Properties

Label 2240.2.a.bj.1.1
Level $2240$
Weight $2$
Character 2240.1
Self dual yes
Analytic conductor $17.886$
Analytic rank $0$
Dimension $2$
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] = [2240,2,Mod(1,2240)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2240, 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("2240.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2240 = 2^{6} \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2240.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: \(17.8864900528\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{17}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 4 \) 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 1120)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-1.56155\) of defining polynomial
Character \(\chi\) \(=\) 2240.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.56155 q^{3} -1.00000 q^{5} +1.00000 q^{7} -0.561553 q^{9} +O(q^{10})\) \(q-1.56155 q^{3} -1.00000 q^{5} +1.00000 q^{7} -0.561553 q^{9} -1.56155 q^{11} -3.56155 q^{13} +1.56155 q^{15} +0.438447 q^{17} -1.56155 q^{21} +1.00000 q^{25} +5.56155 q^{27} -6.68466 q^{29} +3.12311 q^{31} +2.43845 q^{33} -1.00000 q^{35} +1.12311 q^{37} +5.56155 q^{39} +2.00000 q^{41} +4.00000 q^{43} +0.561553 q^{45} -0.684658 q^{47} +1.00000 q^{49} -0.684658 q^{51} -13.1231 q^{53} +1.56155 q^{55} +6.24621 q^{59} +2.00000 q^{61} -0.561553 q^{63} +3.56155 q^{65} +10.2462 q^{67} +4.00000 q^{71} -0.246211 q^{73} -1.56155 q^{75} -1.56155 q^{77} -7.80776 q^{79} -7.00000 q^{81} +10.2462 q^{83} -0.438447 q^{85} +10.4384 q^{87} -4.24621 q^{89} -3.56155 q^{91} -4.87689 q^{93} +12.9309 q^{97} +0.876894 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + q^{3} - 2 q^{5} + 2 q^{7} + 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + q^{3} - 2 q^{5} + 2 q^{7} + 3 q^{9} + q^{11} - 3 q^{13} - q^{15} + 5 q^{17} + q^{21} + 2 q^{25} + 7 q^{27} - q^{29} - 2 q^{31} + 9 q^{33} - 2 q^{35} - 6 q^{37} + 7 q^{39} + 4 q^{41} + 8 q^{43} - 3 q^{45} + 11 q^{47} + 2 q^{49} + 11 q^{51} - 18 q^{53} - q^{55} - 4 q^{59} + 4 q^{61} + 3 q^{63} + 3 q^{65} + 4 q^{67} + 8 q^{71} + 16 q^{73} + q^{75} + q^{77} + 5 q^{79} - 14 q^{81} + 4 q^{83} - 5 q^{85} + 25 q^{87} + 8 q^{89} - 3 q^{91} - 18 q^{93} - 3 q^{97} + 10 q^{99}+O(q^{100}) \) 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
\(3\) −1.56155 −0.901563 −0.450781 0.892634i \(-0.648855\pi\)
−0.450781 + 0.892634i \(0.648855\pi\)
\(4\) 0 0
\(5\) −1.00000 −0.447214
\(6\) 0 0
\(7\) 1.00000 0.377964
\(8\) 0 0
\(9\) −0.561553 −0.187184
\(10\) 0 0
\(11\) −1.56155 −0.470826 −0.235413 0.971895i \(-0.575644\pi\)
−0.235413 + 0.971895i \(0.575644\pi\)
\(12\) 0 0
\(13\) −3.56155 −0.987797 −0.493899 0.869520i \(-0.664429\pi\)
−0.493899 + 0.869520i \(0.664429\pi\)
\(14\) 0 0
\(15\) 1.56155 0.403191
\(16\) 0 0
\(17\) 0.438447 0.106339 0.0531695 0.998586i \(-0.483068\pi\)
0.0531695 + 0.998586i \(0.483068\pi\)
\(18\) 0 0
\(19\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(20\) 0 0
\(21\) −1.56155 −0.340759
\(22\) 0 0
\(23\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) 5.56155 1.07032
\(28\) 0 0
\(29\) −6.68466 −1.24131 −0.620655 0.784084i \(-0.713133\pi\)
−0.620655 + 0.784084i \(0.713133\pi\)
\(30\) 0 0
\(31\) 3.12311 0.560926 0.280463 0.959865i \(-0.409512\pi\)
0.280463 + 0.959865i \(0.409512\pi\)
\(32\) 0 0
\(33\) 2.43845 0.424479
\(34\) 0 0
\(35\) −1.00000 −0.169031
\(36\) 0 0
\(37\) 1.12311 0.184637 0.0923187 0.995730i \(-0.470572\pi\)
0.0923187 + 0.995730i \(0.470572\pi\)
\(38\) 0 0
\(39\) 5.56155 0.890561
\(40\) 0 0
\(41\) 2.00000 0.312348 0.156174 0.987730i \(-0.450084\pi\)
0.156174 + 0.987730i \(0.450084\pi\)
\(42\) 0 0
\(43\) 4.00000 0.609994 0.304997 0.952353i \(-0.401344\pi\)
0.304997 + 0.952353i \(0.401344\pi\)
\(44\) 0 0
\(45\) 0.561553 0.0837114
\(46\) 0 0
\(47\) −0.684658 −0.0998677 −0.0499338 0.998753i \(-0.515901\pi\)
−0.0499338 + 0.998753i \(0.515901\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) −0.684658 −0.0958714
\(52\) 0 0
\(53\) −13.1231 −1.80260 −0.901299 0.433198i \(-0.857385\pi\)
−0.901299 + 0.433198i \(0.857385\pi\)
\(54\) 0 0
\(55\) 1.56155 0.210560
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 6.24621 0.813187 0.406594 0.913609i \(-0.366716\pi\)
0.406594 + 0.913609i \(0.366716\pi\)
\(60\) 0 0
\(61\) 2.00000 0.256074 0.128037 0.991769i \(-0.459132\pi\)
0.128037 + 0.991769i \(0.459132\pi\)
\(62\) 0 0
\(63\) −0.561553 −0.0707490
\(64\) 0 0
\(65\) 3.56155 0.441756
\(66\) 0 0
\(67\) 10.2462 1.25177 0.625887 0.779914i \(-0.284737\pi\)
0.625887 + 0.779914i \(0.284737\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 4.00000 0.474713 0.237356 0.971423i \(-0.423719\pi\)
0.237356 + 0.971423i \(0.423719\pi\)
\(72\) 0 0
\(73\) −0.246211 −0.0288168 −0.0144084 0.999896i \(-0.504587\pi\)
−0.0144084 + 0.999896i \(0.504587\pi\)
\(74\) 0 0
\(75\) −1.56155 −0.180313
\(76\) 0 0
\(77\) −1.56155 −0.177955
\(78\) 0 0
\(79\) −7.80776 −0.878442 −0.439221 0.898379i \(-0.644746\pi\)
−0.439221 + 0.898379i \(0.644746\pi\)
\(80\) 0 0
\(81\) −7.00000 −0.777778
\(82\) 0 0
\(83\) 10.2462 1.12467 0.562334 0.826910i \(-0.309904\pi\)
0.562334 + 0.826910i \(0.309904\pi\)
\(84\) 0 0
\(85\) −0.438447 −0.0475563
\(86\) 0 0
\(87\) 10.4384 1.11912
\(88\) 0 0
\(89\) −4.24621 −0.450097 −0.225049 0.974348i \(-0.572254\pi\)
−0.225049 + 0.974348i \(0.572254\pi\)
\(90\) 0 0
\(91\) −3.56155 −0.373352
\(92\) 0 0
\(93\) −4.87689 −0.505710
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 12.9309 1.31293 0.656465 0.754356i \(-0.272051\pi\)
0.656465 + 0.754356i \(0.272051\pi\)
\(98\) 0 0
\(99\) 0.876894 0.0881312
\(100\) 0 0
\(101\) 11.3693 1.13129 0.565645 0.824649i \(-0.308627\pi\)
0.565645 + 0.824649i \(0.308627\pi\)
\(102\) 0 0
\(103\) 7.31534 0.720802 0.360401 0.932797i \(-0.382640\pi\)
0.360401 + 0.932797i \(0.382640\pi\)
\(104\) 0 0
\(105\) 1.56155 0.152392
\(106\) 0 0
\(107\) 15.1231 1.46201 0.731003 0.682374i \(-0.239053\pi\)
0.731003 + 0.682374i \(0.239053\pi\)
\(108\) 0 0
\(109\) −9.80776 −0.939413 −0.469707 0.882823i \(-0.655640\pi\)
−0.469707 + 0.882823i \(0.655640\pi\)
\(110\) 0 0
\(111\) −1.75379 −0.166462
\(112\) 0 0
\(113\) 19.3693 1.82211 0.911056 0.412283i \(-0.135268\pi\)
0.911056 + 0.412283i \(0.135268\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 2.00000 0.184900
\(118\) 0 0
\(119\) 0.438447 0.0401924
\(120\) 0 0
\(121\) −8.56155 −0.778323
\(122\) 0 0
\(123\) −3.12311 −0.281601
\(124\) 0 0
\(125\) −1.00000 −0.0894427
\(126\) 0 0
\(127\) −6.24621 −0.554262 −0.277131 0.960832i \(-0.589384\pi\)
−0.277131 + 0.960832i \(0.589384\pi\)
\(128\) 0 0
\(129\) −6.24621 −0.549948
\(130\) 0 0
\(131\) −14.2462 −1.24470 −0.622349 0.782740i \(-0.713821\pi\)
−0.622349 + 0.782740i \(0.713821\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) −5.56155 −0.478662
\(136\) 0 0
\(137\) 0.246211 0.0210352 0.0105176 0.999945i \(-0.496652\pi\)
0.0105176 + 0.999945i \(0.496652\pi\)
\(138\) 0 0
\(139\) 19.1231 1.62200 0.811000 0.585046i \(-0.198923\pi\)
0.811000 + 0.585046i \(0.198923\pi\)
\(140\) 0 0
\(141\) 1.06913 0.0900370
\(142\) 0 0
\(143\) 5.56155 0.465080
\(144\) 0 0
\(145\) 6.68466 0.555131
\(146\) 0 0
\(147\) −1.56155 −0.128795
\(148\) 0 0
\(149\) 8.24621 0.675556 0.337778 0.941226i \(-0.390325\pi\)
0.337778 + 0.941226i \(0.390325\pi\)
\(150\) 0 0
\(151\) 12.6847 1.03226 0.516131 0.856509i \(-0.327372\pi\)
0.516131 + 0.856509i \(0.327372\pi\)
\(152\) 0 0
\(153\) −0.246211 −0.0199050
\(154\) 0 0
\(155\) −3.12311 −0.250854
\(156\) 0 0
\(157\) 22.4924 1.79509 0.897545 0.440922i \(-0.145349\pi\)
0.897545 + 0.440922i \(0.145349\pi\)
\(158\) 0 0
\(159\) 20.4924 1.62515
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) 8.87689 0.695292 0.347646 0.937626i \(-0.386981\pi\)
0.347646 + 0.937626i \(0.386981\pi\)
\(164\) 0 0
\(165\) −2.43845 −0.189833
\(166\) 0 0
\(167\) 11.8078 0.913712 0.456856 0.889541i \(-0.348975\pi\)
0.456856 + 0.889541i \(0.348975\pi\)
\(168\) 0 0
\(169\) −0.315342 −0.0242570
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) −8.43845 −0.641563 −0.320782 0.947153i \(-0.603946\pi\)
−0.320782 + 0.947153i \(0.603946\pi\)
\(174\) 0 0
\(175\) 1.00000 0.0755929
\(176\) 0 0
\(177\) −9.75379 −0.733140
\(178\) 0 0
\(179\) 2.24621 0.167890 0.0839449 0.996470i \(-0.473248\pi\)
0.0839449 + 0.996470i \(0.473248\pi\)
\(180\) 0 0
\(181\) 21.1231 1.57007 0.785034 0.619453i \(-0.212645\pi\)
0.785034 + 0.619453i \(0.212645\pi\)
\(182\) 0 0
\(183\) −3.12311 −0.230867
\(184\) 0 0
\(185\) −1.12311 −0.0825724
\(186\) 0 0
\(187\) −0.684658 −0.0500672
\(188\) 0 0
\(189\) 5.56155 0.404543
\(190\) 0 0
\(191\) −22.4384 −1.62359 −0.811795 0.583943i \(-0.801509\pi\)
−0.811795 + 0.583943i \(0.801509\pi\)
\(192\) 0 0
\(193\) 0.246211 0.0177227 0.00886134 0.999961i \(-0.497179\pi\)
0.00886134 + 0.999961i \(0.497179\pi\)
\(194\) 0 0
\(195\) −5.56155 −0.398271
\(196\) 0 0
\(197\) 24.7386 1.76255 0.881277 0.472599i \(-0.156684\pi\)
0.881277 + 0.472599i \(0.156684\pi\)
\(198\) 0 0
\(199\) 16.0000 1.13421 0.567105 0.823646i \(-0.308063\pi\)
0.567105 + 0.823646i \(0.308063\pi\)
\(200\) 0 0
\(201\) −16.0000 −1.12855
\(202\) 0 0
\(203\) −6.68466 −0.469171
\(204\) 0 0
\(205\) −2.00000 −0.139686
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 0 0
\(210\) 0 0
\(211\) −6.43845 −0.443241 −0.221620 0.975133i \(-0.571135\pi\)
−0.221620 + 0.975133i \(0.571135\pi\)
\(212\) 0 0
\(213\) −6.24621 −0.427983
\(214\) 0 0
\(215\) −4.00000 −0.272798
\(216\) 0 0
\(217\) 3.12311 0.212010
\(218\) 0 0
\(219\) 0.384472 0.0259802
\(220\) 0 0
\(221\) −1.56155 −0.105041
\(222\) 0 0
\(223\) 3.80776 0.254987 0.127493 0.991839i \(-0.459307\pi\)
0.127493 + 0.991839i \(0.459307\pi\)
\(224\) 0 0
\(225\) −0.561553 −0.0374369
\(226\) 0 0
\(227\) −17.5616 −1.16560 −0.582801 0.812615i \(-0.698043\pi\)
−0.582801 + 0.812615i \(0.698043\pi\)
\(228\) 0 0
\(229\) 16.2462 1.07358 0.536790 0.843716i \(-0.319637\pi\)
0.536790 + 0.843716i \(0.319637\pi\)
\(230\) 0 0
\(231\) 2.43845 0.160438
\(232\) 0 0
\(233\) 0.630683 0.0413174 0.0206587 0.999787i \(-0.493424\pi\)
0.0206587 + 0.999787i \(0.493424\pi\)
\(234\) 0 0
\(235\) 0.684658 0.0446622
\(236\) 0 0
\(237\) 12.1922 0.791971
\(238\) 0 0
\(239\) −15.8078 −1.02252 −0.511260 0.859426i \(-0.670821\pi\)
−0.511260 + 0.859426i \(0.670821\pi\)
\(240\) 0 0
\(241\) −10.8769 −0.700642 −0.350321 0.936630i \(-0.613928\pi\)
−0.350321 + 0.936630i \(0.613928\pi\)
\(242\) 0 0
\(243\) −5.75379 −0.369106
\(244\) 0 0
\(245\) −1.00000 −0.0638877
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) −16.0000 −1.01396
\(250\) 0 0
\(251\) 1.36932 0.0864305 0.0432153 0.999066i \(-0.486240\pi\)
0.0432153 + 0.999066i \(0.486240\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 0 0
\(255\) 0.684658 0.0428750
\(256\) 0 0
\(257\) 2.49242 0.155473 0.0777365 0.996974i \(-0.475231\pi\)
0.0777365 + 0.996974i \(0.475231\pi\)
\(258\) 0 0
\(259\) 1.12311 0.0697864
\(260\) 0 0
\(261\) 3.75379 0.232354
\(262\) 0 0
\(263\) 17.3693 1.07104 0.535519 0.844523i \(-0.320116\pi\)
0.535519 + 0.844523i \(0.320116\pi\)
\(264\) 0 0
\(265\) 13.1231 0.806146
\(266\) 0 0
\(267\) 6.63068 0.405791
\(268\) 0 0
\(269\) −10.4924 −0.639734 −0.319867 0.947462i \(-0.603638\pi\)
−0.319867 + 0.947462i \(0.603638\pi\)
\(270\) 0 0
\(271\) −17.3693 −1.05511 −0.527555 0.849521i \(-0.676891\pi\)
−0.527555 + 0.849521i \(0.676891\pi\)
\(272\) 0 0
\(273\) 5.56155 0.336600
\(274\) 0 0
\(275\) −1.56155 −0.0941652
\(276\) 0 0
\(277\) 4.24621 0.255130 0.127565 0.991830i \(-0.459284\pi\)
0.127565 + 0.991830i \(0.459284\pi\)
\(278\) 0 0
\(279\) −1.75379 −0.104997
\(280\) 0 0
\(281\) 2.68466 0.160153 0.0800766 0.996789i \(-0.474483\pi\)
0.0800766 + 0.996789i \(0.474483\pi\)
\(282\) 0 0
\(283\) −6.43845 −0.382726 −0.191363 0.981519i \(-0.561291\pi\)
−0.191363 + 0.981519i \(0.561291\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 2.00000 0.118056
\(288\) 0 0
\(289\) −16.8078 −0.988692
\(290\) 0 0
\(291\) −20.1922 −1.18369
\(292\) 0 0
\(293\) −14.6847 −0.857887 −0.428943 0.903331i \(-0.641114\pi\)
−0.428943 + 0.903331i \(0.641114\pi\)
\(294\) 0 0
\(295\) −6.24621 −0.363668
\(296\) 0 0
\(297\) −8.68466 −0.503935
\(298\) 0 0
\(299\) 0 0
\(300\) 0 0
\(301\) 4.00000 0.230556
\(302\) 0 0
\(303\) −17.7538 −1.01993
\(304\) 0 0
\(305\) −2.00000 −0.114520
\(306\) 0 0
\(307\) 23.8078 1.35878 0.679390 0.733777i \(-0.262244\pi\)
0.679390 + 0.733777i \(0.262244\pi\)
\(308\) 0 0
\(309\) −11.4233 −0.649848
\(310\) 0 0
\(311\) −28.4924 −1.61566 −0.807829 0.589418i \(-0.799357\pi\)
−0.807829 + 0.589418i \(0.799357\pi\)
\(312\) 0 0
\(313\) −16.9309 −0.956989 −0.478495 0.878090i \(-0.658817\pi\)
−0.478495 + 0.878090i \(0.658817\pi\)
\(314\) 0 0
\(315\) 0.561553 0.0316399
\(316\) 0 0
\(317\) −10.0000 −0.561656 −0.280828 0.959758i \(-0.590609\pi\)
−0.280828 + 0.959758i \(0.590609\pi\)
\(318\) 0 0
\(319\) 10.4384 0.584441
\(320\) 0 0
\(321\) −23.6155 −1.31809
\(322\) 0 0
\(323\) 0 0
\(324\) 0 0
\(325\) −3.56155 −0.197559
\(326\) 0 0
\(327\) 15.3153 0.846940
\(328\) 0 0
\(329\) −0.684658 −0.0377464
\(330\) 0 0
\(331\) 24.4924 1.34623 0.673113 0.739540i \(-0.264957\pi\)
0.673113 + 0.739540i \(0.264957\pi\)
\(332\) 0 0
\(333\) −0.630683 −0.0345612
\(334\) 0 0
\(335\) −10.2462 −0.559810
\(336\) 0 0
\(337\) −30.9848 −1.68785 −0.843926 0.536460i \(-0.819761\pi\)
−0.843926 + 0.536460i \(0.819761\pi\)
\(338\) 0 0
\(339\) −30.2462 −1.64275
\(340\) 0 0
\(341\) −4.87689 −0.264099
\(342\) 0 0
\(343\) 1.00000 0.0539949
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 5.75379 0.308880 0.154440 0.988002i \(-0.450643\pi\)
0.154440 + 0.988002i \(0.450643\pi\)
\(348\) 0 0
\(349\) 19.3693 1.03682 0.518408 0.855133i \(-0.326525\pi\)
0.518408 + 0.855133i \(0.326525\pi\)
\(350\) 0 0
\(351\) −19.8078 −1.05726
\(352\) 0 0
\(353\) 30.6847 1.63318 0.816590 0.577218i \(-0.195862\pi\)
0.816590 + 0.577218i \(0.195862\pi\)
\(354\) 0 0
\(355\) −4.00000 −0.212298
\(356\) 0 0
\(357\) −0.684658 −0.0362360
\(358\) 0 0
\(359\) −30.7386 −1.62232 −0.811162 0.584822i \(-0.801164\pi\)
−0.811162 + 0.584822i \(0.801164\pi\)
\(360\) 0 0
\(361\) −19.0000 −1.00000
\(362\) 0 0
\(363\) 13.3693 0.701707
\(364\) 0 0
\(365\) 0.246211 0.0128873
\(366\) 0 0
\(367\) 34.0540 1.77760 0.888802 0.458292i \(-0.151539\pi\)
0.888802 + 0.458292i \(0.151539\pi\)
\(368\) 0 0
\(369\) −1.12311 −0.0584665
\(370\) 0 0
\(371\) −13.1231 −0.681318
\(372\) 0 0
\(373\) −6.49242 −0.336165 −0.168082 0.985773i \(-0.553757\pi\)
−0.168082 + 0.985773i \(0.553757\pi\)
\(374\) 0 0
\(375\) 1.56155 0.0806382
\(376\) 0 0
\(377\) 23.8078 1.22616
\(378\) 0 0
\(379\) 13.7538 0.706485 0.353242 0.935532i \(-0.385079\pi\)
0.353242 + 0.935532i \(0.385079\pi\)
\(380\) 0 0
\(381\) 9.75379 0.499702
\(382\) 0 0
\(383\) −14.2462 −0.727947 −0.363974 0.931409i \(-0.618580\pi\)
−0.363974 + 0.931409i \(0.618580\pi\)
\(384\) 0 0
\(385\) 1.56155 0.0795841
\(386\) 0 0
\(387\) −2.24621 −0.114181
\(388\) 0 0
\(389\) −27.5616 −1.39743 −0.698713 0.715402i \(-0.746244\pi\)
−0.698713 + 0.715402i \(0.746244\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0 0
\(393\) 22.2462 1.12217
\(394\) 0 0
\(395\) 7.80776 0.392851
\(396\) 0 0
\(397\) −27.1771 −1.36398 −0.681989 0.731362i \(-0.738885\pi\)
−0.681989 + 0.731362i \(0.738885\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −6.68466 −0.333816 −0.166908 0.985972i \(-0.553378\pi\)
−0.166908 + 0.985972i \(0.553378\pi\)
\(402\) 0 0
\(403\) −11.1231 −0.554081
\(404\) 0 0
\(405\) 7.00000 0.347833
\(406\) 0 0
\(407\) −1.75379 −0.0869321
\(408\) 0 0
\(409\) 12.7386 0.629885 0.314942 0.949111i \(-0.398015\pi\)
0.314942 + 0.949111i \(0.398015\pi\)
\(410\) 0 0
\(411\) −0.384472 −0.0189646
\(412\) 0 0
\(413\) 6.24621 0.307356
\(414\) 0 0
\(415\) −10.2462 −0.502967
\(416\) 0 0
\(417\) −29.8617 −1.46234
\(418\) 0 0
\(419\) −16.0000 −0.781651 −0.390826 0.920465i \(-0.627810\pi\)
−0.390826 + 0.920465i \(0.627810\pi\)
\(420\) 0 0
\(421\) 24.9309 1.21506 0.607528 0.794298i \(-0.292161\pi\)
0.607528 + 0.794298i \(0.292161\pi\)
\(422\) 0 0
\(423\) 0.384472 0.0186937
\(424\) 0 0
\(425\) 0.438447 0.0212678
\(426\) 0 0
\(427\) 2.00000 0.0967868
\(428\) 0 0
\(429\) −8.68466 −0.419299
\(430\) 0 0
\(431\) −20.6847 −0.996345 −0.498172 0.867078i \(-0.665995\pi\)
−0.498172 + 0.867078i \(0.665995\pi\)
\(432\) 0 0
\(433\) 12.2462 0.588515 0.294258 0.955726i \(-0.404928\pi\)
0.294258 + 0.955726i \(0.404928\pi\)
\(434\) 0 0
\(435\) −10.4384 −0.500485
\(436\) 0 0
\(437\) 0 0
\(438\) 0 0
\(439\) 11.1231 0.530877 0.265438 0.964128i \(-0.414483\pi\)
0.265438 + 0.964128i \(0.414483\pi\)
\(440\) 0 0
\(441\) −0.561553 −0.0267406
\(442\) 0 0
\(443\) 24.4924 1.16367 0.581835 0.813307i \(-0.302335\pi\)
0.581835 + 0.813307i \(0.302335\pi\)
\(444\) 0 0
\(445\) 4.24621 0.201290
\(446\) 0 0
\(447\) −12.8769 −0.609056
\(448\) 0 0
\(449\) 26.6847 1.25933 0.629663 0.776868i \(-0.283193\pi\)
0.629663 + 0.776868i \(0.283193\pi\)
\(450\) 0 0
\(451\) −3.12311 −0.147061
\(452\) 0 0
\(453\) −19.8078 −0.930650
\(454\) 0 0
\(455\) 3.56155 0.166968
\(456\) 0 0
\(457\) 14.4924 0.677927 0.338963 0.940800i \(-0.389924\pi\)
0.338963 + 0.940800i \(0.389924\pi\)
\(458\) 0 0
\(459\) 2.43845 0.113817
\(460\) 0 0
\(461\) −15.3693 −0.715820 −0.357910 0.933756i \(-0.616511\pi\)
−0.357910 + 0.933756i \(0.616511\pi\)
\(462\) 0 0
\(463\) −12.8769 −0.598440 −0.299220 0.954184i \(-0.596726\pi\)
−0.299220 + 0.954184i \(0.596726\pi\)
\(464\) 0 0
\(465\) 4.87689 0.226161
\(466\) 0 0
\(467\) −26.9309 −1.24621 −0.623106 0.782137i \(-0.714129\pi\)
−0.623106 + 0.782137i \(0.714129\pi\)
\(468\) 0 0
\(469\) 10.2462 0.473126
\(470\) 0 0
\(471\) −35.1231 −1.61839
\(472\) 0 0
\(473\) −6.24621 −0.287201
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 7.36932 0.337418
\(478\) 0 0
\(479\) 34.7386 1.58725 0.793624 0.608408i \(-0.208192\pi\)
0.793624 + 0.608408i \(0.208192\pi\)
\(480\) 0 0
\(481\) −4.00000 −0.182384
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) −12.9309 −0.587161
\(486\) 0 0
\(487\) 31.6155 1.43264 0.716318 0.697774i \(-0.245826\pi\)
0.716318 + 0.697774i \(0.245826\pi\)
\(488\) 0 0
\(489\) −13.8617 −0.626850
\(490\) 0 0
\(491\) −34.9309 −1.57641 −0.788204 0.615414i \(-0.788989\pi\)
−0.788204 + 0.615414i \(0.788989\pi\)
\(492\) 0 0
\(493\) −2.93087 −0.132000
\(494\) 0 0
\(495\) −0.876894 −0.0394135
\(496\) 0 0
\(497\) 4.00000 0.179425
\(498\) 0 0
\(499\) −14.0540 −0.629142 −0.314571 0.949234i \(-0.601861\pi\)
−0.314571 + 0.949234i \(0.601861\pi\)
\(500\) 0 0
\(501\) −18.4384 −0.823769
\(502\) 0 0
\(503\) 13.1771 0.587537 0.293768 0.955877i \(-0.405091\pi\)
0.293768 + 0.955877i \(0.405091\pi\)
\(504\) 0 0
\(505\) −11.3693 −0.505928
\(506\) 0 0
\(507\) 0.492423 0.0218693
\(508\) 0 0
\(509\) 4.73863 0.210036 0.105018 0.994470i \(-0.466510\pi\)
0.105018 + 0.994470i \(0.466510\pi\)
\(510\) 0 0
\(511\) −0.246211 −0.0108917
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) −7.31534 −0.322352
\(516\) 0 0
\(517\) 1.06913 0.0470203
\(518\) 0 0
\(519\) 13.1771 0.578410
\(520\) 0 0
\(521\) 39.8617 1.74637 0.873187 0.487385i \(-0.162049\pi\)
0.873187 + 0.487385i \(0.162049\pi\)
\(522\) 0 0
\(523\) −28.9848 −1.26742 −0.633709 0.773571i \(-0.718468\pi\)
−0.633709 + 0.773571i \(0.718468\pi\)
\(524\) 0 0
\(525\) −1.56155 −0.0681518
\(526\) 0 0
\(527\) 1.36932 0.0596484
\(528\) 0 0
\(529\) −23.0000 −1.00000
\(530\) 0 0
\(531\) −3.50758 −0.152216
\(532\) 0 0
\(533\) −7.12311 −0.308536
\(534\) 0 0
\(535\) −15.1231 −0.653829
\(536\) 0 0
\(537\) −3.50758 −0.151363
\(538\) 0 0
\(539\) −1.56155 −0.0672608
\(540\) 0 0
\(541\) −2.19224 −0.0942516 −0.0471258 0.998889i \(-0.515006\pi\)
−0.0471258 + 0.998889i \(0.515006\pi\)
\(542\) 0 0
\(543\) −32.9848 −1.41552
\(544\) 0 0
\(545\) 9.80776 0.420118
\(546\) 0 0
\(547\) 24.8769 1.06366 0.531830 0.846851i \(-0.321505\pi\)
0.531830 + 0.846851i \(0.321505\pi\)
\(548\) 0 0
\(549\) −1.12311 −0.0479330
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) −7.80776 −0.332020
\(554\) 0 0
\(555\) 1.75379 0.0744442
\(556\) 0 0
\(557\) −13.1231 −0.556044 −0.278022 0.960575i \(-0.589679\pi\)
−0.278022 + 0.960575i \(0.589679\pi\)
\(558\) 0 0
\(559\) −14.2462 −0.602551
\(560\) 0 0
\(561\) 1.06913 0.0451387
\(562\) 0 0
\(563\) −36.9848 −1.55873 −0.779363 0.626573i \(-0.784457\pi\)
−0.779363 + 0.626573i \(0.784457\pi\)
\(564\) 0 0
\(565\) −19.3693 −0.814873
\(566\) 0 0
\(567\) −7.00000 −0.293972
\(568\) 0 0
\(569\) 32.2462 1.35183 0.675916 0.736979i \(-0.263748\pi\)
0.675916 + 0.736979i \(0.263748\pi\)
\(570\) 0 0
\(571\) −40.4924 −1.69456 −0.847278 0.531150i \(-0.821760\pi\)
−0.847278 + 0.531150i \(0.821760\pi\)
\(572\) 0 0
\(573\) 35.0388 1.46377
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) 33.8078 1.40744 0.703718 0.710480i \(-0.251522\pi\)
0.703718 + 0.710480i \(0.251522\pi\)
\(578\) 0 0
\(579\) −0.384472 −0.0159781
\(580\) 0 0
\(581\) 10.2462 0.425084
\(582\) 0 0
\(583\) 20.4924 0.848709
\(584\) 0 0
\(585\) −2.00000 −0.0826898
\(586\) 0 0
\(587\) −0.492423 −0.0203245 −0.0101622 0.999948i \(-0.503235\pi\)
−0.0101622 + 0.999948i \(0.503235\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) 0 0
\(591\) −38.6307 −1.58905
\(592\) 0 0
\(593\) 5.31534 0.218275 0.109137 0.994027i \(-0.465191\pi\)
0.109137 + 0.994027i \(0.465191\pi\)
\(594\) 0 0
\(595\) −0.438447 −0.0179746
\(596\) 0 0
\(597\) −24.9848 −1.02256
\(598\) 0 0
\(599\) −24.1922 −0.988468 −0.494234 0.869329i \(-0.664551\pi\)
−0.494234 + 0.869329i \(0.664551\pi\)
\(600\) 0 0
\(601\) 30.4924 1.24381 0.621906 0.783092i \(-0.286359\pi\)
0.621906 + 0.783092i \(0.286359\pi\)
\(602\) 0 0
\(603\) −5.75379 −0.234312
\(604\) 0 0
\(605\) 8.56155 0.348077
\(606\) 0 0
\(607\) 16.6847 0.677209 0.338605 0.940929i \(-0.390045\pi\)
0.338605 + 0.940929i \(0.390045\pi\)
\(608\) 0 0
\(609\) 10.4384 0.422987
\(610\) 0 0
\(611\) 2.43845 0.0986490
\(612\) 0 0
\(613\) −26.0000 −1.05013 −0.525065 0.851062i \(-0.675959\pi\)
−0.525065 + 0.851062i \(0.675959\pi\)
\(614\) 0 0
\(615\) 3.12311 0.125936
\(616\) 0 0
\(617\) −22.9848 −0.925335 −0.462668 0.886532i \(-0.653108\pi\)
−0.462668 + 0.886532i \(0.653108\pi\)
\(618\) 0 0
\(619\) 32.9848 1.32577 0.662886 0.748720i \(-0.269331\pi\)
0.662886 + 0.748720i \(0.269331\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) −4.24621 −0.170121
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 0.492423 0.0196342
\(630\) 0 0
\(631\) 7.80776 0.310822 0.155411 0.987850i \(-0.450330\pi\)
0.155411 + 0.987850i \(0.450330\pi\)
\(632\) 0 0
\(633\) 10.0540 0.399610
\(634\) 0 0
\(635\) 6.24621 0.247873
\(636\) 0 0
\(637\) −3.56155 −0.141114
\(638\) 0 0
\(639\) −2.24621 −0.0888587
\(640\) 0 0
\(641\) −4.24621 −0.167715 −0.0838576 0.996478i \(-0.526724\pi\)
−0.0838576 + 0.996478i \(0.526724\pi\)
\(642\) 0 0
\(643\) 6.43845 0.253908 0.126954 0.991909i \(-0.459480\pi\)
0.126954 + 0.991909i \(0.459480\pi\)
\(644\) 0 0
\(645\) 6.24621 0.245944
\(646\) 0 0
\(647\) 33.7538 1.32700 0.663499 0.748177i \(-0.269071\pi\)
0.663499 + 0.748177i \(0.269071\pi\)
\(648\) 0 0
\(649\) −9.75379 −0.382870
\(650\) 0 0
\(651\) −4.87689 −0.191141
\(652\) 0 0
\(653\) −14.8769 −0.582178 −0.291089 0.956696i \(-0.594018\pi\)
−0.291089 + 0.956696i \(0.594018\pi\)
\(654\) 0 0
\(655\) 14.2462 0.556646
\(656\) 0 0
\(657\) 0.138261 0.00539406
\(658\) 0 0
\(659\) −11.3153 −0.440783 −0.220392 0.975411i \(-0.570734\pi\)
−0.220392 + 0.975411i \(0.570734\pi\)
\(660\) 0 0
\(661\) −17.1231 −0.666012 −0.333006 0.942925i \(-0.608063\pi\)
−0.333006 + 0.942925i \(0.608063\pi\)
\(662\) 0 0
\(663\) 2.43845 0.0947014
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 0 0
\(668\) 0 0
\(669\) −5.94602 −0.229887
\(670\) 0 0
\(671\) −3.12311 −0.120566
\(672\) 0 0
\(673\) −7.75379 −0.298887 −0.149443 0.988770i \(-0.547748\pi\)
−0.149443 + 0.988770i \(0.547748\pi\)
\(674\) 0 0
\(675\) 5.56155 0.214064
\(676\) 0 0
\(677\) −33.8078 −1.29934 −0.649669 0.760217i \(-0.725092\pi\)
−0.649669 + 0.760217i \(0.725092\pi\)
\(678\) 0 0
\(679\) 12.9309 0.496241
\(680\) 0 0
\(681\) 27.4233 1.05086
\(682\) 0 0
\(683\) 15.1231 0.578670 0.289335 0.957228i \(-0.406566\pi\)
0.289335 + 0.957228i \(0.406566\pi\)
\(684\) 0 0
\(685\) −0.246211 −0.00940725
\(686\) 0 0
\(687\) −25.3693 −0.967900
\(688\) 0 0
\(689\) 46.7386 1.78060
\(690\) 0 0
\(691\) −4.49242 −0.170900 −0.0854499 0.996342i \(-0.527233\pi\)
−0.0854499 + 0.996342i \(0.527233\pi\)
\(692\) 0 0
\(693\) 0.876894 0.0333105
\(694\) 0 0
\(695\) −19.1231 −0.725381
\(696\) 0 0
\(697\) 0.876894 0.0332147
\(698\) 0 0
\(699\) −0.984845 −0.0372503
\(700\) 0 0
\(701\) −10.1922 −0.384955 −0.192478 0.981301i \(-0.561652\pi\)
−0.192478 + 0.981301i \(0.561652\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) 0 0
\(705\) −1.06913 −0.0402658
\(706\) 0 0
\(707\) 11.3693 0.427587
\(708\) 0 0
\(709\) 42.3002 1.58862 0.794308 0.607515i \(-0.207833\pi\)
0.794308 + 0.607515i \(0.207833\pi\)
\(710\) 0 0
\(711\) 4.38447 0.164431
\(712\) 0 0
\(713\) 0 0
\(714\) 0 0
\(715\) −5.56155 −0.207990
\(716\) 0 0
\(717\) 24.6847 0.921865
\(718\) 0 0
\(719\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(720\) 0 0
\(721\) 7.31534 0.272438
\(722\) 0 0
\(723\) 16.9848 0.631673
\(724\) 0 0
\(725\) −6.68466 −0.248262
\(726\) 0 0
\(727\) −32.9848 −1.22334 −0.611670 0.791113i \(-0.709502\pi\)
−0.611670 + 0.791113i \(0.709502\pi\)
\(728\) 0 0
\(729\) 29.9848 1.11055
\(730\) 0 0
\(731\) 1.75379 0.0648662
\(732\) 0 0
\(733\) 23.1771 0.856065 0.428033 0.903763i \(-0.359207\pi\)
0.428033 + 0.903763i \(0.359207\pi\)
\(734\) 0 0
\(735\) 1.56155 0.0575987
\(736\) 0 0
\(737\) −16.0000 −0.589368
\(738\) 0 0
\(739\) 26.9309 0.990668 0.495334 0.868703i \(-0.335046\pi\)
0.495334 + 0.868703i \(0.335046\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 30.2462 1.10963 0.554813 0.831975i \(-0.312790\pi\)
0.554813 + 0.831975i \(0.312790\pi\)
\(744\) 0 0
\(745\) −8.24621 −0.302118
\(746\) 0 0
\(747\) −5.75379 −0.210520
\(748\) 0 0
\(749\) 15.1231 0.552586
\(750\) 0 0
\(751\) −5.06913 −0.184975 −0.0924876 0.995714i \(-0.529482\pi\)
−0.0924876 + 0.995714i \(0.529482\pi\)
\(752\) 0 0
\(753\) −2.13826 −0.0779225
\(754\) 0 0
\(755\) −12.6847 −0.461642
\(756\) 0 0
\(757\) −2.38447 −0.0866651 −0.0433326 0.999061i \(-0.513797\pi\)
−0.0433326 + 0.999061i \(0.513797\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 11.3693 0.412137 0.206069 0.978538i \(-0.433933\pi\)
0.206069 + 0.978538i \(0.433933\pi\)
\(762\) 0 0
\(763\) −9.80776 −0.355065
\(764\) 0 0
\(765\) 0.246211 0.00890179
\(766\) 0 0
\(767\) −22.2462 −0.803264
\(768\) 0 0
\(769\) −22.0000 −0.793340 −0.396670 0.917961i \(-0.629834\pi\)
−0.396670 + 0.917961i \(0.629834\pi\)
\(770\) 0 0
\(771\) −3.89205 −0.140169
\(772\) 0 0
\(773\) 6.19224 0.222719 0.111360 0.993780i \(-0.464479\pi\)
0.111360 + 0.993780i \(0.464479\pi\)
\(774\) 0 0
\(775\) 3.12311 0.112185
\(776\) 0 0
\(777\) −1.75379 −0.0629168
\(778\) 0 0
\(779\) 0 0
\(780\) 0 0
\(781\) −6.24621 −0.223507
\(782\) 0 0
\(783\) −37.1771 −1.32860
\(784\) 0 0
\(785\) −22.4924 −0.802789
\(786\) 0 0
\(787\) −22.0540 −0.786139 −0.393070 0.919509i \(-0.628587\pi\)
−0.393070 + 0.919509i \(0.628587\pi\)
\(788\) 0 0
\(789\) −27.1231 −0.965608
\(790\) 0 0
\(791\) 19.3693 0.688694
\(792\) 0 0
\(793\) −7.12311 −0.252949
\(794\) 0 0
\(795\) −20.4924 −0.726791
\(796\) 0 0
\(797\) 11.0691 0.392089 0.196044 0.980595i \(-0.437190\pi\)
0.196044 + 0.980595i \(0.437190\pi\)
\(798\) 0 0
\(799\) −0.300187 −0.0106198
\(800\) 0 0
\(801\) 2.38447 0.0842512
\(802\) 0 0
\(803\) 0.384472 0.0135677
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 16.3845 0.576761
\(808\) 0 0
\(809\) 4.05398 0.142530 0.0712651 0.997457i \(-0.477296\pi\)
0.0712651 + 0.997457i \(0.477296\pi\)
\(810\) 0 0
\(811\) −29.8617 −1.04859 −0.524294 0.851537i \(-0.675671\pi\)
−0.524294 + 0.851537i \(0.675671\pi\)
\(812\) 0 0
\(813\) 27.1231 0.951249
\(814\) 0 0
\(815\) −8.87689 −0.310944
\(816\) 0 0
\(817\) 0 0
\(818\) 0 0
\(819\) 2.00000 0.0698857
\(820\) 0 0
\(821\) 21.4233 0.747678 0.373839 0.927494i \(-0.378041\pi\)
0.373839 + 0.927494i \(0.378041\pi\)
\(822\) 0 0
\(823\) 6.63068 0.231131 0.115566 0.993300i \(-0.463132\pi\)
0.115566 + 0.993300i \(0.463132\pi\)
\(824\) 0 0
\(825\) 2.43845 0.0848958
\(826\) 0 0
\(827\) 48.1080 1.67288 0.836439 0.548061i \(-0.184634\pi\)
0.836439 + 0.548061i \(0.184634\pi\)
\(828\) 0 0
\(829\) 11.7538 0.408226 0.204113 0.978947i \(-0.434569\pi\)
0.204113 + 0.978947i \(0.434569\pi\)
\(830\) 0 0
\(831\) −6.63068 −0.230016
\(832\) 0 0
\(833\) 0.438447 0.0151913
\(834\) 0 0
\(835\) −11.8078 −0.408625
\(836\) 0 0
\(837\) 17.3693 0.600371
\(838\) 0 0
\(839\) −16.9848 −0.586382 −0.293191 0.956054i \(-0.594717\pi\)
−0.293191 + 0.956054i \(0.594717\pi\)
\(840\) 0 0
\(841\) 15.6847 0.540850
\(842\) 0 0
\(843\) −4.19224 −0.144388
\(844\) 0 0
\(845\) 0.315342 0.0108481
\(846\) 0 0
\(847\) −8.56155 −0.294178
\(848\) 0 0
\(849\) 10.0540 0.345052
\(850\) 0 0
\(851\) 0 0
\(852\) 0 0
\(853\) 27.7538 0.950272 0.475136 0.879912i \(-0.342399\pi\)
0.475136 + 0.879912i \(0.342399\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −20.7386 −0.708418 −0.354209 0.935166i \(-0.615250\pi\)
−0.354209 + 0.935166i \(0.615250\pi\)
\(858\) 0 0
\(859\) 39.2311 1.33855 0.669273 0.743016i \(-0.266606\pi\)
0.669273 + 0.743016i \(0.266606\pi\)
\(860\) 0 0
\(861\) −3.12311 −0.106435
\(862\) 0 0
\(863\) 42.3542 1.44175 0.720876 0.693064i \(-0.243740\pi\)
0.720876 + 0.693064i \(0.243740\pi\)
\(864\) 0 0
\(865\) 8.43845 0.286916
\(866\) 0 0
\(867\) 26.2462 0.891368
\(868\) 0 0
\(869\) 12.1922 0.413593
\(870\) 0 0
\(871\) −36.4924 −1.23650
\(872\) 0 0
\(873\) −7.26137 −0.245760
\(874\) 0 0
\(875\) −1.00000 −0.0338062
\(876\) 0 0
\(877\) −43.3693 −1.46448 −0.732239 0.681048i \(-0.761525\pi\)
−0.732239 + 0.681048i \(0.761525\pi\)
\(878\) 0 0
\(879\) 22.9309 0.773439
\(880\) 0 0
\(881\) 5.12311 0.172602 0.0863009 0.996269i \(-0.472495\pi\)
0.0863009 + 0.996269i \(0.472495\pi\)
\(882\) 0 0
\(883\) 57.8617 1.94720 0.973601 0.228255i \(-0.0733021\pi\)
0.973601 + 0.228255i \(0.0733021\pi\)
\(884\) 0 0
\(885\) 9.75379 0.327870
\(886\) 0 0
\(887\) −12.4924 −0.419454 −0.209727 0.977760i \(-0.567258\pi\)
−0.209727 + 0.977760i \(0.567258\pi\)
\(888\) 0 0
\(889\) −6.24621 −0.209491
\(890\) 0 0
\(891\) 10.9309 0.366198
\(892\) 0 0
\(893\) 0 0
\(894\) 0 0
\(895\) −2.24621 −0.0750826
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) −20.8769 −0.696283
\(900\) 0 0
\(901\) −5.75379 −0.191686
\(902\) 0 0
\(903\) −6.24621 −0.207861
\(904\) 0 0
\(905\) −21.1231 −0.702156
\(906\) 0 0
\(907\) 18.2462 0.605856 0.302928 0.953014i \(-0.402036\pi\)
0.302928 + 0.953014i \(0.402036\pi\)
\(908\) 0 0
\(909\) −6.38447 −0.211760
\(910\) 0 0
\(911\) 38.7386 1.28347 0.641734 0.766927i \(-0.278215\pi\)
0.641734 + 0.766927i \(0.278215\pi\)
\(912\) 0 0
\(913\) −16.0000 −0.529523
\(914\) 0 0
\(915\) 3.12311 0.103247
\(916\) 0 0
\(917\) −14.2462 −0.470451
\(918\) 0 0
\(919\) −6.05398 −0.199702 −0.0998511 0.995002i \(-0.531837\pi\)
−0.0998511 + 0.995002i \(0.531837\pi\)
\(920\) 0 0
\(921\) −37.1771 −1.22503
\(922\) 0 0
\(923\) −14.2462 −0.468920
\(924\) 0 0
\(925\) 1.12311 0.0369275
\(926\) 0 0
\(927\) −4.10795 −0.134923
\(928\) 0 0
\(929\) −34.1080 −1.11905 −0.559523 0.828815i \(-0.689016\pi\)
−0.559523 + 0.828815i \(0.689016\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 0 0
\(933\) 44.4924 1.45662
\(934\) 0 0
\(935\) 0.684658 0.0223907
\(936\) 0 0
\(937\) 54.6847 1.78647 0.893235 0.449590i \(-0.148430\pi\)
0.893235 + 0.449590i \(0.148430\pi\)
\(938\) 0 0
\(939\) 26.4384 0.862786
\(940\) 0 0
\(941\) 5.50758 0.179542 0.0897709 0.995962i \(-0.471386\pi\)
0.0897709 + 0.995962i \(0.471386\pi\)
\(942\) 0 0
\(943\) 0 0
\(944\) 0 0
\(945\) −5.56155 −0.180917
\(946\) 0 0
\(947\) −2.63068 −0.0854857 −0.0427429 0.999086i \(-0.513610\pi\)
−0.0427429 + 0.999086i \(0.513610\pi\)
\(948\) 0 0
\(949\) 0.876894 0.0284652
\(950\) 0 0
\(951\) 15.6155 0.506368
\(952\) 0 0
\(953\) −4.24621 −0.137548 −0.0687741 0.997632i \(-0.521909\pi\)
−0.0687741 + 0.997632i \(0.521909\pi\)
\(954\) 0 0
\(955\) 22.4384 0.726091
\(956\) 0 0
\(957\) −16.3002 −0.526910
\(958\) 0 0
\(959\) 0.246211 0.00795058
\(960\) 0 0
\(961\) −21.2462 −0.685362
\(962\) 0 0
\(963\) −8.49242 −0.273664
\(964\) 0 0
\(965\) −0.246211 −0.00792582
\(966\) 0 0
\(967\) 29.8617 0.960289 0.480144 0.877189i \(-0.340584\pi\)
0.480144 + 0.877189i \(0.340584\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 2.13826 0.0686200 0.0343100 0.999411i \(-0.489077\pi\)
0.0343100 + 0.999411i \(0.489077\pi\)
\(972\) 0 0
\(973\) 19.1231 0.613059
\(974\) 0 0
\(975\) 5.56155 0.178112
\(976\) 0 0
\(977\) −27.8617 −0.891376 −0.445688 0.895188i \(-0.647041\pi\)
−0.445688 + 0.895188i \(0.647041\pi\)
\(978\) 0 0
\(979\) 6.63068 0.211918
\(980\) 0 0
\(981\) 5.50758 0.175843
\(982\) 0 0
\(983\) −2.05398 −0.0655116 −0.0327558 0.999463i \(-0.510428\pi\)
−0.0327558 + 0.999463i \(0.510428\pi\)
\(984\) 0 0
\(985\) −24.7386 −0.788238
\(986\) 0 0
\(987\) 1.06913 0.0340308
\(988\) 0 0
\(989\) 0 0
\(990\) 0 0
\(991\) 42.2462 1.34200 0.670998 0.741460i \(-0.265866\pi\)
0.670998 + 0.741460i \(0.265866\pi\)
\(992\) 0 0
\(993\) −38.2462 −1.21371
\(994\) 0 0
\(995\) −16.0000 −0.507234
\(996\) 0 0
\(997\) −14.6847 −0.465068 −0.232534 0.972588i \(-0.574702\pi\)
−0.232534 + 0.972588i \(0.574702\pi\)
\(998\) 0 0
\(999\) 6.24621 0.197621
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 2240.2.a.bj.1.1 2
4.3 odd 2 2240.2.a.bc.1.2 2
8.3 odd 2 1120.2.a.t.1.1 yes 2
8.5 even 2 1120.2.a.r.1.2 2
40.19 odd 2 5600.2.a.ba.1.2 2
40.29 even 2 5600.2.a.bh.1.1 2
56.13 odd 2 7840.2.a.bh.1.1 2
56.27 even 2 7840.2.a.bc.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1120.2.a.r.1.2 2 8.5 even 2
1120.2.a.t.1.1 yes 2 8.3 odd 2
2240.2.a.bc.1.2 2 4.3 odd 2
2240.2.a.bj.1.1 2 1.1 even 1 trivial
5600.2.a.ba.1.2 2 40.19 odd 2
5600.2.a.bh.1.1 2 40.29 even 2
7840.2.a.bc.1.2 2 56.27 even 2
7840.2.a.bh.1.1 2 56.13 odd 2