Properties

Label 6840.2.a.bm.1.2
Level $6840$
Weight $2$
Character 6840.1
Self dual yes
Analytic conductor $54.618$
Analytic rank $0$
Dimension $3$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [6840,2,Mod(1,6840)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(6840, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 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("6840.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 6840 = 2^{3} \cdot 3^{2} \cdot 5 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6840.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: \(54.6176749826\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.316.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 4x + 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 760)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(2.34292\) of defining polynomial
Character \(\chi\) \(=\) 6840.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.00000 q^{5} +1.19656 q^{7} +O(q^{10})\) \(q+1.00000 q^{5} +1.19656 q^{7} -4.97858 q^{11} -6.63565 q^{13} -1.48929 q^{17} +1.00000 q^{19} +0.510711 q^{23} +1.00000 q^{25} +7.88240 q^{29} -2.97858 q^{31} +1.19656 q^{35} -7.14637 q^{37} -1.66442 q^{41} -6.39312 q^{43} +9.95715 q^{47} -5.56825 q^{49} +11.4219 q^{53} -4.97858 q^{55} +11.8396 q^{59} +3.66442 q^{61} -6.63565 q^{65} +7.61423 q^{67} +13.8396 q^{73} -5.95715 q^{77} +12.6858 q^{79} +8.68585 q^{83} -1.48929 q^{85} +4.87819 q^{89} -7.93994 q^{91} +1.00000 q^{95} -6.81079 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 3 q^{5} - q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 3 q + 3 q^{5} - q^{7} - 11 q^{13} + 3 q^{17} + 3 q^{19} + 9 q^{23} + 3 q^{25} + 7 q^{29} + 6 q^{31} - q^{35} - 20 q^{37} + 22 q^{41} - 10 q^{43} + 12 q^{49} + 7 q^{53} - 11 q^{59} - 16 q^{61} - 11 q^{65} - q^{67} - 5 q^{73} + 12 q^{77} + 26 q^{79} + 14 q^{83} + 3 q^{85} + 6 q^{89} + 29 q^{91} + 3 q^{95} + 8 q^{97}+O(q^{100}) \) 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
\(3\) 0 0
\(4\) 0 0
\(5\) 1.00000 0.447214
\(6\) 0 0
\(7\) 1.19656 0.452256 0.226128 0.974098i \(-0.427393\pi\)
0.226128 + 0.974098i \(0.427393\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) −4.97858 −1.50110 −0.750549 0.660815i \(-0.770211\pi\)
−0.750549 + 0.660815i \(0.770211\pi\)
\(12\) 0 0
\(13\) −6.63565 −1.84040 −0.920200 0.391449i \(-0.871974\pi\)
−0.920200 + 0.391449i \(0.871974\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −1.48929 −0.361206 −0.180603 0.983556i \(-0.557805\pi\)
−0.180603 + 0.983556i \(0.557805\pi\)
\(18\) 0 0
\(19\) 1.00000 0.229416
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 0.510711 0.106491 0.0532453 0.998581i \(-0.483043\pi\)
0.0532453 + 0.998581i \(0.483043\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 7.88240 1.46373 0.731863 0.681452i \(-0.238651\pi\)
0.731863 + 0.681452i \(0.238651\pi\)
\(30\) 0 0
\(31\) −2.97858 −0.534968 −0.267484 0.963562i \(-0.586192\pi\)
−0.267484 + 0.963562i \(0.586192\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 1.19656 0.202255
\(36\) 0 0
\(37\) −7.14637 −1.17486 −0.587428 0.809277i \(-0.699859\pi\)
−0.587428 + 0.809277i \(0.699859\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −1.66442 −0.259939 −0.129970 0.991518i \(-0.541488\pi\)
−0.129970 + 0.991518i \(0.541488\pi\)
\(42\) 0 0
\(43\) −6.39312 −0.974941 −0.487470 0.873139i \(-0.662080\pi\)
−0.487470 + 0.873139i \(0.662080\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 9.95715 1.45240 0.726200 0.687483i \(-0.241285\pi\)
0.726200 + 0.687483i \(0.241285\pi\)
\(48\) 0 0
\(49\) −5.56825 −0.795464
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 11.4219 1.56892 0.784458 0.620182i \(-0.212941\pi\)
0.784458 + 0.620182i \(0.212941\pi\)
\(54\) 0 0
\(55\) −4.97858 −0.671311
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 11.8396 1.54138 0.770690 0.637211i \(-0.219912\pi\)
0.770690 + 0.637211i \(0.219912\pi\)
\(60\) 0 0
\(61\) 3.66442 0.469181 0.234591 0.972094i \(-0.424625\pi\)
0.234591 + 0.972094i \(0.424625\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −6.63565 −0.823052
\(66\) 0 0
\(67\) 7.61423 0.930226 0.465113 0.885251i \(-0.346014\pi\)
0.465113 + 0.885251i \(0.346014\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(72\) 0 0
\(73\) 13.8396 1.61980 0.809899 0.586570i \(-0.199522\pi\)
0.809899 + 0.586570i \(0.199522\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −5.95715 −0.678881
\(78\) 0 0
\(79\) 12.6858 1.42727 0.713635 0.700518i \(-0.247048\pi\)
0.713635 + 0.700518i \(0.247048\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 8.68585 0.953395 0.476698 0.879067i \(-0.341834\pi\)
0.476698 + 0.879067i \(0.341834\pi\)
\(84\) 0 0
\(85\) −1.48929 −0.161536
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 4.87819 0.517087 0.258544 0.966000i \(-0.416757\pi\)
0.258544 + 0.966000i \(0.416757\pi\)
\(90\) 0 0
\(91\) −7.93994 −0.832332
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 1.00000 0.102598
\(96\) 0 0
\(97\) −6.81079 −0.691531 −0.345765 0.938321i \(-0.612381\pi\)
−0.345765 + 0.938321i \(0.612381\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 2.29273 0.228135 0.114068 0.993473i \(-0.463612\pi\)
0.114068 + 0.993473i \(0.463612\pi\)
\(102\) 0 0
\(103\) −6.51806 −0.642243 −0.321122 0.947038i \(-0.604060\pi\)
−0.321122 + 0.947038i \(0.604060\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −7.71462 −0.745800 −0.372900 0.927872i \(-0.621637\pi\)
−0.372900 + 0.927872i \(0.621637\pi\)
\(108\) 0 0
\(109\) 15.5468 1.48912 0.744558 0.667558i \(-0.232660\pi\)
0.744558 + 0.667558i \(0.232660\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 0.753250 0.0708598 0.0354299 0.999372i \(-0.488720\pi\)
0.0354299 + 0.999372i \(0.488720\pi\)
\(114\) 0 0
\(115\) 0.510711 0.0476241
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) −1.78202 −0.163357
\(120\) 0 0
\(121\) 13.7862 1.25329
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 1.00000 0.0894427
\(126\) 0 0
\(127\) 10.4177 0.924419 0.462210 0.886771i \(-0.347057\pi\)
0.462210 + 0.886771i \(0.347057\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 17.9572 1.56892 0.784462 0.620177i \(-0.212939\pi\)
0.784462 + 0.620177i \(0.212939\pi\)
\(132\) 0 0
\(133\) 1.19656 0.103755
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 9.83956 0.840650 0.420325 0.907374i \(-0.361916\pi\)
0.420325 + 0.907374i \(0.361916\pi\)
\(138\) 0 0
\(139\) −0.978577 −0.0830018 −0.0415009 0.999138i \(-0.513214\pi\)
−0.0415009 + 0.999138i \(0.513214\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 33.0361 2.76262
\(144\) 0 0
\(145\) 7.88240 0.654598
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 8.24989 0.675857 0.337928 0.941172i \(-0.390274\pi\)
0.337928 + 0.941172i \(0.390274\pi\)
\(150\) 0 0
\(151\) 13.8568 1.12765 0.563824 0.825895i \(-0.309330\pi\)
0.563824 + 0.825895i \(0.309330\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) −2.97858 −0.239245
\(156\) 0 0
\(157\) −15.7073 −1.25358 −0.626788 0.779190i \(-0.715631\pi\)
−0.626788 + 0.779190i \(0.715631\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 0.611096 0.0481611
\(162\) 0 0
\(163\) 5.80765 0.454891 0.227445 0.973791i \(-0.426963\pi\)
0.227445 + 0.973791i \(0.426963\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 13.8322 1.07037 0.535184 0.844735i \(-0.320242\pi\)
0.535184 + 0.844735i \(0.320242\pi\)
\(168\) 0 0
\(169\) 31.0319 2.38707
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) −14.8108 −1.12604 −0.563022 0.826442i \(-0.690361\pi\)
−0.563022 + 0.826442i \(0.690361\pi\)
\(174\) 0 0
\(175\) 1.19656 0.0904513
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) −15.6644 −1.17081 −0.585407 0.810740i \(-0.699065\pi\)
−0.585407 + 0.810740i \(0.699065\pi\)
\(180\) 0 0
\(181\) −14.7862 −1.09905 −0.549526 0.835477i \(-0.685192\pi\)
−0.549526 + 0.835477i \(0.685192\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) −7.14637 −0.525411
\(186\) 0 0
\(187\) 7.41454 0.542205
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 6.36748 0.460735 0.230367 0.973104i \(-0.426007\pi\)
0.230367 + 0.973104i \(0.426007\pi\)
\(192\) 0 0
\(193\) −19.5970 −1.41062 −0.705312 0.708897i \(-0.749193\pi\)
−0.705312 + 0.708897i \(0.749193\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −10.7862 −0.768487 −0.384244 0.923232i \(-0.625538\pi\)
−0.384244 + 0.923232i \(0.625538\pi\)
\(198\) 0 0
\(199\) 2.80344 0.198731 0.0993654 0.995051i \(-0.468319\pi\)
0.0993654 + 0.995051i \(0.468319\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 9.43175 0.661979
\(204\) 0 0
\(205\) −1.66442 −0.116248
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) −4.97858 −0.344375
\(210\) 0 0
\(211\) −11.2541 −0.774764 −0.387382 0.921919i \(-0.626621\pi\)
−0.387382 + 0.921919i \(0.626621\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) −6.39312 −0.436007
\(216\) 0 0
\(217\) −3.56404 −0.241943
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 9.88240 0.664762
\(222\) 0 0
\(223\) 17.4966 1.17166 0.585831 0.810433i \(-0.300768\pi\)
0.585831 + 0.810433i \(0.300768\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 3.84942 0.255495 0.127748 0.991807i \(-0.459225\pi\)
0.127748 + 0.991807i \(0.459225\pi\)
\(228\) 0 0
\(229\) −14.6430 −0.967637 −0.483818 0.875168i \(-0.660750\pi\)
−0.483818 + 0.875168i \(0.660750\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −11.9572 −0.783339 −0.391670 0.920106i \(-0.628102\pi\)
−0.391670 + 0.920106i \(0.628102\pi\)
\(234\) 0 0
\(235\) 9.95715 0.649533
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) −15.5897 −1.00841 −0.504206 0.863583i \(-0.668215\pi\)
−0.504206 + 0.863583i \(0.668215\pi\)
\(240\) 0 0
\(241\) −16.0575 −1.03436 −0.517178 0.855878i \(-0.673018\pi\)
−0.517178 + 0.855878i \(0.673018\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) −5.56825 −0.355742
\(246\) 0 0
\(247\) −6.63565 −0.422217
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) −23.9143 −1.50946 −0.754729 0.656037i \(-0.772232\pi\)
−0.754729 + 0.656037i \(0.772232\pi\)
\(252\) 0 0
\(253\) −2.54262 −0.159853
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 24.4324 1.52405 0.762025 0.647548i \(-0.224206\pi\)
0.762025 + 0.647548i \(0.224206\pi\)
\(258\) 0 0
\(259\) −8.55104 −0.531336
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 14.4935 0.893707 0.446854 0.894607i \(-0.352544\pi\)
0.446854 + 0.894607i \(0.352544\pi\)
\(264\) 0 0
\(265\) 11.4219 0.701641
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 14.0000 0.853595 0.426798 0.904347i \(-0.359642\pi\)
0.426798 + 0.904347i \(0.359642\pi\)
\(270\) 0 0
\(271\) −6.76060 −0.410677 −0.205339 0.978691i \(-0.565830\pi\)
−0.205339 + 0.978691i \(0.565830\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −4.97858 −0.300219
\(276\) 0 0
\(277\) −25.4292 −1.52789 −0.763947 0.645279i \(-0.776741\pi\)
−0.763947 + 0.645279i \(0.776741\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) −9.07896 −0.541605 −0.270803 0.962635i \(-0.587289\pi\)
−0.270803 + 0.962635i \(0.587289\pi\)
\(282\) 0 0
\(283\) 12.0147 0.714199 0.357100 0.934066i \(-0.383766\pi\)
0.357100 + 0.934066i \(0.383766\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) −1.99158 −0.117559
\(288\) 0 0
\(289\) −14.7820 −0.869531
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) −23.6718 −1.38292 −0.691460 0.722415i \(-0.743032\pi\)
−0.691460 + 0.722415i \(0.743032\pi\)
\(294\) 0 0
\(295\) 11.8396 0.689326
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) −3.38890 −0.195985
\(300\) 0 0
\(301\) −7.64973 −0.440923
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 3.66442 0.209824
\(306\) 0 0
\(307\) −16.0246 −0.914570 −0.457285 0.889320i \(-0.651178\pi\)
−0.457285 + 0.889320i \(0.651178\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 27.5468 1.56204 0.781019 0.624508i \(-0.214700\pi\)
0.781019 + 0.624508i \(0.214700\pi\)
\(312\) 0 0
\(313\) 10.1751 0.575133 0.287566 0.957761i \(-0.407154\pi\)
0.287566 + 0.957761i \(0.407154\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −29.6289 −1.66413 −0.832063 0.554681i \(-0.812840\pi\)
−0.832063 + 0.554681i \(0.812840\pi\)
\(318\) 0 0
\(319\) −39.2432 −2.19719
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) −1.48929 −0.0828662
\(324\) 0 0
\(325\) −6.63565 −0.368080
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 11.9143 0.656857
\(330\) 0 0
\(331\) 30.4679 1.67467 0.837333 0.546694i \(-0.184114\pi\)
0.837333 + 0.546694i \(0.184114\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 7.61423 0.416010
\(336\) 0 0
\(337\) 26.1396 1.42392 0.711958 0.702222i \(-0.247808\pi\)
0.711958 + 0.702222i \(0.247808\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 14.8291 0.803039
\(342\) 0 0
\(343\) −15.0386 −0.812010
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 7.31415 0.392644 0.196322 0.980539i \(-0.437100\pi\)
0.196322 + 0.980539i \(0.437100\pi\)
\(348\) 0 0
\(349\) 0.628308 0.0336325 0.0168163 0.999859i \(-0.494647\pi\)
0.0168163 + 0.999859i \(0.494647\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 8.23267 0.438181 0.219090 0.975705i \(-0.429691\pi\)
0.219090 + 0.975705i \(0.429691\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −12.1323 −0.640318 −0.320159 0.947364i \(-0.603736\pi\)
−0.320159 + 0.947364i \(0.603736\pi\)
\(360\) 0 0
\(361\) 1.00000 0.0526316
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 13.8396 0.724396
\(366\) 0 0
\(367\) 17.9572 0.937356 0.468678 0.883369i \(-0.344730\pi\)
0.468678 + 0.883369i \(0.344730\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 13.6669 0.709552
\(372\) 0 0
\(373\) 29.6289 1.53413 0.767064 0.641571i \(-0.221717\pi\)
0.767064 + 0.641571i \(0.221717\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −52.3049 −2.69384
\(378\) 0 0
\(379\) −6.80344 −0.349469 −0.174735 0.984616i \(-0.555907\pi\)
−0.174735 + 0.984616i \(0.555907\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 20.2253 1.03347 0.516733 0.856147i \(-0.327148\pi\)
0.516733 + 0.856147i \(0.327148\pi\)
\(384\) 0 0
\(385\) −5.95715 −0.303605
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) −0.393115 −0.0199317 −0.00996587 0.999950i \(-0.503172\pi\)
−0.00996587 + 0.999950i \(0.503172\pi\)
\(390\) 0 0
\(391\) −0.760597 −0.0384650
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 12.6858 0.638294
\(396\) 0 0
\(397\) 17.4292 0.874748 0.437374 0.899280i \(-0.355909\pi\)
0.437374 + 0.899280i \(0.355909\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −13.1281 −0.655585 −0.327792 0.944750i \(-0.606305\pi\)
−0.327792 + 0.944750i \(0.606305\pi\)
\(402\) 0 0
\(403\) 19.7648 0.984555
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 35.5787 1.76357
\(408\) 0 0
\(409\) −18.7862 −0.928919 −0.464460 0.885594i \(-0.653751\pi\)
−0.464460 + 0.885594i \(0.653751\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 14.1667 0.697098
\(414\) 0 0
\(415\) 8.68585 0.426371
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 5.22219 0.255121 0.127560 0.991831i \(-0.459285\pi\)
0.127560 + 0.991831i \(0.459285\pi\)
\(420\) 0 0
\(421\) −7.68164 −0.374380 −0.187190 0.982324i \(-0.559938\pi\)
−0.187190 + 0.982324i \(0.559938\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) −1.48929 −0.0722411
\(426\) 0 0
\(427\) 4.38469 0.212190
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 17.7073 0.852929 0.426465 0.904504i \(-0.359759\pi\)
0.426465 + 0.904504i \(0.359759\pi\)
\(432\) 0 0
\(433\) 17.9901 0.864551 0.432275 0.901742i \(-0.357711\pi\)
0.432275 + 0.901742i \(0.357711\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 0.510711 0.0244306
\(438\) 0 0
\(439\) 14.8438 0.708454 0.354227 0.935159i \(-0.384744\pi\)
0.354227 + 0.935159i \(0.384744\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −22.4078 −1.06463 −0.532314 0.846547i \(-0.678677\pi\)
−0.532314 + 0.846547i \(0.678677\pi\)
\(444\) 0 0
\(445\) 4.87819 0.231249
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 40.7434 1.92280 0.961400 0.275156i \(-0.0887295\pi\)
0.961400 + 0.275156i \(0.0887295\pi\)
\(450\) 0 0
\(451\) 8.28646 0.390194
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) −7.93994 −0.372230
\(456\) 0 0
\(457\) 8.01721 0.375029 0.187515 0.982262i \(-0.439957\pi\)
0.187515 + 0.982262i \(0.439957\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 25.1281 1.17033 0.585166 0.810914i \(-0.301029\pi\)
0.585166 + 0.810914i \(0.301029\pi\)
\(462\) 0 0
\(463\) −31.7795 −1.47692 −0.738459 0.674298i \(-0.764446\pi\)
−0.738459 + 0.674298i \(0.764446\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −20.9210 −0.968110 −0.484055 0.875038i \(-0.660837\pi\)
−0.484055 + 0.875038i \(0.660837\pi\)
\(468\) 0 0
\(469\) 9.11087 0.420701
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 31.8286 1.46348
\(474\) 0 0
\(475\) 1.00000 0.0458831
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) −27.7220 −1.26665 −0.633324 0.773886i \(-0.718310\pi\)
−0.633324 + 0.773886i \(0.718310\pi\)
\(480\) 0 0
\(481\) 47.4208 2.16220
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) −6.81079 −0.309262
\(486\) 0 0
\(487\) 3.20390 0.145183 0.0725914 0.997362i \(-0.476873\pi\)
0.0725914 + 0.997362i \(0.476873\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 21.5212 0.971238 0.485619 0.874171i \(-0.338594\pi\)
0.485619 + 0.874171i \(0.338594\pi\)
\(492\) 0 0
\(493\) −11.7392 −0.528706
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) −23.1709 −1.03727 −0.518637 0.854995i \(-0.673560\pi\)
−0.518637 + 0.854995i \(0.673560\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 1.88240 0.0839322 0.0419661 0.999119i \(-0.486638\pi\)
0.0419661 + 0.999119i \(0.486638\pi\)
\(504\) 0 0
\(505\) 2.29273 0.102025
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) −5.46365 −0.242172 −0.121086 0.992642i \(-0.538638\pi\)
−0.121086 + 0.992642i \(0.538638\pi\)
\(510\) 0 0
\(511\) 16.5598 0.732564
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) −6.51806 −0.287220
\(516\) 0 0
\(517\) −49.5725 −2.18019
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) −11.2222 −0.491653 −0.245827 0.969314i \(-0.579059\pi\)
−0.245827 + 0.969314i \(0.579059\pi\)
\(522\) 0 0
\(523\) −29.1867 −1.27624 −0.638122 0.769935i \(-0.720289\pi\)
−0.638122 + 0.769935i \(0.720289\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 4.43596 0.193233
\(528\) 0 0
\(529\) −22.7392 −0.988660
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 11.0445 0.478392
\(534\) 0 0
\(535\) −7.71462 −0.333532
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 27.7220 1.19407
\(540\) 0 0
\(541\) 20.2499 0.870611 0.435305 0.900283i \(-0.356640\pi\)
0.435305 + 0.900283i \(0.356640\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 15.5468 0.665953
\(546\) 0 0
\(547\) 42.8830 1.83355 0.916773 0.399409i \(-0.130785\pi\)
0.916773 + 0.399409i \(0.130785\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 7.88240 0.335802
\(552\) 0 0
\(553\) 15.1793 0.645491
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −28.2583 −1.19734 −0.598671 0.800995i \(-0.704304\pi\)
−0.598671 + 0.800995i \(0.704304\pi\)
\(558\) 0 0
\(559\) 42.4225 1.79428
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 20.2253 0.852396 0.426198 0.904630i \(-0.359853\pi\)
0.426198 + 0.904630i \(0.359853\pi\)
\(564\) 0 0
\(565\) 0.753250 0.0316895
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 31.1365 1.30531 0.652655 0.757655i \(-0.273655\pi\)
0.652655 + 0.757655i \(0.273655\pi\)
\(570\) 0 0
\(571\) 27.4868 1.15029 0.575143 0.818053i \(-0.304947\pi\)
0.575143 + 0.818053i \(0.304947\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 0.510711 0.0212981
\(576\) 0 0
\(577\) 23.7820 0.990058 0.495029 0.868876i \(-0.335157\pi\)
0.495029 + 0.868876i \(0.335157\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 10.3931 0.431179
\(582\) 0 0
\(583\) −56.8647 −2.35510
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −24.0147 −0.991192 −0.495596 0.868553i \(-0.665050\pi\)
−0.495596 + 0.868553i \(0.665050\pi\)
\(588\) 0 0
\(589\) −2.97858 −0.122730
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) −46.6148 −1.91424 −0.957121 0.289688i \(-0.906449\pi\)
−0.957121 + 0.289688i \(0.906449\pi\)
\(594\) 0 0
\(595\) −1.78202 −0.0730557
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 44.6002 1.82231 0.911156 0.412061i \(-0.135191\pi\)
0.911156 + 0.412061i \(0.135191\pi\)
\(600\) 0 0
\(601\) −17.5787 −0.717051 −0.358526 0.933520i \(-0.616720\pi\)
−0.358526 + 0.933520i \(0.616720\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 13.7862 0.560490
\(606\) 0 0
\(607\) 37.2467 1.51180 0.755899 0.654688i \(-0.227200\pi\)
0.755899 + 0.654688i \(0.227200\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) −66.0722 −2.67300
\(612\) 0 0
\(613\) −14.8866 −0.601265 −0.300632 0.953740i \(-0.597198\pi\)
−0.300632 + 0.953740i \(0.597198\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −3.70727 −0.149249 −0.0746245 0.997212i \(-0.523776\pi\)
−0.0746245 + 0.997212i \(0.523776\pi\)
\(618\) 0 0
\(619\) 17.7648 0.714028 0.357014 0.934099i \(-0.383795\pi\)
0.357014 + 0.934099i \(0.383795\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 5.83704 0.233856
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 10.6430 0.424364
\(630\) 0 0
\(631\) 40.3074 1.60461 0.802307 0.596912i \(-0.203606\pi\)
0.802307 + 0.596912i \(0.203606\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 10.4177 0.413413
\(636\) 0 0
\(637\) 36.9490 1.46397
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 27.6728 1.09301 0.546506 0.837455i \(-0.315958\pi\)
0.546506 + 0.837455i \(0.315958\pi\)
\(642\) 0 0
\(643\) 19.0277 0.750379 0.375190 0.926948i \(-0.377578\pi\)
0.375190 + 0.926948i \(0.377578\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 32.0403 1.25964 0.629818 0.776743i \(-0.283130\pi\)
0.629818 + 0.776743i \(0.283130\pi\)
\(648\) 0 0
\(649\) −58.9442 −2.31376
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −25.3142 −0.990619 −0.495310 0.868716i \(-0.664945\pi\)
−0.495310 + 0.868716i \(0.664945\pi\)
\(654\) 0 0
\(655\) 17.9572 0.701644
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 0.860981 0.0335391 0.0167695 0.999859i \(-0.494662\pi\)
0.0167695 + 0.999859i \(0.494662\pi\)
\(660\) 0 0
\(661\) 4.55356 0.177113 0.0885564 0.996071i \(-0.471775\pi\)
0.0885564 + 0.996071i \(0.471775\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 1.19656 0.0464005
\(666\) 0 0
\(667\) 4.02563 0.155873
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) −18.2436 −0.704287
\(672\) 0 0
\(673\) −11.7465 −0.452795 −0.226398 0.974035i \(-0.572695\pi\)
−0.226398 + 0.974035i \(0.572695\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 22.2854 0.856497 0.428248 0.903661i \(-0.359131\pi\)
0.428248 + 0.903661i \(0.359131\pi\)
\(678\) 0 0
\(679\) −8.14950 −0.312749
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 7.23833 0.276967 0.138483 0.990365i \(-0.455777\pi\)
0.138483 + 0.990365i \(0.455777\pi\)
\(684\) 0 0
\(685\) 9.83956 0.375950
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) −75.7917 −2.88743
\(690\) 0 0
\(691\) 32.7434 1.24562 0.622809 0.782374i \(-0.285992\pi\)
0.622809 + 0.782374i \(0.285992\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −0.978577 −0.0371195
\(696\) 0 0
\(697\) 2.47881 0.0938915
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 29.4208 1.11121 0.555604 0.831447i \(-0.312487\pi\)
0.555604 + 0.831447i \(0.312487\pi\)
\(702\) 0 0
\(703\) −7.14637 −0.269530
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 2.74338 0.103176
\(708\) 0 0
\(709\) −5.32885 −0.200129 −0.100065 0.994981i \(-0.531905\pi\)
−0.100065 + 0.994981i \(0.531905\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) −1.52119 −0.0569691
\(714\) 0 0
\(715\) 33.0361 1.23548
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) −8.45317 −0.315250 −0.157625 0.987499i \(-0.550384\pi\)
−0.157625 + 0.987499i \(0.550384\pi\)
\(720\) 0 0
\(721\) −7.79923 −0.290459
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 7.88240 0.292745
\(726\) 0 0
\(727\) 2.56825 0.0952511 0.0476256 0.998865i \(-0.484835\pi\)
0.0476256 + 0.998865i \(0.484835\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 9.52119 0.352154
\(732\) 0 0
\(733\) 35.5212 1.31201 0.656003 0.754759i \(-0.272246\pi\)
0.656003 + 0.754759i \(0.272246\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −37.9080 −1.39636
\(738\) 0 0
\(739\) −37.0508 −1.36294 −0.681468 0.731848i \(-0.738658\pi\)
−0.681468 + 0.731848i \(0.738658\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) −23.2039 −0.851269 −0.425634 0.904895i \(-0.639949\pi\)
−0.425634 + 0.904895i \(0.639949\pi\)
\(744\) 0 0
\(745\) 8.24989 0.302252
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) −9.23098 −0.337293
\(750\) 0 0
\(751\) 51.2285 1.86935 0.934677 0.355499i \(-0.115689\pi\)
0.934677 + 0.355499i \(0.115689\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 13.8568 0.504299
\(756\) 0 0
\(757\) −43.3717 −1.57637 −0.788185 0.615438i \(-0.788979\pi\)
−0.788185 + 0.615438i \(0.788979\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −24.7753 −0.898104 −0.449052 0.893506i \(-0.648238\pi\)
−0.449052 + 0.893506i \(0.648238\pi\)
\(762\) 0 0
\(763\) 18.6027 0.673462
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) −78.5632 −2.83675
\(768\) 0 0
\(769\) −11.4893 −0.414314 −0.207157 0.978308i \(-0.566421\pi\)
−0.207157 + 0.978308i \(0.566421\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) −27.0863 −0.974227 −0.487113 0.873339i \(-0.661950\pi\)
−0.487113 + 0.873339i \(0.661950\pi\)
\(774\) 0 0
\(775\) −2.97858 −0.106994
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) −1.66442 −0.0596342
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) −15.7073 −0.560616
\(786\) 0 0
\(787\) 13.4563 0.479666 0.239833 0.970814i \(-0.422907\pi\)
0.239833 + 0.970814i \(0.422907\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 0.901307 0.0320468
\(792\) 0 0
\(793\) −24.3158 −0.863481
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −30.4496 −1.07858 −0.539290 0.842120i \(-0.681307\pi\)
−0.539290 + 0.842120i \(0.681307\pi\)
\(798\) 0 0
\(799\) −14.8291 −0.524615
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) −68.9013 −2.43147
\(804\) 0 0
\(805\) 0.611096 0.0215383
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 29.7047 1.04436 0.522182 0.852834i \(-0.325118\pi\)
0.522182 + 0.852834i \(0.325118\pi\)
\(810\) 0 0
\(811\) 32.2902 1.13386 0.566931 0.823765i \(-0.308130\pi\)
0.566931 + 0.823765i \(0.308130\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 5.80765 0.203433
\(816\) 0 0
\(817\) −6.39312 −0.223667
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −9.06427 −0.316345 −0.158173 0.987411i \(-0.550560\pi\)
−0.158173 + 0.987411i \(0.550560\pi\)
\(822\) 0 0
\(823\) −7.35448 −0.256361 −0.128181 0.991751i \(-0.540914\pi\)
−0.128181 + 0.991751i \(0.540914\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −2.02204 −0.0703132 −0.0351566 0.999382i \(-0.511193\pi\)
−0.0351566 + 0.999382i \(0.511193\pi\)
\(828\) 0 0
\(829\) −36.3675 −1.26309 −0.631547 0.775337i \(-0.717580\pi\)
−0.631547 + 0.775337i \(0.717580\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 8.29273 0.287326
\(834\) 0 0
\(835\) 13.8322 0.478683
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) −25.8223 −0.891486 −0.445743 0.895161i \(-0.647061\pi\)
−0.445743 + 0.895161i \(0.647061\pi\)
\(840\) 0 0
\(841\) 33.1323 1.14249
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 31.0319 1.06753
\(846\) 0 0
\(847\) 16.4960 0.566810
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) −3.64973 −0.125111
\(852\) 0 0
\(853\) 8.54262 0.292494 0.146247 0.989248i \(-0.453281\pi\)
0.146247 + 0.989248i \(0.453281\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −27.8322 −0.950730 −0.475365 0.879789i \(-0.657684\pi\)
−0.475365 + 0.879789i \(0.657684\pi\)
\(858\) 0 0
\(859\) 33.4868 1.14255 0.571277 0.820757i \(-0.306448\pi\)
0.571277 + 0.820757i \(0.306448\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 29.2614 0.996071 0.498036 0.867157i \(-0.334055\pi\)
0.498036 + 0.867157i \(0.334055\pi\)
\(864\) 0 0
\(865\) −14.8108 −0.503582
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) −63.1575 −2.14247
\(870\) 0 0
\(871\) −50.5254 −1.71199
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 1.19656 0.0404510
\(876\) 0 0
\(877\) −46.7852 −1.57982 −0.789911 0.613221i \(-0.789873\pi\)
−0.789911 + 0.613221i \(0.789873\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 30.7581 1.03627 0.518133 0.855300i \(-0.326627\pi\)
0.518133 + 0.855300i \(0.326627\pi\)
\(882\) 0 0
\(883\) 0.435961 0.0146713 0.00733563 0.999973i \(-0.497665\pi\)
0.00733563 + 0.999973i \(0.497665\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 2.15310 0.0722939 0.0361469 0.999346i \(-0.488492\pi\)
0.0361469 + 0.999346i \(0.488492\pi\)
\(888\) 0 0
\(889\) 12.4653 0.418074
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 9.95715 0.333203
\(894\) 0 0
\(895\) −15.6644 −0.523604
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) −23.4783 −0.783047
\(900\) 0 0
\(901\) −17.0105 −0.566701
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) −14.7862 −0.491511
\(906\) 0 0
\(907\) −36.9002 −1.22525 −0.612626 0.790373i \(-0.709887\pi\)
−0.612626 + 0.790373i \(0.709887\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 36.9504 1.22422 0.612111 0.790772i \(-0.290321\pi\)
0.612111 + 0.790772i \(0.290321\pi\)
\(912\) 0 0
\(913\) −43.2432 −1.43114
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 21.4868 0.709556
\(918\) 0 0
\(919\) 19.1537 0.631823 0.315911 0.948789i \(-0.397690\pi\)
0.315911 + 0.948789i \(0.397690\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) −7.14637 −0.234971
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 56.9185 1.86744 0.933718 0.358009i \(-0.116544\pi\)
0.933718 + 0.358009i \(0.116544\pi\)
\(930\) 0 0
\(931\) −5.56825 −0.182492
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 7.41454 0.242481
\(936\) 0 0
\(937\) −24.1176 −0.787888 −0.393944 0.919135i \(-0.628890\pi\)
−0.393944 + 0.919135i \(0.628890\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) −3.19656 −0.104205 −0.0521024 0.998642i \(-0.516592\pi\)
−0.0521024 + 0.998642i \(0.516592\pi\)
\(942\) 0 0
\(943\) −0.850040 −0.0276811
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −3.63504 −0.118123 −0.0590614 0.998254i \(-0.518811\pi\)
−0.0590614 + 0.998254i \(0.518811\pi\)
\(948\) 0 0
\(949\) −91.8345 −2.98107
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 48.0821 1.55753 0.778766 0.627315i \(-0.215846\pi\)
0.778766 + 0.627315i \(0.215846\pi\)
\(954\) 0 0
\(955\) 6.36748 0.206047
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 11.7736 0.380189
\(960\) 0 0
\(961\) −22.1281 −0.713809
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) −19.5970 −0.630850
\(966\) 0 0
\(967\) −23.4637 −0.754540 −0.377270 0.926103i \(-0.623137\pi\)
−0.377270 + 0.926103i \(0.623137\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) −17.4868 −0.561177 −0.280589 0.959828i \(-0.590530\pi\)
−0.280589 + 0.959828i \(0.590530\pi\)
\(972\) 0 0
\(973\) −1.17092 −0.0375381
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −20.2316 −0.647266 −0.323633 0.946183i \(-0.604904\pi\)
−0.323633 + 0.946183i \(0.604904\pi\)
\(978\) 0 0
\(979\) −24.2865 −0.776199
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) −28.1249 −0.897046 −0.448523 0.893771i \(-0.648050\pi\)
−0.448523 + 0.893771i \(0.648050\pi\)
\(984\) 0 0
\(985\) −10.7862 −0.343678
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) −3.26504 −0.103822
\(990\) 0 0
\(991\) −33.9718 −1.07915 −0.539576 0.841937i \(-0.681415\pi\)
−0.539576 + 0.841937i \(0.681415\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 2.80344 0.0888751
\(996\) 0 0
\(997\) −23.8223 −0.754461 −0.377231 0.926119i \(-0.623124\pi\)
−0.377231 + 0.926119i \(0.623124\pi\)
\(998\) 0 0
\(999\) 0 0
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 6840.2.a.bm.1.2 3
3.2 odd 2 760.2.a.i.1.1 3
12.11 even 2 1520.2.a.q.1.3 3
15.2 even 4 3800.2.d.n.3649.6 6
15.8 even 4 3800.2.d.n.3649.1 6
15.14 odd 2 3800.2.a.w.1.3 3
24.5 odd 2 6080.2.a.bx.1.3 3
24.11 even 2 6080.2.a.br.1.1 3
60.59 even 2 7600.2.a.bp.1.1 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
760.2.a.i.1.1 3 3.2 odd 2
1520.2.a.q.1.3 3 12.11 even 2
3800.2.a.w.1.3 3 15.14 odd 2
3800.2.d.n.3649.1 6 15.8 even 4
3800.2.d.n.3649.6 6 15.2 even 4
6080.2.a.br.1.1 3 24.11 even 2
6080.2.a.bx.1.3 3 24.5 odd 2
6840.2.a.bm.1.2 3 1.1 even 1 trivial
7600.2.a.bp.1.1 3 60.59 even 2