Properties

Label 960.4.a.bf.1.1
Level $960$
Weight $4$
Character 960.1
Self dual yes
Analytic conductor $56.642$
Analytic rank $0$
Dimension $1$
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] = [960,4,Mod(1,960)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(960, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0, 0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("960.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 960 = 2^{6} \cdot 3 \cdot 5 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 960.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: \(56.6418336055\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 120)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 960.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+3.00000 q^{3} +5.00000 q^{5} +4.00000 q^{7} +9.00000 q^{9} +O(q^{10})\) \(q+3.00000 q^{3} +5.00000 q^{5} +4.00000 q^{7} +9.00000 q^{9} -72.0000 q^{11} +6.00000 q^{13} +15.0000 q^{15} +38.0000 q^{17} -52.0000 q^{19} +12.0000 q^{21} +152.000 q^{23} +25.0000 q^{25} +27.0000 q^{27} +78.0000 q^{29} +120.000 q^{31} -216.000 q^{33} +20.0000 q^{35} +150.000 q^{37} +18.0000 q^{39} +362.000 q^{41} +484.000 q^{43} +45.0000 q^{45} +280.000 q^{47} -327.000 q^{49} +114.000 q^{51} +670.000 q^{53} -360.000 q^{55} -156.000 q^{57} -696.000 q^{59} -222.000 q^{61} +36.0000 q^{63} +30.0000 q^{65} +4.00000 q^{67} +456.000 q^{69} +96.0000 q^{71} +178.000 q^{73} +75.0000 q^{75} -288.000 q^{77} -632.000 q^{79} +81.0000 q^{81} +612.000 q^{83} +190.000 q^{85} +234.000 q^{87} +994.000 q^{89} +24.0000 q^{91} +360.000 q^{93} -260.000 q^{95} +1634.00 q^{97} -648.000 q^{99} +O(q^{100})\)

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
\(3\) 3.00000 0.577350
\(4\) 0 0
\(5\) 5.00000 0.447214
\(6\) 0 0
\(7\) 4.00000 0.215980 0.107990 0.994152i \(-0.465559\pi\)
0.107990 + 0.994152i \(0.465559\pi\)
\(8\) 0 0
\(9\) 9.00000 0.333333
\(10\) 0 0
\(11\) −72.0000 −1.97353 −0.986764 0.162160i \(-0.948154\pi\)
−0.986764 + 0.162160i \(0.948154\pi\)
\(12\) 0 0
\(13\) 6.00000 0.128008 0.0640039 0.997950i \(-0.479613\pi\)
0.0640039 + 0.997950i \(0.479613\pi\)
\(14\) 0 0
\(15\) 15.0000 0.258199
\(16\) 0 0
\(17\) 38.0000 0.542138 0.271069 0.962560i \(-0.412623\pi\)
0.271069 + 0.962560i \(0.412623\pi\)
\(18\) 0 0
\(19\) −52.0000 −0.627875 −0.313937 0.949444i \(-0.601648\pi\)
−0.313937 + 0.949444i \(0.601648\pi\)
\(20\) 0 0
\(21\) 12.0000 0.124696
\(22\) 0 0
\(23\) 152.000 1.37801 0.689004 0.724757i \(-0.258048\pi\)
0.689004 + 0.724757i \(0.258048\pi\)
\(24\) 0 0
\(25\) 25.0000 0.200000
\(26\) 0 0
\(27\) 27.0000 0.192450
\(28\) 0 0
\(29\) 78.0000 0.499456 0.249728 0.968316i \(-0.419659\pi\)
0.249728 + 0.968316i \(0.419659\pi\)
\(30\) 0 0
\(31\) 120.000 0.695246 0.347623 0.937634i \(-0.386989\pi\)
0.347623 + 0.937634i \(0.386989\pi\)
\(32\) 0 0
\(33\) −216.000 −1.13942
\(34\) 0 0
\(35\) 20.0000 0.0965891
\(36\) 0 0
\(37\) 150.000 0.666482 0.333241 0.942842i \(-0.391858\pi\)
0.333241 + 0.942842i \(0.391858\pi\)
\(38\) 0 0
\(39\) 18.0000 0.0739053
\(40\) 0 0
\(41\) 362.000 1.37890 0.689450 0.724333i \(-0.257852\pi\)
0.689450 + 0.724333i \(0.257852\pi\)
\(42\) 0 0
\(43\) 484.000 1.71650 0.858248 0.513236i \(-0.171553\pi\)
0.858248 + 0.513236i \(0.171553\pi\)
\(44\) 0 0
\(45\) 45.0000 0.149071
\(46\) 0 0
\(47\) 280.000 0.868983 0.434491 0.900676i \(-0.356928\pi\)
0.434491 + 0.900676i \(0.356928\pi\)
\(48\) 0 0
\(49\) −327.000 −0.953353
\(50\) 0 0
\(51\) 114.000 0.313004
\(52\) 0 0
\(53\) 670.000 1.73644 0.868222 0.496175i \(-0.165263\pi\)
0.868222 + 0.496175i \(0.165263\pi\)
\(54\) 0 0
\(55\) −360.000 −0.882589
\(56\) 0 0
\(57\) −156.000 −0.362504
\(58\) 0 0
\(59\) −696.000 −1.53579 −0.767894 0.640577i \(-0.778695\pi\)
−0.767894 + 0.640577i \(0.778695\pi\)
\(60\) 0 0
\(61\) −222.000 −0.465970 −0.232985 0.972480i \(-0.574849\pi\)
−0.232985 + 0.972480i \(0.574849\pi\)
\(62\) 0 0
\(63\) 36.0000 0.0719932
\(64\) 0 0
\(65\) 30.0000 0.0572468
\(66\) 0 0
\(67\) 4.00000 0.00729370 0.00364685 0.999993i \(-0.498839\pi\)
0.00364685 + 0.999993i \(0.498839\pi\)
\(68\) 0 0
\(69\) 456.000 0.795593
\(70\) 0 0
\(71\) 96.0000 0.160466 0.0802331 0.996776i \(-0.474434\pi\)
0.0802331 + 0.996776i \(0.474434\pi\)
\(72\) 0 0
\(73\) 178.000 0.285388 0.142694 0.989767i \(-0.454424\pi\)
0.142694 + 0.989767i \(0.454424\pi\)
\(74\) 0 0
\(75\) 75.0000 0.115470
\(76\) 0 0
\(77\) −288.000 −0.426242
\(78\) 0 0
\(79\) −632.000 −0.900070 −0.450035 0.893011i \(-0.648589\pi\)
−0.450035 + 0.893011i \(0.648589\pi\)
\(80\) 0 0
\(81\) 81.0000 0.111111
\(82\) 0 0
\(83\) 612.000 0.809346 0.404673 0.914461i \(-0.367385\pi\)
0.404673 + 0.914461i \(0.367385\pi\)
\(84\) 0 0
\(85\) 190.000 0.242452
\(86\) 0 0
\(87\) 234.000 0.288361
\(88\) 0 0
\(89\) 994.000 1.18386 0.591931 0.805988i \(-0.298366\pi\)
0.591931 + 0.805988i \(0.298366\pi\)
\(90\) 0 0
\(91\) 24.0000 0.0276471
\(92\) 0 0
\(93\) 360.000 0.401401
\(94\) 0 0
\(95\) −260.000 −0.280794
\(96\) 0 0
\(97\) 1634.00 1.71039 0.855194 0.518309i \(-0.173438\pi\)
0.855194 + 0.518309i \(0.173438\pi\)
\(98\) 0 0
\(99\) −648.000 −0.657843
\(100\) 0 0
\(101\) −890.000 −0.876815 −0.438407 0.898776i \(-0.644457\pi\)
−0.438407 + 0.898776i \(0.644457\pi\)
\(102\) 0 0
\(103\) −524.000 −0.501274 −0.250637 0.968081i \(-0.580640\pi\)
−0.250637 + 0.968081i \(0.580640\pi\)
\(104\) 0 0
\(105\) 60.0000 0.0557657
\(106\) 0 0
\(107\) −932.000 −0.842055 −0.421027 0.907048i \(-0.638330\pi\)
−0.421027 + 0.907048i \(0.638330\pi\)
\(108\) 0 0
\(109\) −446.000 −0.391918 −0.195959 0.980612i \(-0.562782\pi\)
−0.195959 + 0.980612i \(0.562782\pi\)
\(110\) 0 0
\(111\) 450.000 0.384794
\(112\) 0 0
\(113\) −786.000 −0.654342 −0.327171 0.944965i \(-0.606095\pi\)
−0.327171 + 0.944965i \(0.606095\pi\)
\(114\) 0 0
\(115\) 760.000 0.616264
\(116\) 0 0
\(117\) 54.0000 0.0426692
\(118\) 0 0
\(119\) 152.000 0.117091
\(120\) 0 0
\(121\) 3853.00 2.89482
\(122\) 0 0
\(123\) 1086.00 0.796108
\(124\) 0 0
\(125\) 125.000 0.0894427
\(126\) 0 0
\(127\) 716.000 0.500273 0.250137 0.968211i \(-0.419524\pi\)
0.250137 + 0.968211i \(0.419524\pi\)
\(128\) 0 0
\(129\) 1452.00 0.991019
\(130\) 0 0
\(131\) 808.000 0.538895 0.269448 0.963015i \(-0.413159\pi\)
0.269448 + 0.963015i \(0.413159\pi\)
\(132\) 0 0
\(133\) −208.000 −0.135608
\(134\) 0 0
\(135\) 135.000 0.0860663
\(136\) 0 0
\(137\) −1770.00 −1.10381 −0.551903 0.833909i \(-0.686098\pi\)
−0.551903 + 0.833909i \(0.686098\pi\)
\(138\) 0 0
\(139\) 924.000 0.563832 0.281916 0.959439i \(-0.409030\pi\)
0.281916 + 0.959439i \(0.409030\pi\)
\(140\) 0 0
\(141\) 840.000 0.501708
\(142\) 0 0
\(143\) −432.000 −0.252627
\(144\) 0 0
\(145\) 390.000 0.223364
\(146\) 0 0
\(147\) −981.000 −0.550418
\(148\) 0 0
\(149\) 3198.00 1.75832 0.879162 0.476522i \(-0.158103\pi\)
0.879162 + 0.476522i \(0.158103\pi\)
\(150\) 0 0
\(151\) −3384.00 −1.82375 −0.911874 0.410470i \(-0.865365\pi\)
−0.911874 + 0.410470i \(0.865365\pi\)
\(152\) 0 0
\(153\) 342.000 0.180713
\(154\) 0 0
\(155\) 600.000 0.310924
\(156\) 0 0
\(157\) 3302.00 1.67852 0.839262 0.543727i \(-0.182987\pi\)
0.839262 + 0.543727i \(0.182987\pi\)
\(158\) 0 0
\(159\) 2010.00 1.00254
\(160\) 0 0
\(161\) 608.000 0.297622
\(162\) 0 0
\(163\) −2252.00 −1.08215 −0.541074 0.840975i \(-0.681982\pi\)
−0.541074 + 0.840975i \(0.681982\pi\)
\(164\) 0 0
\(165\) −1080.00 −0.509563
\(166\) 0 0
\(167\) −184.000 −0.0852596 −0.0426298 0.999091i \(-0.513574\pi\)
−0.0426298 + 0.999091i \(0.513574\pi\)
\(168\) 0 0
\(169\) −2161.00 −0.983614
\(170\) 0 0
\(171\) −468.000 −0.209292
\(172\) 0 0
\(173\) 2646.00 1.16284 0.581421 0.813603i \(-0.302497\pi\)
0.581421 + 0.813603i \(0.302497\pi\)
\(174\) 0 0
\(175\) 100.000 0.0431959
\(176\) 0 0
\(177\) −2088.00 −0.886688
\(178\) 0 0
\(179\) 608.000 0.253877 0.126939 0.991911i \(-0.459485\pi\)
0.126939 + 0.991911i \(0.459485\pi\)
\(180\) 0 0
\(181\) −2246.00 −0.922342 −0.461171 0.887311i \(-0.652570\pi\)
−0.461171 + 0.887311i \(0.652570\pi\)
\(182\) 0 0
\(183\) −666.000 −0.269028
\(184\) 0 0
\(185\) 750.000 0.298060
\(186\) 0 0
\(187\) −2736.00 −1.06993
\(188\) 0 0
\(189\) 108.000 0.0415653
\(190\) 0 0
\(191\) −3848.00 −1.45776 −0.728878 0.684643i \(-0.759958\pi\)
−0.728878 + 0.684643i \(0.759958\pi\)
\(192\) 0 0
\(193\) 2058.00 0.767555 0.383777 0.923426i \(-0.374623\pi\)
0.383777 + 0.923426i \(0.374623\pi\)
\(194\) 0 0
\(195\) 90.0000 0.0330515
\(196\) 0 0
\(197\) 3838.00 1.38805 0.694026 0.719950i \(-0.255835\pi\)
0.694026 + 0.719950i \(0.255835\pi\)
\(198\) 0 0
\(199\) −1992.00 −0.709594 −0.354797 0.934943i \(-0.615450\pi\)
−0.354797 + 0.934943i \(0.615450\pi\)
\(200\) 0 0
\(201\) 12.0000 0.00421102
\(202\) 0 0
\(203\) 312.000 0.107872
\(204\) 0 0
\(205\) 1810.00 0.616663
\(206\) 0 0
\(207\) 1368.00 0.459336
\(208\) 0 0
\(209\) 3744.00 1.23913
\(210\) 0 0
\(211\) −4764.00 −1.55435 −0.777174 0.629286i \(-0.783347\pi\)
−0.777174 + 0.629286i \(0.783347\pi\)
\(212\) 0 0
\(213\) 288.000 0.0926452
\(214\) 0 0
\(215\) 2420.00 0.767640
\(216\) 0 0
\(217\) 480.000 0.150159
\(218\) 0 0
\(219\) 534.000 0.164769
\(220\) 0 0
\(221\) 228.000 0.0693979
\(222\) 0 0
\(223\) 4092.00 1.22879 0.614396 0.788998i \(-0.289400\pi\)
0.614396 + 0.788998i \(0.289400\pi\)
\(224\) 0 0
\(225\) 225.000 0.0666667
\(226\) 0 0
\(227\) −468.000 −0.136838 −0.0684191 0.997657i \(-0.521795\pi\)
−0.0684191 + 0.997657i \(0.521795\pi\)
\(228\) 0 0
\(229\) 5586.00 1.61194 0.805968 0.591959i \(-0.201645\pi\)
0.805968 + 0.591959i \(0.201645\pi\)
\(230\) 0 0
\(231\) −864.000 −0.246091
\(232\) 0 0
\(233\) −1058.00 −0.297476 −0.148738 0.988877i \(-0.547521\pi\)
−0.148738 + 0.988877i \(0.547521\pi\)
\(234\) 0 0
\(235\) 1400.00 0.388621
\(236\) 0 0
\(237\) −1896.00 −0.519656
\(238\) 0 0
\(239\) 6840.00 1.85123 0.925613 0.378472i \(-0.123550\pi\)
0.925613 + 0.378472i \(0.123550\pi\)
\(240\) 0 0
\(241\) −6430.00 −1.71864 −0.859321 0.511437i \(-0.829113\pi\)
−0.859321 + 0.511437i \(0.829113\pi\)
\(242\) 0 0
\(243\) 243.000 0.0641500
\(244\) 0 0
\(245\) −1635.00 −0.426352
\(246\) 0 0
\(247\) −312.000 −0.0803728
\(248\) 0 0
\(249\) 1836.00 0.467276
\(250\) 0 0
\(251\) 6352.00 1.59735 0.798675 0.601763i \(-0.205535\pi\)
0.798675 + 0.601763i \(0.205535\pi\)
\(252\) 0 0
\(253\) −10944.0 −2.71954
\(254\) 0 0
\(255\) 570.000 0.139980
\(256\) 0 0
\(257\) 1422.00 0.345144 0.172572 0.984997i \(-0.444792\pi\)
0.172572 + 0.984997i \(0.444792\pi\)
\(258\) 0 0
\(259\) 600.000 0.143947
\(260\) 0 0
\(261\) 702.000 0.166485
\(262\) 0 0
\(263\) 7224.00 1.69373 0.846865 0.531808i \(-0.178487\pi\)
0.846865 + 0.531808i \(0.178487\pi\)
\(264\) 0 0
\(265\) 3350.00 0.776562
\(266\) 0 0
\(267\) 2982.00 0.683504
\(268\) 0 0
\(269\) −3186.00 −0.722133 −0.361067 0.932540i \(-0.617587\pi\)
−0.361067 + 0.932540i \(0.617587\pi\)
\(270\) 0 0
\(271\) −256.000 −0.0573834 −0.0286917 0.999588i \(-0.509134\pi\)
−0.0286917 + 0.999588i \(0.509134\pi\)
\(272\) 0 0
\(273\) 72.0000 0.0159620
\(274\) 0 0
\(275\) −1800.00 −0.394706
\(276\) 0 0
\(277\) 5942.00 1.28888 0.644441 0.764654i \(-0.277090\pi\)
0.644441 + 0.764654i \(0.277090\pi\)
\(278\) 0 0
\(279\) 1080.00 0.231749
\(280\) 0 0
\(281\) 3202.00 0.679770 0.339885 0.940467i \(-0.389612\pi\)
0.339885 + 0.940467i \(0.389612\pi\)
\(282\) 0 0
\(283\) −3940.00 −0.827593 −0.413796 0.910370i \(-0.635797\pi\)
−0.413796 + 0.910370i \(0.635797\pi\)
\(284\) 0 0
\(285\) −780.000 −0.162117
\(286\) 0 0
\(287\) 1448.00 0.297814
\(288\) 0 0
\(289\) −3469.00 −0.706086
\(290\) 0 0
\(291\) 4902.00 0.987493
\(292\) 0 0
\(293\) −1826.00 −0.364082 −0.182041 0.983291i \(-0.558270\pi\)
−0.182041 + 0.983291i \(0.558270\pi\)
\(294\) 0 0
\(295\) −3480.00 −0.686825
\(296\) 0 0
\(297\) −1944.00 −0.379806
\(298\) 0 0
\(299\) 912.000 0.176396
\(300\) 0 0
\(301\) 1936.00 0.370728
\(302\) 0 0
\(303\) −2670.00 −0.506229
\(304\) 0 0
\(305\) −1110.00 −0.208388
\(306\) 0 0
\(307\) −6580.00 −1.22326 −0.611629 0.791144i \(-0.709486\pi\)
−0.611629 + 0.791144i \(0.709486\pi\)
\(308\) 0 0
\(309\) −1572.00 −0.289411
\(310\) 0 0
\(311\) −5728.00 −1.04439 −0.522195 0.852826i \(-0.674887\pi\)
−0.522195 + 0.852826i \(0.674887\pi\)
\(312\) 0 0
\(313\) −1742.00 −0.314580 −0.157290 0.987552i \(-0.550276\pi\)
−0.157290 + 0.987552i \(0.550276\pi\)
\(314\) 0 0
\(315\) 180.000 0.0321964
\(316\) 0 0
\(317\) −8746.00 −1.54960 −0.774802 0.632204i \(-0.782150\pi\)
−0.774802 + 0.632204i \(0.782150\pi\)
\(318\) 0 0
\(319\) −5616.00 −0.985692
\(320\) 0 0
\(321\) −2796.00 −0.486160
\(322\) 0 0
\(323\) −1976.00 −0.340395
\(324\) 0 0
\(325\) 150.000 0.0256015
\(326\) 0 0
\(327\) −1338.00 −0.226274
\(328\) 0 0
\(329\) 1120.00 0.187683
\(330\) 0 0
\(331\) 2564.00 0.425771 0.212885 0.977077i \(-0.431714\pi\)
0.212885 + 0.977077i \(0.431714\pi\)
\(332\) 0 0
\(333\) 1350.00 0.222161
\(334\) 0 0
\(335\) 20.0000 0.00326184
\(336\) 0 0
\(337\) −4166.00 −0.673402 −0.336701 0.941612i \(-0.609311\pi\)
−0.336701 + 0.941612i \(0.609311\pi\)
\(338\) 0 0
\(339\) −2358.00 −0.377785
\(340\) 0 0
\(341\) −8640.00 −1.37209
\(342\) 0 0
\(343\) −2680.00 −0.421885
\(344\) 0 0
\(345\) 2280.00 0.355800
\(346\) 0 0
\(347\) 9444.00 1.46104 0.730519 0.682892i \(-0.239278\pi\)
0.730519 + 0.682892i \(0.239278\pi\)
\(348\) 0 0
\(349\) 9218.00 1.41383 0.706917 0.707296i \(-0.250085\pi\)
0.706917 + 0.707296i \(0.250085\pi\)
\(350\) 0 0
\(351\) 162.000 0.0246351
\(352\) 0 0
\(353\) −4698.00 −0.708355 −0.354177 0.935178i \(-0.615239\pi\)
−0.354177 + 0.935178i \(0.615239\pi\)
\(354\) 0 0
\(355\) 480.000 0.0717627
\(356\) 0 0
\(357\) 456.000 0.0676025
\(358\) 0 0
\(359\) −6056.00 −0.890316 −0.445158 0.895452i \(-0.646852\pi\)
−0.445158 + 0.895452i \(0.646852\pi\)
\(360\) 0 0
\(361\) −4155.00 −0.605773
\(362\) 0 0
\(363\) 11559.0 1.67132
\(364\) 0 0
\(365\) 890.000 0.127629
\(366\) 0 0
\(367\) −8228.00 −1.17029 −0.585147 0.810927i \(-0.698963\pi\)
−0.585147 + 0.810927i \(0.698963\pi\)
\(368\) 0 0
\(369\) 3258.00 0.459633
\(370\) 0 0
\(371\) 2680.00 0.375037
\(372\) 0 0
\(373\) −5954.00 −0.826505 −0.413253 0.910616i \(-0.635607\pi\)
−0.413253 + 0.910616i \(0.635607\pi\)
\(374\) 0 0
\(375\) 375.000 0.0516398
\(376\) 0 0
\(377\) 468.000 0.0639343
\(378\) 0 0
\(379\) −5284.00 −0.716150 −0.358075 0.933693i \(-0.616567\pi\)
−0.358075 + 0.933693i \(0.616567\pi\)
\(380\) 0 0
\(381\) 2148.00 0.288833
\(382\) 0 0
\(383\) 9832.00 1.31173 0.655864 0.754879i \(-0.272305\pi\)
0.655864 + 0.754879i \(0.272305\pi\)
\(384\) 0 0
\(385\) −1440.00 −0.190621
\(386\) 0 0
\(387\) 4356.00 0.572165
\(388\) 0 0
\(389\) 222.000 0.0289353 0.0144677 0.999895i \(-0.495395\pi\)
0.0144677 + 0.999895i \(0.495395\pi\)
\(390\) 0 0
\(391\) 5776.00 0.747071
\(392\) 0 0
\(393\) 2424.00 0.311131
\(394\) 0 0
\(395\) −3160.00 −0.402524
\(396\) 0 0
\(397\) −12098.0 −1.52942 −0.764712 0.644372i \(-0.777119\pi\)
−0.764712 + 0.644372i \(0.777119\pi\)
\(398\) 0 0
\(399\) −624.000 −0.0782934
\(400\) 0 0
\(401\) −5958.00 −0.741966 −0.370983 0.928640i \(-0.620979\pi\)
−0.370983 + 0.928640i \(0.620979\pi\)
\(402\) 0 0
\(403\) 720.000 0.0889969
\(404\) 0 0
\(405\) 405.000 0.0496904
\(406\) 0 0
\(407\) −10800.0 −1.31532
\(408\) 0 0
\(409\) 1930.00 0.233331 0.116665 0.993171i \(-0.462779\pi\)
0.116665 + 0.993171i \(0.462779\pi\)
\(410\) 0 0
\(411\) −5310.00 −0.637282
\(412\) 0 0
\(413\) −2784.00 −0.331699
\(414\) 0 0
\(415\) 3060.00 0.361951
\(416\) 0 0
\(417\) 2772.00 0.325529
\(418\) 0 0
\(419\) −4744.00 −0.553125 −0.276563 0.960996i \(-0.589195\pi\)
−0.276563 + 0.960996i \(0.589195\pi\)
\(420\) 0 0
\(421\) −1614.00 −0.186845 −0.0934223 0.995627i \(-0.529781\pi\)
−0.0934223 + 0.995627i \(0.529781\pi\)
\(422\) 0 0
\(423\) 2520.00 0.289661
\(424\) 0 0
\(425\) 950.000 0.108428
\(426\) 0 0
\(427\) −888.000 −0.100640
\(428\) 0 0
\(429\) −1296.00 −0.145854
\(430\) 0 0
\(431\) 9296.00 1.03892 0.519458 0.854496i \(-0.326134\pi\)
0.519458 + 0.854496i \(0.326134\pi\)
\(432\) 0 0
\(433\) −3494.00 −0.387785 −0.193893 0.981023i \(-0.562111\pi\)
−0.193893 + 0.981023i \(0.562111\pi\)
\(434\) 0 0
\(435\) 1170.00 0.128959
\(436\) 0 0
\(437\) −7904.00 −0.865216
\(438\) 0 0
\(439\) −12584.0 −1.36811 −0.684056 0.729429i \(-0.739786\pi\)
−0.684056 + 0.729429i \(0.739786\pi\)
\(440\) 0 0
\(441\) −2943.00 −0.317784
\(442\) 0 0
\(443\) 12852.0 1.37837 0.689184 0.724586i \(-0.257969\pi\)
0.689184 + 0.724586i \(0.257969\pi\)
\(444\) 0 0
\(445\) 4970.00 0.529440
\(446\) 0 0
\(447\) 9594.00 1.01517
\(448\) 0 0
\(449\) 14458.0 1.51963 0.759816 0.650138i \(-0.225289\pi\)
0.759816 + 0.650138i \(0.225289\pi\)
\(450\) 0 0
\(451\) −26064.0 −2.72130
\(452\) 0 0
\(453\) −10152.0 −1.05294
\(454\) 0 0
\(455\) 120.000 0.0123641
\(456\) 0 0
\(457\) −4310.00 −0.441167 −0.220583 0.975368i \(-0.570796\pi\)
−0.220583 + 0.975368i \(0.570796\pi\)
\(458\) 0 0
\(459\) 1026.00 0.104335
\(460\) 0 0
\(461\) −5338.00 −0.539296 −0.269648 0.962959i \(-0.586907\pi\)
−0.269648 + 0.962959i \(0.586907\pi\)
\(462\) 0 0
\(463\) 1156.00 0.116034 0.0580171 0.998316i \(-0.481522\pi\)
0.0580171 + 0.998316i \(0.481522\pi\)
\(464\) 0 0
\(465\) 1800.00 0.179512
\(466\) 0 0
\(467\) −5948.00 −0.589380 −0.294690 0.955593i \(-0.595216\pi\)
−0.294690 + 0.955593i \(0.595216\pi\)
\(468\) 0 0
\(469\) 16.0000 0.00157529
\(470\) 0 0
\(471\) 9906.00 0.969096
\(472\) 0 0
\(473\) −34848.0 −3.38755
\(474\) 0 0
\(475\) −1300.00 −0.125575
\(476\) 0 0
\(477\) 6030.00 0.578815
\(478\) 0 0
\(479\) 6888.00 0.657037 0.328519 0.944498i \(-0.393451\pi\)
0.328519 + 0.944498i \(0.393451\pi\)
\(480\) 0 0
\(481\) 900.000 0.0853149
\(482\) 0 0
\(483\) 1824.00 0.171832
\(484\) 0 0
\(485\) 8170.00 0.764908
\(486\) 0 0
\(487\) 2892.00 0.269095 0.134547 0.990907i \(-0.457042\pi\)
0.134547 + 0.990907i \(0.457042\pi\)
\(488\) 0 0
\(489\) −6756.00 −0.624779
\(490\) 0 0
\(491\) −4096.00 −0.376476 −0.188238 0.982123i \(-0.560278\pi\)
−0.188238 + 0.982123i \(0.560278\pi\)
\(492\) 0 0
\(493\) 2964.00 0.270775
\(494\) 0 0
\(495\) −3240.00 −0.294196
\(496\) 0 0
\(497\) 384.000 0.0346575
\(498\) 0 0
\(499\) −11060.0 −0.992212 −0.496106 0.868262i \(-0.665237\pi\)
−0.496106 + 0.868262i \(0.665237\pi\)
\(500\) 0 0
\(501\) −552.000 −0.0492246
\(502\) 0 0
\(503\) −9648.00 −0.855235 −0.427617 0.903960i \(-0.640647\pi\)
−0.427617 + 0.903960i \(0.640647\pi\)
\(504\) 0 0
\(505\) −4450.00 −0.392124
\(506\) 0 0
\(507\) −6483.00 −0.567890
\(508\) 0 0
\(509\) 10062.0 0.876209 0.438104 0.898924i \(-0.355650\pi\)
0.438104 + 0.898924i \(0.355650\pi\)
\(510\) 0 0
\(511\) 712.000 0.0616380
\(512\) 0 0
\(513\) −1404.00 −0.120835
\(514\) 0 0
\(515\) −2620.00 −0.224177
\(516\) 0 0
\(517\) −20160.0 −1.71496
\(518\) 0 0
\(519\) 7938.00 0.671367
\(520\) 0 0
\(521\) −7966.00 −0.669859 −0.334930 0.942243i \(-0.608713\pi\)
−0.334930 + 0.942243i \(0.608713\pi\)
\(522\) 0 0
\(523\) −7668.00 −0.641106 −0.320553 0.947231i \(-0.603869\pi\)
−0.320553 + 0.947231i \(0.603869\pi\)
\(524\) 0 0
\(525\) 300.000 0.0249392
\(526\) 0 0
\(527\) 4560.00 0.376920
\(528\) 0 0
\(529\) 10937.0 0.898907
\(530\) 0 0
\(531\) −6264.00 −0.511929
\(532\) 0 0
\(533\) 2172.00 0.176510
\(534\) 0 0
\(535\) −4660.00 −0.376578
\(536\) 0 0
\(537\) 1824.00 0.146576
\(538\) 0 0
\(539\) 23544.0 1.88147
\(540\) 0 0
\(541\) −6590.00 −0.523708 −0.261854 0.965107i \(-0.584334\pi\)
−0.261854 + 0.965107i \(0.584334\pi\)
\(542\) 0 0
\(543\) −6738.00 −0.532514
\(544\) 0 0
\(545\) −2230.00 −0.175271
\(546\) 0 0
\(547\) 4700.00 0.367381 0.183691 0.982984i \(-0.441196\pi\)
0.183691 + 0.982984i \(0.441196\pi\)
\(548\) 0 0
\(549\) −1998.00 −0.155323
\(550\) 0 0
\(551\) −4056.00 −0.313596
\(552\) 0 0
\(553\) −2528.00 −0.194397
\(554\) 0 0
\(555\) 2250.00 0.172085
\(556\) 0 0
\(557\) 15766.0 1.19933 0.599665 0.800251i \(-0.295300\pi\)
0.599665 + 0.800251i \(0.295300\pi\)
\(558\) 0 0
\(559\) 2904.00 0.219725
\(560\) 0 0
\(561\) −8208.00 −0.617722
\(562\) 0 0
\(563\) −22788.0 −1.70586 −0.852930 0.522025i \(-0.825177\pi\)
−0.852930 + 0.522025i \(0.825177\pi\)
\(564\) 0 0
\(565\) −3930.00 −0.292631
\(566\) 0 0
\(567\) 324.000 0.0239977
\(568\) 0 0
\(569\) −3358.00 −0.247407 −0.123704 0.992319i \(-0.539477\pi\)
−0.123704 + 0.992319i \(0.539477\pi\)
\(570\) 0 0
\(571\) 11444.0 0.838733 0.419366 0.907817i \(-0.362252\pi\)
0.419366 + 0.907817i \(0.362252\pi\)
\(572\) 0 0
\(573\) −11544.0 −0.841636
\(574\) 0 0
\(575\) 3800.00 0.275602
\(576\) 0 0
\(577\) −10622.0 −0.766377 −0.383189 0.923670i \(-0.625174\pi\)
−0.383189 + 0.923670i \(0.625174\pi\)
\(578\) 0 0
\(579\) 6174.00 0.443148
\(580\) 0 0
\(581\) 2448.00 0.174802
\(582\) 0 0
\(583\) −48240.0 −3.42692
\(584\) 0 0
\(585\) 270.000 0.0190823
\(586\) 0 0
\(587\) 6588.00 0.463230 0.231615 0.972808i \(-0.425599\pi\)
0.231615 + 0.972808i \(0.425599\pi\)
\(588\) 0 0
\(589\) −6240.00 −0.436528
\(590\) 0 0
\(591\) 11514.0 0.801392
\(592\) 0 0
\(593\) −11362.0 −0.786815 −0.393408 0.919364i \(-0.628704\pi\)
−0.393408 + 0.919364i \(0.628704\pi\)
\(594\) 0 0
\(595\) 760.000 0.0523646
\(596\) 0 0
\(597\) −5976.00 −0.409684
\(598\) 0 0
\(599\) 1624.00 0.110776 0.0553880 0.998465i \(-0.482360\pi\)
0.0553880 + 0.998465i \(0.482360\pi\)
\(600\) 0 0
\(601\) −14950.0 −1.01468 −0.507340 0.861746i \(-0.669371\pi\)
−0.507340 + 0.861746i \(0.669371\pi\)
\(602\) 0 0
\(603\) 36.0000 0.00243123
\(604\) 0 0
\(605\) 19265.0 1.29460
\(606\) 0 0
\(607\) 8244.00 0.551258 0.275629 0.961264i \(-0.411114\pi\)
0.275629 + 0.961264i \(0.411114\pi\)
\(608\) 0 0
\(609\) 936.000 0.0622802
\(610\) 0 0
\(611\) 1680.00 0.111237
\(612\) 0 0
\(613\) −6698.00 −0.441321 −0.220660 0.975351i \(-0.570821\pi\)
−0.220660 + 0.975351i \(0.570821\pi\)
\(614\) 0 0
\(615\) 5430.00 0.356030
\(616\) 0 0
\(617\) 22670.0 1.47919 0.739595 0.673053i \(-0.235017\pi\)
0.739595 + 0.673053i \(0.235017\pi\)
\(618\) 0 0
\(619\) 10060.0 0.653224 0.326612 0.945159i \(-0.394093\pi\)
0.326612 + 0.945159i \(0.394093\pi\)
\(620\) 0 0
\(621\) 4104.00 0.265198
\(622\) 0 0
\(623\) 3976.00 0.255690
\(624\) 0 0
\(625\) 625.000 0.0400000
\(626\) 0 0
\(627\) 11232.0 0.715411
\(628\) 0 0
\(629\) 5700.00 0.361326
\(630\) 0 0
\(631\) 10240.0 0.646035 0.323017 0.946393i \(-0.395303\pi\)
0.323017 + 0.946393i \(0.395303\pi\)
\(632\) 0 0
\(633\) −14292.0 −0.897403
\(634\) 0 0
\(635\) 3580.00 0.223729
\(636\) 0 0
\(637\) −1962.00 −0.122037
\(638\) 0 0
\(639\) 864.000 0.0534888
\(640\) 0 0
\(641\) 13218.0 0.814477 0.407238 0.913322i \(-0.366492\pi\)
0.407238 + 0.913322i \(0.366492\pi\)
\(642\) 0 0
\(643\) 23412.0 1.43589 0.717946 0.696098i \(-0.245082\pi\)
0.717946 + 0.696098i \(0.245082\pi\)
\(644\) 0 0
\(645\) 7260.00 0.443197
\(646\) 0 0
\(647\) −15264.0 −0.927496 −0.463748 0.885967i \(-0.653496\pi\)
−0.463748 + 0.885967i \(0.653496\pi\)
\(648\) 0 0
\(649\) 50112.0 3.03092
\(650\) 0 0
\(651\) 1440.00 0.0866944
\(652\) 0 0
\(653\) −1482.00 −0.0888134 −0.0444067 0.999014i \(-0.514140\pi\)
−0.0444067 + 0.999014i \(0.514140\pi\)
\(654\) 0 0
\(655\) 4040.00 0.241001
\(656\) 0 0
\(657\) 1602.00 0.0951293
\(658\) 0 0
\(659\) 18920.0 1.11839 0.559195 0.829036i \(-0.311110\pi\)
0.559195 + 0.829036i \(0.311110\pi\)
\(660\) 0 0
\(661\) 24218.0 1.42507 0.712535 0.701637i \(-0.247547\pi\)
0.712535 + 0.701637i \(0.247547\pi\)
\(662\) 0 0
\(663\) 684.000 0.0400669
\(664\) 0 0
\(665\) −1040.00 −0.0606458
\(666\) 0 0
\(667\) 11856.0 0.688255
\(668\) 0 0
\(669\) 12276.0 0.709443
\(670\) 0 0
\(671\) 15984.0 0.919606
\(672\) 0 0
\(673\) 890.000 0.0509762 0.0254881 0.999675i \(-0.491886\pi\)
0.0254881 + 0.999675i \(0.491886\pi\)
\(674\) 0 0
\(675\) 675.000 0.0384900
\(676\) 0 0
\(677\) −29250.0 −1.66052 −0.830258 0.557380i \(-0.811807\pi\)
−0.830258 + 0.557380i \(0.811807\pi\)
\(678\) 0 0
\(679\) 6536.00 0.369409
\(680\) 0 0
\(681\) −1404.00 −0.0790035
\(682\) 0 0
\(683\) −14580.0 −0.816820 −0.408410 0.912799i \(-0.633917\pi\)
−0.408410 + 0.912799i \(0.633917\pi\)
\(684\) 0 0
\(685\) −8850.00 −0.493637
\(686\) 0 0
\(687\) 16758.0 0.930651
\(688\) 0 0
\(689\) 4020.00 0.222278
\(690\) 0 0
\(691\) −23668.0 −1.30300 −0.651500 0.758649i \(-0.725860\pi\)
−0.651500 + 0.758649i \(0.725860\pi\)
\(692\) 0 0
\(693\) −2592.00 −0.142081
\(694\) 0 0
\(695\) 4620.00 0.252153
\(696\) 0 0
\(697\) 13756.0 0.747555
\(698\) 0 0
\(699\) −3174.00 −0.171748
\(700\) 0 0
\(701\) −32402.0 −1.74580 −0.872901 0.487898i \(-0.837764\pi\)
−0.872901 + 0.487898i \(0.837764\pi\)
\(702\) 0 0
\(703\) −7800.00 −0.418467
\(704\) 0 0
\(705\) 4200.00 0.224370
\(706\) 0 0
\(707\) −3560.00 −0.189374
\(708\) 0 0
\(709\) 30626.0 1.62226 0.811131 0.584865i \(-0.198852\pi\)
0.811131 + 0.584865i \(0.198852\pi\)
\(710\) 0 0
\(711\) −5688.00 −0.300023
\(712\) 0 0
\(713\) 18240.0 0.958055
\(714\) 0 0
\(715\) −2160.00 −0.112978
\(716\) 0 0
\(717\) 20520.0 1.06881
\(718\) 0 0
\(719\) 13440.0 0.697117 0.348559 0.937287i \(-0.386671\pi\)
0.348559 + 0.937287i \(0.386671\pi\)
\(720\) 0 0
\(721\) −2096.00 −0.108265
\(722\) 0 0
\(723\) −19290.0 −0.992258
\(724\) 0 0
\(725\) 1950.00 0.0998913
\(726\) 0 0
\(727\) −24820.0 −1.26619 −0.633097 0.774073i \(-0.718217\pi\)
−0.633097 + 0.774073i \(0.718217\pi\)
\(728\) 0 0
\(729\) 729.000 0.0370370
\(730\) 0 0
\(731\) 18392.0 0.930578
\(732\) 0 0
\(733\) −21986.0 −1.10787 −0.553937 0.832559i \(-0.686875\pi\)
−0.553937 + 0.832559i \(0.686875\pi\)
\(734\) 0 0
\(735\) −4905.00 −0.246155
\(736\) 0 0
\(737\) −288.000 −0.0143943
\(738\) 0 0
\(739\) −4420.00 −0.220017 −0.110008 0.993931i \(-0.535088\pi\)
−0.110008 + 0.993931i \(0.535088\pi\)
\(740\) 0 0
\(741\) −936.000 −0.0464033
\(742\) 0 0
\(743\) 34560.0 1.70644 0.853219 0.521553i \(-0.174647\pi\)
0.853219 + 0.521553i \(0.174647\pi\)
\(744\) 0 0
\(745\) 15990.0 0.786347
\(746\) 0 0
\(747\) 5508.00 0.269782
\(748\) 0 0
\(749\) −3728.00 −0.181867
\(750\) 0 0
\(751\) −24792.0 −1.20462 −0.602312 0.798261i \(-0.705754\pi\)
−0.602312 + 0.798261i \(0.705754\pi\)
\(752\) 0 0
\(753\) 19056.0 0.922230
\(754\) 0 0
\(755\) −16920.0 −0.815605
\(756\) 0 0
\(757\) 2166.00 0.103996 0.0519978 0.998647i \(-0.483441\pi\)
0.0519978 + 0.998647i \(0.483441\pi\)
\(758\) 0 0
\(759\) −32832.0 −1.57013
\(760\) 0 0
\(761\) −10622.0 −0.505975 −0.252988 0.967470i \(-0.581413\pi\)
−0.252988 + 0.967470i \(0.581413\pi\)
\(762\) 0 0
\(763\) −1784.00 −0.0846463
\(764\) 0 0
\(765\) 1710.00 0.0808172
\(766\) 0 0
\(767\) −4176.00 −0.196593
\(768\) 0 0
\(769\) 29826.0 1.39864 0.699319 0.714809i \(-0.253487\pi\)
0.699319 + 0.714809i \(0.253487\pi\)
\(770\) 0 0
\(771\) 4266.00 0.199269
\(772\) 0 0
\(773\) −6386.00 −0.297139 −0.148570 0.988902i \(-0.547467\pi\)
−0.148570 + 0.988902i \(0.547467\pi\)
\(774\) 0 0
\(775\) 3000.00 0.139049
\(776\) 0 0
\(777\) 1800.00 0.0831076
\(778\) 0 0
\(779\) −18824.0 −0.865776
\(780\) 0 0
\(781\) −6912.00 −0.316685
\(782\) 0 0
\(783\) 2106.00 0.0961204
\(784\) 0 0
\(785\) 16510.0 0.750659
\(786\) 0 0
\(787\) −3516.00 −0.159253 −0.0796263 0.996825i \(-0.525373\pi\)
−0.0796263 + 0.996825i \(0.525373\pi\)
\(788\) 0 0
\(789\) 21672.0 0.977875
\(790\) 0 0
\(791\) −3144.00 −0.141325
\(792\) 0 0
\(793\) −1332.00 −0.0596478
\(794\) 0 0
\(795\) 10050.0 0.448348
\(796\) 0 0
\(797\) 25030.0 1.11243 0.556216 0.831038i \(-0.312253\pi\)
0.556216 + 0.831038i \(0.312253\pi\)
\(798\) 0 0
\(799\) 10640.0 0.471109
\(800\) 0 0
\(801\) 8946.00 0.394621
\(802\) 0 0
\(803\) −12816.0 −0.563221
\(804\) 0 0
\(805\) 3040.00 0.133101
\(806\) 0 0
\(807\) −9558.00 −0.416924
\(808\) 0 0
\(809\) 7962.00 0.346019 0.173009 0.984920i \(-0.444651\pi\)
0.173009 + 0.984920i \(0.444651\pi\)
\(810\) 0 0
\(811\) 34668.0 1.50106 0.750529 0.660837i \(-0.229799\pi\)
0.750529 + 0.660837i \(0.229799\pi\)
\(812\) 0 0
\(813\) −768.000 −0.0331303
\(814\) 0 0
\(815\) −11260.0 −0.483952
\(816\) 0 0
\(817\) −25168.0 −1.07774
\(818\) 0 0
\(819\) 216.000 0.00921569
\(820\) 0 0
\(821\) −250.000 −0.0106274 −0.00531368 0.999986i \(-0.501691\pi\)
−0.00531368 + 0.999986i \(0.501691\pi\)
\(822\) 0 0
\(823\) −6388.00 −0.270561 −0.135280 0.990807i \(-0.543194\pi\)
−0.135280 + 0.990807i \(0.543194\pi\)
\(824\) 0 0
\(825\) −5400.00 −0.227883
\(826\) 0 0
\(827\) −3932.00 −0.165331 −0.0826657 0.996577i \(-0.526343\pi\)
−0.0826657 + 0.996577i \(0.526343\pi\)
\(828\) 0 0
\(829\) 25906.0 1.08535 0.542673 0.839944i \(-0.317412\pi\)
0.542673 + 0.839944i \(0.317412\pi\)
\(830\) 0 0
\(831\) 17826.0 0.744136
\(832\) 0 0
\(833\) −12426.0 −0.516849
\(834\) 0 0
\(835\) −920.000 −0.0381292
\(836\) 0 0
\(837\) 3240.00 0.133800
\(838\) 0 0
\(839\) −9944.00 −0.409184 −0.204592 0.978847i \(-0.565587\pi\)
−0.204592 + 0.978847i \(0.565587\pi\)
\(840\) 0 0
\(841\) −18305.0 −0.750543
\(842\) 0 0
\(843\) 9606.00 0.392465
\(844\) 0 0
\(845\) −10805.0 −0.439886
\(846\) 0 0
\(847\) 15412.0 0.625221
\(848\) 0 0
\(849\) −11820.0 −0.477811
\(850\) 0 0
\(851\) 22800.0 0.918418
\(852\) 0 0
\(853\) 14630.0 0.587247 0.293623 0.955921i \(-0.405139\pi\)
0.293623 + 0.955921i \(0.405139\pi\)
\(854\) 0 0
\(855\) −2340.00 −0.0935980
\(856\) 0 0
\(857\) 478.000 0.0190527 0.00952635 0.999955i \(-0.496968\pi\)
0.00952635 + 0.999955i \(0.496968\pi\)
\(858\) 0 0
\(859\) −24132.0 −0.958525 −0.479263 0.877672i \(-0.659096\pi\)
−0.479263 + 0.877672i \(0.659096\pi\)
\(860\) 0 0
\(861\) 4344.00 0.171943
\(862\) 0 0
\(863\) 15776.0 0.622273 0.311136 0.950365i \(-0.399290\pi\)
0.311136 + 0.950365i \(0.399290\pi\)
\(864\) 0 0
\(865\) 13230.0 0.520039
\(866\) 0 0
\(867\) −10407.0 −0.407659
\(868\) 0 0
\(869\) 45504.0 1.77631
\(870\) 0 0
\(871\) 24.0000 0.000933650 0
\(872\) 0 0
\(873\) 14706.0 0.570129
\(874\) 0 0
\(875\) 500.000 0.0193178
\(876\) 0 0
\(877\) 33542.0 1.29149 0.645743 0.763555i \(-0.276548\pi\)
0.645743 + 0.763555i \(0.276548\pi\)
\(878\) 0 0
\(879\) −5478.00 −0.210203
\(880\) 0 0
\(881\) 22858.0 0.874127 0.437063 0.899431i \(-0.356019\pi\)
0.437063 + 0.899431i \(0.356019\pi\)
\(882\) 0 0
\(883\) −2764.00 −0.105341 −0.0526704 0.998612i \(-0.516773\pi\)
−0.0526704 + 0.998612i \(0.516773\pi\)
\(884\) 0 0
\(885\) −10440.0 −0.396539
\(886\) 0 0
\(887\) 6216.00 0.235302 0.117651 0.993055i \(-0.462464\pi\)
0.117651 + 0.993055i \(0.462464\pi\)
\(888\) 0 0
\(889\) 2864.00 0.108049
\(890\) 0 0
\(891\) −5832.00 −0.219281
\(892\) 0 0
\(893\) −14560.0 −0.545612
\(894\) 0 0
\(895\) 3040.00 0.113537
\(896\) 0 0
\(897\) 2736.00 0.101842
\(898\) 0 0
\(899\) 9360.00 0.347245
\(900\) 0 0
\(901\) 25460.0 0.941394
\(902\) 0 0
\(903\) 5808.00 0.214040
\(904\) 0 0
\(905\) −11230.0 −0.412484
\(906\) 0 0
\(907\) 18884.0 0.691326 0.345663 0.938359i \(-0.387654\pi\)
0.345663 + 0.938359i \(0.387654\pi\)
\(908\) 0 0
\(909\) −8010.00 −0.292272
\(910\) 0 0
\(911\) −15232.0 −0.553961 −0.276981 0.960876i \(-0.589334\pi\)
−0.276981 + 0.960876i \(0.589334\pi\)
\(912\) 0 0
\(913\) −44064.0 −1.59727
\(914\) 0 0
\(915\) −3330.00 −0.120313
\(916\) 0 0
\(917\) 3232.00 0.116390
\(918\) 0 0
\(919\) 7744.00 0.277966 0.138983 0.990295i \(-0.455617\pi\)
0.138983 + 0.990295i \(0.455617\pi\)
\(920\) 0 0
\(921\) −19740.0 −0.706249
\(922\) 0 0
\(923\) 576.000 0.0205409
\(924\) 0 0
\(925\) 3750.00 0.133296
\(926\) 0 0
\(927\) −4716.00 −0.167091
\(928\) 0 0
\(929\) 22266.0 0.786355 0.393177 0.919463i \(-0.371376\pi\)
0.393177 + 0.919463i \(0.371376\pi\)
\(930\) 0 0
\(931\) 17004.0 0.598586
\(932\) 0 0
\(933\) −17184.0 −0.602978
\(934\) 0 0
\(935\) −13680.0 −0.478485
\(936\) 0 0
\(937\) 16202.0 0.564884 0.282442 0.959284i \(-0.408856\pi\)
0.282442 + 0.959284i \(0.408856\pi\)
\(938\) 0 0
\(939\) −5226.00 −0.181623
\(940\) 0 0
\(941\) 53494.0 1.85319 0.926596 0.376057i \(-0.122720\pi\)
0.926596 + 0.376057i \(0.122720\pi\)
\(942\) 0 0
\(943\) 55024.0 1.90014
\(944\) 0 0
\(945\) 540.000 0.0185886
\(946\) 0 0
\(947\) 2332.00 0.0800209 0.0400105 0.999199i \(-0.487261\pi\)
0.0400105 + 0.999199i \(0.487261\pi\)
\(948\) 0 0
\(949\) 1068.00 0.0365319
\(950\) 0 0
\(951\) −26238.0 −0.894664
\(952\) 0 0
\(953\) 15414.0 0.523933 0.261967 0.965077i \(-0.415629\pi\)
0.261967 + 0.965077i \(0.415629\pi\)
\(954\) 0 0
\(955\) −19240.0 −0.651929
\(956\) 0 0
\(957\) −16848.0 −0.569089
\(958\) 0 0
\(959\) −7080.00 −0.238400
\(960\) 0 0
\(961\) −15391.0 −0.516633
\(962\) 0 0
\(963\) −8388.00 −0.280685
\(964\) 0 0
\(965\) 10290.0 0.343261
\(966\) 0 0
\(967\) 35012.0 1.16433 0.582167 0.813070i \(-0.302205\pi\)
0.582167 + 0.813070i \(0.302205\pi\)
\(968\) 0 0
\(969\) −5928.00 −0.196527
\(970\) 0 0
\(971\) −11360.0 −0.375448 −0.187724 0.982222i \(-0.560111\pi\)
−0.187724 + 0.982222i \(0.560111\pi\)
\(972\) 0 0
\(973\) 3696.00 0.121776
\(974\) 0 0
\(975\) 450.000 0.0147811
\(976\) 0 0
\(977\) −24586.0 −0.805093 −0.402546 0.915400i \(-0.631875\pi\)
−0.402546 + 0.915400i \(0.631875\pi\)
\(978\) 0 0
\(979\) −71568.0 −2.33639
\(980\) 0 0
\(981\) −4014.00 −0.130639
\(982\) 0 0
\(983\) 8832.00 0.286569 0.143284 0.989682i \(-0.454234\pi\)
0.143284 + 0.989682i \(0.454234\pi\)
\(984\) 0 0
\(985\) 19190.0 0.620756
\(986\) 0 0
\(987\) 3360.00 0.108359
\(988\) 0 0
\(989\) 73568.0 2.36535
\(990\) 0 0
\(991\) −22912.0 −0.734434 −0.367217 0.930135i \(-0.619689\pi\)
−0.367217 + 0.930135i \(0.619689\pi\)
\(992\) 0 0
\(993\) 7692.00 0.245819
\(994\) 0 0
\(995\) −9960.00 −0.317340
\(996\) 0 0
\(997\) 10974.0 0.348596 0.174298 0.984693i \(-0.444234\pi\)
0.174298 + 0.984693i \(0.444234\pi\)
\(998\) 0 0
\(999\) 4050.00 0.128265
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 960.4.a.bf.1.1 1
4.3 odd 2 960.4.a.o.1.1 1
8.3 odd 2 240.4.a.h.1.1 1
8.5 even 2 120.4.a.a.1.1 1
24.5 odd 2 360.4.a.l.1.1 1
24.11 even 2 720.4.a.v.1.1 1
40.3 even 4 1200.4.f.a.49.2 2
40.13 odd 4 600.4.f.i.49.1 2
40.19 odd 2 1200.4.a.k.1.1 1
40.27 even 4 1200.4.f.a.49.1 2
40.29 even 2 600.4.a.l.1.1 1
40.37 odd 4 600.4.f.i.49.2 2
120.29 odd 2 1800.4.a.n.1.1 1
120.53 even 4 1800.4.f.a.649.1 2
120.77 even 4 1800.4.f.a.649.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
120.4.a.a.1.1 1 8.5 even 2
240.4.a.h.1.1 1 8.3 odd 2
360.4.a.l.1.1 1 24.5 odd 2
600.4.a.l.1.1 1 40.29 even 2
600.4.f.i.49.1 2 40.13 odd 4
600.4.f.i.49.2 2 40.37 odd 4
720.4.a.v.1.1 1 24.11 even 2
960.4.a.o.1.1 1 4.3 odd 2
960.4.a.bf.1.1 1 1.1 even 1 trivial
1200.4.a.k.1.1 1 40.19 odd 2
1200.4.f.a.49.1 2 40.27 even 4
1200.4.f.a.49.2 2 40.3 even 4
1800.4.a.n.1.1 1 120.29 odd 2
1800.4.f.a.649.1 2 120.53 even 4
1800.4.f.a.649.2 2 120.77 even 4