Properties

Label 1280.4.d.u.641.1
Level $1280$
Weight $4$
Character 1280.641
Analytic conductor $75.522$
Analytic rank $0$
Dimension $4$
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] = [1280,4,Mod(641,1280)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1280, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1280.641");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1280 = 2^{8} \cdot 5 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1280.d (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: \(75.5224448073\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(i, \sqrt{10})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 25 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{3} \)
Twist minimal: no (minimal twist has level 160)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 641.1
Root \(-1.58114 - 1.58114i\) of defining polynomial
Character \(\chi\) \(=\) 1280.641
Dual form 1280.4.d.u.641.4

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-6.32456i q^{3} -5.00000i q^{5} +18.9737 q^{7} -13.0000 q^{9} +O(q^{10})\) \(q-6.32456i q^{3} -5.00000i q^{5} +18.9737 q^{7} -13.0000 q^{9} -12.6491i q^{11} +38.0000i q^{13} -31.6228 q^{15} +34.0000 q^{17} +101.193i q^{19} -120.000i q^{21} +82.2192 q^{23} -25.0000 q^{25} -88.5438i q^{27} +270.000i q^{29} +341.526 q^{31} -80.0000 q^{33} -94.8683i q^{35} -206.000i q^{37} +240.333 q^{39} +270.000 q^{41} +537.587i q^{43} +65.0000i q^{45} +132.816 q^{47} +17.0000 q^{49} -215.035i q^{51} +258.000i q^{53} -63.2456 q^{55} +640.000 q^{57} -75.8947i q^{59} -250.000i q^{61} -246.658 q^{63} +190.000 q^{65} +815.868i q^{67} -520.000i q^{69} -645.105 q^{71} +1078.00 q^{73} +158.114i q^{75} -240.000i q^{77} +278.280 q^{79} -911.000 q^{81} +1106.80i q^{83} -170.000i q^{85} +1707.63 q^{87} -890.000 q^{89} +720.999i q^{91} -2160.00i q^{93} +505.964 q^{95} -254.000 q^{97} +164.438i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 52 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 52 q^{9} + 136 q^{17} - 100 q^{25} - 320 q^{33} + 1080 q^{41} + 68 q^{49} + 2560 q^{57} + 760 q^{65} + 4312 q^{73} - 3644 q^{81} - 3560 q^{89} - 1016 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1280\mathbb{Z}\right)^\times\).

\(n\) \(257\) \(261\) \(511\)
\(\chi(n)\) \(1\) \(-1\) \(1\)

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
\(3\) − 6.32456i − 1.21716i −0.793492 0.608581i \(-0.791739\pi\)
0.793492 0.608581i \(-0.208261\pi\)
\(4\) 0 0
\(5\) − 5.00000i − 0.447214i
\(6\) 0 0
\(7\) 18.9737 1.02448 0.512241 0.858842i \(-0.328816\pi\)
0.512241 + 0.858842i \(0.328816\pi\)
\(8\) 0 0
\(9\) −13.0000 −0.481481
\(10\) 0 0
\(11\) − 12.6491i − 0.346714i −0.984859 0.173357i \(-0.944539\pi\)
0.984859 0.173357i \(-0.0554614\pi\)
\(12\) 0 0
\(13\) 38.0000i 0.810716i 0.914158 + 0.405358i \(0.132853\pi\)
−0.914158 + 0.405358i \(0.867147\pi\)
\(14\) 0 0
\(15\) −31.6228 −0.544331
\(16\) 0 0
\(17\) 34.0000 0.485071 0.242536 0.970143i \(-0.422021\pi\)
0.242536 + 0.970143i \(0.422021\pi\)
\(18\) 0 0
\(19\) 101.193i 1.22185i 0.791687 + 0.610927i \(0.209203\pi\)
−0.791687 + 0.610927i \(0.790797\pi\)
\(20\) 0 0
\(21\) − 120.000i − 1.24696i
\(22\) 0 0
\(23\) 82.2192 0.745387 0.372693 0.927955i \(-0.378434\pi\)
0.372693 + 0.927955i \(0.378434\pi\)
\(24\) 0 0
\(25\) −25.0000 −0.200000
\(26\) 0 0
\(27\) − 88.5438i − 0.631121i
\(28\) 0 0
\(29\) 270.000i 1.72889i 0.502729 + 0.864444i \(0.332329\pi\)
−0.502729 + 0.864444i \(0.667671\pi\)
\(30\) 0 0
\(31\) 341.526 1.97871 0.989353 0.145537i \(-0.0464908\pi\)
0.989353 + 0.145537i \(0.0464908\pi\)
\(32\) 0 0
\(33\) −80.0000 −0.422006
\(34\) 0 0
\(35\) − 94.8683i − 0.458162i
\(36\) 0 0
\(37\) − 206.000i − 0.915302i −0.889132 0.457651i \(-0.848691\pi\)
0.889132 0.457651i \(-0.151309\pi\)
\(38\) 0 0
\(39\) 240.333 0.986772
\(40\) 0 0
\(41\) 270.000 1.02846 0.514231 0.857652i \(-0.328078\pi\)
0.514231 + 0.857652i \(0.328078\pi\)
\(42\) 0 0
\(43\) 537.587i 1.90654i 0.302117 + 0.953271i \(0.402307\pi\)
−0.302117 + 0.953271i \(0.597693\pi\)
\(44\) 0 0
\(45\) 65.0000i 0.215325i
\(46\) 0 0
\(47\) 132.816 0.412195 0.206097 0.978531i \(-0.433924\pi\)
0.206097 + 0.978531i \(0.433924\pi\)
\(48\) 0 0
\(49\) 17.0000 0.0495627
\(50\) 0 0
\(51\) − 215.035i − 0.590410i
\(52\) 0 0
\(53\) 258.000i 0.668661i 0.942456 + 0.334330i \(0.108510\pi\)
−0.942456 + 0.334330i \(0.891490\pi\)
\(54\) 0 0
\(55\) −63.2456 −0.155055
\(56\) 0 0
\(57\) 640.000 1.48719
\(58\) 0 0
\(59\) − 75.8947i − 0.167469i −0.996488 0.0837343i \(-0.973315\pi\)
0.996488 0.0837343i \(-0.0266847\pi\)
\(60\) 0 0
\(61\) − 250.000i − 0.524741i −0.964967 0.262371i \(-0.915496\pi\)
0.964967 0.262371i \(-0.0845043\pi\)
\(62\) 0 0
\(63\) −246.658 −0.493269
\(64\) 0 0
\(65\) 190.000 0.362563
\(66\) 0 0
\(67\) 815.868i 1.48767i 0.668362 + 0.743837i \(0.266996\pi\)
−0.668362 + 0.743837i \(0.733004\pi\)
\(68\) 0 0
\(69\) − 520.000i − 0.907256i
\(70\) 0 0
\(71\) −645.105 −1.07831 −0.539154 0.842207i \(-0.681256\pi\)
−0.539154 + 0.842207i \(0.681256\pi\)
\(72\) 0 0
\(73\) 1078.00 1.72836 0.864181 0.503182i \(-0.167837\pi\)
0.864181 + 0.503182i \(0.167837\pi\)
\(74\) 0 0
\(75\) 158.114i 0.243432i
\(76\) 0 0
\(77\) − 240.000i − 0.355202i
\(78\) 0 0
\(79\) 278.280 0.396316 0.198158 0.980170i \(-0.436504\pi\)
0.198158 + 0.980170i \(0.436504\pi\)
\(80\) 0 0
\(81\) −911.000 −1.24966
\(82\) 0 0
\(83\) 1106.80i 1.46370i 0.681468 + 0.731848i \(0.261342\pi\)
−0.681468 + 0.731848i \(0.738658\pi\)
\(84\) 0 0
\(85\) − 170.000i − 0.216930i
\(86\) 0 0
\(87\) 1707.63 2.10434
\(88\) 0 0
\(89\) −890.000 −1.06000 −0.529999 0.847998i \(-0.677808\pi\)
−0.529999 + 0.847998i \(0.677808\pi\)
\(90\) 0 0
\(91\) 720.999i 0.830563i
\(92\) 0 0
\(93\) − 2160.00i − 2.40840i
\(94\) 0 0
\(95\) 505.964 0.546430
\(96\) 0 0
\(97\) −254.000 −0.265874 −0.132937 0.991124i \(-0.542441\pi\)
−0.132937 + 0.991124i \(0.542441\pi\)
\(98\) 0 0
\(99\) 164.438i 0.166936i
\(100\) 0 0
\(101\) − 598.000i − 0.589141i −0.955630 0.294570i \(-0.904823\pi\)
0.955630 0.294570i \(-0.0951766\pi\)
\(102\) 0 0
\(103\) −499.640 −0.477971 −0.238985 0.971023i \(-0.576815\pi\)
−0.238985 + 0.971023i \(0.576815\pi\)
\(104\) 0 0
\(105\) −600.000 −0.557657
\(106\) 0 0
\(107\) 626.131i 0.565704i 0.959164 + 0.282852i \(0.0912806\pi\)
−0.959164 + 0.282852i \(0.908719\pi\)
\(108\) 0 0
\(109\) 854.000i 0.750444i 0.926935 + 0.375222i \(0.122433\pi\)
−0.926935 + 0.375222i \(0.877567\pi\)
\(110\) 0 0
\(111\) −1302.86 −1.11407
\(112\) 0 0
\(113\) 1698.00 1.41358 0.706789 0.707424i \(-0.250143\pi\)
0.706789 + 0.707424i \(0.250143\pi\)
\(114\) 0 0
\(115\) − 411.096i − 0.333347i
\(116\) 0 0
\(117\) − 494.000i − 0.390345i
\(118\) 0 0
\(119\) 645.105 0.496947
\(120\) 0 0
\(121\) 1171.00 0.879790
\(122\) 0 0
\(123\) − 1707.63i − 1.25180i
\(124\) 0 0
\(125\) 125.000i 0.0894427i
\(126\) 0 0
\(127\) 234.009 0.163503 0.0817516 0.996653i \(-0.473949\pi\)
0.0817516 + 0.996653i \(0.473949\pi\)
\(128\) 0 0
\(129\) 3400.00 2.32057
\(130\) 0 0
\(131\) − 1732.93i − 1.15578i −0.816116 0.577888i \(-0.803877\pi\)
0.816116 0.577888i \(-0.196123\pi\)
\(132\) 0 0
\(133\) 1920.00i 1.25177i
\(134\) 0 0
\(135\) −442.719 −0.282246
\(136\) 0 0
\(137\) −1546.00 −0.964115 −0.482057 0.876140i \(-0.660110\pi\)
−0.482057 + 0.876140i \(0.660110\pi\)
\(138\) 0 0
\(139\) 328.877i 0.200683i 0.994953 + 0.100342i \(0.0319936\pi\)
−0.994953 + 0.100342i \(0.968006\pi\)
\(140\) 0 0
\(141\) − 840.000i − 0.501708i
\(142\) 0 0
\(143\) 480.666 0.281086
\(144\) 0 0
\(145\) 1350.00 0.773182
\(146\) 0 0
\(147\) − 107.517i − 0.0603258i
\(148\) 0 0
\(149\) − 3246.00i − 1.78472i −0.451328 0.892358i \(-0.649050\pi\)
0.451328 0.892358i \(-0.350950\pi\)
\(150\) 0 0
\(151\) −1505.24 −0.811225 −0.405613 0.914045i \(-0.632942\pi\)
−0.405613 + 0.914045i \(0.632942\pi\)
\(152\) 0 0
\(153\) −442.000 −0.233553
\(154\) 0 0
\(155\) − 1707.63i − 0.884904i
\(156\) 0 0
\(157\) − 1226.00i − 0.623219i −0.950210 0.311610i \(-0.899132\pi\)
0.950210 0.311610i \(-0.100868\pi\)
\(158\) 0 0
\(159\) 1631.74 0.813868
\(160\) 0 0
\(161\) 1560.00 0.763635
\(162\) 0 0
\(163\) − 1448.32i − 0.695960i −0.937502 0.347980i \(-0.886868\pi\)
0.937502 0.347980i \(-0.113132\pi\)
\(164\) 0 0
\(165\) 400.000i 0.188727i
\(166\) 0 0
\(167\) −2333.76 −1.08139 −0.540694 0.841219i \(-0.681838\pi\)
−0.540694 + 0.841219i \(0.681838\pi\)
\(168\) 0 0
\(169\) 753.000 0.342740
\(170\) 0 0
\(171\) − 1315.51i − 0.588300i
\(172\) 0 0
\(173\) − 3098.00i − 1.36148i −0.732524 0.680742i \(-0.761658\pi\)
0.732524 0.680742i \(-0.238342\pi\)
\(174\) 0 0
\(175\) −474.342 −0.204896
\(176\) 0 0
\(177\) −480.000 −0.203836
\(178\) 0 0
\(179\) − 2352.73i − 0.982411i −0.871044 0.491206i \(-0.836556\pi\)
0.871044 0.491206i \(-0.163444\pi\)
\(180\) 0 0
\(181\) − 2182.00i − 0.896060i −0.894019 0.448030i \(-0.852126\pi\)
0.894019 0.448030i \(-0.147874\pi\)
\(182\) 0 0
\(183\) −1581.14 −0.638695
\(184\) 0 0
\(185\) −1030.00 −0.409336
\(186\) 0 0
\(187\) − 430.070i − 0.168181i
\(188\) 0 0
\(189\) − 1680.00i − 0.646572i
\(190\) 0 0
\(191\) −3023.14 −1.14527 −0.572635 0.819810i \(-0.694079\pi\)
−0.572635 + 0.819810i \(0.694079\pi\)
\(192\) 0 0
\(193\) 1298.00 0.484104 0.242052 0.970263i \(-0.422180\pi\)
0.242052 + 0.970263i \(0.422180\pi\)
\(194\) 0 0
\(195\) − 1201.67i − 0.441298i
\(196\) 0 0
\(197\) − 2846.00i − 1.02928i −0.857405 0.514642i \(-0.827925\pi\)
0.857405 0.514642i \(-0.172075\pi\)
\(198\) 0 0
\(199\) 3592.35 1.27967 0.639836 0.768511i \(-0.279002\pi\)
0.639836 + 0.768511i \(0.279002\pi\)
\(200\) 0 0
\(201\) 5160.00 1.81074
\(202\) 0 0
\(203\) 5122.89i 1.77121i
\(204\) 0 0
\(205\) − 1350.00i − 0.459942i
\(206\) 0 0
\(207\) −1068.85 −0.358890
\(208\) 0 0
\(209\) 1280.00 0.423634
\(210\) 0 0
\(211\) 4186.86i 1.36604i 0.730398 + 0.683021i \(0.239334\pi\)
−0.730398 + 0.683021i \(0.760666\pi\)
\(212\) 0 0
\(213\) 4080.00i 1.31247i
\(214\) 0 0
\(215\) 2687.94 0.852631
\(216\) 0 0
\(217\) 6480.00 2.02715
\(218\) 0 0
\(219\) − 6817.87i − 2.10369i
\(220\) 0 0
\(221\) 1292.00i 0.393255i
\(222\) 0 0
\(223\) −4762.39 −1.43010 −0.715052 0.699071i \(-0.753597\pi\)
−0.715052 + 0.699071i \(0.753597\pi\)
\(224\) 0 0
\(225\) 325.000 0.0962963
\(226\) 0 0
\(227\) − 1663.36i − 0.486348i −0.969983 0.243174i \(-0.921811\pi\)
0.969983 0.243174i \(-0.0781886\pi\)
\(228\) 0 0
\(229\) 1050.00i 0.302995i 0.988458 + 0.151498i \(0.0484096\pi\)
−0.988458 + 0.151498i \(0.951590\pi\)
\(230\) 0 0
\(231\) −1517.89 −0.432338
\(232\) 0 0
\(233\) −2778.00 −0.781085 −0.390543 0.920585i \(-0.627713\pi\)
−0.390543 + 0.920585i \(0.627713\pi\)
\(234\) 0 0
\(235\) − 664.078i − 0.184339i
\(236\) 0 0
\(237\) − 1760.00i − 0.482381i
\(238\) 0 0
\(239\) 2555.12 0.691536 0.345768 0.938320i \(-0.387618\pi\)
0.345768 + 0.938320i \(0.387618\pi\)
\(240\) 0 0
\(241\) −5350.00 −1.42997 −0.714987 0.699138i \(-0.753567\pi\)
−0.714987 + 0.699138i \(0.753567\pi\)
\(242\) 0 0
\(243\) 3370.99i 0.889913i
\(244\) 0 0
\(245\) − 85.0000i − 0.0221651i
\(246\) 0 0
\(247\) −3845.33 −0.990577
\(248\) 0 0
\(249\) 7000.00 1.78155
\(250\) 0 0
\(251\) − 5881.84i − 1.47912i −0.673093 0.739558i \(-0.735034\pi\)
0.673093 0.739558i \(-0.264966\pi\)
\(252\) 0 0
\(253\) − 1040.00i − 0.258436i
\(254\) 0 0
\(255\) −1075.17 −0.264039
\(256\) 0 0
\(257\) 1074.00 0.260678 0.130339 0.991469i \(-0.458393\pi\)
0.130339 + 0.991469i \(0.458393\pi\)
\(258\) 0 0
\(259\) − 3908.58i − 0.937711i
\(260\) 0 0
\(261\) − 3510.00i − 0.832427i
\(262\) 0 0
\(263\) −1486.27 −0.348469 −0.174235 0.984704i \(-0.555745\pi\)
−0.174235 + 0.984704i \(0.555745\pi\)
\(264\) 0 0
\(265\) 1290.00 0.299034
\(266\) 0 0
\(267\) 5628.85i 1.29019i
\(268\) 0 0
\(269\) 406.000i 0.0920233i 0.998941 + 0.0460116i \(0.0146511\pi\)
−0.998941 + 0.0460116i \(0.985349\pi\)
\(270\) 0 0
\(271\) −392.122 −0.0878957 −0.0439479 0.999034i \(-0.513994\pi\)
−0.0439479 + 0.999034i \(0.513994\pi\)
\(272\) 0 0
\(273\) 4560.00 1.01093
\(274\) 0 0
\(275\) 316.228i 0.0693427i
\(276\) 0 0
\(277\) − 5934.00i − 1.28715i −0.765385 0.643573i \(-0.777451\pi\)
0.765385 0.643573i \(-0.222549\pi\)
\(278\) 0 0
\(279\) −4439.84 −0.952710
\(280\) 0 0
\(281\) 1870.00 0.396992 0.198496 0.980102i \(-0.436394\pi\)
0.198496 + 0.980102i \(0.436394\pi\)
\(282\) 0 0
\(283\) 4888.88i 1.02690i 0.858118 + 0.513452i \(0.171634\pi\)
−0.858118 + 0.513452i \(0.828366\pi\)
\(284\) 0 0
\(285\) − 3200.00i − 0.665093i
\(286\) 0 0
\(287\) 5122.89 1.05364
\(288\) 0 0
\(289\) −3757.00 −0.764706
\(290\) 0 0
\(291\) 1606.44i 0.323612i
\(292\) 0 0
\(293\) − 5198.00i − 1.03642i −0.855254 0.518209i \(-0.826599\pi\)
0.855254 0.518209i \(-0.173401\pi\)
\(294\) 0 0
\(295\) −379.473 −0.0748942
\(296\) 0 0
\(297\) −1120.00 −0.218818
\(298\) 0 0
\(299\) 3124.33i 0.604297i
\(300\) 0 0
\(301\) 10200.0i 1.95322i
\(302\) 0 0
\(303\) −3782.08 −0.717079
\(304\) 0 0
\(305\) −1250.00 −0.234671
\(306\) 0 0
\(307\) 3750.46i 0.697232i 0.937266 + 0.348616i \(0.113348\pi\)
−0.937266 + 0.348616i \(0.886652\pi\)
\(308\) 0 0
\(309\) 3160.00i 0.581767i
\(310\) 0 0
\(311\) 6261.31 1.14163 0.570814 0.821079i \(-0.306628\pi\)
0.570814 + 0.821079i \(0.306628\pi\)
\(312\) 0 0
\(313\) −2218.00 −0.400539 −0.200270 0.979741i \(-0.564182\pi\)
−0.200270 + 0.979741i \(0.564182\pi\)
\(314\) 0 0
\(315\) 1233.29i 0.220597i
\(316\) 0 0
\(317\) 4134.00i 0.732456i 0.930525 + 0.366228i \(0.119351\pi\)
−0.930525 + 0.366228i \(0.880649\pi\)
\(318\) 0 0
\(319\) 3415.26 0.599429
\(320\) 0 0
\(321\) 3960.00 0.688553
\(322\) 0 0
\(323\) 3440.56i 0.592687i
\(324\) 0 0
\(325\) − 950.000i − 0.162143i
\(326\) 0 0
\(327\) 5401.17 0.913411
\(328\) 0 0
\(329\) 2520.00 0.422286
\(330\) 0 0
\(331\) 11953.4i 1.98495i 0.122443 + 0.992476i \(0.460927\pi\)
−0.122443 + 0.992476i \(0.539073\pi\)
\(332\) 0 0
\(333\) 2678.00i 0.440701i
\(334\) 0 0
\(335\) 4079.34 0.665308
\(336\) 0 0
\(337\) −8014.00 −1.29540 −0.647701 0.761895i \(-0.724269\pi\)
−0.647701 + 0.761895i \(0.724269\pi\)
\(338\) 0 0
\(339\) − 10739.1i − 1.72055i
\(340\) 0 0
\(341\) − 4320.00i − 0.686044i
\(342\) 0 0
\(343\) −6185.42 −0.973706
\(344\) 0 0
\(345\) −2600.00 −0.405737
\(346\) 0 0
\(347\) − 4484.11i − 0.693716i −0.937918 0.346858i \(-0.887248\pi\)
0.937918 0.346858i \(-0.112752\pi\)
\(348\) 0 0
\(349\) 910.000i 0.139574i 0.997562 + 0.0697868i \(0.0222319\pi\)
−0.997562 + 0.0697868i \(0.977768\pi\)
\(350\) 0 0
\(351\) 3364.66 0.511659
\(352\) 0 0
\(353\) 12962.0 1.95438 0.977192 0.212357i \(-0.0681140\pi\)
0.977192 + 0.212357i \(0.0681140\pi\)
\(354\) 0 0
\(355\) 3225.52i 0.482234i
\(356\) 0 0
\(357\) − 4080.00i − 0.604864i
\(358\) 0 0
\(359\) 12193.7 1.79265 0.896325 0.443398i \(-0.146227\pi\)
0.896325 + 0.443398i \(0.146227\pi\)
\(360\) 0 0
\(361\) −3381.00 −0.492929
\(362\) 0 0
\(363\) − 7406.05i − 1.07085i
\(364\) 0 0
\(365\) − 5390.00i − 0.772947i
\(366\) 0 0
\(367\) −3434.23 −0.488462 −0.244231 0.969717i \(-0.578536\pi\)
−0.244231 + 0.969717i \(0.578536\pi\)
\(368\) 0 0
\(369\) −3510.00 −0.495185
\(370\) 0 0
\(371\) 4895.21i 0.685031i
\(372\) 0 0
\(373\) − 4622.00i − 0.641603i −0.947146 0.320802i \(-0.896048\pi\)
0.947146 0.320802i \(-0.103952\pi\)
\(374\) 0 0
\(375\) 790.569 0.108866
\(376\) 0 0
\(377\) −10260.0 −1.40164
\(378\) 0 0
\(379\) − 8449.61i − 1.14519i −0.819838 0.572595i \(-0.805937\pi\)
0.819838 0.572595i \(-0.194063\pi\)
\(380\) 0 0
\(381\) − 1480.00i − 0.199010i
\(382\) 0 0
\(383\) 1815.15 0.242166 0.121083 0.992642i \(-0.461363\pi\)
0.121083 + 0.992642i \(0.461363\pi\)
\(384\) 0 0
\(385\) −1200.00 −0.158851
\(386\) 0 0
\(387\) − 6988.63i − 0.917964i
\(388\) 0 0
\(389\) 11106.0i 1.44755i 0.690037 + 0.723774i \(0.257594\pi\)
−0.690037 + 0.723774i \(0.742406\pi\)
\(390\) 0 0
\(391\) 2795.45 0.361566
\(392\) 0 0
\(393\) −10960.0 −1.40677
\(394\) 0 0
\(395\) − 1391.40i − 0.177238i
\(396\) 0 0
\(397\) − 5754.00i − 0.727418i −0.931513 0.363709i \(-0.881510\pi\)
0.931513 0.363709i \(-0.118490\pi\)
\(398\) 0 0
\(399\) 12143.1 1.52360
\(400\) 0 0
\(401\) −1118.00 −0.139228 −0.0696138 0.997574i \(-0.522177\pi\)
−0.0696138 + 0.997574i \(0.522177\pi\)
\(402\) 0 0
\(403\) 12978.0i 1.60417i
\(404\) 0 0
\(405\) 4555.00i 0.558864i
\(406\) 0 0
\(407\) −2605.72 −0.317348
\(408\) 0 0
\(409\) 11374.0 1.37508 0.687540 0.726146i \(-0.258690\pi\)
0.687540 + 0.726146i \(0.258690\pi\)
\(410\) 0 0
\(411\) 9777.76i 1.17348i
\(412\) 0 0
\(413\) − 1440.00i − 0.171568i
\(414\) 0 0
\(415\) 5533.99 0.654585
\(416\) 0 0
\(417\) 2080.00 0.244264
\(418\) 0 0
\(419\) 12674.4i 1.47777i 0.673832 + 0.738885i \(0.264647\pi\)
−0.673832 + 0.738885i \(0.735353\pi\)
\(420\) 0 0
\(421\) − 1150.00i − 0.133130i −0.997782 0.0665648i \(-0.978796\pi\)
0.997782 0.0665648i \(-0.0212039\pi\)
\(422\) 0 0
\(423\) −1726.60 −0.198464
\(424\) 0 0
\(425\) −850.000 −0.0970143
\(426\) 0 0
\(427\) − 4743.42i − 0.537588i
\(428\) 0 0
\(429\) − 3040.00i − 0.342127i
\(430\) 0 0
\(431\) −1353.45 −0.151261 −0.0756307 0.997136i \(-0.524097\pi\)
−0.0756307 + 0.997136i \(0.524097\pi\)
\(432\) 0 0
\(433\) −7918.00 −0.878787 −0.439394 0.898295i \(-0.644807\pi\)
−0.439394 + 0.898295i \(0.644807\pi\)
\(434\) 0 0
\(435\) − 8538.15i − 0.941087i
\(436\) 0 0
\(437\) 8320.00i 0.910754i
\(438\) 0 0
\(439\) 14217.6 1.54572 0.772858 0.634579i \(-0.218827\pi\)
0.772858 + 0.634579i \(0.218827\pi\)
\(440\) 0 0
\(441\) −221.000 −0.0238635
\(442\) 0 0
\(443\) 10581.0i 1.13480i 0.823441 + 0.567401i \(0.192051\pi\)
−0.823441 + 0.567401i \(0.807949\pi\)
\(444\) 0 0
\(445\) 4450.00i 0.474045i
\(446\) 0 0
\(447\) −20529.5 −2.17229
\(448\) 0 0
\(449\) 4474.00 0.470247 0.235124 0.971965i \(-0.424450\pi\)
0.235124 + 0.971965i \(0.424450\pi\)
\(450\) 0 0
\(451\) − 3415.26i − 0.356582i
\(452\) 0 0
\(453\) 9520.00i 0.987392i
\(454\) 0 0
\(455\) 3605.00 0.371439
\(456\) 0 0
\(457\) −4154.00 −0.425199 −0.212599 0.977139i \(-0.568193\pi\)
−0.212599 + 0.977139i \(0.568193\pi\)
\(458\) 0 0
\(459\) − 3010.49i − 0.306138i
\(460\) 0 0
\(461\) − 11282.0i − 1.13982i −0.821709 0.569908i \(-0.806979\pi\)
0.821709 0.569908i \(-0.193021\pi\)
\(462\) 0 0
\(463\) 5458.09 0.547860 0.273930 0.961750i \(-0.411676\pi\)
0.273930 + 0.961750i \(0.411676\pi\)
\(464\) 0 0
\(465\) −10800.0 −1.07707
\(466\) 0 0
\(467\) 3775.76i 0.374136i 0.982347 + 0.187068i \(0.0598984\pi\)
−0.982347 + 0.187068i \(0.940102\pi\)
\(468\) 0 0
\(469\) 15480.0i 1.52409i
\(470\) 0 0
\(471\) −7753.90 −0.758559
\(472\) 0 0
\(473\) 6800.00 0.661024
\(474\) 0 0
\(475\) − 2529.82i − 0.244371i
\(476\) 0 0
\(477\) − 3354.00i − 0.321948i
\(478\) 0 0
\(479\) −8930.27 −0.851847 −0.425923 0.904759i \(-0.640051\pi\)
−0.425923 + 0.904759i \(0.640051\pi\)
\(480\) 0 0
\(481\) 7828.00 0.742050
\(482\) 0 0
\(483\) − 9866.31i − 0.929467i
\(484\) 0 0
\(485\) 1270.00i 0.118903i
\(486\) 0 0
\(487\) 2422.30 0.225390 0.112695 0.993630i \(-0.464052\pi\)
0.112695 + 0.993630i \(0.464052\pi\)
\(488\) 0 0
\(489\) −9160.00 −0.847095
\(490\) 0 0
\(491\) 8993.52i 0.826623i 0.910590 + 0.413311i \(0.135628\pi\)
−0.910590 + 0.413311i \(0.864372\pi\)
\(492\) 0 0
\(493\) 9180.00i 0.838634i
\(494\) 0 0
\(495\) 822.192 0.0746561
\(496\) 0 0
\(497\) −12240.0 −1.10471
\(498\) 0 0
\(499\) − 3541.75i − 0.317737i −0.987300 0.158868i \(-0.949215\pi\)
0.987300 0.158868i \(-0.0507845\pi\)
\(500\) 0 0
\(501\) 14760.0i 1.31622i
\(502\) 0 0
\(503\) 2384.36 0.211358 0.105679 0.994400i \(-0.466298\pi\)
0.105679 + 0.994400i \(0.466298\pi\)
\(504\) 0 0
\(505\) −2990.00 −0.263472
\(506\) 0 0
\(507\) − 4762.39i − 0.417170i
\(508\) 0 0
\(509\) 2350.00i 0.204640i 0.994752 + 0.102320i \(0.0326266\pi\)
−0.994752 + 0.102320i \(0.967373\pi\)
\(510\) 0 0
\(511\) 20453.6 1.77067
\(512\) 0 0
\(513\) 8960.00 0.771138
\(514\) 0 0
\(515\) 2498.20i 0.213755i
\(516\) 0 0
\(517\) − 1680.00i − 0.142914i
\(518\) 0 0
\(519\) −19593.5 −1.65714
\(520\) 0 0
\(521\) −858.000 −0.0721491 −0.0360745 0.999349i \(-0.511485\pi\)
−0.0360745 + 0.999349i \(0.511485\pi\)
\(522\) 0 0
\(523\) 5799.62i 0.484894i 0.970165 + 0.242447i \(0.0779501\pi\)
−0.970165 + 0.242447i \(0.922050\pi\)
\(524\) 0 0
\(525\) 3000.00i 0.249392i
\(526\) 0 0
\(527\) 11611.9 0.959813
\(528\) 0 0
\(529\) −5407.00 −0.444399
\(530\) 0 0
\(531\) 986.631i 0.0806330i
\(532\) 0 0
\(533\) 10260.0i 0.833790i
\(534\) 0 0
\(535\) 3130.65 0.252991
\(536\) 0 0
\(537\) −14880.0 −1.19575
\(538\) 0 0
\(539\) − 215.035i − 0.0171841i
\(540\) 0 0
\(541\) 20478.0i 1.62739i 0.581292 + 0.813695i \(0.302547\pi\)
−0.581292 + 0.813695i \(0.697453\pi\)
\(542\) 0 0
\(543\) −13800.2 −1.09065
\(544\) 0 0
\(545\) 4270.00 0.335609
\(546\) 0 0
\(547\) 10429.2i 0.815210i 0.913158 + 0.407605i \(0.133636\pi\)
−0.913158 + 0.407605i \(0.866364\pi\)
\(548\) 0 0
\(549\) 3250.00i 0.252653i
\(550\) 0 0
\(551\) −27322.1 −2.11245
\(552\) 0 0
\(553\) 5280.00 0.406019
\(554\) 0 0
\(555\) 6514.29i 0.498228i
\(556\) 0 0
\(557\) − 13194.0i − 1.00368i −0.864962 0.501838i \(-0.832657\pi\)
0.864962 0.501838i \(-0.167343\pi\)
\(558\) 0 0
\(559\) −20428.3 −1.54566
\(560\) 0 0
\(561\) −2720.00 −0.204703
\(562\) 0 0
\(563\) − 9771.44i − 0.731469i −0.930719 0.365734i \(-0.880818\pi\)
0.930719 0.365734i \(-0.119182\pi\)
\(564\) 0 0
\(565\) − 8490.00i − 0.632172i
\(566\) 0 0
\(567\) −17285.0 −1.28025
\(568\) 0 0
\(569\) −4594.00 −0.338472 −0.169236 0.985576i \(-0.554130\pi\)
−0.169236 + 0.985576i \(0.554130\pi\)
\(570\) 0 0
\(571\) − 4389.24i − 0.321688i −0.986980 0.160844i \(-0.948578\pi\)
0.986980 0.160844i \(-0.0514217\pi\)
\(572\) 0 0
\(573\) 19120.0i 1.39398i
\(574\) 0 0
\(575\) −2055.48 −0.149077
\(576\) 0 0
\(577\) −14926.0 −1.07691 −0.538455 0.842654i \(-0.680992\pi\)
−0.538455 + 0.842654i \(0.680992\pi\)
\(578\) 0 0
\(579\) − 8209.27i − 0.589233i
\(580\) 0 0
\(581\) 21000.0i 1.49953i
\(582\) 0 0
\(583\) 3263.47 0.231834
\(584\) 0 0
\(585\) −2470.00 −0.174567
\(586\) 0 0
\(587\) − 8101.76i − 0.569668i −0.958577 0.284834i \(-0.908061\pi\)
0.958577 0.284834i \(-0.0919385\pi\)
\(588\) 0 0
\(589\) 34560.0i 2.41769i
\(590\) 0 0
\(591\) −17999.7 −1.25281
\(592\) 0 0
\(593\) −26958.0 −1.86683 −0.933417 0.358794i \(-0.883188\pi\)
−0.933417 + 0.358794i \(0.883188\pi\)
\(594\) 0 0
\(595\) − 3225.52i − 0.222241i
\(596\) 0 0
\(597\) − 22720.0i − 1.55757i
\(598\) 0 0
\(599\) −6349.85 −0.433135 −0.216568 0.976268i \(-0.569486\pi\)
−0.216568 + 0.976268i \(0.569486\pi\)
\(600\) 0 0
\(601\) −21970.0 −1.49114 −0.745570 0.666427i \(-0.767823\pi\)
−0.745570 + 0.666427i \(0.767823\pi\)
\(602\) 0 0
\(603\) − 10606.3i − 0.716287i
\(604\) 0 0
\(605\) − 5855.00i − 0.393454i
\(606\) 0 0
\(607\) 3876.95 0.259243 0.129622 0.991564i \(-0.458624\pi\)
0.129622 + 0.991564i \(0.458624\pi\)
\(608\) 0 0
\(609\) 32400.0 2.15585
\(610\) 0 0
\(611\) 5047.00i 0.334173i
\(612\) 0 0
\(613\) − 2878.00i − 0.189627i −0.995495 0.0948135i \(-0.969775\pi\)
0.995495 0.0948135i \(-0.0302255\pi\)
\(614\) 0 0
\(615\) −8538.15 −0.559823
\(616\) 0 0
\(617\) −27354.0 −1.78481 −0.892407 0.451231i \(-0.850985\pi\)
−0.892407 + 0.451231i \(0.850985\pi\)
\(618\) 0 0
\(619\) − 12547.9i − 0.814771i −0.913256 0.407386i \(-0.866441\pi\)
0.913256 0.407386i \(-0.133559\pi\)
\(620\) 0 0
\(621\) − 7280.00i − 0.470429i
\(622\) 0 0
\(623\) −16886.6 −1.08595
\(624\) 0 0
\(625\) 625.000 0.0400000
\(626\) 0 0
\(627\) − 8095.43i − 0.515631i
\(628\) 0 0
\(629\) − 7004.00i − 0.443987i
\(630\) 0 0
\(631\) 30876.5 1.94798 0.973988 0.226598i \(-0.0727605\pi\)
0.973988 + 0.226598i \(0.0727605\pi\)
\(632\) 0 0
\(633\) 26480.0 1.66269
\(634\) 0 0
\(635\) − 1170.04i − 0.0731208i
\(636\) 0 0
\(637\) 646.000i 0.0401812i
\(638\) 0 0
\(639\) 8386.36 0.519185
\(640\) 0 0
\(641\) −9430.00 −0.581065 −0.290532 0.956865i \(-0.593832\pi\)
−0.290532 + 0.956865i \(0.593832\pi\)
\(642\) 0 0
\(643\) − 9847.33i − 0.603952i −0.953316 0.301976i \(-0.902354\pi\)
0.953316 0.301976i \(-0.0976462\pi\)
\(644\) 0 0
\(645\) − 17000.0i − 1.03779i
\(646\) 0 0
\(647\) 30048.0 1.82582 0.912911 0.408158i \(-0.133829\pi\)
0.912911 + 0.408158i \(0.133829\pi\)
\(648\) 0 0
\(649\) −960.000 −0.0580636
\(650\) 0 0
\(651\) − 40983.1i − 2.46737i
\(652\) 0 0
\(653\) 18742.0i 1.12317i 0.827418 + 0.561586i \(0.189809\pi\)
−0.827418 + 0.561586i \(0.810191\pi\)
\(654\) 0 0
\(655\) −8664.64 −0.516879
\(656\) 0 0
\(657\) −14014.0 −0.832174
\(658\) 0 0
\(659\) − 8323.11i − 0.491992i −0.969271 0.245996i \(-0.920885\pi\)
0.969271 0.245996i \(-0.0791150\pi\)
\(660\) 0 0
\(661\) − 7630.00i − 0.448975i −0.974477 0.224488i \(-0.927929\pi\)
0.974477 0.224488i \(-0.0720708\pi\)
\(662\) 0 0
\(663\) 8171.33 0.478655
\(664\) 0 0
\(665\) 9600.00 0.559808
\(666\) 0 0
\(667\) 22199.2i 1.28869i
\(668\) 0 0
\(669\) 30120.0i 1.74067i
\(670\) 0 0
\(671\) −3162.28 −0.181935
\(672\) 0 0
\(673\) −10878.0 −0.623055 −0.311528 0.950237i \(-0.600841\pi\)
−0.311528 + 0.950237i \(0.600841\pi\)
\(674\) 0 0
\(675\) 2213.59i 0.126224i
\(676\) 0 0
\(677\) − 126.000i − 0.00715299i −0.999994 0.00357649i \(-0.998862\pi\)
0.999994 0.00357649i \(-0.00113844\pi\)
\(678\) 0 0
\(679\) −4819.31 −0.272383
\(680\) 0 0
\(681\) −10520.0 −0.591964
\(682\) 0 0
\(683\) − 16412.2i − 0.919467i −0.888057 0.459734i \(-0.847945\pi\)
0.888057 0.459734i \(-0.152055\pi\)
\(684\) 0 0
\(685\) 7730.00i 0.431165i
\(686\) 0 0
\(687\) 6640.78 0.368794
\(688\) 0 0
\(689\) −9804.00 −0.542094
\(690\) 0 0
\(691\) − 13193.0i − 0.726319i −0.931727 0.363159i \(-0.881698\pi\)
0.931727 0.363159i \(-0.118302\pi\)
\(692\) 0 0
\(693\) 3120.00i 0.171023i
\(694\) 0 0
\(695\) 1644.38 0.0897483
\(696\) 0 0
\(697\) 9180.00 0.498877
\(698\) 0 0
\(699\) 17569.6i 0.950707i
\(700\) 0 0
\(701\) − 22010.0i − 1.18589i −0.805244 0.592943i \(-0.797966\pi\)
0.805244 0.592943i \(-0.202034\pi\)
\(702\) 0 0
\(703\) 20845.7 1.11837
\(704\) 0 0
\(705\) −4200.00 −0.224370
\(706\) 0 0
\(707\) − 11346.3i − 0.603564i
\(708\) 0 0
\(709\) − 550.000i − 0.0291335i −0.999894 0.0145668i \(-0.995363\pi\)
0.999894 0.0145668i \(-0.00463691\pi\)
\(710\) 0 0
\(711\) −3617.65 −0.190819
\(712\) 0 0
\(713\) 28080.0 1.47490
\(714\) 0 0
\(715\) − 2403.33i − 0.125706i
\(716\) 0 0
\(717\) − 16160.0i − 0.841710i
\(718\) 0 0
\(719\) 17936.4 0.930343 0.465171 0.885221i \(-0.345993\pi\)
0.465171 + 0.885221i \(0.345993\pi\)
\(720\) 0 0
\(721\) −9480.00 −0.489672
\(722\) 0 0
\(723\) 33836.4i 1.74051i
\(724\) 0 0
\(725\) − 6750.00i − 0.345778i
\(726\) 0 0
\(727\) −16728.4 −0.853403 −0.426701 0.904393i \(-0.640324\pi\)
−0.426701 + 0.904393i \(0.640324\pi\)
\(728\) 0 0
\(729\) −3277.00 −0.166489
\(730\) 0 0
\(731\) 18278.0i 0.924808i
\(732\) 0 0
\(733\) 2422.00i 0.122044i 0.998136 + 0.0610222i \(0.0194361\pi\)
−0.998136 + 0.0610222i \(0.980564\pi\)
\(734\) 0 0
\(735\) −537.587 −0.0269785
\(736\) 0 0
\(737\) 10320.0 0.515797
\(738\) 0 0
\(739\) 19555.5i 0.973426i 0.873562 + 0.486713i \(0.161804\pi\)
−0.873562 + 0.486713i \(0.838196\pi\)
\(740\) 0 0
\(741\) 24320.0i 1.20569i
\(742\) 0 0
\(743\) −31059.9 −1.53362 −0.766808 0.641876i \(-0.778156\pi\)
−0.766808 + 0.641876i \(0.778156\pi\)
\(744\) 0 0
\(745\) −16230.0 −0.798149
\(746\) 0 0
\(747\) − 14388.4i − 0.704743i
\(748\) 0 0
\(749\) 11880.0i 0.579554i
\(750\) 0 0
\(751\) −12155.8 −0.590641 −0.295320 0.955398i \(-0.595426\pi\)
−0.295320 + 0.955398i \(0.595426\pi\)
\(752\) 0 0
\(753\) −37200.0 −1.80032
\(754\) 0 0
\(755\) 7526.22i 0.362791i
\(756\) 0 0
\(757\) 19346.0i 0.928854i 0.885611 + 0.464427i \(0.153740\pi\)
−0.885611 + 0.464427i \(0.846260\pi\)
\(758\) 0 0
\(759\) −6577.54 −0.314558
\(760\) 0 0
\(761\) 33078.0 1.57566 0.787830 0.615893i \(-0.211205\pi\)
0.787830 + 0.615893i \(0.211205\pi\)
\(762\) 0 0
\(763\) 16203.5i 0.768816i
\(764\) 0 0
\(765\) 2210.00i 0.104448i
\(766\) 0 0
\(767\) 2884.00 0.135769
\(768\) 0 0
\(769\) 32530.0 1.52544 0.762719 0.646730i \(-0.223864\pi\)
0.762719 + 0.646730i \(0.223864\pi\)
\(770\) 0 0
\(771\) − 6792.57i − 0.317287i
\(772\) 0 0
\(773\) 12002.0i 0.558450i 0.960226 + 0.279225i \(0.0900776\pi\)
−0.960226 + 0.279225i \(0.909922\pi\)
\(774\) 0 0
\(775\) −8538.15 −0.395741
\(776\) 0 0
\(777\) −24720.0 −1.14134
\(778\) 0 0
\(779\) 27322.1i 1.25663i
\(780\) 0 0
\(781\) 8160.00i 0.373864i
\(782\) 0 0
\(783\) 23906.8 1.09114
\(784\) 0 0
\(785\) −6130.00 −0.278712
\(786\) 0 0
\(787\) − 19954.0i − 0.903789i −0.892071 0.451895i \(-0.850748\pi\)
0.892071 0.451895i \(-0.149252\pi\)
\(788\) 0 0
\(789\) 9400.00i 0.424143i
\(790\) 0 0
\(791\) 32217.3 1.44819
\(792\) 0 0
\(793\) 9500.00 0.425416
\(794\) 0 0
\(795\) − 8158.68i − 0.363973i
\(796\) 0 0
\(797\) − 32666.0i − 1.45181i −0.687797 0.725903i \(-0.741422\pi\)
0.687797 0.725903i \(-0.258578\pi\)
\(798\) 0 0
\(799\) 4515.73 0.199944
\(800\) 0 0
\(801\) 11570.0 0.510369
\(802\) 0 0
\(803\) − 13635.7i − 0.599246i
\(804\) 0 0
\(805\) − 7800.00i − 0.341508i
\(806\) 0 0
\(807\) 2567.77 0.112007
\(808\) 0 0
\(809\) 23110.0 1.00433 0.502166 0.864771i \(-0.332537\pi\)
0.502166 + 0.864771i \(0.332537\pi\)
\(810\) 0 0
\(811\) − 35632.5i − 1.54282i −0.636338 0.771411i \(-0.719552\pi\)
0.636338 0.771411i \(-0.280448\pi\)
\(812\) 0 0
\(813\) 2480.00i 0.106983i
\(814\) 0 0
\(815\) −7241.62 −0.311243
\(816\) 0 0
\(817\) −54400.0 −2.32952
\(818\) 0 0
\(819\) − 9372.99i − 0.399901i
\(820\) 0 0
\(821\) 8850.00i 0.376208i 0.982149 + 0.188104i \(0.0602343\pi\)
−0.982149 + 0.188104i \(0.939766\pi\)
\(822\) 0 0
\(823\) 13895.0 0.588519 0.294259 0.955726i \(-0.404927\pi\)
0.294259 + 0.955726i \(0.404927\pi\)
\(824\) 0 0
\(825\) 2000.00 0.0844013
\(826\) 0 0
\(827\) 35841.3i 1.50704i 0.657425 + 0.753520i \(0.271646\pi\)
−0.657425 + 0.753520i \(0.728354\pi\)
\(828\) 0 0
\(829\) − 23034.0i − 0.965023i −0.875890 0.482511i \(-0.839725\pi\)
0.875890 0.482511i \(-0.160275\pi\)
\(830\) 0 0
\(831\) −37529.9 −1.56666
\(832\) 0 0
\(833\) 578.000 0.0240414
\(834\) 0 0
\(835\) 11668.8i 0.483612i
\(836\) 0 0
\(837\) − 30240.0i − 1.24880i
\(838\) 0 0
\(839\) 36960.7 1.52089 0.760444 0.649403i \(-0.224981\pi\)
0.760444 + 0.649403i \(0.224981\pi\)
\(840\) 0 0
\(841\) −48511.0 −1.98905
\(842\) 0 0
\(843\) − 11826.9i − 0.483204i
\(844\) 0 0
\(845\) − 3765.00i − 0.153278i
\(846\) 0 0
\(847\) 22218.2 0.901328
\(848\) 0 0
\(849\) 30920.0 1.24991
\(850\) 0 0
\(851\) − 16937.2i − 0.682254i
\(852\) 0 0
\(853\) 19122.0i 0.767555i 0.923425 + 0.383778i \(0.125377\pi\)
−0.923425 + 0.383778i \(0.874623\pi\)
\(854\) 0 0
\(855\) −6577.54 −0.263096
\(856\) 0 0
\(857\) −17786.0 −0.708936 −0.354468 0.935068i \(-0.615338\pi\)
−0.354468 + 0.935068i \(0.615338\pi\)
\(858\) 0 0
\(859\) 28713.5i 1.14050i 0.821470 + 0.570251i \(0.193154\pi\)
−0.821470 + 0.570251i \(0.806846\pi\)
\(860\) 0 0
\(861\) − 32400.0i − 1.28245i
\(862\) 0 0
\(863\) 23748.7 0.936750 0.468375 0.883530i \(-0.344840\pi\)
0.468375 + 0.883530i \(0.344840\pi\)
\(864\) 0 0
\(865\) −15490.0 −0.608874
\(866\) 0 0
\(867\) 23761.4i 0.930770i
\(868\) 0 0
\(869\) − 3520.00i − 0.137408i
\(870\) 0 0
\(871\) −31003.0 −1.20608
\(872\) 0 0
\(873\) 3302.00 0.128013
\(874\) 0 0
\(875\) 2371.71i 0.0916324i
\(876\) 0 0
\(877\) − 7706.00i − 0.296708i −0.988934 0.148354i \(-0.952602\pi\)
0.988934 0.148354i \(-0.0473975\pi\)
\(878\) 0 0
\(879\) −32875.0 −1.26149
\(880\) 0 0
\(881\) 10410.0 0.398095 0.199048 0.979990i \(-0.436215\pi\)
0.199048 + 0.979990i \(0.436215\pi\)
\(882\) 0 0
\(883\) − 26822.4i − 1.02225i −0.859506 0.511125i \(-0.829229\pi\)
0.859506 0.511125i \(-0.170771\pi\)
\(884\) 0 0
\(885\) 2400.00i 0.0911583i
\(886\) 0 0
\(887\) −21130.3 −0.799873 −0.399937 0.916543i \(-0.630968\pi\)
−0.399937 + 0.916543i \(0.630968\pi\)
\(888\) 0 0
\(889\) 4440.00 0.167506
\(890\) 0 0
\(891\) 11523.3i 0.433273i
\(892\) 0 0
\(893\) 13440.0i 0.503642i
\(894\) 0 0
\(895\) −11763.7 −0.439348
\(896\) 0 0
\(897\) 19760.0 0.735526
\(898\) 0 0
\(899\) 92212.0i 3.42096i
\(900\) 0 0
\(901\) 8772.00i 0.324348i
\(902\) 0 0
\(903\) 64510.5 2.37738
\(904\) 0 0
\(905\) −10910.0 −0.400730
\(906\) 0 0
\(907\) 19220.3i 0.703639i 0.936068 + 0.351819i \(0.114437\pi\)
−0.936068 + 0.351819i \(0.885563\pi\)
\(908\) 0 0
\(909\) 7774.00i 0.283660i
\(910\) 0 0
\(911\) 39402.0 1.43298 0.716491 0.697597i \(-0.245747\pi\)
0.716491 + 0.697597i \(0.245747\pi\)
\(912\) 0 0
\(913\) 14000.0 0.507483
\(914\) 0 0
\(915\) 7905.69i 0.285633i
\(916\) 0 0
\(917\) − 32880.0i − 1.18407i
\(918\) 0 0
\(919\) −42931.1 −1.54099 −0.770493 0.637449i \(-0.779990\pi\)
−0.770493 + 0.637449i \(0.779990\pi\)
\(920\) 0 0
\(921\) 23720.0 0.848643
\(922\) 0 0
\(923\) − 24514.0i − 0.874201i
\(924\) 0 0
\(925\) 5150.00i 0.183060i
\(926\) 0 0
\(927\) 6495.32 0.230134
\(928\) 0 0
\(929\) 34746.0 1.22710 0.613552 0.789654i \(-0.289740\pi\)
0.613552 + 0.789654i \(0.289740\pi\)
\(930\) 0 0
\(931\) 1720.28i 0.0605584i
\(932\) 0 0
\(933\) − 39600.0i − 1.38955i
\(934\) 0 0
\(935\) −2150.35 −0.0752128
\(936\) 0 0
\(937\) −21594.0 −0.752876 −0.376438 0.926442i \(-0.622851\pi\)
−0.376438 + 0.926442i \(0.622851\pi\)
\(938\) 0 0
\(939\) 14027.9i 0.487521i
\(940\) 0 0
\(941\) − 20018.0i − 0.693484i −0.937961 0.346742i \(-0.887288\pi\)
0.937961 0.346742i \(-0.112712\pi\)
\(942\) 0 0
\(943\) 22199.2 0.766601
\(944\) 0 0
\(945\) −8400.00 −0.289156
\(946\) 0 0
\(947\) 46150.3i 1.58361i 0.610771 + 0.791807i \(0.290859\pi\)
−0.610771 + 0.791807i \(0.709141\pi\)
\(948\) 0 0
\(949\) 40964.0i 1.40121i
\(950\) 0 0
\(951\) 26145.7 0.891517
\(952\) 0 0
\(953\) 342.000 0.0116248 0.00581242 0.999983i \(-0.498150\pi\)
0.00581242 + 0.999983i \(0.498150\pi\)
\(954\) 0 0
\(955\) 15115.7i 0.512180i
\(956\) 0 0
\(957\) − 21600.0i − 0.729602i
\(958\) 0 0
\(959\) −29333.3 −0.987718
\(960\) 0 0
\(961\) 86849.0 2.91528
\(962\) 0 0
\(963\) − 8139.70i − 0.272376i
\(964\) 0 0
\(965\) − 6490.00i − 0.216498i
\(966\) 0 0
\(967\) 51728.5 1.72025 0.860123 0.510087i \(-0.170387\pi\)
0.860123 + 0.510087i \(0.170387\pi\)
\(968\) 0 0
\(969\) 21760.0 0.721395
\(970\) 0 0
\(971\) 3099.03i 0.102423i 0.998688 + 0.0512115i \(0.0163083\pi\)
−0.998688 + 0.0512115i \(0.983692\pi\)
\(972\) 0 0
\(973\) 6240.00i 0.205596i
\(974\) 0 0
\(975\) −6008.33 −0.197354
\(976\) 0 0
\(977\) 26226.0 0.858796 0.429398 0.903115i \(-0.358726\pi\)
0.429398 + 0.903115i \(0.358726\pi\)
\(978\) 0 0
\(979\) 11257.7i 0.367516i
\(980\) 0 0
\(981\) − 11102.0i − 0.361325i
\(982\) 0 0
\(983\) −11049.0 −0.358503 −0.179251 0.983803i \(-0.557368\pi\)
−0.179251 + 0.983803i \(0.557368\pi\)
\(984\) 0 0
\(985\) −14230.0 −0.460310
\(986\) 0 0
\(987\) − 15937.9i − 0.513990i
\(988\) 0 0
\(989\) 44200.0i 1.42111i
\(990\) 0 0
\(991\) 45928.9 1.47223 0.736115 0.676856i \(-0.236658\pi\)
0.736115 + 0.676856i \(0.236658\pi\)
\(992\) 0 0
\(993\) 75600.0 2.41601
\(994\) 0 0
\(995\) − 17961.7i − 0.572287i
\(996\) 0 0
\(997\) 31026.0i 0.985560i 0.870154 + 0.492780i \(0.164019\pi\)
−0.870154 + 0.492780i \(0.835981\pi\)
\(998\) 0 0
\(999\) −18240.0 −0.577666
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 1280.4.d.u.641.1 4
4.3 odd 2 inner 1280.4.d.u.641.3 4
8.3 odd 2 inner 1280.4.d.u.641.2 4
8.5 even 2 inner 1280.4.d.u.641.4 4
16.3 odd 4 320.4.a.p.1.1 2
16.5 even 4 160.4.a.f.1.1 2
16.11 odd 4 160.4.a.f.1.2 yes 2
16.13 even 4 320.4.a.p.1.2 2
48.5 odd 4 1440.4.a.v.1.1 2
48.11 even 4 1440.4.a.v.1.2 2
80.19 odd 4 1600.4.a.ch.1.2 2
80.27 even 4 800.4.c.j.449.1 4
80.29 even 4 1600.4.a.ch.1.1 2
80.37 odd 4 800.4.c.j.449.4 4
80.43 even 4 800.4.c.j.449.3 4
80.53 odd 4 800.4.c.j.449.2 4
80.59 odd 4 800.4.a.p.1.1 2
80.69 even 4 800.4.a.p.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
160.4.a.f.1.1 2 16.5 even 4
160.4.a.f.1.2 yes 2 16.11 odd 4
320.4.a.p.1.1 2 16.3 odd 4
320.4.a.p.1.2 2 16.13 even 4
800.4.a.p.1.1 2 80.59 odd 4
800.4.a.p.1.2 2 80.69 even 4
800.4.c.j.449.1 4 80.27 even 4
800.4.c.j.449.2 4 80.53 odd 4
800.4.c.j.449.3 4 80.43 even 4
800.4.c.j.449.4 4 80.37 odd 4
1280.4.d.u.641.1 4 1.1 even 1 trivial
1280.4.d.u.641.2 4 8.3 odd 2 inner
1280.4.d.u.641.3 4 4.3 odd 2 inner
1280.4.d.u.641.4 4 8.5 even 2 inner
1440.4.a.v.1.1 2 48.5 odd 4
1440.4.a.v.1.2 2 48.11 even 4
1600.4.a.ch.1.1 2 80.29 even 4
1600.4.a.ch.1.2 2 80.19 odd 4