Properties

Label 507.4.b.b.337.1
Level $507$
Weight $4$
Character 507.337
Analytic conductor $29.914$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [507,4,Mod(337,507)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(507, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("507.337");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 507 = 3 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 507.b (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: \(29.9139683729\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(i)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 39)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 337.1
Root \(1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 507.337
Dual form 507.4.b.b.337.2

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-3.00000 q^{3} +8.00000 q^{4} -12.0000i q^{5} -2.00000i q^{7} +9.00000 q^{9} +O(q^{10})\) \(q-3.00000 q^{3} +8.00000 q^{4} -12.0000i q^{5} -2.00000i q^{7} +9.00000 q^{9} +36.0000i q^{11} -24.0000 q^{12} +36.0000i q^{15} +64.0000 q^{16} +78.0000 q^{17} +74.0000i q^{19} -96.0000i q^{20} +6.00000i q^{21} +96.0000 q^{23} -19.0000 q^{25} -27.0000 q^{27} -16.0000i q^{28} +18.0000 q^{29} -214.000i q^{31} -108.000i q^{33} -24.0000 q^{35} +72.0000 q^{36} +286.000i q^{37} -384.000i q^{41} -524.000 q^{43} +288.000i q^{44} -108.000i q^{45} -300.000i q^{47} -192.000 q^{48} +339.000 q^{49} -234.000 q^{51} +558.000 q^{53} +432.000 q^{55} -222.000i q^{57} -576.000i q^{59} +288.000i q^{60} +74.0000 q^{61} -18.0000i q^{63} +512.000 q^{64} +38.0000i q^{67} +624.000 q^{68} -288.000 q^{69} -456.000i q^{71} +682.000i q^{73} +57.0000 q^{75} +592.000i q^{76} +72.0000 q^{77} +704.000 q^{79} -768.000i q^{80} +81.0000 q^{81} -888.000i q^{83} +48.0000i q^{84} -936.000i q^{85} -54.0000 q^{87} +1020.00i q^{89} +768.000 q^{92} +642.000i q^{93} +888.000 q^{95} +110.000i q^{97} +324.000i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 6 q^{3} + 16 q^{4} + 18 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 6 q^{3} + 16 q^{4} + 18 q^{9} - 48 q^{12} + 128 q^{16} + 156 q^{17} + 192 q^{23} - 38 q^{25} - 54 q^{27} + 36 q^{29} - 48 q^{35} + 144 q^{36} - 1048 q^{43} - 384 q^{48} + 678 q^{49} - 468 q^{51} + 1116 q^{53} + 864 q^{55} + 148 q^{61} + 1024 q^{64} + 1248 q^{68} - 576 q^{69} + 114 q^{75} + 144 q^{77} + 1408 q^{79} + 162 q^{81} - 108 q^{87} + 1536 q^{92} + 1776 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/507\mathbb{Z}\right)^\times\).

\(n\) \(170\) \(340\)
\(\chi(n)\) \(1\) \(-1\)

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 1.00000 \(0\)
−1.00000 \(\pi\)
\(3\) −3.00000 −0.577350
\(4\) 8.00000 1.00000
\(5\) − 12.0000i − 1.07331i −0.843801 0.536656i \(-0.819687\pi\)
0.843801 0.536656i \(-0.180313\pi\)
\(6\) 0 0
\(7\) − 2.00000i − 0.107990i −0.998541 0.0539949i \(-0.982805\pi\)
0.998541 0.0539949i \(-0.0171955\pi\)
\(8\) 0 0
\(9\) 9.00000 0.333333
\(10\) 0 0
\(11\) 36.0000i 0.986764i 0.869813 + 0.493382i \(0.164240\pi\)
−0.869813 + 0.493382i \(0.835760\pi\)
\(12\) −24.0000 −0.577350
\(13\) 0 0
\(14\) 0 0
\(15\) 36.0000i 0.619677i
\(16\) 64.0000 1.00000
\(17\) 78.0000 1.11281 0.556405 0.830911i \(-0.312180\pi\)
0.556405 + 0.830911i \(0.312180\pi\)
\(18\) 0 0
\(19\) 74.0000i 0.893514i 0.894655 + 0.446757i \(0.147421\pi\)
−0.894655 + 0.446757i \(0.852579\pi\)
\(20\) − 96.0000i − 1.07331i
\(21\) 6.00000i 0.0623480i
\(22\) 0 0
\(23\) 96.0000 0.870321 0.435161 0.900353i \(-0.356692\pi\)
0.435161 + 0.900353i \(0.356692\pi\)
\(24\) 0 0
\(25\) −19.0000 −0.152000
\(26\) 0 0
\(27\) −27.0000 −0.192450
\(28\) − 16.0000i − 0.107990i
\(29\) 18.0000 0.115259 0.0576296 0.998338i \(-0.481646\pi\)
0.0576296 + 0.998338i \(0.481646\pi\)
\(30\) 0 0
\(31\) − 214.000i − 1.23986i −0.784659 0.619928i \(-0.787162\pi\)
0.784659 0.619928i \(-0.212838\pi\)
\(32\) 0 0
\(33\) − 108.000i − 0.569709i
\(34\) 0 0
\(35\) −24.0000 −0.115907
\(36\) 72.0000 0.333333
\(37\) 286.000i 1.27076i 0.772200 + 0.635380i \(0.219156\pi\)
−0.772200 + 0.635380i \(0.780844\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) − 384.000i − 1.46270i −0.682002 0.731350i \(-0.738890\pi\)
0.682002 0.731350i \(-0.261110\pi\)
\(42\) 0 0
\(43\) −524.000 −1.85835 −0.929177 0.369634i \(-0.879483\pi\)
−0.929177 + 0.369634i \(0.879483\pi\)
\(44\) 288.000i 0.986764i
\(45\) − 108.000i − 0.357771i
\(46\) 0 0
\(47\) − 300.000i − 0.931053i −0.885034 0.465527i \(-0.845865\pi\)
0.885034 0.465527i \(-0.154135\pi\)
\(48\) −192.000 −0.577350
\(49\) 339.000 0.988338
\(50\) 0 0
\(51\) −234.000 −0.642481
\(52\) 0 0
\(53\) 558.000 1.44617 0.723087 0.690757i \(-0.242723\pi\)
0.723087 + 0.690757i \(0.242723\pi\)
\(54\) 0 0
\(55\) 432.000 1.05911
\(56\) 0 0
\(57\) − 222.000i − 0.515870i
\(58\) 0 0
\(59\) − 576.000i − 1.27100i −0.772102 0.635498i \(-0.780795\pi\)
0.772102 0.635498i \(-0.219205\pi\)
\(60\) 288.000i 0.619677i
\(61\) 74.0000 0.155323 0.0776617 0.996980i \(-0.475255\pi\)
0.0776617 + 0.996980i \(0.475255\pi\)
\(62\) 0 0
\(63\) − 18.0000i − 0.0359966i
\(64\) 512.000 1.00000
\(65\) 0 0
\(66\) 0 0
\(67\) 38.0000i 0.0692901i 0.999400 + 0.0346451i \(0.0110301\pi\)
−0.999400 + 0.0346451i \(0.988970\pi\)
\(68\) 624.000 1.11281
\(69\) −288.000 −0.502480
\(70\) 0 0
\(71\) − 456.000i − 0.762215i −0.924531 0.381107i \(-0.875543\pi\)
0.924531 0.381107i \(-0.124457\pi\)
\(72\) 0 0
\(73\) 682.000i 1.09345i 0.837311 + 0.546726i \(0.184126\pi\)
−0.837311 + 0.546726i \(0.815874\pi\)
\(74\) 0 0
\(75\) 57.0000 0.0877572
\(76\) 592.000i 0.893514i
\(77\) 72.0000 0.106561
\(78\) 0 0
\(79\) 704.000 1.00261 0.501305 0.865271i \(-0.332853\pi\)
0.501305 + 0.865271i \(0.332853\pi\)
\(80\) − 768.000i − 1.07331i
\(81\) 81.0000 0.111111
\(82\) 0 0
\(83\) − 888.000i − 1.17435i −0.809462 0.587173i \(-0.800241\pi\)
0.809462 0.587173i \(-0.199759\pi\)
\(84\) 48.0000i 0.0623480i
\(85\) − 936.000i − 1.19439i
\(86\) 0 0
\(87\) −54.0000 −0.0665449
\(88\) 0 0
\(89\) 1020.00i 1.21483i 0.794385 + 0.607415i \(0.207793\pi\)
−0.794385 + 0.607415i \(0.792207\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 768.000 0.870321
\(93\) 642.000i 0.715831i
\(94\) 0 0
\(95\) 888.000 0.959020
\(96\) 0 0
\(97\) 110.000i 0.115142i 0.998341 + 0.0575712i \(0.0183356\pi\)
−0.998341 + 0.0575712i \(0.981664\pi\)
\(98\) 0 0
\(99\) 324.000i 0.328921i
\(100\) −152.000 −0.152000
\(101\) 990.000 0.975333 0.487667 0.873030i \(-0.337848\pi\)
0.487667 + 0.873030i \(0.337848\pi\)
\(102\) 0 0
\(103\) −1208.00 −1.15561 −0.577805 0.816175i \(-0.696090\pi\)
−0.577805 + 0.816175i \(0.696090\pi\)
\(104\) 0 0
\(105\) 72.0000 0.0669189
\(106\) 0 0
\(107\) 996.000 0.899878 0.449939 0.893059i \(-0.351446\pi\)
0.449939 + 0.893059i \(0.351446\pi\)
\(108\) −216.000 −0.192450
\(109\) − 1402.00i − 1.23199i −0.787749 0.615997i \(-0.788754\pi\)
0.787749 0.615997i \(-0.211246\pi\)
\(110\) 0 0
\(111\) − 858.000i − 0.733673i
\(112\) − 128.000i − 0.107990i
\(113\) 1926.00 1.60339 0.801694 0.597735i \(-0.203932\pi\)
0.801694 + 0.597735i \(0.203932\pi\)
\(114\) 0 0
\(115\) − 1152.00i − 0.934127i
\(116\) 144.000 0.115259
\(117\) 0 0
\(118\) 0 0
\(119\) − 156.000i − 0.120172i
\(120\) 0 0
\(121\) 35.0000 0.0262960
\(122\) 0 0
\(123\) 1152.00i 0.844491i
\(124\) − 1712.00i − 1.23986i
\(125\) − 1272.00i − 0.910169i
\(126\) 0 0
\(127\) 988.000 0.690321 0.345161 0.938544i \(-0.387824\pi\)
0.345161 + 0.938544i \(0.387824\pi\)
\(128\) 0 0
\(129\) 1572.00 1.07292
\(130\) 0 0
\(131\) −2100.00 −1.40059 −0.700297 0.713851i \(-0.746949\pi\)
−0.700297 + 0.713851i \(0.746949\pi\)
\(132\) − 864.000i − 0.569709i
\(133\) 148.000 0.0964904
\(134\) 0 0
\(135\) 324.000i 0.206559i
\(136\) 0 0
\(137\) 2496.00i 1.55655i 0.627922 + 0.778276i \(0.283906\pi\)
−0.627922 + 0.778276i \(0.716094\pi\)
\(138\) 0 0
\(139\) −2464.00 −1.50355 −0.751776 0.659418i \(-0.770803\pi\)
−0.751776 + 0.659418i \(0.770803\pi\)
\(140\) −192.000 −0.115907
\(141\) 900.000i 0.537544i
\(142\) 0 0
\(143\) 0 0
\(144\) 576.000 0.333333
\(145\) − 216.000i − 0.123709i
\(146\) 0 0
\(147\) −1017.00 −0.570617
\(148\) 2288.00i 1.27076i
\(149\) 216.000i 0.118761i 0.998235 + 0.0593806i \(0.0189125\pi\)
−0.998235 + 0.0593806i \(0.981087\pi\)
\(150\) 0 0
\(151\) 898.000i 0.483962i 0.970281 + 0.241981i \(0.0777971\pi\)
−0.970281 + 0.241981i \(0.922203\pi\)
\(152\) 0 0
\(153\) 702.000 0.370937
\(154\) 0 0
\(155\) −2568.00 −1.33075
\(156\) 0 0
\(157\) −1510.00 −0.767587 −0.383793 0.923419i \(-0.625383\pi\)
−0.383793 + 0.923419i \(0.625383\pi\)
\(158\) 0 0
\(159\) −1674.00 −0.834949
\(160\) 0 0
\(161\) − 192.000i − 0.0939858i
\(162\) 0 0
\(163\) 394.000i 0.189328i 0.995509 + 0.0946640i \(0.0301777\pi\)
−0.995509 + 0.0946640i \(0.969822\pi\)
\(164\) − 3072.00i − 1.46270i
\(165\) −1296.00 −0.611476
\(166\) 0 0
\(167\) − 84.0000i − 0.0389228i −0.999811 0.0194614i \(-0.993805\pi\)
0.999811 0.0194614i \(-0.00619515\pi\)
\(168\) 0 0
\(169\) 0 0
\(170\) 0 0
\(171\) 666.000i 0.297838i
\(172\) −4192.00 −1.85835
\(173\) −1194.00 −0.524729 −0.262365 0.964969i \(-0.584502\pi\)
−0.262365 + 0.964969i \(0.584502\pi\)
\(174\) 0 0
\(175\) 38.0000i 0.0164145i
\(176\) 2304.00i 0.986764i
\(177\) 1728.00i 0.733810i
\(178\) 0 0
\(179\) −3156.00 −1.31782 −0.658912 0.752220i \(-0.728983\pi\)
−0.658912 + 0.752220i \(0.728983\pi\)
\(180\) − 864.000i − 0.357771i
\(181\) 1078.00 0.442691 0.221346 0.975195i \(-0.428955\pi\)
0.221346 + 0.975195i \(0.428955\pi\)
\(182\) 0 0
\(183\) −222.000 −0.0896760
\(184\) 0 0
\(185\) 3432.00 1.36392
\(186\) 0 0
\(187\) 2808.00i 1.09808i
\(188\) − 2400.00i − 0.931053i
\(189\) 54.0000i 0.0207827i
\(190\) 0 0
\(191\) 3192.00 1.20924 0.604620 0.796514i \(-0.293325\pi\)
0.604620 + 0.796514i \(0.293325\pi\)
\(192\) −1536.00 −0.577350
\(193\) − 722.000i − 0.269278i −0.990895 0.134639i \(-0.957012\pi\)
0.990895 0.134639i \(-0.0429875\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 2712.00 0.988338
\(197\) 2796.00i 1.01120i 0.862767 + 0.505601i \(0.168729\pi\)
−0.862767 + 0.505601i \(0.831271\pi\)
\(198\) 0 0
\(199\) 340.000 0.121115 0.0605577 0.998165i \(-0.480712\pi\)
0.0605577 + 0.998165i \(0.480712\pi\)
\(200\) 0 0
\(201\) − 114.000i − 0.0400047i
\(202\) 0 0
\(203\) − 36.0000i − 0.0124468i
\(204\) −1872.00 −0.642481
\(205\) −4608.00 −1.56994
\(206\) 0 0
\(207\) 864.000 0.290107
\(208\) 0 0
\(209\) −2664.00 −0.881688
\(210\) 0 0
\(211\) −1924.00 −0.627742 −0.313871 0.949466i \(-0.601626\pi\)
−0.313871 + 0.949466i \(0.601626\pi\)
\(212\) 4464.00 1.44617
\(213\) 1368.00i 0.440065i
\(214\) 0 0
\(215\) 6288.00i 1.99460i
\(216\) 0 0
\(217\) −428.000 −0.133892
\(218\) 0 0
\(219\) − 2046.00i − 0.631305i
\(220\) 3456.00 1.05911
\(221\) 0 0
\(222\) 0 0
\(223\) 5042.00i 1.51407i 0.653375 + 0.757034i \(0.273352\pi\)
−0.653375 + 0.757034i \(0.726648\pi\)
\(224\) 0 0
\(225\) −171.000 −0.0506667
\(226\) 0 0
\(227\) − 2676.00i − 0.782433i −0.920299 0.391217i \(-0.872054\pi\)
0.920299 0.391217i \(-0.127946\pi\)
\(228\) − 1776.00i − 0.515870i
\(229\) 2410.00i 0.695447i 0.937597 + 0.347723i \(0.113045\pi\)
−0.937597 + 0.347723i \(0.886955\pi\)
\(230\) 0 0
\(231\) −216.000 −0.0615228
\(232\) 0 0
\(233\) −3726.00 −1.04763 −0.523816 0.851831i \(-0.675492\pi\)
−0.523816 + 0.851831i \(0.675492\pi\)
\(234\) 0 0
\(235\) −3600.00 −0.999311
\(236\) − 4608.00i − 1.27100i
\(237\) −2112.00 −0.578857
\(238\) 0 0
\(239\) 1248.00i 0.337767i 0.985636 + 0.168884i \(0.0540162\pi\)
−0.985636 + 0.168884i \(0.945984\pi\)
\(240\) 2304.00i 0.619677i
\(241\) 4210.00i 1.12527i 0.826706 + 0.562635i \(0.190212\pi\)
−0.826706 + 0.562635i \(0.809788\pi\)
\(242\) 0 0
\(243\) −243.000 −0.0641500
\(244\) 592.000 0.155323
\(245\) − 4068.00i − 1.06080i
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) 2664.00i 0.678009i
\(250\) 0 0
\(251\) 7692.00 1.93432 0.967161 0.254165i \(-0.0818007\pi\)
0.967161 + 0.254165i \(0.0818007\pi\)
\(252\) − 144.000i − 0.0359966i
\(253\) 3456.00i 0.858802i
\(254\) 0 0
\(255\) 2808.00i 0.689583i
\(256\) 4096.00 1.00000
\(257\) −1326.00 −0.321843 −0.160921 0.986967i \(-0.551447\pi\)
−0.160921 + 0.986967i \(0.551447\pi\)
\(258\) 0 0
\(259\) 572.000 0.137229
\(260\) 0 0
\(261\) 162.000 0.0384197
\(262\) 0 0
\(263\) −6048.00 −1.41801 −0.709003 0.705205i \(-0.750855\pi\)
−0.709003 + 0.705205i \(0.750855\pi\)
\(264\) 0 0
\(265\) − 6696.00i − 1.55220i
\(266\) 0 0
\(267\) − 3060.00i − 0.701382i
\(268\) 304.000i 0.0692901i
\(269\) 6474.00 1.46739 0.733693 0.679481i \(-0.237795\pi\)
0.733693 + 0.679481i \(0.237795\pi\)
\(270\) 0 0
\(271\) − 5978.00i − 1.33999i −0.742365 0.669996i \(-0.766296\pi\)
0.742365 0.669996i \(-0.233704\pi\)
\(272\) 4992.00 1.11281
\(273\) 0 0
\(274\) 0 0
\(275\) − 684.000i − 0.149988i
\(276\) −2304.00 −0.502480
\(277\) −8750.00 −1.89797 −0.948983 0.315327i \(-0.897886\pi\)
−0.948983 + 0.315327i \(0.897886\pi\)
\(278\) 0 0
\(279\) − 1926.00i − 0.413285i
\(280\) 0 0
\(281\) − 8976.00i − 1.90556i −0.303656 0.952782i \(-0.598207\pi\)
0.303656 0.952782i \(-0.401793\pi\)
\(282\) 0 0
\(283\) 592.000 0.124349 0.0621745 0.998065i \(-0.480196\pi\)
0.0621745 + 0.998065i \(0.480196\pi\)
\(284\) − 3648.00i − 0.762215i
\(285\) −2664.00 −0.553690
\(286\) 0 0
\(287\) −768.000 −0.157957
\(288\) 0 0
\(289\) 1171.00 0.238347
\(290\) 0 0
\(291\) − 330.000i − 0.0664775i
\(292\) 5456.00i 1.09345i
\(293\) 4608.00i 0.918779i 0.888235 + 0.459389i \(0.151932\pi\)
−0.888235 + 0.459389i \(0.848068\pi\)
\(294\) 0 0
\(295\) −6912.00 −1.36418
\(296\) 0 0
\(297\) − 972.000i − 0.189903i
\(298\) 0 0
\(299\) 0 0
\(300\) 456.000 0.0877572
\(301\) 1048.00i 0.200683i
\(302\) 0 0
\(303\) −2970.00 −0.563109
\(304\) 4736.00i 0.893514i
\(305\) − 888.000i − 0.166711i
\(306\) 0 0
\(307\) 3166.00i 0.588577i 0.955717 + 0.294289i \(0.0950827\pi\)
−0.955717 + 0.294289i \(0.904917\pi\)
\(308\) 576.000 0.106561
\(309\) 3624.00 0.667191
\(310\) 0 0
\(311\) −2472.00 −0.450721 −0.225361 0.974275i \(-0.572356\pi\)
−0.225361 + 0.974275i \(0.572356\pi\)
\(312\) 0 0
\(313\) −3094.00 −0.558732 −0.279366 0.960185i \(-0.590124\pi\)
−0.279366 + 0.960185i \(0.590124\pi\)
\(314\) 0 0
\(315\) −216.000 −0.0386356
\(316\) 5632.00 1.00261
\(317\) 2316.00i 0.410345i 0.978726 + 0.205173i \(0.0657756\pi\)
−0.978726 + 0.205173i \(0.934224\pi\)
\(318\) 0 0
\(319\) 648.000i 0.113734i
\(320\) − 6144.00i − 1.07331i
\(321\) −2988.00 −0.519545
\(322\) 0 0
\(323\) 5772.00i 0.994312i
\(324\) 648.000 0.111111
\(325\) 0 0
\(326\) 0 0
\(327\) 4206.00i 0.711292i
\(328\) 0 0
\(329\) −600.000 −0.100544
\(330\) 0 0
\(331\) − 4426.00i − 0.734970i −0.930030 0.367485i \(-0.880219\pi\)
0.930030 0.367485i \(-0.119781\pi\)
\(332\) − 7104.00i − 1.17435i
\(333\) 2574.00i 0.423587i
\(334\) 0 0
\(335\) 456.000 0.0743700
\(336\) 384.000i 0.0623480i
\(337\) −866.000 −0.139982 −0.0699911 0.997548i \(-0.522297\pi\)
−0.0699911 + 0.997548i \(0.522297\pi\)
\(338\) 0 0
\(339\) −5778.00 −0.925716
\(340\) − 7488.00i − 1.19439i
\(341\) 7704.00 1.22345
\(342\) 0 0
\(343\) − 1364.00i − 0.214720i
\(344\) 0 0
\(345\) 3456.00i 0.539318i
\(346\) 0 0
\(347\) −2556.00 −0.395427 −0.197714 0.980260i \(-0.563352\pi\)
−0.197714 + 0.980260i \(0.563352\pi\)
\(348\) −432.000 −0.0665449
\(349\) 11014.0i 1.68930i 0.535318 + 0.844650i \(0.320192\pi\)
−0.535318 + 0.844650i \(0.679808\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) − 9720.00i − 1.46556i −0.680465 0.732781i \(-0.738222\pi\)
0.680465 0.732781i \(-0.261778\pi\)
\(354\) 0 0
\(355\) −5472.00 −0.818095
\(356\) 8160.00i 1.21483i
\(357\) 468.000i 0.0693815i
\(358\) 0 0
\(359\) 2988.00i 0.439277i 0.975581 + 0.219639i \(0.0704879\pi\)
−0.975581 + 0.219639i \(0.929512\pi\)
\(360\) 0 0
\(361\) 1383.00 0.201633
\(362\) 0 0
\(363\) −105.000 −0.0151820
\(364\) 0 0
\(365\) 8184.00 1.17362
\(366\) 0 0
\(367\) −2068.00 −0.294138 −0.147069 0.989126i \(-0.546984\pi\)
−0.147069 + 0.989126i \(0.546984\pi\)
\(368\) 6144.00 0.870321
\(369\) − 3456.00i − 0.487567i
\(370\) 0 0
\(371\) − 1116.00i − 0.156172i
\(372\) 5136.00i 0.715831i
\(373\) 902.000 0.125211 0.0626056 0.998038i \(-0.480059\pi\)
0.0626056 + 0.998038i \(0.480059\pi\)
\(374\) 0 0
\(375\) 3816.00i 0.525486i
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) 12818.0i 1.73725i 0.495473 + 0.868623i \(0.334995\pi\)
−0.495473 + 0.868623i \(0.665005\pi\)
\(380\) 7104.00 0.959020
\(381\) −2964.00 −0.398557
\(382\) 0 0
\(383\) − 1332.00i − 0.177708i −0.996045 0.0888538i \(-0.971680\pi\)
0.996045 0.0888538i \(-0.0283204\pi\)
\(384\) 0 0
\(385\) − 864.000i − 0.114373i
\(386\) 0 0
\(387\) −4716.00 −0.619452
\(388\) 880.000i 0.115142i
\(389\) −3054.00 −0.398056 −0.199028 0.979994i \(-0.563779\pi\)
−0.199028 + 0.979994i \(0.563779\pi\)
\(390\) 0 0
\(391\) 7488.00 0.968502
\(392\) 0 0
\(393\) 6300.00 0.808633
\(394\) 0 0
\(395\) − 8448.00i − 1.07611i
\(396\) 2592.00i 0.328921i
\(397\) − 11162.0i − 1.41110i −0.708663 0.705548i \(-0.750701\pi\)
0.708663 0.705548i \(-0.249299\pi\)
\(398\) 0 0
\(399\) −444.000 −0.0557088
\(400\) −1216.00 −0.152000
\(401\) 14820.0i 1.84557i 0.385310 + 0.922787i \(0.374095\pi\)
−0.385310 + 0.922787i \(0.625905\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 7920.00 0.975333
\(405\) − 972.000i − 0.119257i
\(406\) 0 0
\(407\) −10296.0 −1.25394
\(408\) 0 0
\(409\) − 9682.00i − 1.17052i −0.810844 0.585262i \(-0.800992\pi\)
0.810844 0.585262i \(-0.199008\pi\)
\(410\) 0 0
\(411\) − 7488.00i − 0.898676i
\(412\) −9664.00 −1.15561
\(413\) −1152.00 −0.137255
\(414\) 0 0
\(415\) −10656.0 −1.26044
\(416\) 0 0
\(417\) 7392.00 0.868076
\(418\) 0 0
\(419\) −348.000 −0.0405750 −0.0202875 0.999794i \(-0.506458\pi\)
−0.0202875 + 0.999794i \(0.506458\pi\)
\(420\) 576.000 0.0669189
\(421\) 2486.00i 0.287792i 0.989593 + 0.143896i \(0.0459630\pi\)
−0.989593 + 0.143896i \(0.954037\pi\)
\(422\) 0 0
\(423\) − 2700.00i − 0.310351i
\(424\) 0 0
\(425\) −1482.00 −0.169147
\(426\) 0 0
\(427\) − 148.000i − 0.0167734i
\(428\) 7968.00 0.899878
\(429\) 0 0
\(430\) 0 0
\(431\) − 1812.00i − 0.202508i −0.994861 0.101254i \(-0.967715\pi\)
0.994861 0.101254i \(-0.0322855\pi\)
\(432\) −1728.00 −0.192450
\(433\) 6226.00 0.690999 0.345499 0.938419i \(-0.387710\pi\)
0.345499 + 0.938419i \(0.387710\pi\)
\(434\) 0 0
\(435\) 648.000i 0.0714235i
\(436\) − 11216.0i − 1.23199i
\(437\) 7104.00i 0.777644i
\(438\) 0 0
\(439\) 12544.0 1.36376 0.681882 0.731462i \(-0.261162\pi\)
0.681882 + 0.731462i \(0.261162\pi\)
\(440\) 0 0
\(441\) 3051.00 0.329446
\(442\) 0 0
\(443\) −8556.00 −0.917625 −0.458812 0.888533i \(-0.651725\pi\)
−0.458812 + 0.888533i \(0.651725\pi\)
\(444\) − 6864.00i − 0.733673i
\(445\) 12240.0 1.30389
\(446\) 0 0
\(447\) − 648.000i − 0.0685668i
\(448\) − 1024.00i − 0.107990i
\(449\) − 4116.00i − 0.432619i −0.976325 0.216310i \(-0.930598\pi\)
0.976325 0.216310i \(-0.0694021\pi\)
\(450\) 0 0
\(451\) 13824.0 1.44334
\(452\) 15408.0 1.60339
\(453\) − 2694.00i − 0.279415i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) − 6514.00i − 0.666766i −0.942791 0.333383i \(-0.891810\pi\)
0.942791 0.333383i \(-0.108190\pi\)
\(458\) 0 0
\(459\) −2106.00 −0.214160
\(460\) − 9216.00i − 0.934127i
\(461\) 10500.0i 1.06081i 0.847744 + 0.530405i \(0.177960\pi\)
−0.847744 + 0.530405i \(0.822040\pi\)
\(462\) 0 0
\(463\) 5542.00i 0.556282i 0.960540 + 0.278141i \(0.0897183\pi\)
−0.960540 + 0.278141i \(0.910282\pi\)
\(464\) 1152.00 0.115259
\(465\) 7704.00 0.768311
\(466\) 0 0
\(467\) 5220.00 0.517244 0.258622 0.965979i \(-0.416732\pi\)
0.258622 + 0.965979i \(0.416732\pi\)
\(468\) 0 0
\(469\) 76.0000 0.00748263
\(470\) 0 0
\(471\) 4530.00 0.443166
\(472\) 0 0
\(473\) − 18864.0i − 1.83376i
\(474\) 0 0
\(475\) − 1406.00i − 0.135814i
\(476\) − 1248.00i − 0.120172i
\(477\) 5022.00 0.482058
\(478\) 0 0
\(479\) − 11592.0i − 1.10575i −0.833266 0.552873i \(-0.813532\pi\)
0.833266 0.552873i \(-0.186468\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 0 0
\(483\) 576.000i 0.0542627i
\(484\) 280.000 0.0262960
\(485\) 1320.00 0.123584
\(486\) 0 0
\(487\) 12170.0i 1.13239i 0.824270 + 0.566196i \(0.191586\pi\)
−0.824270 + 0.566196i \(0.808414\pi\)
\(488\) 0 0
\(489\) − 1182.00i − 0.109309i
\(490\) 0 0
\(491\) −1812.00 −0.166547 −0.0832733 0.996527i \(-0.526537\pi\)
−0.0832733 + 0.996527i \(0.526537\pi\)
\(492\) 9216.00i 0.844491i
\(493\) 1404.00 0.128262
\(494\) 0 0
\(495\) 3888.00 0.353036
\(496\) − 13696.0i − 1.23986i
\(497\) −912.000 −0.0823115
\(498\) 0 0
\(499\) − 1330.00i − 0.119317i −0.998219 0.0596583i \(-0.980999\pi\)
0.998219 0.0596583i \(-0.0190011\pi\)
\(500\) − 10176.0i − 0.910169i
\(501\) 252.000i 0.0224721i
\(502\) 0 0
\(503\) −2688.00 −0.238274 −0.119137 0.992878i \(-0.538013\pi\)
−0.119137 + 0.992878i \(0.538013\pi\)
\(504\) 0 0
\(505\) − 11880.0i − 1.04684i
\(506\) 0 0
\(507\) 0 0
\(508\) 7904.00 0.690321
\(509\) 5124.00i 0.446203i 0.974795 + 0.223101i \(0.0716181\pi\)
−0.974795 + 0.223101i \(0.928382\pi\)
\(510\) 0 0
\(511\) 1364.00 0.118082
\(512\) 0 0
\(513\) − 1998.00i − 0.171957i
\(514\) 0 0
\(515\) 14496.0i 1.24033i
\(516\) 12576.0 1.07292
\(517\) 10800.0 0.918730
\(518\) 0 0
\(519\) 3582.00 0.302953
\(520\) 0 0
\(521\) −882.000 −0.0741672 −0.0370836 0.999312i \(-0.511807\pi\)
−0.0370836 + 0.999312i \(0.511807\pi\)
\(522\) 0 0
\(523\) −2320.00 −0.193970 −0.0969852 0.995286i \(-0.530920\pi\)
−0.0969852 + 0.995286i \(0.530920\pi\)
\(524\) −16800.0 −1.40059
\(525\) − 114.000i − 0.00947689i
\(526\) 0 0
\(527\) − 16692.0i − 1.37972i
\(528\) − 6912.00i − 0.569709i
\(529\) −2951.00 −0.242541
\(530\) 0 0
\(531\) − 5184.00i − 0.423666i
\(532\) 1184.00 0.0964904
\(533\) 0 0
\(534\) 0 0
\(535\) − 11952.0i − 0.965851i
\(536\) 0 0
\(537\) 9468.00 0.760846
\(538\) 0 0
\(539\) 12204.0i 0.975257i
\(540\) 2592.00i 0.206559i
\(541\) − 21422.0i − 1.70241i −0.524833 0.851205i \(-0.675872\pi\)
0.524833 0.851205i \(-0.324128\pi\)
\(542\) 0 0
\(543\) −3234.00 −0.255588
\(544\) 0 0
\(545\) −16824.0 −1.32231
\(546\) 0 0
\(547\) 7040.00 0.550290 0.275145 0.961403i \(-0.411274\pi\)
0.275145 + 0.961403i \(0.411274\pi\)
\(548\) 19968.0i 1.55655i
\(549\) 666.000 0.0517745
\(550\) 0 0
\(551\) 1332.00i 0.102986i
\(552\) 0 0
\(553\) − 1408.00i − 0.108272i
\(554\) 0 0
\(555\) −10296.0 −0.787461
\(556\) −19712.0 −1.50355
\(557\) 8400.00i 0.638994i 0.947587 + 0.319497i \(0.103514\pi\)
−0.947587 + 0.319497i \(0.896486\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) −1536.00 −0.115907
\(561\) − 8424.00i − 0.633978i
\(562\) 0 0
\(563\) −19044.0 −1.42559 −0.712797 0.701371i \(-0.752572\pi\)
−0.712797 + 0.701371i \(0.752572\pi\)
\(564\) 7200.00i 0.537544i
\(565\) − 23112.0i − 1.72094i
\(566\) 0 0
\(567\) − 162.000i − 0.0119989i
\(568\) 0 0
\(569\) 4698.00 0.346134 0.173067 0.984910i \(-0.444632\pi\)
0.173067 + 0.984910i \(0.444632\pi\)
\(570\) 0 0
\(571\) 8728.00 0.639677 0.319838 0.947472i \(-0.396371\pi\)
0.319838 + 0.947472i \(0.396371\pi\)
\(572\) 0 0
\(573\) −9576.00 −0.698156
\(574\) 0 0
\(575\) −1824.00 −0.132289
\(576\) 4608.00 0.333333
\(577\) 2018.00i 0.145599i 0.997347 + 0.0727993i \(0.0231933\pi\)
−0.997347 + 0.0727993i \(0.976807\pi\)
\(578\) 0 0
\(579\) 2166.00i 0.155468i
\(580\) − 1728.00i − 0.123709i
\(581\) −1776.00 −0.126817
\(582\) 0 0
\(583\) 20088.0i 1.42703i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 11376.0i 0.799894i 0.916538 + 0.399947i \(0.130971\pi\)
−0.916538 + 0.399947i \(0.869029\pi\)
\(588\) −8136.00 −0.570617
\(589\) 15836.0 1.10783
\(590\) 0 0
\(591\) − 8388.00i − 0.583818i
\(592\) 18304.0i 1.27076i
\(593\) 25596.0i 1.77252i 0.463192 + 0.886258i \(0.346704\pi\)
−0.463192 + 0.886258i \(0.653296\pi\)
\(594\) 0 0
\(595\) −1872.00 −0.128982
\(596\) 1728.00i 0.118761i
\(597\) −1020.00 −0.0699260
\(598\) 0 0
\(599\) 3480.00 0.237377 0.118689 0.992932i \(-0.462131\pi\)
0.118689 + 0.992932i \(0.462131\pi\)
\(600\) 0 0
\(601\) 10010.0 0.679395 0.339698 0.940535i \(-0.389675\pi\)
0.339698 + 0.940535i \(0.389675\pi\)
\(602\) 0 0
\(603\) 342.000i 0.0230967i
\(604\) 7184.00i 0.483962i
\(605\) − 420.000i − 0.0282238i
\(606\) 0 0
\(607\) 3764.00 0.251690 0.125845 0.992050i \(-0.459836\pi\)
0.125845 + 0.992050i \(0.459836\pi\)
\(608\) 0 0
\(609\) 108.000i 0.00718618i
\(610\) 0 0
\(611\) 0 0
\(612\) 5616.00 0.370937
\(613\) 13610.0i 0.896742i 0.893848 + 0.448371i \(0.147996\pi\)
−0.893848 + 0.448371i \(0.852004\pi\)
\(614\) 0 0
\(615\) 13824.0 0.906402
\(616\) 0 0
\(617\) 6408.00i 0.418114i 0.977903 + 0.209057i \(0.0670394\pi\)
−0.977903 + 0.209057i \(0.932961\pi\)
\(618\) 0 0
\(619\) 6694.00i 0.434660i 0.976098 + 0.217330i \(0.0697348\pi\)
−0.976098 + 0.217330i \(0.930265\pi\)
\(620\) −20544.0 −1.33075
\(621\) −2592.00 −0.167493
\(622\) 0 0
\(623\) 2040.00 0.131189
\(624\) 0 0
\(625\) −17639.0 −1.12890
\(626\) 0 0
\(627\) 7992.00 0.509043
\(628\) −12080.0 −0.767587
\(629\) 22308.0i 1.41411i
\(630\) 0 0
\(631\) 27250.0i 1.71918i 0.510981 + 0.859592i \(0.329282\pi\)
−0.510981 + 0.859592i \(0.670718\pi\)
\(632\) 0 0
\(633\) 5772.00 0.362427
\(634\) 0 0
\(635\) − 11856.0i − 0.740931i
\(636\) −13392.0 −0.834949
\(637\) 0 0
\(638\) 0 0
\(639\) − 4104.00i − 0.254072i
\(640\) 0 0
\(641\) 12630.0 0.778245 0.389122 0.921186i \(-0.372778\pi\)
0.389122 + 0.921186i \(0.372778\pi\)
\(642\) 0 0
\(643\) 14798.0i 0.907583i 0.891108 + 0.453792i \(0.149929\pi\)
−0.891108 + 0.453792i \(0.850071\pi\)
\(644\) − 1536.00i − 0.0939858i
\(645\) − 18864.0i − 1.15158i
\(646\) 0 0
\(647\) −26232.0 −1.59395 −0.796976 0.604012i \(-0.793568\pi\)
−0.796976 + 0.604012i \(0.793568\pi\)
\(648\) 0 0
\(649\) 20736.0 1.25417
\(650\) 0 0
\(651\) 1284.00 0.0773025
\(652\) 3152.00i 0.189328i
\(653\) −30390.0 −1.82121 −0.910607 0.413274i \(-0.864385\pi\)
−0.910607 + 0.413274i \(0.864385\pi\)
\(654\) 0 0
\(655\) 25200.0i 1.50328i
\(656\) − 24576.0i − 1.46270i
\(657\) 6138.00i 0.364484i
\(658\) 0 0
\(659\) −28740.0 −1.69886 −0.849432 0.527698i \(-0.823055\pi\)
−0.849432 + 0.527698i \(0.823055\pi\)
\(660\) −10368.0 −0.611476
\(661\) 9214.00i 0.542183i 0.962554 + 0.271092i \(0.0873846\pi\)
−0.962554 + 0.271092i \(0.912615\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) − 1776.00i − 0.103564i
\(666\) 0 0
\(667\) 1728.00 0.100312
\(668\) − 672.000i − 0.0389228i
\(669\) − 15126.0i − 0.874148i
\(670\) 0 0
\(671\) 2664.00i 0.153268i
\(672\) 0 0
\(673\) −16598.0 −0.950677 −0.475339 0.879803i \(-0.657674\pi\)
−0.475339 + 0.879803i \(0.657674\pi\)
\(674\) 0 0
\(675\) 513.000 0.0292524
\(676\) 0 0
\(677\) −8610.00 −0.488788 −0.244394 0.969676i \(-0.578589\pi\)
−0.244394 + 0.969676i \(0.578589\pi\)
\(678\) 0 0
\(679\) 220.000 0.0124342
\(680\) 0 0
\(681\) 8028.00i 0.451738i
\(682\) 0 0
\(683\) − 804.000i − 0.0450428i −0.999746 0.0225214i \(-0.992831\pi\)
0.999746 0.0225214i \(-0.00716938\pi\)
\(684\) 5328.00i 0.297838i
\(685\) 29952.0 1.67067
\(686\) 0 0
\(687\) − 7230.00i − 0.401516i
\(688\) −33536.0 −1.85835
\(689\) 0 0
\(690\) 0 0
\(691\) 2270.00i 0.124971i 0.998046 + 0.0624854i \(0.0199027\pi\)
−0.998046 + 0.0624854i \(0.980097\pi\)
\(692\) −9552.00 −0.524729
\(693\) 648.000 0.0355202
\(694\) 0 0
\(695\) 29568.0i 1.61378i
\(696\) 0 0
\(697\) − 29952.0i − 1.62771i
\(698\) 0 0
\(699\) 11178.0 0.604851
\(700\) 304.000i 0.0164145i
\(701\) −1782.00 −0.0960131 −0.0480066 0.998847i \(-0.515287\pi\)
−0.0480066 + 0.998847i \(0.515287\pi\)
\(702\) 0 0
\(703\) −21164.0 −1.13544
\(704\) 18432.0i 0.986764i
\(705\) 10800.0 0.576953
\(706\) 0 0
\(707\) − 1980.00i − 0.105326i
\(708\) 13824.0i 0.733810i
\(709\) 10690.0i 0.566250i 0.959083 + 0.283125i \(0.0913712\pi\)
−0.959083 + 0.283125i \(0.908629\pi\)
\(710\) 0 0
\(711\) 6336.00 0.334203
\(712\) 0 0
\(713\) − 20544.0i − 1.07907i
\(714\) 0 0
\(715\) 0 0
\(716\) −25248.0 −1.31782
\(717\) − 3744.00i − 0.195010i
\(718\) 0 0
\(719\) −11568.0 −0.600019 −0.300009 0.953936i \(-0.596990\pi\)
−0.300009 + 0.953936i \(0.596990\pi\)
\(720\) − 6912.00i − 0.357771i
\(721\) 2416.00i 0.124794i
\(722\) 0 0
\(723\) − 12630.0i − 0.649675i
\(724\) 8624.00 0.442691
\(725\) −342.000 −0.0175194
\(726\) 0 0
\(727\) 11644.0 0.594019 0.297010 0.954874i \(-0.404011\pi\)
0.297010 + 0.954874i \(0.404011\pi\)
\(728\) 0 0
\(729\) 729.000 0.0370370
\(730\) 0 0
\(731\) −40872.0 −2.06800
\(732\) −1776.00 −0.0896760
\(733\) − 15010.0i − 0.756353i −0.925733 0.378177i \(-0.876551\pi\)
0.925733 0.378177i \(-0.123449\pi\)
\(734\) 0 0
\(735\) 12204.0i 0.612451i
\(736\) 0 0
\(737\) −1368.00 −0.0683730
\(738\) 0 0
\(739\) − 33410.0i − 1.66307i −0.555474 0.831534i \(-0.687463\pi\)
0.555474 0.831534i \(-0.312537\pi\)
\(740\) 27456.0 1.36392
\(741\) 0 0
\(742\) 0 0
\(743\) − 6504.00i − 0.321142i −0.987024 0.160571i \(-0.948666\pi\)
0.987024 0.160571i \(-0.0513336\pi\)
\(744\) 0 0
\(745\) 2592.00 0.127468
\(746\) 0 0
\(747\) − 7992.00i − 0.391448i
\(748\) 22464.0i 1.09808i
\(749\) − 1992.00i − 0.0971777i
\(750\) 0 0
\(751\) 13912.0 0.675973 0.337987 0.941151i \(-0.390254\pi\)
0.337987 + 0.941151i \(0.390254\pi\)
\(752\) − 19200.0i − 0.931053i
\(753\) −23076.0 −1.11678
\(754\) 0 0
\(755\) 10776.0 0.519442
\(756\) 432.000i 0.0207827i
\(757\) −23974.0 −1.15106 −0.575528 0.817782i \(-0.695204\pi\)
−0.575528 + 0.817782i \(0.695204\pi\)
\(758\) 0 0
\(759\) − 10368.0i − 0.495829i
\(760\) 0 0
\(761\) 288.000i 0.0137188i 0.999976 + 0.00685939i \(0.00218343\pi\)
−0.999976 + 0.00685939i \(0.997817\pi\)
\(762\) 0 0
\(763\) −2804.00 −0.133043
\(764\) 25536.0 1.20924
\(765\) − 8424.00i − 0.398131i
\(766\) 0 0
\(767\) 0 0
\(768\) −12288.0 −0.577350
\(769\) 1514.00i 0.0709964i 0.999370 + 0.0354982i \(0.0113018\pi\)
−0.999370 + 0.0354982i \(0.988698\pi\)
\(770\) 0 0
\(771\) 3978.00 0.185816
\(772\) − 5776.00i − 0.269278i
\(773\) 15816.0i 0.735915i 0.929843 + 0.367957i \(0.119943\pi\)
−0.929843 + 0.367957i \(0.880057\pi\)
\(774\) 0 0
\(775\) 4066.00i 0.188458i
\(776\) 0 0
\(777\) −1716.00 −0.0792293
\(778\) 0 0
\(779\) 28416.0 1.30694
\(780\) 0 0
\(781\) 16416.0 0.752126
\(782\) 0 0
\(783\) −486.000 −0.0221816
\(784\) 21696.0 0.988338
\(785\) 18120.0i 0.823861i
\(786\) 0 0
\(787\) − 10154.0i − 0.459912i −0.973201 0.229956i \(-0.926142\pi\)
0.973201 0.229956i \(-0.0738583\pi\)
\(788\) 22368.0i 1.01120i
\(789\) 18144.0 0.818686
\(790\) 0 0
\(791\) − 3852.00i − 0.173150i
\(792\) 0 0
\(793\) 0 0
\(794\) 0 0
\(795\) 20088.0i 0.896161i
\(796\) 2720.00 0.121115
\(797\) 17442.0 0.775191 0.387596 0.921830i \(-0.373306\pi\)
0.387596 + 0.921830i \(0.373306\pi\)
\(798\) 0 0
\(799\) − 23400.0i − 1.03609i
\(800\) 0 0
\(801\) 9180.00i 0.404943i
\(802\) 0 0
\(803\) −24552.0 −1.07898
\(804\) − 912.000i − 0.0400047i
\(805\) −2304.00 −0.100876
\(806\) 0 0
\(807\) −19422.0 −0.847196
\(808\) 0 0
\(809\) −8778.00 −0.381481 −0.190740 0.981641i \(-0.561089\pi\)
−0.190740 + 0.981641i \(0.561089\pi\)
\(810\) 0 0
\(811\) − 430.000i − 0.0186182i −0.999957 0.00930909i \(-0.997037\pi\)
0.999957 0.00930909i \(-0.00296322\pi\)
\(812\) − 288.000i − 0.0124468i
\(813\) 17934.0i 0.773644i
\(814\) 0 0
\(815\) 4728.00 0.203208
\(816\) −14976.0 −0.642481
\(817\) − 38776.0i − 1.66047i
\(818\) 0 0
\(819\) 0 0
\(820\) −36864.0 −1.56994
\(821\) − 32976.0i − 1.40179i −0.713264 0.700895i \(-0.752784\pi\)
0.713264 0.700895i \(-0.247216\pi\)
\(822\) 0 0
\(823\) 1168.00 0.0494701 0.0247351 0.999694i \(-0.492126\pi\)
0.0247351 + 0.999694i \(0.492126\pi\)
\(824\) 0 0
\(825\) 2052.00i 0.0865957i
\(826\) 0 0
\(827\) 17172.0i 0.722042i 0.932558 + 0.361021i \(0.117572\pi\)
−0.932558 + 0.361021i \(0.882428\pi\)
\(828\) 6912.00 0.290107
\(829\) −27146.0 −1.13730 −0.568649 0.822580i \(-0.692534\pi\)
−0.568649 + 0.822580i \(0.692534\pi\)
\(830\) 0 0
\(831\) 26250.0 1.09579
\(832\) 0 0
\(833\) 26442.0 1.09983
\(834\) 0 0
\(835\) −1008.00 −0.0417764
\(836\) −21312.0 −0.881688
\(837\) 5778.00i 0.238610i
\(838\) 0 0
\(839\) − 30696.0i − 1.26310i −0.775334 0.631552i \(-0.782418\pi\)
0.775334 0.631552i \(-0.217582\pi\)
\(840\) 0 0
\(841\) −24065.0 −0.986715
\(842\) 0 0
\(843\) 26928.0i 1.10018i
\(844\) −15392.0 −0.627742
\(845\) 0 0
\(846\) 0 0
\(847\) − 70.0000i − 0.00283970i
\(848\) 35712.0 1.44617
\(849\) −1776.00 −0.0717929
\(850\) 0 0
\(851\) 27456.0i 1.10597i
\(852\) 10944.0i 0.440065i
\(853\) − 24842.0i − 0.997156i −0.866845 0.498578i \(-0.833856\pi\)
0.866845 0.498578i \(-0.166144\pi\)
\(854\) 0 0
\(855\) 7992.00 0.319673
\(856\) 0 0
\(857\) −11406.0 −0.454634 −0.227317 0.973821i \(-0.572995\pi\)
−0.227317 + 0.973821i \(0.572995\pi\)
\(858\) 0 0
\(859\) 20540.0 0.815851 0.407925 0.913015i \(-0.366252\pi\)
0.407925 + 0.913015i \(0.366252\pi\)
\(860\) 50304.0i 1.99460i
\(861\) 2304.00 0.0911964
\(862\) 0 0
\(863\) 9108.00i 0.359258i 0.983734 + 0.179629i \(0.0574898\pi\)
−0.983734 + 0.179629i \(0.942510\pi\)
\(864\) 0 0
\(865\) 14328.0i 0.563198i
\(866\) 0 0
\(867\) −3513.00 −0.137610
\(868\) −3424.00 −0.133892
\(869\) 25344.0i 0.989340i
\(870\) 0 0
\(871\) 0 0
\(872\) 0 0
\(873\) 990.000i 0.0383808i
\(874\) 0 0
\(875\) −2544.00 −0.0982890
\(876\) − 16368.0i − 0.631305i
\(877\) − 24046.0i − 0.925856i −0.886396 0.462928i \(-0.846799\pi\)
0.886396 0.462928i \(-0.153201\pi\)
\(878\) 0 0
\(879\) − 13824.0i − 0.530457i
\(880\) 27648.0 1.05911
\(881\) −7998.00 −0.305856 −0.152928 0.988237i \(-0.548870\pi\)
−0.152928 + 0.988237i \(0.548870\pi\)
\(882\) 0 0
\(883\) −24032.0 −0.915902 −0.457951 0.888978i \(-0.651416\pi\)
−0.457951 + 0.888978i \(0.651416\pi\)
\(884\) 0 0
\(885\) 20736.0 0.787608
\(886\) 0 0
\(887\) −15648.0 −0.592343 −0.296172 0.955135i \(-0.595710\pi\)
−0.296172 + 0.955135i \(0.595710\pi\)
\(888\) 0 0
\(889\) − 1976.00i − 0.0745477i
\(890\) 0 0
\(891\) 2916.00i 0.109640i
\(892\) 40336.0i 1.51407i
\(893\) 22200.0 0.831909
\(894\) 0 0
\(895\) 37872.0i 1.41444i
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) − 3852.00i − 0.142905i
\(900\) −1368.00 −0.0506667
\(901\) 43524.0 1.60932
\(902\) 0 0
\(903\) − 3144.00i − 0.115865i
\(904\) 0 0
\(905\) − 12936.0i − 0.475146i
\(906\) 0 0
\(907\) 808.000 0.0295802 0.0147901 0.999891i \(-0.495292\pi\)
0.0147901 + 0.999891i \(0.495292\pi\)
\(908\) − 21408.0i − 0.782433i
\(909\) 8910.00 0.325111
\(910\) 0 0
\(911\) 39144.0 1.42360 0.711799 0.702383i \(-0.247880\pi\)
0.711799 + 0.702383i \(0.247880\pi\)
\(912\) − 14208.0i − 0.515870i
\(913\) 31968.0 1.15880
\(914\) 0 0
\(915\) 2664.00i 0.0962504i
\(916\) 19280.0i 0.695447i
\(917\) 4200.00i 0.151250i
\(918\) 0 0
\(919\) −38248.0 −1.37289 −0.686445 0.727182i \(-0.740830\pi\)
−0.686445 + 0.727182i \(0.740830\pi\)
\(920\) 0 0
\(921\) − 9498.00i − 0.339815i
\(922\) 0 0
\(923\) 0 0
\(924\) −1728.00 −0.0615228
\(925\) − 5434.00i − 0.193155i
\(926\) 0 0
\(927\) −10872.0 −0.385203
\(928\) 0 0
\(929\) − 54264.0i − 1.91641i −0.286084 0.958205i \(-0.592354\pi\)
0.286084 0.958205i \(-0.407646\pi\)
\(930\) 0 0
\(931\) 25086.0i 0.883094i
\(932\) −29808.0 −1.04763
\(933\) 7416.00 0.260224
\(934\) 0 0
\(935\) 33696.0 1.17859
\(936\) 0 0
\(937\) 12206.0 0.425563 0.212782 0.977100i \(-0.431748\pi\)
0.212782 + 0.977100i \(0.431748\pi\)
\(938\) 0 0
\(939\) 9282.00 0.322584
\(940\) −28800.0 −0.999311
\(941\) 17664.0i 0.611934i 0.952042 + 0.305967i \(0.0989797\pi\)
−0.952042 + 0.305967i \(0.901020\pi\)
\(942\) 0 0
\(943\) − 36864.0i − 1.27302i
\(944\) − 36864.0i − 1.27100i
\(945\) 648.000 0.0223063
\(946\) 0 0
\(947\) 51984.0i 1.78379i 0.452238 + 0.891897i \(0.350626\pi\)
−0.452238 + 0.891897i \(0.649374\pi\)
\(948\) −16896.0 −0.578857
\(949\) 0 0
\(950\) 0 0
\(951\) − 6948.00i − 0.236913i
\(952\) 0 0
\(953\) −13782.0 −0.468460 −0.234230 0.972181i \(-0.575257\pi\)
−0.234230 + 0.972181i \(0.575257\pi\)
\(954\) 0 0
\(955\) − 38304.0i − 1.29789i
\(956\) 9984.00i 0.337767i
\(957\) − 1944.00i − 0.0656642i
\(958\) 0 0
\(959\) 4992.00 0.168092
\(960\) 18432.0i 0.619677i
\(961\) −16005.0 −0.537243
\(962\) 0 0
\(963\) 8964.00 0.299959
\(964\) 33680.0i 1.12527i
\(965\) −8664.00 −0.289020
\(966\) 0 0
\(967\) 14618.0i 0.486125i 0.970011 + 0.243063i \(0.0781521\pi\)
−0.970011 + 0.243063i \(0.921848\pi\)
\(968\) 0 0
\(969\) − 17316.0i − 0.574066i
\(970\) 0 0
\(971\) −18708.0 −0.618299 −0.309149 0.951013i \(-0.600044\pi\)
−0.309149 + 0.951013i \(0.600044\pi\)
\(972\) −1944.00 −0.0641500
\(973\) 4928.00i 0.162368i
\(974\) 0 0
\(975\) 0 0
\(976\) 4736.00 0.155323
\(977\) − 48804.0i − 1.59814i −0.601241 0.799068i \(-0.705327\pi\)
0.601241 0.799068i \(-0.294673\pi\)
\(978\) 0 0
\(979\) −36720.0 −1.19875
\(980\) − 32544.0i − 1.06080i
\(981\) − 12618.0i − 0.410664i
\(982\) 0 0
\(983\) 44736.0i 1.45153i 0.687941 + 0.725766i \(0.258515\pi\)
−0.687941 + 0.725766i \(0.741485\pi\)
\(984\) 0 0
\(985\) 33552.0 1.08534
\(986\) 0 0
\(987\) 1800.00 0.0580493
\(988\) 0 0
\(989\) −50304.0 −1.61737
\(990\) 0 0
\(991\) −21004.0 −0.673274 −0.336637 0.941635i \(-0.609289\pi\)
−0.336637 + 0.941635i \(0.609289\pi\)
\(992\) 0 0
\(993\) 13278.0i 0.424335i
\(994\) 0 0
\(995\) − 4080.00i − 0.129995i
\(996\) 21312.0i 0.678009i
\(997\) 9038.00 0.287098 0.143549 0.989643i \(-0.454149\pi\)
0.143549 + 0.989643i \(0.454149\pi\)
\(998\) 0 0
\(999\) − 7722.00i − 0.244558i
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 507.4.b.b.337.1 2
13.5 odd 4 39.4.a.a.1.1 1
13.8 odd 4 507.4.a.c.1.1 1
13.12 even 2 inner 507.4.b.b.337.2 2
39.5 even 4 117.4.a.a.1.1 1
39.8 even 4 1521.4.a.f.1.1 1
52.31 even 4 624.4.a.g.1.1 1
65.44 odd 4 975.4.a.e.1.1 1
91.83 even 4 1911.4.a.f.1.1 1
104.5 odd 4 2496.4.a.o.1.1 1
104.83 even 4 2496.4.a.f.1.1 1
156.83 odd 4 1872.4.a.m.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
39.4.a.a.1.1 1 13.5 odd 4
117.4.a.a.1.1 1 39.5 even 4
507.4.a.c.1.1 1 13.8 odd 4
507.4.b.b.337.1 2 1.1 even 1 trivial
507.4.b.b.337.2 2 13.12 even 2 inner
624.4.a.g.1.1 1 52.31 even 4
975.4.a.e.1.1 1 65.44 odd 4
1521.4.a.f.1.1 1 39.8 even 4
1872.4.a.m.1.1 1 156.83 odd 4
1911.4.a.f.1.1 1 91.83 even 4
2496.4.a.f.1.1 1 104.83 even 4
2496.4.a.o.1.1 1 104.5 odd 4