.872248i \(-0.662661\pi\)
−0.489063 + 0.872248i \(0.662661\pi\)
\(258\) 0 0
\(259\) 7.93752 0.493214
\(260\) 0 0
\(261\) −9.71895 −0.601588
\(262\) 0 0
\(263\) 6.81953 0.420510 0.210255 0.977647i \(-0.432571\pi\)
0.210255 + 0.977647i \(0.432571\pi\)
\(264\) 0 0
\(265\) 11.4850 0.705516
\(266\) 0 0
\(267\) −21.1687 −1.29550
\(268\) 0 0
\(269\) 13.1168 0.799747 0.399874 0.916570i \(-0.369054\pi\)
0.399874 + 0.916570i \(0.369054\pi\)
\(270\) 0 0
\(271\) −4.88076 −0.296485 −0.148242 0.988951i \(-0.547362\pi\)
−0.148242 + 0.988951i \(0.547362\pi\)
\(272\) 0 0
\(273\) 3.79766 0.229845
\(274\) 0 0
\(275\) 1.23892 0.0747097
\(276\) 0 0
\(277\) 10.6270 0.638517 0.319258 0.947668i \(-0.396566\pi\)
0.319258 + 0.947668i \(0.396566\pi\)
\(278\) 0 0
\(279\) 11.8764 0.711021
\(280\) 0 0
\(281\) −17.4670 −1.04200 −0.520998 0.853558i \(-0.674440\pi\)
−0.520998 + 0.853558i \(0.674440\pi\)
\(282\) 0 0
\(283\) 0.147036 0.00874038 0.00437019 0.999990i \(-0.498609\pi\)
0.00437019 + 0.999990i \(0.498609\pi\)
\(284\) 0 0
\(285\) 7.50073 0.444305
\(286\) 0 0
\(287\) −1.47495 −0.0870636
\(288\) 0 0
\(289\) 20.5599 1.20941
\(290\) 0 0
\(291\) −2.15025 −0.126050
\(292\) 0 0
\(293\) 4.60003 0.268737 0.134368 0.990931i \(-0.457099\pi\)
0.134368 + 0.990931i \(0.457099\pi\)
\(294\) 0 0
\(295\) −4.80965 −0.280028
\(296\) 0 0
\(297\) 6.95091 0.403333
\(298\) 0 0
\(299\) 5.59306 0.323455
\(300\) 0 0
\(301\) −1.00000 −0.0576390
\(302\) 0 0
\(303\) 17.1713 0.986465
\(304\) 0 0
\(305\) 1.61188 0.0922962
\(306\) 0 0
\(307\) −11.9599 −0.682589 −0.341295 0.939956i \(-0.610865\pi\)
−0.341295 + 0.939956i \(0.610865\pi\)
\(308\) 0 0
\(309\) −0.206306 −0.0117364
\(310\) 0 0
\(311\) 18.0828 1.02538 0.512690 0.858574i \(-0.328649\pi\)
0.512690 + 0.858574i \(0.328649\pi\)
\(312\) 0 0
\(313\) −1.38689 −0.0783916 −0.0391958 0.999232i \(-0.512480\pi\)
−0.0391958 + 0.999232i \(0.512480\pi\)
\(314\) 0 0
\(315\) 1.28831 0.0725883
\(316\) 0 0
\(317\) 20.5960 1.15679 0.578395 0.815757i \(-0.303679\pi\)
0.578395 + 0.815757i \(0.303679\pi\)
\(318\) 0 0
\(319\) 9.34633 0.523294
\(320\) 0 0
\(321\) 23.4789 1.31046
\(322\) 0 0
\(323\) 35.1361 1.95502
\(324\) 0 0
\(325\) 2.90271 0.161013
\(326\) 0 0
\(327\) −5.05869 −0.279746
\(328\) 0 0
\(329\) −5.53769 −0.305303
\(330\) 0 0
\(331\) −7.94575 −0.436738 −0.218369 0.975866i \(-0.570074\pi\)
−0.218369 + 0.975866i \(0.570074\pi\)
\(332\) 0 0
\(333\) 10.2260 0.560382
\(334\) 0 0
\(335\) −4.62612 −0.252752
\(336\) 0 0
\(337\) 33.0122 1.79829 0.899145 0.437650i \(-0.144189\pi\)
0.899145 + 0.437650i \(0.144189\pi\)
\(338\) 0 0
\(339\) −1.89926 −0.103154
\(340\) 0 0
\(341\) −11.4210 −0.618484
\(342\) 0 0
\(343\) −1.00000 −0.0539949
\(344\) 0 0
\(345\) −2.52091 −0.135721
\(346\) 0 0
\(347\) 24.5710 1.31904 0.659521 0.751686i \(-0.270759\pi\)
0.659521 + 0.751686i \(0.270759\pi\)
\(348\) 0 0
\(349\) 20.2276 1.08276 0.541379 0.840778i \(-0.317902\pi\)
0.541379 + 0.840778i \(0.317902\pi\)
\(350\) 0 0
\(351\) 16.2856 0.869259
\(352\) 0 0
\(353\) −1.72446 −0.0917839 −0.0458919 0.998946i \(-0.514613\pi\)
−0.0458919 + 0.998946i \(0.514613\pi\)
\(354\) 0 0
\(355\) 14.5579 0.772653
\(356\) 0 0
\(357\) −8.01815 −0.424366
\(358\) 0 0
\(359\) 2.47147 0.130439 0.0652195 0.997871i \(-0.479225\pi\)
0.0652195 + 0.997871i \(0.479225\pi\)
\(360\) 0 0
\(361\) 13.8687 0.729932
\(362\) 0 0
\(363\) 12.3833 0.649954
\(364\) 0 0
\(365\) −3.30509 −0.172996
\(366\) 0 0
\(367\) 34.4696 1.79930 0.899650 0.436612i \(-0.143822\pi\)
0.899650 + 0.436612i \(0.143822\pi\)
\(368\) 0 0
\(369\) −1.90020 −0.0989205
\(370\) 0 0
\(371\) −11.4850 −0.596270
\(372\) 0 0
\(373\) 35.2740 1.82642 0.913210 0.407490i \(-0.133596\pi\)
0.913210 + 0.407490i \(0.133596\pi\)
\(374\) 0 0
\(375\) −1.30831 −0.0675611
\(376\) 0 0
\(377\) 21.8979 1.12780
\(378\) 0 0
\(379\) −7.02417 −0.360807 −0.180404 0.983593i \(-0.557740\pi\)
−0.180404 + 0.983593i \(0.557740\pi\)
\(380\) 0 0
\(381\) −16.7461 −0.857926
\(382\) 0 0
\(383\) −34.4855 −1.76213 −0.881063 0.472999i \(-0.843172\pi\)
−0.881063 + 0.472999i \(0.843172\pi\)
\(384\) 0 0
\(385\) −1.23892 −0.0631412
\(386\) 0 0
\(387\) −1.28831 −0.0654887
\(388\) 0 0
\(389\) 14.9699 0.759004 0.379502 0.925191i \(-0.376095\pi\)
0.379502 + 0.925191i \(0.376095\pi\)
\(390\) 0 0
\(391\) −11.8088 −0.597199
\(392\) 0 0
\(393\) −24.0825 −1.21480
\(394\) 0 0
\(395\) 2.77804 0.139778
\(396\) 0 0
\(397\) 33.5107 1.68186 0.840928 0.541147i \(-0.182010\pi\)
0.840928 + 0.541147i \(0.182010\pi\)
\(398\) 0 0
\(399\) −7.50073 −0.375506
\(400\) 0 0
\(401\) 11.4759 0.573079 0.286539 0.958068i \(-0.407495\pi\)
0.286539 + 0.958068i \(0.407495\pi\)
\(402\) 0 0
\(403\) −26.7588 −1.33295
\(404\) 0 0
\(405\) −3.47531 −0.172689
\(406\) 0 0
\(407\) −9.83395 −0.487451
\(408\) 0 0
\(409\) 28.2267 1.39572 0.697861 0.716233i \(-0.254135\pi\)
0.697861 + 0.716233i \(0.254135\pi\)
\(410\) 0 0
\(411\) −16.9892 −0.838015
\(412\) 0 0
\(413\) 4.80965 0.236667
\(414\) 0 0
\(415\) −8.41048 −0.412854
\(416\) 0 0
\(417\) 10.5377 0.516032
\(418\) 0 0
\(419\) −32.6705 −1.59606 −0.798029 0.602619i \(-0.794124\pi\)
−0.798029 + 0.602619i \(0.794124\pi\)
\(420\) 0 0
\(421\) −12.1217 −0.590774 −0.295387 0.955378i \(-0.595449\pi\)
−0.295387 + 0.955378i \(0.595449\pi\)
\(422\) 0 0
\(423\) −7.13428 −0.346881
\(424\) 0 0
\(425\) −6.12861 −0.297281
\(426\) 0 0
\(427\) −1.61188 −0.0780046
\(428\) 0 0
\(429\) −4.70500 −0.227159
\(430\) 0 0
\(431\) −5.95544 −0.286864 −0.143432 0.989660i \(-0.545814\pi\)
−0.143432 + 0.989660i \(0.545814\pi\)
\(432\) 0 0
\(433\) 16.9181 0.813031 0.406515 0.913644i \(-0.366744\pi\)
0.406515 + 0.913644i \(0.366744\pi\)
\(434\) 0 0
\(435\) −9.86983 −0.473222
\(436\) 0 0
\(437\) −11.0468 −0.528440
\(438\) 0 0
\(439\) −2.35549 −0.112421 −0.0562107 0.998419i \(-0.517902\pi\)
−0.0562107 + 0.998419i \(0.517902\pi\)
\(440\) 0 0
\(441\) −1.28831 −0.0613483
\(442\) 0 0
\(443\) −7.11901 −0.338234 −0.169117 0.985596i \(-0.554092\pi\)
−0.169117 + 0.985596i \(0.554092\pi\)
\(444\) 0 0
\(445\) 16.1801 0.767011
\(446\) 0 0
\(447\) 4.97925 0.235510
\(448\) 0 0
\(449\) −20.9736 −0.989805 −0.494903 0.868949i \(-0.664796\pi\)
−0.494903 + 0.868949i \(0.664796\pi\)
\(450\) 0 0
\(451\) 1.82735 0.0860464
\(452\) 0 0
\(453\) 5.60578 0.263382
\(454\) 0 0
\(455\) −2.90271 −0.136081
\(456\) 0 0
\(457\) 18.8604 0.882251 0.441126 0.897445i \(-0.354579\pi\)
0.441126 + 0.897445i \(0.354579\pi\)
\(458\) 0 0
\(459\) −34.3843 −1.60492
\(460\) 0 0
\(461\) 13.4892 0.628253 0.314127 0.949381i \(-0.398288\pi\)
0.314127 + 0.949381i \(0.398288\pi\)
\(462\) 0 0
\(463\) −21.8444 −1.01520 −0.507598 0.861594i \(-0.669467\pi\)
−0.507598 + 0.861594i \(0.669467\pi\)
\(464\) 0 0
\(465\) 12.0608 0.559305
\(466\) 0 0
\(467\) −4.28941 −0.198490 −0.0992452 0.995063i \(-0.531643\pi\)
−0.0992452 + 0.995063i \(0.531643\pi\)
\(468\) 0 0
\(469\) 4.62612 0.213615
\(470\) 0 0
\(471\) −21.5122 −0.991231
\(472\) 0 0
\(473\) 1.23892 0.0569656
\(474\) 0 0
\(475\) −5.73312 −0.263054
\(476\) 0 0
\(477\) −14.7962 −0.677473
\(478\) 0 0
\(479\) 32.9883 1.50727 0.753637 0.657290i \(-0.228297\pi\)
0.753637 + 0.657290i \(0.228297\pi\)
\(480\) 0 0
\(481\) −23.0403 −1.05055
\(482\) 0 0
\(483\) 2.52091 0.114705
\(484\) 0 0
\(485\) 1.64353 0.0746288
\(486\) 0 0
\(487\) −4.09704 −0.185655 −0.0928273 0.995682i \(-0.529590\pi\)
−0.0928273 + 0.995682i \(0.529590\pi\)
\(488\) 0 0
\(489\) −17.5452 −0.793423
\(490\) 0 0
\(491\) 35.6073 1.60694 0.803468 0.595348i \(-0.202986\pi\)
0.803468 + 0.595348i \(0.202986\pi\)
\(492\) 0 0
\(493\) −46.2338 −2.08227
\(494\) 0 0
\(495\) −1.59612 −0.0717402
\(496\) 0 0
\(497\) −14.5579 −0.653011
\(498\) 0 0
\(499\) −13.9895 −0.626257 −0.313128 0.949711i \(-0.601377\pi\)
−0.313128 + 0.949711i \(0.601377\pi\)
\(500\) 0 0
\(501\) −8.60835 −0.384593
\(502\) 0 0
\(503\) 17.4420 0.777701 0.388850 0.921301i \(-0.372872\pi\)
0.388850 + 0.921301i \(0.372872\pi\)
\(504\) 0 0
\(505\) −13.1247 −0.584043
\(506\) 0 0
\(507\) 5.98458 0.265784
\(508\) 0 0
\(509\) 26.2919 1.16537 0.582683 0.812699i \(-0.302003\pi\)
0.582683 + 0.812699i \(0.302003\pi\)
\(510\) 0 0
\(511\) 3.30509 0.146209
\(512\) 0 0
\(513\) −32.1655 −1.42014
\(514\) 0 0
\(515\) 0.157689 0.00694859
\(516\) 0 0
\(517\) 6.86076 0.301736
\(518\) 0 0
\(519\) −27.8961 −1.22450
\(520\) 0 0
\(521\) −37.7842 −1.65536 −0.827679 0.561202i \(-0.810339\pi\)
−0.827679 + 0.561202i \(0.810339\pi\)
\(522\) 0 0
\(523\) 27.9099 1.22041 0.610207 0.792242i \(-0.291086\pi\)
0.610207 + 0.792242i \(0.291086\pi\)
\(524\) 0 0
\(525\) 1.30831 0.0570995
\(526\) 0 0
\(527\) 56.4969 2.46105
\(528\) 0 0
\(529\) −19.2873 −0.838578
\(530\) 0 0
\(531\) 6.19633 0.268898
\(532\) 0 0
\(533\) 4.28136 0.185446
\(534\) 0 0
\(535\) −17.9459 −0.775869
\(536\) 0 0
\(537\) 8.56120 0.369443
\(538\) 0 0
\(539\) 1.23892 0.0533641
\(540\) 0 0
\(541\) 15.7760 0.678263 0.339132 0.940739i \(-0.389867\pi\)
0.339132 + 0.940739i \(0.389867\pi\)
\(542\) 0 0
\(543\) 2.23432 0.0958840
\(544\) 0 0
\(545\) 3.86657 0.165626
\(546\) 0 0
\(547\) −20.0312 −0.856472 −0.428236 0.903667i \(-0.640865\pi\)
−0.428236 + 0.903667i \(0.640865\pi\)
\(548\) 0 0
\(549\) −2.07661 −0.0886277
\(550\) 0 0
\(551\) −43.2503 −1.84252
\(552\) 0 0
\(553\) −2.77804 −0.118134
\(554\) 0 0
\(555\) 10.3848 0.440809
\(556\) 0 0
\(557\) 26.4546 1.12092 0.560459 0.828182i \(-0.310625\pi\)
0.560459 + 0.828182i \(0.310625\pi\)
\(558\) 0 0
\(559\) 2.90271 0.122772
\(560\) 0 0
\(561\) 9.93385 0.419407
\(562\) 0 0
\(563\) −43.8319 −1.84729 −0.923647 0.383245i \(-0.874806\pi\)
−0.923647 + 0.383245i \(0.874806\pi\)
\(564\) 0 0
\(565\) 1.45169 0.0610729
\(566\) 0 0
\(567\) 3.47531 0.145949
\(568\) 0 0
\(569\) −15.0973 −0.632912 −0.316456 0.948607i \(-0.602493\pi\)
−0.316456 + 0.948607i \(0.602493\pi\)
\(570\) 0 0
\(571\) −22.0890 −0.924395 −0.462197 0.886777i \(-0.652939\pi\)
−0.462197 + 0.886777i \(0.652939\pi\)
\(572\) 0 0
\(573\) 26.2292 1.09574
\(574\) 0 0
\(575\) 1.92684 0.0803547
\(576\) 0 0
\(577\) −27.6953 −1.15297 −0.576485 0.817107i \(-0.695576\pi\)
−0.576485 + 0.817107i \(0.695576\pi\)
\(578\) 0 0
\(579\) −29.4873 −1.22545
\(580\) 0 0
\(581\) 8.41048 0.348925
\(582\) 0 0
\(583\) 14.2289 0.589303
\(584\) 0 0
\(585\) −3.73960 −0.154614
\(586\) 0 0
\(587\) 41.1716 1.69933 0.849667 0.527320i \(-0.176803\pi\)
0.849667 + 0.527320i \(0.176803\pi\)
\(588\) 0 0
\(589\) 52.8511 2.17769
\(590\) 0 0
\(591\) −18.5426 −0.762742
\(592\) 0 0
\(593\) −6.00108 −0.246435 −0.123217 0.992380i \(-0.539321\pi\)
−0.123217 + 0.992380i \(0.539321\pi\)
\(594\) 0 0
\(595\) 6.12861 0.251249
\(596\) 0 0
\(597\) −5.86664 −0.240106
\(598\) 0 0
\(599\) 41.1384 1.68087 0.840436 0.541911i \(-0.182299\pi\)
0.840436 + 0.541911i \(0.182299\pi\)
\(600\) 0 0
\(601\) 8.18862 0.334021 0.167010 0.985955i \(-0.446589\pi\)
0.167010 + 0.985955i \(0.446589\pi\)
\(602\) 0 0
\(603\) 5.95990 0.242706
\(604\) 0 0
\(605\) −9.46508 −0.384810
\(606\) 0 0
\(607\) 2.93184 0.119000 0.0594998 0.998228i \(-0.481049\pi\)
0.0594998 + 0.998228i \(0.481049\pi\)
\(608\) 0 0
\(609\) 9.86983 0.399946
\(610\) 0 0
\(611\) 16.0743 0.650297
\(612\) 0 0
\(613\) 6.90975 0.279082 0.139541 0.990216i \(-0.455437\pi\)
0.139541 + 0.990216i \(0.455437\pi\)
\(614\) 0 0
\(615\) −1.92970 −0.0778130
\(616\) 0 0
\(617\) 17.0412 0.686055 0.343027 0.939325i \(-0.388548\pi\)
0.343027 + 0.939325i \(0.388548\pi\)
\(618\) 0 0
\(619\) 19.1903 0.771323 0.385661 0.922640i \(-0.373973\pi\)
0.385661 + 0.922640i \(0.373973\pi\)
\(620\) 0 0
\(621\) 10.8105 0.433809
\(622\) 0 0
\(623\) −16.1801 −0.648243
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) 9.29280 0.371119
\(628\) 0 0
\(629\) 48.6460 1.93964
\(630\) 0 0
\(631\) −37.3435 −1.48662 −0.743311 0.668946i \(-0.766746\pi\)
−0.743311 + 0.668946i \(0.766746\pi\)
\(632\) 0 0
\(633\) 8.22938 0.327088
\(634\) 0 0
\(635\) 12.7997 0.507941
\(636\) 0 0
\(637\) 2.90271 0.115010
\(638\) 0 0
\(639\) −18.7551 −0.741942
\(640\) 0 0
\(641\) −19.7505 −0.780099 −0.390050 0.920794i \(-0.627542\pi\)
−0.390050 + 0.920794i \(0.627542\pi\)
\(642\) 0 0
\(643\) 38.2547 1.50862 0.754309 0.656519i \(-0.227972\pi\)
0.754309 + 0.656519i \(0.227972\pi\)
\(644\) 0 0
\(645\) −1.30831 −0.0515148
\(646\) 0 0
\(647\) −17.4389 −0.685595 −0.342797 0.939409i \(-0.611374\pi\)
−0.342797 + 0.939409i \(0.611374\pi\)
\(648\) 0 0
\(649\) −5.95877 −0.233902
\(650\) 0 0
\(651\) −12.0608 −0.472699
\(652\) 0 0
\(653\) 29.3538 1.14870 0.574352 0.818609i \(-0.305254\pi\)
0.574352 + 0.818609i \(0.305254\pi\)
\(654\) 0 0
\(655\) 18.4073 0.719233
\(656\) 0 0
\(657\) 4.25800 0.166120
\(658\) 0 0
\(659\) 21.6571 0.843640 0.421820 0.906680i \(-0.361391\pi\)
0.421820 + 0.906680i \(0.361391\pi\)
\(660\) 0 0
\(661\) 35.8839 1.39572 0.697861 0.716233i \(-0.254135\pi\)
0.697861 + 0.716233i \(0.254135\pi\)
\(662\) 0 0
\(663\) 23.2744 0.903902
\(664\) 0 0
\(665\) 5.73312 0.222321
\(666\) 0 0
\(667\) 14.5359 0.562834
\(668\) 0 0
\(669\) −30.9672 −1.19726
\(670\) 0 0
\(671\) 1.99700 0.0770932
\(672\) 0 0
\(673\) 47.5338 1.83229 0.916147 0.400844i \(-0.131283\pi\)
0.916147 + 0.400844i \(0.131283\pi\)
\(674\) 0 0
\(675\) 5.61046 0.215947
\(676\) 0 0
\(677\) −6.05477 −0.232704 −0.116352 0.993208i \(-0.537120\pi\)
−0.116352 + 0.993208i \(0.537120\pi\)
\(678\) 0 0
\(679\) −1.64353 −0.0630729
\(680\) 0 0
\(681\) 29.2497 1.12085
\(682\) 0 0
\(683\) −8.80842 −0.337045 −0.168522 0.985698i \(-0.553900\pi\)
−0.168522 + 0.985698i \(0.553900\pi\)
\(684\) 0 0
\(685\) 12.9856 0.496153
\(686\) 0 0
\(687\) 8.69460 0.331720
\(688\) 0 0
\(689\) 33.3375 1.27006
\(690\) 0 0
\(691\) −36.1107 −1.37372 −0.686858 0.726791i \(-0.741011\pi\)
−0.686858 + 0.726791i \(0.741011\pi\)
\(692\) 0 0
\(693\) 1.59612 0.0606315
\(694\) 0 0
\(695\) −8.05439 −0.305520
\(696\) 0 0
\(697\) −9.03941 −0.342392
\(698\) 0 0
\(699\) −4.85608 −0.183674
\(700\) 0 0
\(701\) −41.7831 −1.57813 −0.789063 0.614312i \(-0.789434\pi\)
−0.789063 + 0.614312i \(0.789434\pi\)
\(702\) 0 0
\(703\) 45.5068 1.71632
\(704\) 0 0
\(705\) −7.24504 −0.272864
\(706\) 0 0
\(707\) 13.1247 0.493607
\(708\) 0 0
\(709\) 4.71335 0.177014 0.0885068 0.996076i \(-0.471790\pi\)
0.0885068 + 0.996076i \(0.471790\pi\)
\(710\) 0 0
\(711\) −3.57898 −0.134222
\(712\) 0 0
\(713\) −17.7627 −0.665217
\(714\) 0 0
\(715\) 3.59623 0.134491
\(716\) 0 0
\(717\) −39.6679 −1.48143
\(718\) 0 0
\(719\) −6.63400 −0.247407 −0.123703 0.992319i \(-0.539477\pi\)
−0.123703 + 0.992319i \(0.539477\pi\)
\(720\) 0 0
\(721\) −0.157689 −0.00587263
\(722\) 0 0
\(723\) −0.973167 −0.0361925
\(724\) 0 0
\(725\) 7.54393 0.280175
\(726\) 0 0
\(727\) −40.1827 −1.49029 −0.745147 0.666900i \(-0.767621\pi\)
−0.745147 + 0.666900i \(0.767621\pi\)
\(728\) 0 0
\(729\) 26.4980 0.981408
\(730\) 0 0
\(731\) −6.12861 −0.226675
\(732\) 0 0
\(733\) −10.5151 −0.388382 −0.194191 0.980964i \(-0.562208\pi\)
−0.194191 + 0.980964i \(0.562208\pi\)
\(734\) 0 0
\(735\) −1.30831 −0.0482579
\(736\) 0 0
\(737\) −5.73140 −0.211119
\(738\) 0 0
\(739\) 35.0044 1.28766 0.643829 0.765169i \(-0.277345\pi\)
0.643829 + 0.765169i \(0.277345\pi\)
\(740\) 0 0
\(741\) 21.7724 0.799831
\(742\) 0 0
\(743\) −19.4515 −0.713606 −0.356803 0.934180i \(-0.616133\pi\)
−0.356803 + 0.934180i \(0.616133\pi\)
\(744\) 0 0
\(745\) −3.80585 −0.139435
\(746\) 0 0
\(747\) 10.8353 0.396444
\(748\) 0 0
\(749\) 17.9459 0.655729
\(750\) 0 0
\(751\) −54.2810 −1.98074 −0.990372 0.138435i \(-0.955793\pi\)
−0.990372 + 0.138435i \(0.955793\pi\)
\(752\) 0 0
\(753\) 1.85715 0.0676784
\(754\) 0 0
\(755\) −4.28473 −0.155937
\(756\) 0 0
\(757\) 31.9821 1.16241 0.581205 0.813757i \(-0.302582\pi\)
0.581205 + 0.813757i \(0.302582\pi\)
\(758\) 0 0
\(759\) −3.12321 −0.113365
\(760\) 0 0
\(761\) −2.47608 −0.0897577 −0.0448789 0.998992i \(-0.514290\pi\)
−0.0448789 + 0.998992i \(0.514290\pi\)
\(762\) 0 0
\(763\) −3.86657 −0.139979
\(764\) 0 0
\(765\) 7.89558 0.285465
\(766\) 0 0
\(767\) −13.9610 −0.504103
\(768\) 0 0
\(769\) −0.599916 −0.0216335 −0.0108168 0.999941i \(-0.503443\pi\)
−0.0108168 + 0.999941i \(0.503443\pi\)
\(770\) 0 0
\(771\) 20.5151 0.738833
\(772\) 0 0
\(773\) 0.234610 0.00843834 0.00421917 0.999991i \(-0.498657\pi\)
0.00421917 + 0.999991i \(0.498657\pi\)
\(774\) 0 0
\(775\) −9.21855 −0.331140
\(776\) 0 0
\(777\) −10.3848 −0.372552
\(778\) 0 0
\(779\) −8.45608 −0.302970
\(780\) 0 0
\(781\) 18.0361 0.645381
\(782\) 0 0
\(783\) 42.3249 1.51257
\(784\) 0 0
\(785\) 16.4427 0.586865
\(786\) 0 0
\(787\) −9.59722 −0.342104 −0.171052 0.985262i \(-0.554717\pi\)
−0.171052 + 0.985262i \(0.554717\pi\)
\(788\) 0 0
\(789\) −8.92209 −0.317635
\(790\) 0 0
\(791\) −1.45169 −0.0516160
\(792\) 0 0
\(793\) 4.67884 0.166150
\(794\) 0 0
\(795\) −15.0259 −0.532915
\(796\) 0 0
\(797\) −24.9670 −0.884377 −0.442188 0.896922i \(-0.645798\pi\)
−0.442188 + 0.896922i \(0.645798\pi\)
\(798\) 0 0
\(799\) −33.9384 −1.20065
\(800\) 0 0
\(801\) −20.8451 −0.736524
\(802\) 0 0
\(803\) −4.09474 −0.144500
\(804\) 0 0
\(805\) −1.92684 −0.0679121
\(806\) 0 0
\(807\) −17.1609 −0.604094
\(808\) 0 0
\(809\) −15.1099 −0.531237 −0.265618 0.964078i \(-0.585576\pi\)
−0.265618 + 0.964078i \(0.585576\pi\)
\(810\) 0 0
\(811\) −15.2016 −0.533799 −0.266899 0.963724i \(-0.585999\pi\)
−0.266899 + 0.963724i \(0.585999\pi\)
\(812\) 0 0
\(813\) 6.38556 0.223952
\(814\) 0 0
\(815\) 13.4106 0.469751
\(816\) 0 0
\(817\) −5.73312 −0.200577
\(818\) 0 0
\(819\) 3.73960 0.130672
\(820\) 0 0
\(821\) 43.9370 1.53341 0.766706 0.641999i \(-0.221895\pi\)
0.766706 + 0.641999i \(0.221895\pi\)
\(822\) 0 0
\(823\) 18.5188 0.645523 0.322762 0.946480i \(-0.395389\pi\)
0.322762 + 0.946480i \(0.395389\pi\)
\(824\) 0 0
\(825\) −1.62090 −0.0564324
\(826\) 0 0
\(827\) 26.0560 0.906056 0.453028 0.891496i \(-0.350344\pi\)
0.453028 + 0.891496i \(0.350344\pi\)
\(828\) 0 0
\(829\) 5.02890 0.174661 0.0873304 0.996179i \(-0.472166\pi\)
0.0873304 + 0.996179i \(0.472166\pi\)
\(830\) 0 0
\(831\) −13.9035 −0.482307
\(832\) 0 0
\(833\) −6.12861 −0.212344
\(834\) 0 0
\(835\) 6.57973 0.227701
\(836\) 0 0
\(837\) −51.7203 −1.78772
\(838\) 0 0
\(839\) 2.62865 0.0907512 0.0453756 0.998970i \(-0.485552\pi\)
0.0453756 + 0.998970i \(0.485552\pi\)
\(840\) 0 0
\(841\) 27.9109 0.962444
\(842\) 0 0
\(843\) 22.8524 0.787078
\(844\) 0 0
\(845\) −4.57426 −0.157359
\(846\) 0 0
\(847\) 9.46508 0.325224
\(848\) 0 0
\(849\) −0.192369 −0.00660210
\(850\) 0 0
\(851\) −15.2943 −0.524282
\(852\) 0 0
\(853\) −32.2934 −1.10570 −0.552852 0.833280i \(-0.686460\pi\)
−0.552852 + 0.833280i \(0.686460\pi\)
\(854\) 0 0
\(855\) 7.38606 0.252598
\(856\) 0 0
\(857\) −30.8305 −1.05315 −0.526575 0.850129i \(-0.676524\pi\)
−0.526575 + 0.850129i \(0.676524\pi\)
\(858\) 0 0
\(859\) −27.1500 −0.926346 −0.463173 0.886268i \(-0.653289\pi\)
−0.463173 + 0.886268i \(0.653289\pi\)
\(860\) 0 0
\(861\) 1.92970 0.0657640
\(862\) 0 0
\(863\) 56.8854 1.93640 0.968201 0.250172i \(-0.0804873\pi\)
0.968201 + 0.250172i \(0.0804873\pi\)
\(864\) 0 0
\(865\) 21.3221 0.724975
\(866\) 0 0
\(867\) −26.8988 −0.913531
\(868\) 0 0
\(869\) 3.44177 0.116754
\(870\) 0 0
\(871\) −13.4283 −0.455001
\(872\) 0 0
\(873\) −2.11738 −0.0716625
\(874\) 0 0
\(875\) −1.00000 −0.0338062
\(876\) 0 0
\(877\) 22.8602 0.771934 0.385967 0.922513i \(-0.373868\pi\)
0.385967 + 0.922513i \(0.373868\pi\)
\(878\) 0 0
\(879\) −6.01828 −0.202992
\(880\) 0 0
\(881\) 4.08133 0.137504 0.0687518 0.997634i \(-0.478098\pi\)
0.0687518 + 0.997634i \(0.478098\pi\)
\(882\) 0 0
\(883\) −11.5304 −0.388030 −0.194015 0.980999i \(-0.562151\pi\)
−0.194015 + 0.980999i \(0.562151\pi\)
\(884\) 0 0
\(885\) 6.29253 0.211521
\(886\) 0 0
\(887\) −2.25829 −0.0758261 −0.0379131 0.999281i \(-0.512071\pi\)
−0.0379131 + 0.999281i \(0.512071\pi\)
\(888\) 0 0
\(889\) −12.7997 −0.429289
\(890\) 0 0
\(891\) −4.30563 −0.144244
\(892\) 0 0
\(893\) −31.7483 −1.06242
\(894\) 0 0
\(895\) −6.54369 −0.218731
\(896\) 0 0
\(897\) −7.31747 −0.244323
\(898\) 0 0
\(899\) −69.5441 −2.31943
\(900\) 0 0
\(901\) −70.3869 −2.34493
\(902\) 0 0
\(903\) 1.30831 0.0435380
\(904\) 0 0
\(905\) −1.70779 −0.0567688
\(906\) 0 0
\(907\) −41.8040 −1.38808 −0.694040 0.719937i \(-0.744171\pi\)
−0.694040 + 0.719937i \(0.744171\pi\)
\(908\) 0 0
\(909\) 16.9088 0.560829
\(910\) 0 0
\(911\) 32.3597 1.07212 0.536062 0.844179i \(-0.319911\pi\)
0.536062 + 0.844179i \(0.319911\pi\)
\(912\) 0 0
\(913\) −10.4199 −0.344849
\(914\) 0 0
\(915\) −2.10885 −0.0697165
\(916\) 0 0
\(917\) −18.4073 −0.607862
\(918\) 0 0
\(919\) 18.7266 0.617733 0.308867 0.951105i \(-0.400050\pi\)
0.308867 + 0.951105i \(0.400050\pi\)
\(920\) 0 0
\(921\) 15.6473 0.515598
\(922\) 0 0
\(923\) 42.2574 1.39092
\(924\) 0 0
\(925\) −7.93752 −0.260984
\(926\) 0 0
\(927\) −0.203152 −0.00667240
\(928\) 0 0
\(929\) −40.2370 −1.32013 −0.660067 0.751206i \(-0.729472\pi\)
−0.660067 + 0.751206i \(0.729472\pi\)
\(930\) 0 0
\(931\) −5.73312 −0.187896
\(932\) 0 0
\(933\) −23.6580 −0.774527
\(934\) 0 0
\(935\) −7.59286 −0.248313
\(936\) 0 0
\(937\) −14.7895 −0.483152 −0.241576 0.970382i \(-0.577664\pi\)
−0.241576 + 0.970382i \(0.577664\pi\)
\(938\) 0 0
\(939\) 1.81449 0.0592136
\(940\) 0 0
\(941\) −32.4761 −1.05869 −0.529346 0.848406i \(-0.677563\pi\)
−0.529346 + 0.848406i \(0.677563\pi\)
\(942\) 0 0
\(943\) 2.84199 0.0925480
\(944\) 0 0
\(945\) −5.61046 −0.182508
\(946\) 0 0
\(947\) −25.3125 −0.822546 −0.411273 0.911512i \(-0.634916\pi\)
−0.411273 + 0.911512i \(0.634916\pi\)
\(948\) 0 0
\(949\) −9.59373 −0.311426
\(950\) 0 0
\(951\) −26.9461 −0.873787
\(952\) 0 0
\(953\) 22.3834 0.725068 0.362534 0.931970i \(-0.381912\pi\)
0.362534 + 0.931970i \(0.381912\pi\)
\(954\) 0 0
\(955\) −20.0481 −0.648742
\(956\) 0 0
\(957\) −12.2279 −0.395273
\(958\) 0 0
\(959\) −12.9856 −0.419326
\(960\) 0 0
\(961\) 53.9817 1.74135
\(962\) 0 0
\(963\) 23.1199 0.745030
\(964\) 0 0
\(965\) 22.5384 0.725536
\(966\) 0 0
\(967\) −8.68590 −0.279320 −0.139660 0.990200i \(-0.544601\pi\)
−0.139660 + 0.990200i \(0.544601\pi\)
\(968\) 0 0
\(969\) −45.9691 −1.47674
\(970\) 0 0
\(971\) 29.4274 0.944371 0.472185 0.881499i \(-0.343465\pi\)
0.472185 + 0.881499i \(0.343465\pi\)
\(972\) 0 0
\(973\) 8.05439 0.258212
\(974\) 0 0
\(975\) −3.79766 −0.121622
\(976\) 0 0
\(977\) −31.3083 −1.00164 −0.500820 0.865551i \(-0.666968\pi\)
−0.500820 + 0.865551i \(0.666968\pi\)
\(978\) 0 0
\(979\) 20.0459 0.640669
\(980\) 0 0
\(981\) −4.98136 −0.159043
\(982\) 0 0
\(983\) −33.2231 −1.05965 −0.529827 0.848106i \(-0.677743\pi\)
−0.529827 + 0.848106i \(0.677743\pi\)
\(984\) 0 0
\(985\) 14.1729 0.451587
\(986\) 0 0
\(987\) 7.24504 0.230612
\(988\) 0 0
\(989\) 1.92684 0.0612699
\(990\) 0 0
\(991\) −12.8580 −0.408448 −0.204224 0.978924i \(-0.565467\pi\)
−0.204224 + 0.978924i \(0.565467\pi\)
\(992\) 0 0
\(993\) 10.3955 0.329892
\(994\) 0 0
\(995\) 4.48412 0.142156
\(996\) 0 0
\(997\) −40.9955 −1.29834 −0.649170 0.760643i \(-0.724884\pi\)
−0.649170 + 0.760643i \(0.724884\pi\)
\(998\) 0 0
\(999\) −44.5332 −1.40897
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 6020.2.a.k.1.5 13
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
6020.2.a.k.1.5 13 1.1 even 1 trivial