Properties

Label 1600.3.h.n.1599.5
Level $1600$
Weight $3$
Character 1600.1599
Analytic conductor $43.597$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1600,3,Mod(1599,1600)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1600, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0, 1]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1600.1599");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1600 = 2^{6} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 1600.h (of order \(2\), degree \(1\), not minimal)

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: no
Analytic conductor: \(43.5968422976\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\Q(\zeta_{20})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - x^{6} + x^{4} - x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{17} \)
Twist minimal: no (minimal twist has level 20)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 1599.5
Root \(-0.587785 + 0.809017i\) of defining polynomial
Character \(\chi\) \(=\) 1600.1599
Dual form 1600.3.h.n.1599.6

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+2.35114 q^{3} +5.25731 q^{7} -3.47214 q^{9} +O(q^{10})\) \(q+2.35114 q^{3} +5.25731 q^{7} -3.47214 q^{9} -19.9192i q^{11} -8.47214i q^{13} +11.8885i q^{17} -15.2169i q^{19} +12.3607 q^{21} -0.555029 q^{23} -29.3238 q^{27} -10.9443 q^{29} -8.29451i q^{31} -46.8328i q^{33} +18.3607i q^{37} -19.9192i q^{39} -14.5836 q^{41} -22.2703 q^{43} -53.3902 q^{47} -21.3607 q^{49} +27.9516i q^{51} -66.3607i q^{53} -35.7771i q^{57} +17.4370i q^{59} -90.1378 q^{61} -18.2541 q^{63} +50.2220 q^{67} -1.30495 q^{69} -80.7868i q^{71} +5.55418i q^{73} -104.721i q^{77} +13.8448i q^{79} -37.6950 q^{81} +76.2155 q^{83} -25.7315 q^{87} +111.443 q^{89} -44.5407i q^{91} -19.5016i q^{93} -92.8328i q^{97} +69.1621i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 8 q^{9} - 80 q^{21} - 16 q^{29} - 224 q^{41} + 8 q^{49} - 256 q^{61} + 240 q^{69} - 552 q^{81} + 176 q^{89}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1600\mathbb{Z}\right)^\times\).

\(n\) \(577\) \(901\) \(1151\)
\(\chi(n)\) \(-1\) \(1\) \(-1\)

Coefficient data

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



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 2.35114 0.783714 0.391857 0.920026i \(-0.371833\pi\)
0.391857 + 0.920026i \(0.371833\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) 5.25731 0.751044 0.375522 0.926813i \(-0.377463\pi\)
0.375522 + 0.926813i \(0.377463\pi\)
\(8\) 0 0
\(9\) −3.47214 −0.385793
\(10\) 0 0
\(11\) − 19.9192i − 1.81084i −0.424522 0.905418i \(-0.639558\pi\)
0.424522 0.905418i \(-0.360442\pi\)
\(12\) 0 0
\(13\) − 8.47214i − 0.651703i −0.945421 0.325851i \(-0.894349\pi\)
0.945421 0.325851i \(-0.105651\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 11.8885i 0.699326i 0.936876 + 0.349663i \(0.113704\pi\)
−0.936876 + 0.349663i \(0.886296\pi\)
\(18\) 0 0
\(19\) − 15.2169i − 0.800890i −0.916321 0.400445i \(-0.868856\pi\)
0.916321 0.400445i \(-0.131144\pi\)
\(20\) 0 0
\(21\) 12.3607 0.588604
\(22\) 0 0
\(23\) −0.555029 −0.0241317 −0.0120659 0.999927i \(-0.503841\pi\)
−0.0120659 + 0.999927i \(0.503841\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) −29.3238 −1.08606
\(28\) 0 0
\(29\) −10.9443 −0.377389 −0.188694 0.982036i \(-0.560426\pi\)
−0.188694 + 0.982036i \(0.560426\pi\)
\(30\) 0 0
\(31\) − 8.29451i − 0.267565i −0.991011 0.133782i \(-0.957288\pi\)
0.991011 0.133782i \(-0.0427123\pi\)
\(32\) 0 0
\(33\) − 46.8328i − 1.41918i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 18.3607i 0.496235i 0.968730 + 0.248117i \(0.0798118\pi\)
−0.968730 + 0.248117i \(0.920188\pi\)
\(38\) 0 0
\(39\) − 19.9192i − 0.510748i
\(40\) 0 0
\(41\) −14.5836 −0.355697 −0.177849 0.984058i \(-0.556914\pi\)
−0.177849 + 0.984058i \(0.556914\pi\)
\(42\) 0 0
\(43\) −22.2703 −0.517915 −0.258957 0.965889i \(-0.583379\pi\)
−0.258957 + 0.965889i \(0.583379\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −53.3902 −1.13596 −0.567981 0.823042i \(-0.692275\pi\)
−0.567981 + 0.823042i \(0.692275\pi\)
\(48\) 0 0
\(49\) −21.3607 −0.435932
\(50\) 0 0
\(51\) 27.9516i 0.548071i
\(52\) 0 0
\(53\) − 66.3607i − 1.25209i −0.779788 0.626044i \(-0.784673\pi\)
0.779788 0.626044i \(-0.215327\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) − 35.7771i − 0.627668i
\(58\) 0 0
\(59\) 17.4370i 0.295543i 0.989022 + 0.147771i \(0.0472100\pi\)
−0.989022 + 0.147771i \(0.952790\pi\)
\(60\) 0 0
\(61\) −90.1378 −1.47767 −0.738834 0.673887i \(-0.764623\pi\)
−0.738834 + 0.673887i \(0.764623\pi\)
\(62\) 0 0
\(63\) −18.2541 −0.289748
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 50.2220 0.749582 0.374791 0.927109i \(-0.377715\pi\)
0.374791 + 0.927109i \(0.377715\pi\)
\(68\) 0 0
\(69\) −1.30495 −0.0189123
\(70\) 0 0
\(71\) − 80.7868i − 1.13784i −0.822392 0.568921i \(-0.807361\pi\)
0.822392 0.568921i \(-0.192639\pi\)
\(72\) 0 0
\(73\) 5.55418i 0.0760846i 0.999276 + 0.0380423i \(0.0121122\pi\)
−0.999276 + 0.0380423i \(0.987888\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) − 104.721i − 1.36002i
\(78\) 0 0
\(79\) 13.8448i 0.175251i 0.996154 + 0.0876253i \(0.0279278\pi\)
−0.996154 + 0.0876253i \(0.972072\pi\)
\(80\) 0 0
\(81\) −37.6950 −0.465371
\(82\) 0 0
\(83\) 76.2155 0.918260 0.459130 0.888369i \(-0.348161\pi\)
0.459130 + 0.888369i \(0.348161\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) −25.7315 −0.295765
\(88\) 0 0
\(89\) 111.443 1.25217 0.626083 0.779757i \(-0.284657\pi\)
0.626083 + 0.779757i \(0.284657\pi\)
\(90\) 0 0
\(91\) − 44.5407i − 0.489458i
\(92\) 0 0
\(93\) − 19.5016i − 0.209694i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) − 92.8328i − 0.957039i −0.878077 0.478520i \(-0.841174\pi\)
0.878077 0.478520i \(-0.158826\pi\)
\(98\) 0 0
\(99\) 69.1621i 0.698607i
\(100\) 0 0
\(101\) −64.1115 −0.634767 −0.317383 0.948297i \(-0.602804\pi\)
−0.317383 + 0.948297i \(0.602804\pi\)
\(102\) 0 0
\(103\) 137.769 1.33757 0.668783 0.743458i \(-0.266816\pi\)
0.668783 + 0.743458i \(0.266816\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −51.3320 −0.479739 −0.239869 0.970805i \(-0.577105\pi\)
−0.239869 + 0.970805i \(0.577105\pi\)
\(108\) 0 0
\(109\) 133.469 1.22449 0.612243 0.790669i \(-0.290267\pi\)
0.612243 + 0.790669i \(0.290267\pi\)
\(110\) 0 0
\(111\) 43.1685i 0.388906i
\(112\) 0 0
\(113\) − 170.721i − 1.51081i −0.655259 0.755404i \(-0.727441\pi\)
0.655259 0.755404i \(-0.272559\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 29.4164i 0.251422i
\(118\) 0 0
\(119\) 62.5018i 0.525225i
\(120\) 0 0
\(121\) −275.774 −2.27912
\(122\) 0 0
\(123\) −34.2881 −0.278765
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) 198.637 1.56407 0.782035 0.623235i \(-0.214182\pi\)
0.782035 + 0.623235i \(0.214182\pi\)
\(128\) 0 0
\(129\) −52.3607 −0.405897
\(130\) 0 0
\(131\) − 7.77041i − 0.0593161i −0.999560 0.0296580i \(-0.990558\pi\)
0.999560 0.0296580i \(-0.00944183\pi\)
\(132\) 0 0
\(133\) − 80.0000i − 0.601504i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 0.832816i 0.00607895i 0.999995 + 0.00303947i \(0.000967496\pi\)
−0.999995 + 0.00303947i \(0.999033\pi\)
\(138\) 0 0
\(139\) − 237.658i − 1.70977i −0.518817 0.854885i \(-0.673627\pi\)
0.518817 0.854885i \(-0.326373\pi\)
\(140\) 0 0
\(141\) −125.528 −0.890269
\(142\) 0 0
\(143\) −168.758 −1.18013
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) −50.2220 −0.341646
\(148\) 0 0
\(149\) −36.9706 −0.248125 −0.124062 0.992274i \(-0.539592\pi\)
−0.124062 + 0.992274i \(0.539592\pi\)
\(150\) 0 0
\(151\) 282.723i 1.87234i 0.351552 + 0.936168i \(0.385654\pi\)
−0.351552 + 0.936168i \(0.614346\pi\)
\(152\) 0 0
\(153\) − 41.2786i − 0.269795i
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) − 204.748i − 1.30413i −0.758165 0.652063i \(-0.773904\pi\)
0.758165 0.652063i \(-0.226096\pi\)
\(158\) 0 0
\(159\) − 156.023i − 0.981279i
\(160\) 0 0
\(161\) −2.91796 −0.0181240
\(162\) 0 0
\(163\) −107.235 −0.657885 −0.328943 0.944350i \(-0.606692\pi\)
−0.328943 + 0.944350i \(0.606692\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −33.2090 −0.198856 −0.0994280 0.995045i \(-0.531701\pi\)
−0.0994280 + 0.995045i \(0.531701\pi\)
\(168\) 0 0
\(169\) 97.2229 0.575284
\(170\) 0 0
\(171\) 52.8352i 0.308978i
\(172\) 0 0
\(173\) − 226.361i − 1.30844i −0.756303 0.654222i \(-0.772996\pi\)
0.756303 0.654222i \(-0.227004\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 40.9969i 0.231621i
\(178\) 0 0
\(179\) 224.337i 1.25328i 0.779308 + 0.626641i \(0.215571\pi\)
−0.779308 + 0.626641i \(0.784429\pi\)
\(180\) 0 0
\(181\) −86.2229 −0.476370 −0.238185 0.971220i \(-0.576552\pi\)
−0.238185 + 0.971220i \(0.576552\pi\)
\(182\) 0 0
\(183\) −211.927 −1.15807
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 236.810 1.26636
\(188\) 0 0
\(189\) −154.164 −0.815683
\(190\) 0 0
\(191\) − 31.0198i − 0.162407i −0.996698 0.0812036i \(-0.974124\pi\)
0.996698 0.0812036i \(-0.0258764\pi\)
\(192\) 0 0
\(193\) − 110.223i − 0.571103i −0.958363 0.285552i \(-0.907823\pi\)
0.958363 0.285552i \(-0.0921768\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 172.525i 0.875760i 0.899033 + 0.437880i \(0.144271\pi\)
−0.899033 + 0.437880i \(0.855729\pi\)
\(198\) 0 0
\(199\) 272.208i 1.36788i 0.729538 + 0.683940i \(0.239735\pi\)
−0.729538 + 0.683940i \(0.760265\pi\)
\(200\) 0 0
\(201\) 118.079 0.587457
\(202\) 0 0
\(203\) −57.5374 −0.283436
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 1.92714 0.00930984
\(208\) 0 0
\(209\) −303.108 −1.45028
\(210\) 0 0
\(211\) 205.266i 0.972826i 0.873729 + 0.486413i \(0.161695\pi\)
−0.873729 + 0.486413i \(0.838305\pi\)
\(212\) 0 0
\(213\) − 189.941i − 0.891743i
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) − 43.6068i − 0.200953i
\(218\) 0 0
\(219\) 13.0586i 0.0596285i
\(220\) 0 0
\(221\) 100.721 0.455753
\(222\) 0 0
\(223\) 235.731 1.05709 0.528545 0.848905i \(-0.322738\pi\)
0.528545 + 0.848905i \(0.322738\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) −58.5165 −0.257782 −0.128891 0.991659i \(-0.541142\pi\)
−0.128891 + 0.991659i \(0.541142\pi\)
\(228\) 0 0
\(229\) 162.721 0.710574 0.355287 0.934757i \(-0.384383\pi\)
0.355287 + 0.934757i \(0.384383\pi\)
\(230\) 0 0
\(231\) − 246.215i − 1.06586i
\(232\) 0 0
\(233\) − 319.050i − 1.36931i −0.728867 0.684656i \(-0.759953\pi\)
0.728867 0.684656i \(-0.240047\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 32.5511i 0.137346i
\(238\) 0 0
\(239\) − 236.810i − 0.990837i −0.868654 0.495419i \(-0.835015\pi\)
0.868654 0.495419i \(-0.164985\pi\)
\(240\) 0 0
\(241\) −0.917961 −0.00380897 −0.00190448 0.999998i \(-0.500606\pi\)
−0.00190448 + 0.999998i \(0.500606\pi\)
\(242\) 0 0
\(243\) 175.287 0.721347
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) −128.920 −0.521942
\(248\) 0 0
\(249\) 179.193 0.719653
\(250\) 0 0
\(251\) 136.690i 0.544582i 0.962215 + 0.272291i \(0.0877813\pi\)
−0.962215 + 0.272291i \(0.912219\pi\)
\(252\) 0 0
\(253\) 11.0557i 0.0436985i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) − 274.944i − 1.06982i −0.844908 0.534911i \(-0.820345\pi\)
0.844908 0.534911i \(-0.179655\pi\)
\(258\) 0 0
\(259\) 96.5278i 0.372694i
\(260\) 0 0
\(261\) 38.0000 0.145594
\(262\) 0 0
\(263\) 406.385 1.54519 0.772596 0.634899i \(-0.218958\pi\)
0.772596 + 0.634899i \(0.218958\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 262.018 0.981339
\(268\) 0 0
\(269\) −348.525 −1.29563 −0.647816 0.761797i \(-0.724317\pi\)
−0.647816 + 0.761797i \(0.724317\pi\)
\(270\) 0 0
\(271\) − 247.849i − 0.914571i −0.889320 0.457286i \(-0.848822\pi\)
0.889320 0.457286i \(-0.151178\pi\)
\(272\) 0 0
\(273\) − 104.721i − 0.383595i
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) 54.7539i 0.197667i 0.995104 + 0.0988337i \(0.0315112\pi\)
−0.995104 + 0.0988337i \(0.968489\pi\)
\(278\) 0 0
\(279\) 28.7997i 0.103225i
\(280\) 0 0
\(281\) −50.3607 −0.179220 −0.0896098 0.995977i \(-0.528562\pi\)
−0.0896098 + 0.995977i \(0.528562\pi\)
\(282\) 0 0
\(283\) 147.336 0.520621 0.260310 0.965525i \(-0.416175\pi\)
0.260310 + 0.965525i \(0.416175\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) −76.6705 −0.267145
\(288\) 0 0
\(289\) 147.663 0.510943
\(290\) 0 0
\(291\) − 218.263i − 0.750045i
\(292\) 0 0
\(293\) 178.859i 0.610441i 0.952282 + 0.305220i \(0.0987301\pi\)
−0.952282 + 0.305220i \(0.901270\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 584.105i 1.96668i
\(298\) 0 0
\(299\) 4.70228i 0.0157267i
\(300\) 0 0
\(301\) −117.082 −0.388977
\(302\) 0 0
\(303\) −150.735 −0.497475
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) −284.550 −0.926873 −0.463436 0.886130i \(-0.653384\pi\)
−0.463436 + 0.886130i \(0.653384\pi\)
\(308\) 0 0
\(309\) 323.915 1.04827
\(310\) 0 0
\(311\) 282.199i 0.907392i 0.891157 + 0.453696i \(0.149895\pi\)
−0.891157 + 0.453696i \(0.850105\pi\)
\(312\) 0 0
\(313\) 567.548i 1.81325i 0.421935 + 0.906626i \(0.361351\pi\)
−0.421935 + 0.906626i \(0.638649\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) − 161.141i − 0.508331i −0.967161 0.254165i \(-0.918199\pi\)
0.967161 0.254165i \(-0.0818008\pi\)
\(318\) 0 0
\(319\) 218.001i 0.683389i
\(320\) 0 0
\(321\) −120.689 −0.375978
\(322\) 0 0
\(323\) 180.907 0.560083
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) 313.805 0.959647
\(328\) 0 0
\(329\) −280.689 −0.853158
\(330\) 0 0
\(331\) − 331.966i − 1.00292i −0.865181 0.501459i \(-0.832797\pi\)
0.865181 0.501459i \(-0.167203\pi\)
\(332\) 0 0
\(333\) − 63.7508i − 0.191444i
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) − 269.108i − 0.798541i −0.916833 0.399271i \(-0.869263\pi\)
0.916833 0.399271i \(-0.130737\pi\)
\(338\) 0 0
\(339\) − 401.390i − 1.18404i
\(340\) 0 0
\(341\) −165.220 −0.484516
\(342\) 0 0
\(343\) −369.908 −1.07845
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −503.075 −1.44978 −0.724892 0.688863i \(-0.758110\pi\)
−0.724892 + 0.688863i \(0.758110\pi\)
\(348\) 0 0
\(349\) −0.504658 −0.00144601 −0.000723006 1.00000i \(-0.500230\pi\)
−0.000723006 1.00000i \(0.500230\pi\)
\(350\) 0 0
\(351\) 248.435i 0.707791i
\(352\) 0 0
\(353\) 335.994i 0.951824i 0.879493 + 0.475912i \(0.157882\pi\)
−0.879493 + 0.475912i \(0.842118\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 146.950i 0.411626i
\(358\) 0 0
\(359\) − 98.4859i − 0.274334i −0.990548 0.137167i \(-0.956200\pi\)
0.990548 0.137167i \(-0.0437997\pi\)
\(360\) 0 0
\(361\) 129.446 0.358576
\(362\) 0 0
\(363\) −648.384 −1.78618
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) −498.473 −1.35824 −0.679118 0.734029i \(-0.737638\pi\)
−0.679118 + 0.734029i \(0.737638\pi\)
\(368\) 0 0
\(369\) 50.6362 0.137226
\(370\) 0 0
\(371\) − 348.879i − 0.940374i
\(372\) 0 0
\(373\) 600.354i 1.60953i 0.593594 + 0.804765i \(0.297709\pi\)
−0.593594 + 0.804765i \(0.702291\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 92.7214i 0.245945i
\(378\) 0 0
\(379\) 303.490i 0.800765i 0.916348 + 0.400383i \(0.131123\pi\)
−0.916348 + 0.400383i \(0.868877\pi\)
\(380\) 0 0
\(381\) 467.023 1.22578
\(382\) 0 0
\(383\) −332.583 −0.868362 −0.434181 0.900826i \(-0.642962\pi\)
−0.434181 + 0.900826i \(0.642962\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 77.3256 0.199808
\(388\) 0 0
\(389\) 392.354 1.00862 0.504312 0.863522i \(-0.331746\pi\)
0.504312 + 0.863522i \(0.331746\pi\)
\(390\) 0 0
\(391\) − 6.59849i − 0.0168759i
\(392\) 0 0
\(393\) − 18.2693i − 0.0464868i
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) − 334.190i − 0.841789i −0.907110 0.420895i \(-0.861716\pi\)
0.907110 0.420895i \(-0.138284\pi\)
\(398\) 0 0
\(399\) − 188.091i − 0.471407i
\(400\) 0 0
\(401\) 121.003 0.301753 0.150877 0.988553i \(-0.451790\pi\)
0.150877 + 0.988553i \(0.451790\pi\)
\(402\) 0 0
\(403\) −70.2722 −0.174373
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 365.730 0.898599
\(408\) 0 0
\(409\) 607.410 1.48511 0.742555 0.669785i \(-0.233614\pi\)
0.742555 + 0.669785i \(0.233614\pi\)
\(410\) 0 0
\(411\) 1.95807i 0.00476415i
\(412\) 0 0
\(413\) 91.6718i 0.221966i
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) − 558.768i − 1.33997i
\(418\) 0 0
\(419\) − 466.760i − 1.11398i −0.830518 0.556992i \(-0.811955\pi\)
0.830518 0.556992i \(-0.188045\pi\)
\(420\) 0 0
\(421\) 73.0883 0.173606 0.0868031 0.996225i \(-0.472335\pi\)
0.0868031 + 0.996225i \(0.472335\pi\)
\(422\) 0 0
\(423\) 185.378 0.438246
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) −473.882 −1.10979
\(428\) 0 0
\(429\) −396.774 −0.924881
\(430\) 0 0
\(431\) − 463.630i − 1.07571i −0.843038 0.537853i \(-0.819235\pi\)
0.843038 0.537853i \(-0.180765\pi\)
\(432\) 0 0
\(433\) 99.8359i 0.230568i 0.993333 + 0.115284i \(0.0367778\pi\)
−0.993333 + 0.115284i \(0.963222\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 8.44582i 0.0193268i
\(438\) 0 0
\(439\) − 374.086i − 0.852133i −0.904692 0.426066i \(-0.859899\pi\)
0.904692 0.426066i \(-0.140101\pi\)
\(440\) 0 0
\(441\) 74.1672 0.168180
\(442\) 0 0
\(443\) 290.100 0.654854 0.327427 0.944877i \(-0.393818\pi\)
0.327427 + 0.944877i \(0.393818\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) −86.9231 −0.194459
\(448\) 0 0
\(449\) −299.921 −0.667976 −0.333988 0.942577i \(-0.608394\pi\)
−0.333988 + 0.942577i \(0.608394\pi\)
\(450\) 0 0
\(451\) 290.493i 0.644109i
\(452\) 0 0
\(453\) 664.721i 1.46738i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 822.328i 1.79941i 0.436503 + 0.899703i \(0.356217\pi\)
−0.436503 + 0.899703i \(0.643783\pi\)
\(458\) 0 0
\(459\) − 348.617i − 0.759513i
\(460\) 0 0
\(461\) 456.885 0.991075 0.495537 0.868587i \(-0.334971\pi\)
0.495537 + 0.868587i \(0.334971\pi\)
\(462\) 0 0
\(463\) 400.249 0.864469 0.432234 0.901761i \(-0.357725\pi\)
0.432234 + 0.901761i \(0.357725\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 913.145 1.95534 0.977672 0.210139i \(-0.0673915\pi\)
0.977672 + 0.210139i \(0.0673915\pi\)
\(468\) 0 0
\(469\) 264.033 0.562969
\(470\) 0 0
\(471\) − 481.391i − 1.02206i
\(472\) 0 0
\(473\) 443.607i 0.937858i
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 230.413i 0.483047i
\(478\) 0 0
\(479\) 526.131i 1.09840i 0.835692 + 0.549198i \(0.185067\pi\)
−0.835692 + 0.549198i \(0.814933\pi\)
\(480\) 0 0
\(481\) 155.554 0.323397
\(482\) 0 0
\(483\) −6.86054 −0.0142040
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) −443.541 −0.910762 −0.455381 0.890297i \(-0.650497\pi\)
−0.455381 + 0.890297i \(0.650497\pi\)
\(488\) 0 0
\(489\) −252.125 −0.515594
\(490\) 0 0
\(491\) 287.163i 0.584854i 0.956288 + 0.292427i \(0.0944628\pi\)
−0.956288 + 0.292427i \(0.905537\pi\)
\(492\) 0 0
\(493\) − 130.111i − 0.263918i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) − 424.721i − 0.854570i
\(498\) 0 0
\(499\) 810.936i 1.62512i 0.582876 + 0.812561i \(0.301927\pi\)
−0.582876 + 0.812561i \(0.698073\pi\)
\(500\) 0 0
\(501\) −78.0789 −0.155846
\(502\) 0 0
\(503\) 642.471 1.27728 0.638639 0.769506i \(-0.279498\pi\)
0.638639 + 0.769506i \(0.279498\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 228.585 0.450858
\(508\) 0 0
\(509\) −915.050 −1.79774 −0.898870 0.438216i \(-0.855611\pi\)
−0.898870 + 0.438216i \(0.855611\pi\)
\(510\) 0 0
\(511\) 29.2000i 0.0571429i
\(512\) 0 0
\(513\) 446.217i 0.869818i
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 1063.49i 2.05704i
\(518\) 0 0
\(519\) − 532.206i − 1.02544i
\(520\) 0 0
\(521\) 1006.98 1.93279 0.966396 0.257058i \(-0.0827533\pi\)
0.966396 + 0.257058i \(0.0827533\pi\)
\(522\) 0 0
\(523\) 774.173 1.48025 0.740127 0.672467i \(-0.234765\pi\)
0.740127 + 0.672467i \(0.234765\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 98.6096 0.187115
\(528\) 0 0
\(529\) −528.692 −0.999418
\(530\) 0 0
\(531\) − 60.5437i − 0.114018i
\(532\) 0 0
\(533\) 123.554i 0.231809i
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 527.449i 0.982214i
\(538\) 0 0
\(539\) 425.487i 0.789401i
\(540\) 0 0
\(541\) 259.115 0.478955 0.239477 0.970902i \(-0.423024\pi\)
0.239477 + 0.970902i \(0.423024\pi\)
\(542\) 0 0
\(543\) −202.722 −0.373337
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) −149.818 −0.273890 −0.136945 0.990579i \(-0.543728\pi\)
−0.136945 + 0.990579i \(0.543728\pi\)
\(548\) 0 0
\(549\) 312.971 0.570074
\(550\) 0 0
\(551\) 166.538i 0.302247i
\(552\) 0 0
\(553\) 72.7864i 0.131621i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) − 511.698i − 0.918668i −0.888264 0.459334i \(-0.848088\pi\)
0.888264 0.459334i \(-0.151912\pi\)
\(558\) 0 0
\(559\) 188.677i 0.337526i
\(560\) 0 0
\(561\) 556.774 0.992467
\(562\) 0 0
\(563\) −490.726 −0.871627 −0.435814 0.900037i \(-0.643539\pi\)
−0.435814 + 0.900037i \(0.643539\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) −198.175 −0.349514
\(568\) 0 0
\(569\) 232.748 0.409047 0.204523 0.978862i \(-0.434436\pi\)
0.204523 + 0.978862i \(0.434436\pi\)
\(570\) 0 0
\(571\) 210.755i 0.369098i 0.982823 + 0.184549i \(0.0590824\pi\)
−0.982823 + 0.184549i \(0.940918\pi\)
\(572\) 0 0
\(573\) − 72.9318i − 0.127281i
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) 341.712i 0.592222i 0.955154 + 0.296111i \(0.0956898\pi\)
−0.955154 + 0.296111i \(0.904310\pi\)
\(578\) 0 0
\(579\) − 259.150i − 0.447581i
\(580\) 0 0
\(581\) 400.689 0.689654
\(582\) 0 0
\(583\) −1321.85 −2.26733
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 618.412 1.05351 0.526756 0.850016i \(-0.323408\pi\)
0.526756 + 0.850016i \(0.323408\pi\)
\(588\) 0 0
\(589\) −126.217 −0.214290
\(590\) 0 0
\(591\) 405.630i 0.686345i
\(592\) 0 0
\(593\) − 120.663i − 0.203478i −0.994811 0.101739i \(-0.967559\pi\)
0.994811 0.101739i \(-0.0324407\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 640.000i 1.07203i
\(598\) 0 0
\(599\) 849.927i 1.41891i 0.704751 + 0.709455i \(0.251059\pi\)
−0.704751 + 0.709455i \(0.748941\pi\)
\(600\) 0 0
\(601\) −11.3576 −0.0188978 −0.00944890 0.999955i \(-0.503008\pi\)
−0.00944890 + 0.999955i \(0.503008\pi\)
\(602\) 0 0
\(603\) −174.378 −0.289183
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) 1115.12 1.83710 0.918550 0.395305i \(-0.129361\pi\)
0.918550 + 0.395305i \(0.129361\pi\)
\(608\) 0 0
\(609\) −135.279 −0.222132
\(610\) 0 0
\(611\) 452.329i 0.740309i
\(612\) 0 0
\(613\) 499.475i 0.814805i 0.913249 + 0.407402i \(0.133565\pi\)
−0.913249 + 0.407402i \(0.866435\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 545.935i 0.884822i 0.896813 + 0.442411i \(0.145877\pi\)
−0.896813 + 0.442411i \(0.854123\pi\)
\(618\) 0 0
\(619\) 455.011i 0.735075i 0.930009 + 0.367537i \(0.119799\pi\)
−0.930009 + 0.367537i \(0.880201\pi\)
\(620\) 0 0
\(621\) 16.2755 0.0262086
\(622\) 0 0
\(623\) 585.889 0.940432
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) −712.650 −1.13660
\(628\) 0 0
\(629\) −218.282 −0.347030
\(630\) 0 0
\(631\) 267.706i 0.424257i 0.977242 + 0.212128i \(0.0680395\pi\)
−0.977242 + 0.212128i \(0.931960\pi\)
\(632\) 0 0
\(633\) 482.610i 0.762417i
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 180.971i 0.284098i
\(638\) 0 0
\(639\) 280.503i 0.438971i
\(640\) 0 0
\(641\) −418.571 −0.652997 −0.326499 0.945198i \(-0.605869\pi\)
−0.326499 + 0.945198i \(0.605869\pi\)
\(642\) 0 0
\(643\) −439.339 −0.683265 −0.341633 0.939834i \(-0.610980\pi\)
−0.341633 + 0.939834i \(0.610980\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −419.644 −0.648600 −0.324300 0.945954i \(-0.605129\pi\)
−0.324300 + 0.945954i \(0.605129\pi\)
\(648\) 0 0
\(649\) 347.331 0.535179
\(650\) 0 0
\(651\) − 102.526i − 0.157490i
\(652\) 0 0
\(653\) − 370.085i − 0.566746i −0.959010 0.283373i \(-0.908547\pi\)
0.959010 0.283373i \(-0.0914534\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) − 19.2849i − 0.0293529i
\(658\) 0 0
\(659\) − 322.823i − 0.489868i −0.969540 0.244934i \(-0.921234\pi\)
0.969540 0.244934i \(-0.0787664\pi\)
\(660\) 0 0
\(661\) 812.735 1.22955 0.614777 0.788701i \(-0.289246\pi\)
0.614777 + 0.788701i \(0.289246\pi\)
\(662\) 0 0
\(663\) 236.810 0.357180
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 6.07439 0.00910703
\(668\) 0 0
\(669\) 554.237 0.828456
\(670\) 0 0
\(671\) 1795.47i 2.67581i
\(672\) 0 0
\(673\) − 467.378i − 0.694469i −0.937778 0.347235i \(-0.887121\pi\)
0.937778 0.347235i \(-0.112879\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) − 548.237i − 0.809803i −0.914360 0.404902i \(-0.867306\pi\)
0.914360 0.404902i \(-0.132694\pi\)
\(678\) 0 0
\(679\) − 488.051i − 0.718779i
\(680\) 0 0
\(681\) −137.580 −0.202027
\(682\) 0 0
\(683\) −23.9663 −0.0350898 −0.0175449 0.999846i \(-0.505585\pi\)
−0.0175449 + 0.999846i \(0.505585\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) 382.581 0.556886
\(688\) 0 0
\(689\) −562.217 −0.815989
\(690\) 0 0
\(691\) 186.981i 0.270595i 0.990805 + 0.135298i \(0.0431990\pi\)
−0.990805 + 0.135298i \(0.956801\pi\)
\(692\) 0 0
\(693\) 363.607i 0.524685i
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) − 173.378i − 0.248748i
\(698\) 0 0
\(699\) − 750.130i − 1.07315i
\(700\) 0 0
\(701\) 706.636 1.00804 0.504020 0.863692i \(-0.331854\pi\)
0.504020 + 0.863692i \(0.331854\pi\)
\(702\) 0 0
\(703\) 279.393 0.397429
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) −337.054 −0.476738
\(708\) 0 0
\(709\) −188.597 −0.266005 −0.133002 0.991116i \(-0.542462\pi\)
−0.133002 + 0.991116i \(0.542462\pi\)
\(710\) 0 0
\(711\) − 48.0710i − 0.0676104i
\(712\) 0 0
\(713\) 4.60369i 0.00645679i
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) − 556.774i − 0.776533i
\(718\) 0 0
\(719\) − 156.085i − 0.217086i −0.994092 0.108543i \(-0.965381\pi\)
0.994092 0.108543i \(-0.0346186\pi\)
\(720\) 0 0
\(721\) 724.296 1.00457
\(722\) 0 0
\(723\) −2.15825 −0.00298514
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) 715.164 0.983719 0.491859 0.870675i \(-0.336317\pi\)
0.491859 + 0.870675i \(0.336317\pi\)
\(728\) 0 0
\(729\) 751.381 1.03070
\(730\) 0 0
\(731\) − 264.762i − 0.362191i
\(732\) 0 0
\(733\) 1233.29i 1.68252i 0.540632 + 0.841259i \(0.318185\pi\)
−0.540632 + 0.841259i \(0.681815\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) − 1000.38i − 1.35737i
\(738\) 0 0
\(739\) 8.55656i 0.0115786i 0.999983 + 0.00578928i \(0.00184280\pi\)
−0.999983 + 0.00578928i \(0.998157\pi\)
\(740\) 0 0
\(741\) −303.108 −0.409053
\(742\) 0 0
\(743\) −1010.56 −1.36011 −0.680053 0.733163i \(-0.738043\pi\)
−0.680053 + 0.733163i \(0.738043\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) −264.631 −0.354258
\(748\) 0 0
\(749\) −269.868 −0.360305
\(750\) 0 0
\(751\) − 1104.31i − 1.47046i −0.677820 0.735228i \(-0.737075\pi\)
0.677820 0.735228i \(-0.262925\pi\)
\(752\) 0 0
\(753\) 321.378i 0.426796i
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 875.633i 1.15671i 0.815783 + 0.578357i \(0.196306\pi\)
−0.815783 + 0.578357i \(0.803694\pi\)
\(758\) 0 0
\(759\) 25.9936i 0.0342471i
\(760\) 0 0
\(761\) −647.207 −0.850470 −0.425235 0.905083i \(-0.639808\pi\)
−0.425235 + 0.905083i \(0.639808\pi\)
\(762\) 0 0
\(763\) 701.688 0.919644
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 147.729 0.192606
\(768\) 0 0
\(769\) −631.430 −0.821106 −0.410553 0.911837i \(-0.634664\pi\)
−0.410553 + 0.911837i \(0.634664\pi\)
\(770\) 0 0
\(771\) − 646.433i − 0.838434i
\(772\) 0 0
\(773\) − 421.522i − 0.545306i −0.962112 0.272653i \(-0.912099\pi\)
0.962112 0.272653i \(-0.0879011\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 226.950i 0.292086i
\(778\) 0 0
\(779\) 221.917i 0.284874i
\(780\) 0 0
\(781\) −1609.21 −2.06044
\(782\) 0 0
\(783\) 320.927 0.409869
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) 838.633 1.06561 0.532804 0.846239i \(-0.321138\pi\)
0.532804 + 0.846239i \(0.321138\pi\)
\(788\) 0 0
\(789\) 955.469 1.21099
\(790\) 0 0
\(791\) − 897.535i − 1.13468i
\(792\) 0 0
\(793\) 763.659i 0.963001i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) − 1213.57i − 1.52268i −0.648354 0.761339i \(-0.724542\pi\)
0.648354 0.761339i \(-0.275458\pi\)
\(798\) 0 0
\(799\) − 634.732i − 0.794408i
\(800\) 0 0
\(801\) −386.944 −0.483076
\(802\) 0 0
\(803\) 110.635 0.137777
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) −819.431 −1.01540
\(808\) 0 0
\(809\) 229.214 0.283330 0.141665 0.989915i \(-0.454754\pi\)
0.141665 + 0.989915i \(0.454754\pi\)
\(810\) 0 0
\(811\) − 454.225i − 0.560080i −0.959988 0.280040i \(-0.909652\pi\)
0.959988 0.280040i \(-0.0903478\pi\)
\(812\) 0 0
\(813\) − 582.728i − 0.716762i
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 338.885i 0.414792i
\(818\) 0 0
\(819\) 154.651i 0.188829i
\(820\) 0 0
\(821\) −1130.90 −1.37747 −0.688733 0.725015i \(-0.741833\pi\)
−0.688733 + 0.725015i \(0.741833\pi\)
\(822\) 0 0
\(823\) 780.148 0.947931 0.473966 0.880543i \(-0.342822\pi\)
0.473966 + 0.880543i \(0.342822\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 209.175 0.252932 0.126466 0.991971i \(-0.459636\pi\)
0.126466 + 0.991971i \(0.459636\pi\)
\(828\) 0 0
\(829\) −508.525 −0.613419 −0.306710 0.951803i \(-0.599228\pi\)
−0.306710 + 0.951803i \(0.599228\pi\)
\(830\) 0 0
\(831\) 128.734i 0.154915i
\(832\) 0 0
\(833\) − 253.947i − 0.304859i
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 243.226i 0.290593i
\(838\) 0 0
\(839\) 274.028i 0.326613i 0.986575 + 0.163306i \(0.0522159\pi\)
−0.986575 + 0.163306i \(0.947784\pi\)
\(840\) 0 0
\(841\) −721.223 −0.857578
\(842\) 0 0
\(843\) −118.405 −0.140457
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) −1449.83 −1.71172
\(848\) 0 0
\(849\) 346.407 0.408018
\(850\) 0 0
\(851\) − 10.1907i − 0.0119750i
\(852\) 0 0
\(853\) − 1583.28i − 1.85613i −0.372416 0.928066i \(-0.621471\pi\)
0.372416 0.928066i \(-0.378529\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) − 1007.38i − 1.17547i −0.809054 0.587735i \(-0.800020\pi\)
0.809054 0.587735i \(-0.199980\pi\)
\(858\) 0 0
\(859\) − 76.6086i − 0.0891835i −0.999005 0.0445917i \(-0.985801\pi\)
0.999005 0.0445917i \(-0.0141987\pi\)
\(860\) 0 0
\(861\) −180.263 −0.209365
\(862\) 0 0
\(863\) −255.450 −0.296002 −0.148001 0.988987i \(-0.547284\pi\)
−0.148001 + 0.988987i \(0.547284\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) 347.175 0.400433
\(868\) 0 0
\(869\) 275.777 0.317350
\(870\) 0 0
\(871\) − 425.487i − 0.488504i
\(872\) 0 0
\(873\) 322.328i 0.369219i
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) − 601.522i − 0.685886i −0.939356 0.342943i \(-0.888576\pi\)
0.939356 0.342943i \(-0.111424\pi\)
\(878\) 0 0
\(879\) 420.523i 0.478411i
\(880\) 0 0
\(881\) 237.850 0.269977 0.134989 0.990847i \(-0.456900\pi\)
0.134989 + 0.990847i \(0.456900\pi\)
\(882\) 0 0
\(883\) 1.30294 0.00147559 0.000737794 1.00000i \(-0.499765\pi\)
0.000737794 1.00000i \(0.499765\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 536.353 0.604682 0.302341 0.953200i \(-0.402232\pi\)
0.302341 + 0.953200i \(0.402232\pi\)
\(888\) 0 0
\(889\) 1044.30 1.17469
\(890\) 0 0
\(891\) 750.855i 0.842710i
\(892\) 0 0
\(893\) 812.433i 0.909780i
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 11.0557i 0.0123252i
\(898\) 0 0
\(899\) 90.7773i 0.100976i
\(900\) 0 0
\(901\) 788.932 0.875618
\(902\) 0 0
\(903\) −275.276 −0.304846
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) 332.159 0.366217 0.183108 0.983093i \(-0.441384\pi\)
0.183108 + 0.983093i \(0.441384\pi\)
\(908\) 0 0
\(909\) 222.604 0.244889
\(910\) 0 0
\(911\) − 1450.06i − 1.59172i −0.605478 0.795862i \(-0.707018\pi\)
0.605478 0.795862i \(-0.292982\pi\)
\(912\) 0 0
\(913\) − 1518.15i − 1.66282i
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) − 40.8514i − 0.0445490i
\(918\) 0 0
\(919\) − 814.405i − 0.886186i −0.896476 0.443093i \(-0.853881\pi\)
0.896476 0.443093i \(-0.146119\pi\)
\(920\) 0 0
\(921\) −669.017 −0.726403
\(922\) 0 0
\(923\) −684.437 −0.741535
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) −478.353 −0.516023
\(928\) 0 0
\(929\) −400.039 −0.430612 −0.215306 0.976547i \(-0.569075\pi\)
−0.215306 + 0.976547i \(0.569075\pi\)
\(930\) 0 0
\(931\) 325.043i 0.349134i
\(932\) 0 0
\(933\) 663.489i 0.711135i
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 249.279i 0.266039i 0.991113 + 0.133020i \(0.0424673\pi\)
−0.991113 + 0.133020i \(0.957533\pi\)
\(938\) 0 0
\(939\) 1334.39i 1.42107i
\(940\) 0 0
\(941\) −724.229 −0.769638 −0.384819 0.922992i \(-0.625736\pi\)
−0.384819 + 0.922992i \(0.625736\pi\)
\(942\) 0 0
\(943\) 8.09432 0.00858358
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −1141.54 −1.20542 −0.602712 0.797959i \(-0.705913\pi\)
−0.602712 + 0.797959i \(0.705913\pi\)
\(948\) 0 0
\(949\) 47.0557 0.0495845
\(950\) 0 0
\(951\) − 378.865i − 0.398386i
\(952\) 0 0
\(953\) − 1295.33i − 1.35921i −0.733579 0.679604i \(-0.762152\pi\)
0.733579 0.679604i \(-0.237848\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 512.551i 0.535581i
\(958\) 0 0
\(959\) 4.37837i 0.00456556i
\(960\) 0 0
\(961\) 892.201 0.928409
\(962\) 0 0
\(963\) 178.232 0.185080
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) 398.477 0.412075 0.206037 0.978544i \(-0.433943\pi\)
0.206037 + 0.978544i \(0.433943\pi\)
\(968\) 0 0
\(969\) 425.337 0.438945
\(970\) 0 0
\(971\) 928.093i 0.955811i 0.878411 + 0.477906i \(0.158604\pi\)
−0.878411 + 0.477906i \(0.841396\pi\)
\(972\) 0 0
\(973\) − 1249.44i − 1.28411i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) − 1378.05i − 1.41049i −0.708962 0.705247i \(-0.750836\pi\)
0.708962 0.705247i \(-0.249164\pi\)
\(978\) 0 0
\(979\) − 2219.85i − 2.26747i
\(980\) 0 0
\(981\) −463.423 −0.472398
\(982\) 0 0
\(983\) 311.291 0.316675 0.158337 0.987385i \(-0.449387\pi\)
0.158337 + 0.987385i \(0.449387\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) −659.939 −0.668631
\(988\) 0 0
\(989\) 12.3607 0.0124982
\(990\) 0 0
\(991\) 961.147i 0.969876i 0.874549 + 0.484938i \(0.161158\pi\)
−0.874549 + 0.484938i \(0.838842\pi\)
\(992\) 0 0
\(993\) − 780.498i − 0.786000i
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) 1089.68i 1.09296i 0.837473 + 0.546479i \(0.184032\pi\)
−0.837473 + 0.546479i \(0.815968\pi\)
\(998\) 0 0
\(999\) − 538.404i − 0.538943i
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 1600.3.h.n.1599.5 8
4.3 odd 2 inner 1600.3.h.n.1599.4 8
5.2 odd 4 1600.3.b.s.1151.2 4
5.3 odd 4 320.3.b.c.191.3 4
5.4 even 2 inner 1600.3.h.n.1599.3 8
8.3 odd 2 100.3.d.b.99.8 8
8.5 even 2 100.3.d.b.99.2 8
15.8 even 4 2880.3.e.e.2431.1 4
20.3 even 4 320.3.b.c.191.2 4
20.7 even 4 1600.3.b.s.1151.3 4
20.19 odd 2 inner 1600.3.h.n.1599.6 8
24.5 odd 2 900.3.f.e.199.7 8
24.11 even 2 900.3.f.e.199.1 8
40.3 even 4 20.3.b.a.11.3 4
40.13 odd 4 20.3.b.a.11.4 yes 4
40.19 odd 2 100.3.d.b.99.1 8
40.27 even 4 100.3.b.f.51.2 4
40.29 even 2 100.3.d.b.99.7 8
40.37 odd 4 100.3.b.f.51.1 4
60.23 odd 4 2880.3.e.e.2431.2 4
80.3 even 4 1280.3.g.e.1151.5 8
80.13 odd 4 1280.3.g.e.1151.3 8
80.43 even 4 1280.3.g.e.1151.4 8
80.53 odd 4 1280.3.g.e.1151.6 8
120.29 odd 2 900.3.f.e.199.2 8
120.53 even 4 180.3.c.a.91.1 4
120.59 even 2 900.3.f.e.199.8 8
120.77 even 4 900.3.c.k.451.4 4
120.83 odd 4 180.3.c.a.91.2 4
120.107 odd 4 900.3.c.k.451.3 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
20.3.b.a.11.3 4 40.3 even 4
20.3.b.a.11.4 yes 4 40.13 odd 4
100.3.b.f.51.1 4 40.37 odd 4
100.3.b.f.51.2 4 40.27 even 4
100.3.d.b.99.1 8 40.19 odd 2
100.3.d.b.99.2 8 8.5 even 2
100.3.d.b.99.7 8 40.29 even 2
100.3.d.b.99.8 8 8.3 odd 2
180.3.c.a.91.1 4 120.53 even 4
180.3.c.a.91.2 4 120.83 odd 4
320.3.b.c.191.2 4 20.3 even 4
320.3.b.c.191.3 4 5.3 odd 4
900.3.c.k.451.3 4 120.107 odd 4
900.3.c.k.451.4 4 120.77 even 4
900.3.f.e.199.1 8 24.11 even 2
900.3.f.e.199.2 8 120.29 odd 2
900.3.f.e.199.7 8 24.5 odd 2
900.3.f.e.199.8 8 120.59 even 2
1280.3.g.e.1151.3 8 80.13 odd 4
1280.3.g.e.1151.4 8 80.43 even 4
1280.3.g.e.1151.5 8 80.3 even 4
1280.3.g.e.1151.6 8 80.53 odd 4
1600.3.b.s.1151.2 4 5.2 odd 4
1600.3.b.s.1151.3 4 20.7 even 4
1600.3.h.n.1599.3 8 5.4 even 2 inner
1600.3.h.n.1599.4 8 4.3 odd 2 inner
1600.3.h.n.1599.5 8 1.1 even 1 trivial
1600.3.h.n.1599.6 8 20.19 odd 2 inner
2880.3.e.e.2431.1 4 15.8 even 4
2880.3.e.e.2431.2 4 60.23 odd 4